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Vol. 9, Issue 1 pp.1-168
Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations
Anthony Gruber, Max Gunzburger, Lili Ju, Yuankai Teng & Zhu Wang
10.4208/nmtma.OA-2021-0062
Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 839-861.
Preview Purchase PDF 341 29172 Abstract
A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.
Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients
Xinliang Liu, Lei Zhang & Shengxin Zhu
We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.
A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations
Xian-Ming Gu, Yong-Liang Zhao, Xi-Le Zhao, Bruno Carpentieri & Yu-Yun Huang
The $p$-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald et al. [36] and Lin and Ng [32]). However, when the BDF$p$ (2 ≤ $p$ ≤ 6) method is used to solve time-dependent PDEs, the generalization of these studies is not straightforward as $p$-step BDF is not selfstarting for $p$ ≥ 2. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system. We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.
Order Reduced Schemes for the Fourth Order Eigenvalue Problems on Multi-Connected Planar Domains
Yingxia Xi, Xia Ji & Shuo Zhang
In this paper, we study the order reduced finite element method for the fourth order eigenvalue problems on multi-connected planar domains. Particularly, we take the biharmonic and the Helmholtz transmission eigenvalue problems as model problems, present for each an equivalent order reduced formulation and a corresponding stable discretization scheme, and present rigorous theoretical analysis. The schemes are readily fit for multilevel correction algorithms with optimal computational costs. Numerical experiments are given for verifications.
Stability and Convergence Analyses of the FDM Based on Some L-Type Formulae for Solving the Subdiffusion Equation
Reza Mokhtari, Mohadese Ramezani & Gundolf Haase
Some well-known L-type formulae, i.e., L1, L1-2, and L1-2-3 formulae, are usually employed to approximate the Caputo fractional derivative of order α ∈ (0, 1). In this paper, we aim to elaborate on the stability and convergence analyses of some finite difference methods (FDMs) for solving the subdiffusion equation, i.e., a diffusion equation which exploits the Caputo time-fractional derivative of order $α$. In fact, the FDMs considered here are based on the usual central difference scheme for the spatial derivative, and the Caputo derivative is approximated by using methods such as the L1, L1-2, and L1-2-3 formulae. Thanks to a specific type of the discrete version of the Gronwall inequality, we show that the FDMs are unconditionally stable in the maximum norm and also discrete $H^1$ norm. Then, we prove that the finite difference method which uses the L1, L1-2, and L1-2-3 formulae has the global order of convergence $2−α$, $3−α$, and 3, respectively. Finally, some numerical tests confirm the theoretical results. A brief conclusion finishes the paper.
A Hybrid WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems
Zhuang Zhao, Yibing Chen & Jianxian Qiu
In this paper, we develop a simplified hybrid weighted essentially non-oscillatory (WENO) method combined with the modified ghost fluid method (MGFM) [31] to simulate the compressible two-medium flow problems. The MGFM can turn the two-medium flow problems into two single-medium cases by defining the ghost fluids state in terms of the predicted the interface state, which makes the material interface "invisible". For the single medium flow case, we adapt between the linear upwind scheme and the WENO scheme automatically by identifying the regions of the extreme points for the reconstruction polynomial as same as the hybrid WENO scheme [55]. Instead of calculating their exact locations, we only need to know the regions of the extreme points based on the zero point existence theorem, which is simpler for implementation and saves computation time. Meanwhile, it still keeps the robustness and has high efficiency. Extensive numerical results for both one and two dimensional two-medium flow problems are performed to demonstrate the good performances of the proposed method.
Tensor Bi-CR Methods for Solutions of High Order Tensor Equation Accompanied by Einstein Product
Masoud Hajarian
Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 998-1016.
Tensors have a wide application in control systems, documents analysis, medical engineering, formulating an $n$-person noncooperative game and so on. It is the purpose of this paper to explore two efficient and novel algorithms for computing the solutions $\mathcal{X}$ and $\mathcal{Y}$ of the high order tensor equation $\mathcal{A}*_P\mathcal{X}*_Q\mathcal{B}+\mathcal{C}*_P\mathcal{Y}*_Q\mathcal{D}=\mathcal{H}$ with Einstein product. The algorithms are, respectively, based on the Hestenes-Stiefel (HS) and the Lanczos types of bi-conjugate residual (Bi-CR) algorithm. The theoretical results indicate that the algorithms terminate after finitely many iterations with any initial tensors. The resulting algorithms are easy to implement and simple to use. Finally, we present two numerical examples that confirm our analysis and illustrate the efficiency of the algorithms.
Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization
Xue Li, Zhenwei Zhang, Huibin Chang & Yuping Duan
Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 1017-1041.
We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.
A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form
Ruo Li & Fanyi Yang
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.
On a Parabolic Sine-Gordon Model
Xinyu Cheng, Dong Li, Chaoyu Quan & Wen Yang
We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.
Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients
Xu Yang & Weidong Zhao
This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.
A General Cavitation Model for the Highly Nonlinear Mie-Grüneisen Equation of State
Meiyan Fu & Tiao Lu
A general one-fluid cavitation model is proposed for a family of Mie-Grüneisen equations of state (EOS), which can provide a wide application of cavitation flows, such as liquid-vapour transformation and underwater explosion. An approximate Riemann problem and its approximate solver for the general cavitation model are developed. The approximate solver, which provides the interface pressure and normal velocity by an iterative method, is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian grids. Several numerical examples, including Riemann problems and underwater explosion applications, are presented to validate the cavitation model and the corresponding approximate solver.
Author Index to Volume 14 | CommonCrawl |
An Application of theMelitz Model to Chinese Firms
Sun, Churen and Tian, Guoqiang and Zhang, Tao (2012): An Application of theMelitz Model to Chinese Firms. Published in: Review of Development Economics , Vol. 3, No. 17 (May 2013): pp. 494-509.
This is the latest version of this item.
When the Melitz model is implemented in practice, the industrial productivity distribution is often assumed to be of Pareto form. In this case, a fundamental relationship $\kappa>\sigma-1$ must hold to guarantee the convergence of the industrial average productivity, where $\kappa$ is the concentration degree of the industrial productivity Pareto distribution and $\sigma$ is the substitution elasticity across varieties in the industry. This paper estimates the concentration degrees of the Pareto distribution in industrial productivity and industrial substitution elasticities using firm-level data of 40 Chinese manufacturing industries from 1998 and 2007. However, the paper shows that the above fundamental assumption $\kappa>\sigma-1$ does not hold for nearly all the industries for Chinese firm-level data. An explanation is proposed due to the distorted firm size and productivity for Chinese characteristics.
Melitz model, Pareto distribution, productivity heterogeneity, export
D - Microeconomics > D2 - Production and Organizations > D23 - Organizational Behavior ; Transaction Costs ; Property Rights
F - International Economics > F1 - Trade > F12 - Models of Trade with Imperfect Competition and Scale Economies ; Fragmentation
Churen Sun
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Available Versions of this Item
When Pareto meets Melitz: the inapplicability of the Melitz-Pareto model for Chinese firms. (deposited 27 Dec 2011 21:38)
An Application of theMelitz Model to Chinese Firms. (deposited 14 Aug 2013 15:07) [Currently Displayed] | CommonCrawl |
Time-continuous production networks with random breakdowns
The needle problem approach to non-periodic homogenization
December 2011, 6(4): 715-753. doi: 10.3934/nhm.2011.6.715
Ginzburg-Landau model with small pinning domains
Mickaël Dos Santos 1, and Oleksandr Misiats 2,
Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne, France
Department of Mathematics, The Pennsylvania State University, University Park PA 16802, United States
Received March 2011 Revised October 2011 Published December 2011
We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.
Keywords: Ginzburg-Landau Functional, vortices, pinning domain, degree..
Mathematics Subject Classification: Primary: 49K20, 35J66, 35J50; Secondary: 47H1.
Citation: Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
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Mickaël Dos Santos Oleksandr Misiats | CommonCrawl |
Annals of Human Genetics (1)
Bulletin of Entomological Research (1)
High Power Laser Science and Engineering (1)
International Psychogeriatrics (1)
Journal of Mechanics (1)
Twin Research and Human Genetics (1)
International Psychogeriatric Association (1)
International Soc for Twin Studies (1)
Quantum electrodynamics experiments with colliding petawatt laser pulses
I. C. E. Turcu, B. Shen, D. Neely, G. Sarri, K. A. Tanaka, P. McKenna, S. P. D. Mangles, T.-P. Yu, W. Luo, X.-L. Zhu, Y. Yin
Published online by Cambridge University Press: 14 February 2019, e10
A new generation of high power laser facilities will provide laser pulses with extremely high powers of 10 petawatt (PW) and even 100 PW, capable of reaching intensities of $10^{23}~\text{W}/\text{cm}^{2}$ in the laser focus. These ultra-high intensities are nevertheless lower than the Schwinger intensity $I_{S}=2.3\times 10^{29}~\text{W}/\text{cm}^{2}$ at which the theory of quantum electrodynamics (QED) predicts that a large part of the energy of the laser photons will be transformed to hard Gamma-ray photons and even to matter, via electron–positron pair production. To enable the investigation of this physics at the intensities achievable with the next generation of high power laser facilities, an approach involving the interaction of two colliding PW laser pulses is being adopted. Theoretical simulations predict strong QED effects with colliding laser pulses of ${\geqslant}10~\text{PW}$ focused to intensities ${\geqslant}10^{22}~\text{W}/\text{cm}^{2}$ .
Direct Imaging of Electron Transfer and Its Influence on Superconducting Pairing at FeSe/SrTiO3 Interface
Lijun Wu, Mingda Li, W. Zhao, C-Z. Chang, J. S. Moodera, M.H.W. Chan, Yimei Zhu
A novel miRNA, miR-13664, targets CpCYP314A1 to regulate deltamethrin resistance in Culex pipiens pallens
X. H. Sun, N. Xu, Y. Xu, D. Zhou, Y. Sun, W. J. Wang, L. Ma, C. L. Zhu, B. Shen
Journal: Parasitology / Volume 146 / Issue 2 / February 2019
Extensive insecticide use has led to the resistance of mosquitoes to these insecticides, posing a major barrier to mosquito control. Previous Solexa high-throughput sequencing of Culex pipiens pallens in the laboratory has revealed that the abundance of a novel microRNA (miRNA), miR-13664, was higher in a deltamethrin-sensitive (DS) strain than a deltamethrin-resistant (DR) strain. Real-time quantitative PCR revealed that the miR-13664 transcript level was lower in the DR strain than in the DS strain. MiR-13664 oversupply in the DR strain increased the susceptibility of these mosquitoes to deltamethrin, whereas inhibition of miR-13664 made the DS strain more resistant to deltamethrin. Results of bioinformatic analysis, quantitative reverse-transcriptase polymerase chain reaction, luciferase assay and miR mimic/inhibitor microinjection revealed CpCYP314A1 to be a target of miR-13664. In addition, downregulation of CpCYP314A1 expression in the DR strain reduced the resistance of mosquitoes to deltamethrin. Taken together, our results indicate that miR-13664 could regulate deltamethrin resistance by interacting with CpCYP314A1, providing new insights into mosquito resistance mechanisms.
Antibiotic-induced alterations of the gut microbiota and microbial fermentation in protein parallel the changes in host nitrogen metabolism of growing pigs
Y. Pi, K. Gao, Y. Peng, C. L. Mu, W. Y. Zhu
Gut microbes, especially those in the large intestine, are actively involved in nutrient metabolism; however, their impact on host nitrogen (N) metabolism remains largely unknown. This study was designed to investigate the effects of feeding a cocktail of antibiotics (AGM) (ampicillin, gentamycin and metronidazole) on intestinal microbiota, N utilization efficiency, and amino acid (AA) digestibility in cannulated pigs, with the aim of exploring the impact of gut microbiota on host N metabolism. In total, 16 piglets were surgically fitted with a simple distal ileal T-cannula and a jugular venous catheter. The pigs were fed a basal diet without antibiotics (control; CON) or with antibiotics (antibiotic; ANTI), for 2 weeks. The results showed that feeding AGM did not affect weight gain or digestive enzyme activity. The antibiotics increased the concentration of urea N (P<0.05). However, they reduced N utilization, and the total tract apparent digestibility of isoleucine, methionine, valine, tyrosine and total AA (P<0.05). Furthermore, the antibiotics increased the terminal ileum apparent digestibility of CP, phenylalanine, valine, alanine, tyrosine and total AA (P<0.05). AGM markedly altered the composition of the microbiota in the ileum and feces, with a reduction in populations of Bifidobacterium, Lactobacillus and Ruminococcus, and an increase in the abundance of Escherichia coli (P<0.05). The antibiotics also significantly increased the concentration of cadaverine and ammonia, both in ileal digesta and feces (P<0.05), suggesting a marked impact on N metabolism in the intestine. The analyses indicated that the alteration of gut microbiota was correlated with the apparent digestibility of CP and AA in the intestine. These findings suggest that the AGM-induced alteration of gut microbiota may contribute to the change in intestinal N metabolism, and consequently, N excretion from the body. These results also suggest that antibiotics could have a significant effect on host N metabolism. The present study contributes to our understanding of the effects of antibiotics and provides a rational scientific basis for diet formulation during AGM use.
Structure of triplite LiFeSO4F powder synthesized through an ambient two-step solid-state route
F.-F. Ma, J.-W. Mao, G.-Q. Shao, S.-H. Fan, C. Zhu, A.-L. Zhang, G.-Z. Xie, J.-N. Gu, J.-L. Yan
Journal: Powder Diffraction / Volume 33 / Issue 1 / March 2018
The triplite LiFeSO4F displays both the highest potential ever reported for an Fe-based compound, as well as a comparable specific energy with that of popular LiFePO4. The synthesis is still a challenge because the present approaches are connected with long time, special equipments or organic reagents, etc. In this work, the triplite LiFeSO4F powder was synthesized through an ambient two-step solid-state route. The reaction process and phase purity were analyzed, coupled with structure refinement and electrochemical test.
New microsatellites revealed strong gene flow among populations of a new outbreak pest, Athetis lepigone (Möschler)
W.-C. Zhu, J.-T. Sun, J. Dai, J.-R. Huang, L. Chen, X.-Y. Hong
Journal: Bulletin of Entomological Research / Volume 108 / Issue 5 / October 2018
Published online by Cambridge University Press: 27 November 2017, pp. 636-644
Athetis lepigone (Möschler) (Lepidoptera: Noctuidae) is a new outbreak pest in China. Consequently, it is unclear whether the emergence and spread of the outbreak of this pest are triggered by rapid in situ population size increases in each outbreak area, or by immigrants from a potential source area in China. In order to explore the outbreak process of this pest through a population genetics approach, we developed ten novel polymorphic expressed sequence tags (EST)-derived microsatellites. These new microsatellites had moderately high levels of polymorphism in the tested population. The number of alleles per locus ranged from 3 to 19, with an average of 8.6, and the expected heterozygosity ranged from 0.269 to 0.783. A preliminary population genetic analysis using these new microsatellites revealed a lack of population genetic structure in natural populations of A. lepigone. The estimates of recent migration rate revealed strong gene flow among populations. In conclusion, our study developed the first set of EST-microsatellite markers and shed a new light on the population genetic structure of this pest in China.
Experimental Study on the Shear Adhesion Strength Between the Ice and Substrate in Icing Wind Tunnel
C. X. Zhu, C. L. Zhu, W. W. Zhao, M. J. Tao
Journal: Journal of Mechanics / Volume 34 / Issue 2 / April 2018
Print publication: April 2018
The icing wind tunnel can simulate the air flow at a high altitude; such an air flow contains supercooled droplets moving at certain velocities. An integrated experiment method was proposed, and it included the icing test and shear stress measurements in the simulated environment of the icing wind tunnel. The error caused by the change in experimental environments was completely eliminated with this novel method. Thus, there was no discrepancy between the real-time and experimental values of shear stress between the ice and substrate. The experiments of icing and shear stress measurements are carried out by varying the following parameters: icing temperature, mean volume diameter (MVD) of droplets, and surface roughness of the substrate. The results indicate that the shear stress between the ice and the substrate increases with the decrease in temperature provided the temperature is relatively high. When the MVD value is 22 μm, the liquid water content is about 1 g/m3 and surface roughness is 2 μm. Under these conditions, the shear stress reaches its maximum value at a temperature of –15°C. The shear stress is also affected by the MVD values of droplets, and the surface roughness of substrate.
A cross-sectional study of acute diarrhea in Pudong, Shanghai, China: prevalence, risk factors, and healthcare-seeking practices
J.-X. YU, W.-P. ZHU, C.-C. YE, C.-Y. XUE, S.-J. LAI, H.-L. ZHANG, Z.-K. ZHANG, Q.-B. GENG, W.-Z. YANG, Q. SUN, Z.-J. LI
Journal: Epidemiology & Infection / Volume 145 / Issue 13 / October 2017
Diarrhea is a common cause of morbidity and mortality and the incidence of diarrhea in the world has changed little over the past four decades. To assess the prevalence of and healthcare practices for diarrhea, a cross-sectional study was conducted in Pudong, Shanghai, China. In October 2014, a total of 5324 community residents were interviewed. Respondents were asked if they had experienced diarrhea (defined as ⩾3 passages of watery, loose, bloody, or mucoid stools within a 24-h period) in the previous month prior to the interview. The monthly prevalence of diarrhea was 4·1% (95% CI: 3·3–4·8), corresponding to an incidence rate of 0·54 episodes per person-year. The proportion of individuals with diarrhea who sought healthcare was 21·2% (95% CI: 13·4–29·0). Diarrhea continues to impose a considerable burden on the community and healthcare system in Pudong. Young age and travel were identified as predictors of increased diarrhea occurrence.
Selective functional disconnection of the orbitofrontal subregions in schizophrenia
Y. Xu, W. Qin, C. Zhuo, L. Xu, J. Zhu, X. Liu, C. Yu
Journal: Psychological Medicine / Volume 47 / Issue 9 / July 2017
As a disconnection syndrome, schizophrenia has shown impaired resting-state functional connectivity (rsFC) in the orbitofrontal cortex (OFC); however, the OFC is a rather heterogeneous region and the rsFC changes in the OFC subregions remain unknown.
A total of 98 schizophrenia patients and 102 healthy controls underwent resting-state functional MRI using a sensitivity-encoded spiral-in imaging sequence (SENSE-SPIRAL) to reduce susceptibility-induced signal loss and distortion. The OFC subregions were defined according to a previous parcellation study that divided the OFC into the anterior (OFCa), medial (OFCm), posterior (OFCp), intermediate (OFCi), and lateral (OFCl) subregions. The rsFC was compared using two-way repeated-measures ANOVA.
Whether or not global signal regression, compared with healthy controls, schizophrenia patients consistently exhibited decreased rsFC between the left OFCi and the left middle temporal gyrus and the right middle frontal gyrus (MFG), between the right OFCi and the right MFG and the left inferior frontal gyrus, between the right OFCm and the middle cingulate cortex and the left Rolandic operculum. These rsFC changes still remained significant even after cortical atrophy correction.
These findings suggest a selective functional disconnection of the OFC subregions in schizophrenia, and provide more precise information about the functional disconnections of the OFC in this disorder.
Photo-transmutation of long-lived radionuclide 135Cs by laser–plasma driven electron source
X.-L. Wang, Z.-Y. Tan, W. Luo, Z.-C. Zhu, X.-D. Wang, Y.-M. Song
Journal: Laser and Particle Beams / Volume 34 / Issue 3 / September 2016
Laser-driven relativistic electrons can be focused onto a high-Z convertor for generating high-brightness γ-rays, which in turn can be used to induce photonuclear reactions. In this work, photo-transmutation of long-lived radionuclide 135Cs induced by laser–plasma–interaction-driven electron source is demonstrated using Geant4 simulation (Agostinelli et al., 2003 Nucl. Instrum. Meth. A 506, 250). High-energy electron generation, bremsstrahlung, as well as photonuclear reaction are observed at four different laser intensities: 1020, 5 × 1020, 1021, and 5 × 1021 W/cm2. The transmutation efficiency depends on the laser intensity and target size. An optimum laser intensity, namely 1021 W/cm2, was found, with the corresponding photonuclear reaction yield reaching 108 J−1 of the laser energy. Laser-generated electrons can therefore be a promising tool for transmutation reactions. Potential application in nuclear waste management is suggested.
Studying Perovskite-based Solar Cells with Correlative In-Situ Microscopy
J. A. Aguiar, S. Wozny, W. Zhou, H. Guthrey, H. Moutinho, A. G. Norman, C. S. Jiang, J. Berry, K. Zhu, T. Holesinger, M. M. Al-Jassim
Molecular identification and seasonal infections of species of Fasciola in ruminants from two provinces in China
W. Yuan, J.-M. Liu, K. Lu, H. Li, M.-M. Duan, J.-T. Feng, Y. Hong, Y.-P. Liu, Y. Zhou, L.-B. Tong, J. Lu, C.-G. Zhu, Y.-M. Jin, G.-F. Cheng, J.-J. Lin
Journal: Journal of Helminthology / Volume 90 / Issue 3 / May 2016
We determined the prevalence and seasonality of infections by Fasciola of goats and bovine species (cattle and water buffalo) in Hubei and Anhui provinces of China. Faecal samples were collected at 2- to 3-month intervals from 200 goats in Hubei province and from 152 bovine species in Anhui province. All faecal samples were examined for the presence of parasites. We determined the nucleotide sequences of the first and second internal transcribed spacers (ITS-1 and ITS-2) of the nuclear ribosomal DNA (rDNA) of 39 Fasciola worms from Anhui province. The prevalence of Fasciola infection in goats ranged between 3.5 and 37.0%, with mean eggs per gram (EPG) ranging between 29.0 and 166.0. Prevalence and EPG exhibited downward trends over time with significant differences. The prevalence of Fasciola infection in cattle ranged between 13.3 and 46.2% (mean EPG, 36.4–100.0), and that of water buffalo ranged between 10.3 and 35.4% (mean EPG, 25.0–89.6), with a higher prevalence of infection and EPG from June to October compared with December to March. Analysis of ITS-1 and ITS-2 sequences revealed that F. hepatica and F. gigantica were present in all bovine species of Anhui province and that F. gigantica mainly infected water buffalo. This is the first demonstration of Fasciola infection in Hubei province and detection of F. hepatica and F. gigantica in Anhui province. The present study of Hubei province shows that mass treatment of livestock with closantel sodium injections in April and August/September controlled Fasciola infection effectively.
Multilocus variable-number tandem-repeat analysis of Neisseria meningitidis serogroup C in China
X. Y. SHAN, H. J. ZHOU, J. ZHANG, B. Q. ZHU, L. XU, Z. XU, G. C. HU, A. Y. BAI, Y. W. SHI, B. F. JIANG, Z. J. SHAO
This study characterized Neisseria meningitidis serogroup C strains in China in order to establish their genetic relatedness and describe the use of multilocus variable-number tandem-repeat (VNTR) analysis (MLVA) to provide useful epidemiological information. A total of 215 N. meningitidis serogroup C strains, obtained from 2003 to 2012 in China, were characterized by MLVA with different published schemes as well as multilocus sequence typing. (i) Based on the MLVA scheme with a combination of five highly variable loci, 203 genotypes were identified; this level of discrimination supports its use for resolving closely related isolates. (ii) Based on a combination of ten low variable loci, clear phylogenetic relationships were established within sequence type complexes. In addition, there was evidence of microevolution of VNTR loci over the decade as strain lineages spread from Anhui to other provinces, the more distant the provinces from Anhui, the higher the genetic variation.
By Julie M. Allen, Marina S. Ascunce, Ahidjo Ayouba, David Bass, Frida Ben-Ami, Frédéric Bordes, Bret M. Boyd, Rodney A. Bray, Aurélie Chambouvet, Philippe Christe, Julien Claude, Yves Desdevises, Carl W. Dick, Katharina Dittmar, Ashley Dowling, Bryan G. Falk, Martín García-Varela, Rebecca Rose Gray, Michael W. Hastriter, Hadas Hawlena, Tine Huyse, James C. Iles, Tania Jenkins, Boris R. Krasnov, Armand M. Kuris, Tommy L. F. Leung, D. Timothy J. Littlewood, Peter V. Markov, Camilo Mora, Serge Morand, Solon F. Morse, Steve Nadler, Sigrid Neuhauser, Roderic Page, Bruce D. Patterson, Martine Peeters, Gerardo Pérez-Ponce de León, Susan L. Perkins, Timothée Poisot, Robert Poulin, Oliver G. Pybus, David L. Reed, Thomas A. Richards, Klaus Rohde, Lajos Rózsa, Andrea Šimková, Arne Skorping, Melissa A. Toups, Piotr Tryjanowski, Maarten P. M. Vanhove, Zoltán Vas, Andrea Waeschenbach, Lucy A. Weinert, Michael F. Whiting, Quin Zhu
Edited by Serge Morand, Université de Montpellier II, Boris R. Krasnov, Ben-Gurion University of the Negev, Israel, D. Timothy J. Littlewood, Natural History Museum, London
Book: Parasite Diversity and Diversification
Print publication: 26 February 2015, pp viii-xii
A Twin Study of Breastfeeding With a Preliminary Genome-Wide Association Scan
Lucia Colodro-Conde, Gu Zhu, Robert A. Power, Anjali Henders, Andrew C. Heath, Pamela A. F. Madden, Grant W. Montgomery, Sarah Medland, Juan R. Ordoñana, Nicholas G. Martin
Journal: Twin Research and Human Genetics / Volume 18 / Issue 1 / February 2015
Published online by Cambridge University Press: 05 December 2014, pp. 61-72
Breastfeeding has been an important survival trait during human history, though it has long been recognized that individuals differ in their exact breastfeeding behavior. Here our aims were, first, to explore to what extent genetic and environmental influences contributed to the individual differences in breastfeeding behavior; second, to detect possible genetic variants related to breastfeeding; and lastly, to test if the genetic variants associated with breastfeeding have been previously found to be related with breast size. Data were collected from a large community-based cohort of Australian twins, with 3,364 women participating in the twin modelling analyses and 1,521 of them included in the genome-wide association study (GWAS). Monozygotic (MZ) twin correlations (r MZ = 0.52, 95% CI 0.46–0.57) were larger than dizygotic (DZ) twin correlations (r DZ = 0.35, 95% CI 0.25–0.43) and the best-fitting model was the one composed by additive genetics and unique environmental factors, explaining 53% and 47% of the variance in breastfeeding behavior, respectively. No breastfeeding-related genetic variants reached genome-wide significance. The polygenic risk score analyses showed no significant results, suggesting breast size does not influence breastfeeding. This study confers a replication of a previous one exploring the sources of variance of breastfeeding and, to our knowledge, is the first one to conduct a GWAS on breastfeeding and look at the overlap with variants for breast size.
Case-control study of risk factors for human infection with avian influenza A(H7N9) virus in Shanghai, China, 2013
J. LI, J. CHEN, G. YANG, Y. X. ZHENG, S. H. MAO, W. P. ZHU, X. L. YU, Y. GAO, Q. C. PAN, Z. A. YUAN
Published online by Cambridge University Press: 04 December 2014, pp. 1826-1832
The first human infection with avian influenza A(H7N9) virus was reported in Shanghai, China in March 2013. An additional 32 cases of human H7N9 infection were identified in the following months from March to April 2013 in Shanghai. Here we conducted a case-control study of the patients with H7N9 infection (n = 25) using controls matched by age, sex, and residence to determine risk factors for H7N9 infection. Our findings suggest that chronic disease and frequency of visiting a live poultry market (>10 times, or 1–9 times during the 2 weeks before illness onset) were likely to be significantly associated with H7N9 infection, with the odds ratios being 4·07 [95% confidence interval (CI) 1·32–12·56], 10·61 (95% CI 1·85–60·74), and 3·76 (95% CI 1·31–10·79), respectively. Effective strategies for live poultry market control should be reinforced and ongoing education of the public is warranted to promote behavioural changes that can help to eliminate direct or indirect contact with influenza A(H7N9) virus.
Clinical epidemiology and molecular profiling of human bocavirus in faecal samples from children with diarrhoea in Guangzhou, China
D.-M. ZHANG, M.-M. MA, W.-T. WEN, X. ZHU, L. XU, Z.-J. HE, X. HE, J.-H. WU, Y.-W. HU, Y. ZHENG, Y. DENG, C.-J. LIN, J.-H. LU, M.-F. LI, K.-Y. CAO
To understand the clinical epidemiology and molecular characteristics of human bocavirus (HBoV) infection in children with diarrhoea in Guangzhou, South China, we collected 1128 faecal specimens from children with diarrhoea from July 2010 to December 2012. HBoV and five other major enteric viruses were examined using real-time polymerase chain reaction. Human rotavirus (HRV) was the most prevalent pathogen, detected in 250 (22·2%) cases, followed by enteric adenovirus (EADV) in 76 (6·7%) cases, human astrovirus (HAstV) in 38 (3·4%) cases, HBoV in 17 (1·5%) cases, sapovirus (SaV) in 14 (1·2%) cases, and norovirus (NoV) in 9 (0·8%) cases. Co-infections were identified in 3·7% of the study population and 23·5% of HBoV-positive specimens. Phylogenetic analysis revealed 14 HBoV strains to be clustered into species HBoV1 with only minor variations among them. Overall, the detection of HBoV appears to partially contribute to the overall detection gap for enteric infections, single HBoV infection rarely results in severe clinical outcomes, and HBoV sequencing data appears to support conserved genomes across strains identified in this study.
Pig slurry characteristics, nutrient balance and biogas production as affected by separation and acidification
S. G. SOMMER, M. HJORTH, J. J. LEAHY, K. ZHU, W. CHRISTEL, C. G. SØRENSEN, SUTARYO
Journal: The Journal of Agricultural Science / Volume 153 / Issue 1 / January 2015
Animal slurry is separated in order to avoid excessive nitrogen, phosphorus and potassium (NPK) fertilization of crops in the field. To enhance fertilizer efficiency further, slurry and its separation products may be acidified, for instance in animal houses. The current study quantified the effects of these treatments, both individually and in combination, on fertilizer efficiency, energy production and heavy metal accumulation as a result of manure management. Acidification increased the availability of N to plants in the manure applied, and provided a better match between plant-available NPK in the manure and separation fraction applied to fields and crop need. Total biogas production was not affected by separation, whereas acidification reduced biogas production because the process was inhibited by a low pH and a high sulphur concentration. The amount of copper applied per hectare in the liquid manure to the wheat field was lower than the amount taken up and more zink and copper was applied in the solid fraction to maize field than taken up. The transportation and field application of solids and liquids did not increase management costs when compared to the transportation of slurry alone, but the investment and running costs of separators and manure acidification increased overall management costs.
Effects of long-term fertilization on soil organic carbon content and aggregate composition under continuous maize cropping in Northeast China
Z. W. SONG, P. ZHU, H. J. GAO, C. PENG, A. X. DENG, C. Y. ZHENG, M. A. MANNAF, M. N. ISLAM, W. J. ZHANG
Journal: The Journal of Agricultural Science / Volume 153 / Issue 2 / March 2015
Fertilizer application can play an important role in soil organic carbon (SOC) retention and dynamics. The mechanisms underlying long-term accumulation and protection of SOC in intensive maize cropping systems, however, have not been well documented for cool high-latitude rainfed areas. Based on a 23-year fertilization experiment under a continuous maize cropping system at Gongzhuling, Jilin Province, China, the effects of fertilization regimes on SOC content and soil aggregate-associated carbon (C) composition were investigated. Results showed that, within the 0–1·0 m soil profile, SOC contents decreased significantly with soil depth in all treatments. In the topsoil layer (0–0·2 m), SOC concentrations in balanced inorganic fertilizers plus farmyard manure (MNPK), fallow system (FAL) and balanced inorganic fertilizers plus maize straw residue (SNPK) treatments were significantly greater than initial levels by 61·0, 34·1 and 20·1%, respectively. The MNPK and SNPK treatments increased SOC content by 50·7 and 12·4% compared to the unfertilized control in the topsoil layer, whereas no significant differences were found between balanced inorganic nitrogen, phosphorus and potassium fertilizers (NPK) and the unfertilized control treatment. There were no significant differences in aggregate-size distribution among the unfertilized control, NPK and MNPK treatments, whereas the SNPK treatment significantly enhanced the formation of micro-aggregates (53–250 μm) and decreased the formation of silt+clay aggregates (<53 μm) compared to the unfertilized control, NPK and MNPK treatments. Moreover, SOC concentrations in all aggregate fractions in the MNPK treatment were the highest among treatments. Furthermore, the MNPK treatment significantly increased SOC stock in micro- and silt+clay aggregates, which may slow down C decomposition in the soil. These results indicate that long-term manure amendment can benefit SOC sequestration and stability in the black soil of Northeast China. | CommonCrawl |
Prediction of the potency of mammalian cyclooxygenase inhibitors with ensemble proteochemometric modeling
Thérèse E Malliavin1
Cyclooxygenases (COX) are present in the body in two isoforms, namely: COX-1, constitutively expressed, and COX-2, induced in physiopathological conditions such as cancer or chronic inflammation. The inhibition of COX with non-steroideal anti-inflammatory drugs (NSAIDs) is the most widely used treatment for chronic inflammation despite the adverse effects associated to prolonged NSAIDs intake. Although selective COX-2 inhibition has been shown not to palliate all adverse effects (e.g. cardiotoxicity), there are still niche populations which can benefit from selective COX-2 inhibition. Thus, capitalizing on bioactivity data from both isoforms simultaneously would contribute to develop COX inhibitors with better safety profiles. We applied ensemble proteochemometric modeling (PCM) for the prediction of the potency of 3,228 distinct COX inhibitors on 11 mammalian cyclooxygenases. Ensemble PCM models (\(R_{0\ test}^{2}=0.65\), and RMSEtest = 0.71) outperformed models exclusively trained on compound (\(R_{0\ test}^{2}=0.17\), and RMSEtest = 1.09) or protein descriptors (\(R_{0\ test}^{2}=0.16\) and RMSEtest = 1.10) on the test set. Moreover, PCM predicted COX potency for 1,086 selective and non-selective COX inhibitors with \(R_{0\ test}^{2}=0.59\) and RMSEtest = 0.76. These values are in agreement with the maximum and minimum achievable \(R_{0\ test}^{2}\) and RMSEtest values of approximately 0.68 for both metrics. Confidence intervals for individual predictions were calculated from the standard deviation of the predictions from the individual models composing the ensembles. Finally, two substructure analysis pipelines singled out chemical substructures implicated in both potency and selectivity in agreement with the literature.
Prediction of uncorrelated bioactivity profiles for mammalian COX inhibitors with Ensemble Proteochemometric Modeling.
Cyclooxygenases (EC 1.14.99.1), also known as endoperoxidases, prostaglandin G/H synthases or simply COX, are involved in the biosynthesis of prostaglandin H2 from arachidonic acid [1]. Prostaglandin H2 is further converted into prostanoids which play a key role in inflammation. Thus, since the development of aspirin®; in 1899 [2], the inhibition of the cyclooxygenase activity with non-steroidal anti-inflammatory drugs (NSAIDs) has been exploited to treat inflammation. Nonetheless, kidney failure and gastrointestinal side-effects, such as peptic ulcer, have been correlated to long-term intake of NSAIDs [3]. Until 1991, only one form of the enzyme (COX-1) was thought to be responsible for both the constitutive and the local biosynthesis of prostaglandins. In that year [4], an inducible cyclooxygenase (COX-2) was discovered and the different roles of both isoenzymes were revealed. There does exist however some overlap: COX-1 is constitutively expressed serving as the source of housekeeping prostaglandins, whereas the expression of COX-2 increases in pathophysiological situations such as acute pain, inflammation or cancer [5]. From this it is thought that efficacy and side-effects can, to some extent, be delineated when blocking the prostaglandin synthesis pathway associated with inflammation and pain.
In the last two decades, research in both the pharmaceutical industry and academic laboratories has been driven by the hypothesis that selective COX-2 inhibitors would exhibit strong anti-inflammatory and analgesic properties without leading to the unwanted gastrointestinal side effects [6]. Nevertheless, a few organs, e.g. the brain cortex and renal glomeruli, express COX-2 constitutively [1]. The association between the inhibition of COX-2 in these organs with cardiovascular hazard (CVH) was ratified in 2004 and 2005 [7,8]. These findings led the US Food and Drug Agency (FDA) to retrieve rofecoxib (Vioxx) and valdecoxib (Bextra) from the market, and to include boxed warnings for all selective COX-2 inhibitors. Higher risk of heart attack and hypertension have also been reported for non-selective NSAIDs, thus highlighting that cardiovascular risk might not be related to the degree of COX selectivity [9]. In 2012, Yu et al. [10] demonstrated that the cardiovascular risk originates from COX-2 inhibition by selective and not selective NSAIDs and is taking place in blood vessels. These authors have shown that COX-2 inhibition leads to a decrease in prostaglandin (mainly PGI2) and to increased nitric oxide (NO) production which is sufficient to increase the risk of heart failure, hypertension and thrombosis [10].
Nevertheless, there are still niche populations which can benefit from selective COX-2 inhibitors, e.g. patients who cannot afford to take non-selective COX inhibitors, due to an increased risk of peptic ulcers or cancer. In addition, selective COX-2 inhibitors continue to be the common treatment for chronic inflammatory and pain disorders [3,11], and NSAIDs are known to reduce the risk of (among others) [12-15]: colon cancer [16-19], Alzheimer's disease, and platelet aggregation [5,20]. Overall, NSAIDs are still one of the most commonly prescribed drugs in the world [21], and this trend is likely to increase owing to the aging of the population. Therefore, the administration of NSAIDs in clinics is currently subject to a benefit-risk assessment between the patients clinical profile and potential drugs side-effects [22], always aiming at optimizing both the dosage and the duration of the drug regimen [3].
The isoform selectivity of COX inhibitors stems from a structural difference in the binding site. The binding site of both cyclooxygenases is highly conserved except for the substitution of an isoleucine at position 523 in COX-1 with a valine in COX-2 [23]. This substitution results in a larger binding site in COX-2, as the smaller size of valine allows access to a side-pocket. This structural difference has been exploited for the rational design of potent and selective COX-2 inhibitors by both medicinal and computational chemistry [23-25]. To date, a plethora of in silico studies have been published with the aim of better understanding and predicting the potency of COX inhibitors on either COX-1 or COX-2 using molecular docking and QSAR models [26-30]. Nonetheless, none of these studies was able to integrate bioactivity information from multiple mammalian COX in the frame of a single machine learning model. Given that the bioactivity profiles of selective COX inhibitors on COX-1 and COX-2 are highly uncorrelated, thus presenting high selectivity ratios [24,25], only a predictive model trained on both the chemical and the target space would be able to simultaneously predict compound potency on a panel of cyclooxygenases, as well as to predict the activity of a given compound on a yet untested isoform. In that way, new potent, selective and safe COX inhibitors could be discovered.
Proteochemometrics (PCM) constitutes as an approach capable to simultaneously relate the chemical and the target space in single machine learning models in order to predict the bioactivity for a set of compounds against a panel of (related) biomolecular targets [31-33]. This integration of chemical and biological information enables, within the limits of the data presented to the model, the inter- and extrapolation on both the chemical and the target spaces to predict the potency of (novel) compounds on a panel of (novel) targets.
Therefore, the bioactivity of new compounds on yet untested targets can be predicted. These features of PCM make it different from both chemogenomics and QSAR, thus allowing [34,35]: (i) the inclusion of bioactivity information from orthologuous targets [34], (ii) bioactivity prediction for emergent viral mutations [35], or (iii) the design of personalized medicine for e.g. cancer treatment [33].
In this contribution, we apply the principles of PCM to model the potency of 3,228 compounds on 11 mammalian cyclooxygenases. To this aim, we have trained PCM models with different machine learning algorithms on public IC50 values from ChEMBL 16 [36], including data on human COX-1, COX-2, and on 9 orthologues. In an attempt to increase model performance, these models have been combined in ensembles (ensemble modeling), thus constituting the first PCM study where ensemble PCM modeling is applied. Additionally, the description of compounds with keyed fingerprints has enabled the deconvolution of the chemical space to rationalize both the potency and the selectivity of COX inhibitors towards a particular isoenzyme.
IC50 values for 11 mammalian cyclooxygenases, listed in Table 1, were retrieved from ChEMBL 16 [36]. To ensure the reliability of the bioactivity values, only IC50 values corresponding to small molecules and satisfying the following criteria were kept: (i) activity relationship equal to ' =', (ii) assay score confidence ≥ 8, and (iii) activity unit equal to 'nM'. The average pIC50 value was calculated when multiple IC50 values were annotated on the same compound-target combination. The application of these filters led to a final dataset composed of 3,228 distinct compounds and 11 sequences, being the total number of datapoints 4,937 (13.9% matrix completeness). The negative logarithm with base 10 of the IC50 values (pIC50) was used as the response variable to train all models. We decided to mix bioactivity data from different assays given that Kalliokoski et al. [37] reported that the standard deviation of public IC50 data is 25% larger than the standard deviation corresponding to public Ki data, and thus mixing IC50 data from different assays adds a moderate level of noise. The crystallographic structure of the ovine COX-1 complexed with celecoxib (PDB [38] ID: 3KK6 [39]) was used to extract the residues in the binding site. Those residues within a sphere of radius equal to 10Å centered in the ligand were selected.
Table 1 Composition of the COX dataset
The corresponding residues for the other 10 sequences were identified by multiple sequence alignment [40]. The sequence alignment as well as the final residue selection are provided in the supplementary information.
Chemical structures were standardized with the function StandardiseMolecules from the R package camb [41] with the following options: (i) inorganic molecules were removed, and (ii) molecules were selected irrespectively of the number of fluorines, chlorines, bromines or iodines present in their structure, or of their molecular mass. Morgan fingerprints [42,43] were calculated using RDkit (release version 2013.03.02) [44,45]. For the calculation of unhashed Morgan fingerprints [45], each compound substructure in the dataset, with a maximal diameter of four bonds, was assigned to an unambiguous identifier. Subsequently, substructures were mapped into an unhashed (keyed) array of counts. Physicochemical descriptors (PaDEL) [46] were calculated with the function GeneratePadelDescriptors from the R package camb. The R package vegan was used to generate the distributions of pairwise compound similarities (Jaccard distance) [47].
The amino acids composing the binding site of the mammalian cyclooxygenases considered in this study (Table 1), were described with five amino acid extended principal property scales (5 z-scales) [48]. Z-scales were calculated with the R package camb [41].
Machine learning models were built in the R statistical programming language using the packages caret [49] and camb [41]. Model ensembles were created with the help of the R package caretEnsemble [50]. Both the dataset (Additional file 1) and the modeling pipeline coded in R is available in the documentation of the R package camb [41].
Model generation
Descriptors with a variance close to zero were removed with the function RemoveNearZeroVarianceFeatures from the R package camb using a cut-off value equal to 30/1 [41,49,51]. Subsequently, the remaining descriptors were centered to zero mean and scaled to unit variance with the function PreProcess from the R package camb.
The values of the model parameters were optimized by grid search and 5-fold cross validation (CV) [52]. The whole dataset was split into 6 folds by stratified sampling of the pIC50 values. One fold, 1/6, constituted the test set. The remaining folds, 5/6, were used to optimize the values of the parameters in the following way. For each combination of parameters, a model was trained on 4 folds, and the values for the remaining fold were then predicted. This procedure was repeated 5 times, each time holding out a different fold. The values of the parameters exhibiting the lowest average RMSE value along the 5 folds was considered as optimal. Subsequently, a model was trained on the whole training set, 5/6, using the optimized values for the parameters. The predictive power of this model was assessed on the test set, 1/6. To significantly compare the quality of the modeling with different machine learning algorithms, the same folds were used to train all models.
Both single PCM models and model ensembles were used to predict the bioactivities for the test set, and their error in prediction compared. The bioactivity values corresponding to the datapoints in the test set were not considered when building neither the single PCM models not the model ensembles.
In order to assess whether merging the chemical and the target space in a single PCM model enhances model performance, we trained two Random Forest (RF) models using either: (i) only compound descriptors (Family Quantitative Structure-Activity Relationship -QSAR-) [53], or (ii) only target descriptors (Family Quantitative Sequence-Activity Modeling -QSAM-) [53]. Obtaining a high performance with a Family QSAR model would indicate that the bioactivities of a given compound on different targets are correlated. Thus, target descriptors would not contribute to increase model performance. On the other hand, high performance observed for a Family QSAM model would indicate that the bioactivity values only depend on the targets and not on the compounds, i.e. the bioactivities of a set of diverse compounds are correlated on a given target. In this case, compound descriptors would not be required to predict compounds affinity, as target descriptors alone would be sufficient.
Both internal and external validation were performed according to the criteria proposed by Tropsha et al. [54-56], and to the RMSE values (Equation 1). The formulae of the statistical metrics used in the internal \(\left (\text {RMSE}_{\text {int}} \;\text {and}\; q_{\textit {int}}^{2}\right)\) and the external \(\left (\text {RMSE}_{\text {test}}, q_{\textit {test}}^{2}\; \text {and}\; R_{test\ 0}^{2}\right)\) validation are:
$$ RMSE = \sqrt { \frac {\left(y - \widetilde{y}\right)^{2}} {N} } $$
((1))
$$ q^{2} = 1 - \frac {\sum_{i=1}^{N} \left(y_{i} - \widetilde{y}_{i}\right)^{2}} {\sum_{i=1}^{N} \left(y_{i} - \bar{y}\right)^{2}} $$
$$ {R_{0}^{2}} = 1 - \frac {\sum_{i=1}^{N} \left(y_{i} - \widetilde{y}_{i}^{r0}\right)^{2}} {\sum_{i=1}^{N} \left(y_{i} - \bar{y}\right)^{2}} $$
where N represents the size of the training or test set, y i the observed bioactivity values, \(\widetilde {y}_{i}\) the predicted bioactivity values, and \(\bar {y}\) the average values of the response variable for those datapoints included into either the training or the test set, and \(\widetilde {y}^{r0} = s \widetilde {y}\), with \(s = \frac {\sum {y_{i} \widetilde {y}_{i}}} {\sum {\widetilde {y}_{i}^{2}}} \).
Generally, to consider a model as statistically sound, the statistical metrics must satisfy the following criteria: (i) \(q_{\textit {int}}^{2} >\) 0.5, and (ii) \(q_{\textit {test}}^{2}\) and \(R_{test\ 0}^{2} >\) 0.6. \(R_{test\ 0}^{2}\) imposes the regression line to pass through the origin (intercept equal to zero). Although we follow these requirements here, as they serve as general guidelines for the evaluation of model predictive ability [54-56], the cut-off values for \(q_{\textit {int}}^{2}\), \(q_{\textit {test}}^{2}\) and \(R_{test\ 0}^{2}\) should be adjusted and tailored in other studies depending on the dataset to be modeled.
To further assess the reliability of the models in the light of the uncertainty of the bioactivity values [37,57,58], we established the maximum \(R^{2}_{0\ test}\) and \(q^{2}_{\textit {test}}\), and minimum RMSEtest values achievable given: (i) the uncertainty (experimental error) of public IC50 data, and (ii) the number of datapoints in both the training and the test set. The distributions of minimum RMSEtest, and maximum \(q_{\textit {test}}^{2}\), and \(R_{0\:test}^{2}\) values were calculated in the following way.
Firstly, a random sample, A, was generated from the pIC50 values with a size equal to the test set. Secondly, the sample A noisy was calculated by adding to A random noise with mean zero and standard deviation equal to the experimental error. The experimental error required to define the random samples A noisy was taken as 0.68 pIC50 unit, which corresponds to the average standard deviation value for public IC50 datasets, as estimated by Kalliokoski et al. [37] Then, the statistical metrics were calculated for A with respect to A noisy . These steps were repeated 1,000 times, which permitted to define the distributions for the statistical metrics.
The maximum and minimum values of respectively \(R^{2}_{0\ test}/q^{2}_{\textit {test}}\) and RMSEtest were then used to validate model performance on the test set.
If the obtained metrics were beyond the maximum values (for \(q_{\textit {test}}^{2}\) and \(R_{0\:test}^{2}\)) or the minimum values (for RMSEtest) of the corresponding distributions, the model is likely to be over-optimistic [52]. This estimation of the maximum achievable model performance takes into account the range and distribution of the bioactivities present in the data. This is of particular importance as it has been recently reported by Sheridan [59] that (i) certain bioactivity ranges are better predicted than others, and (ii) \({R_{0}^{2}}\) values might be very low if the bioactivity range considered is too narrow, even if the predictions closely match the observed values.
Ensemble modeling
Gradient-boosting machines (GBM) [60], Random Forest (RF) [61], and Support Vector Machines (SVM) [62] were implemented to train a model library. The resulting models were combined in model ensembles using two techniques, namely: greedy optimization and model stacking. Depending on the models considered when training an ensemble, two types of model ensembles were defined: (i) homo-ensembles: the same algorithm was used to train all models composing the ensemble, though the parameter values were different in each model, (ii) hetero-ensembles: the number of distinct algorithms used to train the models combined in the ensemble was greater or equal than 2.
Greedy optimization
Greedy optimization, based on the work of Caruana et al. [63], optimizes the RMSE on the cross-validation predictions on the hold-out folds. These predictions were calculated for all models in the model library. These models were trained on a training set with identical fold composition. Each model was assigned a weight in the following manner. Initially, all models had a weight equal to zero. Afterwards, the weight of a given model was repeatedly incremented by 1 if the subsequent normalized weight vector allowed a closer match between the weighted combination of cross-validated predictions and the observed pIC50 values. This repetition was carried out n times, n=1000 in the present work, and the resulting weight vector was normalized to obtain the final models weighting. The predicted activity for a given compound corresponds to the weighted sum (using the optimized model weight vector) of the predictions generated by the individual models. The final model ensemble was used to predict the activities on the test set, and the error in prediction compared to that of single PCM models on the same set.
Model stacking (MS)
The concept of model stacking is illustrated in Figure 1. In this case, the predictions on the training set calculated with the model library during cross-validation served as descriptors. Thus, a training matrix was defined where rows were indexed by the datapoints in the training set used to train the model library, and columns by the models in the aforesaid library.
Ensemble modeling with model stacking. A. A set of models are trained with diverse machine learning algorithms (Model1.. Model n in the Figure). The predictions of these models on each datapoint in the training set calculated during cross validation, are used as descriptors to create a new training matrix, which rows are indexed by the datapoints in the training set and columns by the models in the library. A machine learning model is trained on this matrix. The resulting model is the model ensemble. B. The model ensemble is then applied on the test set.
A machine learning model was trained on this matrix, irrespective of the algorithms used to generate the model library. This model is then used to predict the bioactivities for the test set, and the RMSE value compared to that of single PCM models on the test set. To predict the activity for a compound from the test set, the individual models composing the ensemble are used to predict its activity (pIC50). These activities are then used as input features to the model stacking ensemble, which will output the predicted pIC50 value by the ensemble. The bioactivity values corresponding to the datapoints in the test set are not considered when building the ensemble. If the selected algorithm has the inherent capability to determine the importance of each descriptor, as for Elastic Net, a vector of weights for the models can be defined. Given that each descriptor corresponds to a particular model, this vector will determine its contribution to the generated ensemble. In the present study we used the following algorithms: linear model, Elastic Net, SVM with linear and radial kernels, and RF.
Estimation of the error of individual predictions
In order to estimate errors for individual predictions, we used the standard deviation of the predictions of the individual models composing a given model ensemble, i.e. ensemble standard deviation (Estd). Previous studies [59,64-66] have highlighted the usefulness of considering the ensemble standard deviation as a domain applicability (DA) measure, specially in the case of RF models, where the calculation of the standard deviation along the trees is straightforward [59,64]. Here, we extend this idea to ensembles composed of models trained with different algorithms (hetero-ensembles). For each datapoint in either the test set or in the hold-out fold in the case of cross-validation, we calculated the standard deviation of the predictions generated with each model conforming the model ensemble. Subsequently, the ensemble standard deviation was scaled with the parameter β. This permits to obtain individual confidence intervals for each prediction, which are thus defined as:
$$ IC = \widetilde{y} \pm E_{std}\ \beta \ \left\{\beta \in \mathbb{R} \ | \ \beta >0\right\} $$
To assess the practical usefulness of the derived confidence intervals, the percentage of datapoints for which the predicted values lied within IC (0<β<4) was calculated. Both the predictions calculated during model training (using the optimal parameter values), i.e. cross-validated predictions, as well as the predictions on the test set were used.
Interpretation of compound substructures
The contribution of chemical substructures to bioactivity on human cyclooxygenases was deconvoluted using a predictive and a Student's method (Figure 2):
Interpretation of compound substructures. A. Predictive method. The average influence on bioactivity of a given substructure is calculated as the difference between the distributions corresponding to: (i) the predicted bioactivity for all compounds containing that substructure, and (ii) the predicted bioactivity using PCM for these compounds, from which that substructure was virtually removed by setting its count to zero. B. Student's method. In this case, the average substructure influence on bioactivity is evaluated as the difference between the mean values of the pIC50 distributions for those compounds presenting and not presenting a given substructure. The statistical significance of this difference was evaluated with a Student's t-test.
Prediction of bioactivity values with and without each compound substructure (predictive method)
This first technique quantifies the contribution of each chemical substructure to bioactivity by calculating the distribution of differences between (i) the predicted bioactivity for all compounds containing a given substructure, and (ii) the predicted bioactivity using PCM for these compounds, from which that substructure was virtually removed [67-72].
To virtually remove a substructure, we iteratively set its count equal to zero in all compound descriptors presenting it. The difference between the predicted bioactivity values in the presence or absence of a given substructure was then calculated. The average value of these differences, weighted by the number of counts of the feature in each compound, corresponds to the average contribution of that feature to bioactivity [68]. The contribution was estimated for all compound features considered in the model. The sign of the difference ({+/-}) indicates whether the feature is respectively beneficial or deleterious for compound bioactivity.
Statistical significance between bioactivity distributions with and without each compound substructure (Student's method)
In order to identify chemical substructures that might not be recognized by the predictive method due to moderate PCM model performance, we also deconvoluted the chemical space in a model-independent way. We created two bioactivity sets, each containing the pIC50 values for either human COX-1 or human COX-2. For each of these bioactivity sets and for each substructure, we defined two distributions, namely:
(i) the distribution A of pIC50 values corresponding to the compounds presenting a given substructure in a given bioactivity set, and (ii) the distribution B of pIC50 values for those compounds not presenting that substructure in the same bioactivity set. The normality of these distributions was assessed with the Shapiro-Wilk test (α=0.05). If both distributions, A and B, followed the Gaussian distribution, a two-tailed t-test for independent samples (α=0.05) was applied to statistically evaluate the difference between them. If the difference was significant, we assumed that the considered substructure has an influence on bioactivity on the isoenzyme associated to the bioactivity set considered.
The sign of the difference between the mean value of A and B indicates whether the presence of the substructure hampers or fosters compound bioactivity on that isoenzyme. Therefore, each substructure was assigned a label, 'deleterious' or 'beneficial', depending on its influence on bioactivity on either COX-1 and COX-2.
Finally, we intended to assess which substructures always increase or decrease compound bioactivity on human COX-1 and COX-2. In that way, substructures identified in the previous step are finally identified as: (i) increasing or decreasing bioactivity on human COX-1, (ii) increasing or decreasing bioactivity on human COX-2, and (iii) increasing or decreasing bioactivity on both human COX-1 and COX-2.
Analysis of the chemical and the target space
Target space
The PCA analysis of the amino acid descriptors of the binding site of the 11 mammalian cyclooxygenases (Table 1) is shown in Additional file 2: Figure S1. Orthologue sequences COX1 and COX2 define two distant clusters. As paralogues display more sequence variability than orthologues, and as small molecules tend to display similar binding within orthologues [73], we hypothesize that merging bioactivities from orthologues and paralogues will lead to more predictive models. In addition, these results indicate that the amino acid descriptors account for structural differences between COX-1 and COX-2.
Chemical space
The initial bioactivity selection from ChEMBL 16 [36], consisted of 6,804 datapoints. As previously highlighted [57], a large number of target-compound combinations in ChEMBL are annotated with more than one bioactivity value, hence the total number of different compound-target combinations after duplicate removal was 4,937.
The standard deviations for the bioactivity values annotated on the same compound-target combination are in less than 2% of the cases higher than two pIC50 units (Additional file 3: Figure S2A), whereas more than 90% of the repeated bioactivities exhibit a standard deviation close to zero (Additional file 3: Figure S2B). Consequently, we decided to take the average of these repeated values instead of the median value: this latter value would be more suitable only if outliers were more aboundant.
Selectivity dataset
As stated in the introduction, the main advantage of a PCM model applied to mammalian cyclooxygenases would be to anticipate the potency of a given compound towards a particular isoenzyme. To ensure that our dataset covered chemical entities with diverse bioactivity profiles on COX-1 and COX-2, we selected all compounds annotated on both human cyclooxygenases. This resulted in a selection of 1,086 compounds, out of a total of 3,228 distinct inhibitors present in the dataset. The scatterplot of the bioactivities of these compounds on human COX-1 against human COX-2 (Figure 3A) reveals that the difference in bioactivity for some compounds depending on the isoenzyme is higher than 4 pIC50 units (upper left corner of Figure 3A). RMSE and \({R_{0}^{2}}\) values for the bioactivities on COX-1 with respect to COX-2 are, respectively, 1.69 pIC50 units and -0.42. As the area above the diagonal of Figure 3A is more populated, there are more compounds with higher activity on COX-2 than on COX-1. Therefore, these data let us conclude that the dataset comprises compounds exhibiting high selectivity towards COX-2. In addition, the overlap between the datapoints in the PCA of the compound descriptors (Additional file 4: Figure S3) indicates that the compounds annotated on the COX targets cover the same regions of the chemical space.
COX inhibitors selectivity on human COX-1 and COX-2. A. Scatterplot corresponding to the comparison of bioactivities against human COX-1 and COX-2 for 1,288 compounds. A large proportion of the compounds present a COX-2/COX-1 selectivity ratio between 2 and 4 pIC50 units. Therefore, the present dataset includes COX inhibitors with highly divergent bioactivity profiles for COX-1 and COX-2 (\({R^{2}_{0}} = -0.42\)). B. Scatterplot of the observed against the predicted pIC50 values for the compounds described in A. Blue squares correspond to the activity on COX-1, whereas orange squares correspond to the activity on COX-2. The PCM models explain more than 59% of the variance (\({R^{2}_{0}} = 0.59\)), thus highlighting the ability of the PCM models to predict the potency of compounds displaying uncorrelated bioactivity profiles on human cyclooxygenases.
PCM validation
Overall, the models obtained with GBM, RF, and SVM (Table 2A and Figure 4) satisfied our model validation criteria, described in Materials and methods (Equations (1) to (3)), namely: \(q_{\textit {int}}^{2} >\) 0.5 and, \(q_{\textit {test}}^{2}\) and \(R_{test\ 0}^{2} >\) 0.6. The performance of the three algorithms is comparable since \(R^{2}_{0\ test}\) values range from 0.60 to 0.61, and RMSEtest from 0.76 to 0.79 pIC50 units between the different models. Interestingly, the predictive power did not vary when using hashed or unhashed fingerprints, being the \(R^{2}_{0\ test}\) and RMSEtest differences smaller than 0.01 in both cases (data not shown). Thus, we decided to rather use unhashed fingerprints as this choice enables an interpretation of the models according to chemical substructures.
Model performance on the test set. RMSEtest (upper panel) and \(R^{2}_{0\ test}\) (lower panel) values for the following models: (group A) single PCM, (group B) Family QSAR and Family QSAM, (group C) individual QSAR, (group D) model ensembles comprising those single PCM models exhibiting the highest predictive power, and (group E) model ensembles comprising the whole model library. Bars are colored according to the groups defined in Table 2. Confidence intervals correspond to the mean value +/- one standard deviation calculated with bootstrapping [74].
Table 2 Internal and external validation metrics (mean values +/- one stardard deviation) for the PCM (A), Family QSAM (B), Family QSAR (B), Individual QSAR models (C), Ensemble PCM models combining the most predictive models (D), and Ensemble PCM models combining the whole model library (E)
To ensure that our modeling results did not arise from chance correlations, we trained models with an increasingly bigger fraction of the bioactivity values randomized (y-scrambling) [77]. The representation of model performance as a function of the percentage of randomized bioactivities is given in Additional file 5: Figure S4. When approximately 35% of the bioactivity values are randomized, \(R^{2}_{0\ test}\) values become negative, which indicates that the relationships found by our models between both the chemical and the target space, and the bioactivity values are not spurious [77].
PCM models are in agreement with the maximum achievable performance
The distributions of the respectively maximum and minimum achievable \(R^{2}_{0\ test}\) and RMSEtest values are depicted in Figure 5. The maximum correlation values \(R^{2}_{0\ test}\) are far from 1, which agrees with observations previously reported for public data [68,78]. The mean of the minimum theoretical RMSEtest values lies between 0.68 and 0.69, which is comparable to the level of uncertainty in public IC50 data reported by Kalliokoski et al. [37] The mean of the distribution of theoretical \(R^{2}_{0\ test}\) values is between 0.67 and 0.69. The minimum RMSEtest and maximum \(R^{2}_{0\ test}\) values obtained with the individual models, 0.76 and 0.61 respectively (Table 2A and Figure 4), thus appear consistent with the underlying uncertainty in the present dataset.
Distribution of theoretical \(R_{0\ test}^{2}\) (A) and RMSE test (B) values. The mean of the \(R_{0\ test}^{2}\) distribution, 0.68, highlights that the uncertainty in public bioactivity data does not permit models displaying \(R_{0\ test}^{2}\) values close to 1. Similar results were obtained for \(q^{2}_{\textit {test}}\). From these data we conclude that the minimum RMSEtest value that a model can achieve without exhibiting overfitting is close to the experimental uncertainty.
PCM outperforms both family QSAR and family QSAM on this dataset
Interestingly, neither the Family QSAR nor the Family QSAM model alone could infer the relationships in the dataset, as the respective \(R^{2}_{0\ test}\) and RMSEtest values were: (i) for Family QSAR: 0.17 and 1.09 pIC50 units, and (ii) for Family QSAM: 0.16 and 1.10 pIC50 units (Table 2B and Figure 4). Taken together, these results suggest that: (i) compound bioactivities on different targets are not correlated, as indicated by the low performance of the Family QSAR model, and (ii) compound bioactivities depend on compounds structure, as highlighted by the low performance of the QSAM model.
PCM outperforms individual QSAR models
We then evaluated on individual targets the usefulness of PCM in comparison with QSAR models (Table 2C and Figure 4). Independent QSAR models for those targets with more than 100 bioactivities, namely: human COX-1 and COX-2, ovine COX-1 and COX-2, and mouse COX-2. The human COX-2 model exhibits a RMSEtest value of 0.78 pIC50 units, which is 0.03 pIC50 units larger than the RMSEtest value for the datapoints annotated on human COX-2 averaged over ten PCM models, namely 0.76 +/- 0.04 pIC50 units. By contrast, the \(R^{2}_{0\ test}\) value drops to 0.54, indicating the higher peformance of PCM. Better correlations are obtained for the individual QSAR models corresponding to both the mouse and the ovine COX-2, for which the \(R^{2}_{0\ test}\) values are 0.57 in both cases, whereas the RMSEtest values are 0.81 and 0.79 pIC50 unit. In contrast, the human and the ovine COX-1 QSAR models cannot relate the descriptor space to the bioactivity values in a statistically sound manner, as they exhibit respective \(R^{2}_{0\ test}\) values of 0.30 and 0.36.
Altogether, these data evidence the versatility of PCM to integrate incomplete information from different protein targets. Furthermore, PCM strongly outperforms one-target and one-space models (Family QSAR, individual QSAR, and Family QSAM) [33].
Model ensembles exhibit higher performance than single PCM models
As the most predictive PCM model exhibited moderately high \(R^{2}_{0\ test}\) and \(q^{2}_{\textit {test}}\) values, as well as moderately low RMSEtest values (Table 2A and Figure 4), we explored the possibility of enhancing model performance by combining different models into a more predictive model ensemble (Table 2D, E and Figure 4). Two ensemble techniques were implemented, namely: greedy optimization and model stacking (MS), previously described in section "Ensemble modeling". To gather a library of diverse models, we trained a total of 282 GBM, RF and SVM models. Each of these models was trained with different parameter values. Hence, the performance of single models ranged from very poor to that of the individual models described above (Table 2A and Figure 4).
Initially, we created ensembles using only the most predictive GBM, RF and SVM models (Table 2D and Figure 4). Overall, all model ensembles (Table 2D) exhibited higher predictive power than single models (Table 2A). The best \(R^{2}_{0\ test}\) value, 0.63, was obtained with the greedy and the MS linear ensemble. The weights for the three models in the greedy ensemble were: (i) GBM: 0.35, (ii) RF: 0.12, and (iii) SVM: 0.53. The MS Elastic Net ensemble displayed the highest predictive power, with a RMSEtest value of 0.72 (Table 2D and Figure 4). The small differences in performance observed between ensembles, with the exception of the RF ensemble are negligible, since, in the experience of the authors [68], the standard deviation observed for the \(R^{2}_{0\ test}\) and RMSEtest values when using different samples during model training are between 0.1 and 0.3. The only model that led to worse results was the RF ensemble, with \(R^{2}_{0\ test}\) and RMSEtest values of 0.58 and 0.77 pIC 50 unit, respectively.
In a second step, ensembles were optimized using all models in the model library, namely 282 (Table 2E and Figure 4). Interestingly, the values of the statistical metrics for all ensembles increased.
The MS SVM ensemble with radial kernel displayed the highest predictive ability, with \(R^{2}_{0\ test}\) and RMSEtest of 0.65 and 0.71 pIC50 unit, which only differs marginally from the minimum theoretical RMSEtest value, namely 0.68 (Figure 5).
Worthy of mention is the lack of performance improvement (data not shown) of homo-ensembles (i.e ensembles created with models trained with the same algorithm but with different parameter values) with respect to the most predictive single models (Table 2A and Figure 4), as the difference in \(R^{2}_{0\ test}\) and RMSEtest values was below 0.01 for both metrics. By contrast, the ensembles exhibiting the highest predictive power on the test set were obtained when combining models with high and low predictive ability. This increase in performance is likely to arise from the fact that these models display uncorrelated resampling profiles, i.e. the predictions calculated on the hold-out folds during cross-validation are not correlated (Figure 6).
Pairwise Pearson correlation for the cross-validation predictions across the model library. The predictive power across the model library is not uniformly distributed, as the predicted values for a large fraction of model pairs are uncorrelated (yellow areas). Therefore, the combination of these models in a model ensemble is expected to lead to higher predictive power than individual models ("wisdom of crowds").
Overall, these data underline the highest predictive power of hetero-ensembles generated with a model library displaying a comprehensive range of predictive abilities.
The ensemble standard deviation enables the definition of informative confidence intervals
Figure 7 displays the percentage of datapoints which predicted values lie within confidence intervals calculated with increasingly larger β values (Equation 4). The ensemble model exhibiting the highest predictive power (RMSEtest: 0.71; \(R^{2}_{0\ test}\): 0.65), namely MS SVM Radial Ensemble, was used to make the predictions and to calculate the confidence intervals. Confidence intervals calculated for the cross-validated predictions (shown as squares in Figure 7) require larger β values to reach a given level of confidence when compared to those calculated on the test set (shown as triangles in Figure 7). This can be seen as the percentage of datapoints for which the true value is within the confidence interval (β=1) for the cross-validated predictions is 40%, whereas this value increases till 70% in the case of the test set. This difference might be due to the fact that predictions on the test set are made with models trained on a larger fraction of the dataset. Nevertheless, the error in prediction on the test set might increase if the compounds present therein were structurally dissimilar. In those cases, a larger β value would be required, with respect to that for the training set, to reach a given confidence level.
Confidence intervals calculated from the ensemble standard deviation of the models present in the model ensembles. The percentage of datapoints which predicted bioactivities lie within confidence intervals calculated with increasingly larger β values (Equation 4), is shown for: (i) the cross validated predictions calculated during model training (Training in the Figure), and (ii) for the predictions on the test set (Test in the Figure) calculated with the most predictive model ensemble, namely "Stacking SVM Radial Ensemble". The percentage of true values lying within the confidence interval derived for a given β value increases with the number of datapoints available during model training. Overall, the confidence intervals derived from the ensemble standard deviation provide an estimation of the reliability of individual predictions, as in practice, this plot can be used to determine the β value corresponding to a given confidence level.
Overall, the percentage of true values lying within the confidence interval derived for a given β value is expected to increase with model performance. Figure 7 can be used to determine the β value corresponding to the confidence interval required by the user.
Ensemble modeling enables the prediction of uncorrelated human COX inhibitor bioactivity profiles
As previously stated, selectivity is a crucial aspect in the discovery and optimization of COX inhibitors. To assess whether PCM models were able to predict the pIC50 values for compounds displaying uncorrelated bioactivity profiles on human COX-1 and COX-2, we predicted the bioactivity values for the 1,086 compounds annotated on both human COX-1 and COX-2. Figure 3B, which displays the observed against the predicted pIC50 values for these compounds, shows that PCM models are able to predict the potency for compounds displaying uncorrelated bioactivity profiles on human cyclooxygenases. Indeed, the \(R^{2}_{0\ test}\) and RMSEtest values calculated for the observed pIC50 values with respect to those predicted by the PCM model are, respectively, 0.59 and 0.76 pIC50 unit.
Subsequently, we analyzed the capability of PCM models to correctly predict the bioactivity for both selective and non-selective compounds. A compound was considered as selective or non selective if the absolute value of the difference between its bioactivity on COX-1 and COX-2 is larger or smaller than 2 pIC50 units. On this basis, 226 compounds were considered as selective, and 860 as non selective. The error in prediction for the non selective compounds was lower than 1 pIC50 unit in 85.4% of the cases, and lower than 0.5 pIC50 unit for 55.6% thereof. On the other hand, the error in prediction was lower than 1 pIC50 unit for 73.23% of the selective compounds, and lower than 0.5 pIC50 unit for 42.9% thereof. When considering a more stringent selectivity cut-off value, namely 3 pIC50 units, we obtained a set of 61 compounds. The error in prediction for this set was lower than 1 pIC50 unit in 66.4% of the cases, and lower than 0.5 pIC50 unit for 40.2% thereof.
Consequently, these data indicate that PCM models are capable to predict the potency for both selective and non selective compounds on human COX-1 and COX-2. In addition, we anticipate that model performance is likely to increase with the inclusion of more bioactivity data in the models.
Model performance per target is related to compound diversity
To further assess model performance on a per target basis, we generated 10 RF models each one trained on a different subset of the whole dataset.
The variation of performance across the 11 cyclooxygenases considered can be related to the compound diversity (Additional file 6: Figure S5).
Human cyclooxygenases, with the highest number of annotated compounds (Table 1), exhibited average RMSEtest values between 0.74 and 0.76 pIC50 unit. For these proteins, the distributions of pairwise compound similarity (Additional file 6: Figure S5) are skewed towards high similarity values, with mean values between 0.75 and 0.85.
Likewise, mouse COX-2 and ovine COX-1 display average RMSEtest values of 0.70 and 0.73 pIC50 unit probably related to the smaller number of compounds annotated on these proteins (Table 1). High predictive ability on mouse COX-2 was expected given the high \(R^{2}_{0\ test}\) value, 0.57, obtained with the individual QSAR model, whereas low performance was expected for ovine COX-1, as the individual QSAR model displayed a \(R^{2}_{0\ test}\) value of 0.36. Unsurprisingly, skewed distributions in compound diversity are observed for mouse COX-2 and ovine COX-1 (Figure 8).
Target-averaged model performance. The number of datapoints is displayed through the size of the squares. Targets annotated with less than 30 compounds or with chemical structures displaying high structural diversity (Oryctolagus cuniculus COX-1, Rattus norvegicus COX-1, Bos taurus COX-1, and Bos taurus) are produced with high mean RMSEtest values. These observations indicate that PCM models are not always able to extrapolate in the chemical or the target space if a given target or compound family is not sufficiently represented in the dataset.
Conversely, ovine COX-2, with 341 annotated compounds, displayed a worse average RMSEtest value, within the 0.80-0.85 pIC 50 unit range (Figure 8). This decrease in performance for ovine COX-2 might be ascribed to the higher dispersion of the pairwise compound similarity distribution with respect to those observed for mouse COX-2 and ovine COX-1 (Additional file 6: Figure S5).
The dependency of model performance on compound diversity is even more contrasted for targets with less than 100 annotated bioactivities. Indeed, the average RMSEtest value for mouse COX-1, with 50 compounds, lies within the 0.57-0.62 range of pIC50 unit and the distribution of compunds diversity is skewed towards high similarity values (Additional file 6: Figure S5). However, the average RMSEtest value increases till 0.80-0.90 pIC50 unit for bovine COX-1 (Additional file 6: Figure S5), annotated with 48 bioactivities and for which the pairwise compound similarity distribution presents several peaks, thus highlighting the structural diversity of the compounds. Finally, targets with less than 30 annotated compounds exhibit multimodal pairwise similarity distributions and, consequently, model performance is low, with standard deviations in the 0.50-1.00 range of pIC50 unit (Figure 8).
Overall, chemical diversity in the training set contributes to enhance the applicability of a PCM model. Nonetheless, a balance needs to be established between this diversity and the number of datapoints to ensure model convergence.
Predictive method
The usage of unhashed fingerprints permitted the deconvolution of the chemical space to determine the influence of compound substructures on bioactivity. Two substructure analysis methodologies were implemented, as described in the section "Interpretation of Compound Substructures". The first approach, predictive method, relies on the PCM model to correctly predict the bioactivity for a compound when a given substructure is virtually removed from a compound descriptor. The second approach, Student's method, is a pipeline designed to statistically assess how the presence of a given substructure influences, on average, bioactivity on the compounds.
Figure 9 shows the contribution to bioactivity of each substructure considered in the model on human COX-1 and COX-2 calculated with the predictive method. Red and blue areas correspond respectively to substructures that, on average, enhance or decrease compound bioactivity. Representative substructures either deleterious or beneficial for bioactivity are also shown. Generally, substructures shown to have an influence on bioactivity display an opposite behaviour depending on the isoenzyme type. For example, a pyrrole ring with aryl substituents in the 2,3-positions (substructure c in Figure 9) is predicted to have a high influence on bioactivity, increasing it on COX-2 and decreasing it on COX-1. This observation is in agreement with the literature as the 2,3-diarylpyrrole series with an halogen substituent in the 5-position acting as electron withdrawing group have been found as selective COX-2 inhibitors [79,80]. The pyrrole moiety with a radical in the 1-position is also found as a selectivity feature towards COX-2 (substructure b in Figure 9). This agrees with the discovery by Khanna et al. [81] of a series of 1,2-diarylpyrroles as potent and selective COX-2 inhibitors.
Influence of compound substructures on potency and selectivity on human COX-1 and COX-2. Rows in the heatmap are indexed by the isoenzyme type whereas columns correspond to compound substructures. Substructures are depicted in red within arbitrary molecules presenting it. The color represents the average influence (pIC50 units) of each substructure on bioactivity. Red corresponds to an average increase in bioactivity, whereas blue indicates a deleterious effect. Well-known chemical moieties, e.g. pyrrole rings (c), were singled out as selectivity determinants. For instance, substructure d is present in sulfonamides such as diflumidone, and substructure b in selective 1,2-diarylpyrroles COX-2 inhibitors.
On the other hand, substructures conferring a deleterious effect could also be identified. substructure e in Figure 9 is represented within compound 3-(1H-indol-5-yloxy)-5,5-dimethyl-4-(4-methylsulfonylphenyl)furan-2-one (CHEMBL322276). This compound is part of a series of 3-heteroaryloxy-4-phenyl-2(5H)-furanones reported as selective COX-2 inhibitors by Lau et al. [82]. Its COX-1/COX-2 selectivity ratio is larger than 4.17, which agrees with the prediction of decreasing bioactivity on COX-1. In general, substructures decreasing bioactivity tend to be small and less informative (e.g. single atoms or substructures with two heavy atoms), than those fostering compound potency.
Student's method
The implementation of the Student's method to deconvolute the chemical space (Figure 10), which evaluates the statistical significance between bioactivity distributions in the presence or absence of each compound substructure, led to the following observations: (i) 74 substructures increase bioactivity on COX-2, (ii) 64 substructures decrease bioactivity on COX-2, (iii) 9 substructures increase bioactivity on COX-1, (iv) 2 substructures decrease bioactivity on COX-1, (v) 1 substructure increases bioactivity on both COX-1 and COX-2, and (vi) 6 substructures decrease bioactivity on both COX-1 and COX-2.
Volcano plots corresponding to the results of the Student's method applied on human COX-1 (A) and COX-2 (B). The size of the points is proportional to the number of molecules in the dataset containing a given substructure. Significant P values are shown in red (two-tailed t-test, α=0.05).
Well-known chemical moieties conferring selectivity to COX-2 were present in this substructure selection. Additional file 7: Figure S6 shows the 20 substructures predicted to have the highest influence to increase bioactivity on human COX-2. For instance, substructures containing thiazole, pyrrole, pyrazole and oxazole rings were enriched for COX-2 [24,25]. Likewise, tri-fluorometil and sulfonamide radicals, which appear in e.g. celecoxib, were also enriched [24]. Substructures predicted to influence in the same way the compound bioactivity on both COX-1 and COX-2 are small, which makes difficult to extract medicinal chemistry knowledge therefrom (Additional file 8: Figure S7).
It is nevertheless remarkable that the output of both methods is contradictory for some substructures. By way of example, substructure d in Figure 9 is considered as deleterious for bioactivity on COX-1 by the predictive method, whereas it is regarded as beneficial by the Student's method. Dannhardt et al. [83] highlighted the key role of the carbonyl moiety for the potency of a series of diarylmethanone compounds on both COX isoenzymes. Nonetheless, Scholz et al. [84] have recently reported a series of ortho-carbaborane derivatives of indomethacin as selective COX-2 inhibitors. Furthermore, substructure d also appears in a series of [2-[(4-substituted or 4,5-disubstituted)-pyridin-2-yl]carbonyl-(5- or 6-substituted or 5,6-disubstituted)-1H-indol-3-yl]acetic acid analogues identified as COX-2 inhibitors [85]. Plausible reasons for this divergence are analyzed in the Discussion section.
Overall, both substructure analysis pipelines have proven to be able to highlight chemical moieties conferring or decreasing potency and selectivity in agreement with the literature.
In this contribution two ensemble modeling techniques, namely greedy optimization and model stacking, have been presented and benchmarked on a PCM dataset comprising the bioactivities of COX inhibitors on 11 mammalian cyclooxygenases (Table 1). PCM has been shown to relate the target and the chemical spaces to bioactivity in a statistically sound manner (Table 2) [54-56]. Family QSAR as well as Family QSAM displayed poor performance (Table 2B and Figure 4).
Three machine learning algorithms (GBM, RF and SVM) have been implemented individually and combined in model ensembles. The application of ensemble modeling has been shown to outperform single machine learning models, the improvement being larger if the three most predictive GBM, RF and SVM models are combined in the same ensemble (Table 2D and Figure 4). Nonetheless, the model stacking (MS) SVM radial kernel model trained on the predictions of a library of 282 single PCM models (Table 2E and Figure 4) displayed the lowest RMSEtest and the highest \(R^{2}_{0\ test}\) values. This non-linear model combination led to a RMSEtest value comparable to the experimental uncertainty of public pIC50 data [37]. It is noteworthy to mention that this ensemble was obtained by combining several hundreds of poor and highly predictive models instead of only the most predictive models of each class, namely GBM, RF and SVM (Table 2D and Figure 4). Therefore, these results suggest that if sufficient computing resources are available, higher predictive ability can be obtained with a large and diverse model library. Given that the ensemble concept is not restricted to any particular machine learning algorithm, the pipeline proposed in this study can be further explored.
The variability in the predictions of the individual models composing model ensembles, quantified by the ensemble standard deviation, served to define informative confidence intervals. Previous studies highlighted the usefulness of this variability as an applicability domain metric [59,64-66]. Here, we have extended this concept to ensembles of models trained on different algorithms (Figure 7). The higher performance of model ensembles has already been observed [86,87]. This phenomenon, usually termed 'wisdom of crowds', arises from the fact that different models provide complementary information. Moreover, the combination of a number of models palliates the effect of extreme predictions by averaging them (regression to the mean), and the chances of obtaining erroneous predicitons with a single model decrease. Interestingly, it has been recently reported in the context of cell line sensitivity prediction [87] that higher performance was obtained by combining moderate predictive models, instead of the most predictive models of each class. This observation has been corroborated in the present study (Table 2E and Figure 4). Overall, the application of ensemble modeling with a model library trained with either the same algorithm but different parameter values (homo-ensemble), or with different algorithms (hetero-ensemble) constitutes a promising alternative to single models in the context of predictive bioactivity modeling.
High predictive ability for compounds displaying uncorrelated bioactivity profiles on COX-1 and COX-2 was attained with both single models and model ensembles (Figure 3B). Therefore, the present study illustrates how the combination of the target and the chemical spaces in a single PCM model improves the prediction of compound potency in the context of multi-target systems. The implications of COX-2 in widespread diseases, e.g. cancer, has prompted the design of potent and selective COX-2 inhibitors since the early 1990s [24,25]. Thus, the suitability of PCM to predict COX inhibitor potency and to integrate multispecies bioactivity data opens new avenues for the design of cyclooxygenase inhibitors.
The two approaches presented in this study for the deconvolution of the chemical space, namely: (i) bioactivity prediction with and without a given compound substructure (predictive method), and (ii) assessment of the statistical difference between the bioactivity distributions corresponding to compounds presenting or not a given compound substructure (Student's method), singled out chemical moieties responsible for COX-2 selectivity in agreement with the medicinal chemistry literature.
The divergent results described for substructure d in Figure 9, plausibly arise from the following properties of the two methods.
As in the predictive method the bioactivity is predicted by calculating the average difference between the predicted value for a compound with and without a given substructure, the (potentially non-linear) relationships between the substructures present in a molecule can be established, and the dependence of bioactivity on additional substructures or scaffolds present in the molecule accounted. On the other hand, the Student's method considers the substructures as independent. The two methods can thus give contrasted results for example in the following case. We can envision a compound, A, presenting a substructure, S 1, having no effect on bioactivity, and a second substructure, S 2, strongly fostering bioactivity on the studied biomolecular target. Additionaly, we consider compound B, which only harbors substructure S 2. Contradictory results would be given by the two methods with respect to the influence of substructure S 1 on bioactivity.
The predictive method would predict a similar bioactivity value for compound A with and without substructure S 1, as the bioactivity depends on substructure S 2. By contrast, the Student's method would consider substructure S 1 as relevant for bioactivity given that the difference between the bioactivities of compounds A and B, i.e. either presenting or not substructure S 1, would be significant. It follows from the preceeding that the predictive method is best suited to give insight into the contribution of single substructures to the bioactivity of individual compounds, whereas the Student's method is more suited for the identification of the general relevance of the substructures to bioactivity. Another important consideration is the presence of substructures whose effects on bioactivity are correlated. In the situation where a compound presents two substructures whose influences on bioactivity are correlated, the predictive method would likely predict a similar activity when either of them is deleted. Covering diverse structures in the dataset might alleviate this issue, as the probability of finding repeated substructure pairs is likely to decrease with chemical diversity and dataset size. Overall, if the general influence of a substructure on bioactivity is assessed with the predictive method, both the mean value and the standard deviation of the differences between the predicted bioactivity values with and without a given substructure should be reported, as the standard deviation indicates whether the influence of that substructure to bioactivity depends on other substructures or not [68].
In the Student's method, the pIC50 difference associated to a significant p-value might be negligible from a medicinal chemistry standpoint. In addition, the capability of the t-test to identify significant differences depends on the sample size. Thus, a small pIC50 difference can be detected as significant if the sample size is large, whereas it might not be detected for smaller samples. Therefore, the conclusions extracted from the application of the Student's method depend on the analyzed dataset, whereas the predictive method might be less dependent on the dataset composition if the models are applied within their applicability domain. In the present study, we have not applied any method to control the family-wise error rate which comes from the multiple comparisons problem [88]. However, we anticipate that in other studies comprising a larger number of substructures, it would be advisable to control this problem. For a recent and detailed discussion of the application of the student t-test to assess the statistical significance of bioactivity differences in the context of Matched Molecular Pair Analysis (MMPA), the reader is referred to Kramer et al. [89]. In summary, the application of both methods can help to unravel whether the contribution of a given substructure to compound bioactivity depends exclusively on itself, or on the presence of other substructures or chemical scaffolds [90].
Ensemble modeling has been introduced in the context of PCM to predict the potency of mammalian cyclooxygenase inhibitors. The combination of single models in model ensembles has led to increased predictive ability, as well as to the definition of confidence intervals for individual predictions. PCM has been shown to enable the prediction of the potency for compounds exhibiting uncorrelated bioactivity profiles with high confidence. Finally, the implementation of two different substructure analysis pipelines, which reliability for different purposes has been pointed out, has permitted the recognition of chemical moieties implicated in potency and selectivity in agreement with the medicinal chemistry literature.
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ICC thanks the Paris-Pasteur International PhD Programme for funding. ICC and TM thank CNRS and Institut Pasteur for funding. DM thanks Unilever for funding. GvW thanks EMBL (EIPOD) and Marie Curie (COFUND) for funding. AB thanks Unilever and the European Research Commission (Starting Grant ERC-2013-StG 336159 MIXTURE) for funding. The authors acknowledge the three anonymous reviewers that contributed to improve the manuscript with their constructive suggestions and comments.
Département de Biologie Structurale et Chimie, Institut Pasteur, Unité de Bioinformatique Structurale; CNRS UMR 3825, 25, rue du Dr Roux, Paris, 75015, France
Isidro Cortes-Ciriano & Thérèse E Malliavin
Centre for Molecular Science Informatics, Department of Chemistry, University of Cambridge, Cambridge, UK
European Molecular Biology Laboratory European Bioinformatics Institute Wellcome Trust Genome Campus, Hinxton, Cambridge, CB10 1SD, UK
Thérèse E Malliavin
Correspondence to Thérèse E Malliavin.
ICC, AB and TM conceived and designed the study. ICC gathered the dataset, trained the models, analyzed the results and produced the figures. DM provided analytical tools for ensemble modeling. ICC, DM, GvW, AB and TM wrote the paper. All authors read and approved the final manuscript.
Additional file 1
Data set S1. The complete dataset used in this study containing: (i) compound descriptors, (ii) amino acid descriptors, and (iii) compound bioactivities.
Figure S1. PCA analysis of the target space. PCA analysis was applied on the binding site descriptors used to train the models. The first two principal components explained more than 80% of the variance, thus indicating that there are mainly two sources of variability in the descriptor space, namely the isoenzyme type. This fact can be seen as COX-1 (triangles) and COX-2 (squares) define two distant clusters. Overall, the binding sites of orthologue cyclooxygenases are more similar than those of paralog sequences. These results also indicate that the amino acid descriptors account for structural differences between COX-1 and COX-2, which can be learnt by the models. Thus, it is expected that merging orthologues and paralogues will lead to more predictive models.
Figure S2. Statistiscs of the repeated bioactivities for each compound-target combination. A. The abcissa represents the mean value for the bioactivities repeated for each compound-target combination with more than one annotated bioactivity. The ordinate represents their standard deviations. Repeated bioactivities are equally distributed for low, moderate and high affinity COX inhibitors. B. Histogram of the standard deviation of the repeated bioactivities. The distribution is strongly skewed towards 0, thus indicating that the differences between repeated bioactivities are generally negligible.
Figure S3. PCA of the compound descriptors used to train the PCM models. The PCA was performed on the pairwise Pearson rank correlation matrix calculated with the compound descriptors used to train the models. The two first principal components (PC) explain 58.03% of the variance. COX-1 and COX-2 are represented with squares and triangles respectively. Overall, the overlap between the datapoints indicate that the compounds annotated on different targets cover the same regions of the chemical space.
Figure S4. Y-scrambling. Scatterplots corresponding to the percentage of bioactivities randomized, against (A) \(R^{2}_{0\ test}\) and (B) RMSEtest values. The intercept in A becomes negative when 25-50% of the bioactivity variable is randomized. This finding indicates that PCM performance is not the consequency of spurious correlations in the descriptor space.
Figure S5. Jaccard pairwise similarity distributions for the compounds annotated on each target. Compounds annotated on the human cyclooxygenases (annotated with a star in the plots) display compound similarity distributions with mean values skewed towards 1. By contrast, compounds annotated on targets with less than 30 annotated bioactivities display multimodal similarity distributions. A correlation between model performance and both the number of datapoints and chemical diversity was established (see main text). Distributions were calculated with the same descriptors than the ones used to train the PCM models.
Figure S6. Compound substructures predicted to increase the bioactivity on human COX-2. The 20 substructures predicted to have the highest influence on bioactivity on human COX-2 (P35354) are plotted. Known chemical moieties such as pyrrole rings (1), aryl substituents (e.g. 4 and 5) or benzylsulfonamide (17) are represented. These substructures appear in diverse NSAIDs such as rofecoxib or etericoxib, as well as in chemical families of COX-2 inhibitors based on e.g. 1,5-diarylpyrazoles or 3,4-diaryl-substituted furans [23-25].
Figure S7. Compound substructures predicted to have the same influence on human COX-1 and COX-2. Substrucutures predicted to decrease bioactivity are accompanied by a blue arrow, whereas that predicted to increase bioactivity are followed by a red arrow. Smaller substructures are found in this case, predominating substituents on the benzene ring. Therefore, substructure-activity relationships are difficult to be determined.
Cortes-Ciriano, I., Murrell, D.S., van Westen, G.J. et al. Prediction of the potency of mammalian cyclooxygenase inhibitors with ensemble proteochemometric modeling. J Cheminform 7, 1 (2015). https://doi.org/10.1186/s13321-014-0049-z
DOI: https://doi.org/10.1186/s13321-014-0049-z
Cyclooxygenases | CommonCrawl |
POPL 2021 (series) / CPP 2021 (series) / Certified Programs and Proofs /
An Anti-Locally-Nameless Approach to Formalizing Quantifiers
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Sun 17 Jan 2021 18:00 - 18:15 at CPP - Logic, Set Theory, and Category Theory Chair(s): Yannick Forster
We investigate the possibility of formalizing quantifiers in proof theory while avoiding, as far as possible, the use of true binding structures, $\alpha$-equivalence or variable renamings. We propose a solution with two kinds of variables in terms and formulas, as originally done by Gentzen. In this way formulas are first-order structures, and we are able to avoid capture problems in substitutions. However at the level of proofs and proof manipulations, some binding structure seems unavoidable. We give a representation with de Bruijn indices for proof rules which does not impact the formula representation and keeps the whole set of definitions first-order.
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Olivier LaurentCNRS & ENS Lyon
Pre-print Media Attached
The Generalised Continuum Hypothesis Implies the Axiom of Choice in Coq
Dominik KirstSaarland University, Felix RechSaarland University
Formalizing Category Theory in Agda
Jason Z.S. HuMcGill University, Jacques CaretteMcMaster University
Olivier Laurent
CNRS & ENS Lyon | CommonCrawl |
BMC Evolutionary Biology
Methodology article
An invariants-based method for efficient identification of hybrid species from large-scale genomic data
Laura S. Kubatko ORCID: orcid.org/0000-0002-5215-71441,2 &
Julia Chifman3
BMC Evolutionary Biology volume 19, Article number: 112 (2019) Cite this article
Coalescent-based species tree inference has become widely used in the analysis of genome-scale multilocus and SNP datasets when the goal is inference of a species-level phylogeny. However, numerous evolutionary processes are known to violate the assumptions of a coalescence-only model and complicate inference of the species tree. One such process is hybrid speciation, in which a species shares its ancestry with two distinct species. Although many methods have been proposed to detect hybrid speciation, only a few have considered both hybridization and coalescence in a unified framework, and these are generally limited to the setting in which putative hybrid species must be identified in advance.
Here we propose a method that can examine genome-scale data for a large number of taxa and detect those taxa that may have arisen via hybridization, as well as their potential "parental" taxa. The method is based on a model that considers both coalescence and hybridization together, and uses phylogenetic invariants to construct a test that scales well in terms of computational time for both the number of taxa and the amount of sequence data. We test the method using simulated data for up 20 taxa and 100,000bp, and find that the method accurately identifies both recent and ancient hybrid species in less than 30 s. We apply the method to two empirical datasets, one composed of Sistrurus rattlesnakes for which hybrid speciation is not supported by previous work, and one consisting of several species of Heliconius butterflies for which some evidence of hybrid speciation has been previously found.
The proposed method is powerful for detecting hybridization for both recent and ancient hybridization events. The computations required can be carried out rapidly for a large number of sequences using genome-scale data, and the method is appropriate for both SNP and multilocus data.
Large-scale genomic data present many challenges in the inference of the evolutionary history of a collection of species. The most notable of these is the development of methods for inferring species-level phylogenetic relationships from multiple gene alignments that simultaneously incorporate the evolutionary processes that are known to contribute to variability in histories for the individual genes. Two important processes are incomplete lineage sorting (ILS) and hybridization [1]. ILS results when two gene copies fail to coalesce in the most recent ancestral population and is commonly modeled by the coalescent process, which provides a link between the species tree and the gene trees that represent the phylogenetic history for each gene [2–4]. In particular, multispecies coalescent theory models probabilities of rooted gene tree topologies within a given rooted species tree topology and has been used to derive the various probability distributions on gene trees given a particular species tree [5–10]. To date, many methods have been proposed for estimation of species phylogeny from multi-locus data based on the coalescent process (e.g., BEST [11], *BEAST [12], STEM [13], MP-EST [14], SNAPP [15], SVDquartets [16] (now implemented in PAUP* [17]), ASTRAL [18], among others).
Hybridization is another evolutionary process that can cause variability in gene trees within the containing species tree. It generally refers to the interbreeding of individuals from distinct populations, resulting in the production of a hybrid species that shares genetic information with both parental species. Hybridization between distinct species can occur for many generations with fertile offspring, making it possible for a new species to be formed. If the hybridization does not result in the formation of a new lineage, the process is termed introgression or introgressive hybridization [19–28]. Despite the earlier belief that hybridization was rare, numerous recent studies have shown that hybrid speciation occurs in both plants and animals [27, 29–38]. Hybridization has been recognized as an important mechanism for the evolution of new species and recent estimates indicate that approximately 25% of plants and 10% of animals hybridize [26, 27, 27, 28, 39]. However, inference of hybridization cannot be based solely on observed gene tree variability since other processes (e.g., incomplete lineage sorting and gene duplication and loss) may contribute to disagreements in single-gene phylogenies [1].
Several models and methods have been developed to detect hybridization. Here we focus on methods specific to gene flow between species (hybridization) and not on methods that are concerned with gene flow within one species (admixture). One group of methods for detecting hybridization involves the identification and removal of hybrids prior to phylogenetic analysis, with the hybrids added to the inferred tree by connecting them to their parental species [40–42]. Joly et al. (2009) [43] developed a method and software (JML; [44]) for identifying introgressed sequences by proposing that for some hybridization events the minimum distance between two sequences will be smaller than for incomplete lineage sorting. Another test that was originally developed to test ancient admixture is based on a relative abundance of ABBA or BABA single nucleotide patterns that can be evaluated using Patterson's D-statistic [45–47]. However, Eaton and Ree (2013) [48] noted that Patterson's D-statistic does not utilize all the information from incongruent allele patterns in multiple taxa and proposed an extension to the method, which they termed partitioned D-statistic. Meng and Kubatko (2009) [49] proposed a model for detecting hybridization under the coalescent model and used both a maximum likelihood and a Bayesian framework for inference. An extension to that model was later provided by Kubatko (2009) [50] by utilizing gene tree densities for inference. Yu et al. (2014) [51] also proposed a likelihood method that accounts for both reticulate evolutionary events and incomplete lineage sorting by providing methods for computing the likelihood of a phylogenetic network under the coalescent model. This method, as well as some earlier variations of it, is implemented in the software PhyloNet [52].
In this paper we develop a method for detecting and quantifying the extent of hybridization using a coalescent-based model that is fast and accurate. At the heart of our method are special relations called phylogenetic invariants, which are functions (usually polynomials) in the site pattern probabilities that evaluate to zero on any probability distribution that is consistent with the tree topology and associated model. Invariants have been introduced by Cavender and Felsenstein (1987) [53] and Lake (1987) [54] as a means for phylogenetic reconstruction, and have recently been gaining popularity for use in phylogenetic tree inference [16, 55, 56]. Here we propose using a ratio between two linear invariants in site pattern probabilities to develop statistics that accurately identify hybrid taxa. Because these statistics are functions of site pattern probabilities across multi-locus or SNP data, they can be rapidly computed. In addition, we can derive the mean, variance, and asymptotic distribution of these invariants, enabling development of a hypothesis test for hybridization when the number of sites is large. We begin by giving the theoretical details of our model, and then evaluate the performance of several possible invariants-based statistics for four-taxon trees using simulation. The best-performing of these statistics, which we call the Hils statistic, is then evaluated for larger trees using simulation, with hybridization events at various "depths" of the tree (i.e., hybridization between tip species and hybridization between ancestral species). Finally, we apply our method to several empirical data sets, including the Sistrurus rattlesnakes and Heliconius butterflies.
A Coalescent-based Model for Hybridization
We consider here the model originally proposed by Meng and Kubatko (2009) in which data arise along a phylogenetic species tree via an evolutionary process that allows for the possibility of both hybridization and incomplete lineage sorting, as modeled by the coalescent process. Hybridization cannot be modeled by a bifurcating phylogenetic tree, thus it is common to represent hybridization on a phylogeny by a horizontal line connecting two lineages of an otherwise-bifurcating phylogeny (Fig. 1 the leftmost panel). This network represents the evolutionary history of the species as a whole, and depicts a hybrid origin for taxon H. We refer to species H as the hybrid species, and to species P1 and P2 as the parental species. The times labeled by τi are speciation times, and in general we refer to the species network Sγ together with its vector of speciation times τ by (Sγ,τ). The data arising along this phylogenetic species network are a collection of site patterns. Letting XY∈{A,C,G,T} denote the nucleotide observed for species Y at a specific location in the DNA sequence, we define a site pattern \(\mathbf {X} = X_{O} X_{P_{1}} X_{H} X_{P_{2}}\phantom {\dot {i}\!}\) as an assignment of nucleotides to all species. We represent the site pattern probability on the species network (Sγ,τ) for a particular observation ijkl at the tips of the network by
$$ p_{ijkl | (S_{\gamma},\boldsymbol{\tau})} = P(X_{O}=i, X_{P_{1}}=j, X_{H}=k, X_{P_{2}}=l | (S_{\gamma},\boldsymbol{\tau})) $$
Coalescent model with hybridization. Model for the species-level relationships among four taxa under the coalescent model with hybridization. Here taxon H is a hybrid of taxa P1 and P2
for i,j,k,l∈{A,C,G,T}.
Our model defines the probability distribution on the space of all 44=256 site patterns under a model that allows both ILS and hybridization via a three-stage process. First, the hybrid species is assigned one of its two putative parents, with probability γ of selecting parental species P2 and probability 1−γ of selecting parental species P1 (resulting in trees S1 and S2 in Fig. 1 being the "parental species trees", respectively). Next, a gene tree is generated along the parental species tree from step 1 through the standard coalescent process (see, e.g., [2–5, 7, 9, 57, 58]). Finally, a site pattern is generated along the gene tree from step 2 according to one of the standard Markov substitution models (e.g., the GTR+I+ Γ model [59] or one of its sub-models). Combining steps 2 and 3, we see that the probability for site pattern ijkl for a given species tree Si, i∈{1,2}, is given by
$$ p_{ijkl | (S_{i},\boldsymbol{\tau})} = \sum\limits_{G} \int_{\mathbf{t}} p_{ijkl | (G, \mathbf{t})} f((G, \mathbf{t}) | (S_{i},\boldsymbol{\tau})) d \mathbf{t}, $$
where (G,t) represents a gene tree with topology G and branch lengths t, pijkl|(G,t) is the probability of the particular observation ijkl at the tips of gene tree (G,t), and f((G,t)|(Si,τ)) is the joint density of (G,t) conditional on the species tree (Si,τ). A full description of the computations required for this model are given in Chifman and Kubatko (2015) [60], and we do not review them here. Finally, we write the site pattern probability on a hybrid species network as
$$ p_{ijkl | (S_{\gamma},\boldsymbol{\tau})} = \gamma p_{ijkl | (S_{1}, \boldsymbol{\tau})} + (1-\gamma) p_{ijkl | (S_{2}, \boldsymbol{\tau})}. $$
For our purposes, it suffices to view the collection of site patterns observed in an empirical data set as a sample of observations from the probability distribution defined by the \(\{p_{ijkl | (S_{\gamma },\boldsymbol {\tau })} | i,j,k,l \in \{A,C,G,T\} \}\). We call data generated in this way "coalescent independent sites" and refer to this model as the "coalescent independent sites model".
Let NX be the number of sites with site pattern X observed in a sample of N sites generated from hybrid species network (Sγ,τ) under this coalescent-with-hybridization model. Define \(\mathbf {p} = (p_{AAAA|(S_{\gamma },\boldsymbol {\tau })}, p_{AAAC|(S_{\gamma },\boldsymbol {\tau })}, \ldots, p_{TTTT|(S_{\gamma },\boldsymbol {\tau })})\) and \(\hat {\mathbf {p}} = (\hat {p}_{AAAA}, \hat {p}_{AAAC}, \ldots,\hat {p}_{TTTT})\), where \(\hat {p}_{\mathbf {X}} = \frac {N_{\mathbf {X}}}{N}\). The vector \(N\hat {\mathbf {p}}\) then gives the observed counts of the 256 possible site patterns in the sample, and thus
$$ N\hat{\mathbf{p}} \sim \textup{Multinomial}(N; \mathbf{p}). $$
When N is large, the \(\hat {p}_{\mathbf {X}}\) are approximately normally distributed, and thus the sampling distributions of statistics based on the \(\hat {p}_{\mathbf {X}}\) can be derived. We next describe how these ideas can be used to build tests for hybridization.
Invariants-based Hypothesis Tests for Hybridization
As mentioned in the Introduction, our tests are based on phylogenetic invariants, which are polynomials in the site patterns that evaluate to zero on one tree topology but do not evaluate to zero for at least one tree of a different topology.
Consider four linear relationships that arise on the hybrid phylogenetic species network (Sγ,τ) as described in the previous section:
$$\begin{array}{*{20}l} f_{1} &=p_{iijj| (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})}, & f_{3} &= p_{ijii| (S_{\gamma},\boldsymbol{\tau})} - p_{iiji| (S_{\gamma}, \boldsymbol{\tau})}, \\ f_{2} &=p_{ijji | (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})}, & f_{4} &= p_{iiij| (S_{\gamma},\boldsymbol{\tau})} - p_{iiji| (S_{\gamma},\boldsymbol{\tau})}, \end{array} $$
where i≠j∈{A,C,G,T}. It can be shown that f2 and f4 are zero when evaluated on site pattern probabilities that correspond to the species tree S1, while f1 and f3 are non-zero (see [60] for details). Similarly, f1 and f3 are zero when evaluated on site pattern probabilities that correspond to tree S2, while f2 and f4 are not. However, when the site pattern probabilities correspond to the species network (Sγ,τ) with γ∈(0,1), none of the four linear relations are zero.
What is special about these functions is that their ratio is a function of γ∈(0,1):
$$ {{} \begin{aligned} \frac{f_{1}}{f_{2}}&=\frac{p_{iijj| (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})} }{p_{ijji | (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})} }\\ &=\frac{\gamma \left(p_{iijj| (S_{1}, \boldsymbol{\tau})}-p_{ijij| (S_{1}, \boldsymbol{\tau})}\right) + (1-\gamma) \left(p_{iijj | (S_{2}, \boldsymbol{\tau})}-p_{ijij| (S_{2}, \boldsymbol{\tau})}\right)}{\gamma \left(p_{ijji| (S_{1}, \boldsymbol{\tau})}-p_{ijij| (S_{1}, \boldsymbol{\tau})}\right) + (1-\gamma) \left(p_{ijji | (S_{2}, \boldsymbol{\tau})}-p_{ijij| (S_{2}, \boldsymbol{\tau})}\right)}\\ &=\frac{\gamma \left(p_{iijj| (S_{1}, \boldsymbol{\tau})}-p_{ijij| (S_{1}, \boldsymbol{\tau})}\right) + (1-\gamma)(0)}{\gamma (0)+(1-\gamma) \left(p_{ijji | (S_{2}, \boldsymbol{\tau})}-p_{ijij| (S_{2}, \boldsymbol{\tau})}\right)} \\ &=\frac{\gamma}{1-\gamma}. \end{aligned}} $$
Notice that the last equality holds because \(p_{ijji | (S_{2}, \boldsymbol {\tau })}-p_{ijij| (S_{2}, \boldsymbol {\tau })}=p_{iijj| (S_{1}, \boldsymbol {\tau })}-p_{ijij| (S_{1}, \boldsymbol {\tau })}\), which results from the symmetric roles of P1 and P2 leading to \(p_{ijji | (S_{2}, \boldsymbol {\tau })} = p_{iijj| (S_{1}, \boldsymbol {\tau })}\) and \(p_{ijij| (S_{2}, \boldsymbol {\tau })} = p_{ijij| (S_{1}, \boldsymbol {\tau })}\). A full explanation about linear relations under the coalescent model on species trees that satisfy the molecular clock is provided in Chifman and Kubatko (2015), Section 3.1 [60]. Using a similar argument we find that
$$ \frac{f_{3}}{f_{4}}=\frac{\gamma}{1-\gamma} \quad \text{and} \quad \frac{f_{1} + f_{3}}{f_{2} + f_{4}}=\frac{\gamma}{1-\gamma}. $$
If we consider cumulative site pattern probabilities then the results in Eqs. (4) and (5) still hold. By a cumulative site pattern we mean, for example, \(p_{ijji|(S_{\gamma }, \boldsymbol {\tau })} = {\sum \nolimits }_{x \neq y \in \{A, C, G, T\}} p_{xyyx| (S_{\gamma }, \boldsymbol {\tau })}\). Under the JC69 model [61], each of the terms in the sum will have the same value, regardless of the choice of x and y; under more complex models, these probabilities will vary depending on the particular x and y. We implement the JC69 version of the test here, though we use simulation to assess the performance under more complicated models.
Using the ratios in Eqs. (4) and (5) we construct formal significance tests of the following hypotheses:
$$H_{0}: \gamma = 0 \text{ vs.}\ H_{1}: \gamma > 0. $$
Here we consider the ratio \(\frac {f_{1}}{f_{2}}\) to illustrate the procedure. First, we estimate this ratio using the site pattern probabilities observed in the sample,
$$ \frac{\hat{f}_{1}}{\hat{f}_{2}} = \frac{\hat{p}_{iijj} - \hat{p}_{ijij}}{\hat{p}_{ijji} - \hat{p}_{ijij} }. $$
To use this estimator as a test statistic in a hypothesis test, we need the distribution of the statistic when the null hypothesis is true. We first consider distributional results for the numerator and denominator separately. Using standard results for the multinomial distribution, we have
$$\begin{array}{@{}rcl@{}} \mu_{f_{1}} & := & E(\hat{p}_{iijj} - \hat{p}_{ijij}) = p_{iijj| (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \mu_{f_{2}} & := & E(\hat{p}_{ijji} - \hat{p}_{ijij}) = p_{ijji | (S_{\gamma},\boldsymbol{\tau})} - p_{ijij| (S_{\gamma},\boldsymbol{\tau})}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma^{2}_{f_{1}} & := & Var(\hat{p}_{iijj} - \hat{p}_{ijij}) = \frac{1}{N}(p_{iijj| (S_{\gamma},\boldsymbol{\tau})}(1-p_{iijj| (S_{\gamma},\boldsymbol{\tau})}) \\ & &\!\!\!\!\!\!\!\!\!\!\! + p_{ijij| (S_{\gamma},\boldsymbol{\tau})}(1-p_{ijij| (S_{\gamma},\boldsymbol{\tau})}) + 2p_{iijj| (S_{\gamma},\boldsymbol{\tau})} p_{ijij| (S_{\gamma},\boldsymbol{\tau})}), \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma^{2}_{f_{2}} & := & Var(\hat{p}_{ijji} - \hat{p}_{ijij}) = \frac{1}{N} (p_{ijji | (S_{\gamma},\boldsymbol{\tau})}(1-p_{ijji | (S_{\gamma},\boldsymbol{\tau})}) \\ & & \!\!\!\!\!\!\!\!\!\!\!\!+ p_{ijij| (S_{\gamma},\boldsymbol{\tau})}(1-p_{ijij| (S_{\gamma},\boldsymbol{\tau})}) + 2p_{ijji | (S_{\gamma},\boldsymbol{\tau})} p_{ijij| (S_{\gamma},\boldsymbol{\tau})}), \end{array} $$
$$\begin{array}{@{}rcl@{}} \sigma_{f_{1},f_{2}} & := & cov(\hat{p}_{iijj} - \hat{p}_{ijij}, \hat{p}_{ijji} - \hat{p}_{ijij}) \\ & & = \frac{1}{N}(-p_{iijj| (S_{\gamma},\boldsymbol{\tau})} p_{ijji | (S_{\gamma},\boldsymbol{\tau})} + p_{iijj| (S_{\gamma},\boldsymbol{\tau})} p_{ijij| (S_{\gamma},\boldsymbol{\tau})} \\ & & \!\!\!\!\!\!\!\!\!\!\!+ p_{ijji | (S_{\gamma},\boldsymbol{\tau})} p_{ijij| (S_{\gamma},\boldsymbol{\tau})} + p_{ijij| (S_{\gamma},\boldsymbol{\tau})}(1-p_{ijij| (S_{\gamma},\boldsymbol{\tau})})). \end{array} $$
Now, using the fact that when the sample size N is large we have \(\hat {f}_{1} \sim N(\mu _{f_{1}}, \sigma ^{2}_{f_{1}})\) and \(\hat {f_{2}} \sim N(\mu _{f_{2}}, \sigma ^{2}_{f_{2}})\), we apply the Geary-Hinkley transformation [62, 63] to the ratio \(\frac {\hat {f}_{1}}{\hat {f}_{2}}\) to get
$$ \frac{\left(\mu_{f_{2}} \frac{\hat{f}_{1}}{\hat{f}_{2}} - \mu_{f_{1}}\right)}{\sqrt{\sigma_{f_{2}}^{2} \left(\frac{\hat{f}_{1}}{\hat{f}_{2}}\right)^{2}-2\sigma_{f_{1},f_{2}}\frac{\hat{f}_{1}}{\hat{f}_{2}}+\sigma_{f_{1}}^{2}}} \sim N(0,1). $$
The terms in the denominator on the left-hand side of the above equation depend on several unknown quantities, which we estimate by substituting the observed site pattern frequencies into Eqs. (7) - (11). We also multiply the expression in Eq. (12) by \(\frac {\hat {f_{2}}}{\mu _{f_{2}}}\) (which converges in probability to 1, and thus does not change the asymptotic distribution) to obtain the test statistic
$$ H := \frac{\hat{f}_{2}(\frac{\hat{f}_{1}}{\hat{f}_{2}} - \frac{\mu_{f_{1}}}{\mu_{f_{2}}})}{\sqrt{\hat{\sigma}_{f_{2}}^{2} (\frac{\hat{f}_{1}}{\hat{f}_{2}})^{2}-2\hat{\sigma}_{f_{1},f_{2}}\frac{\hat{f}_{1}}{\hat{f}_{2}}+\hat{\sigma}_{f_{1}}^{2}}}. $$
We call the statistic H the Hils statistic, in honor of Professor Matthew H. Hils a. Under the null hypothesis that γ=0, the term \(\frac {\mu _{f_{1}}}{\mu _{f_{2}}}\) in the numerator of (13) is 0, and the hypothesis test can be carried out by comparing the observed value of the test statistic computed with \(\frac {\mu _{f_{1}}}{\mu _{f_{2}}}=0\) to a standard normal distribution. Tests based on the ratios \(\frac {f_{3}}{f_{4}}\) and \(\frac {f_{1}+f_{3}}{f_{2}+f_{4}}\) can be derived analogously.
We note that γ=1 also implies the absence of hybridization, and thus our hypothesis test should consider this situation as well. In fact, the symmetry in the model in Fig. 1 means that this case is already covered by the test above. To see this, note that when γ=0, \(\hat {f}_{1}\) is close to 0, and the hypothesis test will fail to reject the null hypothesis, as could be expected from inspection of Eq. (13). When γ=1, then \(\hat {f}_{2}\) is close to 0. It is not obvious from Eq. (13) that the test statistic would be expected to be close to 0 in this case, but if one multiples both the numerator and the denominator of the test statistic in Eq. (13) by \(\frac {\hat {f}_{2}}{\hat {f}_{1}}\), it can be observed that there is an equivalent version of the test statistic with \(\hat {f}_{2}\), rather than \(\hat {f}_{1}\), in the numerator. Note that our condition \(\hat {p}_{ijij}>max\{\hat {p}_{iijj}, \hat {p}_{ijji} \}\) (see below) ensures that both \(\hat {f}_{1}\) and \(\hat {f}_{2}\) are positive. Thus the test given above is sufficient to test for hybridization with either γ=0 or γ=1.
Extension to Larger Species Networks
The hypothesis test derived in the previous section deals with the case in which four taxa are specified, with one of the four taxa identified as the putative hybrid species. In many settings, however, primary interest is in searching over a large collection of species with the goal of identifying which species might have arisen via a process that involved hybridization at some point in the past. To address this, we consider a large collection of sequences, and suppose that an outgroup sequence can be identified. For each subset of four sequences consisting of three sequences plus the outgroup, we carry out the above test of hybridization for different assignments of the three ingroup sequences to the hybrid and parental taxa. Of the three possible choices for the hybrid taxon, we consider only two of those, eliminating from consideration the one for which \(\hat {p}_{ijij}>max\{\hat {p}_{iijj}, \hat {p}_{ijji} \}\), since this implies that the two parental taxa are more closely related than either is to the putative hybrid. For a data set of n+1 sequences with one outgroup sequence, this results in \(\binom {n}{3}\times 2\) hypothesis tests. To handle the issue of multiple comparisons, we use the Bonferroni correction, which is conservative in this case because the tests are correlated. Thus, if an overall α-level test is desired, we report significant evidence of hybridization when the p-value computed for a particular comparison is smaller than \(\frac {\alpha }{ \binom {n}{3} \times 2}\).
The simulation design for each study is described in the "Methods" section and all code used to carry out the simulations and empirical analyses in this paper is available at https://github.com/lkubatko/HilsTest.
Four-taxon simulation studies
Our results for the four-taxon simulation studies establish that the various tests behaved as we have expected (Fig. 2 and Table 1). First, in all of the cases considered, the power increases as the sample size increases, reaching near 100% when alignments of length 500,000bp were used for many of the simulation conditions (Fig. 2). Second, we note that as the value of γ increases from 0 (no hybridization) to 0.5 (equal contribution from both parental species), the power to detect hybridization increases as well, with near 100% power for the "long" branch length setting when γ≥0.3 for all three of the tests considered. Third, we note that all of the tests are more powerful for data simulated under the "long" branch length setting (Fig. 2e, f, g, and h) than for data generated under the "short" branch length setting (Fig. 2a, b, c, and d). Finally, we note that all tests appear to achieve the nominal 0.05 level when data are simulated under the null hypothesis (γ=0). The ABBA-BABA test (Fig. 2d and h) shows power similar to our test based on the ratio \(\frac {f_{1}}{f_{2}}\) (Fig. 2a and e).
Power Plots. Results of the power simulations for the four-taxon hybrid species network in Fig. 1. Plots (a), (b), (c), and (d) correspond to data simulated for the "short" branch length setting, and plots (e), (f), (g), and (h) correspond to data simulated for the "long" branch length settings. Plots (a) and (e) give results for the test based on \(\frac {f_{1}}{f_{2}}\); plots (b) and (f) give results for the test based \(\frac {f_{3}}{f_{4}}\); plots (c) and (g) give results for the test based on \(\frac {f_{1}+f_{3}}{f_{2}+f_{4}}\); and plots (d) and (h) give results for the ABBA-BABA test
Table 1 Estimates of the parameter γ using the ratio \(\frac {f_{1} }{ f_{2}}\) for data simulated on the four-taxon hybrid species network in Fig. 1 with the "short" and "long" branch lengths settings
One unexpected result of the simulations designed to address the power was that the test based on \(\frac {f_{1}}{f_{2}}\) is more powerful than the tests based on \(\frac {f_{3}}{f_{4}}\) and \(\frac {f_{1}+f_{3}}{f_{2}+f_{4}}\). This is most likely due to the variance associated with estimating the various site pattern probabilities that contribute to each invariant. We return to this point in the discussion. Based on this observation, we report results for only the ratio \(\frac {f_{1}}{f_{2}}\) in what follows.
The results of the four-taxon simulation studies designed to estimate γ using the ratio \(\frac {f_{1}}{f_{2}}\) also matched our intuition about how the method should perform (Table 1). As the sample size increases, the estimates become closer to the true values used to generate the data, and the variance decreases as the sample size increases. In general, the estimates obtained from the "long" branch length setting are slightly better than those obtained from data generated under the "short" branch length setting. Overall, the method seems to provide very reasonable estimates of γ.
The results of the second set of simulation studies are shown in Fig. 3. The results are in general consistent with the results of the first simulation study. In particular, the power increases as γ gets closer to 0.5 and as the sample size increases, and both tests are more powerful when the branch lengths are longer. The Hils test is slightly more powerful than the ABBA-BABA test over most of the simulation conditions examined, but from a practical viewpoint, little difference in performance of the two methods would be expected. While both tests show some decrease in power resulting from the violation of the molecular clock, both still perform well, particularly with sufficient data, suggesting that these methods have some degree of robustness to violation of the assumption of a molecular clock.
Power Plots. Results of the second set of power simulations for the four-taxon hybrid species network in Fig. 1. Plots (a) and (b) correspond to data simulated for the "short" branch length setting, and plots (c) and (d) correspond to data simulated for the "long" branch length setting. Plots (a) and (c) correspond to data simulated under a model species tree in which the length of the branch leading to species P1 is doubled; plots (b) and (d) correspond to data simulated under a model species tree in which the length of the branch leading to species H is doubled. In each plot, the solid lines show results for the Hils test, while the dotted lines show results for the ABBA-BABA test
Simulation studies for larger species networks
For the 9-taxon simulations (Fig. 4 and Table 2), we note first that for data generated under the coalescent independent sites model, when γ=0 approximately 5% of the data sets give significant results, and thus the test appears to attain the desired significance level in this case. For the multilocus data sets, however, the type I error rate is larger than the specified 0.05 level, and thus the test appears to reject the null hypothesis more often than it should. When γ>0, we see that the test is powerful for both the shallow and the deep hybridization events and for both types of data, with the power above 90% in both cases when γ≥0.2. Furthermore, the test almost always selects the correct assignment of hybrid and parental taxa, with the proportion of times that this is exclusively generated increasing toward 100% as γ increases for the coalescent independent sites data. One observation we made that is not reflected in the results in Table 2 is that for data simulated from the network involving the deep hybridization event, many sets appear as significant when some true relationship is detected. For example, it is common to have the hybrid correctly assigned, but the parental species assigned as belonging to a taxon from the sister clade of the true parent. This is especially true for the multilocus data sets with the deep hybridization event. In other words, this test is good at picking out the hybrid taxon, but not as good at unambiguously picking out its parents when the hybridization event occurs deeper in the network. This was not the case for the shallow event, where it often got exactly the correct relationships and only those in most cases.
Trees for Simulation Study. Model networks with 9 and 20 taxa and with either shallow or deep hybridization used for the simulation studies. a 9-taxon shallow hybridization b 9-taxon deep hybridization c 20-taxon shallow hybridization d 20-taxon deep hybridization
Table 2 Results of the simulation study for 9 taxa
The results for the 20-taxon networks are largely the same (Fig. 4 and Table 3). The test still demonstrates good power to detect the hybridization event, though the power does not rise above 90% for all settings until γ≥0.3, rather than 0.2 as in the 9-taxon case. In addition, the proportion of data sets with "Correct Sets" decreases for the shallow hybridization events in this case, meaning that when a hybridization event is identified, it nearly always involved correct identification of which species was the hybrid and which were the parental species. Though there is a hint of an elevated type I error rate when multilocus data were simulated, the problem is not as dramatic as in the 9-taxon case. Overall, the method maintains its good ability to detect hybrid species.
Table 3 Results of the simulation study for 20 taxa
Empirical data: Sistrurus rattlesnakes
Recall that this dataset contains two species, each containing three subspecies, as well as two outgroup species, for a total of eight tips in the species phylogeny of interest. When analyzing empirical data of this nature, for which several individuals are sampled within each species, our main interest will be in detecting individuals that show evidence of hybrid origin from parental individuals that are members of two different species. The current version of our software will output the test statistic for all assignments of hybrid and parental taxa for a given outgroup, but this output can easily be examined to consider only the comparisons of interest. For the rattlesnake data for a particular choice of outgroup, we can consider all choices of one individual allele from each of three subspecies, and for each such choice, one individual will be assigned to be the hybrid and the other two assigned to be the parental taxa. For example, we can select one Sca individual, one Sce individual, and one Sct individual, and carry out the Hils test for each possible choice of hybrid among these three. Thus, for our particular data set consisting of 18 Sca alleles, 8 Sce alleles, 10 Sct alleles, 2 Smm alleles, 6 Smb alleles, and 4 Sms alleles, there will be \({\sum \nolimits }_{n_{i} \in \{0,1\}, \sum \nolimits n_{i}=3} \binom {18}{n_{1}}\binom {8}{n_{2}}\binom {10}{n_{3}}\binom {2}{n_{4}}\binom {6}{n_{5}}\binom {4}{n_{6}} = 7,840\) possible choices of three alleles, and two test statistics will be computed on each, resulting in 2∗7840=15,680 possible comparisons for each choice of outgroup sequence. We carry out the Bonferroni correction within the analysis for each outgroup, and thus each comparison uses significance level α=0.05/15680=0.0000032.
An additional practical issue that arose with our empirical data but was not observed with simulated data was that for some choices of three alleles, one or more of the site pattern frequencies piijj,pijij, and pijji was observed to be 0. To correct for this, we added a small count (0.005) to each observed site pattern count in all cases before computing estimated site pattern frequencies and carrying out the test. With this modification, we find no evidence of hybrid origin for any of the sequences with any choice of outgroup sequence, consistent with other analyses in this group [64, 65].
Empirical data: Heliconius butterflies
This dataset consists of 3 species with 4 individuals sampled per species, plus an outgroup. Thus, the number of comparisons of interest is 4·4·4·2=128 and the Bonferroni-corrected level of the tests is 0.05/128=0.00039. The analysis of all possible hybrid/parental combinations for the alignment of length ≈ 248 million bp took 16 min on a 2 × Quad Core Xeon E5520 / 2.26GHz / 32GB desktop linux machine. All comparisons were statistically significant at the 0.00039 level. This result is not surprising, given the previous evidence of hybridization as described in Martin et al. (2013), and given the large sample size. What is interesting, however, is the strength of the evidence for hybridization. For example, across all comparisons in which an H. m. rosina individual was specified as the hybrid, the smallest test statistic was 172.6143, indicating overwhelming evidence for hybridization (recall that we are comparing to a standard normal distribution). In contrast, when one of the other species was identified as the hybrid and H. m. rosina was (incorrectly) identified as a parental taxon, the values of the test statistic ranged from ∼ 55 to 76, again indicating strongly significant deviation from the expected patterns under no gene flow, but not as strong as the case in which the hybrid is correctly identified as H. m. rosina. Overall, these results are in agreement with the work of Martin et al. (2013) on this group, and demonstrate the utility of our method in rapidly identifying hybrid taxa from genome-scale data.
We have proposed a method for detecting hybrid species using a model of hybrid speciation that incorporates coalescent stochasticity. The test is based on observed site pattern frequencies, which leads to several convenient properties. First, the computations required for the test can be carried out very rapidly, as all that is required is to obtain counts of observed site pattern frequencies for four taxa of interest. This computation is so rapid that there are essentially no limits on the length of sequences that can be handled by the method, and it is thus appropriate for genome-scale data. Second, observed site pattern frequencies arise from a multinomial distribution under the coalescent hybridization model used here, which allows derivation of the asymptotic distribution of the estimators of the site pattern frequencies. This ultimately leads to a null distribution for testing the hypothesis of interest that is asymptotically normally distributed which provides a straightforward test of the hypothesis of interest. Finally, we note that our method is derived under the assumption that each site has its own underlying gene tree, an experimental design that we call "coalescent independent sites". The method is thus clearly appropriate for genome-wide SNP data, whether biallelic or not. We argue that the method is also appropriate for multilocus data, in that as the number of loci becomes large and provided that alignment lengths are not biased toward certain gene tree topologies, the proportion of sites observed from a particular gene tree will approach the proportion expected under the coalescent independent sites model. We thus carry out simulations for both multilocus and coalescent independent sites data, and we test our method on an empirical multilocus dataset.
Our simulations show that the method is powerful for detecting hybridization for both recent and ancient hybridization events, although for ancient hybridization events it may be more difficult to pinpoint the precise parental species for the detected hybrids. In addition, the proportional contribution of the two parental species to the genome of the hybrid species can be estimated accurately and unbiasedly. The simulations also show that the method scales extremely well: for 20-taxon networks with 100,000 sites, computations can be completed in less than 30 s, while for a dataset with 13 sequences and over 248 million sites, the analysis took less than 20 min on an older desktop linux machine. While these analyses demonstrate that sequence length is not a computationally-limiting factor, they also suggest that larger numbers of taxa will be similarly unproblematic. Although adding taxa increases the number of hypothesis tests to be carried out, these are each done very rapidly (e.g., for 20 taxa, there are over 2200 tests being done in less than 30 s), and they could easily be carried out on separate processors, if necessary. To the extent of our knowledge, this method is thus the only technique available for exploratory hybrid identification for large numbers of sequences using genome-scale data.
The method is based on phylogenetic invariants, and we note that the particular choice of invariants used here was somewhat arbitrary. Indeed, the ABBA-BABA test [45–47] is based on the difference of ABBA and BABA patterns similar to our invariant f2 and it too is useful in detecting hybridization. However their statistic is normalized by the total number of observations whereas our method is based on the ratio of two linear invariants leading to a function that depends only on the mixing parameter γ. Based on this crucial observation we were able to derive the Hils statistic for accurate detection of hybridization. We have also noticed that the ratio between f3 and f4 was not as powerful, thus it is possible that other invariants may be identified that work as well or better than the ones we have chosen here. It is also possible that invariants that operate on more than four taxa at a time could be determined, with potential improvements in the localization of hybrid and parental taxa for more ancient hybridization events. There is also a possibility that a set of linear invariants specific to species trees under the coalescent exists and can be classified, and if such a set exists, these species invariants may improve the performance. We suggest that exploring these directions is appealing, as site pattern-based methods provide the possibility of both rapid computation and convenient asymptotic distributions, making them suitable for processing the large genome-scale datasets that are becoming increasingly available. In fact, the performance of these methods improves with sequence length, since site pattern probabilities can be more accurately estimated, with little associated computational cost.
Classification of organisms and estimation of their phylogenetic relationships is central to many areas of biological research, but inference of these relationships comes with several challenges. Most notable are computational challenges arising from the abundance of available DNA sequence data and the need to model organismal evolution at two distinct levels – individual genes, and species as a whole, where the evolutionary histories of genes are constrained by the evolutionary history of the species. Additionally, several processes, such as incomplete lineage sorting (deep coalescence), hybridization, horizontal gene transfer, and gene duplication and loss, lead to the potential for incongruence in the evolutionary histories of the individual genes. The multispecies coalescent is commonly used to model incomplete lineage sorting and provides a model for the generation of gene trees within the containing species tree. We used this model to develop a method for detecting species that have arisen via hybridization and for quantifying the extent of hybridization in a formal statistical framework. We demonstrated the performance of our method using both simulated and empirical data. Our method is capable of processing genome-scale sequence datasets consisting of many taxa in a computationally efficient manner, thus providing researchers with an effective exploratory tool for hybrid identification.
Simulation Studies
Four-taxon species networks
Our first set of simulation studies involves assessing the level and the power of the tests under various choices of the sample size, species trees branch lengths, and value of γ for four-taxon trees. We used a custom python program (available at https://github.com/lkubatko/HilsTest) to simulate gene trees from the two parental species trees in Fig. 1 with γ values of 0, 0.1, 0.2, 0.3, 0.4, and 0.5 and for two sets of speciation times: τ1=0.25,τ2=0.5,τ3=1.0 (the "short" setting) and τ1=0.5,τ2=1.0,τ3=2.0 (the "long" setting). For each setting, we simulated N=50,000,100,000,250,000 and 500,000 coalescent independent sites under the GTR+I+ Γ model using Seq-Gen [66](Seq-Gen options: -mGTR -r 1.0 0.2 10.0 0.75 3.2 1.6 -f 0.15 0.35 0.15 0.35 -i 0.2 -a 5.0 -g 3). For each parameter setting, we generated 500 replicate data sets.
For each simulated data set, we tested the null hypothesis that γ=0 using the test statistics corresponding to the ratios in Eqs. (4) and (5) at level α=0.05. We also applied the ABBA-BABA test [46]. We estimate the power of each test as the proportion of the 500 replicates for which the null hypothesis was rejected (when γ=0, this gives an estimate of the level of the test). We also considered using each of the statistics to estimate the true hybridization parameter, γ. We report the mean of the estimated γ values, as well as the standard deviation and the mean squared error, for each parameter setting.
To evaluate the sensitivity of our test to the assumption of a molecular clock, we carried out a second set of simulations using model trees that violated the clock assumption. We considered violating the molecular clock in two ways. First, we extended the branch leading to species P1 by doubling its length, for both the short and the long branch length settings described above. Second, we extended the branch leading to the hybrid species by doubling its length, again for both branch length settings. As in the first set of simulation studies, we evaluate the power of our test and compare its performance to the ABBA-BABA test. Here, however, we consider only the Hils test based on the ratio \(\frac {f_{1}}{f_{2}}\), since this statistic showed superior performance in the first set of simulations.
Larger species networks
To examine the performance of our method for larger taxon samples, we considered networks containing 8 species and an outgroup, and networks containing 19 species and an outgroup. We also considered both recent hybridization and more ancient hybridization in each case (Fig. 4). For each model network, we generated 125 data sets containing 100,000 coalescent independent sites for γ=0,0.1,0.2,0.3,0.4, and 0.5 as follows. First, 100000γ gene trees were generated from the species tree formed by connecting the hybrid taxon to the "left" parental lineage, and 100000(1−γ) gene trees were generated from the species tree formed by connecting the hybrid taxon to the "right" parental lineage. For each gene tree, one coalescent independent site was generated using Seq-Gen [66] under the GTR+I+ Γ model (Seq-Gen options: -mGTR -r 1.0 0.2 10.0 0.75 3.2 1.6 -f 0.15 0.35 0.15 0.35 -i 0.2 -a 5.0 -g 3). Each simulated data set was then given to our program with the outgroup specified, and the Hils statistic was computed for each possible combination of parents and hybrids. A cut-off for significance was determined using a Bonferroni correction with base level α=0.05, and the putative hybrid and parents were reported for any statistic whose p-value fell below α/M, where M was the total number of comparisons. We summarized results by counting the number of "True Positives" (data sets for which the true hybrid and parental taxa are correctly identified), "True Sets" (data sets for which the true hybrid and parental taxa are identified, but their assignment to which is the hybrid and which are the parental taxa is ambiguous), and "False Positives" (data sets for which an incorrect set of taxa are identified as being subject to hybridization).
Because many of the genome-scale datasets being generated today are multilocus datasets (rather than being generated under the coalescent independent sites model used here), we also simulated data under multilocus n. These simulations proceeded exactly as described above, except that rather than simulating 100,000 coalescent independent sites, we simulated 1000 genes each of length 100bp. This choice was made to mimic the short read lengths generated by next-gen sequencing methods. We summarized these results in the same manner as described above. We justify application of our methodology to multilocus data in the Discussion section.
Empirical examples
We have also explored the performance of our method on two empirical data sets; the Sistrurus rattlesnakes and Heliconius butterflies. The Sistrurus rattlesnakes are found across North America and are currently classified into two species, Sistrurus catenatus and S. miliarius, each with three putative subspecies. The dataset consists of 19 genes sampled from 26 rattlesnakes: 18 individuals within the species Sistrurus catenatus (with subspecies S. c. catenatus (Sca, 9 individuals), S. c. edwardsii (Sce, 4 individuals), and S. c. tergeminus (Sct, 5 individuals)); six within species Sistrurus miliarius (with subspecies S. m. miliarius (Smm, 1 individual), S. m. barbouri (Smb, 3 individuals), and S. m. streckeri (Sms, 2 individuals)); and two outgroup species, Agkistrodon contortrix and A. piscivorus. These data were originally analyzed by [67] to determine species-level phylogenetic relationships. Prior to this analysis, the sequences were computationally phased, resulting in 52 sequences and 8,466 aligned nucleotide positions (data are available at TreeBase ID 11174). These data have been subsequently reanalyzed in several ways. For example, [16] used different methodology to infer the species phylogeny, and found agreement with the original analysis of Kubatko et al. (2011). Gerard et al. (2011) used a subset of the data to examine whether several specimens collected in Missouri and assigned to subspecies S. c. catenatus were actually hybrid species. They did not find evidence of hybridization, in agreement with other results using different data [64].
The Heliconius butterflies are a diverse group of tropical butterflies in the family Heliconii that are found throughout the southern United States and in Central and South America. We consider the study of Martin et al. (2013) [68] in which genome-scale data for 31 individuals from seven distinct species were collected and evidence for gene flow between various species was assessed. We examine a subset of these data consisting of four individuals from each of the species Heliconius cydno, H. melpomene rosina, and H. m. melpomene, as well as one individual from the outgroup species H. hecale. Martin et al. (2013) found evidence that H. m. rosina is a hybrid of H. m. melpomene and H. cydno. We obtained the aligned genome-wide data from the complete study of Martin et al. (2013) from Dryad (http://datadryad.org/resource/doi:10.5061/dryad.dk712) [69], and extracted the 13 sequences of interest. The resulting aligned sequences consisted of 248,822,400 base pairs.
ABBA-BABA:
Patterson's D-statistic to test ancient admixture
ASTRAL:
Accurate Species TRee ALgorithm
BEAST:
Bayesian Evolutionary Analysis Sampling Trees
Bayesian Estimation of Species Trees
GTR+I+ Γ :
General time-reversible model of Tavaré1986 with site-specific rate variation, and invariable sites
ILS:
Incomplete Lineage Sorting
JC69:
the Jukes and Cantor 1969 model of DNA evolution
JML:
Testing hybridization from species trees
MP-EST:
Maximum Pseudo-likelihood for Estimating Species Trees
PAUP*:
Phylogenetic Analysis Using Parsimony *and other methods Seq-Gen: Sequence-Generator
SNAP:
SNP and AFLP Package for Phylogenetic analysis
SNP:
SVDquartets:
Singular Value Decomposition Scores for Species Quartets
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We thank the anonymous reviewers for helpful comments and suggestions.
a Matthew H. Hils was a Professor of Biology at Hiram College until his untimely death in June 2014. He served as academic advisor and research mentor to L.K. during her undergraduate studies, and contributed to her decision to pursue interdisciplinary graduate study tied to the biological sciences. See http://news.hiram.edu/?p=10502.
This work was supported in part by the National Science Foundation under award DMS-1106706 (J.C., L.K.) and NIH Cancer Biology Training Grant T32-CA079448 at Wake Forest School of Medicine (J.C.). The funding agencies played no role in the design of the study, analysis, simulations and interpretation of data and in writing the manuscript.
Datasets used in this article are all publicly available as described in "Methods" section.
Department of Statistics, The Ohio State University, Columbus, USA
Laura S. Kubatko
Department of Evolution, Ecology, and Organismal Biology, The Ohio State University, Columbus, USA
Department of Mathematics and Statistics, American University, Washington, DC, USA
Julia Chifman
Search for Laura S. Kubatko in:
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LK and JC conceived of the study, model development, mathematical and statistical methods and wrote the manuscript. LK designed and executed all simulations. Both authors read and approved the final manuscript.
Correspondence to Laura S. Kubatko.
L.K. serves as Section Editor for the Theory and "Methods" section of BMC Evolutionary Biology.
Kubatko, L.S., Chifman, J. An invariants-based method for efficient identification of hybrid species from large-scale genomic data. BMC Evol Biol 19, 112 (2019) doi:10.1186/s12862-019-1439-7
ABBA-BABA
Coalescence
Phylogenetic invariants
Phylogenetics and phylogeography | CommonCrawl |
How many logical qubits are needed to run Shor's algorithm efficiently on large integers ($n > 2^{1024}$)?
First, I know there are differences in logical qubits and physical qubits. It takes more physical qubits for each logical qubit due to quantum error.
Wikipedia states that it takes quantum gates of order $\mathcal{O}((\log N)^2(\log \log N)(\log \log \log N)$ using fast multiplication for Shor's Algorithm. That comes out to $1,510,745$ gates for $2^{1024}$. Further down the article, it says that it usually take $n^3$ gates for $n$ qubits. This would mean it would take ~$115$ qubits.
However, I've run Shor's Algorithm as implemented in Q# samples using Quantum Phase Estimation and it comes out to $1025$ qubits.
shors-algorithm
MartinQuantum
LeWoodyLeWoody
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$\begingroup$ Don't get misled by something saying it requires order something. You can't use that to calculate gate counts or numbers of qubits, because (i) there are arbitrary constants that are suppressed by the notation that could change orders of magnitude very easily, and (ii) scaling descriptions like that are only true in the large n limit. For any finite n, behaviour could be a bit different, dominated by a term with a weaker scaling but larger unknown factor. $\endgroup$
– DaftWullie
The question is about how many logical qubits it takes to implement Shor's algorithm for factoring an integer $N$ of bit-size $n$, i.e., a non-negative integer $N$ such that $1 \leq N \leq 2^n{-}1$. The question is a poignant one and not easy to answer as there are various tradeoffs possible (e.g., between number of qubits and circuit size).
Executive Summary Answer: $2n{+}2$ qubits which leads to a quantum circuit implementation that has less than $448 n^3 \log_2(n)$ number of $T$-gates. For a bit-size of $n=1,024$, this would work out to be $2050$ logical qubits and $4.81 \cdot 10^{12}$ $T$-gates.
As mentioned in the question, one can apply fast methods such as Schoenhage-Strassen's algorithm for fast multiplication to implement the modular arithmetic asymptotically in $O(n^2 \log(n) \log \log(n))$ primitive operations (say, over the Clifford$+T$ gate set). This has been discussed for instance in Zalka's paper. However, it should be pointed out that this is indeed (i) only a statement about asymptotic cost and (ii) only a statement about the number of operations required and does not imply the number of qubits.
Regarding (i), the constant that is hidden in the "O-notation" can be prohibitively large. To the best of my knowledge, it has not been attempted to construct a quantum circuit to implement Shor's algorithm based on Schoenhage-Strassen, so we do not even know upper bounds on what that constant is. The other catch, (ii), is that it is not straightforward to relate the number of qubits and the gate cost as seems to be suggested in the question. Besides the fact that we do not know the constant, there is another issue, namely that a straightforward implementation of Schoenhage-Strassen via Bennett's method would lead to a very large number of logical qubits required. Therefore, even as there are faster methods available for integer multiplication than the simple method of n additions, these are much more non-trivial to code in quantum programming languages such as LIQUi|> and Q#.
In terms of concrete resource estimates for Shor's algorithm, the paper by Haener et al might be a good entry point which implemented the arithmetic in terms of so-called Toffoli gates which have the advantage of being testable at scale on classical input vectors. It is shown in that paper that $2n{+}2$ logical qubits are sufficient to implement Shor's algorithm for factoring integers using a circuit that uses $64 n^3 \log_2(n)$ Toffoli gates which yields $448 n^3 \log_2(n)$ primitive gates (this latter number refers to the number of $T$-gates and ignores that number of Clifford gates as these are significantly more easy to implement fault-tolerantly).
The currently available Q# implementation of Shor's algorithm (see the IntegerFactorization sample at https://github.com/microsoft/quantum) is based on another way of implementing the arithmetic, namely based on Draper's method to implement additions using the Fourier basis, see also here. This implementation follows Beauregard's paper and requires $2n{+}3$ logical qubits in total. A recent improvement has been obtained by Gidney who reduced the number of clean qubits to $2n{+}1$ (of which only $n{+}2$ have to be "clean" qubits, i.e., initialized in a known state. The rest can be "dirty" qubits that can be used and returned in their (unknown) state). Finally, there is an interesting claim by Zalka that the number of qubits can be reduced to $1.5n{+}2$ (and perhaps even further), however, his proposed solution comes at a dramatic increase of circuit size as it involves inversions and, to my knowledge, has not been verified nor implemented in a programmatic way.
answered Dec 26, 2018 at 19:42
MartinQuantumMartinQuantum
$\begingroup$ I wouldn't call my paper "an improvement of the gate count". It has a gate count orders of magnitude worse to save that one qubit. Fun though. $\endgroup$
– Craig Gidney
Dec 27, 2018 at 3:10
$\begingroup$ there you go, fixed it. $\endgroup$
– MartinQuantum
$\begingroup$ You say $448 n^3 \log_2(n)$ in one place and $448 n^2 \log_2(n)$ in another. $\endgroup$
$\begingroup$ Thanks, @CraigGidney! There was indeed a mixup: the cost is for integer factorization was upper bounded in arxiv.org/pdf/1611.07995.pdf by $64 n^3 \log_2(n)$, plus lower order terms, the cost for elliptic curve dlog was upper bounded in arxiv.org/pdf/1706.06752.pdf by $448 n^3 \log_2(n)$, plus lower order terms. $\endgroup$
$\begingroup$ We're slowly getting there. Thanks again and sorry for confusing the numbers. You are absolutely right: $64 n^3 \log_2(n)$ Toffoli gates for factoring, which, using the deterministic circuit identity for Toffoli (=7 T gates per Toffoli), shakes out to be $448 n^3 \log_2(n)$ T-gates after all. Ironically, this $448 n^3 \log_2(n)$ expression also occurs in the estimates for Shor ECC but there is refers to Toffolis, i.e., the cost for same bit size is about a factor 7 higher for Shor ECC than for Shor factoring. Hope it makes sense now. $\endgroup$
How to show that amount of qubits needed to crack the RSA-2048 protocol using Shor's algorithm?
Are there any other published quantum factoring algorithms that are simpler or more efficient than Shor's?
HHL algorithm, how to decide n qubits to prepare for expressing eigenvalue of A?
Quantum algorithm for linear systems of equations (HHL09): Step 1 - Number of qubits needed
How to measure entanglement in an algorithm?
What is the general method for creating real gate sequences from mathematical algorithms?
"Classical" phase estimation versus iterative phase estimation
Register size in factoring 15 using Shor's algorithm
Is it possible to turn modular multiplication into in-place operation?
Can numbers be factored by using a reverse multiplication circuit on a quantum computer? | CommonCrawl |
Performance demonstration of the PEnELOPE main amplifier HEPA I using broadband nanosecond pulses
HPL_EP HEDP and High Power Laser 2018
D. Albach, M. Loeser, M. Siebold, U. Schramm
Published online by Cambridge University Press: 27 December 2018, e1
We report on the energetic and beam quality performance of the second to the last main amplifier section HEPA I of the PEnELOPE laser project. A polarization coupled double-12-pass scheme to verify the full amplification capacity of the last two amplifiers HEPA I and II was used. The small signal gain for a narrow-band continuous wave laser was 900 and 527 for a broadband nanosecond pulse, demonstrating 12.6 J of output pulse energy. Those pulses, being spectrally wide enough to support equivalent 150 fs long ultrashort pulses, are shown with an excellent spatial beam quality. A first active correction of the wavefront using a deformable mirror resulted in a Strehl ratio of 76% in the single-12-pass configuration for HEPA I.
Experimental platform for the investigation of magnetized-reverse-shock dynamics in the context of POLAR
HPL Laboratory Astrophysics
B. Albertazzi, E. Falize, A. Pelka, F. Brack, F. Kroll, R. Yurchak, E. Brambrink, P. Mabey, N. Ozaki, S. Pikuz, L. Van Box Som, J. M. Bonnet-Bidaud, J. E. Cross, E. Filippov, G. Gregori, R. Kodama, M. Mouchet, T. Morita, Y. Sakawa, R. P. Drake, C. C. Kuranz, M. J.-E. Manuel, C. Li, P. Tzeferacos, D. Lamb, U. Schramm, M. Koenig
The influence of a strong external magnetic field on the collimation of a high Mach number plasma flow and its collision with a solid obstacle is investigated experimentally and numerically. The laser irradiation ( $I\sim 2\times 10^{14}~\text{W}\cdot \text{cm}^{-2}$ ) of a multilayer target generates a shock wave that produces a rear side plasma expanding flow. Immersed in a homogeneous 10 T external magnetic field, this plasma flow propagates in vacuum and impacts an obstacle located a few mm from the main target. A reverse shock is then formed with typical velocities of the order of 15–20 $\pm$ 5 km/s. The experimental results are compared with 2D radiative magnetohydrodynamic simulations using the FLASH code. This platform allows investigating the dynamics of reverse shock, mimicking the processes occurring in a cataclysmic variable of polar type.
The Recent Lightcurve of 3C 345
K.J. Schramm, U. Borgeest, J. V. Linde, S.J. Wagner, J. Heidt
We present the lightcurve of 3C 345 (1641+399, z = 0.595) in Johnson R. The data until summer 1992 are analysed and discussed in detail in Schramm et al. (A&A, Nov. 1993). The more recent lightcurve is almost flat (R ≃ 16.9), giving new constraints on variability models, see Camenzind, this proceedings.
Variability characteristics of Blazar 0J 287
L.O. Takalo, A. Sillanpää, T. Pursimo, H.J. Lehto, K. Nilsson, P. Teerikorpi, P. Heinämäki, M. Kidger, J.A. DE DIEGO, T. Mahoney, J.-M. Rodríguez-Espinosa, J.N. González-Pérez, P. Boltwood, D. Dultzin-Hacyan, E. Benitez, G. Turner, J. Robertson, R. Honeycut, Yu.S. Efimov, N. Shakhovskoy, P. A. Charles, D. Kühl, K.J. Schramm, U. Borgeest, J.V. Linde, W. Weneit, T. Schramm, A. Sadun, R. Grashuis, J. Heidt, H. Bock, S. Wagner, M. Kümmel, A. Heines, M. Fiorucci, G. Tosti, C. Raiteri, M. Villata, G. Latini, S. Bosio, G. Ghisellini, G. De Francesco
Blazar OJ 287 is one of the best observed extragalactic objects. It's historical light curve goes back to 1890′s. Based on the historical behaviour Sillanpää et al. (1988) showed that OJ 287 displays large periodic outbursts, with a period of 11.7 years. We have monitored OJ 287 intensively for two years, during the OJ-94 project. This project was created for monitoring OJ 287 during its predicted new outburst in 1994. In the data archive we have over 7000 observations on OJ 287, in the radio, infrared and optical bands. This data archive contains the best ever obtained light curves for any extragalactic object. The optical light curve shows continuous variability down to time scales of tens of minutes. The variability observed in OJ 287 can be broken down to (at least) four different categories:
Foreground Galaxies And The Variability Of Luminous Quasars
J. von Linde, U. Borgeest, J. Schramm, S. Refsdal, E. Van Drom
In order to look for an amplification bias (AB) by gravitational lensing caused by medium redshift (0.2≲ z ≲ 0.8) clusters or groups of galaxies, we compare galaxy counts in deep CCD images of highly luminous, high redshift QSOs with those in nearby control fields at a distance of 1 deg at the same galactic latitude. The total sample contains 37 objects up to now, from which one field had to be excluded because of a seeing difference between the QSO and control fields.
A Dedicated Quasar Monitoring Telescope
U. Borgeest, K.-J. Schramm, J. Von Linde
Under the auspices of Sjur Refsdal, 25 astrophysicists and engineers from Germany and Scandinavia have founded a non-profit association, aiming at the use of an intelligent telescope for quasar monitoring in the optical (Borgeest et al. 1993). Beyond a better understanding of the physics in quasars, the scientific goals are determining the cosmic distance scale at large redshifts and constraining the nature of Dark Matter, both using the gravitational lens effect. Thus, targets of special interest are the multiply lensed quasars and some well-known violently variable blazars. The optical photometry will in part be carried out simultaneously to observations with, e.g., ISO, ROSAT, CGRO and various radio telescopes. For the first time, a complete quasar sample will be monitored continuously, namely a sub-sample of the all-sky 1 Jy catalogue (5 GHz). Since we will collect about 106 photometric data points during the programme, Megaphot has been chosen as name for the association. Members from Hamburg and Bochum intend to test the 1.5 m Hexapod Telescope (HPT) astronomically in the very near future. The HPT hardware was developed and built by Vertex Antennentechnik, Duisburg together with the Ruhr-Universität, Bochum and Carl Zeiss, Jena; the intelligent software and weather control requires still some work. When working well, the system will be placed at a site with excellent astronomical conditions. After a few years of exclusive quasar monitoring, it will be used as a German photometry telescope.
Foreground Galaxies around Luminous Quasars
J. Von Linde, U. Borgeest, S. Refsdal, K.-J. Schramm, E. Van Drom
We compare galaxy counts in deep R-band exposures of the fields of 36 highly luminous, high redshift QSOs to those in control fields at a distance of 1 deg. We find indication for a weak overdensity of galaxies in the foreground of QSOs on scales of arcminutes on a low significance level. Counts inside rings around the quasars, stars in the quasar fields and stars in the control fields show evidence for an excess of galaxies on scales of several arcseconds around the quasars as well as for a stronger clustering of galaxies in the QSO fields than in the control fields. We interpret this in terms of an amplification bias by gravitational lensing.
Inverse Compton backscattering source driven by the multi-10 TW laser installed at Daresbury
G. Priebe, D. Laundy, M.A. Macdonald, G.P. Diakun, S.P. Jamison, L.B. Jones, D.J. Holder, S.L. Smith, P.J. Phillips, B.D. Fell, B. Sheehy, N. Naumova, I.V. Sokolov, S. Ter-Avetisyan, K. Spohr, G.A. Krafft, J.B. Rosenzweig, U. Schramm, F. Grüner, G.J. Hirst, J. Collier, S. Chattopadhyay, E.A. Seddon
Published online by Cambridge University Press: 20 November 2008, pp. 649-660
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Inverse Compton scattering is a promising method to implement a high brightness, ultra-short, energy tunable X-ray source at accelerator facilities. We have developed an inverse Compton backscattering X-ray source driven by the multi-10 TW laser installed at Daresbury. Hard X-rays, with spectral peaks ranging from 15 to 30 keV, depending on the scattering geometry, will be generated through the interaction of laser pulses with electron bunches delivered by the energy recovery linac machine, initially known as energy recovery linac prototype and subsequently renamed accelerators and lasers in combined experiments. X-ray pulses containing 9 × 107 photons per pulse will be created from head-on collisions, with a pulse duration comparable to the incoming electron bunch length. For transverse collisions 8 × 106 photons per pulse will be generated, where the laser pulse transit time defines the X-ray pulse duration. The peak spectral brightness is predicted to be ~1021 photons/(s mm2 mrad2 0.1% Δλ/λ).
EuroSOMNET – a European database of long-term experiments on soil organic matter: the WWW metadatabase
P. SMITH, P. D. FALLOON, M. KÖRSCHENS, L. K. SHEVTSOVA, U. FRANKO, V. ROMANENKOV, K. COLEMAN, V. RODIONOVA, J. U. SMITH, G. SCHRAMM
Journal: The Journal of Agricultural Science / Volume 138 / Issue 2 / March 2002
Since 1997, the EuroSOMNET project, funded by the EU-ENRICH programme, has assembled a metadatabase, and separate experimental databases, of European long-term experiments that investigate changes in soil organic matter. In this paper, we describe the WWW-based metadatabase, which is a product of this project. The database holds detailed records of 110 long-term soil organic matter experiments, giving a wide geographical coverage of Europe, and includes experiments from the European part of the former Soviet Union, many of which have not been available previously. For speed of access, records are stored as hyper-text mark-up language (HTML) files. In this paper, we describe the metadatabase, the experiments for which records are held, the information stored about each experiment, and summarize the main characteristics of these experiments. Details from the metadatabase have already been used to examine regional trends in soil organic matter in Germany and eastern Europe, to construct and calibrate a regional statistical model of humus balance in Russia, to examine the effects of climatic conditions on soil organic matter dynamics, to estimate the potential for carbon sequestration in agricultural soils in Europe, and to test and improve soil organic matter models. The EuroSOMNET metadatabase provides information applicable to a wide range of agricultural and environmental questions and can be accessed freely via the EuroSOMNET home page at URL: http://www.iacr.bbsrc.ac.uk/aen/eusomnet/index.htm.
Particle physics with petawatt class lasers
S. KARSCH, D. HABS, T. SCHÄTZ, U. SCHRAMM, P.G. THIROLF, J. MEYER-TER-VEHN, A. PUKHOV
Journal: Laser and Particle Beams / Volume 17 / Issue 3 / July 1999
With a Petawatt class CPA laser of the LLNL Livermore or the proposed GSI (Darmstadt) type laser, interactions with matter can be studied in the upper 1020 W/cm2 regime. With such a laser focused into an underdense plasma, strong electron bursts with energies up to several 100 MeV are ejected in a forward direction. This leads to a comparable burst of bremsstrahlung radiation in the presence of high-Z material. Here, we discuss the corresponding γ-induced nuclear reactions including secondary particle production, including pions. Due to the threshold behavior in the production, and the advantage of delayed detection, we propose to employ these reactions for probing the initial plasma conditions.
The Classification of Functional Psychoses and its Implications for Prognosis
H. J. Möller, M. Hohe-Schramm, C. Cording-Tömmel, W. Schmid-Bode, H. U. Wittchen, M. Zaudig, D. Von Zressen
Journal: The British Journal of Psychiatry / Volume 154 / Issue 4 / April 1989
Print publication: April 1989
One hundred and eighty-three patients suffering from functional psychoses were diagnosed according to ICD–8, RDC, and DSM–III criteria, and the concordance rates for the diagnoses compared. The heterogeneity of the diagnosis 'schizoaffective psychosis' as defined by these systems became clear. With respect to prognosis, the DSM–III diagnosis of schizophrenia was most closely related to poor outcome. Affective psychoses and schizoaffective psychoses, as well as DSM–III 'schizophreniform disorders', demonstrated a favourable prognosis. | CommonCrawl |
Volume 60 Issue 4: 2nd Radiocarbon in the Environm...
14C in Urban Secondary Carbonate Deposits: a New Tool...
Site and samples
Conclusion and perspectives
14C in Urban Secondary Carbonate Deposits: a New Tool for Environmental Study
Published online by Cambridge University Press: 10 April 2018
E Pons-Branchu ,
L Bergonzini ,
N Tisnérat-Laborde ,
P Branchu ,
E Dumont ,
M Massault ,
G Bultez ,
D Malnar ,
E Kaltnecker ,
JP Dumoulin ,
A Noret ,
N Pelletier and
M Roy-Barman
E Pons-Branchu
Laboratoire des Sciences du Climat et de l'Environnement, LSCE/IPSL, CEA-CNRS-UVSQ, Université Paris Saclay, Gif-sur-Yvette, France
L Bergonzini
GEOPS, Université Paris-Sud, CNRS UMR 8148, Université Paris-Saclay, F-91405 Orsay, France
N Tisnérat-Laborde
P Branchu
CEREMA: 12 Rue Teisserenc de Bort, 78197 TRAPPES-en-Yvelines Cedex France, and Rue de l'Egalité Prolongée - BP 134, 93352 LE BOURGET Cedex 319, France
E Dumont
M Massault
G Bultez
Château de Versailles: Etablissement Public du château, du musée et du domaine national de Versailles. RP 834 - 78008 Versailles cedex, France
D Malnar
E Kaltnecker
JP Dumoulin
A Noret
N Pelletier
[email protected].
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Secondary carbonate deposits (similar to speleothems) in urban undergrounds, have been recently highlighted as powerful archives for reconstruction of the historical anthropogenic imprint on the environment. The precise chronology of these secondary carbonate deposits is a key issue for the accurate time reconstruction of environmental conditions. We present three 14C data sets for urban speleothem-like deposits that developed in contrasted man made environments. The first one was sampled in an underground technical gallery of the Palace of Versailles (France), and the other two in a manhole (Saint-Martin spring) of a historical underground aqueduct in Paris (France). The comparison of these records with the bomb peak and relative chronology (laminae counting) allowed us to identify: i) fast carbon transfer from the atmosphere to the urban underground; ii) a high proportion of dead carbon and a high damping effect in relation to possible old carbon stored within urban soils and/or the influence of local fossil carbon burning. This study also shows that the lamination of these deposits is bi-annual in these highly urbanized sites.
radiocarbon pulse bomb speleothems urban hydrology
Water, Sediment, Karst
Radiocarbon , Volume 60 , Issue 4: 2nd Radiocarbon in the Environment Conference Debrecen, Hungary, July 3–7, 2017 Part 1 of 2 , August 2018 , pp. 1269 - 1281
DOI: https://doi.org/10.1017/RDC.2018.25[Opens in a new window]
© 2018 by the Arizona Board of Regents on behalf of the University of Arizona
Speleothem-like deposits in urban areas, a natural archive that was poorly known until very recently, are a powerful record of the historical anthropogenic imprint on the environment. The study of key geochemical tracers such as lead or sulfur isotopes within these carbonate deposits can enable the sources of contaminants contained within the water generating their deposition to be identified (Pons-Branchu et al. Reference Pons-Branchu, Ayrault, Roy-Barman, Bordier, Borst, Branchu, Douville and Dumont2015, Reference Pons-Branchu, Roy-Barman, Jean-Soro, Guillerme, Branchu, Fernandez, Dumont, Douville, Michelot and Phillips2017).
The precise chronology of these secondary carbonate deposits is a key issue for the accurate time reconstruction of environmental conditions. In a few favorable cases, these deposits can be dated using the uranium-thorium (230Th/234U) chronometer, with an appropriate correction for detrital thorium, but in many cases, their low uranium content and high detrital thorium prevent this reconstruction. Surprisingly, the study of lamination on two of these deposits from an underground aqueduct in Paris, coupled with U/Th dating, revealed that in this case, lamination is bi-annual, (Pons-Branchu et al. Reference Pons-Branchu, Douville, Roy-Barman, Dumont, Branchu, Thil, Frank, Bordier and Borst2014), as observed in speleothems from natural caves (Allison Reference Allison1926; Broecker et al. Reference Broecker, Olson and Orr1960; Baker et al. Reference Baker, Smart, Edwards and Richards1993; Shopov et al. Reference Shopov, Ford and Schwarz1994). This implies that, at least in some cases, lamination could have the same origin (organic matter content, and/or CaCO3 porosity, facies or mineralogical differences) in natural cave systems and in underground urban structures, but this has to be demonstrated from other cases before generalization. Unfortunately, urban speleothem-like deposits are not always laminated, and new dating tools have to be investigated. As for speleothems from natural caves, 14C chronology in these deposits is hampered by the presence of dead carbon from calcareous host rock or old organic matter contained within the soil (Goslar et al. Reference Goslar, Hercman and Pazdur2000; Griffiths et al. Reference Griffiths, Fohlmeister, Drysdale, Hua, Johnson, Hellstrom and Zhao2012; Noronha et al. Reference Noronha, Johnson, Hu, Ruan, Southon and Ferguson2014). The purpose of this paper is to identify the 14C bomb pulse in recent urban speleothems, as previously observed in speleothems from caves (e.g. Delibrias et al. Reference Delibrias, Guillier and Labeyrie1969; Genty and Massault Reference Genty and Massault1997; Hodge et al. Reference Hodge, McDonald, Fischer, Redwood, Hua, Levchenko, Drysdale, Waring and Fink2011; Hua et al. Reference Hua, McDonald, Redwood, Drysdale, Lee, Fallon and Hellstrom2012) and discuss its applicability to these natural archives and the information provided about carbon transfer within an urban/anthropic context.
The two sites studied are located in the Paris region ca. 20 km apart, in two contrasting urbanized zones: the first one under a street in Versailles, beside Versailles Palace, and the second in the northeastern part of Paris. Both sites are built on Oligocene sedimentary deposits (sand stone in Versailles, sandstone and limestone id Paris) overlain by anthropogenic backfill.
Palace of Versailles Technical Gallery
Ever since the 1600s, water supply has been an important issue for the inhabitants of Versailles and for the famous fountains of the palace gardens. Works undertaken to bring large amounts of water to the Palace and the gardens included diverting water from two rivers (the Seine and the Bièvre), the construction of artificial ponds, some of them several tens of kilometers away from Versailles, and the construction of kilometers of aqueducts and ducts (Barbet Reference Barbet1907; Soullard Reference Soullard1997). At Versailles, numerous underground galleries were built under the gardens and the fountains in order to distribute the waters to the fountains and play the "Grandes Eaux" performance. These historical galleries contain pipes that were maintained over the centuries. Speleothem-like deposits are found in some of them, deposited by water dripping from the roof of the galleries (either from rainwater or leakages directly from the fountains). The F4 samples (see Figure 1) were taken during 2013 from the wall of a technical gallery that connects Versailles Palace with the "Service des fontaines" above the René de Cotte street, ca. 2 m below the street. In this gallery, the only water source is water infiltrating from rainfall. The host rock of the gallery is Fontainebleau Sandstone, overlain by anthropogenic backfill (thickness estimated between 1.5 and 3 m according to nearby geological drilling reported by BRGM/Infoterre).
Figure 1 a) Location b) F4 sample from Versailles underground gallery before sampling (top), and picture of the thin section showing laminae (with the black vertical bar representing 2 mm); c) Saint Martin spring and the CaCO3 crust sampled (arrow), SM B core (with black bar representing 20 mm) and picture of thin section within the laminated level (with the black vertical bar representing 2 mm).
This 32 mm thick deposit displays lamination (see Figure 1b).
Saint Martin Spring
In the northeastern part of Paris, in the Malassis plateau, two perched aquifers developed within Fontainebleau sandstone and Brie limestone (Oligocene). These groundwaters have been drained since the 1100s by religious communities for their needs. In the following centuries, an extensive network of drains leading to several underground aqueducts was developed by the City of Paris. Known as the "Northern Springs", many vestiges of this network still exist, and water still flows from some of the springs. During field work in underground galleries and manholes, speleothems -like deposits on the roof, the walls and the floor (mainly sodastraws and flowstones, and in some places stalagmites) were identified. This is the case of the Saint-Martin manholes, which host calcareous deposits. The Saint-Martin spring drains water from a small urbanized watershed that has been heavily impacted by human activities since the 1800's (road, building constructions, etc.). These water flow in a small gallery with an irregular slope, where CaCO3 deposits, similar to flowstones in natural caves (see picture in Figure 1c). The water that permitted the deposition of these calcareous crusts originates from the spring, because no dripping water from the roof was observed during field work. The crusts was cored (in 2012) at two locations 10 cm apart. SM-A is a 10 cm high core. The first 4/5 mm from the top are laminated, followed by an 8/9 mm thick porous zone, and a second laminated level. At the base (from ca. 4.5 cm to the base), the core is made of building stone (limestone).The top most section of SM A was already studied for lead isotopes (Pons-Branchu et al. Reference Pons-Branchu, Ayrault, Roy-Barman, Bordier, Borst, Branchu, Douville and Dumont2015).
SM-B is a 5 cm long core sampled ca. 10 cm away from SM-A. The top 7 mm are laminated, followed by 18 mm of porous CaCO3, and a 6/7 mm laminated level. The base of the sample (from ca. 30 to ca. 50 mm) is made of concrete. A thin CaCO3 layer suspended in the water (SM fl) was also collected.
Laminae Counting
Polished sections of SM-A, SM-B and F4 were observed and photographed under a stereo microscope (LEICA S6D) using a video camera (Sony e SSC-DC14/14P/18P) at CEREMA laboratory. One (for SM-A and SM-B) to four (for F4) transects of each section were photographed. Lamina counting was performed on these images. Several counting using several images of the same depth range (for F4 sample) or using different portions of the same image (for SM-A and SM-B) were performed. The difference in the number of laminae between the various counts was used as the error bar for derived age.
Stable Isotope Analyses (δ 18 O-CaCO 3 )
Sixteen samples were taken as powder along the growth axis of SM A and SM B (including the base of SM-A made of building stone), and a small piece of SM fl, for O analysis.
δ18O in calcite was obtained from a few mg of CaCO3 powder which was reacted (at 25°C during 24 hr) with H3PO4 to give CO2(g). The gas was used for isotope (δ18O) measurements on a VG SIRA 10 mass spectrometer. The stable isotope analyses (isotope ratios) were measured at the GEOPS laboratory (Orsay, France) and are expressed in delta notation per mil versus V-PDB. They were determined with inter-laboratory analytical precision of 0.2 ‰.
14C Analysis
Along the growth axis of the three speleothem-like deposits, 21 samples (10 to 15 mg) were taken as fragment for 14C measurements: 13 for SM A and SM B cores, and 8 from F4 sample, The topmost section of SM-A was not sampled for 14C analysis.
Pure calcite samples were prepared according to the protocol described by (Tisnérat-Laborde et al. Reference Tisnérat-Laborde, Poupeau, Tannau and Paterne2001; Dumoulin et al. Reference Dumoulin, Comby-Zerbino, Delqué-Količ, Moreau, Caffy, Hain, Perron, Thellier, Setti, Berthier and Beck2017). The calcite was reacted with orthophosphoric acid (pure H3PO4, heated previously for 3 days at 105°C) under vacuum and the CO2 produced was converted to graphite (Vogel et al. Reference Vogel, Southon, Nelson and Brown1984; Dumoulin et al. Reference Dumoulin, Comby-Zerbino, Delqué-Količ, Moreau, Caffy, Hain, Perron, Thellier, Setti, Berthier and Beck2017) and then measured using the accelerator mass spectrometer (LMC14 - Artemis) at CEA Saclay (Cottereau et al. Reference Cottereau, Arnold, Moreau, Baqué, Bavay, Caffy, Comby, Dumoulin, Hain, Perron, Salomon and V Setti2007; Moreau et al. Reference Moreau, Caffy, Comby, Delqué-Količ, Dumoulin, Hain, Quiles, Setti, Souprayen and Thellier2013) in the framework of INSU national service.
Saint-Martin Deposits: δ18O Results and Depth Scale Adjustment
δ18O measured on SM-A and SM-B are presented in table 1 and figure 2a. A slight shift (ca. 0.3 ‰) toward lower δ18O values is observed for the samples corresponding to porous levels. Considering that i) the values for the topmost levels (laminated) and for the porous levels are comparable in the two cores; ii) that similar facies (porous vs laminated) are present in both cores but with a different development; iii) they belong to the same CaCO3 crust; we suggest that similar mechanisms (including potential isotopic fractionation) and environmental factors drive their development, and that they differ by different growth rates (twice as high for SM-B) and with an earlier growth start for SM-A. In order to compare 14C results of the two cores, an adjusted depth scale is proposed for SM-A: the facies (laminated/porous) transitions and the isotopic shift has been aligned to the SM-B's one. This corresponds to a higher growth rate (by a factor of ca. 2) in SM-A compared to SM-B. Using this adjusted depth scale, porous zones in the two cores are superposed, and δ18O values change similarly in both cores for the different facies.
Figure 2 δ18O measured on SM-A and SM-B with proper depth scale (a) and adjusted depth scale (b); 14C values according to adjusted depth scale.
In a location nearby at the Belleville main aqueduct, which is only 500 m away from the Saint-Martin spring, the lamination in urban speleothem-like deposits has been shown to be bi-annual (Pons-Branchu et al. Reference Pons-Branchu, Douville, Roy-Barman, Dumont, Branchu, Thil, Frank, Bordier and Borst2014). The laminae were therefore also assumed to be bi-annual within the three samples studied here.
F4 (Versailles): 136.5±4 laminae are visible on the whole section, corresponding to a 68.2±2 years deposition, and a growth start in 1945±2 AD.
SM-B (Paris): 52±2 laminae are visible on the laminated portion at the top of the core, 14±1 within the porous zone and 50±2 laminae for the deepest laminated level (just above the concrete). Assuming that lamination is bi-annual and that there were laminae deposits every year (including during the deposition of the porous level), the base of this core is 58±2.5 years old, with the start of growth in 1954±2.5 AD. This laminae counting suggests that the growth rate was higher during porous level deposition than during the other periods.
SM-A (Paris): 54±2 laminae are visible on the laminated portion at the top of the core. No laminae were distinguished within the porous zone (see Pons-Branchu et al. Reference Pons-Branchu, Ayrault, Roy-Barman, Bordier, Borst, Branchu, Douville and Dumont2015). The lamination in the lowest portion of the core was poorly visible and were not counted.
With the same number of laminae, the deposition of the laminated levels at the top of the two cores from Saint-Martin manhole could thus be contemporaneous, with deposition starting 26 to 27 years before sampling (1985–1986 AD), just after deposition of the porous level (between 1985/1986 and 1978/1979 AD). For the levels older than the porous level, age determination using laminae counting was possible only for the topmost section.
Taking into account the sampling thickness for 14C analysis (sampling fragments and not powders), the error on laminae-derived ages for 14C analysis is ca. 4 years for the three samples.
Table 1 δ18O (‰ vs PDB) analyses on CaCO3 samples from Saint-Martin spring manhole (Paris). SM-A 47-49 is the base of the core (construction rock).
14C results, reported as Fraction modern, Fm, are presented in table 2. They range between 0.73 and 0.93. Results for the F4 speleothem-like deposit (Versailles) are presented according to laminae-counting derived age (Figure 3).
Figure 3 14C trend vs laminae counting derived age for F4 sample (Versailles gallery)
Table 2 14C measurements on CaCO3 samples from Saint-Martin spring manhole (Paris, SM A and SM B samples) and Versailles underground gallery (F4 sample).
In this sample, 14C activity strongly increase from 1953±4 years AD to 1960±4 AD and decrease until the level representing the year 2011.
Results for the SM-A and SM-B cores are presented according to the adjusted depth scale (Figure 2c). 14C activity increase between 34 and 17 mm (adjusted depth scale) and decrease between 17 mm and the top most level.
This decrease is not contemporaneous with the transition between porous and laminated CaCO3, indicating that the difference of the laminae structure has no influence (or impact) on the 14C record.
For SM-A, using the laminae-derived chronology, this decrease corresponds to the period 1970–2010 (years AD).
Radiocarbon Recording within Urban Speleothems
The general trend for the F4 sample is similar to the atmospheric 14C bomb curve, with a significant and rapid 14C increase during the 1960s, and a gradual fall toward younger levels (Figure 4). Similarly, the trend observed for SM cores with low 14C within the oldest levels (SM-B) and an increase of 14C around ~1960–1970 reminds the atmospheric trend.
Figure 4 Top: Comparison between Saint Martin and Versailles 14C records. Bottom: Comparison between atmospheric (Hua et al. Reference Hua, Barbetti and Rakowski2013) and speleothem 14C records.
However, the maximum 14C reached within the speleothems differs from one site to the other, and is significantly lower than that of the atmosphere (maximum of 0.93 fm and 0.86 fm for the F4 speleothem from Versailles and the Parisian cores, respectively).
The radiocarbon bomb pulse has already been observed within young speleothems from natural caves, with very different shapes and intensities (e.g. Genty and Massault Reference Genty and Massault1997, Reference Genty and Massault1999; Mattey et al. Reference Mattey, Lowry, Duffet, Fisher, Hodge and Frisia2008; Smith et al. Reference Smith, Fairchild, Spötl, Frisia, Borsato, Moreton and Wynn2009; Hua et al. Reference Hua, McDonald, Redwood, Drysdale, Lee, Fallon and Hellstrom2012; Hodge et al. Reference Hodge, McDonald, Fischer, Redwood, Hua, Levchenko, Drysdale, Waring and Fink2011; Fohlmeister et al. Reference Fohlmeister, Kromer and Mangini2011).
This "bomb peak" can i) be lowered by the presence of dead carbon, from geological origin and/or old organic matter stored in the soil resulting in a low (with respect to the atmosphere) 14C peak; ii) be delayed with respect to the atmosphere due to C time transfer from atmosphere/soil to dripping water in the cave; iii) show an amplitude attenuation (or damping effect) of the bomb pulse recorded within speleothems compared to the atmospheric one.
The F4 and SM speleothems come from very different sites, both in an urban environment. The development of F4 is due to water percolating from the outside (rainfall), across a ca. 2 m thick "soil" overlain by a heterogeneous surface (mainly tarmac and paving stones) and across an artificial gallery (calcareous rock). The F4 host rock of the gallery is not calcareous. The SM crust deposits are due to water flowing from the Saint-Martin spring that drains waters from a small intensively urbanized watershed (buildings, tarmac, paving stones, and very few gardens). Host rocks are limestone and sandstone. Chemical analysis of Saint-Martin waters suggests a rapid flow (some weeks) of anthropogenic tracers (e.g. salts, CEREMA, personal communication).
The comparison between atmospheric and speleothem 14C suggests common features for the two studied sites: i) the lack of delay between the atmospheric 14C bomb pulse and its appearance within urban speleothems suggests a very fast (less than 2 years) C transfer; ii) this peak is significantly lower within the two speleothems suggesting the addition of non-atmospheric 14C; iii) the shape of the speleothem 14C record after the maximum displays a slow decrease (buffering effect) compared to the atmospheric 14C, suggesting a pool of longer time transfer C.
The parameters traditionally used for radiocarbon studies within "natural" speleothems are calculation of the dead carbon proportion (or DCP, carbon from host rock and old organic matter), and of the damping effect (attenuation of the atmospheric signal).
DCP were calculated using pre-bomb pulse 14C (atmospheric levels and older CaCO3 levels), following Genty and Massault (Reference Genty and Massault1997). We obtain DCP=17.2±0.3 % for F4 (Versailles) and DCP=21.7±0.2 % for SM-A c (Paris), assuming a "pre bomb age" around 1950 for this level.
Following Genty and Massault's (Reference Genty and Massault1999), Rudzka-Phillips, et al. Reference Rudzka-Phillips, McDermott, Jackson and Fleitmann2013 and Lechleitner et al. Reference Lechleitner, Baldini, Breitenbach, Fohlmeister, McIntyre, Goswami, Jamieson, van der Voort, Prifer, Marwan, Culleton, Kennett, Asmeron, Polyak and Eglinton2016, the damping effect (DE) was calculated using the following equation:
$${\rm DE}{\equals}\left[ {1-\left( {{\rm a}^{{14}} {\rm C}_{{{\rm int}{\rm .max}}} {\rm - a}^{{14}} {\rm C}_{{{\rm int}{\rm .min}}} } \right)/\left( {{\rm a}^{{14}} {\rm C}_{{{\rm atm}{\rm .1964}}} -{\rm a}^{{14}} {\rm C}_{{{\rm atm}{\rm .}}} _{{1950}} } \right)} \right]{\asterisk}100{\rm }\,\%\,.$$
With i) a14Cint.max and a14Cint.min respectively the maximum and minimum 14C initial activities (measured corrected for radioactive decay) within the CaCO3 deposits and ii) a14Catm.1964 and a14Catm.1950 the atmospheric 14C activities for respectively the years 1964 (14C maximum) and 1950. DE is 86.9 % for F4 (using data from levels at 23 and 28 mm). This parameter has to be used with caution in our case, because the "real" 14C maximum could be missing, due to the method of sampling. For SM cores (Paris), the lack of precise chronology for the oldest levels (before 1970 AD) and the low resolution sampling make the DE calculation difficult.
A comparative study of these parameters in speleothems from European natural caves (Rudzka-Phillips et al. Reference Rudzka-Phillips, McDermott, Jackson and Fleitmann2013) showed that high damping effects are found in stalagmites from sites characterized by a thick soil cover and dense, well developed vegetation, under a humid climate and high mean annual air temperatures. In these natural sites with high DE, the carbon incorporated in the stalagmites originates predominantly from old recalcitrant organic matter, which can be mixed with young atmospheric carbon. High DCP has been related to natural sites with a dense vegetation cover (such as forests), and related to intense host rock dissolution due to soil activity (roots and microbial organic matter decomposition, Genty and Massault (Reference Genty and Massault1997), or to old organic matter incorporation.
The urban sites studied here are not covered by dense vegetation, but old organic matter may be stored within the Parisian site at least, since before its urbanization during the 1800s, cultivated fields were present within the watershed of the spring (Huard Reference Huard2011; DelaGrive Reference DelaGrive1870) and wastes from the inhabitants of Paris were used as fertilizer (Delamare 1722–Reference Delamare1738; Barles Reference Barles1999). It may seem contradictory to have a high DE and a fast transfer of C from the atmosphere to the speleothem (as suggested by the lack of delay, within uncertainties, between the 14C rise within CaCO3 and the atmospheric 14C bomb pulse). This can be explained if during the fast transfer of the atmospheric carbon to the speleothem there is a continuous (and fast) mixing with some non-atmospheric carbon (most likely a mixture of carbon derived from soils and limestones when they occur). The signature (a14Cna) and the fraction (fna) of this non-atmospheric carbon can be estimated by assuming that they remain identical before and during the bomb pulse, while the atmospheric signature was different before (a14Ca_1950) and during the pulse (a14Ca_pulse).
The 14C signature of the speleothem before and after the bomb pulse is given by:
$${\rm a}^{{14}} {\rm C}_{{{\rm spel}\_1950}} {\equals}{\rm f}_{{{\rm na}}} {\times}{\rm a}^{{14}} {\rm C}_{{{\rm na}}} {\plus}\left( {1{\minus}{\rm f}_{{{\rm na}}} } \right){\times}{\rm a}^{{14}} {\rm C}_{{{\rm a}\_1950}} \left( {{\rm before}\,{\rm the}\,{\rm bomb}\,{\rm pulse}} \right)$$
$${\rm a}^{{14}} {\rm C}_{{{\rm spel \_pulse}}} {\rm {\equals}f}_{{{\rm na}}} {\times}{\rm a}^{{14}} {\rm C}_{{{\rm na}}} {\plus}\left( {1{\rm {\minus}f}_{{{\rm na}}} } \right){\times}{\rm a}^{{14}} {\rm C}_{{{\rm a\_ pulse}}} \left( {{\rm during}\,{\rm the}\,{\rm bomb}\,{\rm pulse}} \right)$$
it follows that:
$${\rm f}_{{{\rm na}}} {\rm {\equals}}\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_}1950}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_}1950}} } \right){\rm /}\left( {{\rm a}^{{14}} {\rm C}_{{{\rm na}}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_}1950}} } \right)\,\left( {{\rm before}\,{\rm the}\,{\rm bomb}\,{\rm pulse}} \right)$$
$${\rm f}_{{{\rm na}}} {\rm {\equals}}\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_ pulse}}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_ pulse}}} } \right)/\left( {{\rm a}^{{14}} {\rm C}_{{{\rm na}}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_ pulse}}} } \right)\,\left( {{\rm during}\,{\rm the}\,{\rm bomb}\,{\rm pulse}} \right)$$
Combining these 2 equations, we deduce:
$$\eqalignno{ {\rm a}^{{14}} {\rm C}_{{{\rm na}}} & {\equals}\left\{ {{\rm a}^{{14}} {\rm C}_{{{\rm a\_}1950}} {\rm \,/\,}\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_}1950}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_}1950}} } \right){\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_pulse}}} /\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_pulse}}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_pulse}}} } \right)} \right\} \cr & /\left\{ {1\,/\,\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_}1950}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_}1950}} } \right){\rm {\minus}}1/\left( {{\rm a}^{{14}} {\rm C}_{{{\rm spel\_pulse}}} {\rm {\minus}a}^{{14}} {\rm C}_{{{\rm a\_ pulse}}} } \right)} \right\} $$
Using the following numerical values (for F4): a14Cspel_1950=0.81, a14Cspel_pulse=0.93, a14Ca_1950=1.0, a14Ca_pulse= 1.3 -2.0 (to take into account the uncertainties on the chronology), we obtain:
$${\rm a}^{{14}} {\rm C}_{{{\rm na}}} {\equals}0.73\,\pm\,0.5$$
$${\rm f}_{{{\rm na}}} {\equals}74\,\pm\,15\,\%\,$$
Clearly, a14Cna does not correspond to a simple dead carbon reservoir and must contain some relatively young soil carbon.
High DE and fast atmospheric C transfer could thus be attributed to a mixing between "fast C" (fast turn over within urban soil) and a small fraction of "long time C" (from old organic matter stored within urban soils), as observed for some natural caves (e.g. Rudzka-Phillips, et al. Reference Rudzka-Phillips, McDermott, Jackson and Fleitmann2013). A second aspect of our urban sites is the possible incorporation of old anthropogenic carbon (as particles or as reduced atmospheric 14C) due to fossil carbon (gasoline/fuel/coal) burning. A local Suess effect, with urban atmospheric 14C lower than "general" trends, has been reported in several industrial metropolises or regions (Awsiuk and Pazdur Reference Awsiuk and Pazdur1986; Quarta et al. Reference Quarta, D'Elia, Rizzo and Calcagnile2005; Rakowski et al. Reference Rakowski, Nakamura, Pazdur, Charro, Villanueva and Piotrowska2010; Svetlik et al. Reference Svetlik, Povinec, Molnár, Vána, Šivo and Bujtás2010) and could cause apparent DCP and DE increases when compared with clean-air 14C curves. DCP is higher at Saint-Martin spring (Paris), than at the Versailles site. Possible explanations for this higher values could be i) a higher Suess effect, as Paris is subjected to higher levels of fossil carbon burning (more populous); ii) a longer water-transfer time, with higher exchange with host rock for Saint-Martin (Paris) site respect to Versailles; iii) more old carbon stored within the soils in Paris.
Further work will be undertaken to characterize the carbon from present day waters and work at a higher spatial resolution for CaCO3 analysis, but also to analyze new urban sites.
Lamination and Porous Level within Urban Speleothem-like Deposits
For the Saint Martin spring flowstone, we found the same facies change and the alternation of laminated and porous levels with the same chronology, in two locations close to each other (SM A and SM B), despite different growth rates at the two locations. Even if very poorly defined within the porous level, the chronology derived from laminae counting along the cores (assuming two laminae per year) is coherent with the 14C bomb pulse record. The porous level identified within the two cores is characterized by a higher growth rate and lower δ18O values than the laminated levels. Further work will determine the origin of this change in relation with environmental / anthropic factors (water pathway and quality, site ventilation, impact of urbanization on water circulation, etc.). The comparison between laminae counting and the 14C bomb pulse record confirms the bi-annual rate of laminae deposition in the Versailles underground (F4 sample). As mentioned previously, the Versailles (F4 sample) and Paris sites (SM-A and SM-B) present contrasting contexts, particularly for the water sources and pathways: water from a perched aquifer (small watershed infiltration and potential water leaks from drinking and/or waste water, see Pons-Branchu et al. Reference Pons-Branchu, Douville, Roy-Barman, Dumont, Branchu, Thil, Frank, Bordier and Borst2014) for Paris (SM samples), and precipitation infiltrating urban soil for the Versailles gallery (F4). Despite these different water pathways (and possibly different time transfer), bi-annual lamination was found in both sites. In natural sites, the lamination (visible or UV-luminescent) has been characterized by density or textural/mineralogical differences, and/or different organic matter content (e.g. Shopov et al. Reference Shopov, Ford and Schwarz1994; Borsato et al. Reference Borsato, Frisia, Fairchild, Somogyi and Susini2007; Baker et al. Reference Baker, Smith, Jex, Fairchild, Genty and Fuller2008). This lamination could be caused by different factors: seasonal variations in drip rate, seasonal variations in water supersaturation, cave ventilation (relative humidity, CO2), organic matter flushed from the soil during autumn causing the formation of a thin brown and UV-luminescent layer during this period (Baker et al. Reference Baker, Smith, Jex, Fairchild, Genty and Fuller2008 and references therein; Borsato et al. Reference Borsato, Frisia, Fairchild, Somogyi and Susini2007).
Further work on urban speleothem-like deposits will focus on characterization of the visible laminae in urban sites, with no (or very rare) vegetation cover, but also on organic matter characterization within these natural archives in order to understand their formation and the link with environmental parameters.
Three speleothem-like deposits from two historical urban undergrounds in Versailles and Paris (France) were studied for their 14C content, and compared, when possible, with lamination counting. The radiocarbon bomb pulse recorded in these urban deposits is in agreement with a bi-annual lamination, with at least for one site, no delay between atmospheric pulse and the CaCO3 record. For both sites, comparison between CaCO3 urban deposits and the atmospheric record (Hua et al. Reference Hua, Barbetti and Rakowski2013) highlight high dead carbon proportion (between 17.2±0.3 % and 21.7±0.2 % for respectively Versailles and Paris), and a high damping effect (86.9 %) for Versailles site, suggesting i) fast carbon transfer from the atmosphere to the urban underground, ii) the influence of old "geological" carbon (calcareous host rock and/or construction stones or backfills), and/or the possible influence of old recalcitrant organic matter, causing a damping effect and high dead carbon proportion; iii) the possible influence of local anthropic carbon (black carbon and Suess effect).
This study has shown that 14C analysis can be used for chronological purposes in these very poorly-studied natural archives in urban sites, and opens up new perspectives for the study of those without lamination. These archives have a very high potential as historical records of past water quality and the influence of urbanization on the urban water cycle, and the coupling between precise chronology and isotopic tracers of the water cycle and/or human activities (e.g. Sr or Pb isotopes) offers new perspectives for historical reconstructions.
We thank the LMC14 staff (Laboratoire de Mesure du Carbone-14), ARTEMIS national facility, UMS 2572 CNRS-CEA-IRD-IRSN-MCC, for the results obtained with the accelerator mass spectroscopy method.
This work was financed by Paris municipality (Paris 2030 call "Histoires d'eau souterraine" project), by the "Fondation des Sciences du Patrimoine/LabEx Patrima" (ANR-10-LABX-0094-01) and by the CNRS INSU institute for 14C analyses (INSU/ARTEMIS national call).
The authors thank the ASNEP Association (Association Sources du Nord – Etudes et Préservation) and the City of Paris (Direction des Affaires Culturelles) for access to the Belleville aqueduct, sampling facilities and historical information, and F Barbecot for help during SM sampling.
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Implications of unprovability of $P\neq NP$
I was reading "Is P Versus NP Formally Independent?" but I got puzzled.
It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is not provable (say in $ZFC$). (Let's assume that we only find out that $\mathsf{P} \neq \mathsf{NP}$ is independent from $ZFC$ but no further information about how this is proven.)
What will be the implications of this statement? More specifically,
Assuming that $\mathsf{P}$ capture the efficient algorithms (Cobham–Edmonds thesis) and $\mathsf{P} \neq \mathsf{NP}$, we prove $\mathsf{NP\text{-}hardness }$ results to imply that they are beyond the present reach of our efficient algorithms. If we prove the separation, $\mathsf{NP\text{-}hardness}$ means that there is no polynomial time algorithm. But what does an $\mathsf{NP\text{-}hardness }$ result mean if the separation is not provable? What will happen to these results?
efficient algorithms
Does unprovability of the separation mean that we need to change our definition of efficient algorithms?
cc.complexity-theory np-hardness big-picture proofs p-vs-np
Kaveh
$\begingroup$ The first thing you need to ask is: formally independent of what? In mathematical logic, there are many sets of axioms people have considered. The default one is ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice. What it means to be independent of ZFC is that neither P=NP or P!=NP can be proved from these axioms. $\endgroup$ – Peter Shor Feb 2 '12 at 15:38
$\begingroup$ If you want to know what a proof for a statement of the form "whether X or not is independent of axiomatic system Y" looks like, why don't you just read some examples? The independence of the Axiom of Choice from the Zermelo-Fraenkel set theory is a famous example. I voted to close as not a real question by mistake, but I meant to vote to close as off topic. $\endgroup$ – Tsuyoshi Ito Feb 2 '12 at 16:01
$\begingroup$ Did you to read the very good and freely available Scott Aaronson's paper; "Is P Versus NP Formally Independent?" (scottaaronson.com/papers/pnp.pdf) $\endgroup$ – Marzio De Biasi Feb 2 '12 at 16:02
$\begingroup$ The question "if X is proved independent of ZFC, and we have some theorems of the form X $\rightarrow$ Y, what happens to these theorems?" seems well-posed, and is the question that I believe the OP is asking. The answer would seem to be: in some axiom systems, such as ZFC + X, we then have Y holding, while in ZFC + $\lnot$X we have no information about Y. As such, these conditional theorems would still have some value. In fact, they would have more value in this situation than if $\lnot$X were to be proved to be a theorem. $\endgroup$ – András Salamon Feb 7 '12 at 10:15
$\begingroup$ The ZFC unprovability of P vs NP would probably have a lot more implication for Set Theory than Complexity Theory. $\endgroup$ – David Harris Feb 8 '12 at 20:21
Your question might better be phrased, "How would complexity theory be affected by the discovery of a proof that P = NP is formally independent of some strong axiomatic system?"
It's a little hard to answer this question in the abstract, i.e., in the absence of seeing the details of the proof. As Aaronson mentions in his paper, proving the independence of P = NP would require radically new ideas, not just about complexity theory, but about how to prove independence statements. How can we predict the consequences of a radical breakthrough whose shape we currently can't even guess at?
Still, there are a couple of observations we can make. In the wake of the proof of the independence of the continuum hypothesis from ZFC (and later from ZFC + large cardinals), a sizable number of people have come around to the point of view that the continuum hypothesis is neither true nor false. We could ask whether people will similarly come to the conclusion that P = NP is "neither true nor false" in the wake of an independence proof (for the sake of argument, let's suppose that P = NP is proved independent of ZFC + any large cardinal axiom). My guess is not. Aaronson basically says that he wouldn't. Goedel's 2nd incompleteness theorem hasn't led anyone that I know of to argue that "ZFC is consistent" is neither true nor false. P = NP is essentially an arithmetical statement, and most people have strong intuitions that arithmetical statements—or at least arithmetical statements as simple as "P = NP" is—must be either true or false. An independence proof would just be interpreted as saying that we have no way of determining which of P = NP and P $\ne$ NP is the case.
One can also ask whether people would interpret this state of affairs as telling us that there is something "wrong" with our definitions of P and NP. Perhaps we should then redo the foundations of complexity theory with new definitions that are more tractable to work with? At this point I think we are in the realm of wild and unfruitful speculation, where we're trying to cross bridges that we haven't gotten to and trying to fix things that ain't broke yet. Furthermore, it's not even clear that anything would be "broken" in this scenario. Set theorists are perfectly happy assuming any large cardinal axioms that they find convenient. Similarly, complexity theorists might also, in this hypothetical future world, be perfectly happy assuming any separation axioms that they believe are true, even though they're provably unprovable.
In short, nothing much follows logically from an independence proof of P = NP. The face of complexity theory might change radically in the light of such a fantastic breakthrough, but we'll just have to wait and see what the breakthrough looks like.
Timothy ChowTimothy Chow
$\begingroup$ @vzn: Your examples aren't just "arguably" arithmetical; they're unquestionably arithmetical. But I'm not sure what your point is. Take some diophantine equation $E$ with the property that "$E$ has no solutions" is undecidable in ZFC. My point is that everyone I know believes that either $E$ has solutions or it doesn't, and that we just can't prove it one way or the other. Do you believe that there is no fact of the matter about whether $E$ has solutions—that $E$ neither has nor doesn't have solutions? $\endgroup$ – Timothy Chow Feb 7 '12 at 15:13
$\begingroup$ @vzn: I think you completely miss the point. The question is not whether a particular statement is undecidable, but whether it is neither true nor false. The two concepts are entirely distinct. Would you say, for example, that ZFC is neither consistent nor inconsistent? Everyone (else) that I know believes that either ZFC is consistent, or it isn't, even though we may have no way of determining which is the case. $\endgroup$ – Timothy Chow Feb 8 '12 at 1:11
$\begingroup$ "this sounds like religion to me and not mathematics" — Welcome to metamathematics. Perhaps a less objectionable way of saying "X is neither true nor false" is that we have no a priori reason to prefer an axiomatic system in which X is true over an axiomatic system in which X is false. We have an (almost) universally agreed standard model of arithmetic; as a social convention, we accept arithmetic statements that hold in that model as being really, actually true. The same cannot be said for set theory. $\endgroup$ – Jeffε Feb 9 '12 at 13:02
$\begingroup$ See also consc.net/notes/continuum.html and mathoverflow.net/questions/14338/… — Each mathematician's personal mix of formalism, platonism, and intuitionism is essentially a religious conviction. $\endgroup$ – Jeffε Feb 9 '12 at 13:06
$\begingroup$ @vzn: You still miss the point. Even if we grant you your personal religious beliefs, all you're saying is that you wouldn't join Aaronson and the rest of the world in declaring arithmetical sentences to be either true or false. We all agree that there's no way to tell from the form of a statement whether it's undecidable, but that's not the claim. The claim is that almost everyone except you does have strong intuitions that arithmetical statements are either true or false. Just because you don't share that conviction doesn't mean that others don't have it. $\endgroup$ – Timothy Chow Feb 13 '12 at 18:03
This is a valid question, even though perhaps a little unfortunately phrased. The best answer I can give is this reference:
Scott Aaronson: Is P versus NP formally independent. Bulletin of the European Association for Theoretical Computer Science, 2003, vol. 81, pages 109-136.
Abstract: This is a survey about the title question, written for people who (like the author) see logic as forbidding, esoteric, and remote from their usual concerns. Beginning with a crash course on Zermelo Fraenkel set theory, it discusses oracle independence; natural proofs; independence results of Razborov, Raz, DeMillo-Lipton, Sazanov, and others; and obstacles to proving P vs. NP independent of strong logical theories. It ends with some philosophical musings on when one should expect a mathematical question to have a definite answer.
Andrej BauerAndrej Bauer
$\begingroup$ Uh, I totally missed the fact that Aaronson's paper was already mentioned in the comments. My apologies. $\endgroup$ – Andrej Bauer Feb 4 '12 at 15:18
As Timothy Chow explains, just knowing that a theorem is independent from a theory doesn't say much about the truth/falsity of that statement. Most non-experts confuse formal unprovability in a fixed theory (like $[ZFC][1]$) with impossibility of knowing that answer to the truth/falsity of the statement (or sometimes meaninglessness of the statement). Independence and formal unprovability always means independence/unprovability in a theory. It simply means that the theory can prove neither the statement nor its negation. It doesn't mean that the statement does not have a truth value, it doesn't mean that we cannot know the truth value of the statement, we might be able to add new reasonable axioms that will make the theory strong enough to be able to prove the statements or its negation. At the end, provability in a theory is a formal abstract concept. It is related to our real world experience only as a model.
Same applies to the thesis that efficient computation is captured by complexity class $\mathsf{P}$. See this post.
Now you can ask if it is possible for a formal statement to not have a truth value. Generally in practice in principle, we can affirm $\Sigma_1$ (a.k.a. r.e.) properties and refute $\Pi_1$ (a.k.a. co-r.e.) properties by observations. Any statement more complex than this is not directly observable, i.e. no (finite) observation will allow you to affirm or refute the statement. However we can look at the observable logical consequences of these statements and try to use them to decide whether a statement is true or false. (For more on finitely observable properties see Samson Abramsky's Ph.D. thesis "Domain Theory and the Logic of Observable Properties", 1987 and Steven Vickers' "Topology via Logic", 1996.)
For most mathematicians statements of higher logical complexity are also meaningful and have a truth value, but this goes into the philosophical issues in mathematics. Almost all mathematicians believe that statements in the arithmetical hierarchy are meaningful and have definite truth values, and in some sense they view the truth value of these statements to be more definite than statements of higher logical complexity (like CH). The statement $\mathsf{P} \neq \mathsf{NP}$ can be stated as a $\Sigma_2$ statement and therefore is an arithmetical statement. As such, almost all mathematicians would believe that it is meaningful and has a definite truth value. You may want have a look at this MO question, and search the posts on FOM mailing list.
KavehKaveh
Just some rambling thoughts about this. Feel free to criticize.
Let Q = [cannot prove (P = NP) and cannot prove (P /= NP)]. Suppose Q for a contradiction. I will also assume that all known discoveries about P vs NP are still viable. In particular, all NP problems are equivalent in the sense that if you can solve one of them in polynomial time, you can solve all others in polynomial time. So let W be an NP complete problem; W equally represents all problems in NP. Because of Q, one cannot obtain an algotithm A to solve W in polynomial time. Otherwise we have proof that P = NP, which contradicts Q (1)(*). Note that all algorithms are computable by definition. So saying that A cannot exist implies that there is no way to compute W in polynomial time. But this contradicts Q (2). We are left with rejecting either (1) xor rejecting (2). Either case leads to a condradiction. Thus Q is a contradiction, which means that the proof of whether or not (N = NP) must exist.
(*) You might say, "Aha! A might exist, but we just cannot find it". Well, if A existed, we can enumerate through all programs to find A by enumerating from smaller programs to larger programs, starting with the empty program. A must be finite because it is an algorithm, so if A exists, then the enumeration program to find it must terminate.
Thomas EdingThomas Eding
$\begingroup$ @Victor: Good point. I imagine that if A exists, then one can simply analyze each enumerated program to see if it indeed solves an NP complete problem in polynomial time. I believe that since one is working with a finite instruction set (given by some universal computer) that A can be identified. But I'm no expert. $\endgroup$ – Thomas Eding Feb 10 '12 at 20:12
$\begingroup$ Also... If A exists, then let N be the size of A. Let T be the set of all program of size <= N. One can simultaneously run W on all A' in T. As each A' terminates, run the output O through a program that checks to see if O solves W. (Note that any so-called 'solution' to an NP complete problem can be verified in polynomial time.) If O is a correct answer, shut off all other computers and return O. Keep in mind that not every A' must terminate because A is one of them and will output a correct O in polynomial time. Thus one does not need to even prove that A solves P=NP. N exists by definition. $\endgroup$ – Thomas Eding Feb 10 '12 at 20:27
$\begingroup$ I fail to see the problem. When A' terminates, check its output O. If O is valid, stop all other A' and return O. O can be verified on the machine that A' terminates on, so you will not get a bunch of queued verifying programs. The only problem I see with this approach is getting a good N. I think it might be enough to say that N is finite (which it must be) to disprove Q. $\endgroup$ – Thomas Eding Feb 10 '12 at 20:46
$\begingroup$ All NP complete are search problems such that proposed solutions can be verified in polynomial time (acquiring such a proposed solution is allowed to be "difficult" though). O is not intended to prove if P=NP or not. O is simply a proposed solution to a particular instance W of an NP complete problem. For example, if I give you a tour for a particular travelling salesman problem, you can check to see if it is a shortest tour in polynomial time. $\endgroup$ – Thomas Eding Feb 10 '12 at 21:20
$\begingroup$ "P = NP is independent of ZFC" is not the same as "we cannot find an algorithm to solve any problem in NP in deterministic polynomial time", as Victor has pointed out. The precise definitions of these classes are rather important when dealing with notions such as independence with respect to a theory. $\endgroup$ – András Salamon Feb 12 '12 at 19:40
Not the answer you're looking for? Browse other questions tagged cc.complexity-theory np-hardness big-picture proofs p-vs-np or ask your own question.
Do we know that the P vs. NP question isn't affected by Gödels incompleteness theorem?
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Rigour leading to insight
Do the undecidable attributes of P pose an obstruction to deciding P versus NP? (answer: maybe)
Logic capturing automorphism-invariant $\mathsf{AC^0}$ properties
Does P contain incomprehensible languages? (TCS community wiki)
Does P contain languages whose existence is independent of PA or ZFC? (TCS community wiki)
Mathematical implications of complexity theory conjectures outside TCS
Can on every instance P = NP?
Undecidable Single Programs
Quantum Hardness of Finding Nash Equilibria | CommonCrawl |
Picking a station
Written by Colin+ in hypothesis testing, statistics.
There's a lot of dead time to fill up when you're covering the Boat Race. Once you've interviewed one gaggle of inebriated revellers from each of the two universities, you've interviewed them all. Once you've explained that the best strategy is to go faster than the other boat, there's not much left to say on a technical front1 So you're left with... statistics.
Now, the Boat Race has been going an awfully long time -- 150 years or so -- and it's always the same two teams, so there are plenty of easy-to-understand statistics knocking about. One that flashed up on the screen showed the two stations -- which side of the river you start from, Middlesex (to the north) or Surrey (to the south) -- and how many times the team that picked that station had won the race. I forget which way around, but I remember the numbers: it was 76-74, one way or the other.
The commentators talked about the '76' side conferring a clear advantage, and I almost caught a crab2 You see: 76-74 is as close as you can get to a dead-heat over 150 races without actually being one.
Naturally, this calls for an experiment. I pick a null hypothesis: that the station picked confers no advantage -- that is to say, the probability of (say) the Surrey station team winning is 0.5. I could then toss 150 coins over and over again, and count how often I got more than 76 heads.
I'm not going to do that, I have writing to do.
Instead, I'm going to do some statistics, and assume that the results are binomially distributed. If you toss 150 fair coins, you expect to get 75 heads, on average. But what about the variance? Won't someone think of the variance? The variance of a binomial distribution is $npq$, where $n$ is how many times you run the experiment, $p$ is the probability of your event coming off and $q$ is the probability of it not happening -- giving a variance of $150 \times 0.5 \times 0.5 = 37.5$. Its standard deviation, then, is 6.12 or so.
Why is that useful? It's because when you have even a moderately big binomial distribution (with $n > 30$ or so), you can approximate it as a normal distribution. That means, we can do $z$-score tests and work out how unusual our results are.3
The $z$-score is how many standard deviations above the mean your observation is -- in this case, it's $\frac{1}{\sqrt{37.5}}$, or about 0.16. That's a very small $z$-score: the probability of getting 76 or more heads out of 150 is only 56 or 57% -- it's not at all an unusual result.
Before your average statistician calls a difference significant, s/he would usually require a $z$-score of at least 2. Had the Surrey station won somewhere around 87 or 88 races, then you'd be able to say with reasonable confidence that there was an advantage to choosing it. If you were a physicist, on the other hand, requiring a 'six-sigma' level, you'd only be happy to say there was an advantage if one of the stations had won something like 111 or 112 races.
Statistics and the court of law
The Mathematical Ninja and the Poisson Distribution
The Names of the Isle of Portland
Ask Uncle Colin: Cambridge!
Poissons and binomials
Really? That's very interesting. You should go on the telly or something. [↩]
see, I do know something about rowing. [↩]
In this article, I haven't adjusted scores for continuity, for the sake of simplicity. If I had, the results would be even less significant. [↩]
One comment on "Picking a station"
Picking a station http://t.co/4Q6YFbO2rU | CommonCrawl |
The Maths of Star Trek: The Original Series (Part III)
This is the third in a series of posts about the maths of Star Trek. Part I covered the probability of survival while wearing a red shirt, and Part II discussed the mathematics of alien biology.
Are we alone in the galaxy? Drake's Equation
In the episode 'Balance of Terror', Captain Kirk and Doctor McCoy find themselves in reflective mood and the question of our uniqueness in the galaxy is discussed:
McCoy counsels Kirk
KIRK: I wish I were on a long sea voyage somewhere. Not too much deck tennis, no frantic dancing, and no responsibility. Why me? I look around that Bridge, and I see the men waiting for me to make the next move. And Bones, what if I'm wrong?
MCCOY: Captain, I…
KIRK: No, I don't really expect an answer.
MCCOY: But I've got one. Something I seldom say to a customer, Jim. In this galaxy, there's a mathematical probability of three million Earth-type planets. And in all of the universe, three million million galaxies like this. And in all of that, and perhaps more, only one of each of us. Don't destroy the one named Kirk.
Here, McCoy is applying an idea similar to one used by astronomers today to estimate the number of active alien civilisations in our galaxy that might have radio technology, which we can then detect with our large radio receivers.
Frank Drake
To estimate this number, in 1961 astrophysicist Frank Drake came up with the following formula:
\[N= R_* \times f_p \times n_e \times f_\ell \times f_i \times f_c \times L\]
with parameters;
$R_*$, the average rate of star formation per year in our galaxy. Drake and his colleagues originally gave this a conservative estimate of 1 star per year. Current estimates from NASA would now put this at around 7 per year.
$f_p$, the fraction of those stars that have planets. Drake originally estimated this fraction to be around 20%-50%. Current estimates are closer to 100%, in that stars without planets appear to be the exception rather than the rule.
$n_e$, the average number of planets that can potentially support life per star that has planets. Drake estimated this to be around 1-5 planets per star. NASA have said 5.4% of stars may host a terrestrial planet, making 0.054 planets per star.
$f_\ell$, the fraction of the above that actually go on to develop planetary life at some point. Drake put this value at 100%, and it is still thought to be very high.
$f_i$, the fraction of the above that actually go on to develop intelligent life. Again, Drake estimated this to be 100%, but the true value for this parameter is unknown.
$f_c$, the fraction of civilizations that develop a technology that releases detectable signs of their existence into space. Drake's estimate was 10%-20%, although any sufficiently advanced civilisation may be thought of as 'broadcasting' even if not intentionally.
$L$, the length of time for which such civilizations release detectable signals into space. Drake and his colleagues estimated this to be between 1000 and 1,000,000,000 years. The average human civilisation has only lasted 420 years, but an advanced society might be able survive perpetually.
Drake and his colleagues concluded that, given the uncertainties, $N \approx L$, meaning there were probably between 1000 and 100,000,000 civilizations in the galaxy which we could detect.
Our current estimates put $N \approx \frac{L}{5}$, which means the determining factor for communication is still $L$, the length of time we can expect an advanced civilisation with broadcasting technology to last.
The original intention of the formula was to promote discussion between Drake and his colleagues, and many of the variables of Drake's equation are open to conjecture. Estimates for the number of active civilisations with radio communication in our galaxy has ranged from the tens of millions to Earth being alone in the galaxy. One thing that can be agreed on is that the existence of human civilisation proves it's not zero.
Drake's equation was used by Gene Roddenberry in 1964 in his pitch for Star Trek to justify the large number of inhabited planets in the show. He did not have a copy of the equation so he made up his own. Roddenberry's formula was as follows:
$$F f^2 (MgE) – C^1 Ri^1 \times M = L / So$$
No explanation of the parameters of Roddenberry's equation were given, however both forms of the formula may be seen on a poster as a background detail in a later episode of Star Trek: Voyager.
It is said that Drake himself visited the set one day, and even viewed Roddenberry's "second variation", before gently pointing out that a value raised to the first power is merely the value itself.
A similar mistake later made it to the screen in one of Star Trek's most notorious mathematical gaffs. In 'Court Martial' Kirk searches for a missing crewmember by removing the remainder of the crew and using the ship's sensors to pick up a heartbeat:
1 × 1 × 1 × 1 = 1
KIRK: Ready, Mister Spock?
SPOCK: Affirmative, Captain.
KIRK: Gentlemen, this computer has an auditory sensor. It can, in effect, hear sounds. By installing a booster, we can increase that capability on the order of one to the fourth power. The computer should bring us every sound occurring on the ship.
Unfortunately, no one points out to Kirk that 1 to the fourth power is still 1.
In recent years, some wags have repurposed Drake's equation, not to work out the probability of finding intelligent life, but to work out the probability they will find a girlfriend. Parameters include rate of people formation (birth rate), fraction of people who are women (51%), and fraction of age-appropriate women the author finds attractive (various estimates). When applied, the author invariably discovers that the probability is very low.
Next we will consider a different type of intelligent civilisation – android civilisation. It's life Jim, but not as we know it.
20th century mathematicians and 23rd century androids – The Liar's Paradox
Throughout the original series, Captain Kirk talked a computer to death on no less than four different occasions, through the use of paradoxes or other dilemmas.
For example, in 'I Mudd' Kirk and co are kidnapped and held by a planet of androids. Here, after seeding confusion by miming an explosion, Kirk and the rogue Harry Mudd introduce a paradox to Norman, the android leader:
Norman expires
NORMAN: But there was no explosion.
MUDD: I lied.
NORMAN: What?
KIRK: He lied. Everything Harry tells you is a lie. Remember that. Everything Harry tells you is a lie.
MUDD: Listen to this carefully, Norman. I am lying.
NORMAN: You say you are lying, but if everything you say is a lie then you are telling the truth, but you cannot tell the truth because everything you say is a lie. You lie. You tell the truth. But you cannot for. Illogical! Illogical! Please explain.
(Smoke comes out of Norman's head.)
Unable to resolve the paradox, Norman goes blank, leaving the crew of the Enterprise to escape. This is known as the Liar's Paradox and may be more simply stated as: "This sentence is false".
It turns out that the Liar's Paradox is central to one of the greatest results in 20th century mathematics – but a result that led to many mathematicians with smoke coming out of their head.
Mathematics is built from a number of basic truths called 'axioms', from which we prove other results. For example, some axioms of arithmetic might be that $0$ is a natural number, and all natural numbers have a successor. From these basic truths we can prove such results as $1 + 1 = 2$. In fact, this result was proved in 1910 by Alfred North Whitehead and Bertrand Russell and appears on page 86, Volume II of their book Principia Mathematica.
The above proposition is occasionally useful
At the beginning of the 20th century, logicians such as Whitehead, Russell, and David Hilbert wanted to worked out these basic truths of mathematics, from which all other mathematical results can be proven. Importantly, the system of axioms would need to be 'complete', so they could be used to prove all true results, and 'consistent', so we always get the same result no matter how we choose to prove it. In other words, we don't find anything contradictory like $1 = 2$.
In 1931, the Austrian American mathematician, Kurt Gödel rocked the mathematical community with the proof of what is now known as Gödel's Incompleteness Theorems. In his proof, Gödel could represent statements with numbers, so any system that can be used to prove facts of arithmetic, may also be used to prove facts about its own statements.
Kurt Gödel. I call his hair Lokai
Now replace the sentence "This sentence is false" in the Liar's Paradox, with the sentence "This sentence is not provable". Gödel showed that a number representing such a statement can always be constructed, meaning the system could not be both complete and consistent. In other words, some results will be true but not provable within that framework.
The use of paradoxes against artificial life must have become required reading at Starfleet academy. Later, Captain Picard and the crew of the Enterprise D come up with a similar tactic of using an Escher-style picture to destroy the Borg… but that's an essay for another time.
Tags: Alfred North Whitehead, axioms, Bertrand Russell, David Hilbert, Drake's Equation, foundations, Frank Drake, Gödel's Incompleteness Theorems, Kurt Gödel, Liar's Paradox, one to the fourth power, Roddenberry's equation, Star Trek
11 Responses to "The Maths of Star Trek: The Original Series (Part III)"
Marek Bernat May 3rd, 2013
Order of one doesn't mean 1. It usually (at least in physics) means the most significant digit. So, Spock could have very well meant that the capability was 2, 3 or even 4.9 (5 might be considered an order of ten though).
Rob May 3rd, 2013
But it's not necessarily "on the order of 1" "to the power of 4". I would say it's more intuitive to be read as "on the order of" "one to the power of 4". If it was correctly written as "on the order of ten to the power of 4" you would consider that "on the order of ten thousand" not "somewhere between 5^4=625 and 15^4=50625".
Geoff Robbins (@_TheGeoff) May 3rd, 2013
That's generally referred to as an order of magnitude (powers of ten to non-physicists), however, but yes, agree "of the same order of magnitude" in a physics context can mean up to 9x bigger. Don't think that was the scriptwriters intention though.
antialiasis May 3rd, 2013
In that case he could still have just said "on the order of one" rather than "on the order of one to the fourth power".
Scott May 13th, 2013
Guys, guys. I've discovered the solution to the 1x1x1x1 problem.
What Kirk meant to say was:
"By installing a booster we can increase that capability on the order of one, to the fourth power."
(I've moved the comma, Kirk had the comma in his script in the wrong place.) Now it reads as meaning "after we've ordered [purchased] a booster, the
capability is increased to the fourth power."
Christian Perfect May 13th, 2013
So you reckon the "on the order of one" bit was just redundant verbiage? I'm not convinced.
Anthony February 19th, 2016
What application does Godel's liars paradox proof have–beyond destroying optimistic dreams of prooving all maths as complete and consistent? Like is this the reason that 9.999repeating is actually 10?
France's Yozawitz April 3rd, 2017
I Love Star Trek.
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Impact of alternative materials to plasticized PVC infusion tubings on drug sorption and plasticizer release
N. Tokhadze ORCID: orcid.org/0000-0001-7448-37951,
P. Chennell1,
L. Bernard1,
C. Lambert2,
B. Pereira2,
B. Mailhot-Jensen3 &
V. Sautou1
Medical tubings in plasticized polyvinylchloride (PVC) are widely used for the infusion of medications but are known in some cases to cause content-container interactions (drug sorption and plasticizer release). The aim of this study was to assess interactions between drugs and five alternative materials to a reference plasticized PVC intravenous (IV) infusion tubing: three were PVC coextruded with polyethylene (PE), polyurethane (PU) or a thermoplastic elastomer (Styrene-EthyleneButadiene-Styrene (SEBS)) and two were SEBS or thermoplastic olefin (TPO) monolayer tubings. Diazepam and insulin were chosen as respective reference of absorption and adsorption while paracetamol acted as a negative control. The concentration of each drug was quantified with liquid chromatography to evaluate a potential loss after a static contact condition and simulated infusion at 1 mL/h and 10 mL/h dynamic condition by an electric syringe pump. A characterization of each material's surface was performed by Fourier transform infrared spectroscopy in attenuated total reflection mode (ATR-FTIR) and by measurement of surface zeta potential. Plasticizer release was quantified by gas chromatography coupled with mass spectrometry (GC-MS). For all tubings except PVC/PU, no loss of paracetamol was observed in any condition. Diazepam sorption appeared to be less important with PVC/PE, PVC/SEBS, SEBS and TPO tubings than with PVC, but was more important when using PVC/PU tubings. PVC tubings induced the least loss of insulin amongst all the studied materials. Surface analysis by ATR-FTIR highlighted the presence of a plasticizer (that could be attributed to Tris (2-Ethylhexyl) Trimellitate (TOTM)) in the coextruded SEBS layer of PVC/SEBS, which could have influenced drug sorption, probably as a consequence of a migration from the PVC layer. Coextruded PVC/SEBS and PVC/PE presented the lowest zeta potential of all studied materials with respective values of −39 mV and −36 mV and were related to the highest sorption of insulin while PVC/PU with the highest zeta potential (about −9 mV) presented the highest absorption of diazepam. Coextruded layered materials appeared to have a lower plasticizer release than PVC alone. As a conclusion, PVC/PE and thermoplastic elastomers alone or coextruded with PVC could be interesting alternatives to PVC tubings with regards to sorption phenomena and plasticizer release.
Because of its good mechanical properties combined with a low cost of fabrication, PVC has been widely used for the manufacture of IV tubings. Yet this material is not completely inert when infusing drug solutions. It can affect drug solutions by releasing compounds into the infusate or by retaining the drug (sorption) thus potentially affecting infusion safety and effectiveness. Sorption phenomena can cause a loss of active pharmaceutical ingredient (API)1,2 or protective excipients3 and are mediated by different physicochemical parameters/properties4,5,6 that are still incorrectly evaluated. The phenomena can be detailed in two steps: adsorption then absorption. Adsorption is the result of a weak interaction between a compound in solution and a surface. This phenomenon is fast and reversible. Absorption corresponds to the diffusion of a molecule inside the material. It is slower and comes after adsorption. As Peterfreund et al. reported7, sorption related loss of drug is underappreciated. This issue was reported with PVC bags and tubings from the 80's4 with different drugs such as diazepam8,9, amiodarone10, isosorbide dinitrate11, insulin12,13. More recently, many studies also highlighted losses of drug during administration with PVC but also with non-PVC based catheters or IV tubings2,14,15,16,17. Even though the sorption issue has been known for a long time, the mechanisms involved during this phenomenon are not completely elucidated.
Moreover when using PVC based IV tubings, plasticizer leaching is a major concern. Plasticizers are compounds added to the PVC to make it more flexible, and these products can be released into the infused medication and then in the bloodstream. Until about ten years ago, the most used plasticizer was the Di(ethylhexyl)phthalate (DEHP) which was regulated because of its toxicity. Alternative plasticizers like Diisononyl Phtalate (DINP), Di-(2-Ethylhexyl) Phtalate (DEHT), 1,2-cyclohexane dicarboxylic acid diisononyl ester (DINCH) or Tris(2-Ethylhexyl) trimellitate (TOTM) were then used in the manufacturing of medical devices18, yet all those alternatives plasticizers can potentially migrate from the PVC matrix. Recently, our research team has shown that the addition of a coextruded inner layer of polyethylene (PE) in PVC infusion tubings appears to reduce plasticizer release19, but the effect of other coextruded materials like polyurethane on plasticizer release has not yet been studied. In addition to their potential toxic effects, plasticizers have also been shown to have an influence on drug sorption20,21.
Factors affecting sorption are related to the physicochemical properties of the drug itself (lipophilicity, pKa, isoelectric point, steric hindrance, concentration), but are also related to the excipient composition, infusing parameters (flowrate, medical devices length) and the physicochemical properties of the polymer constituting the IV tubings. Identifying and evaluating material related factors appears as a challenging way to provide information to better understand drug sorption, and could help identify at-risk situations and select the best material for IV-tubings, and thus improve the control of the administered dose to the patient.
The aim of this study was to assess the sorption in conditions simulating the clinical use of three drugs with PVC and 5 alternative materials (co-extruded with an inner layer of polyethylene (PE), polyurethane (PU) and styrene-ethylenebutadiene-styrene block copolymer (SEBS), bulk SEBS and a bulk thermoplastic olefin (TPO)). For each material, the influence of surface physicochemical properties in the sorption process was investigated. The three molecules that were studied were diazepam which is an API that has been known for many years to absorb into medical tubings20,22, insulin which is subject to adsorption only13,17 and paracetamol which was used as a negative control (not reported to be sensitive to sorption phenomena). In order to provide additional information about mechanisms involved in the sorption process, the inner surface of each IV tubings was also characterized. The majority plasticizer migration potential was also evaluated to assess the impact of the coextruded layer on plasticizer migration.
The tubings used are described in Table 1.
Table 1 Description of IV-tubings (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin)
For PVC, PVC/PE, PVC/PU and PVC/SEBS, the PVC matrix was plasticized with plasticizer tris(2-ethylhexyl) trimellitate (TOTM) (manufacturer information).
50 mL Luer Lock polypropylene syringes (BD Plastipak®, reference 300865 (Becton Dickinson, France), batches 1712226 and 1701265 P, expiring respectively 11/2022 and 12/2021)) were also used as storage container during infusion simulation.
The following marketed medications were used:
VALIUM® (Diazepam) 10 mg/2 mL (Roche, Rosny-sous-Bois, France; batch F1126F01, expiring 09/2020).
NOVORAPID® (Insulin aspart) 100 UI/mL (Novo Nordisk, Courbevoie, France; batch HS65E14, expiring 01/2020). Insulin aspart will henceforth be referred to as insulin.
Paracetamol B BRAUN® (paracetamol) 10 mg/mL (B. Braun, Saint Cloud, France; batch 18105452, expiring 02/2020 and 18141450, expiring 03/2020).
Physicochemical properties, and Van der Waals volume, of each API were obtained from Chemicalize®23. For diazepam, insulin and paracetamol, partition coefficient (logP) were respectively of 3.08, −18.85 and 0.91 and Van der Waals volume were respectively of 242.85Å3, 3123.51Å3 and 138.08Å3.
The following reagents were used for chromatographic separation: acetonitrile (ACN) 99% purity (Fisher Chemical, United Kingdom); methanol 99% purity (Fisher Chemical, United Kingdom); formic acid 98% purity (Fluka, Germany), trifluoroacetic acid (TFA) (Sigma-Aldrich, Germany); monobasic potassium phosphate (Sigma-Aldrich, Germany). All reagents were of certified HPLC grade.
Sorption phenomena were evaluated by quantification of the API after static and dynamic contact between all medications and IV tubings. Static condition simulated a worst-case scenario in which drugs and materials were put in contact during 96 hours without renewal of the solution. Dynamic contact aimed to simulate an 8 hours infusion at two different flowrates.
Each analysis was performed in triplicate. All devices and equipment used for the preparation of the drugs, the conditioning and the sample withdrawal were chosen so as to avoid any kind of content-container interactions, and the physicochemical inertia relative to the sorption phenomena was preliminarily checked.
The studied solutions were prepared as follow, by dilution from the commercial medications and in accordance with the summary of product recommendations:
Paracetamol: diluted to 1 mg/mL in a 0.9% sodium chloride solution Versylene® (Fresenius Kabi, France).
Diazepam: diluted to 0.2 mg/mL in a 5% glucose solution B. Braun® solution (B.Braun, Germany)
Insulin: diluted to 0.1 UI/mL in a 0.9% sodium chloride solution Versylene® (Fresenius Kabi, France.
The final concentrations were chosen to be representative of clinically used concentrations.
Static study
For each condition, the IV-tubings were filled with each drug at the studied concentration. After filling, the tubings were clamped and stored in standardized conditions in a validated climate chamber (Binder, model KBF240, GmbH Tuttligen, Germany) at 25 °C ± 2 °C and 60% humidity, in the dark.
For each sample, in the syringe before tubing filling (initial time Ti), then from the tubings right after filling and purge (T0) and after 24 and 96 hours of contact (further referred to as T0, T24 and T96) a visual control and API quantification was performed. For each analytical time, three different tubings (n = 3) were used: the tubings were fully emptied and discarded after analysis. Thus, for the static study a total of 9 units were used.
Dynamic study
A simulation of an IV infusion using an electric syringe pump (Orchestra® DPS modules, Fresenius, France) was performed at two different flowrates: 1 mL/h and 10 mL/h, which are flowrates commonly used for IV drug infusion. The experimental setup is presented in Fig. 1.
Picture of the experimental setup in dynamic condition (A) electric syringe pump; (B) 50 mL syringe; (C) infusion tubing; (D) withdrawing site at the tip of the tube).
For each condition, a sample of the drug solution was collected from the tip of the syringe before contact with the tubing (Ti), then at T0 at the tip of the tubing, after purging. Other samples were collected at the tip of the tubing without stopping the infusion, after 1, 2, 4 and 8 hours of simulated infusion. An approximate volume of 150 µL was collected for each analysis time (minimum volume needed to perform the quantitative analysis) and thus the sampling time was dependent of the flowrate (about 1 min and 10 min respectively for the 10 mL/h and 1 mL/h condition). Visual control and API quantification were performed on the samples.
After preparation, visual examination of diluted solutions was performed and pH was measured. For all analytical time, API was quantified after separation by HPLC.
Evaluation of the physicochemical properties of the tubings inner surface was performed by ATR-FTIR spectroscopy and zeta potential measurement.
For PVC or coextruded PVC tubings, the released plasticizer was identified and its migration quantified.
Visual examination and pH measurement
Each collected sample was visually controlled and compared to a freshly prepared sample. Immediately after preparation pH was measured using a SevenMultiTM pH-meter with an InLab MicroPro glass electrode (Mettler-Toledo, Viroflay, France).
API quantification
At each analytical time, API was quantified using one of the following high-pressure liquid chromatography systems and integrated data treatment software:
AS4150 autosampler, PU4180 pump, CO-4061 oven, and MD-2018 diode array detector (Jasco, Bouguenay, France)
LC-2010HT compact system (Shimadzu, France).
The in-house chromatography methods used for the quantification of the API are presented in Table 2.
Table 2 Chromatography methods used for quantification of paracetamol, diazepam and insulin. TFA: trifluoroacetic acid.
The linearity of the method was verified through the analysis of 3 independent calibration ranges performed on solutions for each API on 3 different days. The mean accuracy, the repeatability and the intermediate precision were calculated through repeated quantitative analysis of 6 independent solutions, repeated on 3 different days. For this study, the limit of detection was validated as the lowest point of the calibration curve, except for insulin for which an optimized quantification limit was researched.
All samples were diluted to within theoretical calibration curve range, and if beneath quantification limit the samples were reanalyzed after adapting the dilution.
Expression of the results of API quantification
For all three tested API, the results were expressed as the percentage of the initial concentration (measured at Ti). Error bars expressed the 95% confidence interval of the mean value.
In order to make the results of API quantification comparable from one molecule to one another and from one tubing to one another, the percentage of the initial concentration was divided by the surface contact area of the tubings. Sorption rates were calculated with Eq. 1, and expressed as a percentage of sorption per square centimeter of tubing.
Equation 1: Calculation of the sorption rate standardized by area of contact between drug solution and tubings inner material
$$Sorption=\frac{Ci-Cf}{Ci}\,\times \frac{1}{S}\times 100$$
Ci: initial concentration (mM)
Cf: final concentration (mM)
S: inner surface area (cm2)
All statistical analyses were performed using Stata statistical software (version 13, StataCorp, College Station, US). The tests were two-sided, with a type I error set at 5%. Continuous parameters were expressed as mean ± standard-error of mean (SEM) according to statistical distribution (assumption of normality studied using Shapiro-Wilk's test).
To study longitudinal evolution, correlated repeated data were analyzed using linear mixed models. This approach seems more relevant rather than usual statistical tests because assumption concerning independence of data is not met. The (fixed) effects group, time-point evaluation and their interactions time x flow rate were studied. The normality of residuals from these models has been studied using the Shapiro-Wilk test. When appropriate, a logarithmic transformation was proposed to achieve the normality of dependent data. A Sidak's correction of the type I error was applied to take into account multiple comparisons. Finally, Bayesian Information Criterion (BIC) was estimated to determine the most appropriate model, notably concerning the covariance structure for the random-effects due to repeated measures across the time and consequently to the autocorrelation.
Concerning comparisons involving non-repeated data, the quantitative variables were compared between independent groups by ANOVA or Student t-test. The assumptions of ANOVA and t-test were evaluated. More precisely, the homoscedasticity was analyzed using the Bartlett test. Furthermore, when appropriate, post-hoc tests were performed to take into account multiple comparisons (Tukey-Kramer post ANOVA and Dunn after Kruskal-Wallis). Hedges' g effect sizes (ES)24, calculated as presented in Equation 2, and 95% confidence intervals (CI) were calculated at T8 between PVC and each alternative tubings. Effect size can be interpreted according to Cohen's recommendations25. A negative effect size is indicative of the material inducing a higher sorption rate than the PVC reference material, whilst a positive effect size was indicative of a lower tendency to promote sorption of the tested material compared to PVC tubings. Forest-plots were used to represent graphically these results.
Equation 2: Hedge's effect sizes calculation
$$ES=\,\frac{{m}_{1}-{m}_{2}}{S{D}_{pooled}}=\frac{{m}_{1}-{m}_{2}}{\sqrt{\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}}}$$
m1 and m2: mean at T8 for PVC (m1) and alternative tubing (m2)
n1 and n2: sample sizes
s1 and s2: standard deviation
ATR-FTIR spectra of the inner surface of each tubing were acquired with a spectrum 100 spectrometer (PerkinElmer) equipped with an ATR diamond crystal. All spectra were acquired from 3500 to 650 cm−1, using 16 scans with a 2 cm−1 resolution.
Surface Zeta potential measurements
In contact with an aqueous solution, a solid surface assumes a surface charge. The Zeta potential (or electrokinetic potential) describes the charging behavior at interfaces. Surface Zeta potential is representative of the electric charge at the shear plane between the diffuse layer and the immobile layer of a material. The surface Zeta potential of the inner surface (before any drug administration) of all tested IV-tubings was assessed by measuring the streaming potential with a Surpass 3 (Anton Paar, France) equipped with a tubing cell analysis system, in a 1 mmol/L potassium chloride solution at pH 5 before analysis in order to standardize the conditions.
Plasticizer quantification
The amount of the plasticizer in the PVC matrix (for the PVC containing tubings) was quantified by gas chromatography coupled to a mass spectrometer (GC-MS) with the chromatographic method and extraction process developed by Bourdeaux et al.26. The plasticizer migration was assessed following the model published by Bernard et al.27.
Validation of API quantification
The analytical method validation data of each API is presented in Table 3.
Table 3 Analytical method validation data. CV: coefficient of variation.
The mean coefficients of variation are under 5%, and mean coefficients of determination above 0.99 for paracetamol and diazepam. A slightly more important variability was noted for insulin. The methods can therefore be considered as linear, accurate, true and repeatable for the tested conditions.
The limit of quantification of insulin was fixed at 0.03 UI/mL, limit at which the mean coefficients of repeatability, intermediate precision and relative trueness bias were of 4.08%, 8.31% and 12.27%, respectively. Insulin concentrations between 0.03 and 0.06 UI/mL were taken into account for their indicative value only.
At the initial time of every studied condition, paracetamol concentrations in solution were comprised between 0.913 mg/mL and 1.045 mg/mL and pH was of 5.30. Paracetamol percentages of initial concentrations measured after static or dynamic contract with IV-tubings are presented in Fig. 2.
Evolution of paracetamol concentrations compared to initial concentration in static condition (A); 1 mL/h dynamic (B) and 10 mL/h dynamic condition (C) for every studied tubing (n = 3, mean ± standard error of mean). (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin).
Paracetamol concentrations did not vary by more than 10% of Ti concentration for every condition except for PVC/PU IV-tubings in static condition for which paracetamol concentration decreased to 81.61% ± 0.57% at T24 and 81.23% ± 0.08% at T96.
Paracetamol sorption relative to contact surface area (in cm²) showed a significant increase with time (p < 0.001) only for PVC/PU samples, respectively +2.39 ± 0.01%/cm² at T96 in static condition and +0.20 ± 0.06%/cm² at T8 in 1 mL/h dynamic condition (see details in Supplementary Data, Fig. A).
For all materials and flowrate conditions, diazepam concentrations in solution before infusion were comprised between 0.18 mg/mL and 0.21 mg/mL and pH was of 5.38. The variation from diazepam Ti concentrations is shown in Fig. 3.
Evolution of diazepam concentrations compared to initial concentration in static condition (A); 1 mL/h dynamic (B) and 10 mL/h dynamic condition (C) for every studied tubing (n = 3, mean ± standard error of mean). (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin).
In static condition, diazepam concentrations decreased for every tested material after 24 hours contact (T24) and at T96, with most important losses noticed for PVC and PVC/PU (residual diazepam percentages being of 1.29% ± 0.00% and 1.88% ± 0.03% respectively for PVC and PVC/PU tubings after 96 hours of contact). The materials inducing the least loss were PVC/PE and SEBS (residual diazepam concentrations of respectively 47.52% ± 1.45% and 49.86% ± 0.02% at T96).
During the 1 mL/h dynamic condition, a decrease of diazepam concentrations was observed right from T1 for every material samples, with concentrations ranging from 13% to 92% of initial concentration, function of the material. In correlation with the result of static condition, groups with different behaviors were observed. PVC and PVC/PU tubings induced the most important loss right from T1 (residual concentrations of about 15%) then remained stable throughout the rest of the study. For PVC/PE and SEBS tubings, the remaining concentrations remained over 85% for all analytical times. With PVC/SEBS and TPO tubings, the loss of diazepam was progressive until T4 (about 50% of initial concentration), but at T8 the decrease continued to up to 40% with PVC/SEBS tubings, while diazepam concentration in TPO tubings increased to 92%.
During the 10 mL/h dynamic condition, diazepam concentrations decreased less, with concentrations comprised between 77% and 97% for each tested material, except for PVC tubings where a minimum of 17.41% ± 0.21% was reached at T1, before raising up again to 36.10% ± 0.27% at T8.
For all analytical time, conditions and materials, diazepam sorption/cm² was statistically significantly higher (p < 0.001) when compared to sorption at T0 (Supplementary Data, Fig. B), indicating that observed decrease in concentrations is significant.
Initial concentrations of all insulin samples were comprised between 0.083 and 0.115 IU/mL and pH was of 6.44. As shown in Fig. 4, insulin concentrations varied differently between static and dynamic condition.
Evolution of insulin concentration compared to initial concentration in static condition (A); 1 mL/h dynamic (B) and 10 mL/h dynamic condition (C) for every studied tubings (n = 3, mean ± standard error of mean). (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin).
In static condition, insulin concentrations decreased to around 40% at T24 then remained almost stable for all tested materials except for PVC/PU tubings where they remained almost stable from T0 to T96.
During the 1 mL/h dynamic condition, insulin concentrations decreased at T1 then remained stable for the other analytical times for every tubing. However, three groups with different behaviors were observed: the loss was most important for PVC/PE and PVC/SEBS tubings (about 75% loss), SEBS and TPO had an intermediate behavior (about 60% decrease), while PVC and PVC/PU were the tubings that induced the least loss of insulin, as about 50% of initial concentration remained.
With a 10 mL/h flowrate, two behaviors were observed. On one side, PVC, PVC/PE, PVC/SEBS and TPO tubings presented a loss of insulin at T1, but concentrations then returned to about 100% of initial values from T2 to T8. On the other side, contact with PVC/PU and SEBS samples induced a decrease of about 20% of initial concentration from T1 to T8.
For all any analytical times, insulin sorption relative to contact surface area (sorption/cm2) in static condition was statistically significantly increased (p < 0.001) when compared to T0 for all tubings except for PVC/PU (see details in Supplementary Data, Fig. C). In the 1 mL/h dynamic condition, sorption/cm² was significantly (p < 0.001) increased for all analytical times when compared to T0 for all IV tubings. In the 10 mL/h dynamic condition, insulin sorption/cm² at T1 was different from T0 (p < 0.001), but not at any other analytical time for PVC, PVC/PE, PVC/SEBS and PVC/TPO tubings. For other tubings, insulin sorption/cm² was significantly different from T0 at all analytical time (except for PVC/SEBS at T4).
PVC was chosen as the reference tubing material and all the other tubings were compared to this reference at the final analytical time (T8). A comparison of effect sizes is presented in Fig. 5.
Effect size of the comparison of each material to PVC after an 8 h infusion at 1 ml/h and 10 ml/h. (A) Paracetamol; (B) Diazepam; (C) Insulin (mean ± confidence interval of 95%). (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin).
For all molecules, absolute value of effect size was reduced when the flow rate was increased. During paracetamol contact, only the PVC/PU tubings at a flow rate of 1 mL/h appeared to be significantly different from 0 thus different from PVC since it was comprised between −6.80 and −1.00. During diazepam infusion, only the PVC/PU samples had a negative effect size (for both 1 mL/h and 10 mL/h conditions). PVC/SEBS tubings had an ES comprised between 0.37 and 4.58, which was significantly higher than PVC but also significantly lower than PVC/PE, SEBS and TPO tubings. These last three tubings had a positive effect (less sorption) but not significantly different from one another. During insulin infusion, all materials appeared to have a negative effect size (exhibiting higher sorption levels) at a flow rate of 1 mL/h, and were significantly different from PVC.
The inner surface of each studied extension set was analyzed by FTIR spectroscopy. SEBS and PVC/SEBS spectra are presented in Fig. 6. Bulk SEBS and coextruded SEBS spectra showed many similarities, indicating close chemical composition. However, coextruded SEBS presented additional bands (1730, 1304, 1282 1233 and 1114 cm−1), that could be attributed to TOTM (see TOTM reference spectra presented Fig. 6(C). Based on this result, it is possible that TOTM is present in the analyzed surface. PVC, PU, PE and TPO spectra are presented in Supplementary Data (Fig. D) and conformed to reference spectra.
FTIR spectra of the inner surface before infusion of SEBS (A) and PVC/SEBS (B) tubings compared to TOTM FTIR spectrum (C) (PVC: Polyvinylchloride; SEBS: Styrene-Ethylenebutadiene-Styrene).
Surface zeta potential
The inner surface charge of every tubing was estimated by measuring its surface zeta potential (Table 4).
Table 4 zeta potential (mV) of the inner surface of every tubings at pH = 5.0 (PVC: Polyvinylchloride; PE: Polyethylene; PU: Polyurethane; SEBS: Styrene-Ethylenebutadiene-Styrene; TPO: Thermoplastic olefin).
Only the PVC/PU tubing presented a surface zeta potential different from the other tubings (−9.5 mV). All the other five tubings had surface zeta potential ranging from −27.4 (PVC) to −39.6 mV (PVC/SEBS).
Plasticizer migration
The majority plasticizer present in the PVC and PVC coextruded with PE, PU and SEBS was TOTM. As shown in Table 5, TOTM was quantified in the PVC tubings and coextruded PVC and its migration was estimated. Percentages of TOTM in plasticized PVC tubings were not statistically different from those in the external PVC layer of coextruded tubings. However, the release of TOTM was significantly lower with coextruded tubings than with PVC alone with an < 0.01 p-value (T-test) for all coextruded materials. TOTM release from PVC/PU tubings was more important than from PVC/PE tubings (T-test, p < 0.01) and from PVC/SEBS tubings (T-test, p < 0.01).
Table 5 Quantification of plasticizer (Tris(2-ethylheyl) trimellitate: TOTM) in Polyvinyl Chloride (PVC) and PVC coextruded tubings and plasticizer migration assay (n = 3, mean ± standard error of mean).
Our study presents new results that bring complementary information about the interactions between medications and new alternative to PVC polymer materials used for drug infusions in conditions simulating clinical administration. The materials interacted differently with active ingredients depending on the characteristics of the drugs and the flow rate. The overall result of effect size calculation based on the comparison of sorption rates at T8 between PVC and alternative tubings (taking into account surface contact area of the tubings) highlighted that the PVC/PU IV tubings were more prone to drug sorption than PVC with all tested molecules while PE and thermoplastic elastomers (PVC/SEBS, SEBS and TPO) had a better behavior than PVC when in contact with diazepam. An adsorption phenomenon was observed for all IV tubings when in contact with insulin, yet differences between each material were less important than for other tested drugs. This study also highlights that the analysis of material surface physicochemical properties by zeta potential measurement was an innovative and interesting approach for the characterization of medication mediated content-container interaction and brought information about factors involved in drug sorption.
Three medications were chosen as models because of their different behavior to sorption. Paracetamol acted as a negative control because it has no known tendency to interact with materials, diazepam as a reference of absorption and insulin as marker of adsorption only. Paracetamol is a slightly lipophilic drug (logP = 0.91) with a low Van der Waals volume (representing the volume occupied by one single molecule) and is under a non-ionized form at the studied pH. The slight lipophilic properties of paracetamol coupled with the relatively high concentration at which it is administered might explain that to our knowledge no studies have reported a drug loss due to a sorption phenomenon. Diazepam is a highly lipophilic drug (logP = 3.08), with a still relatively low Van der Waals volume yet bigger than paracetamol (242.85 versus 138.08 Å3). Diazepam solutions were studied at pH = 5.3, at which diazepam was completely under its non-ionized form. Insulin is a peptide, with a much higher Van der Waals volume (3123.51 Å3) and positively charged at the pH of injectable forms (pH 6.4). The presence of a positive charge could explain that insulin has a tendency to adsorb to the material's surface (by a weak charge interaction). But the combination of charge and important steric hindrance is not in favor of its diffusion inside the polymer material.
Since the length of the tested tubes was different from one to another, the straight reading of the loss of concentration of the API did not allow direct comparison. The effect size was therefore calculated with sorption rates expressed as percentage/cm² in order to compare the influence of materials for each drug sorption. The expression of the effect size allowed us to compare how much sorption with alternative tubings was different from sorption with PVC at a given time (T8 in this study). In clinical research, effect sizes are usually interpreted according to Cohen's rules defined as follow: small (ES = 0.2), medium (ES = 0.5) and large (ES = 0.8: grossly perceptible and therefore large). In this study, several effect sizes were much larger than 0.8 implying certainly relevant differences from a pharmacological point of view.
The static condition (flow rate = 0 mL/h) was studied in order to create a "worst-case" condition, in which contact between drugs and surface will be at its maximum. On the contrary, dynamic conditions at 1 mL/h and 10 mL/h were simulating clinical use situation. For paracetamol and diazepam, the loss of active product ingredient was more important in static than in both dynamic conditions. Variation during infusions of insulin low concentrations and low flowrate has already been observed28, and were imputed to an adsorption phenomena. In our case, insulin loss was less for a flow rate of 10 ml/h than for 1 mL/h, this could be explained by the fact that a faster flowrate would induce a faster saturation of the binding sites. Once all the binding sites were occupied an equilibrium state was reached and the concentrations converge to the T0 concentrations. The interaction of insulin with a saturated surface is not known, if no interactions occur the concentrations would be the same as T0, or the potential loss of insulin could be counterbalanced by the possible desorption of API from the saturated surface. Another possible explanation of this phenomenon could be that at a faster flowrate, the contact time between insulin and material was shorter and thus led to a fewer loss due to adsorption. A possible competition between the sorption interactions and flow driven interactions could also explained the flowrate dependent equilibrium. However, PVC/PU tubings did not induce any API loss during static contact with insulin solutions, while a loss of API was noticed during dynamic contact with a flow rate of 1 mL/h. This could possibly be explained by a competition between the phenolic excipients (phenol and metacresol) entering in the composition of Novorapid® and insulin. According to the Van der Waals volume (90.52 Å3 and 107.31 Å3) and logP (1.67 and 2.18) of phenol and metacresol, the adsorption could possibly be followed by an absorption phenomenon, thus inducing a difference in the sorption kinetics between insulin and excipient. Insulin adsorption appeared to be a fast phenomenon, highlighted by the interaction in dynamic condition. When increasing contact time in static condition, excipients with a different sorption behavior and kinetics could shift the sorption equilibrium, decrease insulin affinity for the tubing surface, comparatively to dynamic conditions. This hypothesis is in good agreement with data reported by Masse et al.3, who showed a sorption phenomenon involving metacresol and phenol of a Novorapid diluted solution (1 UI/mL) when in contact with PVC tubings. Based on this result, drugs with an adsorption only profile such as therapeutic peptides or monoclonal antibodies should not be tested in static condition, as a dynamic test at a low flow rate appeared to be more suitable. When increasing the flow rate, the percentage of API lost decreased thus concentrations remained close to the initial ones. With a high flow rate, the volume of solution is higher than with a low flow rate, thus the total quantity of API in contact with the material is also increased. This increase could be in favor of a saturation of the tubing surface, decreasing the tendency to adsorption, or could also cause a faster renewal of the solution which gives less time for the molecules to adsorb onto the tubing wall. Similar results to ours have also been reported for diazepam8 and insulin29 infusion (less drug loss for faster infusion rates), thus limiting the potential clinical impact for the patient.
The physicochemical characterization of each material was performed by assessing the qualitative composition of the surface in contact with the medication by FTIR spectroscopy, and by measuring the charge (estimated by zeta potential) that could interact with non-ionized or ionized drugs.
PVC was chosen for reference material as it is widely used in IV tubings manufacturing due to its very good mechanical properties (transparency, flexibility) and its low cost. As it has already been observed8,9,12,13,22, our results show that PVC had a high tendency to absorb diazepam (at 1 mL/h, the loss was comprised between 85.58% and 93.91% of initial concentration) and also induced insulin adsorption (loss of 32.56% to 43.53% of initial concentration at 1 mL/h), but which was however the least loss amongst all alternative materials for insulin.
The PVC/PU tubings appeared to have a high tendency for sorption phenomena. Compared to PVC, PVC/PU had a negative effect size (indicating a significantly higher tendency for sorption) for the three studied drugs at a flowrate of 1 mL/h. Moreover, PVC/PU had the closest to 0 zeta potential of all the studied materials and could be correlated with its higher tendency to absorb diazepam, but not adsorb insulin. As diazepam was under its non-ionized form, a low surface charge could promote sorption phenomenon and on the other hand this slightly negative charge could interact with positively charged molecules such as insulin. As both paracetamol and diazepam were non-ionized in the condition of this study, a low charge surface could have been favorable for interaction between drug and material. However, PU tubings have been shown to behave very differently depending on the nature of the PU. In a recent study, Foinard et al.2 highlighted that thermoplastic PU were more prone to absorption of diazepam and isosorbide dinitrate than thermosetting PU. The polyurethane used in this study was of thermoplastic nature, and also showed a high tendency to promote diazepam sorption, which is coherent with their results. It is therefore possible that using a thermosetting PU could yield different sorption results, however it would not be able to be used as a coextruding material.
The PVC/PE was not completely inert as it induced a slight loss of diazepam (ranging from 8.95% to 15.25% at 1 mL/h), but interacted much more with insulin (losses ranging from 70.38% to 75.09% at 1 mL/h). Like for PVC/SEBS tubings, PVC/PE tubings presented the most important loss of insulin compared to PVC alone, this observation could be related to zeta potential measurement as PVC/SEBS and PVC presented also the lower zeta potential. Insulin is infused at a pH of 6.1 and at this pH is present in a positively charged form, thus interaction between the positive charge of the drug and the negative charge of the surface could have been promoted. The impact of zeta potential could be more accurately estimated in further studies by assessing the zeta potential as a function of pH. The results presented here can be correlated with previous data already reported by other authors indicating interactions between insulin and PE tubings14,17,30,31.
As expected, coextruded SEBS and SEBS were both styrenic thermoplastic elastomers with a very similar composition as shown by FTIR spectroscopy. PVC/SEBS samples had the lowest zeta potential and yet were more prone to absorption of diazepam than PE, SEBS and TPO, as shown by effect size results. Surface charge is not the only factor affecting drug sorption. Based on the FTIR result, it can be hypothesized that TOTM was present in the SEBS analyzed layer. As TOTM was used as a plasticizer in the external PVC layer, it was not supposed to be present in the SEBS layer. The presence of TOTM could have therefore modified the surface properties of the coextruded SEBS and allowed diazepam to absorb more easily. The impact of the plasticizer's amount in the sorption process has already been shown for PVC by Treleano et al.21 and by Al Salloum et al.20, but this is to our knowledge the first published example of its influence on promoting sorption phenomena in other materials. Monolayered PE tubings available on the market were not selected for this study as they are generally not considered to be adequate for infusion medical tubings as their rigidity is too high and they cannot be clamped without altering the tubing. In the field of infusion, manufacturers prefer to associate the PE with PVC in order to maintain the flexibility of tubings, particularly infusion sets. According to its mechanical properties, TPO was therefore chosen as a PVC free alternative. TPO is an olefin thermoplastic elastomer whose exact chemical structure is not publically available. Its behavior was close to that of SEBS and PE, but diazepam sorption was more important in static condition with TPO tubings. Moreover, even if the effect size calculated at the final analytical time gave a higher value than SEBS, the evolution overtime was different and showed a higher loss of diazepam.
In summary, PVC/PE and a thermoplastic elastomer alternative (SEBS) alone or coextruded with PVC presented a better behavior than PVC alone, as absorption was decreased, especially when in contact with diazepam solutions. The loss was less important with these 3 materials even at a high flow rate of 10 mL/h. However, PVC seemed to behave least badly than other studied tubings with regards to insulin adsorption.
Measuring the surface zeta potential was an innovative approach to explain drug sorption phenomenon, and the results obtained in this study are promising but further analysis needs to be performed to assess if materials' surface charge has a critical influence on sorption phenomenon or not. In order to ensure comparison between materials, the surface zeta potential of the IV tubings was measured only at pH = 5, yet the diluted drug solutions that were administered in a simulated clinical setting were at various pH. This study focused on three drugs, in respect with the recommendation for their administration. In such conditions, the API were positively charged or neutral. No pH adjustments were made in order to assess the phenomenon as it can occur during clinical use. Changing the pH of each drug solution to assess the sorption profiles at extreme pH where the drugs would be charged differently could help better understand the mechanisms involved in the sorption phenomenon, but is experimentally difficult to undertake due to the potential instability of the drugs at these pH. Further studies with negatively charged drugs (like zoledronic acid) would be of interest to determine the impact of surface charge potential in drug sorption. Also, additional analyses of the materials could also be performed at multiple pH magnitudes in order to get a zeta potential profile that will help to evaluate the usefulness of surface zeta potential to estimate the sorption tendency of drugs with materials.
Even though none of the coextruded material completely prevented plasticizer migration, the release of TOTM appeared to be decreased with all coextruded materials compared to PVC tubings. Adding a coextruded inner layer to PVC tubings can decrease TOTM migration19 and reduce absorption of small drugs, which is also what we confirmed but with results varying with the nature of the coextruded material. Amongst all coextruded alternatives, PVC/PU tubings was the one with the least protective impact on TOTM release. The data presented in our study is in favor of the presence of TOTM at the surface of the SEBS coextruded layer, possibly caused by either a migration from the PVC matrix or by surface contamination during the manufacturing process, but further studies throughout the whole SEBS layer need to be performed to be able confirm or not these hypotheses. As SEBS is a styrenic based block copolymer, the aromatic ring in the styrene function could present an affinity for TOTM which also possesses an aromatic ring. The presence of TOTM between the SEBS polymer chains could have modified the matrix structure and have potentially promoted drug sorption into the coextruded inner layer. Based on this result, non-coextruded thermoplastic elastomers appear as an interesting alternative for the manufacturing of infusion tubings as they could combine a limited tendency to promote drug sorption and would be plasticizer free, limiting the potential clinical impact for the patient of both content-container interaction (sorption and release). However, leachables and extractibles originating from the elastomers were not assessed in this study and should be evaluated in order to perform a complete characterization of the material.
This study was performed with commercial medications, following recommendations for medical devices use and drugs reconstitution at clinical used concentrations. Yet, a high concentration could have masked slight variation of API concentration. Commercial medications are composed of API and excipients which are diluted or not in a dilution solvent (0.9% NaCl or 5% glucose). This study has shown the variation of API concentration but did not assess the potential variation of excipient concentration or impact of dilution solvent. Excipient could have been in competition with API leading to an underestimation of the loss or on the opposite could have promoted API sorption.
Sorption is a complex process involving several parameters of the material and the drug at the same time making it very complex to predict. None of the studied materials was inert with all drugs but SEBS and TPO along with PVC/PE appeared to induce less absorption phenomena and thus represented very interesting alternatives to PVC tubings. Moreover, the use of PVC based coextruded alternatives also decreased the ability of plasticizer to migrate from the PVC matrix migration, especially for PVC coextruded with PE.
The measure of the zeta potential appeared to be an interesting tool to characterize the inner surface of the tubings by highlighting that differences in zeta potential could be related to different sorption behavior. Further studies will also be necessary to precise the impact of plasticizer migration in the coextruded layer upon sorption phenomena.
The data that support the findings of this study are available on request from the corresponding author, P. Chennell.
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The authors thank Renate Kohl and Christine Koerner from Anton Paar for their help in zeta potential measurement. The authors also thank Pauline Plaidy, Marie-Lyne Pradal and Elora Richard for their technical help during this study. This study was supported by Cair LGL and Wittenburg BV, who validated the research protocol and approved the final manuscript.
Universite Clermont Auvergne, CHU Clermont-Ferrand, CNRS, SIGMA Clermont, ICCF, F-63000, Clermont-Ferrand, France
N. Tokhadze
, P. Chennell
, L. Bernard
& V. Sautou
Unité De Biostatistiques (Délégation à La Recherche Clinique Et à l'Innovation), CHU de Clermont-Ferrand, 63000, Clermont-Ferrand, France
C. Lambert
& B. Pereira
Universite Clermont Auvergne, CNRS, SIGMA Clermont, ICCF, F-63000, Clermont-Ferrand, France
B. Mailhot-Jensen
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N. Tokhadze carried out the lab work, participated in data analysis and drafted the manuscript. P. Chennell elaborated the study design, participated in data analysis and drafted the manuscript. L. Bernard participated in the lab work and data analysis, and helped draft the manuscript. C. Lambert carried out statistical analysis, participated in data analysis, and helped draft the manuscript. B. Pereira carried out statistical analysis, participated in data analysis, and helped draft the manuscript. B. Mailhot-Jensen carried out material analysis and participated in data analysis, and helped draft the manuscript. V. Sautou elaborated the study design, coordinated the study and helped draft the manuscript. All authors gave final approval for publication.
Correspondence to P. Chennell.
Tokhadze, N., Chennell, P., Bernard, L. et al. Impact of alternative materials to plasticized PVC infusion tubings on drug sorption and plasticizer release. Sci Rep 9, 18917 (2019). https://doi.org/10.1038/s41598-019-55113-x | CommonCrawl |
Numerical approximation of an optimization problem to reduce leakage in water distribution systems
MCRF Home
The simplest semilinear parabolic equation of normal type
June 2012, 2(2): 121-140. doi: 10.3934/mcrf.2012.2.121
On the control of some coupled systems of the Boussinesq kind with few controls
Enrique Fernández-Cara 1, and Diego A. Souza 2,
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
Dpto. de Matemática, Universidade Federal da Paraba, 58051-900, João Pessoa, Brazil
Received February 2011 Revised September 2011 Published May 2012
This paper is devoted to prove the local exact controllability to the trajectories for a coupled system, of the Boussinesq kind, with a reduced number of controls. In the state system, the unknowns are the velocity field and pressure of the fluid $(\mathbf{y},p)$, the temperature $\theta$ and an additional variable $c$ that can be viewed as the concentration of a contaminant solute. We prove several results, that essentially show that it is sufficient to act locally in space on the equations satisfied by $\theta$ and $c$.
Keywords: Navier-Stokes and Boussinesq-like systems, control reduction., controllability.
Mathematics Subject Classification: Primary: 35B37, 93B05; Secondary: 35Q3.
Citation: Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control & Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121
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Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021
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Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
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Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
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Enrique Fernández-Cara Diego A. Souza | CommonCrawl |
» Core Mathematics
» Calculus
Maclaurin Series
Contents Toggle Main Menu 1 Definition 2 Worked Example 3 Video Examples3.1 Maclaurin Series Expansions3.2 Using Maclaurin Series to find a Limit 4 Workbook 5 See Also 6 External Resources
The Maclaurin series is a special case of the Taylor series, centred at $c=0$. The Maclaurin series of a function $f(x)$ is
\[\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n=f(0)=f'(0)x+\frac{f''(0)}{2!}+\frac{f'''(0)}{3!}+\ldots.\]
The prime notation denotes the derivative of $f$ with respect to $x$.
Maclaurin series expansions have many applications, including evaluating definite integrals, finding the limit of a function or approximating the value of an expression.
Use a Maclaurin series expansion to evaluate $\displaystyle \lim_{x\to0}\left[\frac{\cos{x}-1}{x^2}\right]$.
Note that this limit cannot be evaluated by substituting in $x=0$, as then the denominator would equal zero and the function would be undefined.
The limit can be found by expressing $\cos{x}$ as a Maclaurin series expansion. Only the first few terms are necessary; since the limit is to be evaluated as $x$ tends to zero, the higher terms in the expansion will tend to zero very rapidly and can therefore be suppressed.
Recall that the derivative of $\cos{x}$ is $-\sin{x}$, and that the derivative of $\sin{x}$ is $\cos{x}$. The Maclaurin series expansion of $\cos{x}$ is therefore given by
\begin{align} \cos{x} &= f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^2+\frac{f^{\prime\prime\prime}(0)}{3!}x^3+\frac{f^{iv}(0)}{4!}x^3+ \ldots \\ &= \cos{0}-(\sin{0})x-\frac{\cos{0} }{2!}x^2+\frac{\sin{0} }{3!}x^3 + \frac{\cos{0} }{4!}x^4+\ldots \\ &= 1 - 0\cdot x - \frac{1}{2}x^2 + 0\cdot x^3 + \frac{1}{4}x^4 + \ldots \\ &= 1 - \frac{1}{2}x^2 + \frac{1}{4}x^4 + \ldots \end{align}
The limit therefore becomes
\begin{align} \lim_{x\to0}\left[\frac{\cos{x}-1}{x^2}\right] &= \lim_{x\to0}\left[\frac{(1 - \frac{1}{2}x^2 + \frac{1}{4}x^4 + \ldots) - 1}{x^2}\right], \\ &= \lim_{x\to0}\left[\frac{ - \frac{1}{2}x^2 + \frac{1}{4}x^4 + \ldots}{x^2}\right] \\ &= \lim_{x\to0}\left[-\frac{1}{2}+\frac{1}{4}x^2+\ldots\right], \\ \end{align} As $x$ tends to $0$ the terms involving powers of $x$ vanish, leaving us with
\[\lim_{x\to0}\left[\frac{\cos{x}-1}{x}\right]=-\frac{1}{2}.\]
Maclaurin Series Expansions
Prof. Robin Johnson finds the first three terms in the Maclaurin expansion of $\sin{(2x)}$.
Prof. Robin Johnson finds the first three terms in the Maclaurin expansion of $\ln{\Bigl(1-\dfrac{1}{2}x\Bigl)}$.
Prof. Robin Johnson finds the first three terms in the Maclaurin expansion of $\dfrac{1}{2-3x}$.
Prof. Robin Johnson finds the first three terms in the Maclaurin expansion of $\sqrt{3+2x}$.
Prof Robin Johnson finds the first three terms in the Maclaurin expansion of $\dfrac{\sqrt{1-x}}{1+x}$.
Prof. Robin Johnson finds the first four terms of the Maclaurin expansion of $(2-3x)^{^1/_2}$.
Using Maclaurin Series to find a Limit
Prof. Robin Johnson uses the Maclaurin series to find $\begin{align}\lim_{x \to 0} \left[\dfrac{e^{12\large{x}}-(1+2x)^{-1}}{\cos{(3x)}-(1-x^2)^{-2}}\right]\end{align}$.
This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.
Maclaurin and Taylor series
Taylor Series
Maclaurin and Taylor series videos at Khan Academy.
Pure Maths | CommonCrawl |
npj science of food
An enzymatically controlled mucoadhesive system for enhancing flavour during food oral processing
Vlad Dinu1,2,
Arthur Gadon2,
Katherine Hurst2,
Mui Lim2,
Charfedinne Ayed2,
Richard B. Gillis ORCID: orcid.org/0000-0002-7607-68083,
Gary G. Adams ORCID: orcid.org/0000-0002-0335-21623,
Stephen E. Harding1,4 &
Ian D. Fisk ORCID: orcid.org/0000-0001-8448-31232
npj Science of Food volume 3, Article number: 11 (2019) Cite this article
Biomaterials – proteins
While a good mucoadhesive biopolymer must adhere to a mucus membrane, it must also have a good unloading ability. Here, we demonstrate that the biopolymer pullulan is partially digested by human salivary α-amylase, thus acting as a controlled release system, in which the enzyme triggers an increased release of flavour. Our oral processing simulations have confirmed an increase in the bioavailability of aroma and salt compounds as a function of oral pullulan degradation, although the release kinetics suggest a rather slow process. One of the greatest challenges in flavour science is to retain and rapidly unload the bioactive aroma and taste compounds in the oral cavity before they are ingested. By developing a cationic pullulan analogue we have, in theory, addressed the "loss through ingestion" issue by facilitating the adhesion of the modified polymer to the oral mucus, to retain more of the flavour in the oral cavity. Dimethylaminoethyl pullulan (DMAE-pullulan) was synthesised for the first time, and shown to bind submaxillary mucin, while still retaining its susceptibility to α-amylase hydrolysis. Although DMAE-pullulan is not currently food grade, we suggest that the synthesis of a sustainable food grade alternative would be a next generation mucoadhesive targeted for the oral cavity.
Mucadhesion describes the ability of a biochemical material to adhere to a mucosal membrane1,2 and has been a subject of research for the food industry and academia in recent years.3,4,5 However, it still remains a loosely understood and poorly applied concept. Several theories were proposed to describe mucoadhesion, particularly the wetting theory, mechanical interlocking, electron transfer, adsorption and fracture theories.6 Its main clinical relevance is to enhance drug loading capacity and residence time at tissues of interest. However, a good mucoadhesive must also have a very good unloading capability at the site of action.2 Oral mucoadhesion has recently attracted the attention of the food industry with regard to flavour maximisation during oral processing, particularly in "healthy" low calorie reformulated foods. This is because sensory properties, such as texture and flavour are the two most important factors impacting consumer choice, after cost7,8 however, mucoadhesion targeted at the oral cavity does not come without complications. The fate of the flavour perceived during oral processing will be determined by the rate of release of aroma and taste compounds from the salivary bolus and availability at aroma and taste receptors. The vast majority of flavour, alongside other bioactive compounds, are rapidly lost through ingestion and are therefore not available for perception.
Food grade biopolymers have always been an attractive option for the food industry. Many of them are anionic polysaccharides which have widely been applied as stabilising agents and thickeners. While many research groups have tried to characterise and compare their mucoadhesive properties,4,5 there is limited evidence to suggest that anionic polysaccharides are, in chemical terms, mucoadhesive. In order to understand the fundamental molecular processes involved in adhesion, we need to better understand the physico-chemical composition of mucus, which consists of an anionic mucin glycoprotein as its main structural component. Within the saliva, mucin is identified as the second most abundant component, after salivary α-amylase which varies upon stimulation.9 It is characterised as having much lower molecular weights (<500 kilo Daltons, kDa) and lower degree of glycosylation (~60%) as compared to gastric, intestinal or colonic mucins. Oral and salivary mucins consist of gel forming mucins derived from the MUC5B gene and low molecular weight, soluble mucins encoded by MUC7 genes.10 However, it is very difficult to obtain human salivary mucins in any useful quantity for performing mucoadhesive experiments. Therefore, in our study, as in most oral formulation research, mucin from bovine salivary/submaxillary glands (BSM) is employed as a close surrogate for its human equivalent.
Like other mucins, submaxillary mucins have an amino acid domain rich in serine and threonine that forms a bridge with the hydroxyl groups of the N-acetylgalactosamine residues of the carbohydrate fraction.11,12 The carbohydrate region consists of up to five different monosaccharides, such as sialic acids, galactose, fucose, N-acetylglucosamine and N-acetylgalactosamine. They can form weak hydrophobic interactions at their carbonyl and methyl groups, or can form electrostatic interactions via carboxylic acids or through the sulphate groups of the protein region.2,11,12 As a result of the negatively charged overall configuration of mucins, neutral and anionic food biopolymers such as hydroxypropylmethyl cellulose (HPMC), carboxymethyl cellulose (CMC), pectin, alginate, guar, carrageenan or xanthan cannot chemically interact with mucus under the dilute solution conditions of the bolus. While these polysaccharides can mix in the aqueous environment, and physically interact to define the rheology and tribology of the bolus, they lack the molecular ability to bind to the mucus membranes. Yet, various food polysaccharides are still considered mucoadhesive because they have been shown to extend the residence time and release kinetics of bioactive compounds, as a result of the physical and chemical properties of the food thickener i.e. high viscosity, gelation. However, the process by which these anionic hydrocolloids increase flavour intensity is still a matter of debate, whether it is a chemical or a purely rheological mechanism.
By contrast, polycations such as chitosan (pKa ~5.5–7), have extensively been studied for their ability to form strong mucoadhesive electrostatic interactions with mucins, however chitosan applications are limited.13 While chitosan was found useful in mucoadhesive applications targeted at the gastro-intestinal region, its applications in the oral cavity are restricted, as chitosan is so strongly charged that it can precipitate mucins and other functional glycoproteins present in the saliva.3,13 Protein precipitation is also attributed to an unpleasant and astringent mouthfeel response, thereby negatively modifying the organoleptic properties of food.14 Besides, chitosan mucoadhesives are limited in their ability to "unload", since a large proportion of the bioactive molecule remains trapped in the mucus/chitosan complex and is passed along the alimentary canal. Thiomers or thiolated polymers are an example of a more recent development of mucoadhesive formulations. They are principally synthesised by coupling thiol containing functional groups (SH), capable of forming stable hydrogen bonds with sulphate rich protein domains in mucin.15 However, the use or sulphur containing polymers is limited in flavour applications.
For food applications, there is a need to develop a tasteless, non-toxic and milder mucoadhesive, which has a good loading capacity, but which must also be able to unload the flavour compounds during mastication or during ingestion (via retronasal olfaction). Diethylaminoethyl-dextran or DEAE-Dextran, is an example of a much milder mucoadhesive that was shown to interact with mucin.16 However, its mucoadhesive properties were too modest considering the high charge density of the modified polymer. It was suggested that the α(1–3) branches of dextran and the presence of ethyl groups limit the access of the charged amino groups for the sialic acid groups of mucin, due to steric hindrance.
The polysaccharide pullulan, is produced by bacterial fermentation using Aureobasidium pullulans17 and is particularly used in Asia, as a partial replacement for starch as a low calorie ingredient in food and drink.18 It forms clear, odourless and tasteless solutions which do not gel, but can form transparent and oxygen impermeable films upon drying. Due to its film forming properties, it has been extensively used as a coating agent in confectionery, edible films, as well as a replacement for gelatin in medicinal applications.18 The use of pullulan has strong potential for encapsulation and release of flavour compounds due to its quick dissolution properties. For example, used as a breath freshener due to its ability to dissolve rapidly on consumption, and release the bound menthol molecules.18
It is a linear polymer consisting of α(1–4) linked maltotriose and infrequent maltotetraose units, linked together by α(1–6) glycosidic bonds. Previous studies suggested that some α-amylases are able to digest the polysaccharide at its maltotetraose units, thus rendering the polymer partially hydrolysed.19 It has previously been established that at equivalent polymer viscosities, starch thickened products have a good flavour and taste profile, and this is partially due to the decrease in viscosity in the mouth, resulting from salivary α-amylase digestion. In a comparable way, it is suggested that the partial in-vivo degradation of pullulan would result in an increased release of flavour, similar to starch based ingredients, which has been shown previously to enhance perception as a result of an increase in the concentration of volatile aroma compounds reaching the olfactory receptors.20 Thus, the polymer is expected to act as a controlled release excipient of aroma and taste molecules, in which salivary α-amylase releases actives close to the point of perception. Our hypothesis is that the synthesis of a mild pullulan mucoadhesive would reduce the loss of flavour through ingestion by increasing adhesion to the oral surface along with associated flavour compounds, provided the cationic polymer does not interfere with the normal functioning of the enzyme.
In the present study, we tested whether pullulan can be hydrolysed by human salivary α-amylase. Then, we evaluated its ability to modify flavour and salt release from model and real food systems by using Gas Chromatography-Mass Spectrometry (GC–MS), Atmospheric Pressure Chemical Ionization- Mass Spectrometry (APCI-MS), and also conductivity analysis using the INSENTTM electronic tongue tasting system (E-tongue) and a standard conductivity probe. Then we synthesised a cationic pullulan analogue, dimethylaminoethyl pullulan (DMAE-pullulan), which was confirmed by Fourier-transform infrared spectroscopy (FT-IR). The newly synthesised polymer was subsequently evaluated for its mucoadhesive ability using a range of matrix/column free hydrodynamic techniques such as: Viscometry, Dynamic Light Scattering (DLS) and Sedimentation Velocity- Analytical ultracentrifugation (SV-AUC). To the best of the authors' knowledge, this is the first synthesis of DMAE-pullulan. The advantage of dimethylaminoethyl compared to previously characterised diethylaminoethyl (DEAE) synthesis, is that the shorter methyl groups, as opposed to the ethyl groups, may increase the availability of the positively charged amino groups to the negatively charged carbohydrate residues of mucin.
The impact of pullulan hydrolysis by α-amylase on flavour release
Pullulan consists primarily of α(1–4) linked trisaccharide units linked together by α(1–6) glyosidic bonds (Fig. 1). However, depending on the fermentation conditions, the linear polymer has been found to contain up to 6% tetrasaccharide units, allowing access to the active site of α-amylase to hydrolyse the polymer19 (Fig. 1-top). We employed an SV-AUC experiment to analyse the resulting interaction between human salivary α-amylase (HSA) and pullulan. The analysis was performed using highly purified 200 kDa molecular weight pullulan standard, which yielded a single, monodisperse peak at ~4.6S (Fig. 1b). The addition of α-amylase revealed the formation of two distinct degradation fragments corresponding to a major peak at ~2S and a minor peak at ~0.8S (Fig. 1c). Note that a proportion of the monodisperse peak at ~0.8S is partly due to the presence of the smaller component present in the α-amylase control (Fig. 1a). The relative molecular weight for the digested fractions are approximated using 10 and 49 kDa molecular weight pullulan standards. For the rest of the investigation, we used an unfractionated, food grade 200 kDa commercially available source of pullulan. Next, an experiment was employed to determine the ability of HSA to digest the commercial product. For this we used dynamic light scattering (DLS) to examine changes in the molecular hydrodynamic size of pullulan before and after the addition of α-amylase.
Structural representation of pullulan showing the its tetrasaccharide units which can be hydrolysed by α-amylase (top); and the sedimentation velocity- c(s) analysis (bottom), showing the sedimentation coefficient distributions of α-amylase a, 200 kDa pullulan standard b, and the result of their interaction c. A constant concentration of 1 mgmL−1 was used for the α-amylase and pullulan controls, and the mixture. Note that some of the material at ~0.8 S is also present in the α-amylase control, therefore some of it will contribute to an overestimate of peak ~0.8 S in the mixture. Rotor speed: 45,000 rpm (130,000 × g), 20.0 °C
Undigested pullulan showed an z- average apparent hydrodynamic radius, rz, of ~8.5 nm, which was not visible upon the addition of α-amylase, resulting in a z- average of ~5 nm (Fig. 2a). The addition of HSA also resulted in a two fold decrease in viscosity, as indicated by the Solomon-Ciuta extrapolation for intrinsic viscosity, [η] (Fig. 2b). Taken together, we can confirm that human salivary α-amylase is capable of partially digesting pullulan, producing smaller fragments of lower molecular weights and lower viscosity. The next important question became whether partial polymer hydrolysis correlates to an increase in the release of flavour from dilute systems. It is interesting to suggest that in thicker systems, such as starchy food, a decrease in the in mouth viscosity generated by the action of HSA is directly related to an enhanced flavour perception through a cross-modal interactions related to the perceived changes in mechanical stress.20 We therefore sought to analyse the release of taste and aroma compounds as a function of pullulan degradation. A selection of volatile aroma molecules used in this analysis were ethyl butyrate, hexanal, linalool, citral and α-ionone, while model taste compounds included sodium (Na+) and potassium (K+) ions. The results in Fig. 3a illustrate the release intensity and persistence of α-ionone from model solutions, before and after pullulan digestion. In the presence of undegraded pullulan solutions (4 mgmL−1), the headspace concentration for the majority of volatile aroma compounds reached a plateau, while it continued to increase for an additional ~20 s when the pullulan was digested by HSA. Although the time scale of this analysis is not representative of the very short amount of time needed to consume food and drink, the model system confirms the effect of pullulan digestion on aroma release. However, the rate of release may be increased during oral processing, unlike the current simulated in-vitro conditions. This is because, under real in mouth conditions, constant salivary secretion accompanied by mechanical changes due to mastication may enhance aroma release.21
Changes in the apparent z-average hydrodynamic radius of pullulan before and after the addition of HSA a and Solomon–Ciuta results showing a change in the intrinsic viscosity of pullulan upon the addition of HSA b. DLS size distributions are given as an average of three measurements. Experiments performed at 20.0 °C, concentration of pullulan was 5 mgmL−1
Results from APCI-MS, Na+ conductivity analysis and E-tongue showing the impact of polymer hydrolysis on the release of flavour from model solutions of pullulan, showing the real time data for α-ionone a, sodium ions b and potassium ions c, respectively; d GC–MS results showing the effect of pullulan hydrolysis on the relative headspace concentration of volatile aroma compounds from in Robinson's orange squash, where 'R' represents the standard squash dilution, 'A1' and 'A2' are increasing α-amylase concentrations of 0.1 and 1 mgmL−1, and 'P' represents pullulan at a constant concentration of 2 mgmL−1; and e APCI in vivo analysis showing the comparative release of aroma compound ethyl butyrate from model drink solutions containing either pullulan or carboxy-methyl cellulose (CMC). Values are expressed as mean ± SD (n = 3)
A similar trend is observed in Fig. 3b, in which the conductivity analysis indicated an increase in the rate of release of sodium ions. This suggests that the availability of sodium can be increased by the oral degradation of pullulan. Similarly, in the next step we evaluated the intensity of potassium ions before and after enzyme hydrolysis, using the taste evaluation INSENT E-tongue (Fig. 3c). Although not statistically significant, results indicate that increasing the α-amylase concentration can increase the availability of K+ ions. The hypothesis was further tested in the presence of a commercial fruit drink. For its simplicity, we have chosen to analyse the effect of α-amylase hydrolysis on the release of aroma compounds from a dilute orange squash preparation 'R' in the presence and absence of pullulan 'P' (Fig. 3d). Interestingly, α-amylase (A), which is naturally present in saliva, reduced the headspace concentrations of the compounds, in a positive concentration dependent manner (A1, A2 in Fig. 3d). However, if pullulan is present, the aroma suppression of the orange squash is mitigated and the volatile aroma compounds are released into the headspace at higher concentrations. Furthermore, we performed in vivo simulations looking at the release of ethyl butyrate from a model drink containing sucrose and citric acid, in which we compared carboxymethylcellulose (CMC) with pullulan, corrected for viscosity (Fig. 3e). The retronasal ion intensity was recorded after swallowing the model drinks. For the drinks containing pullulan, it was observed that a late swallow (20 s) was correlated with a higher intensity of ethyl butyrate, compared to drinks containing CMC. Although results are not significantly different, this confirms our hypothesis that it is possible to maximise the release of flavour as a function of matrix viscosity, even in dilute solution conditions.
Given that α-amylase can be secreted to elevated concentrations during oral processing of food, our ex-vivo and in-vivo results are in excellent agreement and suggest that the release of aroma compounds can be enhanced in the presence of pullulan, despite the presence of other food constituents which might interfere with the normal functioning of the enzyme i.e. citric acids. However, as shown in Fig. 3, there are limitations in whether enzyme hydrolysis can significantly increase aroma release and perception in-vivo, in time for ingestion, which for some products, such as cordials or soft drinks, corresponds to only a couple of seconds. These simulations form the basis for our development of a mucoadhesive polymer system, which can be initiated by the action of the enzymes naturally present in the saliva. We suggest that by modifying the chemical properties of the polymer, such that it becomes adhesive towards the oral mucus, the loss of bioactive associated with the rapid ingestion can be mitigated.
Developing a functional mucoadhesive pullulan analogue
Initially our studies began with the coupling of amino functional groups onto the polysaccharide backbone to produce a functional cationic pullulan analogue, without impeding access for enzyme hydrolysis. The most promising candidate was for dimethylaminoethyl-pullulan (DMAE-pullulan), synthesised, as shown in Fig. 4a. The polymer was purified and the resulting material analysed by FT-IR (Fig. 4b). In comparison with the unmodified pullulan spectra, the absorption bands detected at ~900 and ~3050 cm−1 correspond to stretching and wagging vibrations of the amino group while the strong absorption at 1390 and 1460 cm−1 correspond to the CH2 and CH3 vibrations of the dimethylaminoethyl chain. A characteristic CO group is observed around 1720 cm−1, while broader and weaker vibrations are observed in the region 1800–2500 cm−1, which indicate the presence of the CN bonds of the amino group (Fig. 4b). Therefore, the results indicate that the DMAE group was grafted onto the pullulan backbone.
Schematic representation of the chemical modification of pullulan showing the addition of the tertiary amine, dimethylaminoethyl (DMAE) chloride a and FT-IR spectra of pullulan before after synthesis highlighting the qualitative changes in the spectral intensity correlating to the new functional groups b. The reaction was performed using an adapted version from San Juan et al.31 Five gram of Pullulan (Carbosynth, 200 kDa) was dissolved in 25 ml of distilled water and mixed with a 25 mL 10 M sodium hydroxide solution to activate the pullulan hydroxyl functions. Then, 35gm of 2-chloro-N,N dimethylethylamine hydrochloride was added to the mixture and left stirring at 60 °C for 1 h. After the reaction was completed, the mixture was washed four times with 50 ml diethyl ether and after was diluted in water to a concentration of 10 mgmL−1 and adjusted to pH 7 using HCl
Then, our next goal was to evaluate the ability of the newly modified cationic pullulan to interact with our two main salivary components, mucin and α-amylase. First, a combined viscosity and particle size analysis approach has confirmed the ability of α-amylase to reduce the hydrodynamic particle size (radius) of the newly modified polymer from ~8 to ~5 nm (Fig. 5a-top). In the presence of submaxillary mucin (~6.5 nm) an increase in a particle size distribution was observed, suggesting mucoadhesive phenomena, corresponding to a z-average hydrodynamic radius of rz of ~12 nm (Fig. 5a-bottom). Similarly, we evaluated changes in the intrinsic viscosity (Fig. 5b), which corresponded to a 32% decrease upon the addition of α-amylase. By contrast, the intrinsic viscosity of the DMAE-mucin mixture was 23% higher than the viscosity of submaxillary mucin (Fig. 5b).
Results showing changes in the apparent z-average hydrodynamic radii of DMAE-pullulan, mucin, α-amylase, and the result of their interactions a, and viscosity results showing the Solomon-Ciuta estimations of the intrinsic viscosities of DMAE-pullulan, mucin, α-amylase, and their mixtures b. The concentrations represent dilutions of each sample. DLS size distributions are given as an average of three measurements. Performed at 20.0 °C, macromolecular concentrations were in a ratio of 1:1
The interaction analysis was further reinforced by a SV–AUC interactions experiment which allowed us to directly monitor changes in the sedimentation coefficient distribution of DMAE-pullulan upon the addition of α-amylase and submaxillary mucin (Fig. 6). By itself, DMAE-pullulan revealed a rather broad macromolecular sedimentation profile, indicative of a heterogeneous composition, but nearly identical to the native unfractionated food grade pullulan used for the synthesis, which confirms that the chemical synthesis did not cause the polymer to degrade. At a constant concentration of 0.5 mgmL−1, the sedimentation coefficient distribution ranged from 1S to ~12S (Fig. 6b). An initial assessment reveals an increase in the sedimentation coefficient distribution to ~25S upon the addition of mucin, indicative of an interaction, although a large proportion of sedimentation species (~70%) remained the same (Fig. 6a). One of the methods previously used to assess for mucoadhesion is measuring the sedimentation coefficient distribution ratio of the mucin/polymer complex to that of the mucin (scomplex/smucin)2. Our results showed that the ratio ranged from 1.1 to 2 (Fig. 6a). These values are similar to DEAE-dextran which are still fairly modest compared to stronger mucoadhesive polymers such as chitosan, which has been shown to give sedimentation ratios of up to ~40. However, chitosan mucoadhesion is an extreme example which would not only lead to the precipitation of mucin glycoproteins, but also other anionic glycoproteins present in the saliva, causing a very unpleasant astringent sensation. Overall, our results demonstrate that up to 30% of DMAE-pullulan can bind mucin, as given by the area under the sedimentation curve (Fig. 6a).
Sedimentation velocity, g(s) analysis showing the sedimentation coefficient distributions for DMAE-pullulan at 0.5 mgmL−1 b and the result of the addition of mucin at 0.5 mgmL−1 a and α-amylase at 0.1 mgmL−1 c; and the GC–MS volatile analysis from modified and unmodified pullulan and solutions upon the addition of saliva d. Rotor speed: 45,000 rpm (130,000 × g), 20.0 °C. The distributions reflect the real time migration of molecules driven by the centrifugal force. For the same type of macromolecule, i.e. DMAE-pullulan, a larger S value corresponds to a larger molecular weight
By contrast, the addition of α-amylase caused a reduction in the sedimentation coefficient distribution of the modified polymer from ~12S to ~9S (Fig. 6c). This translates to a ~25% loss in higher z-average molecular weight fractions, and an increase in the concentration of lower molecular weight DMAE-pullulan fractions. These values are qualitatively consistent with the results from viscosity and DLS (Fig. 5). In addition, we have compared the aroma release ability of the modified polymer to its native pullulan counterpart in a model ex-vivo system containing saliva and aroma compounds (Fig. 6d). To our surprise, it was shown that the release of the volatiles was significantly increased in the presence of DMAE-pullulan, as opposed to pullulan. The additional increase can be explained by the loss in molecular weight and viscosity of the modified polymer, and perhaps due to a reduction in the damping effects other proteins present in the saliva.
It is worth mentioning that our preliminary results tentatively indicate that the interaction mechanisms of DMAE-pullulan with saliva may lead to very minor changes in the in mouth rheology of the bolus, since the viscosity increase due to adhesive interactions is counterbalanced by the degree of hydrolysis. As a result, the sensory properties of the food/saliva mixture, i.e. mouthfeel, are expected to be the same. However, we would further need to perform in-vivo trials and take into account factors such as mastication and salivation, which have been shown to play a key role in the release of volatile aroma compounds.21 Although this would be more applicable to solid food systems where the breakdown of the food structure which can influence the rate of release of aroma compounds, as well as altering the proportions of hydrophilic compounds.21 Similarly, mucoadhesion may play a role in the after taste, by increasing the residence time of flavour compounds onto the oral surface. Though in order to analyse this effect we would first require an approved food grade cationic pullulan analogue.
Research is currently being undertaken to identify greener ways to produce cationic pullulan analogues that would meet the required quality and purity criteria of food ingredients, but we suggest that a food grade cationic pullulan could become one of the next generation mucoadhesive biopolymer candidates targeted at the oral cavity. Regardless of the final chemical product and instrumental analysis, we must remember that flavour is not just a group of attributes or a group of chemicals, but a perceptual phenomenon that will strongly depend on the physiological status of the individual.
The oral processing simulation experiments have shown that pullulan can be used for the targeted release of bioactive flavour compounds, as a result its partial in-vivo digestion. The time scale of polymer hydrolysis was over 20 s, however for a lot of liquid and semi-liquid foods such as juices or yoghurts, the oral transit time is no longer than a few seconds which results in a rapid loss of flavour through ingestion. To address this issue, we have synthesised a cationic pullulan analogue, DMAE-pullulan, which was assessed for its mucoadhesive ability, whilst ensuring it retains its inherent susceptibility to α-amylase hydrolysis. We have shown that the cationic polymer binds to submaxillary mucin, aimed at increasing the oral retention of flavour compounds. Then, we have shown that the release of flavour compounds can be enhanced through the action of salivary α-amylase, which partially degrades the modified polymer. Once a food grade cationic pullulan becomes available, sensory experiments would add to our analysis and provide a broader explanation of its impact on flavour perception.
To conclude, we developed a unique concept of a controlled release mucoadhesive system targeted for the oral cavity which may have strong resonances for enhancing the release of flavour and other bioactive compounds during oral processing.
Bovine submaxillary mucin (type I-S, M3895), human salivary α-amylase (type IX-A, A0521), 200 kDa pullulan standard (01615) and volatile aroma compounds used in this study were purchased from Sigma Aldrich (Dorset, UK). The food grade 200 kDa sample was purchased from Carbosynth, UK. The 0.1 M phosphate buffered saline (PBS) was made according to Green (1933),22 (Fisher Scientific, UK). Saliva samples were from the Centre for Biomolecular Sciences, University of Nottingham. All samples were collected in accordance with the ethical approval R12122013, Faculty of Medicine and Health Sciences Research Ethics Committee, Queens Medical Centre, Nottingham University Hospitals.23 Participation was voluntary and informed written consent was obtained. All data were held in accordance with the Data Protection Act. The pooled samples were centrifuged (6000 g), dialysed against 0.1 M phosphate chloride bugger using a 14 kDa dialysis membrane and filtered through a 0.45 µm membrane filter to remove larger aggregates, such as gelled mucus and small molecular weight peptides, respectively, then stored at −80 °C until use. Loading and unloading of samples was carried out in a Level 2 microbiological safety cabinet.
Robinson's sugar free orange squash concentrate was purchased from the local supermarket. Final samples used for the GC analysis were diluted according to the manufacturer, one part concentrate and four parts water/ solution. The samples were mixed with the polymer solutions such that the concentration of squash is always constant. Highly purified RO (reverse osmosis) water was used throughout the sample preparation.
Sedimentation Velocity-Analytical ultracentrifugation (SV-AUC)
Experiments were performed at 20.0 °C using the Optima XL-I analytical ultracentrifuge (Beckman, Palo Alto, USA) equipped with Rayleigh interference optics. Samples of 395 μL (and 405 μL solvent) were injected into the 12 mm double sector epoxy cells with sapphire windows and run at 40,000 rpm (120,000 × g). Scans were taken at 2 min intervals. The interference system produced data derived by recording changes in concentration (in fringe units) versus radial displacement. The results were analysed in SEDFIT using the least squares ls-g*(s) or 'g(s)' and the diffusion corrected c(s) processing methods (the latter valid because of the high degree of fractionation/low polydispersity of the P200 pullulan), by generating sedimentation coefficient distributions, s20,w (in Svedberg units, S = 10–13 s) normalised to standard conditions (viscosity & density of 0.1 M PBS at 20.0 °C).24,25,26
Gas chromatography–mass spectrometry (GC–MS)
The Trace 1300 series Gas Chromatograph coupled with the single-quadrupole mass spectrometer (Thermo Fisher Scientific, Hemel Hempstead, UK) was used. Samples were incubated at 37.0 °C for 20 min with intermittent stirring. Then, the solid phase microextraction (SPME) fibre (50/30 μm DVB/CAR/PDMS, Supelco, Sigma Aldrich, UK) was used to extract for 40 min then desorb for 1 min. Separation was carried out by a ZB-WAX capillary gas chromatography column (length 30 m, internal diameter 1 mm, 1.00 μm film thickness). The column temperature was initially at 40.0 °C for 2 min, then increased by 6.0 °C every minute up until 250.0 °C and held for 5 min. Full scan mode was chosen to measure volatile compounds (mass range from 20 to 300 Da). A splitless mode was used, and a constant carrier pressure of 18 psi was applied. Volatiles were identified by comparison of each mass spectrum with either the spectra from the NIST Mass Spectral Library.
Atmospheric Pressure Chemical Ionization-Mass Spectrometry (APCI-MS)
The APCI-MS (Platform II, Micromass, Manchester) was used to analyse the real time concentration of volatile compounds under static conditions. A final concentration of ~10–50 ppm (parts per million) was sampled with an air flow adjusted to 50 ml/min. The instrument was set in Selective Ion Recording (SIR) mode to monitor the selected mass to charge ions (m/z). The ion intensity was measured at cone voltage of 50 V, source temperature of 75 °C and dwell time of 0.02 s. The in-vivo analysis shown in Fig. 3d was performed by consuming model drink solutions of sucrose, citric acid in which pullulan or CMC were added, and the retronasal ion intensity was captured by exhaling into the MS-NOSE interface of the APCI.
Sampling took place until the signal plateaued and started to decrease. The curves were integrated in Mass LynxTM (Waters, UK).
Dynamic Light Scattering (DLS)
The experiments were performed using the Zetasizer Nano-ZS detector and low volume (ZEN0112) disposable sizing cuvettes (Malvern Instruments Ltd, Malvern, UK). The samples were measured at (20.00 ± 0.01)°C using the 173° scattering angle collected for 3 runs of 10 s. For polydisperse particles, DLS can provide useful information about the size (radius) of molecules by calculating an estimate for z-average hydrodynamic radius, rz, and translational diffusion coefficient, Dtrans, via the Stokes–Einstein equation:
$${\mathrm{D}}_{{\mathrm{trans}}} = \frac{{k_BT}}{{3\pi \eta d}}$$
where the hydrodynamic diameter d = 2rz, kB is the Boltzmann constant, η is the solvent viscosity, T is absolute temperature (K), and Dtrans (cm2s−1). The contribution of rotational diffusion effects to the autocorrelation function is assumed negligible (see Burchard, 1992).7
Capillary viscometry
Flow times of solvent (t0) and solutions (ts) were measured using a semi-automated (Schott Geräte, Hofheim, Germany) U-tube Ostwald capillary viscometer immersed in a temperature controlled water bath at 20.0 °C. A constant volume of 2.0 ml was sampled at a series of mucin concentrations (0.2–1.0 mgmL−1), sufficiently low to allow the assumption that no correction was needed for solution density, assuming ηs/η0 is equal to ts/t0. The intrinsic viscosity [η] plot is shown as according to the Solomon–Ciuta equation and extrapolated to zero concentration to account for non-ideality.27,28
E-tongue
Digested and undigested pullulan solutions were made in 30 mM potassium chloride buffer and poured into the sample cups for the electronic tongue in triplicate (Taste Sensing System TS-5000Z). Manufactures guidelines were used for analysis and data extraction. The experimental design was kindly performed by New Food Innovation specialists as in previous studies.29,30
Conductivity metre
Dissolution of sodium was evaluated using a Mettler Toledo conductivity metre (Ohio, USA). A 1 ml sodium chloride solution (0.1 mgmL−1) was dissolved in a beaker containing normal and digested pullulan solution. Data were recorded every 2 s until a plateau was reached (~20 s). For this analysis, a 1 mL solution of sodium was added at a concentration of 1 mgmL−1 into a 50 ml pullulan solution, equivalent to the dissolution of 0.2 mg of sodium, under constant magnetic stirring and maintained at 25.0 °C. Three replicates were performed and normalised by conductivity.
Synthesis of dimethylaminoethyl (DMAE) pullulan
The reaction was performed using an adapted version from San Juan et al.31 Five gram of Pullulan (Carbosynth, 200 kDa) was dissolved in 25 ml of distilled water and mixed with a 25 mL 10 M sodium hydroxide solution to activate the pullulan hydroxyl functions. Then, 35gm of 2-chloro-N,N dimethylethylamine hydrochloride was added to the mixture and left stirring at 60 °C for 1 h. After the reaction was completed, the mixture was washed four times with 50 mL diethyl ether and after was diluted in water to a concentration of 10 mgmL−1 and adjusted to pH 7 using HCl. The solution was further cleaned of organic solvents and concentrated in a rotary evaporator, after which was dialysed in PBS buffer on a 14,000 Da (g/mol) membrane for two days. The resulting solution was freeze-dried which resulted in the formation of white odourless powder. The powder was stored at 4 °C until needed.
Fourier-transform infrared spectroscopy (FT-IR)
The resulting powder was subjected to FT-IR analysis. Measurements were performed in transmission mode on an IRAFFINITY-1S spectrometer equipped with an A219653 attenuated total reflection (ATR) module (Shimadzu, Japan). For each sample, the spectrum was taken as the average of three different measurements at various sites of the dry sample Spectra were acquired between 500 and 3500 cm−1 at a resolution of 4 cm−1. Dry pullulan samples were pressed against the diamond surface to ensure good contact. Measurements were repeated twice for reliability.
GC–MS and conductivity samples were analysed in triplicate in a randomised sample order, and the error is given as a as mean ± SD (n = 3). Figures were made in Origin 7.5 (OriginLab, USA).
Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
The data that support the findings of this study are available from the corresponding author upon request.
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Harding, S. E. Trends in mucoadhesive analysis. Trends Food Sci. Technol. 17, 255–262 (2006).
Mackie, A. R. et al. Innovative methods and applications in mucoadhesion research. Macromol. Biosci. 17, 1600534 (2017).
Cook, S. L. et al. Mucoadhesive polysaccharides modulate sodium retention, release and taste perception. Food Chem. 240, 482–489 (2018).
Cook, S. L. et al. Polysaccharide food matrices for controlling the release, retention and perception of favours. Food Hydrocoll. 79, 253–261 (2018).
Mathiowitx, E. & Chickering, D. E. in Bioadhesive Drug Delivery Systems: Fundamentals, Novel Approaches and Development (eds Mathiowitz, E., Chickering, D. E. & Lehr, C. M.) 1–10 (Marcel Decker Inc., New York, 1999).
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Dinu, V. et al. Analytical ultracentrifugation in saliva research: Impact of green tea astringency and its significance on the in-vivo aroma release. Sci. Rep. 8, 13350 (2018).
Tekahara, S. et al. Degradation of MUC7 and MUC5B in human saliva. PLoS ONE 8, e69059 (2013).
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Cook, S. L. et al. Mucoadhesion: a food perspective. Food Hydrocoll. 72, 281–296 (2017).
Cook, M. T. et al. Synthesis of mucoadhesive thiol-bearing microgels from 2-(acetylthio)ethylacrylate and 2-hydroxyethylmethacrylate: novel drug delivery systems for chemotherapeutic agents to the bladder. J. Mater. Chem. B. 3, 6599 (2015).
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This work was supported by the Biotechnology and Biological Sciences Research Council (grant number BB/N021126/1) and the Engineering and Physical Sciences Research Council (grant number EP/L015633/1). The work was carried out in the Flavour Research Group, Division of Food Sciences and the National Centre for Macromolecular Hydrodynamics, in the School of Biosciences, University of Nottingham. We would also like to thank the team from the Centre for Biomolecular Sciences for their supply of human saliva and New Food Innovation for their support on the E-tongue.
National Centre for Macromolecular Hydrodynamics, School of Biosciences, University of Nottingham, Sutton Bonington Campus, Leicestershire, UK
Vlad Dinu & Stephen E. Harding
Division of Food, Nutrition and Dietetics, School of Biosciences, University of Nottingham, Sutton Bonington Campus, Leicestershire, UK
Vlad Dinu, Arthur Gadon, Katherine Hurst, Mui Lim, Charfedinne Ayed & Ian D. Fisk
School of Health Sciences, Faculty of Medicine and Health Sciences, Queen's Medical Centre, Clifton Boulevard, Nottingham, UK
Richard B. Gillis & Gary G. Adams
Universitetet i Oslo, Postboks 6762, St. Olavs plass, 0130, Oslo, Norway
Stephen E. Harding
Vlad Dinu
Arthur Gadon
Katherine Hurst
Mui Lim
Charfedinne Ayed
Richard B. Gillis
Gary G. Adams
Ian D. Fisk
V.D. conceived the idea, performed the experiments and directed the research project. A.G. performed the chemical synthesis. K.H. helped with the conductivity metre experiments. M.L. developed the in-vivo analysis on the APCI. C.A. helped run the GC–MS and APCI-MS analysis and aided in sample preparation. R.B.G. helped with the analysis and interpretation of the FT-IR and contributed to the writing of the manuscript. G.G.A., S.E.H., and I.F. have supervised and co-investigated the project.
Correspondence to Ian D. Fisk.
Dinu, V., Gadon, A., Hurst, K. et al. An enzymatically controlled mucoadhesive system for enhancing flavour during food oral processing. npj Sci Food 3, 11 (2019). https://doi.org/10.1038/s41538-019-0043-y
Accepted: 30 April 2019
DOI: https://doi.org/10.1038/s41538-019-0043-y
Understanding the lost functionality of ethanol in non-alcoholic beer using sensory evaluation, aroma release and molecular hydrodynamics
Imogen Ramsey
, Vlad Dinu
, Rob Linforth
, Gleb E. Yakubov
, Stephen E. Harding
, Qian Yang
, Rebecca Ford
& Ian Fisk
Scientific Reports (2020)
Mucin immobilization in calcium alginate: A possible mucus mimetic tool for evaluating mucoadhesion and retention of flavour
, Mui Lim
, Katherine Hurst
, Gary G. Adams
& Ian D. Fisk
International Journal of Biological Macromolecules (2019)
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WizEdu
What is the electric potential energy of the group of charges in (Figure 1)? Assume that q = -5.5 nC.
In: Physics
U= (value) (units)
Concepts and reason
The concepts used to solve this problem are electric potential energy and Pythagoras theorem. Initially use the Pythagoras theorem to calculate the hypotenuse distance. Then use the pairs of charges and the distance between them in the potential energy formula. Finally add all the potential energy terms to calculate the total potential energy of the group of charges.
The electric potential energy of a group of charges is,
$$ U=\sum_{i, j, i * j} \frac{1}{4 \pi \varepsilon_{0}} \frac{q_{i} q_{j}}{r_{i j}} $$
Here, \(\varepsilon_{0}\) is the permittivity of the free space, \(q\) is the charge, and \(r\) is distance between the two charges, \(U\) is the
potential energy of the arrangement, and \(i\) and \(j\) are indices of charges. Pythagoras theorem gives the hypotenuse distance in a right-angled triangle. \(c=\sqrt{a^{2}+b^{2}}\)
\(a, b,\) and \(c\) Here, \(a, b,\) and \(c\) are the sides of the triangle.
Use the Pythagoras theorem to find the distance between the two positive charges.
Substitute \(4.0 \mathrm{~cm}\) for \(a,\) and \(3.0 \mathrm{~cm}\) for \(b\) in the equation \(c=\sqrt{a^{2}+b^{2}}\)
$$ \begin{aligned} c &=\sqrt{(4.0 \mathrm{~cm})^{2}+(3.0 \mathrm{~cm})^{2}} \\ &=\sqrt{16.0 \mathrm{~cm}^{2}+9.0 \mathrm{~cm}^{2}} \end{aligned} $$
\(=5.0 \mathrm{~cm}\)
Use the potential energy formula.
\(\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}} \quad \sum_{\text {for } i, j, i * j} \frac{1}{4 \pi \varepsilon_{0}} \frac{q_{i} q_{j}}{r_{i j}}\) in the equation.
\(U=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}}\)
Refer the figure, the three charges make a right-angled triangle at negative charge. There are three charges so
\(1 \leq i, j \leq 3, i \neq j .\) Here the charge 1 is the bottom positive charge and charge 2 is the negative charge, and charge 3
is the positive charge on the right. The distance between the charge 1 and 2 is \(r_{12}\). The distance between the charge
2 and 3 is \(r_{23}\) The distance between the charge 1 and 3 is \(r_{13}\).
Use the electric potential energy equation.
Substitute \(8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\) for \(\frac{1}{4 \pi \varepsilon_{0}}, 3.0 \mathrm{nC}\) for \(q_{1},-5.5 \mathrm{nC}\) for \(q_{2}, 3.0 \mathrm{nC}\) for \(q_{3}, 4.0 \mathrm{~cm}\) for \(r_{12}\)
\(3.0 \mathrm{~cm}\) for \(r_{23}\), and \(5.0 \mathrm{~cm}\) for \(r_{13}\) in the equation \(U=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r_{12}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{3}}{r_{13}}+\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{2} q_{3}}{r_{23}}\)
The electric potential energy of the group of charges is \(-7.0 \times 10^{-6} \mathrm{~J}\).
Calculate the electric potential by substituting the value of charges and distances in the expression of electric potential energy.
Dr. OWL answered 4 weeks ago
What is the electric potential energy of the group of charges in the figure?
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What is the electric potential at the point indicated with the dot in the figure?
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Rapid optimization of spore production from Bacillus amyloliquefaciens in submerged cultures based on dipicolinic acid fluorimetry assay
Hang Ren1,
Ya-ting Su1 and
Xiao-hua Guo1Email author
Some optimization techniques have been widely applied for spore fermentation based on the plate counting. This study optimized the culture medium for the spore production of Bacillus amyloliquefaciens BS-20 and investigated the feasibility of using a dipicolonic acid (DPA) fluorimetry assay as a simpler alternative to plate counting for evaluating spore yields. Through the single-factor experiment, the metal ions and agro-industrial raw materials that significantly enhanced spore production were determined. After conducting a response surface methodology (RSM) analysis of several metal ions, the combined use of optimum concentrations of Mn2+, Fe2+, and Ca2+ in culture media produced a 3.4-fold increase in spore yields. Subsequently, supplementing soybean meal and corn meal with optimum concentrations determined by another RSM analysis produced an 8.8-fold increase. The final spore concentration from a culture medium incorporating optimum concentrations of the metal ions and raw materials mentioned above was verified to reach (8.05 ± 0.70) × 109 CFU/mL by both DPA fluorimetry and plate counting. The results suggest that the use of DPA fluorescence intensity as an alternative value to colony counting provides a general method for assessing spore yields with less work and shorter time.
Bacillus amyloliquefaciens
Spore yields
Response surface methodology
Dipicolinic acid
Bacillus species are aerobic or facultative anaerobic, sporulating, rod-shaped bacteria (Driks 2002). They can form protective endospores that allow them to tolerate harsh environmental stress, such as heat, radiation, desiccation, freezing and chemical disinfectants (Setlow 2006). The Bacillus spores can survive through the digestive process and germinate within the digestive tract (Casula and Cutting 2002). As a common source of probiotic supplements, Bacillus is often used in animal feeds, human dietary supplements and even in medicines (Cutting 2011).
Endospores of Bacillus are formed after the exponential phase of vegetative cell growth as a result of nutrient depletion and cell accumulation (Driks 2002). In the commercialization of Bacillus-based bio-products, high spore yields in bioreaction with less cost are preferred in industrial exploitation (Chen et al. 2010; Khardziani et al. 2017; Lalloo et al. 2009; Posada-Uribe et al. 2015). The regulation of sporulation parameters in fermentation was often carefully considered for enhanced spore production (Monteiro et al. 2005; Rao et al. 2007).
In early reports, the optimization of culture media and culture conditions was largely studied for higher spore yields for particular Bacillus strains, since each strain differed from different nutrient requirements and culture conditions (Chen et al. 2010; Khardziani et al. 2017; Posada-Uribe et al. 2015; Setlow 2006; Shi and Zhu 2007). In whichever reports, the spore concentrations were all quantified by plate counting assay, which were generally time-consuming and quite tedious (Hazan et al. 2012). Some alternative techniques on specific detection of spores were developed (Ai et al. 2009; He et al. 2003; Hindle and Hall 1999; Pellegrino et al. 2002), one of which was fluorimetry assay based on dipicolinic acid (DPA). DPA is a universal and specific component of bacterial spores and the limit of detection (LOD) on spores based on the DPA fluorimetry assay has reached 1000 spores/mL in the report of Pellegrino et al. (2002). The fluorimetry assay is comparatively simple, time-saving, and especially suitable for the simultaneous detection with many analytes (Pellegrino et al. 2002). However, till now, no reports were focused on spore production based on the specific DPA fluorimetry assay during the optimization procedure.
Using a statistical experiment design, this study determined the optimum concentrations of metal ions and raw materials to enhance the spore production of the potentially probiotic strain of Bacillus amyloliquefaciens BS-20 under submerged fermentation. In addition, the feasibility of using a DPA fluorimetry assay to quantify spore yields as the response variable in the optimization procedures was investigated.
Bacterial strains and culture condition
Bacillus amyloliquefaciens BS-20, previously screened as probiotics from Bacillus species, was used as the starter cultures in submerged fermentation, and the isolate was deposited in the China Center for Type Culture Collection (CCTCC) as No. M 2017587. The strain was maintained at − 80 °C in 20% sterile glycerol until needed. The medium was initially developed for the maximum cell growth based on Luria–Bertani (LB) broth and was composed of: glucose 8 g/L, beef extract 7.2 g/L, NaCl 10 g/L, pH 7.0. The medium was autoclaved at 121 °C for 15 min and then used as the initial broth for the strain's growth and spore production. The culture was kept in a 250 mL Erlenmeyer flask containing 50 mL of broth. After inoculating 2% freshly prepared culture with an initial cell concentration of approximately 2 × 107 cells/mL, spore fermentation began. All experiments were carried out in a rotating shaker at 200 rpm and 37 °C. The samples were cultured for 48 h and then harvested.
Spore detection
The spore concentration was quantified using the DPA marker in spores based on a technique described in previous reports with some modifications (Hindle and Hall 1999; Pellegrino et al. 2002). The principle of detection is that in the presence of the chelating agent cyclohexanediamine tetraacetic acid (CyDTA), DPA and the lanthanide metal europium produce a specific fluorescence excited by ultraviolet light, the intensity of which is in proportion to the concentration of DPA.
Specifically, the spores from the fermentation broth were harvested for analysis by centrifugation (2500×g for 10 min) and washed twice, and suspended in sterile Tris–HCl (50 mM, pH 8.0). The spore suspensions were then treated at 121 °C for 5–10 min for the full release of DPA into the buffer based on based on an earlier study (data not shown). The DPA-containing supernatants were collected after being centrifuged at 2500×g for 10 min. With a certain dilution, the supernatants were assayed for fluorescence intensity by mixing EuCl3 (2 mM) and CyDTA (2 mM) with the proportion 1:4.5:4.5 by a vortex oscillator. Meanwhile, in the fluorescent complex, DPA supernatants were replaced by isometric Tris–HCl buffer to serve as a blank control. A Hitachi F-7000 spectrofluorophotometer (Hitachi Ltd., Tokyo, Japan) was used to detect the fluorescence intensity at the excitation wavelength of 272 nm and emission wavelength of 619 nm. The scanning speed was pre-set to 3000 nm/min, the slit to 5 nm/10 nm, the photo-multiplier tube (PMT) voltage to 700 V, and the responding time to 0.08 s. In order to keep the accuracy of measurement, the DPA samples were serially diluted to make the light output in arbitrary units on a scale from 0 to 1000.
A traditional plate-counting assay was performed to verify the reliability of the DPA fluorimetry assay on spore detection. Spores were counted by heating dilutions of the culture at 80 °C for 15 min to kill vegetative cells before they were plated onto an LB agar medium. The colonies were counted after cultivation at 37 °C for 24 h, and the final results were expressed as colony-forming units per mL (CFU/mL).
The spore suspensions with the initial optical density about OD600 nm of 1.0 were twofold serially diluted and the ensuing DPA fluorescence intensity was detected. The concentration of spore suspensions was detected by plate counting and DPA fluorimetry assay, respectively. The linear correlation between spore concentrations (CFU/mL) and the fluorescence intensity (AU) was built.
Screening of significant metal ions for spore production
Six metal ions, Mn2+, Fe3+, Fe2+, Ca2+, Mg2+ and Zn2+, were identified as key factors in improving spore production based on previous reports (Granger et al. 2011; Kihm et al. 1988; Kolodziej and Slepecky 1964; Oh and Freese 1976). A single-factor experiment was carried out by adding metal ions into the autoclaved basal medium, which contained 8 g/L of glucose and 7.2 g/L of beef extract. The metal ions were filter-sterilized and added into the basal medium to reach the final concentrations listed in Table 1. The initial broth (glucose 8 g/L, beef extract 7.2 g/L, NaCl 10 g/L) served as a control. Both media were inoculated with B. amyloliquefaciens BS-20 and cultivated for 48 h. The harvested culture was immediately treated and quantified by DPA fluorimetry assay. The results were expressed as the means of fluorescence intensity and their standard deviations (SDs) based on three replicates. The data were analyzed by Student's t test in the JMP11.0 software (SAS Institute Inc., USA). P values less than 0.05 were regarded as a significant difference. The metal ions that showed a significant positive influence on spore production were selected for optimization by a central composite design (CCD) experiment and response surface methodology (RSM) analysis.
Effects of metal ions with different concentrations on the spore yields of B. amyloliquefaciens BS-20 detected by the fluorometric assay
Concentration (mM)
Fluorescence intensity (AU)
Fe3+
Mn2+
Mg2+
Ca2+
Zn2+
996.7 ± 48.5
996.7 ± 48.5a
1080.0 ± 112.8a
1073.0 ± 26.5
1661.5 ± 102.5b
1184.5 ± 71.4b
1067.0 ± 98.9ab
285.0 ± 18.4b
1526.5 ± 64.3c
226.0 ± 29.7bc
175.0 ± 15.6c
1097.5 ± 3.5bc
1219.0 ± 4.2b
65.0 ± 9.9d
1136.0 ± 90.5a
1068.5 ± 62.9bc
Mean values in the same column with different letters (a, b, c, d) are significantly different (P < 0.05). The final results are expressed as the mean ± standard deviation (n = 3) of 100-fold diluted spore samples
Ion optimization by central composite design
A CCD and RSM analysis were employed to investigate the optimal combination of the metal ions. The RSM was applied through the statistical software JMP 11 (SAS Institute Inc., USA). The optimal concentrations of the key metal ions identified by the single-factor experiment were determined by studying each factor at five different levels: −a, −, 0, +, A (Table 2), which represented low star point, low central point, center point, high central point and high star point, respectively. For each factor, the central coded value was considered as zero, and the concentrations at the zero points were the values that significantly contributed to the highest fluorescence intensity in the single-factor experiment. The axial value was set as 1.483. The CCD was undertaken in 27 runs including 3 replicates of central point. The fluorescence intensity produced by the harvested spores was used as the response value for experimental analyses. The quadratic models for RSM were used to predict the co-effect of metal ions. The optimum concentration points for maximum spore production was determined based on the quadratic Eq. (1).
$$y = \beta_{0} + \sum\limits_{i = 1}^{k} {\beta_{i} {\text{x}}_{i} } + \sum\limits_{i = 1}^{k} {\beta_{ii} {\text{x}}_{i}^{2} } + \sum\limits_{i < j}^{k} {\beta_{ij} {\text{x}}_{i} {\text{x}}_{j} }$$
Central composite design for metal ion factors associated with spore density by the fluorometric assay
Metal ions concentration (mM)
Mn2+ (x1)
Fe2+ (x2)
Ca2+ (x3)
Mg2+ (x4)
−−++
+−++
−+++
−−−+
++−−
++−+
−−+−
+−−+
−+−−
+−+−
+−−−
+++−
−−−−
280.1 ± 4.7
−++−
−+−+
The experimental results are the means of two replicates of 1000-fold diluted spore samples. The symbols in the model column mean each factor at five different levels (−a, −, 0, +, A). The variables at a central coded value are considered at zero
Selection of significant raw materials for spore enhancement
Different agro-industrial materials including corn meal, soybean meal, wheat bran and molasses (about 48% sugars) were bought locally. In a similar way as the metal ions were optimized, the raw materials were added to the medium containing optimized ions and further analyzed by another single factor experiment with the same design as that described in the previous section (Table 3). The basal medium that contained these raw materials were autoclaved at 121 °C for 15 min, and the optimized ions were then added after filter-sterilization. The basal medium that only contained the optimized metal ions was used as a control.
Effects of different raw materials on the spore yields of B. amyloliquefaciens BS-20 detected by the fluorometric assay
Concentration (g/L)
Soybean meal
Mean values in the same column with different letters (a, b, c) are significantly different (P < 0.05). The final results are expressed as the mean ± standard deviation (n = 3) of 1000-fold diluted spore samples
Raw materials optimization by central composite design
The single-factor experiment identified the key raw materials to include for enhancing spore yields. To determine the optimum combination of raw materials, similar procedures to those used for optimizing the ions by a CCD and an RSM analysis were carried out. Similar procedures as described in ion optimization by CCD and RSM were carried out.
Validation of the optimization procedures
After optimizing the ions and raw materials, verification experiments were carried out to check whether the spore concentrations quantified by the fluorimetry and plate counting assay were consistent. The initial broth was used as a control. The results were expressed as the means of fluorescence intensity or CFU/mL and their standard deviations (SD) based on three replicated experiments.
DPA fluorimetry assay for quantifying the spore concentration
Figure 1 shows the good linear correlation between the spore concentrations varying from 8 × 103 to 8 × 106 CFU/mL, and corresponding DPA fluorescence intensity (coefficient R2 = 0.9999). The limit of detection (LOD) reached 8000 spores/mL. As a result, the fluorimetry assay was used in the following optimization procedures for spore production.
Calibration curves of spore counts of B. amyloliquefaciens BS-20 and their fluorescence intensity. The spores of B. amyloliquefaciens BS-20 with 1.34 × 108 CFU/mL were twofold serially diluted and treated for the detection of fluorescence intensity
Effect of metal ions on spore yields
Of the six metal ions, four ions including Mn2+, Fe2+, Ca2+, and Mg2+ showed significant positive influence on the enhancement of sporulation compared with the control (P < 0.05) (Table 1). The optimum concentrations of metal ions were 1.0 mM of Mn2+, 3.0 mM of Fe2+, 2.0 mM of Ca2+, and 3.0 mM of Mg2+, respectively.
Ion optimization by a response surface methodology analysis
The significant metal ions chosen from the one-factor experiment, Mn2+ (x1), Fe2+ (x2), Ca2+ (x3) and Mg2+ (x4) were included in the CCD for the determination of their optimum concentrations, and the results are listed in Table 2. As observed from Table 2, the response variable was analyzed through RSM and a standard analysis of variance (ANOVA) (Table 5). The dataset could be fitted with a regression quadratic equation as described in Eq. (2).
$$\begin{aligned} {\text{Y }} &= { 198}.0 8 { } + { 38}. 8 8x_{ 1} + { 28}. 3x_{ 2} + { 36}. 5 7x_{ 3} + { 1}. 5x_{ 4} \\ & \quad + { 3}. 1 4x_{ 1} x_{ 2} + { 3}. 6 6x_{ 1} x_{ 3} - { 3}. 7 8x_{ 2} x_{ 3} + { 4}. 6 4x_{ 1} x_{ 4} - \, 0. 1 7x_{ 2} x_{ 4} \\ & \quad + { 1}. 9 7x_{ 3} x_{ 4} - { 33}. 3 7x_{ 1}^{ 2} - { 3}. 8 4x_{ 2}^{ 2} - { 8}.0 2x_{ 3}^{ 2} - { 2}.0 6x_{ 4}^{ 2} \hfill \\ \end{aligned}$$
The model showed the optimization was successful in improving spore production since the coefficient of determination, R2, and adjusted determination coefficient Adj. R2 were 0.94 and 0.87, respectively. The value of "P > F" was less than 0.05, indicating that the model was significant. The terms \(x_{1}^{2}\), \(x_{3}^{2}\), x3, x2x3, x2, \(x_{2}^{2}\), x1 and x1x4 (arranged by ascending P values) were found to be significant (P < 0.05). For the other model terms associated with the variable Mg2+ (i.e. \(x_{4}^{2}\), x4, x2x4), the P values were 0.1026, 0.8547 and 0.8588, respectively. Therefore, Mg2+ (x4) might play less roles in interacting with other metal ions in sporulation. A complementary experiment was undertaken to test the effect of the ion-optimized medium in the presence or absence of Mg2+. No significant difference in spore yields was observed (data were not shown). In order to lower the number of variable in final medium, Mg2+ was not considered in the further study.
Response surface plots were drawn to study the interactive effects of metal ions on sporulation and to determine their optimum concentrations for maximum possible spore yields (Fig. 2a–c). The response surface and contour plots indicated that the interactions between the independent variables Mn2+ (x1), Fe2+ (x2) and Ca2+ (x3) were significant. All three response surface plots had a convex surface with a downward opening shown in Fig. 2. Therefore, the response surface maximal point (300.02 AU) was obtained when the optimal significant variables were at the following levels: Mn2+ (x1) = 1.0 mM, Fe2+ (x2) = 3.0 mM, Ca2+ (x3) = 2.1 mM.
Response surface plots for spore production caused by metal ions. The interaction between a Mn2+ and Fe2+, b Fe2+ and Ca2+, c Mn2+ and Ca2+, respectively
Effect of raw materials on spore yields
On the basal medium containing the optimized concentration of metal ions, the effects of four main raw materials on the spore yields conducted in a one-factor experiment are presented in Table 3. Corn meal and soybean meal positively influenced spore production (P < 0.05). However, no significant effect was found from wheat bran and molasses (P > 0.05). The co-effect of corn meal and soybean meal was further studied in a CCD and RSM analysis over 11 runs, including 3 replicates of central point.
Raw materials optimization by response surface methodology
The design and result of the CCD from the corn and soybean meal variables are presented in Table 4, and the RSM analysis and ANOVA are presented in Table 5. The quadratic regression is described in Eq. (3).
$${\text{Y }} = \, - 5 5 8 8. 8 4 { } + { 559}. 1x_{ 5} + { 812}. 4 8x_{ 6} - { 6}. 7 3x_{5} x_{6} - 2 7. 4 8x_{ 5}^{ 2} - { 39}. 4 3x_{ 6}^{ 2}$$
Central composite design for soybean meal and corn meal associated with spore density by the fluorometric assay
Corn meal (x5)
Soybean meal (x6)
−−
Analysis of variance (ANOVA) for response surface quadratic models for spore production based on DPA florescence detection by metal ion-optimized RSM and sequential raw material-optimized RSM in submerged fermentation
Metal ion-optimized RSM
Raw material-optimized RSM
P > F
< 0.0001
Adj. R2
Root mean square error
Response surface solution
The value of "P > F" less than 0.05 indicates the model terms are significant
The optimization of the raw materials were also successful and greatly increased the spore yields with the value of "P > F" = 0.0059. The R2 and Adj. R2 were 0.9328 and 0.8656, respectively. The model terms x6 and \(x_{6}^{2}\) were found to be significant (P = 0.0298 and 0.0195, respectively). The response surface plots had a downward opening convex showed the response surface maximal point was 802.03 AU (Fig. 3), which was about 2.7 times of the value in the ion-optimized RSM. The critical variable concentrations for predicted maximum spore yields were as follows: corn meal (x5) = 9.0 g/L and soybean meal (x6) = 9.5 g/L, respectively.
Response surface plots for spore production caused by soybean meal and corn meal
Verification for spore production after optimization
The spore production results were verified to check the accuracy of the models over three replicates (Table 6). The results showed that the experimental values were very close to the predicted values, and the optimization models were validated. Moreover, the calculated colony concentrations based on the standard curves in Fig. 1 were also close to the practical measured colony concentrations (Table 6). The results indicated that the spore yield detected by fluorimetry assay were consistent to that by plate counting assay.
Verification for spore production after two-step RSM optimization procedures
Optimization procedures
DPA fluorimetry assay
Plate counting assay
Predicted fluorescence intensity (AU)
Observed fluorescence intensity (AU)
Calculated colony concentrations (CFU/mL)
Measured colony concentrations (CFU/mL)
(9.01 ± 0.03) × 108
From the verification experiments, the optimized media (glucose 8 g/L, beef extract 7.2 g/L, corn meal 9.0 g/L, soybean meal 9.5 g/L, Mn2+ 1.0 mM, Fe2+ 3.0 mM and Ca2+ 2.1 mM) gave an 8.8-fold increase in the spore yield compared with the control (glucose 8 g/L, beef extract 7.2 g/L, NaCl 10 g/L). The experimental values measured by plate counting assay reached (8.05 ± 0.70) × 109 CFU/mL (n = 3).
Several studies have been performed on the enhancement of spore production, and the top 2 highest documented spore concentrations of Bacillus undergoing submerged fermentation were 1.56 × 1010 CFU/mL (Chen et al. 2010) and 7 × 1010 CFU/mL (Khardziani et al. 2017), respectively. Both of these high spore yields were observed in the fermentation of B. subtilis. The spore yields obtained in this study are the highest levels in B. amyloliquefaciens fermentation compared to other reports, whose yields range from 5.93 × 108 CFU/mL (Rao et al. 2007) to 3.82 × 109 CFU/mL (Tzeng et al. 2008). Moreover, higher spore yield could be achieved by optimizing the culture or fermentation conditions in bioreactors with better ventilation and agitation using an optimized medium as a base (Khardziani et al. 2017).
This study focused on factors that previous reports had suggested to influence spore production (Chen et al. 2010; Khardziani et al. 2017; Kihm et al. 1988; Shi and Zhu 2007). The final result in the study showed that optimizing the type and concentration of metal ions and raw materials improved spore yields by 3.4- and 8.8-fold, respectively (Table 6). The metal ions likely played a role in activating enzyme systems necessary for sporulation (Kolodziej and Slepecky 1964). Manganese and iron are indispensable for sporulation and participate in the synthesis of Bacillus's secondary metabolites, such as antibiotics and peptides (Granger et al. 2011; Greene and Slepecky 1972; Oh and Freese 1976). Calcium acts as an important component of spores by chelating with DPA (Ca-DPA) and helps to improve heat resistance (Levinson et al. 1961). This study found similar results on metal ions' contribution to spore production (see Table 1). The single-factor experiment identified Mn2+, Fe2+ and Ca2+ as having a significantly positive effect on spore production. In contrast with another report (Kihm et al. 1988), the inclusion of zinc had a significantly negative effect on sporulation in the present study (P < 0.05). The results suggest that different strains might have different response to metal ions the in medium and using a thorough screening procedure is important before optimizing the concentration of metal ions. The inclusion of raw materials in the medium greatly improved spore yields both in the current study and other reports (Chen et al. 2010; Khardziani et al. 2017; Posada-Uribe et al. 2015). Generally, proteinase and amylase activity are similar across Bacillus species, and B. amyloliquefaciens BS-20 showed more enzyme activity than other Bacillus probiotics in our previous studies (data not shown). The gradually hydrolyzed substrates from protein and starch in the raw materials provides nutrients for Bacillus growth and spore production, which could also alleviate possible catabolite repression on sporulation caused by glucose (Chen et al. 2010; Shi and Zhu 2007).
More importantly, the current study demonstrated the use of DPA fluorimetry assays as an alternative to traditional plate counting for quantifying spore concentration in the optimization procedures. From the linear curve in Fig. 1, it can be seen that the LOD in this study (8000 spores/mL) was close to the lowest LOD (1000 spores/mL) identified in the literature (Pellegrino et al. 2002). The LOD was low enough to allow for the quantification of spore concentrations since spore yields in fermented cultures are often above 108 spores/mL. Moreover, the DPA fluorimetry assay used in this study is very simple, and the fluorescent complex was produced by just mixing the diluted DPA samples, europium, and the chelating agent CyDTA. The fluorescence intensity was readily measured by a fluorescence spectrophotometer or microplate readers (Pellegrino et al. 2002). Additionally, DPA fluorimetry assay allowed fast and synchronous detection of many samples in the statistical optimization experiments. For example, in the ion-optimized RSM experiment of this study, 27 runs with 2 replicates were carried out simultaneously and all the 54 samples could be detected in 1 h by the fluorescence spectrophotometer. However, in the plate counting assay, the spore concentration of one sample was achieved by plating three tenfold dilutions of spore suspensions with at least three replicates for each dilution. Therefore, at least 3 × 3 × 54 plates were required and the colonies were finally counted after at least 24 h cultivation. Based on the results found by the DPA fluorimetry assay (presented in Table 6), it was demonstrated that the optimization techniques described in this paper provided an easy and feasible way to enhance spore production. Finally, from the optimized and verified results in this study, a DPA fluorimetry assay was successfully applied and provided a general analytical method for assessing spore concentrations with less work and time than a plate-counting assay would require.
arbitrary units
CCTCC:
China Center for Type Culture Collection
CCD:
CFU:
colony-forming units
CyDTA:
cyclohexanediamine tetraacetic acid
DPA:
Luria–Bertani
LOD:
limit of detection
optical density
PMT:
photomultiplier tube
RSM:
SD:
Planning and designing of the study: XHG; experimentation: HY and YTS; data analysis: XHG; manuscript drafting: XHG. All authors read and approved the final manuscript.
We would like to thank Xiaosheng Liang and Li Zhang for their technical suggestions.
The data supporting the conclusions of this article are all included within the article.
All authors gave their consent for publication.
This work was financially supported by the National Natural Science Foundation of China (No. 31672455) and the Fundamental Research Funds for the Central Universities (CZT18002).
Provincial Key Laboratory for Protection and Application of Special Plants in Wuling Area of China, College of Life Science, South-Central University for Nationalities, No. 182, Minyuan Road, Hongshan District, Wuhan, 430074, Hubei, China
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Deep learning approach to security enforcement in cloud workflow orchestration
Hadeel T. El-Kassabi1,
Mohamed Adel Serhani2,
Mohammad M. Masud3,
Khaled Shuaib3 &
Khaled Khalil4
Supporting security and data privacy in cloud workflows has attracted significant research attention. For example, private patients' data managed by a workflow deployed on the cloud need to be protected, and communication of such data across multiple stakeholders should also be secured. In general, security threats in cloud environments have been studied extensively. Such threats include data breaches, data loss, denial of service, service rejection, and malicious insiders generated from issues such as multi-tenancy, loss of control over data and trust. Supporting the security of a cloud workflow deployed and executed over a dynamic environment, across different platforms, involving different stakeholders, and dynamic data is a difficult task and is the sole responsibility of cloud providers. Therefore, in this paper, we propose an architecture and a formal model for security enforcement in cloud workflow orchestration. The proposed architecture emphasizes monitoring cloud resources, workflow tasks, and the data to detect and predict anomalies in cloud workflow orchestration using a multi-modal approach that combines deep learning, one class classification, and clustering. It also features an adaptation scheme to cope with anomalies and mitigate their effect on the workflow cloud performance. Our prediction model captures unsupervised static and dynamic features as well as reduces the data dimensionality, which leads to better characterization of various cloud workflow tasks, and thus provides better prediction of potential attacks. We conduct a set of experiments to evaluate the proposed anomaly detection, prediction, and adaptation schemes using a real COVID-19 dataset of patient health records. The results of the training and prediction experiments show high anomaly prediction accuracy in terms of precision, recall, and F1 scores. Other experimental results maintained a high execution performance of the cloud workflow after applying adaptation strategy to respond to some detected anomalies. The experiments demonstrate how the proposed architecture prevents unnecessary wastage of resources due to anomaly detection and prediction.
Cloud computing has emerged as a promising and powerful paradigm for managing and delivering computations, applications, and services over the Internet [1]. The processing power provided by the cloud covers a wide landscape of services, including storage, processing, and application services. This computational processing power has enabled researchers to use various computationally intensive and scientific workflows to perform vast experiments that were impossible to implement using local servers. This trend has significantly decreased the total cost incurred by related software systems [1] and is, in fact, a promising design paradigm for workflow deployment, processing, and orchestration. A typical large-scale scientific workflow comprises a set of interrelated tasks that are inherently complex, fault tolerant, and dynamically executed and orchestrated to produce scientific results. However, a cloud workflow refers to a workflow that is deployed and executed on the cloud. Cloud workflow features are classified as transparent, scalable, multi-tenant, and monitored in real-time [2].
With virtually infinite computing resources, cloud computing meets the needs of complex scientific data-intensive workflow tasks and releases cloud workflows from the burden of planning for resource provisioning. However, many research challenges need to be addressed before this potential can be fully realized. Such challenges include cloud security threats against integrity, authorization, availability, reliability, and trust. These challenges also apply to workflow security and privacy enforcement in the cloud environment, which is characterized by complexity, dynamicity, and multi-dimensional aspects. Supporting secure access, deployment, execution, and management of workflows over cloud platforms is of prime importance to both cloud providers and consumers. Providers must ensure that the resources they make available for workflow execution are not hacked, misused, or damaged. Similarly, workflow customers must be assured that their workflows and associated data are secured and protected from any outsider attacks.
Several cloud computing security threats have been identified [3] and extensively studied [4,5,6,7,8]. Such threats, which include data breaches, data leaks, data loss, denial of service (DoS), and malicious insiders, are generated from issues such as multi-tenancy, loss of control over data, and breaches of trust. Supporting security in a dynamic environment, across different platforms, different stakeholders, and various processes requires the involvement of various entities besides the cloud providers. A comprehensive security solution that considers security enforcement, trust chain in clouds, and ensures policies and regulations to guarantee security and privacy across multi-participants and heterogeneous environments is of paramount importance.
Emerging security issues in a cloud workflow have motivated researchers to address various research problems related to the security enforcement in a dynamically executed and orchestrated cloud workflow. However, most existing research on workflow management primarily tackled mainly aspects related to 1) anomaly and error detection [9], 2) workflow task scheduling [10], and 3) autonomic workflow resource provisioning and management [11]. In these studies, the main purpose was to avoid failure or resource contention and ensure efficient deployment, execution, and performance guarantee of these workflows [12]. Such initiatives neglect cloud workflow security enforcement which may strengthen the aspects mentioned above and fill the gap in handling security and data integrity of dynamic cloud workflows. Very few studies have exclusively tackled the security of cloud workflow orchestration, management, and enforcement [13]. Such studies typically focus on anomaly detection and prediction using various techniques such as HTM [14], statistical-clustering [8, 15, 16], regression [17, 18], and unsupervised machine learning (ML) [19]. They neither clearly define the security attributes nor specify the cloud workflow characteristics, which can be described as resource-aware, time-series, and highly dynamic. In addition, they focus on a specific security dimension, i.e., data, task, or resource of the workflow, and ignore other dimensions that will lead to better security enforcement when combined. Furthermore, the anomaly prediction and detection schemes proposed in most of these studies rely only on the dynamic features of the cloud workflow and neglect the static features that generally capture important workflow information and its hosting environment. Finally, most previous attempts primarily focused on security/anomaly detection and prediction and ignored the resources and workflow adaptation strategies that should be undertaken to mitigate security threats and possible attacks.
Therefore, to address some of the limitations of previous studies, we emphasize security anomaly/attack detection and prediction in a cloud workflow orchestration environment. We propose an adaptation scheme to cope with possible vulnerabilities and mitigate their effects on cloud workflow execution. In such an environment, the attack could target different entities and components such as workflow data, tasks, resources, monitoring, and adaptation. Our proposed model contributes to the state-of-art literature on cloud workflow security by including the following:
A multidimensional security enforcement emphasizing cloud workflow security at various levels: the task, data, resources, and monitoring scheme.
A scheme combining static and dynamic features for anomaly/attacks prediction, which is a unique way to model features that provide better anomaly detection, i.e., features that capture all stakeholders' needs and all aspects of the cloud workflow. This scheme also applies deep learning autoencoder-based dimensionality reduction for the dynamic data, which will lead to better characterization of workflow tasks and will thus provide better attack prediction.
An unsupervised learning technique that does not require class labeling, no additional work, and no manual intervention from experts, thus making it convenient and more realistic to deal with unknown anomalies.
An adaptation model that accommodates the flexible representation and planning of resource requirements over time and over the various phases of the cloud workflow execution cycle.
The remainder of this paper is organized as follows: In Section 2 related work is discussed and compared and their limitations are identified. Section 3 presents a case study using a workflow based on a COVID-19 dataset. An architecture to enforce end-to-end security in a cloud workflow orchestration is proposed in Section 4. In section 5, a detailed cloud workflow security enforcement model formulation and the associated learning pipeline algorithms are presented. Section 6 presents the implementation details, conducted scenarios, experiments, and a discussion of the obtained results. Finally, conclusions and suggestions for future work are presented in Section 7.
Security in cloud computing ecosystem is a comprehensive field that attracted significant attention as the cloud services hype grew exponentially, and cloud security threats and vulnerabilities also evolved over time. To help advance stat-of-the-art security solutions, first, related risks in emerging cloud services need to be identified. Such security risks are initiated by three types of attack vectors: external users, internal users, and cloud providers [20, 21]. Security threats in a cloud service environment are emerging over time. The most common threats include data breaches, data loss, DoS, malicious insiders, service traffic hijacking, shared technology vulnerabilities, malware, cyber-attacks, network intrusion, VM-level threats and data transparency [22,23,24,25].
Recently, deep learning approaches for cloud security threat detection have been proposed by several researchers. However, these approaches are unable to deliver a comprehensive solution for all security threats. However, they only address and detect patterns for a particular threat only in single deployment. The authors in [26] used a multi-layer neural network to detect and recognize malicious behaviors exhibited by users. They converted user behavior data into an understandable format and classified the malicious behavior for detection and recognition.
In [27], the authors proposed PredictDeep, a security analytical framework for known and unknown anomaly detection and prediction in Big Data systems. PredictDeep is proposed as a service to be offered to cloud users. The framework comprises three main modules, namely: a graph model designer, a feature extractor, and an anomaly predictor. PredictDeep is scalable and works well in a dynamic environment to monitor anomalies in real-time systems. However, with PredictDeep, anomaly detection assumes that all log files are accurate and no fake data that could compromise the accuracy of the prediction model has been injected. In addition, the proposed approach assumes the integrity of the infrastructure used for deployment.
Intrusion detection systems (IDS) are considered important tools for monitoring networks, services, and workflows for violations or malicious activities in cloud services orchestration. Detecting novel attacks in such scenarios is a challenging task. Deep learning-based intrusion detection techniques have yielded encouraging results relative to predicting unknown attacks and detection mechanisms. The authors in [28] presented an IDS using deep reinforcement learning-based architecture that can address and classify new and complex attacks. They employed a reward vector such that a classifier giving an identical result is awarded a positive point otherwise a negative point is obtained. In addition, the authors in [29] addressed a multi-cloud cooperative intrusion detection system and enhanced decision making in a real-time environment using a deep neural network (DNN) model. They employed historical feedback data to predict suspicious intrusions. Furthermore, a deep learning-based IDS has been proposed [25] to detect suspicious attacks in a cloud computing environment by monitoring network traffic. This system employs a self-adaptive genetic algorithm (SAGA) that automatically creates a DNN-based anomaly network IDS and demonstrated high detection rates, high accuracy, and low false alarm rates.
Other research initiatives have addressed cloud workflow security enforcement and several researchers and industries have proposed possible solutions to enhance the security of cloud services and cloud workflow orchestration. For Example, in [30], the developers currently working on the European-funded ASCLEPIOS (Advanced Secure Cloud Encrypted Platform for Internationally Orchestrated Solutions in Healthcare) project have exploited various cryptographic and access control techniques to protect user data privacy and provide protection against other security breaches as part of a cloud-based eHealth framework. One objective of this project is to enable various healthcare stakeholders to share data securely while preserving participant privacy. The proposed architecture consists of seven layers to provide security and analytic features to support data privacy and access control. However, the utilization of ML techniques for the detection, evaluation, and mitigation of potential anomalies in data or attacks on certain components of the overall system such as available resources was not incorporated into the proposed architecture.
In the PICASO project [31], a framework was proposed to enable cross-organization sharing of electronic health records using a cloud-based solution. This project aims to provide the required security and privacy measures in addition to service orchestration and data capture and management. Here, security features were implemented via separate subsystems to guarantee the privacy of patient records, user authentication, transaction information traceability, and enforcement of access control policies. However, the project did not implement any mechanisms for to detect of anomalies within the proposed framework.
Authors in [32] provided a literature review on security in FaaS orchestration systems [32]. They have classified the existing works considering various criteria including the protected asset, the cause of threats, and the protection approach. They concluded that most of the work focused on data confidentiality, however, data integrity is less considered. Function flows and platforms misconfiguration are also considered in most of the reviewed works. Moreover, authors in [33] provided another classification for existing works using Machine learning and Deep learning techniques for online malware detection in cloud. They classified the malware detection approaches into static analysis that is offline with no monitoring required, and dynamic analysis, where it requires real-time monitoring and use neural networks to predict when a virtual machine might be infected, this is more appropriate for cloud environment. The experimentation showed that Deep learning techniques provide good accuracy while detecting malware. However, this work did not analyze end-to-end security enforcement for workflows over cloud environment.
Zarca et al. proposed a semantic-aware and policy-driven security orchestration framework for autonomic security orchestration in IoT systems to detect semantic conflicts during the orchestration. The authors also proposed an optimized Service Function Chaining algorithm, which maximizes the QoS, security aspects and resources usage during Virtual network Security Functions allocation. However, they only detect anomalies but no prediction of anomalies is proposed [34].
Behavior attacks detection were proposed in [35] where different machine learning techniques for supervised classification were analyzed and compared. The study concluded that neural network models have the best performance in terms of accuracy to detect the impact malware on the process level features of virtual machines in the cloud. They collect different system features such as memory, CPU, and input/output from all process that are running on the VM at certain times. Similar work was proposed in [36] concentrating on different Convolutional Neural Networks (CNNs) for online detection of malware in cloud IaaS in real-time. The work focused on behavioral data using process level performance metrics including CPU, memory, and disk usage. Although their solution provide high accuracy, their proposed malware detection system is limited to a single virtual machine and does not support features such as auto-scaling [36].
Table 1 summarizes the state-of-the-art research with respect to anomaly detection in cloud environment. In this table we compare existing systems against the anomaly type they detect, anomaly detection technique adopted, data collected for anomaly detection, and other supported features such as real-time and orchestration. None of the surveyed systems provided anomaly detection from an end-to-end workflow orchestration perspective, rather a few systems detect anomalies in single VMs or single processes within a workflow [30, 34].
Table 1 Overview and classification of existing work on security in cloud
With the rapid increase in the use of cloud computing for electronic health records, issues related to security and privacy are critical for establishing participant trust in the deployed system. Thus, complex data-intensive cloud workflows must provide trustworthy results that enforce secure input and output data free of unauthorized access and malicious manipulation. Challenges related to handling security breaches in a cloud workflow orchestration system include the ability to identify the properties of each specific task comprising the workflow and each physical resource allocated in this dynamic infrastructure as well as integrate the collected information to detect anomalies and malicious actions. Anomaly detection must be timely, and an appropriate remedial action must be selected and executed before damage occurs. Moreover, the security process must have a minimal effect on the computing environment to maintain workflow execution performance. In other words, we must guarantee smooth and efficient handling of all security breaches including identification, prediction, and remediation.
To support security enforcement of cloud workflows and address some of the above abovementioned research challenges, we approach this problem from different dimensions including multi-level security enforcement, pre-evaluation of various prediction models for security threat detection and prediction, combining static and dynamic features for anomaly detection, and adaptation strategies to mitigate various security risks. Before we detail the features of our proposed cloud workflow security enforcement methodology, we illustrate the concept with an example cloud workflow handling COVID-19 dataset and identify potential security threats the cloud workflow may encounter that our proposed approach is expected to help detect. In addition, we predict some of these breaches and propose adaptation actions to protect against them.
Case study: COVID-19 cloud workflow
The effectiveness of healthcare systems worldwide has been challenged recently owing to the outbreak of the Novel Coronavirus (COVID-19) which was declared a pandemic by the WHO in March 2020. The impact of such new strains of viruses has been demonstrated to defeat all expectations of any healthcare system. This pandemic has strained involved entities, working to find a cure or vaccine, including healthcare providers, government agencies, and research facilities. Such pressures have led to proper protection of facilities, confidential data, and workflows from possible vicious attacks that could easily compromise the integrity of the overall process [38]. During COVID-19 pandemic, the reliance on online resources and cloud-based infrastructure systems has increased drastically due to lockdowns, contact-tracing applications, and increased use of remote working and distance-learning platforms. This has caused a huge leap in cyber-attacks and data confidentiality and integrity breaches [38]. To illustrate the applicability and usefulness of the security enforcement architecture and identify the main security threats in cloud workflow orchestration, we describe a case study involving a cloud workflow encompassing the composition of tasks handling a COVID-19 dataset.
Cloud workflow and COVID-19 dataset
Figure 1 shows the health monitoring cloud workflow we developed using the epidemiological data from a COVID-19 outbreak dataset that employs a deep learning model to predict the length of hospital stay of COVID-19 patients [39]. The dataset was collected and curated from national, provincial, and municipal health reports, as well as other online reports. The data are geocoded and include symptoms, key dates (date of onset, admission, and confirmation), and travel histories of different patients [40]. We used data collected up to the June 20, 2020, including 2,500,000 records, each of which represents an individual patient case. The dataset includes 33 columns including patient ID, age, gender, date_onset_symptoms, date_admission_hospital, date_confirmation, additional_information, chronic_disease_binary, chronic_disease, symptoms, and outcome. The explanation for each field is provided in [39]. We adopted this cloud workflow example to identify and evaluate different security breaches that could be encountered and therefore mitigate their effects. The workflow was deployed on a Docker Swarm Cluster and the data were loaded automatically from database tables to satisfy the service tasks outlined in the workflow. The workflow comprises a set of both sequential and parallel tasks. The sequential tasks include retrieving data from the database and conducting data processing, while the parallel tasks include training, prediction, and validation tasks.
COVID-19 patient health monitoring workflow example
Security threats
Different security issues in a cloud-based infrastructure were addressed in the literature, examples of breaches include for instance insider attacks, data loss, and DoS attacks. In this section, we focus on anomaly detection in a cloud workflow orchestration setting. In such an environment, an attack could target different entities and components including workflow data, tasks, resources, monitoring, and adaptation components. In what follows, we describe a few examples of security breaches in a cloud workflow.
Cloud workflow data attack
Some examples of data attacks involve data injection attacks that intend to corrupt the dataset or compromise it through, for example, suspicious sharing or downloads. Other anomalies include unauthorized data access and anomalous admin user activities. For example, in our cloud workflow, the attacker may inject redundant and fabricated data into the workflow to tamper with training and prediction processes which will affect the quality of the prediction model and may cause critical problems (e.g. patient death) or overburden the ML training process, thereby falsely activating Quality of Service (QoS) degradation and triggering unnecessary workflow adaptation.
Cloud workflow task attack
Cloud workflow is composed of many different tasks that can run in parallel or sequentially with different dependency levels. Workflow task attacks include a wide range of different anomalies including malware infection, query injection, and DoS. Furthermore, an attacker can maximize damage, by targeting sensitive processes or tasks (e.g., tasks on which many other tasks depend).
Resource attack
Resources such as cloud VMs, CPUs, memory, and networks can also be the target of different types of attacks, including unauthorized resource access, or overwhelming service requests. Such attacks could arise by falsely reporting resource overload/overutilization in monitoring logs, which will cause the compromised node to trigger unnecessary and costly workflow adaptation processes.
Monitoring and adaptation component attack
Monitoring and adaptation component attacks are very crucial in any cloud workflow orchestration environment because these components are crucial to resource management and performance optimization. In this workflow example, an attack against a monitoring system can force the compromised monitoring task to generate false resource underutilization logs, to avoid necessary adaptation and thus, causing performance degradation leading to a DoS. Another example of such an attack is automatic system reconfiguration which can cause a compromised node to falsely identify a problem and trigger unnecessary adaptation actions.
The aforementioned attack types negatively impact the performance and integrity of a cloud workflow orchestration system. In this work, we focus on anomaly detection in cloud workflow data, cloud resources, tasks, and monitoring components. Hence, we propose to monitor resources, such as utilization of CPU, memory, I/O, and network, as well as task profile, and task performance. In the following section, we present our proposed security enforcement for cloud workflow orchestration.
End-to-end security enforcement in cloud workflow orchestration
In this section, we design and describe our end-to-end security enforcement architecture as depicted in Fig. 2. It consists of two main modules: a workflow deployment module and a security enforcement module. Both modules use the underlying processing and storage resources (e.g., VMs, GPUs, Storage) from the cloud infrastructure to dispatch various storage and processing tasks. Security enforcement events implemented within our architecture are applied to four main entities: the user, resources, workflow tasks, and data.
An architecture for security enforcement in cloud workflow orchestration
In the following, we describe each component of the architecture in detail and highlight the security features that enhance security, data integrity, and authentication.
Entities interact with the two modules of the architecture to ensure various security boundaries including authentication and identity management for users interacting with the architecture, access and connectivity management of the employed resources, security enforcement of cloud workflow tasks, and workflow data access and integrity.
Workflow deployment module
This module involves two sub-components, the workflow specification, and the workflow deployment components. The workflow specification component builds the functional and non-functional (quality and security) requirements of the workflow and creates profiles for entities, such as tasks, data, and resources. The workflow deployment component manages the workflow deployment and execution lifecycle over the cloud infrastructure. The output of this module is a running workflow monitored by the security enhancement module to detect and/or predict encountered security threats and adopt the necessary adaptation action to mitigate it.
Security enforcement module
This module is composed of three sub-components: monitoring, Multi-Modal Deep Learning Autoencoder (MMDLA) based prediction and adaptation sub-modules. These sub-modules mutually interact to achieve a complete scenario of cloud workflow monitoring, anomaly detection, and prediction. Finally, these submodules apply an adaptation strategy to mitigate risks identified through various anomaly evaluations.
Monitoring sub-module
This submodule is responsible for continuous data collection and monitoring. Various runtime data/logs are collected from monitored entities including tasks, data, and resources. The collected data are used for the training, and prediction purposes and are stored in a historical database for further analysis.
MMDLA sub-module
This module uses the data collected from the monitoring submodule as an input to train a multi-modal deep learning autoencoder model for dimension reduction and trains a profile matching classification model using the dimensionally reduced data to predict anomalies. Training of the MMDLA model employs a combination of the input data generated from the entities profiling module (static) and monitoring time-series real-time logs data (dynamic). The resultant MMDLA model reduces the data dimension to increase efficiency and efficacy and provides reduced dimensional data as input to an anomaly detection ML algorithm to accomplish anomaly detection. If an anomaly is detected, then anomaly evaluation is performed to determine the type and threat level of the detected anomaly. Then, the anomaly evaluation information is passed as input to the risk estimation process and eventually stored in a database for expert validation (e.g., to identify suspicious user behavior). A detailed description and implementation of the key component's features of this module is given in subsequent sections.
Adaptation sub-module
This submodule implements adaptation strategies to proactively react to security threats before they occur and propagate. This begins by estimating the risk of each anomaly detected/predicted by the previous module to ultimately apply a mitigation strategy which may involve a redeployment of the cloud workflow to handle the employed adaptations. Such an adaptation may involve securing access to cloud workflow resource execution, guaranteeing legitimate additional resource allocation or deallocation, and terminating compromised tasks.
This serves the architecture requirements in terms of the various resources needed to process and store data. Processing tasks include MMDLA model training for dimension reduction, anomaly detection model training and classification, and data storage monitoring.
Cloud workflow security enforcement module
In this section, we detail the working principle of the MMDLA prediction-based security enforcement module. First, we define essential terms in understanding the prediction model, and then we discuss problem formulation. Finally, we describe the learning pipelines algorithms used for the solution approach.
Definition 1 (Task) A task \(\mathcal{T}\) is an operational unit consisting of one or more instructions, and can be dependent on one or more other tasks. Each task \({\mathcal{T}}_i\) runs on a designated container \({\mathcal{C}}_i\) such as a virtual machine.
Definition 2( Workflow) A workflow \(\mathcal{W}\) is a collection of tasks { \({\mathcal{T}}_1\), ..., \({\mathcal{T}}_n\) } performed according to a schedule \(\mathcal{S}\) toward achieving a specific work (e.g., patient classification).
Definition 3 (Task Profile) A Task profile \({\mathcal{P}}_i\) of a task \({\mathcal{T}}_i\) is the tuple (δ i , \({\mathcal{R}}_i\) ) where δ i is the unique id of the task and \({\mathcal{R}}_i\) is the task runtime data, to be defined next.
Definition 4 (Task runtime data) Task runtime data \({\mathcal{R}}_i\) of task \({\mathcal{T}}_i\) consists of both static and dynamic runtime data, which can be represented as a tuple (D, λ, η, μ, Θ) i. The static runtime data are composed of first four items of the tuple, namely:
Di: The task duration in seconds.
λi: The task category (e.g. preprocessing, training, evaluation).
ηi: The input size in bytes.
μi: The output size in bytes.
The dynamic runtime data Θi is a multivariate time-series data produced by a task monitoring system for task \({\mathcal{T}}_i\) which consists of periodical observation of six different runtime parameters, namely, CPU utilization, memory consumption, network input, network output, disk read, and disk write. Therefore, Θi can be defined as the tuple (Π, \(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{B}\), \(\mathcal{D}\), \(\mathcal{E}\)) i, as explained below.
Each observation of dynamic runtime data is performed every τ seconds (a system parameter). Therefore, the total number of such observations for \({\mathcal{T}}_i\) is
$${k}_i=\frac{D_i}{\tau }$$
where Di is the task duration as mentioned above. The six time-series variables are as follows:
Πi: The CPU utilization observations, which generate a time-series data such that, Πi={πi[1], ... πi[ki]}, where πi[j] is the j-th observation of CPU utilization for task 𝒯𝑖.
ℳ𝑖: The series of memory usage observations performed every τ seconds, Mi = { mi[1],..., mi[ki]}.
𝒜𝑖: The series of (cumulative) network input volume (in KB) observations for Container \(\mathcal{C}_{\boldsymbol{i}}\) performed every τ seconds, \(\mathcal{A}_{\boldsymbol{i}}=\left\{{\boldsymbol{\alpha}}_{\boldsymbol{i}}\left[\textbf{1}\right],\dots, {\boldsymbol{\alpha}}_{\boldsymbol{i}}\left[{\boldsymbol{k}}_{\boldsymbol{i}}\right]\right\}\)
\(\mathcal{B}_{\boldsymbol{i}}\): The series of (cumulative) network output volume (in KB) observations for Container \(\mathcal{C}_{\boldsymbol{i}}\) performed every τ seconds, \(\mathcal{B}_{\boldsymbol{i}}=\left\{{\boldsymbol{\beta}}_{\boldsymbol{i}}\left[\textbf{1}\right],\dots, {\boldsymbol{\beta}}_{\boldsymbol{i}}\left[{\boldsymbol{k}}_{\boldsymbol{i}}\right]\right\}\)
\(\mathcal{D}_{\boldsymbol{i}}\): The series of (cumulative) disk read volume (in KB) observations for Container \(\mathcal{C}_{\boldsymbol{i}}\) performed every τ seconds, \(\mathcal{D}_{\boldsymbol{i}}=\left\{{\boldsymbol{d}}_{\boldsymbol{i}}\left[\textbf{1}\right],\dots, {\boldsymbol{d}}_{\boldsymbol{i}}\left[{\boldsymbol{k}}_{\boldsymbol{i}}\right]\right\}\)
\(\mathcal{E}_{\boldsymbol{i}}\): The series of (cumulative) disk write volume (in KB) observations for Container \(\mathcal{C}_{\boldsymbol{i}}\) performed every τ seconds, \(\mathcal{E}_{\boldsymbol{i}}=\left\{{\boldsymbol{e}}_{\boldsymbol{i}}\left[\textbf{1}\right],\dots, {\boldsymbol{e}}_{\boldsymbol{i}}\left[{\boldsymbol{k}}_{\boldsymbol{i}}\right]\right\}\)
Therefore, dynamic runtime data Θi can be expressed as the following two-dimensional matrix:
$${\Theta}_i=\left\{{\Pi}_i,{\mathcal{M}}_i,{\mathcal{A}}_i,{\mathcal{B}}_i,{\mathcal{D}}_i,{\mathcal{E}}_i\right\}=\left(\begin{array}{llllll}{\pi}_i\left[1\right]& {m}_i\left[1\right]& {\alpha}_i\left[1\right]& {\beta}_i\left[1\right]& {d}_i\left[1\right]& {e}_i\left[1\right]\\ {}{\pi}_i\left[2\right]& {m}_i\left[1\right]& {\alpha}_i\left[2\right]& {\beta}_i\left[2\right]& {d}_i\left[2\right]& {e}_i\left[2\right]\\ {}\dots & \dots & \dots & \dots & \dots & \dots \\ {}{\pi}_i\left[{k}_i\right]& {m}_i\left[1\right]& {\alpha}_i\left[{k}_i\right]& {\beta}_i\left[{k}_i\right]& {d}_i\left[{k}_i\right]& {e}_i\left[{k}_i\right]\\ {}& & & & & \end{array}\right)$$
Problem formulation
Let \(R=\left\{{\mathcal{R}}_1,\dots {\mathcal{R}}_i,\dots, {\mathcal{R}}_n\right\}\) be the set of all task runtime information under a normal scenario, i.e., all runtime scenarios without any attacks. We assume that any attack would cause at least one running task \({\mathcal{T}}^{\prime }\) to behave in a manner that would generate the corresponding task runtime information \({\mathcal{R}}^{\prime }\) such that
$${\mathcal{R}}^{\prime}\notin R$$
Therefore, relation 3 is a necessary and sufficient condition for \({\mathcal{T}}^{\prime }\) being affected by an attack. So, the problem is to learn a model \(\mathcal{H}\left(\mathcal{R}\right)\) that will predict whether any given task runtime information \(\mathcal{R}\) has been generated by a task affected by an attack. Formally, the model \(\mathcal{H}\), given input \(\mathcal{R}\), outputs true or false such that:
$$\mathcal{H}\left(\mathcal{R}\right)=\left\{\begin{array}{l} true, if\ \mathcal{R}\notin R\\ {} false, otherwise\end{array}\right.$$
In other words, \(\mathcal{H}\left(\mathcal{R}\right)\) will hold true if and only if \(\mathcal{R}\) belongs to a task affected by an attack.
Solution approach
To learn model \(\mathcal{H}\) according to condition (4) above, we must train \(\mathcal{H}\) with the generalized description of R, i.e., the set of all possible task runtime information generated by tasks not affected by any attack. Here, we employ an unsupervised technique for training, where we attempt to learn \(\mathcal{H}\) from a subset of R, i.e., the set of all normal task runtime information. We collect the normal data from workflows running under normal scenarios, i.e., scenarios known to have no attacks. This data is then used to learn the desired model using one-class classifier learning techniques, including one-class SVM, isolation forest, elliptic envelope, and local outlier factor. We also use different clustering algorithms to learn clusters or normal data.
Learning pipeline algorithms and descriptions
The learning process requires several steps in the learning pipeline, namely, monitoring data collection from logs, feature extraction and feature vector generation, feature dimension reduction, training, and classification. The following subsections describe these processes in detail.
Monitoring data collection
For each workflow, logs are generated by the task monitor of each container \({\mathcal{C}}_i\) of task \({\mathcal{T}}_i\); these logs are collected and processed for training. The logs are primarily represented in an unstructured text format, which must be processed and converted into a structured format.
Feature extraction and feature vector generation
The processed logs are then used to extract the task profile, which includes the task id, static runtime data and dynamic runtime data, as explained above. The extracted task profiles are then used to generate two types of feature vectors for each task.
The static feature vector Si = (D, λ, η, μ) i consists of the static runtime data of task \({\mathcal{T}}_i\), and the dynamic feature vector, (i.e., the feature matrix) is essentially the dynamic runtime data of \({\mathcal{T}}_i\), i.e., Θi. Therefore, the combined feature vector for task \({\mathcal{T}}_i\) is essentially the task runtime data Ri, as defined previously.
A training dataset Xtrain is built by collecting task runtime information from n tasks. In other words, \({X}_{train}={\cup}_{j=1}^n\left\{{\mathcal{T}}_j\right\}\). Therefore, the feature extraction process generates the training feature vector Rtrain, consisting of the feature vectors of all tasks in Xtrain, i.e.,
$${R}_{train}={\left\{{R}_1,\dots, {R}_n\right\}}^T$$
Where Ri is the feature vector of \({\mathcal{T}}_i\). Recall that Ri consists of two types of feature vectors, namely static feature vector (one-dimensional) Si which is duplicated to be concatenated with each row in the dynamic feature vector (2D matrix) Θi. Thus, we can represent Rtrain as a concatenation of two matrices:
$${R}_{train}=\left({S}_{train}\right)\left({\Theta}_{train}\right)={\left\{{S}_1,\dots, {S}_n\right\}}^T{\left\{{\Theta}_1,\dots, {\Theta}_n\right\}}^T=\left(\begin{array}{ll}{S}_1& {\Theta}_1\\ {}\dots & \dots \\ {}{S}_n& {\Theta}_n\\ {}& \end{array}\right)$$
Feature reduction using deep autoencoder
As discussed previously, the feature vector for each task consists of four static features and six time-series features. To train a model that learns from two-dimensional feature vectors, we need to flatten the time-series feature matrix to a one-dimensional feature vector and combine it with the static features. However, this can cause the formation of a very high-dimensional feature vector. In particular, the total features in the feature vector would be 4 + 6 ki, where ki is the number of observations of the dynamic features. For example, if ki = 100, the total number of flattened features would be 604. Therefore, we must adopt a feature reduction technique. Here, we reduce the number of features using an unsupervised deep learning technique called AutoEncoder [41]. Although there are many alternative feature reduction or feature selection techniques, we employ the AutoEncoder technique for two main reasons:
First, AutoEncoder can perform unsupervised feature reduction, which is an important aspect of our proposed model.
Second, we propose multi-modal deep learning (MMDLA) based AutoEncoder model by combining long short term memory (LSTM) (a specific type of recurrent neural network (RNN)) [42] with a Deep Feed Forward network (DFN). This MMDLA model facilitates the feature reduction process to learn from the temporal relationships among time-series features and combine it with static features, rather than implementing a feature reduction process that flattens all the time-series features and loses the temporal information contained in the feature set.
Figure 3 shows the high-level architecture of this feature reduction, training, and prediction technique.
High level diagram of the proposed approach
Here, we describe the proposed AutoEncoder-based model which will be referred to as MMDLA. It consists of two main components, namely, the Encoder, and the Decoder in detail. The Encoder consists of two LSTM layers, a concatenation layer, and three fully connected layers as shown in Fig. 3. The purpose of the Encoder is to take feature vector Rtrain as input, then output a reduced dimensional feature vector (also known as embedding) Rϵ.
The Decoder has a network concept similar to that of the Encoder. The purpose of the Decoder is to take the embedding Rϵ as input and reconstruct the original feature vector. Thus, the output of the decoder is R′train = (S′train)(Θ′train), such that the matrix dimensions of (S′train) and (Θ′train) are the same as those of (Strain) and (Θtrain), respectively. Essentially, (S′train) and (Θ′train) are approximations of (Strain) and (Θtrain), respectively. Therefore, the learning objective of the AutoEncoder is to minimize the loss, i.e., the difference between the input and reconstructed output. Therefore, the AutoEncoder loss \(\mathcal{L}\) can be represented as the sum of the loss of the static runtime data (\({\mathcal{L}}_{stat}\)) and dynamic runtime data (\({\mathcal{L}}_{dyn}\)):
$$\mathcal{L}={\mathcal{L}}_{stat}+{\mathcal{L}}_{dyn}={\Sigma}_{i=1}^n{\left({S}_i-S{\prime}_i\right)}^2+{\Sigma}_{i=1}^n{\left({\Theta}_i-\Theta {\prime}_i\right)}^2$$
After training the AutoEncoder model, we take the reduced dimensional feature vector, i.e., embedding Rϵ as the new feature vector and train an unsupervised anomaly detection model (e.g., one class classifier).
Classification and prediction
The embedded feature vector Rϵ is used to train an anomaly detection model \(\mathcal{H}\) as expressed in equation 4. The learning algorithm is assumed to be one-class classifier training or unsupervised clustering that only requires normal data for training. Once the clustering or one-class classifier model is trained, it is deployed in the system to detect (i.e., predict) anomalous task runtime data, supposedly generated from tasks affected by an attack.
In this section, we present the algorithms for the learning and prediction processes of the security enforcement model. Algorithm 1 describes the training pipeline of the anomaly detection model. The input to this algorithm is the training data. First, lines 1–4 retrieve monitoring log data, extract features, and generate the feature vector. Then we train the AutoEncoder (lines 6–7). In line 8, we obtain the embedding of the training data from the AutoEncoder, and in line 9, an anomaly detection model is trained with this reduced feature vector.
Algorithm 1 Security Enforcement Model Training (Xtrain)
Algorithm 2 requires three inputs, namely, the task to examine, prediction model \(\mathcal{H}\), and the AutoEncoder model AE. First, we extract features and generate a feature vector from the logs. Line 3 applies the embedding on the task runtime data to obtain a reduced feature vector. Finally, anomaly detection model \(\mathcal{H}\) predicts whether the runtime data is generated by a task affected by an attack.
Algorithm 2 Attack prediction (𝒯𝑖,𝓗,AE)
Adaptation scheme
The monitoring process is performed continuously for all running cloud workflows. The monitoring logs are collected periodically, and the status of all cloud workflows is checked for anomalies or other quality performance issues such as performance degradation and resource over/under utilization. For each running cloud workflow, we inspect its tasks monitoring logs by running the attack prediction algorithm depicted in Section 5.5. Once an attack is detected, we apply the appropriate mitigation strategy including task restart, workflow restart, and reverting to former logs depending on the outcome of the risk estimation process. The risk estimation process evaluates the status of the workflow and cause of the anomaly and recommends mitigation action to prevent or override the attack. First, it checks the anomaly type and the task status, then recommends a set of actions according to the following rules. If the anomaly type is resource over-utilization, additional resources is allocated to the workflow. Otherwise, if the anomaly type is under-utilization, then a resource can be released. Different anomaly types are handled by the risk estimation process according to the predefined rules. The set of recommended actions can also be applied to the tasks in the task dependency list. For example, if a task was attacked, the task dependencies list is checked to decide whether other dependent tasks should be also restarted along with the task under attack. Otherwise, if no attack is detected, the performance recorded values of attributes are checked and if they do not satisfy the required quality thresholds, adaptation actions are applied (e.g., adding a new node if CPUs are over-utilized). Algorithm 3 presents the cloud workflow adaptation after an anomaly is detected. This algorithm takes as input the list of currently running cloud workflows, the anomaly detection model, the trained AutoEncoder, the collected monitoring logs, the desired/acceptable ranges for each performance quality feature, and the list of possible adaptation actions that will maintain the required workflow QoS levels. First, the algorithm applies an anomaly prediction model to each task in the workflow. When an anomaly is detected, a mitigation strategy is applied as explained in Section 4. Otherwise, if monitoring logs show out-of-range values, the regular adaptation mechanisms are applied.
Algorithm 3 Cloud Workflow adaptation with anomaly detection
Implementation and experiment
In this section, we describe the experimental environment. We created a Docker Swarm Cluster comprising one master node and four worker nodes. We deployed the cloud workflow described in Section 3 over a workstation running Linux Ubuntu 18.04 with 24 CPU cores and two NVIDIA GeForce GTX 1080 Ti GPUs with 11 GB GDDR5X memory each, a 1-TB HDD, and 64-GB RAM. Each task in the cloud workflow was created as a Docker container executed using different data input sizes. The Docker swarm cluster had a master node, that performed the orchestration to conserve the required cluster state. The worker nodes received and ran tasks dispatched from the master node. Deploying a workflow to a swarm requires providing service definition to the master node, which accordingly dispatches units of work, called tasks, to the worker nodes. During workflow execution, we collected a live data stream to run task containers to monitor various performance metrics, which are discussed in detail in the following section. Additionally, we ran other mock containers to overload nodes in the cluster to simulate a real environment. The experimental environment is depicted in Fig. 4.
Experimentation environment
We implemented the proposed algorithms in Jupyter Notebook running Python 3.6. The AutoEncoder and SVM algorithms were developed using Pytorch and Scikit-learn, which are open-source Python implementations of machine learning and deep learning neural networks. The experiments were executed on a Mac computer with OS X Catalina 10.15.4 operating system with a 2.8-GHz Quad-Core Intel Core i7 and 16-GB 1600-MHz DDR3 RAM.
In the experiments, we combined two types of data for each running task. We initially defined a static task profile to include various types of information such as task duration, data input size, data output size, and task category (pre-processing, training, or evaluation). The dynamic data comprised live stream data of performance monitoring metrics for each running task. This data consists of time-series records which include the CPU and memory usage by the container, total memory used by the container, size of data sent and received by the container over the underlying network, and the size of read/write data by the container from block devices on the host.
Data preparation and preprocessing
The preprocessing activities primarily focused on converting Docker's generated monitoring statistics. First, the Docker stats were given a format flag to output the exact required container statistics. The output file was then parsed and cleaned using regex to split the column headers appropriately. The data was then converted to a Pandas DataFrame and proper datatypes were assigned to each column (e.g., timestamp column used the datetime datatype). Additionally, the units of the memory utilization columns were all standardized to Bytes.
Deep learning approach for training and anomaly detection
To detect anomalies, we first trained our dataset using a reconstruction AutoEncoder model to reduce the data dimension into a 30-D of embeddings. Afterwards, we input the AutoEncoder model generated output into an anomaly detection model. The following sequence of steps details our implementation: First, we split the dataset into two sets; static profile data and dynamic time-series performance monitoring information. Figure 3 shows the architecture of the encoder-decoder neural network developed for feature learning. The dynamic part of the data is fed into two-layers of a time-series RNN model encoder. This model takes batch size, number of records, and number of features as inputs and returns outputs in the form of a (1, 30) vector which is the final hidden state. The output is concatenated with the static data portion which is fed into three fully connected layers to produce the output shape of a (1, 30) vector. The decoder, on the other hand, uses the (1, 30) vector and passes it to two separate layer sequences, i.e., three fully connected layers and two RNN layers. The fully connected layers decode the static part of the input, while the RNN layers produce the dynamic time-series part. Here, a key aspect is that the encoder always provides the data input length such that the decoder knows how many time-series data points to produce.
The output of the encoder was trained over an anomaly detection model such as a one-class classification or clustering. The one-class classification algorithms are unsupervised learning algorithms that we trained using only non-anomaly data, i.e., the reduced feature set resulted from the aforementioned AutoEncoder algorithm, which can classify anomaly and non-anomaly data. These include one-class SVM, Isolation Forest, Elliptic Envelope, and Local Outlier Factor. In addition, we used different clustering algorithms, i.e., unsupervised learning algorithms trained using both anomaly and normal data. Among which we use k-means, Mini Batch k-means, Mean Shift, and Birch. All models predicted two classes or clusters, i.e., normal or anomaly. However, the performance of each model varied in terms of accuracy, precision, recall, and F1 score. We then selected the best performing model based on the calculated performance metrics for our real-time security enforcement.
Experimental scenarios, evaluation criteria, and fault injection scheme
We conducted several experiments to evaluate our proposed security enforcement and anomaly detection framework. In these experiments, we intended to evaluate the anomaly detection scheme by investigating the performance of different anomaly detection algorithms and models. In addition, we conducted different experiments to evaluate the performance of the cloud workflow within the adopted proposed security enforcement model. We benchmarked the cloud workflow performance based on the previously proposed adaptation strategies [43]. In these experiments, we ran our designed hospital length of stay prediction workflow several times with different patient dataset sizes. The performance of the cloud workflow was continuously monitored, and adaptation strategies were executed when necessary, depending on the decision taken by the adaptation module.
We designed two scenarios for testing the proposed security enforcement model. The first scenario focused on testing the performance and accuracy of our anomaly detection and prediction model, and the second scenario evaluated the overall performance of the cloud workflow. The first scenario was implemented in two stages: First, we used the Deep Learning AutoEncoder (Section 0) to reduce dimension of the dataset containing encodings, which were then fed to the anomaly detection module. The latter implements different ML algorithms, including one-class classification and clustering algorithms. Each algorithm was evaluated and compared in terms of four different performance measures including accuracy, precision, recall, and F1 scores after applying cross fold with k-fold values of 3, 5, and 10.
In the second scenario where we evaluated the overall cloud workflow performance, we considered the CPU utilization, memory usage, network I/O bound, and disk space usage features. The cloud workflow was executed over the implemented Docker swarm environment with different resource load capacities. We compared how the adaptation module behaved in response to the detected anomalies and the performance of the cloud workflow after applying automatic adaptation strategies to respond to anomaly detection with the performance of the cloud workflow without anomaly detection application.
For our AutoEncoder, we employed one of the commonly used time-series prediction models evaluation metrics, which is the Mean Square Error (MSE) defined by the following formula:
$$MSE=\frac{\sum_{t=1}^n{\left({y}_{pt}-{y}_t\right)}^2}{n}$$
where ypt is the predicted value at time t, yt is the actual value at time t, and n is the number of observations [44].
To further evaluate and compare our anomaly prediction models including one-class classification and clustering, we adopted different evaluation criteria including accuracy as the most intuitive measure. However, in some cases, accuracy is not always the best measure for assessing the model performance. Henceforth, we used precision, recall, and F1 score to compare and select the best prediction performance model. Precision is also known as the positive predictive value, which is the ratio of correctly predicted values to the total number of predicted values. Additionally, recall is referred to as the sensitivity measure and it is defined as the ratio of correctly predicted values to the number of correctly predicted values. Moreover, we have used F1 score, which is defined as the weighted average of precision and recall [45]. These common measures well represent the overall performance of our prediction models.
Furthermore, in our experimentations, we define precision as the ratio of the number of correctly predicted anomalies to the total number of correctly predicted anomalies and the normal incorrectly identified as anomalies. We also express recall as the number of anomalies correctly identified over the total number of correctly predicted anomalies and anomalies incorrectly predicted as normal. In addition, F1 score is defined as the weighted average of precision and recall. This is given by the following formulas:
$$Recall=\frac{true\ positives}{true\ positives+ false\ negatives}=\frac{correctly\ predicted\ anomalies}{correctly\ predicted\ anomalies+ anomalies\ incorrectly\ predicted\ as\ normal}$$
$$Precision=\frac{true\ positive s}{true\ positive s+ false\ positive}=\frac{correctly\ predicted\ anomalies}{correctly\ predicted\ anomalies+ normal\ data\ incorrectly\ predicted\ as\ anomalies}$$
$$F1\ Score=\frac{2\ast \left( Recall\ast Precision\right)}{Recall+ Precision}$$
We adopted these measurements for the obtained results to further validate our model.
Anomaly injection techniques
To facilitate the testing and evaluation of our anomaly detection model in consideration of various anomalies, we employed simulation-based fault injection to inject anomalous behaviors in the cloud workflow task as well as injecting false values into the monitoring log files. Existing techniques in software fault injection include runtime injections and compile-time injection [46]. Here, we adopted runtime fault injection techniques such as code insertion to simulate system stress. In this approach, we synthesized and injected different types of anomalies such as code-modification which implements fault injection during runtime and adds instructions to increase the task execution time (e.g., adding infinite loops or time delays). Faults were randomly injected in different task instances to trigger higher CPU consumption and memory usage. The objective of these anomalies was to simulate cloud workflow task attacks and cloud resources attacks. Furthermore, we simulated monitoring component attacks by injecting anomalies into the monitoring logs. These faults included heavy or light CPU utilization, memory usage, disk I/O access, and network latency which were randomly generated to synthesize log anomalies [47]. Then the behavior of the adaptation model under stress was tested to ensure the reliability and overall performance of our proposed model.
Deep learning AutoEncoder model evaluation
In the first stage of anomaly detection, we applied deep learning with AutoEncoder to generate a reduced dimension embedding which served as an input to the anomaly detection algorithm in the second stage. We trained the AutoEncoder with normal data generated by monitoring the execution of the target case study cloud workflow. Subsequently, we selected the model that minimized the reconstruction error in the original AutoEncoder. Here, to determine embedding size, we measured the average loss while using different embedding vector dimensions during the AutoEncoder training phase. The experimental results depicted in Fig. 5 demonstrate that the average AutoEncoder reconstruction loss was reduced with higher embedding dimensions. Thus, we set the dimension of the output embedding to 30 because this provided the smallest loss value. Although higher dimension values provide slightly better loss, we set the encoder generated embedding vector size to 30 embeddings because the main objective was to reduce the dimensionality of the original dataset, which generally leads to improved accuracy. Figure 6 illustrates the AutoEncoder reconstruction loss values based on MSE while generating (1X30) vector embeddings.
Average loss versus embedding dimensions
AutoEncoder reconstruction loss
Anomaly detection model evaluation
The main objective of this experiment was to evaluate the performance of each ML anomaly detection algorithm and select the model best suited for our dataset. We detected and predicted the anomalies in our dataset which comprised the collected cloud workflow monitoring time-series log files and the static task profile dataset. We executed the cloud workflow with normal environment settings to produce a regular dataset under the true positive conditions. Moreover, we synthesized the dataset to reflect different types of anomalies and attacks, such as task, log, or resource anomalies. For example, a task anomaly could alter a task's behavior by increasing or reducing processing time. Whereas a log anomaly could be instantiated by injecting the monitoring logs with contradicting statistics. Furthermore, the resource anomaly included simulation of heavy load exertion on the CPU and memory resources allocated to service the cloud workflow. Here, the total number of records in both the regular and anomaly dataset was 1200 records.
We selected two main ML techniques for anomaly detection: one-class classification and clustering. For one-class classification, we compared the performance of the SVM, Isolation Forest, Elliptic Envelope, and Local Outlier Factor, each of which was subject to substantial hyperparameter tuning. For example, we ran over 800 different combinations of hyperparameter values to automatically tune the SVM model, which is discussed in the following section. All one-class classification models were trained using all regular dataset and tested with a dataset including 50% regular and 50% anomaly. On the other hand, for clustering, we evaluated k-means, Mini Batch k-means, Mean Shift, and Birch algorithms on our dataset. We trained and tested the clustering models using a dataset with a 50% regular and 50% anomaly data. Here, we adopted k-fold cross-validation to evaluate the models including classification and clustering. We applied 3-fold, 5-fold, and 10-fold cross-validation. In the following, we present our testing results.
One-class SVM model tuning
In this experiment, we investigated the effect of hyperparameter tuning on the performance of the one-class SVM model. We automated the hyperparameter tuning process to quickly select the best parameter combination that gave the best accuracy. The main hyperparameters that provide the best accuracy include nu = 0.01, gamma = 0.1, tolerance = 0.001, coefficient = 0, kernel cache size = 200, and degree (for poly) = 3. In addition, kernel selection has the greatest effect on accuracy improvement. Figure 7 depicts the effect of different kernel parameter adoption on the accuracy using 3-fold, 5-fold, and 10-fold cross-validation. As can be seen, the RBF kernel provided the best accuracy value over sigmoid, linear, and polynomial kernels.
One-class classification models evaluation
One-class SVM tuning and accuracy
In these experiments, we compared the performance of four one-class anomaly detection classification methods including SVM, Isolation Forest, Elliptic Envelope, and Local Outlier Factor. Figure 8 depicts the performance of each algorithm in terms of accuracy, precision, recall, and F1 score. As shown, the Isolation Forest technique obtained the highest accuracy of 96.14, precision of 0.93, recall of 0.99, and F1 score of 0.96 using 10-Fold cross validation. Similar results were obtained when using 3-fold, and 5-fold cross-validation indicating that the Isolation Forest algorithm outperformed the other one-class classification algorithms.
Clustering model evaluation
One-class classification performance evaluation
Additionally, we measured the performance of different clustering algorithms namely k-means, Mini Batch k-means, Mean Shift, and Birch. Here, we trained all clustering algorithms to generate two clusters, i.e., one for regular data and another one for anomaly data. Likewise, we evaluated these algorithms with respect to accuracy, precision, recall, and F1 score while performing 3-fold, 5-fold, and 10-fold clustering as shown in Fig. 9. Generally, the k-means algorithm obtained the best results, demonstrating an accuracy of 96.43, precision of 0.94, Recall of 0.99, and F1 score of 0.96 using 10-fold cross-validation, and similar results were obtained with 3-fold and 5-fold cross-validation.
Overall discussion
Clustering performance evaluation
We adopted two approaches for anomaly detection and prediction for security enforcement during cloud workflow execution: one-class classification and clustering. Experimental results demonstrate that clustering provided slightly better performance in terms of accuracy, precision, recall, and F1 scores over one-class classification. The k-means technique outperformed all other clustering algorithms. However, the isolation forest provided the best prediction performance among one-class classification algorithms and gave results that were very close to those of clustering. Considering that one-class classification training is performed using only regular data which is more likely to be the real case scenario for our cloud workflows execution rather than training with 50%:50% regular to anomaly data ratio, therefore, we recommend one-class classification specifically the Isolation Forest. Table 2 gives the anomaly detection performance results.
Table 2 Performance evaluation results of anomaly detection using various Machine Learning Algorithms
Overall cloud workflow performance evaluation
In this section, we evaluate the overall performance of the system when using an anomaly detection approach for security enforcement over the normal adaptation strategies with no anomaly detection.
We monitored CPU utilization and memory usage of cloud workflow tasks executed over multiple nodes in the cluster. Different tasks present different utilization levels according to the nature of the task as defined by its profile. For example, a preprocessing task unitizes more CPU and memory resources than an evaluation task because preprocessing requires iterating through the entire dataset to clean and prepare the data used for training the ML model. In what follows, two experimental scenarios are discussed to demonstrate the performance evaluation of the cloud workflow.
In the first scenario, we executed our cloud workflow while adopting regular quality enforcement adaptation strategies [48]. As illustrated in Fig. 10, the memory and CPU resources required to process the workflow increased over time which involved an adaptation action to add a new node after detecting that the sudden increase in resource usage was caused by an anomaly attack. In this experiment, we synthesized the log anomaly described in Section 3.2, which deceived the adaptation system, thereby resulting in unnecessary addition of resources to maintain the quality of the cloud workflow performance.
CPU utilization and memory usage during an anomaly attack
In the second scenario, we executed the cloud workflow while embracing our new proposed security enforcement extension. Figure 11, shows that the security enforcement module detected the anomaly in task 1, thereby causing it to discard the corrupted logs and issue an action to use an older version of the logs. This action prevented the adaptation module from adding unnecessary resources.
CPU utilization and memory usage with anomaly attack detection
Security enforcement in cloud workflow orchestration is considered a complex research problem because of its dynamicity and changing cloud workflow execution environments. In this paper, we have proposed an architecture for cloud workflow security enforcement. The proposed architecture is applied to four main entities: the user, resources, workflow tasks, and data. A multi-modal approach incorporating deep learning, one-class classification, and clustering applied to training, anomaly detection, and prediction has also been proposed. The proposed model considers both unsupervised static and dynamic features which is a unique way of modeling features that results in better anomaly detection. It also reduces the data dimensionality which leads to better characterization of workflow tasks and thus provides a better attack prediction. Once anomalies are detected and/or predicted, adaptation measures are implemented to secure the cloud workflow execution and ensure performance. The adaptation scheme accommodates a flexible representation and planning of resource requirements over time and over the various phases of the cloud workflow execution cycle.
We conducted a set of experiments to evaluate the various features of our solutions including the application of Multi-Modal training and anomaly detection using a real COVID-19 dataset of patient health records. The proposed Multi-modal approach was formulated and tested in an experimental setup where two main scenarios were used for verification. The first scenario focused on testing the performance and accuracy of our AutoEncoder and anomaly detection model, while the second scenario was used to evaluate the overall cloud workflow performance by assessing adaptation actions taken to respond to injected anomaly detection and their impact on the performance of cloud workflow execution. Two main approaches were adopted for anomaly detection and prediction of security enforcement during the execution of the proposed workflow, i.e., LSTM-based AutoEncoder and an ML model including one-class classification and clustering. The experimental results demonstrate that clustering provides slightly better performance in terms of accuracy, precision, recall, and F1 scores over the one-class classification with k-means outperforming other clustering algorithms. Other experimental results of the adaptation strategy implemented to respond to detected anomalies revealed a high execution performance of the workflow. The experimental results demonstrate that the proposed architecture prevents unnecessary wastage of resources due to anomaly detection and prediction.
We plan to explore other ML algorithms to detect and predict other categories of anomalies and attacks as future work. We also plan to explore ensemble ML and natural language processing algorithms to explore new levels of cloud workflow automation, robustness, and fault tolerance.
The dataset used in this study is not publicly available, however it can be provided up on request.
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D. M. Powers, (2020) "Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation," arXiv Prepr. arXiv2010.16061.
Hsueh M-C, Tsai TK, Iyer RK (1997) Fault injection techniques and tools. Computer (Long Beach Calif) 30(4):75–82
Sauvanaud C, Lazri K, Kaâniche M, Kanoun K (2016) Anomaly detection and root cause localization in virtual network functions. 2016 IEEE 27th International Symposium on Software Reliability Engineering (ISSRE). pp 196–206
El-Kassabi HT, Adel Serhani M, Dssouli R, Navaz AN (2019) Trust enforcement through self-adapting cloud workflow orchestration. Futur Gener Comput Syst 97:462–481. https://doi.org/10.1016/j.future.2019.03.004
This work is supported by both research Grant no: 31R227 and research Grant no: 31R180 from Zayed Center for Health Sciences.
Department of Computer Science and Software Engineering, Gina Cody School of Engineering and Computer Science, Concordia University, Montreal, Canada
Hadeel T. El-Kassabi
College of Computing and Informatics, Sharjah University, Sharjah, UAE
Mohamed Adel Serhani
College of Information Technology, UAEU, Al Ain, Abu Dhabi, UAE
Mohammad M. Masud & Khaled Shuaib
Faculty of Applied Science & Engineering, University of Toronto, Toronto, Ontario, Canada
Khaled Khalil
Mohammad M. Masud
Khaled Shuaib
Hadeel, conceived the main conceptual ideas related to security enforcement in cloud workflow orchestration architecture, literature, and overall implementation/execution of experimentation. Mohamed Adel contributed to the overall architecture of the proposed model, supervised the study, and oversaw overall direction and planning. Mohammad Masud developed the formal model and contributed to the design of the anomaly detection module. Khaled Shuaib contributed to the literature surveys, he provided inputs on overall design and implementation. Khaled Khalil pre-processed the dataset, and was involved in the deployment, and evaluation of the Cloud Workflow orchestration model. All authors contributed to the writing of the manuscript, and revision and proofreading of the final version of the manuscript. The author(s) read and approved the final manuscript.
Correspondence to Mohamed Adel Serhani.
All authors declare they have no competing interests, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
El-Kassabi, H.T., Serhani, M.A., Masud, M.M. et al. Deep learning approach to security enforcement in cloud workflow orchestration. J Cloud Comp 12, 10 (2023). https://doi.org/10.1186/s13677-022-00387-2
Cloud workflow | CommonCrawl |
Flat fading criterion of OFDM subcarrier spacing
I have always thought that the OFDM subcarrier spacing $\Delta f$ is chosen such that
not too small because Doppler spread can destroy subcarrier orthogonality
not too large to avoid Cyclic Prefix (CP) overhead because OFDM symbol period $T_u = 1/\Delta f$ needs to be much larger than $T_{CP}$ which, to avoid OFDM symbol ISI, must be larger than multipath delay spread $\tau_m$ which depends only on the given physical environment.
The second criterion implies that $T_u = 1/\Delta f \gg T_{CP} > \tau_m \implies \Delta f \ll 1/\tau_m \sim B_c$ where $B_c$ is coherence bandwidth, this is flat fading definition. I means that flat fading is a consequence of the design criterion, not a criterion itself.
But in this wikipedia article Wikipedia's Fading Article, it is said that
Since different frequency components of the signal are affected independently, it is highly unlikely that all parts of the signal will be simultaneously affected by a deep fade.Certain modulation schemes such as orthogonal frequency-division multiplexing (OFDM) and code division multiple access (CDMA) are well-suited to employing frequency diversity to provide robustness to fading. OFDM divides the wideband signal into many slowly modulated narrowband subcarriers, each exposed to flat fading rather than frequency selective fading.
It seems that flat fading is the design objective.
If a portion $B_c$-wide is in deep fade, all subcarriers in that $B_c$ are in deep fade, I am not sure that increasing diversity is the purpose. I mean : could 50 errors per 100 bits be better for channel coding than 5 error per 10 bits?
I understand that $\Delta f < B_c$ facilitates channel estimation in frequency domain by adding at least one pilot per $B_c$. Thus my question : is channel estimation the only reason to choose $\Delta f$ as small as possible ?
Now I look at the time domain, $\Delta f > B_c \implies T_u < \tau_m$. As being visualized in figure below, nothing bad happens. We can capture all the energy from all physical paths. Is there any explaination if we look only at the time domain?
Update : I am doing some math to see how the flat fading condition leads to single-tap equalizer.
Let $N$ the number of subcarrier, or FFT size (no zero subcarrier)
$T_s$ is sampling period $\implies N\Delta f = 1/T_s$ or $\Delta f T_s = 1/N$
The signal at TX after IFFT :
$$x(t) = \frac{1}{N}\sum\limits^{N-1}_{k=0} x[k]e^{j2\pi k \Delta f (t - N_{CP}T_s)}\text{ with }0 \leq t < (N_{CP} + N)$$
The received signal from $L$ multipath propagation:
$$y(nT_s) = \sum\limits^{L-1}_{l=0}h_l(n T_s)x(n T_s - l T_s)\text{ with }0 \leq n \leq N_{CP} + N-1$$
Take the part of FFT window as in the figure above, $N_{CP} \leq n \leq N_{CP} + N-1$. Set $m = n - N_{CP}$ and suppose that $h_l(t)$ is time-invariant in $0 \leq t < (N_{CP} + N)$ so that the time index of $h_l(t)$ can be dropped (this is the underspread assumption which is valid for typical channels) :
$$\begin{align} y[m] &= y(m T_s) \\ &= \sum\limits^{L-1}_{l=0}h_l \sum\limits^{N-1}_{k=0} x[k]e^{j2\pi k \Delta f (m - l)T_s}\\ y[m] &= \sum\limits^{L-1}_{l=0}h_l \frac{1}{N}\sum\limits^{N-1}_{k=0} x[k]e^{j2\pi k (m - l) / N} \end{align}$$
with $0 \leq m \leq N - 1$
Note that $T_s\Delta f = 1/N$.
Take FFT of $y[m]$:
$$\begin{align} z[k_0] &= \sum\limits^{N-1}_{m=0} y[m] e^{-j2\pi k_0 m / N} \\ &= \sum\limits^{N-1}_{k=0}x[k] \sum\limits^{L-1}_{l=0} h_l e^{-j2\pi k l/N} \frac{1}{N} \sum\limits^{N-1}_{m=0} e^{j2\pi (k-k_0)m/N} \end{align}$$
This orthogonal property assures that: $$R(k) = \frac{1}{N} \sum\limits^{N-1}_{m=0} e^{j2\pi (k-k_0)m/N} = \delta(k_0)$$
Thus $$z[k_0] = x[k_0] \times \sum\limits^{L-1}_{l=0} h_l e^{-j2\pi k_0 l/N} = x[k_0] \times H(k = k_0)\text{,}$$ where $H(k)$ is DFT of channel impulse reponse $(h_l, 0 \leq l \leq L-1)$ : is this the desired one-tap equalizer ?
Could someone tell me in which step I did use the condition flat fading $\Delta f < B_c \sim 1/\tau_m$ to come to single-tap equalizer model ?
ofdm delay fading-channel
Marcus Müller
AlexTPAlexTP
Your mathematical derivation is correct, your $H[k]$ is the single-tap equalizer (i.e. one tap for each subcarrier, and the subcarriers do not mix with each other. That's the orthogonal in OFDM).
Let me try to explain this a bit more general, without going into coherence bandwidth and flat fading. To my understanding, explaining it with $B_c$ and flat fading is a bit superficial and undergraduate explanation, such that you understand the idea, but can't really prove it works. Even, the coherence bandwidth is not strictly defined, how can one design a system according to that?
Assuming that the channel is time-invariant over all times, the channel is an LTI system. The eigenfunctions of LTI systems are complex exponentials (of all frequencies). Note that these complex exponential range over all times (i.e. they are infinitely long). So, in principle, one could transmit data on all frequencies (infinitely small apart), but one would have to wait infinitely long until the signals arrived. The principle bases on the convolution theorem, i.e. your channel in time-domain does a convolution, so in frequency domain, it's doing elementwise multiplication. That's the continuous-time principle of OFDM. Clearly, not very attractive for practical implementations.
What can you do? We've know that two complex exponentials of different frequency are orthogonal to each other when considering the whole time axis, i.e. $\int_\mathbb{R}\exp(j2\pi f_1 t)\exp(j2\pi f_2 t)dt=\delta(f_1-f_2)$. We want to have orthogonality between the signals, (because we want to have single-tap equalization), but we dont want to wait infinitely long. Unfortunately, the orthogonality between to arbitrary frequency complex exponentials is not given, when the time interval is shorter: $\int_T\exp(j2\pi f_1 t)\exp(j2\pi f_2 t)dt\neq\delta(f_1-f_2)$. However, there are frequencies $f_1, f_2$ such that the orthogonality holds, namely $\int_T\exp(j2\pi f_1 t)\exp(j2\pi (f_1+\frac{n}{T})t)dt=\delta_n$ (here $\delta_n$ is the Kronecker symbol, i.e. 1 for $n=0$, else zero.). I.e. two exponentials that are $n/T$ apart in frequency are orthogonal over a time interval of $T$.
So, in principle, we could use time-intervals of $T$ and use only frequencies that are $1/T$ apart from each other to transmit orthogonal signals. But, there's more. If the orthogonality should hold at the receiver, the received signal needs to be a linear combination of complex exponentials. We know, that complex exponentials are eigenfunctions to the LTI system, so everything should be fine, right? No! Only infinitely long complex exponentials are eigenfunctions. If we transmit $\exp(j2\pi f_1t)\text{rect}(t/T)$, we will not get a complex exponential at the output, so it wont work. So, what to do? Essentially, we'd have to transmit infinitely long complex exponentials with frequency distance $1/T$ to get signals at the receiver, which are orthogonal in the interval $T$. Hm, still doesn't sound very appealing.
Here's the solution: The impulse response of the LTI system (i.e. the channel) is usually not infinitely long, but has a length $T_C$ (at least, we approximate/assume that it's zero for $t>T_C$). If the channel has length $T_C$, then for a given point in time $t$, only the signal times from $t-T_C$ to $t$ have an influence onto the output signal at the current point in time. So, we dont need to transmit infinitely long complex exponentials, but just of length $T+T_C$, and we get orthogonality in the time interval $[T_C,T+T_C]$.
You see where this is leading to, right? The extra time in the beginning is what we denote as the Cyclic prefix. Why do we call it cyclic? Because, the exponentials with frequency distance $1/T$ are periodic with period $T$. So, the signal in time $[0,...,T_C]$ is exactly equal to the signal at $[T,...,T+T_C]$.
To summarize: Transmitting complex exponentials of duration $T+T_C$ over an LTI system with impulse duration $T_C$ yields orthogonal signals over the interval $[T_C,...,T_C+T]$. Orthogonal means single-tap equalization is possible. What is the coefficient for the equalizer? Due to the convolution theorem, it is just the value of the frequency response of the channel at the carrier frequency.
This is the time-continuous explanation, which can become intuitive. However, I personally like more the discrete-time explanation, using linear algebra.
Let $F$ be the N-point Fourier transform matrix, i.e. $FF^H=F^HF=I$. So, the transmitted signal without CP is $y=F^Hx$ where $x$ is the data. Adding a CP is done by $y_{cp}=Cy$ with $C$ being a $N+N_{CP}\times N$ matrix. Then, we the received signal becomes
$$z=HCF^Hx$$ where $H$ is the Toeplitz matrix that does the convolution (I ignore noise here). $z$ is of length $N+N_{CP}+L-1$, where $L$ is the channel length. Then, at the receiver, the CP is removed, by a matrix $D$ of dimension $N\times N+N_{CP}+L-1$, i.e.
$$w=DHCF^Hx$$
Finally, we go to the frequency domain:
$$W=FDHCF^Hx$$
Let's have a look at the $N\times N$ matrix $DHC$: It's a matrix that performs circular convolution with the impulse response of your channel. In finite-discrete time the Convolution theorem states, that circular convolution equals elementwise multiplication of the DFTs. So, $FDHCF^H$ is a diagonal matrix, containing the frequency response of the channel on the diagonal. There you have your single-tap equalizer again (inversion by diagonal matrix is just division by the diagonal).
Maximilian MatthéMaximilian Matthé
$\begingroup$ Reading through your excellent answer.. thanks for pointing out the orthogonality breakdown for non-integer-factor of symbol-duration oscillations! I was clearly oversimplifying in my "pro OFDM is only the efficiency of FFT" argument :) $\endgroup$ – Marcus Müller Apr 12 '17 at 9:13
There's a lot of very valid aspects that you touch, but from what I've learned (and experienced having fun with OFDM SDR transceivers), the main reasoning to do OFDM is having a narrow-channel multicarrier system with low complexity. Let me elaborate:
Multi-carrier is, as you mentioned, very handy because you get a flat channel. That means that you can do single-tap equalization – just bring the magnitude back to the right level and correct the phase shift. That's a multiplication with a single complex number.
OFDM is an especially efficiently implementable multicarrier system because of the FFT. That's all the magic to it – from a robustness against frequency offsets point of view, against Doppler spread and a lot of other aspects (sidelobes etc), a more general filterbank multicarrier system would be preferable. (You can interpret the DFT as sinc-filterbank, and you'll immediately see why the sidelobe energy gets problematic rather quickly if you just shift one carrier by $\frac1{10}\Delta f$).
So the point is not really whether you have 5 in 10 bits wrong or 50 in 100, the point is that not having that single bit wrong gets easier – for the channel state estimate, you'll only need to estimate that single complex number per subcarrier, that's relatively efficient, and even more, the whole equalization process is just as many multiplications as there are subcarriers, each frame, so it's really one multiplication per DFT sample.
Now, if you needed to implement an equalizing FIR, the length of that FIR would contribute quadratically to the complexity – and so, you'd avoid that.
Marcus MüllerMarcus Müller
$\begingroup$ Thank you Marcus. I understand the insight that flat fading leads to single-tap equalizer and CP helps create circular convolution so that equalization becomes multiplication with the inverse of estimated channel. But I don't get it mathematically. I have just updated the question with some math formula. Could you please take a look and tell me in which step I did use the flat fading condition to come to conclusion of single-tap equalizer ? $\endgroup$ – AlexTP Apr 11 '17 at 22:54
$\begingroup$ I think that we have single tap equalization because of circular convolution of added CP. And flat fading condition helps to estimate the single complex number of flat channel. $\endgroup$ – AlexTP Apr 11 '17 at 23:39
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Mode and median
When working with a large numerical data set, it is useful to represent the whole data set with one number. For example, we can use arithmetic mean. Except for mean, other measures of center in a given data set are mode and median.
Mode, denoted by $M_{o}$, is the property of a variable which has the highest frequency. We can say that mode is the property which appears most frequently.
Example 1: The mode of the set $\{1, 2, 3, 4, 2, 2, 5, 2\}$ is number $2$ because $2$ occurs $4$ times, which is more than any other element of a given set.
Example 2: The mode of the set $\{2.5, 2.6, 2.45, 3, 3.1\}$ doesn't exist since each element of the given set is different.
Note: A set can have more than one mode. If the set has two modes, it is bimodal. Furthermore, if it has three modes, it is trimodal and so on.
Example 3: The set $\{8, 8 , 11, 8, 24, 13, 11, 11\}$ is bimodal. In other words, numbers $8$ and $11$ are modes as they each appear three times and no other element appears more than that.
Mode of grouped data
If a distribution of numeric variable is grouped in classes, we define modal class as class with the highest frequency. We can estimate the mode using the following formula:
$$M_{o} = L_{1} + \frac{b – a}{(b-a) + (b-c)}l,$$
where $L_{1}$ is lower class boundary, $l$ its width, $a$ the frequency of the class before the modal class, $b$ the frequency of the modal class and $c$ the frequency of the class after the modal class.
Example 4: Tom wrote the results of the sprint race for $21$ competitors and grouped them in the following table. Calculate the mode.
Modal class is obviously $60 – 64$, which means that
$$L_{1} = 59.5, a = 7, b = 8, c = 4, l = 5.$$
Therefore, the mode is
$$M_{o} = 59.5 + \frac{8-7}{(8 – 7) + (8 – 4)}\cdot 5 = 59.5 + \frac{1}{1 + 4} \cdot 5 = 59.5 + 1 = 60.5.$$
The median is the middle point in a given data set. In other words, half of the data points are smaller than the median and half of them are larger. More formally, the following holds:
Let $y_{1}, \cdots, y_{N}$ be a grouped statistical sequence. Precisely, let $y_{1}, \cdots, y_{N}$ be the values of numeric variable so that $y_{1}\leq \cdots \leq y_{N}$.
Furthermore, let $r = Int \left(\frac{N}{2}\right) + 1$, where Int denotes the whole value of a number without decimals (e.g. Int $7.9=7$).
First case: $N$ is odd
We define median as a value of the r – th member of sequence, $y_{r}$.
Second case: $N$ is even
We define median as $$M_{e} = \frac{y_{r-1}+y_{r}}{2}.$$
Therefore, in order to find the median, we need to arrange points from smallest to largest. If $N$ is odd, the median is the middle data point in the list. If $N$ is even, the median is the average of the two middle data points in the list.
Example 5: Find the median of the following data: $2, 5, 8, 13, 18$.
$$r = Int \left(\frac{5}{2}\right) + 1 = 3 \Rightarrow M_{e} = y_{r} = y_{3} = 8$$
Example 6: Find the mode and median of the following data: $1, 3, 3, 2, 5, 3, 7, 7, 8, 8, 10, 11$.
First, we need to rearrange the data set so that numbers start from the smallest and end with the largest:
$$1, 2, 3, 3, 3, 5, 7, 7, 8, 8, 10, 11$$
$M_{o} = 3$, since number $3$ occurs $3$ times, and no other element occurs more than that.
$N$ is even, so $M_{e} = \frac{y_{6} + y_{7}}{2} = \frac{5 + 7}{2} = 6$.
In this example we can see that median doesn't have to be in the list of given numbers.
Median of grouped data
If a distribution of numeric variable is grouped in classes, we define median class as first class $[L_{1}, L_{2}]$ whose cumulative frequency is greater than or equal to $\frac{N}{2}$. We can calculate the median using the following formula:
$$M_{e} = L_{1} + \frac{\frac{N}{2}- F(L_{1})}{f_{med}}l,$$
where $f_{med}$ is a frequency of median class, $l = L_{2} – L_{1}$ its width and $F(L_{1})$ cumulative frequency.
Example 7: The following table shows the number of unemployed people in Croatia in $1999$. Calculate the median.
$$N = 341730, \frac{N}{2} = 170865$$
The class with the highest frequency ($119819$) is median class. Therefore,
$$L_{1} = 25, f_{med} = 119819, l = 5, F(L_{1}) = 115652.$$
Finally, the median is
$$M_{e} = L_{1} + \frac{\frac{N}{2} – F(25)}{f_{med}}l = 27.304.$$
In conclusion, we can say that the age of the first half of the people which were unemployed was $27$ years or less and the other half were people older than $27$ years.
Note: We can imagine mode as a point in the base of polygon of frequency in which the polygon has the highest value. Furthermore, we can interpret median as a point in the base in which a perpendicular divides the polygon in two parts of equal areas.
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find the slope of the line passing through the points calculator
If the y-values are decreasing, it referred to as the line has a negative slope. The coefficient of x is said to be as 1/2, means the slope of the line is 1/2.
Now, you ought to calculate where the line intersects with the y-axis: You ought to enter one of the coordinates into this slope equation: y – mx = b. Slope tells us the nature of change of function. The formula becomes increasingly useful as the coordinates take on larger values or decimal values. For further assistance, please contact us. Given m, it is possible to determine the direction of the line that m describes based on its sign and value: Slope is essentially change in height over change in horizontal distance, and is often referred to as "rise over run." Yes, slope point calculator helps you in finding the slope and shows you the slope graph corresponding to the given points by using the simple slope equation. Let us the formula to calculate the slope of the line passing through the points $(2,5)$ and $(-5, 1)$; Subtract the second coordinates and first coordinates, this gives us $y_B-y_A=1-5=-4$ and $x_B-x_A=-5-2=-7$; Simplify the fraction to get the slope of $\frac 47$. If you trace the line by using your finger, means from left to right (same like the direction that you read a book), the line will go down to the right. Simply, all you have to remember is that the slope is equal to the tangent of the angle. Remember that there is not a slope for these types of lines.
\( y_2 = y_1 + m \times \frac{d}{\sqrt(1 + m^2)} \).
The formulas to find x and y of the point to the right of the point are as: \( x_2 = x_1 + \frac{d}{\sqrt(1 + m^2)} \) Solution for Find the slope of the line passing through the given points, when possible. Subtract the lower bound from the upper bound. For example, an angle of 30° has a tangent of 0.577.
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These values must be real numbers or parameters; Slope calculator will give the slope of the line that passes through $A$ and $B$. Refer to the Triangle Calculator for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. the slope of the line that passes through the points. From the question.
Second calculator finds the line equation in parametric form, that is,. m = 2. The line is horizontal; the slope is expressed as a 0. The formula to determine the distance (D) between 2 different points is: \( Distance (d) = \sqrt {(x₂ – x₁)^2 + (y₂ – y₁)^2 } \). So, the equation of the line is $x=a$. The slope is an important concept in mathematics that is usually used in basic or advanced graphing like linear regression; the slope is said to be one of the primary numbers in a linear formula. The slope calculator, formula, work with steps and practice problems would be very useful for grade school students (K-12 education) to learn about the concept of line in geometry, how to find the general equation of a line and how to find relation between two lines. Practice Problem 1:Find the slope of the line through $(-1,6)$ and $(-10,15)$. Get check the result and you ought to make sure that this slope make sense by thinking about the points on the coordinate plane. The simple slope calculator is the tool that helps to find slope & distance between two points, slope & angle, x and y intercept, and slope intercept form for a given parameters. In many cases, we can find the slope for a given points by hand, especially for integers. This number is the gradient of the hill if it increases linearly. The slope intercept form calculator will find the slope of the line passing through the two given points, its y-intercept and slope-intercept form of the line, with steps shown. matter!
Objective : It will help you to find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas. If you can only measure the change in x, multiply this value by the gradient to find the change in the y axis. Check out 23 similar coordinate geometry calculators , Input the values into the formula. (If an answer is undefined, enter UNDEFINED.) The run is the change in $x$, $\Delta x$.
Rejecting cookies may impair some of our website's functionality. How To Find The Slope of a Line Given 2 Points? If the y-values are increasing as the x-values increase, it referred to as the line has a positive slope. The slope $m$ of a line $y=mx+b$ can be defined also as the rise divided by the run. Finding the Slope of a Line from the Graph, Finding the Slope of a Line from Two Points, Finding the Slope of a Line from the Equation, Finding the Equation of a Line Given a Point and a Slope, Finding the Equation of a Line Given Two Points. Rejecting cookies may impair some of our website's functionality. To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. To find the slope of a line we need two coordinates on the line.
A line is increasing, and goes upwards from left to right when m > 0, A line is decreasing, and goes downwards from left to right when m < 0, A line has a constant slope, and is horizontal when m = 0. If you want to calculate slope, all you need to divide the different of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa). For any other coordinates of points, just supply four real numbers and click on the "GENERATE WORK" button. 1/4″ per foot pitch equals to 2% (percent), and remember that it is not expressed as 2 degrees.
What is the equation? Solution: Slopes are very important tool to determine whether two lines perpendicular or not.
As we know, the Greek letter $\Delta$, means difference or change. As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you'd like to find the gradient.
The formula becomes increasingly useful as the coordinates take on larger values or decimal values. The equation point slope calculator will find an equation in either slope intercept form or point slope form when given a point and a slope. Message received. Divide m by the new number of the order and put it in front of the new x. The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. Slope definition is very simple; it is said to be a measure of the difference in position between two points on a line. All Rights Reserved. The slope of a roof will change depending on the style and where you live. To find the gradient of other polynomials, you will need to differentiate the function with respect to x. The calculator also has the ability to provide step by step solutions. Remember that if the line is horizontal anytime (flat from left to the right), the slope is said to be zero. Notice that the slope of a line is easily calculated by hand using small, whole number coordinates.
Download Slope Calculator App for Your Mobile, So you can calculate your values in your hand. Answer to: Find the slope of a line passing through the points (-3, 1/2) and (2, 5). No doubt, points on a line can be readily solved given the slope of the line and the distance from another point. Enter coordinates $(x_A,y_A)$ and $(x_B,y_B)$ of two points $A$ and $B$ in the box. According to the mathematician, if the line is plotted on a 2-dimensional graph, then the slope is something that shows how much the line moves along the x-axis and the y-axis between those 2 points. We have the final answer as. The symbol Δ is used to express the delta of x and y, simply, it is the absolute value of the distance between x values or y values of 2 points. This will result in a zero in the numerator of the slope formula. The slope of these points (-10, 1) and (-4, 0) is perpendicular to this line.
In the case of a road the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. if it goes up from left to right; Negative slope $m<0$, if a line $y=mx+b$ is decreasing, i.e. This new value is the length of the slope. The method for finding the slope from an equation will vary depending on the form of the equation in front of you. By using this website, you agree to our Cookie Policy. Free line equation calculator - find the equation of a line step-by-step This website uses cookies to ensure you get the best experience.
If the y-values are not changing as x increases, it is indicated as the line will have a slope of 0. Yes, slope can be determined as a percentage that is calculated in much the same as the gradient.
A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis. where (x1 , y1) and (x2 , y2) are the points. Well, we can easily calculate 'b' from this equation: Now, let's plug-in the values into the above equation: Very next, we plug-in the value of 'b' and the slope into the given equation: Also, you can use the above point slope calculator to perform instant calculations instead of sticking to these manual calculation steps! `\text{Slope} m=\frac{10-2}{7-3}` An online point slope calculator allows you to find the slope or gradient between two points in the Cartesian coordinate system. Write a new line where you add 1 to the order of the x (e.g., x becomes x^2, x^2.5 becomes x^3.5). If the product of slopes of two lines in the plane is $-1$, then the lines are perpendicular and vice-versa. © 2019 Coolmath.com LLC. Also, our slope intercept calculator will also show you the same answer for these given parameters.
It doesn't make any legitimate sense to divide by 0 so it is said that the slope of a vertical line is undefined.
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Quantitative risk assessment of the introduction of rabies into Japan through the importation of dogs and cats worldwide – ADDENDUM
N. C. L. KWAN, K. SUGIURA, Y. HOSOI, A. YAMADA, E. L. SNARY
Journal: Epidemiology & Infection / Volume 146 / Issue 10 / July 2018
Published online by Cambridge University Press: 14 June 2018, p. 1281
Thin films in partial wetting: stability, dewetting and coarsening
A. Alizadeh Pahlavan, L. Cueto-Felgueroso, A. E. Hosoi, G. H. McKinley, R. Juanes
Journal: Journal of Fluid Mechanics / Volume 845 / 25 June 2018
A uniform nanometric thin liquid film on a solid substrate can become unstable due to the action of van der Waals (vdW) forces. The instability leads to dewetting of the uniform film and the formation of drops. To minimize the total free energy of the system, these drops coarsen over time until one single drop remains. Here, using a thermodynamically consistent framework, we derive a new model for thin films in partial wetting with a free energy that resembles the Cahn–Hilliard form with a height-dependent surface tension that leads to a generalized disjoining pressure, and revisit the dewetting problem. Using both linear stability analysis and nonlinear simulations we show that the new model predicts a slightly smaller critical instability wavelength and a significantly (up to six-fold) faster growth rate than the classical model in the spinodal regime; this faster growth rate brings the theoretical predictions closer to published experimental observations. During coarsening at intermediate times, the dynamics become self-similar and model-independent; we therefore observe the same scalings in both the classical (with and without thermal noise) and new models. Both models also lead to a mean-field Lifshitz–Slyozov–Wagner (LSW)-type droplet-size distribution at intermediate times for small drop sizes. We, however, observe a skewed drop-size distribution for larger drops in the new model; while the tail of the distribution follows a Smoluchowski equation, it is not associated with a coalescence-dominated coarsening, calling into question the association made in some earlier experiments. Our observations point to the importance of the height dependence of surface tension in the early and late stages of dewetting of nanometric films and motivate new high-resolution experimental observations to guide the development of improved models of interfacial flows at the nanoscale.
Self-similar kinematics among efficient slender swimmers
A. J. Wiens, A. E. Hosoi
Journal: Journal of Fluid Mechanics / Volume 840 / 10 April 2018
Print publication: 10 April 2018
We present an analysis of efficient undulatory propulsion for slender animals swimming at high Reynolds number. Using Lighthill's large-amplitude elongated-body theory, we show that optimally efficient swimming kinematics can be characterized through a single dimensionless variable $\unicode[STIX]{x1D713}$ . This variable, $\unicode[STIX]{x1D713}$ , is defined by a simple function of a swimming animal's body wave properties. Physically, $\unicode[STIX]{x1D713}$ characterizes how the velocity of an animal's tail varies throughout its swimming stroke. Lighthill's model predicts that swimming efficiency is near optimal in the range $0.3<\unicode[STIX]{x1D713}<1.0$ and peaks at $\unicode[STIX]{x1D713}=0.87$ . At this point, the average magnitude of the tail velocity is minimized and swimming kinematics are tuned such that the thrust coefficient is as close to constant as possible throughout the swimming stroke. We use a compiled dataset of over 250 unique measurements to show that species across a wide range of size and shape fall within the optimal region.
Quantitative risk assessment of the introduction of rabies into Japan through the importation of dogs and cats worldwide
Journal: Epidemiology & Infection / Volume 145 / Issue 6 / April 2017
Published online by Cambridge University Press: 18 January 2017, pp. 1168-1182
Japan has been free from rabies since 1958. A strict import regimen has been adopted since 2004 consisting of identification of an animal with microchip, two-time rabies vaccination, neutralizing antibody titration test and a waiting period of 180 days. The present study aims to quantitatively assess the risk of rabies introduction into Japan through the international importation of dogs and cats and hence provide evidence-based recommendations to strengthen the current rabies prevention system. A stochastic scenario tree model was developed and simulations were run using @RISK. The probability of infection in a single dog or cat imported into Japan is estimated to be 2·16 × 10−9 [90% prediction interval (PI) 6·65 × 10−11–6·48 × 10−9]. The number of years until the introduction of a rabies case is estimated to be 49 444 (90% PI 19 170–94 641) years. The current import regimen is effective in maintaining the very low risk of rabies introduction into Japan and responding to future changes including increases in import level and rabies prevalence in the world. However, non-compliance or smuggling activities could substantially increase the risk of rabies introduction. Therefore, policy amendment which could promote compliance is highly recommended. Scenario analysis demonstrated that the waiting period could be reduced to 90 days and the requirement for vaccination could be reduced to a single vaccination, but serological testing should not be stopped.
Drag kings: characterizing large-scale flows in cycling aerodynamics
A. E. Hosoi
In their recent publication Crouch et al. (J. Fluid Mech., this issue, vol. 748, 2014, pp. 5–35) use wind tunnel experiments to quantify the large-scale vortical structures that develop as a cyclist progresses through a full rotation of the pedals. The authors identify asymmetries in the trailing vortex wake, which intensify as one leg straightens, as the primary source of drag variation over one pedal cycle. These new data suggest that targeted approaches to mitigate asymmetries in the trailing wake present an intriguing opportunity to reduce drag in cycling strategies and technologies.
Coarsening and solidification via solvent-annealing in thin liquid films
Tony S. Yu, Vladimir Bulović, A. E. Hosoi
Journal: Journal of Fluid Mechanics / Volume 723 / 25 May 2013
Published online by Cambridge University Press: 16 April 2013, pp. 69-90
Print publication: 25 May 2013
We examine solidification in thin liquid films produced by annealing amorphous ${\mathrm{Alq} }_{3} $ (tris-(8-hydroxyquinoline) aluminium) in methanol vapour. Micrographs acquired during annealing capture the evolution of the film: the initially-uniform film breaks up into drops that coarsen, and single crystals of ${\mathrm{Alq} }_{3} $ nucleate randomly on the substrate and grow as slender 'needles'. The growth of these needles appears to follow power-law behaviour, where the growth exponent, $\gamma $ , depends on the thickness of the deposited ${\mathrm{Alq} }_{3} $ film. The evolution of the thin film is modelled by a lubrication equation, and an advection–diffusion equation captures the transport of ${\mathrm{Alq} }_{3} $ and methanol within the film. We define a dimensionless transport parameter, $\alpha $ , which is analogous to an inverse Sherwood number and quantifies the relative effects of diffusion- and coarsening-driven advection. For large $\alpha $ -values, the model recovers the theory of one-dimensional, diffusion-driven solidification, such that $\gamma \rightarrow 1/ 2$ . For low $\alpha $ -values, the collapse of drops, i.e. coarsening, drives flow and regulates the growth of needles. Within this regime, we identify two relevant limits: needles that are small compared to the typical drop size, and those that are large. Both scaling analysis and simulations of the full model reveal that $\gamma \rightarrow 2/ 5$ for small needles and $\gamma \rightarrow 0. 29$ for large needles.
Molecular genetic conspecificity of Spiculopteragia houdemeri (Schwartz, 1926) and S. andreevae (Dróżdż, 1965) (Nematoda: Ostertagiinae) from wild ruminants in Japan
K. Sultan, M. Omar, P. Makouloutou, Y. Kaneshiro, E. Saita, M. Yokoyama, K. Suzuki, E. Hosoi, H. Sato
Journal: Journal of Helminthology / Volume 88 / Issue 1 / March 2014
Male dimorphism of the subfamily Ostertagiinae (Nematoda: Trichostrongylidae) is a well-known phenomenon, and two or more morphotypes of a single species have previously been described as different species. Two Spiculopteragia spp., S. houdemeri (syn. S. yamashitai) and S. andreevae (syn. Rinadia andreevae) recorded in Asian cervids and wild bovids, are considered to represent major and minor morphs of S. houdemeri, respectively, based solely on their co-occurrence in the same host individual along with monomorphic females. In this study, males of morph houdemeri ( = S. houdemeri) and morph andreevae ( = S. andreevae) as well as females with three different vulval ornamentations were collected from sika deer (Cervus nippon) and Japanese serows (Capricornis crispus) distributed on the mainland of Japan. Morphologically characterized worms were subjected to molecular genetic analyses based on the internal transcribed spacer region of the ribosomal RNA gene and a partial region of the cytochrome c oxidase subunit I gene of mitochondrial DNA. Of 181 collected sika deer, 177 (97.8%) and 73 (40.3%) deer harboured males of morphs houdemeri and andreevae, respectively. Worm numbers of the former morph were found to range between 1 and 444 per individual, whereas only 1–25 worms per individual were detected for the latter morph. Five out of six serows harboured 47–71 or 2–9 males of morphs houdemeri and andreevae per individual, respectively. Females with one or two vulval flaps were predominant, but there was a substantial presence of flapless females in both host species. All the morphs of male and female adults had an identical genetic background, thus directly confirming the morphological polymorphism of S. houdemeri.
A two-dimensional model of low-Reynolds number swimming beneath a free surface
DARREN CROWDY, SUNGYON LEE, OPHIR SAMSON, ERIC LAUGA, A. E. HOSOI
Journal: Journal of Fluid Mechanics / Volume 681 / 25 August 2011
Published online by Cambridge University Press: 29 June 2011, pp. 24-47
Print publication: 25 August 2011
Biological organisms swimming at low-Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper, we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by the swimmer and the free boundary problem at the fluid surface.
Marangoni convection in droplets on superhydrophobic surfaces
DANIEL TAM, VOLKMAR von ARNIM, G. H. McKINLEY, A. E. HOSOI
We consider a small droplet of water sitting on top of a heated superhydrophobic surface. A toroidal convection pattern develops in which fluid is observed to rise along the surface of the spherical droplet and to accelerate downwards in the interior towards the liquid/solid contact point. The internal dynamics arise due to the presence of a vertical temperature gradient; this leads to a gradient in surface tension which in turn drives fluid away from the contact point along the interface. We develop a solution to this thermocapillary-driven Marangoni flow analytically in terms of streamfunctions. Quantitative comparisons between analytical and experimental results, as well as effective heat transfer coefficients, are presented.
Shape optimization of a sheet swimming over a thin liquid layer
JON WILKENING, A. E. HOSOI
Motivated by the propulsion mechanisms adopted by gastropods, annelids and other invertebrates, we consider shape optimization of a flexible sheet that moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin layer of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler–Lagrange equations to simultaneously optimize swimming speed, efficiency and fluid loss. We find that as the parameters controlling these quantities approach critical values, the optimal solutions become singular in a self-similar fashion and sometimes leave the realm of validity of the lubrication model. We explore these singular limits by computing higher-order corrections to the zeroth order theory and find that wave profiles that develop cusp-like singularities are appropriately penalized, yielding non-singular optimal solutions. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations, and, to the extent possible, using recent rigorous a priori error bounds.
An experimental investigation of the stability of the circular hydraulic jump
JOHN W. M. BUSH, JEFFREY M. ARISTOFF, A. E. HOSOI
Journal: Journal of Fluid Mechanics / Volume 558 / 10 July 2006
We present the results of an experimental investigation of the striking flow structures that may arise when a vertical jet of fluid impinges on a thin fluid layer overlying a horizontal boundary. Ellegaard et al. (Nature, vol. 392, 1998, p. 767; Nonlinearity, vol. 12, 1999, p. 1) demonstrated that the axial symmetry of the circular hydraulic jump may be broken, resulting in steady polygonal jumps. In addition to these polygonal forms, our experiments reveal a new class of steady asymmetric jump forms that include structures resembling cat's eyes, three- and four-leaf clovers, bowties and butterflies. An extensive parameter study reveals the dependence of the jump structure on the governing dimensionless groups. The symmetry-breaking responsible for the asymmetric jumps is interpreted as resulting from a capillary instability of the circular jump. For all steady non-axisymmetric forms observed, the wavelength of instability of the jump is related to the surface tension, $\sigma$, fluid density $\rho$ and speed $U_v$ of the radial outflow at the jump through $\lambda\,{=}\,(74\pm7)\sigma/(\rho U_v^2)$.
Lubrication in a corner
ROMAN STOCKER, A. E. HOSOI
Journal: Journal of Fluid Mechanics / Volume 544 / 10 December 2005
Print publication: 10 December 2005
A mathematical model for the evolution of a thin film in an interior corner region is presented. The model is based on the idea that the film can be considered thin everywhere in the $\eta$-direction if viewed in the new coordinate system $\xi\,{=}\,x^2\,{-}\,y^2$, $\eta\,{=}\,2xy$. Lubrication theory is applied to the governing equations written in this coordinate system. The exact integration of the mass conservation equation for a no-slip boundary condition yields a single evolution equation, which is integrated numerically. The evolution of a thin film driven by surface tension and gravity is predicted as a function of the Bond number and successfully compared to laboratory experiments.
The effect of surface tension on rimming flows in a partially filled rotating cylinder
J. ASHMORE, A. E. HOSOI, H. A. STONE
Journal: Journal of Fluid Mechanics / Volume 479 / 25 March 2003
We study the shape of the interface in a partially filled horizontal cylinder which is rotating about its axis. Two-dimensional steady solutions for the interface height are examined under the assumptions that the filling fraction is small, inertia may be neglected, and the fluid forms a continuous film covering the surface. Three different regimes of steady solutions have been reported in the literature, corresponding to limits in which the ratio of gravitational to viscous forces (as defined in the text) is small, moderate or large. In each case, solutions have only been described analytically in the limit that surface tension effects are negligible everywhere. We use analytical and numerical methods, include surface tension and study steady solutions in a regime when the ratio of gravitational to viscous forces is large. This solution comprises a fluid pool that sits near the bottom of the cylinder and a film that coats the sides and top of the cylinder, the thickness of which can be determined by Landau–Levich–Derjaguin type arguments. We also examine the effect of surface tension on the solutions in the limits of the ratio of gravity to viscous forces being moderate and small.
Evaporative instabilities in climbing films
A. E. HOSOI, JOHN W. M. BUSH
Journal: Journal of Fluid Mechanics / Volume 442 / 10 September 2001
Print publication: 10 September 2001
We consider flow in a thin film generated by partially submerging an inclined rigid plate in a reservoir of ethanol– or methanol–water solution and wetting its surface. Evaporation leads to concentration and surface tension gradients that drive flow up the plate. An experimental study indicates that the climbing film is subject to two distinct instabilities. The first is a convective instability characterized by flattened convection rolls aligned in the direction of flow and accompanied by free-surface deformations; in the meniscus region, this instability gives rise to pronounced ridge structures aligned with the mean flow. The second instability, evident when the plate is nearly vertical, takes the form of transverse surface waves propagating up the plate.
We demonstrate that the observed longitudinal rolls are driven by the combined influence of surface deformations and alcohol concentration gradients. Guided by the observation that the rolls are flattened, we develop a quasi-two-dimensional theoretical model for the instability of the film, based on lubrication theory, which includes the effects of gravity, capillarity and Marangoni stresses at the surface. We develop stability criteria for the film which are in qualitative agreement with our experimental observations. Our analysis yields an equation for the shape of the interface which is solved numerically and reproduces the salient features of the observed flows, including the slow lateral drift and merging of the ridges.
Layer formation in monodispersive suspensions and colloids
A. E. Hosoi, Todd F. Dupont
We present theoretical results on spontaneous stratification of sedimenting suspensions and colloids caused by a lateral temperature gradient. Fluid motion is treated in the Stokes approximation, and motion of suspended particles is described by Burgers equation with convection. The internal structure and interaction of shocks at convection roll boundaries is studied numerically using a reduced one-dimensional model based on a Galerkin approach. Qualitative comparison is made to experimental data.
Computer Interfaced Electron Microscope
Y. Kokubo, S. Moriguchi, J. Hosoi, E. Watanabe, J. Nash
Journal: MRS Online Proceedings Library Archive / Volume 31 / 1983
Published online by Cambridge University Press: 21 February 2011, 23
Some applications of the computer for the electron microscope, in three major areas - 1) control of the microscope,2) image processing, and 3) structure Analysis - are discussed in the present paper. | CommonCrawl |
Some pretty simple inequalities I
Grahame Bennett
Karl-G. Grosse-Erdmann
First Online: 06 June 2014
We show that a great variety of inequalities are best understood via an extrapolation principle: they hold true simply because they are valid on a small set of test sequences (or functions). The real challenge then is to determine the best constant. This invariably leads to interesting discussions of monotonicities of sequences.
Extrapolation principle Cone Sums of powers Boas's inequality Hölder's inequality
The research of the second author was supported by the FNRS, Belgium.
Mathematics Subject Classification (2000)
Appendix: Cases of equality
The method employed in Sect. 2 allows us to determine those sequences \({\varvec{x}}\) for which the inequality
$$\begin{aligned} \Vert A{\varvec{x}}\Vert _q\le K \Vert B{\varvec{x}}\Vert _p,\quad {\varvec{x}}\in {\mathcal {C}} \end{aligned}$$
is actually an equality. Here we speak of an equality only if both sides are finite.
The problem will again be reduced to a consideration of the generators \(\varvec{g}^{(m)}\), \(m\ge 1\), of the cone \({\mathcal {C}}\). Any member \({\varvec{x}}\in {\mathcal {C}}\) can be written in the form
$$\begin{aligned} {\varvec{x}}=\sum _m\lambda _m \varvec{g}^{(m)}\quad \text {with } \lambda _m\ge 0. \end{aligned}$$
Let us suppose that \({\varvec{x}}\) is not a multiple of a generator. Then there are \(\mu ,\nu \ge 1\), \(\mu \ne \nu \), such that \(\lambda _\mu \ne 0\), \(\lambda _\nu \ne 0\). If now \(0<p\le 1\le q\) then a finer analysis of (10) shows that
$$\begin{aligned} \Vert A{\varvec{x}}\Vert _q&= \Big \Vert \lambda _\mu A \varvec{g}^{(\mu )} + \lambda _\nu A \varvec{g}^{(\nu )} +\sum _{m\ne \mu ,\nu }\lambda _m A \varvec{g}^{(m)}\Big \Vert _q\\&\le \Vert \lambda _\mu A \varvec{g}^{(\mu )} + \lambda _\nu A \varvec{g}^{(\nu )}\Vert _q +\sum _{m\ne \mu ,\nu }\lambda _m \Vert A \varvec{g}^{(m)}\Vert _q\\&< \sum _{m}\lambda _m \Vert A \varvec{g}^{(m)}\Vert _q \end{aligned}$$
provided that
$$\begin{aligned} \Vert \lambda _\mu A \varvec{g}^{(\mu )} + \lambda _\nu A \varvec{g}^{(\nu )}\Vert _q < \lambda _\mu \Vert A \varvec{g}^{(\mu )}\Vert _q + \lambda _\nu \Vert A \varvec{g}^{(\nu )}\Vert _q. \end{aligned}$$
The cases of strict inequality in Minkowski's inequality are well known: (70) holds precisely if \(q>1\) and \(A \varvec{g}^{(\mu )}\) and \(A \varvec{g}^{(\nu )}\) are linearly independent.
Continuing then the proof of (10) we arrive at
$$\begin{aligned} \Vert A{\varvec{x}}\Vert _q< K \Vert B{\varvec{x}}\Vert _p. \end{aligned}$$
Alternatively, we obtain strict inequality if \(p<1\) and \(B \varvec{g}^{(\mu )}\) and \(B \varvec{g}^{(\nu )}\) are linearly independent. Thus we have proved the following.
Theorem 17
Suppose that \(0<p\le 1\le q\) and that \(K\) is the best constant in the inequality
$$\begin{aligned} \Vert A{\varvec{x}}\Vert _q\le K \Vert B{\varvec{x}}\Vert _p\quad ({\varvec{x}}\in {\mathcal {C}}). \end{aligned}$$
Let \(\varvec{g}^{(1)},\varvec{g}^{(2)},\ldots \) be generators of the cone \({\mathcal {C}}\). If
$$\begin{aligned} q>1\text { and, for any }\mu \ne \nu , A \varvec{g}^{(\mu )}\text { and }A \varvec{g}^{(\nu )} \text { are linearly independent} \end{aligned}$$
$$\begin{aligned} p<1\text { and, for any }\mu \ne \nu , B \varvec{g}^{(\mu )}\text { and }B \varvec{g}^{(\nu )}\text { are linearly independent} \end{aligned}$$
then equality can hold in (71) only for the multiples of the generators \(\varvec{g}^{(m)},~m\ge 1\).
The situation is slightly more complicated for the inequalities in Theorems 2 and 3. For Theorem 2 we consider the equivalent inequality (17). Writing
$$\begin{aligned} {\varvec{x}}^p = \lambda _\mu \big (\varvec{g}^{(\mu )}\big )^p + \lambda _\nu \big (\varvec{g}^{(\nu )}\big )^p,\quad \lambda _\mu \ne 0, \lambda _\nu \ne 0, \end{aligned}$$
(as above we need only consider sequences of this form) it suffices to ensure that either, for some \(n\ge 1\),
$$\begin{aligned}&\left( \sum _ka_{n,k}\left( \lambda _\mu \big (g_k^{(\mu )}\big )^p + \lambda _\nu \big (g_k^{(\nu )}\big )^p\right) ^{1/p}\right) ^p\\&\quad < \left( \sum _ka_{n,k}\lambda ^{1/p}_\mu g_k^{(\mu )}\right) ^p + \left( \sum _ka_{n,k}\lambda _\nu ^{1/p} g_k^{(\nu )}\right) ^p \end{aligned}$$
$$\begin{aligned}&\left( \sum _n\Big [\left( \sum _ka_{n,k}\lambda _\mu ^{1/p} g_k^{(\mu )}\right) ^p + \left( \sum _ka_{n,k}\lambda _\nu ^{1/p} g_k^{(\nu )}\right) ^p\Big ]^{q/p}\right) ^{p/q}\\&\quad <\left( \sum _n\left( \sum _ka_{n,k}\lambda _\mu ^{1/p} g_k^{(\mu )}\right) ^q\right) ^{p/q} + \left( \sum _n\left( \sum _ka_{n,k}\lambda _\nu ^{1/p} g_k^{(\nu )}\right) ^q\right) ^{p/q}. \end{aligned}$$
This happens precisely when \(p<1\) (respectively \(q>p\)) and the corresponding sequences are linearly independent. We are thus led to the following result.
Suppose that \(0<p\le 1\) and \(q\ge p,\) that \(B\) is a diagonal matrix and that \(K\) is the best constant in the inequality
Let \(\varvec{g}^{(1)},\varvec{g}^{(2)},\ldots \) be \(p\)-spanning generators of the cone \({\mathcal {C}}\). If
$$\begin{aligned}&p<1 \text { and, for any }\mu \ne \nu ,\text { there is some }n\ge 1\text { such that}\\&\quad (a_{n,k} g_k^{(\mu )})_k\text { and }(a_{n,k} g_k^{(\nu )})_k\text { are linearly independent} \end{aligned}$$
$$\begin{aligned} q>p\text { and, for any }\mu \ne \nu , A \varvec{g}^{(\mu )}\text { and }A \varvec{g}^{(\nu )} \text { are linearly independent} \end{aligned}$$
For an application of this result we refer to Theorem 6. In the same way we obtain the following.
Suppose that \(q\ge 1\) and \(0<p\le q,\) that \(A\) is a diagonal matrix and that \(K\) is the best constant in the inequality
Let \(\varvec{g}^{(1)},\varvec{g}^{(2)},\ldots \) be \(q\)-spanning generators of the cone \({\mathcal {C}}\). If
$$\begin{aligned}&q>1\text { and, for any }\mu \ne \nu ,\text { there is some }n\ge 1\text { such that}\\&\quad (b_{n,k} g_k^{(\mu )})_k\text { and }(b_{n,k} g_k^{(\nu )})_k\text { are linearly independent} \end{aligned}$$
$$\begin{aligned} p<q\text { and, for any }\mu \ne \nu , B \varvec{g}^{(\mu )}\text { and }B \varvec{g}^{(\nu )} \text { are linearly independent} \end{aligned}$$
For an application of this result see Theorem 4.
It is instructive to analyse the first condition in Theorems 18 and 19 for the simple cones \({\mathcal {C}}_+\), \({\mathcal {C}}_\downarrow \) and \({\mathcal {C}}_\uparrow \) with their canonical generators (12), (13) and (14). We consider here Theorem 18. The first hypothesis then amounts to the following: for any \(\mu <\nu \) there is some \(n\ge 1\) such that
$$\begin{aligned}&\text {for } {\mathcal {C}}_+:\quad a_{n,\mu }\ne 0 \hbox { and } a_{n,\nu }\ne 0;\\&\text {for } {\mathcal {C}}_\downarrow :\quad \hbox {there are } 1\le j\le \mu < k\le \nu \hbox { such that } a_{n,j}\ne 0 \hbox { and } a_{n,k}\ne 0;\\&\text {for } {\mathcal {C}}_\uparrow :\quad \hbox {there are } \mu \le j< \nu \le k \hbox {such that } a_{n,j}\ne 0 \hbox { and } a_{n,k}\ne 0. \end{aligned}$$
Remark 3
We finally consider the case (of less interest) omitted in Theorems 17–19 where \(p=q=1\). Then clearly one has equality exactly for the non-negative linear combinations
$$\begin{aligned} \sum _{m\in M}\lambda _m\varvec{g}^{(m)},\quad \lambda _m\ge 0 \end{aligned}$$
of all generators \(\varvec{g}^{(m)}\) for which equality holds: \(M=\{m\ge 1: \Vert A\varvec{g}^{(m)}\Vert _1= K \Vert B\varvec{g}^{(m)}\Vert _1\}\).
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© Springer Basel 2014
1.Department of MathematicsIndiana UniversityBloomingtonUSA
2.Département de Mathématique, Institut ComplexysUniversité de MonsMonsBelgium
Bennett, G. & Grosse-Erdmann, KG. Positivity (2015) 19: 251. https://doi.org/10.1007/s11117-014-0294-4
Received 21 December 2013
First Online 06 June 2014
Publisher Name Springer Basel | CommonCrawl |
Queries for Set partitions: search statistic / browse statistics / browse maps from / browse maps to
Definition & Example
A set partition of size $n$ is a partition of the set $\mathcal{S} = \{1,\ldots,n\}$. This is a collection of non-empty pairwise disjoint subsets (parts) of $\mathcal{S}$ whose union is $\mathcal{S}$. In symbols, $\mathcal{P} = \{P_1,\ldots,P_k \}$ such that
$$S = P_1 \sqcup P_2 \sqcup \dots \sqcup P_k, \quad P_i \cap P_j = \emptyset \text{ for all }i \neq j, \quad \emptyset \notin \mathcal{P}.$$
the 5 Set partitions of size 3
{{1,2,3}} {{1,2},{3}} {{1,3},{2}} {{1},{2,3}} {{1},{2},{3}}
Set partitions of size $n$ are graphically represented by drawing the numbers $1$ through $n$ around a circle and then drawing the convex hulls of the blocks.
The number of set partitions of size $n$ is $n$-th Bell number $B_n$ (A000110).
The number of set partitions of size $n$ into $k$ blocks is the Stirling number of the second kind (A008277).
The set partitions of size $n$ form a poset by containment order. This poset is indeed a lattice which is the intersection lattice of the braid arrangement.
A set partition is said to be non-crossing if the graphical representation does not have any crossing blocks. In symbols, this is to say that there does not exist $P_i, P_j \in \mathcal{P}$ which contains elements $a, b \in P_i$ and $x, y \in P_j$ such that $a < x < b < y$. The number of non-crossing set partitions of size $n$ is the n-th Catalan number.
Sage examples
Technical information for database usage
A set partition is represented as a set of disjoint blocks, which are themselves sets.
Set partitions are graded by the size of the ground set.
The database contains all set partitions of size at most 7.
If you want to edit this wiki page, you can download the raw markdown and send your new version to [email protected] | CommonCrawl |
The power calculation indicates a 20% chance of getting useful information. My quasi-experiment has <70% chance of being right, and I preserve a general skepticism about any experiment, even one as well done as the medical student one seems to be, and give that one a <80% chance of being right; so let's call it 70% the effect exists, or 30% it doesn't exist (which is the case in which I save money by dropping fish oil for 10 years).
Kennedy et al. (1990) administered what they termed a grammatical reasoning task to subjects, in which a sentence describing the order of two letters, A and B, is presented along with the letter pair, and subjects must determine whether or not the sentence correctly describes the letter pair. They found no effect of d-AMP on performance of this task.
Googling, you sometimes see correlational studies like Intake of Flavonoid-Rich Wine, Tea, and Chocolate by Elderly Men and Women Is Associated with Better Cognitive Test Performance; in this one, the correlated performance increase from eating chocolate was generally fairly modest (say, <10%), and the maximum effects were at 10g/day of what was probably milk chocolate, which generally has 10-40% chocolate liquor in it, suggesting any experiment use 1-4g. More interesting is the blind RCT experiment Consumption of cocoa flavanols results in acute improvements in mood and cognitive performance during sustained mental effort11, which found improvements at ~1g; the most dramatic improvement of the 4 tasks (on the Threes correct) saw a difference of 2 to 6 at the end of the hour of testing, while several of the other tests converged by the end or saw the controls winning (Sevens correct). Crews et al 2008 found no cognitive benefit, and an fMRI experiment found the change in brain oxygen levels it wanted but no improvement to reaction times.
I bought 500g of piracetam (Examine.com; FDA adverse events) from Smart Powders (piracetam is one of the cheapest nootropics and SP was one of the cheapest suppliers; the others were much more expensive as of October 2010), and I've tried it out for several days (started on 7 September 2009, and used it steadily up to mid-December). I've varied my dose from 3 grams to 12 grams (at least, I think the little scoop measures in grams), taking them in my tea or bitter fruit juice. Cranberry worked the best, although orange juice masks the taste pretty well; I also accidentally learned that piracetam stings horribly when I got some on a cat scratch. 3 grams (alone) didn't seem to do much of anything while 12 grams gave me a nasty headache. I also ate 2 or 3 eggs a day.
Use of prescription stimulants by normal healthy individuals to enhance cognition is said to be on the rise. Who is using these medications for cognitive enhancement, and how prevalent is this practice? Do prescription stimulants in fact enhance cognition for normal healthy people? We review the epidemiological and cognitive neuroscience literatures in search of answers to these questions. Epidemiological issues addressed include the prevalence of nonmedical stimulant use, user demographics, methods by which users obtain prescription stimulants, and motivations for use. Cognitive neuroscience issues addressed include the effects of prescription stimulants on learning and executive function, as well as the task and individual variables associated with these effects. Little is known about the prevalence of prescription stimulant use for cognitive enhancement outside of student populations. Among college students, estimates of use vary widely but, taken together, suggest that the practice is commonplace. The cognitive effects of stimulants on normal healthy people cannot yet be characterized definitively, despite the volume of research that has been carried out on these issues. Published evidence suggests that declarative memory can be improved by stimulants, with some evidence consistent with enhanced consolidation of memories. Effects on the executive functions of working memory and cognitive control are less reliable but have been found for at least some individuals on some tasks. In closing, we enumerate the many outstanding questions that remain to be addressed by future research and also identify obstacles facing this research.
Factor analysis. The strategy: read in the data, drop unnecessary data, impute missing variables (data is too heterogeneous and collected starting at varying intervals to be clean), estimate how many factors would fit best, factor analyze, pick the ones which look like they match best my ideas of what productive is, extract per-day estimates, and finally regress LLLT usage on the selected factors to look for increases.
With all these studies pointing to the nootropic benefits of some essential oils, it can logically be concluded then that some essential oils can be considered "smart drugs." However, since essential oils have so much variety and only a small fraction of this wide range has been studied, it cannot be definitively concluded that absolutely all essential oils have brain-boosting benefits. The connection between the two is strong, however.
Barbaresi WJ, Katusic SK, Colligan RC, Weaver AL, Jacobsen SJ. Modifiers of long-term school outcomes for children with attention-deficit/hyperactivity disorder: Does treatment with stimulant medication make a difference? Results from a population-based study. Journal of Developmental and Behavioral Pediatrics. 2007;28:274–287. doi: 10.1097/DBP.0b013e3180cabc28. [PubMed] [CrossRef]
Smart pills containing Aniracetam may also improve communication between the brain's hemispheres. This benefit makes Aniracetam supplements ideal for enhancing creativity and stabilizing mood. But, the anxiolytic effects of Aniracetam may be too potent for some. There are reports of some users who find that it causes them to feel unmotivated or sedated. Though, it may not be an issue if you only seek the anti-stress and anxiety-reducing effects.
For obvious reasons, it's difficult for researchers to know just how common the "smart drug" or "neuro-enhancing" lifestyle is. However, a few recent studies suggest cognition hacking is appealing to a growing number of people. A survey conducted in 2016 found that 15% of University of Oxford students were popping pills to stay competitive, a rate that mirrored findings from other national surveys of UK university students. In the US, a 2014 study found that 18% of sophomores, juniors, and seniors at Ivy League colleges had knowingly used a stimulant at least once during their academic career, and among those who had ever used uppers, 24% said they had popped a little helper on eight or more occasions. Anecdotal evidence suggests that pharmacological enhancement is also on the rise within the workplace, where modafinil, which treats sleep disorders, has become particularly popular.
Phenserine, as well as the drugs Aricept and Exelon, which are already on the market, work by increasing the level of acetylcholine, a neurotransmitter that is deficient in people with the disease. A neurotransmitter is a chemical that allows communication between nerve cells in the brain. In people with Alzheimer's disease, many brain cells have died, so the hope is to get the most out of those that remain by flooding the brain with acetylcholine.
Though coffee gives instant alertness, the effect lasts only for a short while. People who drink coffee every day may develop caffeine tolerance; this is the reason why it is still important to control your daily intake. It is advisable that an individual should not consume more than 300 mg of coffee a day. Caffeine, the world's favorite nootropic has fewer side effects, but if consumed abnormally in excess, it can result in nausea, restlessness, nervousness, and hyperactivity. This is the reason why people who need increased sharpness would instead induce L-theanine, or some other Nootropic, along with caffeine. Today, you can find various smart drugs that contain caffeine in them. OptiMind, one of the best and most sought-after nootropics in the U.S, containing caffeine, is considered best brain supplement for adults and kids when compared to other focus drugs present in the market today.
(If I am not deficient, then supplementation ought to have no effect.) The previous material on modern trends suggests a prior >25%, and higher than that if I were female. However, I was raised on a low-salt diet because my father has high blood pressure, and while I like seafood, I doubt I eat it more often than weekly. I suspect I am somewhat iodine-deficient, although I don't believe as confidently as I did that I had a vitamin D deficiency. Let's call this one 75%.
Some critics argue that Modafinil is an expression of that, a symptom of a new 24/7 work routine. But what if the opposite is true? Let's say you could perform a task in significantly less time than usual. You could then use the rest of your time differently, spending it with family, volunteering, or taking part in a leisure activity. And imagine that a drug helped you focus on clearing your desk and inbox before leaving work. Wouldn't that help you relax once you get home?
No. There are mission essential jobs that require you to live on base sometimes. Or a first term person that is required to live on base. Or if you have proven to not be as responsible with rent off base as you should be so your commander requires you to live on base. Or you're at an installation that requires you to live on base during your stay. Or the only affordable housing off base puts you an hour away from where you work. It isn't simple. The fact that you think it is tells me you are one of the "dumb@$$es" you are referring to above.
Capsule Connection sells 1000 00 pills (the largest pills) for $9. I already have a pill machine, so that doesn't count (a sunk cost). If we sum the grams per day column from the first table, we get 9.75 grams a day. Each 00 pill can take around 0.75 grams, so we need 13 pills. (Creatine is very bulky, alas.) 13 pills per day for 1000 days is 13,000 pills, and 1,000 pills is $9 so we need 13 units and 13 times 9 is $117.
l-theanine (Examine.com) is occasionally mentioned on Reddit or Imminst or LessWrong32 but is rarely a top-level post or article; this is probably because theanine was discovered a very long time ago (>61 years ago), and it's a pretty straightforward substance. It's a weak relaxant/anxiolytic (Google Scholar) which is possibly responsible for a few of the health benefits of tea, and which works synergistically with caffeine (and is probably why caffeine delivered through coffee feels different from the same amount consumed in tea - in one study, separate caffeine and theanine were a mixed bag, but the combination beat placebo on all measurements). The half-life in humans seems to be pretty short, with van der Pijl 2010 putting it ~60 minutes. This suggests to me that regular tea consumption over a day is best, or at least that one should lower caffeine use - combining caffeine and theanine into a single-dose pill has the problem of caffeine's half-life being much longer so the caffeine will be acting after the theanine has been largely eliminated. The problem with getting it via tea is that teas can vary widely in their theanine levels and the variations don't seem to be consistent either, nor is it clear how to estimate them. (If you take a large dose in theanine like 400mg in water, you can taste the sweetness, but it's subtle enough I doubt anyone can actually distinguish the theanine levels of tea; incidentally, r-theanine - the useless racemic other version - anecdotally tastes weaker and less sweet than l-theanine.)
My intent here is not to promote illegal drugs or promote the abuse of prescription drugs. In fact, I have identified which drugs require a prescription. If you are a servicemember and you take a drug (such as Modafinil and Adderall) without a prescription, then you will fail a urinalysis test. Thus, you will most likely be discharged from the military.
Going back to the 1960s, although it was a Romanian chemist who is credited with discovering nootropics, a substantial amount of research on racetams was conducted in the Soviet Union. This resulted in the birth of another category of substances entirely: adaptogens, which, in addition to benefiting cognitive function were thought to allow the body to better adapt to stress.
I take my piracetam in the form of capped pills consisting (in descending order) of piracetam, choline bitartrate, anhydrous caffeine, and l-tyrosine. On 8 December 2012, I happened to run out of them and couldn't fetch more from my stock until 27 December. This forms a sort of (non-randomized, non-blind) short natural experiment: did my daily 1-5 mood/productivity ratings fall during 8-27 December compared to November 2012 & January 2013? The graphed data28 suggests to me a decline:
"Cavin, you are phemomenal! An incredulous journey of a near death accident scripted by an incredible man who chose to share his knowledge of healing his own broken brain. I requested our public library purchase your book because everyone, those with and without brain injuries, should have access to YOUR brain and this book. Thank you for your legacy to mankind!"
Take at 10 AM; seem a bit more active but that could just be the pressure of the holiday season combined with my nice clean desk. I do the chores without too much issue and make progress on other things, but nothing major; I survive going to The Sitter without too much tiredness, so ultimately I decide to give the palm to it being active, but only with 60% confidence. I check the next day, and it was placebo. Oops.
I almost resigned myself to buying patches to cut (and let the nicotine evaporate) and hope they would still stick on well enough afterwards to be indistinguishable from a fresh patch, when late one sleepless night I realized that a piece of nicotine gum hanging around on my desktop for a week proved useless when I tried it, and that was the answer: if nicotine evaporates from patches, then it must evaporate from gum as well, and if gum does evaporate, then to make a perfect placebo all I had to do was cut some gum into proper sizes and let the pieces sit out for a while. (A while later, I lost a piece of gum overnight and consumed the full 4mg to no subjective effect.) Google searches led to nothing indicating I might be fooling myself, and suggested that evaporation started within minutes in patches and a patch was useless within a day. Just a day is pushing it (who knows how much is left in a useless patch?), so I decided to build in a very large safety factor and let the gum sit for around a month rather than a single day.
Ashwagandha has been shown to improve cognition and motivation, by means of reducing anxiety [46]. It has been shown to significantly reduce stress and anxiety. As measured by cortisol levels, anxiety symptoms were reduced by around 30% compared to a placebo-controlled (double-blind) group [47]. And it may have neuroprotective effects and improve sleep, but these claims are still being researched.
I have personally found that with respect to the NOOTROPIC effect(s) of all the RACETAMS, whilst I have experienced improvements in concentration and working capacity / productivity, I have never experienced a noticeable ongoing improvement in memory. COLURACETAM is the only RACETAM that I have taken wherein I noticed an improvement in MEMORY, both with regards to SHORT-TERM and MEDIUM-TERM MEMORY. To put matters into perspective, the memory improvement has been mild, yet still significant; whereas I have experienced no such improvement at all with the other RACETAMS.
Take at 11 AM; distractions ensue and the Christmas tree-cutting also takes up much of the day. By 7 PM, I am exhausted and in a bad mood. While I don't expect day-time modafinil to buoy me up, I do expect it to at least buffer me against being tired, and so I conclude placebo this time, and with more confidence than yesterday (65%). I check before bed, and it was placebo.
It's been widely reported that Silicon Valley entrepreneurs and college students turn to Adderall (without a prescription) to work late through the night. In fact, a 2012 study published in the Journal of American College Health, showed that roughly two-thirds of undergraduate students were offered prescription stimulants for non-medical purposes by senior year.
First was a combination of L-theanine and aniracetam, a synthetic compound prescribed in Europe to treat degenerative neurological diseases. I tested it by downing the recommended dosages and then tinkering with a story I had finished a few days earlier, back when caffeine was my only performance-enhancing drug. I zoomed through the document with renewed vigor, striking some sentences wholesale and rearranging others to make them tighter and punchier.
…Four subjects correctly stated when they received nicotine, five subjects were unsure, and the remaining two stated incorrectly which treatment they received on each occasion of testing. These numbers are sufficiently close to chance expectation that even the four subjects whose statements corresponded to the treatments received may have been guessing.
Depending on where you live, some nootropics may not be sold over the counter, but they are usually available online. The law regarding nootropics can vary massively around the world, so be sure to do your homework before you purchase something for the first time. Be particularly cautious when importing smart drugs, because quality control and regulations abroad are not always as stringent as they are in the US. Do not put your health at risk if all you are trying to do is gain an edge in a competitive sport.
Taken together, these considerations suggest that the cognitive effects of stimulants for any individual in any task will vary based on dosage and will not easily be predicted on the basis of data from other individuals or other tasks. Optimizing the cognitive effects of a stimulant would therefore require, in effect, a search through a high-dimensional space whose dimensions are dose; individual characteristics such as genetic, personality, and ability levels; and task characteristics. The mixed results in the current literature may be due to the lack of systematic optimization.
Another prescription stimulant medication, modafinil (known by the brand name Provigil), is usually prescribed to patients suffering from narcolepsy and shift-work sleep disorder, but it might turn out to have broader applications. "We have conducted at the University of Cambridge double-blind, placebo-controlled studies in healthy people using modafinil and have found improvements in cognition, including in working memory," Sahakian says. However, she doesn't think everyone should start using the drug off-label. "There are no long-term safety and efficacy studies of modafinil in healthy people, and so it is unclear what the risks might be."
Both nootropics startups provide me with samples to try. In the case of Nootrobox, it is capsules called Sprint designed for a short boost of cognitive enhancement. They contain caffeine – the equivalent of about a cup of coffee, and L-theanine – about 10 times what is in a cup of green tea, in a ratio that is supposed to have a synergistic effect (all the ingredients Nootrobox uses are either regulated as supplements or have a "generally regarded as safe" designation by US authorities)
The chemicals he takes, dubbed nootropics from the Greek "noos" for "mind", are intended to safely improve cognitive functioning. They must not be harmful, have significant side-effects or be addictive. That means well-known "smart drugs" such as the prescription-only stimulants Adderall and Ritalin, popular with swotting university students, are out. What's left under the nootropic umbrella is a dizzying array of over-the-counter supplements, prescription drugs and unclassified research chemicals, some of which are being trialled in older people with fading cognition.
That doesn't necessarily mean all smart drugs – now and in the future – will be harmless, however. The brain is complicated. In trying to upgrade it, you risk upsetting its intricate balance. "It's not just about more, it's about having to be exquisitely and exactly right. And that's very hard to do," says Arnstein. "What's good for one system may be bad for another system," adds Trevor Robbins, Professor of Cognitive Neuroscience at the University of Cambridge. "It's clear from the experimental literature that you can affect memory with pharmacological agents, but the problem is keeping them safe."
Thursday: 3g piracetam/4g choline bitartrate at 1; 1 200mg modafinil at 2:20; noticed a leveling of fatigue by 3:30; dry eyes? no bad after taste or anything. a little light-headed by 4:30, but mentally clear and focused. wonder if light-headedness is due simply to missing lunch and not modafinil. 5:43: noticed my foot jiggling - doesn't usually jiggle while in piracetam/choline. 7:30: starting feeling a bit jittery & manic - not much or to a problematic level but definitely noticeable; but then, that often happens when I miss lunch & dinner. 12:30: bedtime. Can't sleep even with 3mg of melatonin! Subjectively, I toss & turn (in part thanks to my cat) until 4:30, when I really wake up. I hang around bed for another hour & then give up & get up. After a shower, I feel fairly normal, strangely, though not as good as if I had truly slept 8 hours. The lesson here is to pay attention to wikipedia when it says the half-life is 12-15 hours! About 6AM I take 200mg; all the way up to 2pm I feel increasingly less energetic and unfocused, though when I do apply myself I think as well as ever. Not fixed by food or tea or piracetam/choline. I want to be up until midnight, so I take half a pill of 100mg and chew it (since I'm not planning on staying up all night and I want it to work relatively soon). From 4-12PM, I notice that today as well my heart rate is elevated; I measure it a few times and it seems to average to ~70BPM, which is higher than normal, but not high enough to concern me. I stay up to midnight fine, take 3mg of melatonin at 12:30, and have no trouble sleeping; I think I fall asleep around 1. Alarm goes off at 6, I get up at 7:15 and take the other 100mg. Only 100mg/half-a-pill because I don't want to leave the half laying around in the open, and I'm curious whether 100mg + ~5 hours of sleep will be enough after the last 2 days. Maybe next weekend I'll just go without sleep entirely to see what my limits are.
The majority of nonmedical users reported obtaining prescription stimulants from a peer with a prescription (Barrett et al., 2005; Carroll et al., 2006; DeSantis et al., 2008, 2009; DuPont et al., 2008; McCabe & Boyd, 2005; Novak et al., 2007; Rabiner et al., 2009; White et al., 2006). Consistent with nonmedical user reports, McCabe, Teter, and Boyd (2006) found 54% of prescribed college students had been approached to divert (sell, exchange, or give) their medication. Studies of secondary school students supported a similar conclusion (McCabe et al., 2004; Poulin, 2001, 2007). In Poulin's (2007) sample, 26% of students with prescribed stimulants reported giving or selling some of their medication to other students in the past month. She also found that the number of students in a class with medically prescribed stimulants was predictive of the prevalence of nonmedical stimulant use in the class (Poulin, 2001). In McCabe et al.'s (2004) middle and high school sample, 23% of students with prescriptions reported being asked to sell or trade or give away their pills over their lifetime.
What if you could simply take a pill that would instantly make you more intelligent? One that would enhance your cognitive capabilities including attention, memory, focus, motivation and other higher executive functions? If you have ever seen the movie Limitless, you have an idea of what this would look like—albeit the exaggerated Hollywood version. The movie may be fictional but the reality may not be too far behind.
Another interpretation of the mixed results in the literature is that, in some cases at least, individual differences in response to stimulants have led to null results when some participants in the sample are in fact enhanced and others are not. This possibility is not inconsistent with the previously mentioned ones; both could be at work. Evidence has already been reviewed that ability level, personality, and COMT genotype modulate the effect of stimulants, although most studies in the literature have not broken their samples down along these dimensions. There may well be other as-yet-unexamined individual characteristics that determine drug response. The equivocal nature of the current literature may reflect a mixture of substantial cognitive-enhancement effects for some individuals, diluted by null effects or even counteracted by impairment in others.
He used to get his edge from Adderall, but after moving from New Jersey to San Francisco, he says, he couldn't find a doctor who would write him a prescription. Driven to the Internet, he discovered a world of cognition-enhancing drugs known as nootropics — some prescription, some over-the-counter, others available on a worldwide gray market of private sellers — said to improve memory, attention, creativity and motivation.
Two increasingly popular options are amphetamines and methylphenidate, which are prescription drugs sold under the brand names Adderall and Ritalin. In the United States, both are approved as treatments for people with ADHD, a behavioural disorder which makes it hard to sit still or concentrate. Now they're also widely abused by people in highly competitive environments, looking for a way to remain focused on specific tasks.
Blinding stymied me for a few months since the nasty taste was unmistakable and I couldn't think of any gums with a similar flavor to serve as placebo. (The nasty taste does not seem to be due to the nicotine despite what one might expect; Vaniver plausibly suggested the bad taste might be intended to prevent over-consumption, but nothing in the Habitrol ingredient list seemed to be noted for its bad taste, and a number of ingredients were sweetening sugars of various sorts. So I couldn't simply flavor some gum.)
It is known that American college students have embraced cognitive enhancement, and some information exists about the demographics of the students most likely to practice cognitive enhancement with prescription stimulants. Outside of this narrow segment of the population, very little is known. What happens when students graduate and enter the world of work? Do they continue using prescription stimulants for cognitive enhancement in their first jobs and beyond? How might the answer to this question depend on occupation? For those who stay on campus to pursue graduate or professional education, what happens to patterns of use? To what extent do college graduates who did not use stimulants as students begin to use them for cognitive enhancement later in their careers? To what extent do workers without college degrees use stimulants to enhance job performance? How do the answers to these questions differ for countries outside of North America, where the studies of Table 1 were carried out?
A rough translation for the word "nootropic" comes from the Greek for "to bend or shape the mind." And already, there are dozens of over-the-counter (OTC) products—many of which are sold widely online or in stores—that claim to boost creativity, memory, decision-making or other high-level brain functions. Some of the most popular supplements are a mixture of food-derived vitamins, lipids, phytochemicals and antioxidants that studies have linked to healthy brain function. One popular pick on Amazon, for example, is an encapsulated cocktail of omega-3s, B vitamins and plant-derived compounds that its maker claims can improve memory, concentration and focus.
Productivity is the most cited reason for using nootropics. With all else being equal, smart drugs are expected to give you that mental edge over other and advance your career. Nootropics can also be used for a host of other reasons. From studying to socialising. And from exercise and health to general well-being. Different nootropics cater to different audiences.
Many people find it difficult to think clearly when they are stressed out. Ongoing stress leads to progressive mental fatigue and an eventual breakdown. Luckily, there are several ways that nootropics can help relieve stress. One is through the natural promotion of feelings of relaxation and the other is by replenishing the brain chemicals drained by stress.
First off, overwhelming evidence suggests that smart drugs actually work. A meta-analysis by researchers at Harvard Medical School and Oxford showed that Modafinil has significant cognitive benefits for those who do not suffer from sleep deprivation. The drug improves their ability to plan and make decisions and has a positive effect on learning and creativity. Another study, by researchers at Imperial College London, showed that Modafinil helped sleep-deprived surgeons become better at planning, redirecting their attention, and being less impulsive when making decisions.
"Smart Drugs" are chemical substances that enhance cognition and memory or facilitate learning. However, within this general umbrella of "things you can eat that make you smarter," there are many variations as far as methods of action within the body, perceptible (and measurable) effects, potential for use and abuse, and the spillover impact on the body's non-cognitive processes.
But he has also seen patients whose propensity for self-experimentation to improve cognition got out of hand. One chief executive he treated, Ngo said, developed an unhealthy predilection for albuterol, because he felt the asthma inhaler medicine kept him alert and productive long after others had quit working. Unfortunately, the drug ended up severely imbalancing his electrolytes, which can lead to dehydration, headaches, vision and cardiac problems, muscle contractions and, in extreme cases, seizures.
The blood half-life is 12-36 hours; hence two or three days ought to be enough to build up and wash out. A week-long block is reasonable since that gives 5 days for effects to manifest, although month-long blocks would not be a bad choice either. (I prefer blocks which fit in round periods because it makes self-experiments easier to run if the blocks fit in normal time-cycles like day/week/month. The most useless self-experiment is the one abandoned halfway.)
Supplements, medications, and coffee certainly might play a role in keeping our brains running smoothly at work or when we're trying to remember where we left our keys. But the long-term effects of basic lifestyle practices can't be ignored. "For good brain health across the life span, you should keep your brain active," Sahakian says. "There is good evidence for 'use it or lose it.'" She suggests brain-training apps to improve memory, as well as physical exercise. "You should ensure you have a healthy diet and not overeat. It is also important to have good-quality sleep. Finally, having a good work-life balance is important for well-being." Try these 8 ways to get smarter while you sleep.
The general cost of fish oil made me interested in possible substitutes. Seth Roberts uses exclusively flaxseed oil or flaxseed meal, and this seems to work well for him with subjective effects (eg. noticing his Chinese brands seemed to not work, possibly because they were unrefrigerated and slightly rancid). It's been studied much less than fish oil, but omega acids are confusing enough in general (is there a right ratio? McCluskey's roundup gives the impression claims about ratios may have been overstated) that I'm not convinced ALA is a much inferior replacement for fish oil's mixes of EPA & DHA.
(People aged <=18 shouldn't be using any of this except harmless stuff - where one may have nutritional deficits - like fish oil & vitamin D; melatonin may be especially useful, thanks to the effects of screwed-up school schedules & electronics use on teenagers' sleep. Changes in effects with age are real - amphetamines' stimulant effects and modafinil's histamine-like side-effects come to mind as examples.)
Somewhat ironically given the stereotypes, while I was in college I dabbled very little in nootropics, sticking to melatonin and tea. Since then I have come to find nootropics useful, and intellectually interesting: they shed light on issues in philosophy of biology & evolution, argue against naive psychological dualism and for materialism, offer cases in point on the history of technology & civilization or recent psychology theories about addiction & willpower, challenge our understanding of the validity of statistics and psychology - where they don't offer nifty little problems in statistics and economics themselves, and are excellent fodder for the young Quantified Self movement4; modafinil itself demonstrates the little-known fact that sleep has no accepted evolutionary explanation. (The hard drugs also have more ramifications than one might expect: how can one understand the history of Southeast Asia and the Vietnamese War without reference to heroin, or more contemporaneously, how can one understand the lasting appeal of the Taliban in Afghanistan and the unpopularity & corruption of the central government without reference to the Taliban's frequent anti-drug campaigns or the drug-funded warlords of the Northern Alliance?)
Results: Women with high caffeine intakes had significantly higher rates of bone loss at the spine than did those with low intakes (−1.90 ± 0.97% compared with 1.19 ± 1.08%; P = 0.038). When the data were analyzed according to VDR genotype and caffeine intake, women with the tt genotype had significantly (P = 0.054) higher rates of bone loss at the spine (−8.14 ± 2.62%) than did women with the TT genotype (−0.34 ± 1.42%) when their caffeine intake was >300 mg/d…In 1994, Morrison et al (22) first reported an association between vitamin D receptor gene (VDR) polymorphism and BMD of the spine and hip in adults. After this initial report, the relation between VDR polymorphism and BMD, bone turnover, and bone loss has been extensively evaluated. The results of some studies support an association between VDR polymorphism and BMD (23-,25), whereas other studies showed no evidence for this association (26,27)…At baseline, no significant differences existed in serum parathyroid hormone, serum 25-hydroxyvitamin D, serum osteocalcin, and urinary N-telopeptide between the low- and high-caffeine groups (Table 1⇑). In the longitudinal study, the percentage of change in serum parathyroid hormone concentrations was significantly lower in the high-caffeine group than in the low-caffeine group (Table 2⇑). However, no significant differences existed in the percentage of change in serum 25-hydroxyvitamin D
The evidence? A 2012 study in Greece found it can boost cognitive function in adults with mild cognitive impairment (MCI), a type of disorder marked by forgetfulness and problems with language, judgement, or planning that are more severe than average "senior moments," but are not serious enough to be diagnosed as dementia. In some people, MCI will progress into dementia.
Swanson J, Arnold LE, Kraemer H, Hechtman L, Molina B, Hinshaw S, Wigal T. Evidence, interpretation and qualification from multiple reports of long-term outcomes in the Multimodal Treatment Study of Children With ADHD (MTA): Part II. Supporting details. Journal of Attention Disorders. 2008;12:15–43. doi: 10.1177/1087054708319525. [PubMed] [CrossRef]
Learning how products have worked for other users can help you feel more confident in your purchase. Similarly, your opinion may help others find a good quality supplement. After you have started using a particular supplement and experienced the benefits of nootropics for memory, concentration, and focus, we encourage you to come back and write your own review to share your experience with others.
Even if you eat foods that contain these nutrients, Hogan says their beneficial effects are in many ways cumulative—meaning the brain perks don't emerge unless you've been eating them for long periods of time. Swallowing more of these brain-enhancing compounds at or after middle-age "may be beyond the critical period" when they're able to confer cognitive enhancements, he says.
But perhaps the biggest difference between Modafinil and other nootropics like Piracetam, according to Patel, is that Modafinil studies show more efficacy in young, healthy people, not just the elderly or those with cognitive deficits. That's why it's great for (and often prescribed to) military members who are on an intense tour, or for those who can't get enough sleep for physiological reasons. One study, by researchers at Imperial College London, and published in Annals of Surgery, even showed that Modafinil helped sleep-deprived surgeons become better at planning, redirecting their attention, and being less impulsive when making decisions.
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Please browse our website to learn more about how to enhance your memory. Our blog contains informative articles about the science behind nootropic supplements, specific ingredients, and effective methods for improving memory. Browse through our blog articles and read and compare reviews of the top rated natural supplements and smart pills to find everything you need to make an informed decision.
My first dose on 1 March 2017, at the recommended 0.5ml/1.5mg was miserable, as I felt like I had the flu and had to nap for several hours before I felt well again, requiring 6h to return to normal; after waiting a month, I tried again, but after a week of daily dosing in May, I noticed no benefits; I tried increasing to 3x1.5mg but this immediately caused another afternoon crash/nap on 18 May. So I scrapped my cytisine. Oh well.
Dr. Larry Cleary's Lucidal – the critically acclaimed secret formula that has been created, revised, and optimized to the point that it's Dr. Cleary-approved. As a product of Dr. Cleary's extensive years and expertise in the industry, it is his brainchild. Heavily marketed as the pill for reversing memory loss, whilst aiding focus, it's seen some popularity in the last few years. In light of all the hubbub and controversy, we put their claims to the test, to see whether or not Lucidal is able to come forth with flying colors, just as all its acclamation has it to be… Learn More...
Low-dose lithium orotate is extremely cheap, ~$10 a year. There is some research literature on it improving mood and impulse control in regular people, but some of it is epidemiological (which implies considerable unreliability); my current belief is that there is probably some effect size, but at just 5mg, it may be too tiny to matter. I have ~40% belief that there will be a large effect size, but I'm doing a long experiment and I should be able to detect a large effect size with >75% chance. So, the formula is NPV of the difference between taking and not taking, times quality of information, times expectation: \frac{10 - 0}{\ln 1.05} \times 0.75 \times 0.40 = 61.4, which justifies a time investment of less than 9 hours. As it happens, it took less than an hour to make the pills & placebos, and taking them is a matter of seconds per week, so the analysis will be the time-consuming part. This one may actually turn a profit.
A similar pill from HQ Inc. (Palmetto, Fla.) called the CorTemp Ingestible Core Body Temperature Sensor transmits real-time body temperature. Firefighters, football players, soldiers and astronauts use it to ensure that they do not overheat in high temperatures. HQ Inc. is working on a consumer version, to be available in 2018, that would wirelessly communicate to a smartphone app.
When I spoke with Jesse Lawler, who hosts the podcast Smart Drugs Smarts, about breakthroughs in brain health and neuroscience, he was unsurprised to hear of my disappointing experience. Many nootropics are supposed to take time to build up in the body before users begin to feel their impact. But even then, says Barry Gordon, a neurology professor at the Johns Hopkins Medical Center, positive results wouldn't necessarily constitute evidence of a pharmacological benefit.
10:30 AM; no major effect that I notice throughout the day - it's neither good nor bad. This smells like placebo (and part of my mind is going how unlikely is it to get placebo 3 times in a row!, which is just the Gambler's fallacy talking inasmuch as this is sampling with replacement). I give it 60% placebo; I check the next day right before taking, and it is. Man!
The evidence? In small studies, healthy people taking modafinil showed improved planning and working memory, and better reaction time, spatial planning, and visual pattern recognition. A 2015 meta-analysis claimed that "when more complex assessments are used, modafinil appears to consistently engender enhancement of attention, executive functions, and learning" without affecting a user's mood. In a study from earlier this year involving 39 male chess players, subjects taking modafinil were found to perform better in chess games played against a computer.
One symptom of Alzheimer's disease is a reduced brain level of the neurotransmitter called acetylcholine. It is thought that an effective treatment for Alzheimer's disease might be to increase brain levels of acetylcholine. Another possible treatment would be to slow the death of neurons that contain acetylcholine. Two drugs, Tacrine and Donepezil, are both inhibitors of the enzyme (acetylcholinesterase) that breaks down acetylcholine. These drugs are approved in the US for treatment of Alzheimer's disease.
Took pill around 6 PM; I had a very long drive to and from an airport ahead of me, ideal for Adderall. In case it was Adderall, I chewed up the pill - by making it absorb faster, more of the effect would be there when I needed it, during driving, and not lingering in my system past midnight. Was it? I didn't notice any change in my pulse, I yawned several times on the way back, my conversation was not more voluminous than usual. I did stay up later than usual, but that's fully explained by walking to get ice cream. All in all, my best guess was that the pill was placebo, and I feel fairly confident but not hugely confident that it was placebo. I'd give it ~70%. And checking the next morning… I was right! Finally.
Several studies have assessed the effect of MPH and d-AMP on tasks tapping various other aspects of spatial working memory. Three used the spatial working memory task from the CANTAB battery of neuropsychological tests (Sahakian & Owen, 1992). In this task, subjects search for a target at different locations on a screen. Subjects are told that locations containing a target in previous trials will not contain a target in future trials. Efficient performance therefore requires remembering and avoiding these locations in addition to remembering and avoiding locations already searched within a trial. Mehta et al. (2000) found evidence of greater accuracy with MPH, and Elliott et al. (1997) found a trend for the same. In Mehta et al.'s study, this effect depended on subjects' working memory ability: the lower a subject's score on placebo, the greater the improvement on MPH. In Elliott et al.'s study, MPH enhanced performance for the group of subjects who received the placebo first and made little difference for the other group. The reason for this difference is unclear, but as mentioned above, this may reflect ability differences between the groups. More recently, Clatworthy et al. (2009) undertook a positron emission tomography (PET) study of MPH effects on two tasks, one of which was the CANTAB spatial working memory task. They failed to find consistent effects of MPH on working memory performance but did find a systematic relation between the performance effect of the drug in each individual and its effect on individuals' dopamine activity in the ventral striatum.
Ongoing studies are looking into the possible pathways by which nootropic substances function. Researchers have postulated that the mental health advantages derived from these substances can be attributed to their effects on the cholinergic and dopaminergic systems of the brain. These systems regulate two important neurotransmitters, acetylcholine and dopamine. | CommonCrawl |
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
Search for extended sources in the Galactic Plane using 6 years of Fermi-Large Area Telescope Pass 8 data above 10 GeV (1702.00476)
The Fermi LAT Collaboration: M. Ackermann, M. Ajello, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, R. Bellazzini, E. Bissaldi, E. D. Bloom, R. Bonino, E. Bottacini, T. J. Brandt, J. Bregeon, P. Bruel, R. Buehler, R. A. Cameron, M. Caragiulo, P. A. Caraveo, D. Castro, E. Cavazzuti, C. Cecchi, E. Charles, A. Chekhtman, C. C. Cheung, G. Chiaro, S. Ciprini, J.M. Cohen, D. Costantin, F. Costanza, S. Cutini, F. D'Ammando, F. de Palma, R. Desiante, S. W. Digel, N. Di Lalla, M. Di Mauro, L. Di Venere, C. Favuzzi, S. J. Fegan, E. C. Ferrara, A. Franckowiak, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, F. Giordano, M. Giroletti, D. Green, I. A. Grenier, M.-H. Grondin, L. Guillemot, S. Guiriec, A. K. Harding, E. Hays, J.W. Hewitt, D. Horan, X. Hou, G. Jóhannesson, T. Kamae, M. Kuss, G. La Mura, S. Larsson, M. Lemoine-Goumard, J. Li, F. Longo, F. Loparco, P. Lubrano, J. D. Magill, S. Maldera, D. Malyshev, A. Manfreda, M. N. Mazziotta, P. F. Michelson, W. Mitthumsiri, T. Mizuno, M. E. Monzani, A. Morselli, I. V. Moskalenko, M. Negro, E. Nuss, T. Ohsugi, N. Omodei, M. Orienti, E. Orlando, J. F. Ormes, V. S. Paliya, D. Paneque, J. S. Perkins, M. Persic, M. Pesce-Rollins, V. Petrosian, F. Piron, T. A. Porter, G. Principe, S. Raino, R. Rando, M. Razzano, S. Razzaque, A. Reimer, O. Reimer, T. Reposeur, C. Sgro, D. Simone, E. J. Siskind, F. Spada, G. Spandre, P. Spinelli, D. J. Suson, D. Tak, J. B. Thayer, D. J. Thompson, D. F. Torres, G. Tosti, E. Troja, G. Vianello, K. S. Wood, M. Wood
April 11, 2018 astro-ph.HE
The spatial extension of a gamma-ray source is an essential ingredient to determine its spectral properties as well as its potential multi-wavelength counterpart. The capability to spatially resolve gamma-ray sources is greatly improved by the newly delivered Fermi-Large Area Telescope (LAT) Pass 8 event-level analysis which provides a greater acceptance and an improved point spread function, two crucial factors for the detection of extended sources. Here, we present a complete search for extended sources located within 7 degrees from the Galactic plane, using 6 years of LAT data above 10 GeV. We find 46 extended sources and provide their morphological and spectral characteristics. This constitutes the first catalog of hard LAT extended sources, named the Fermi Galactic Extended Source Catalog, which allows a thorough study of the properties of the Galactic plane in the sub-TeV domain.
A wide and collimated radio jet in 3C 84 on the scale of a few hundred gravitational radii (1804.02198)
G. Giovannini, T. Savolainen, M. Orienti, M. Nakamura, H. Nagai, M. Kino, M. Giroletti, K. Hada, G. Bruni, Y.Y. Kovalev, J.M. Anderson, F. D'Ammando, J. Hodgson, M. Honma, T.P. Krichbaum, S.-S. Lee, R. Lico, M.M. Lisakov, A.P. Lobanov, L. Petrov, B.W. Sohn, K.V. Sokolovsky, P.A. Voitsik, J.A. Zensus, S. Tingay
April 6, 2018 astro-ph.GA, astro-ph.HE
Understanding the launching, acceleration, and collimation of jets powered by active galactic nuclei remains an outstanding problem in relativistic astrophysics. This is partly because observational tests of jet formation models suffer from the limited angular resolution of ground-based very long baseline interferometry that has thus far been able to probe the transverse jet structure in the acceleration and collimation zone of only two sources. Here we report radio interferometric observations of 3C 84 (NGC 1275), the central galaxy of the Perseus cluster, made with an array including the orbiting radio telescope of the RadioAstron mission. The obtained image transversely resolves the edge-brightened jet in 3C 84 only 30 microarcseconds from the core, which is ten times closer to the central engine than what has been possible in previous ground-based observations, and it allows us to measure the jet collimation profile from ~ 100 to ~10000 gravitational radii from the black hole. The previously found, almost cylindrical jet profile on scales larger than a few thousand r_g is now seen to continue at least down to a few hundred r_g from the black hole and we find a broad jet with a transverse radius larger than about 250 r_g at only 350 r_g from the core. If the bright outer jet layer is launched by the black hole ergosphere, it has to rapidly expand laterally on scales smaller than 100 r_g. If this is not the case, then this jet sheath is likely launched from the accretion disk.
Science with e-ASTROGAM (A space mission for MeV-GeV gamma-ray astrophysics) (1711.01265)
A. De Angelis, V. Tatischeff, I. A. Grenier, J. McEnery, M. Mallamaci, M. Tavani, U. Oberlack, L. Hanlon, R. Walter, A. Argan, P. Von Ballmoos, A. Bulgarelli, A. Bykov, M. Hernanz, G. Kanbach, I. Kuvvetli, M. Pearce, A. Zdziarski, J. Conrad, G. Ghisellini, A. Harding, J. Isern, M. Leising, F. Longo, G. Madejski, M. Martinez, M. N. Mazziotta, J. M. Paredes, M. Pohl, R. Rando, M. Razzano, A. Aboudan, M. Ackermann, A. Addazi, M. Ajello, C. Albertus, J. M. Alvarez, G. Ambrosi, S. Anton, L. A. Antonelli, A. Babic, B. Baibussinov, M. Balbo, L. Baldini, S. Balman, C. Bambi, U. Barres de Almeida, J. A. Barrio, R. Bartels, D. Bastieri, W. Bednarek, D. Bernard, E. Bernardini, T. Bernasconi, B. Bertucci, A. Biland, E. Bissaldi, M. Boettcher, V. Bonvicini, V. Bosch Ramon, E. Bottacini, V. Bozhilov, T. Bretz, M. Branchesi, V. Brdar, T. Bringmann, A. Brogna, C. Budtz Jorgensen, G. Busetto, S. Buson, M. Busso, A. Caccianiga, S. Camera, R. Campana, P. Caraveo, M. Cardillo, P. Carlson, S. Celestin, M. Cermeno, A. Chen, C. C Cheung, E. Churazov, S. Ciprini, A. Coc, S. Colafrancesco, A. Coleiro, W. Collmar, P. Coppi, R. Curado da Silva, S. Cutini, F. DAmmando, B. De Lotto, D. de Martino, A. De Rosa, M. Del Santo, L. Delgado, R. Diehl, S. Dietrich, A. D. Dolgov, A. Dominguez, D. Dominis Prester, I. Donnarumma, D. Dorner, M. Doro, M. Dutra, D. Elsaesser, M. Fabrizio, A. FernandezBarral, V. Fioretti, L. Foffano, V. Formato, N. Fornengo, L. Foschini, A. Franceschini, A. Franckowiak, S. Funk, F. Fuschino, D. Gaggero, G. Galanti, F. Gargano, D. Gasparrini, R. Gehrz, P. Giammaria, N. Giglietto, P. Giommi, F. Giordano, M. Giroletti, G. Ghirlanda, N. Godinovic, C. Gouiffes, J. E. Grove, C. Hamadache, D. H. Hartmann, M. Hayashida, A. Hryczuk, P. Jean, T. Johnson, J. Jose, S. Kaufmann, B. Khelifi, J. Kiener, J. Knodlseder, M. Kole, J. Kopp, V. Kozhuharov, C. Labanti, S. Lalkovski, P. Laurent, O. Limousin, M. Linares, E. Lindfors, M. Lindner, J. Liu, S. Lombardi, F. Loparco, R. LopezCoto, M. Lopez Moya, B. Lott, P. Lubrano, D. Malyshev, N. Mankuzhiyil, K. Mannheim, M. J. Marcha, A. Marciano, B. Marcote, M. Mariotti, M. Marisaldi, S. McBreen, S. Mereghetti, A. Merle, R. Mignani, G. Minervini, A. Moiseev, A. Morselli, F. Moura, K. Nakazawa, L. Nava, D. Nieto, M. Orienti, M. Orio, E. Orlando, P. Orleanski, S. Paiano, R. Paoletti, A. Papitto, M. Pasquato, B. Patricelli, M. A. PerezGarcia, M. Persic, G. Piano, A. Pichel, M. Pimenta, C. Pittori, T. Porter, J. Poutanen, E. Prandini, N. Prantzos, N. Produit, S. Profumo, F. S. Queiroz, S. Raino, A. Raklev, M. Regis, I. Reichardt, Y. Rephaeli, J. Rico, W. Rodejohann, G. Rodriguez Fernandez, M. Roncadelli, L. Roso, A. Rovero, R. Ruffini, G. Sala, M. A. SanchezConde, A. Santangelo, P. Saz Parkinson, T. Sbarrato, A. Shearer, R. Shellard, K. Short, T. Siegert, C. Siqueira, P. Spinelli, A. Stamerra, S. Starrfield, A. Strong, I. Strumke, F. Tavecchio, R. Taverna, T. Terzic, D. J. Thompson, O. Tibolla, D. F. Torres, R. Turolla, A. Ulyanov, A. Ursi, A. Vacchi, J. Van den Abeele, G. Vankova Kirilovai, C. Venter, F. Verrecchia, P. Vincent, X. Wang, C. Weniger, X. Wu, G. Zaharijas, L. Zampieri, S. Zane, S. Zimmer, A. Zoglauer, the eASTROGAM collaboration
April 5, 2018 hep-ex, astro-ph.SR, astro-ph.IM, astro-ph.HE
e-ASTROGAM (enhanced ASTROGAM) is a breakthrough Observatory space mission, with a detector composed by a Silicon tracker, a calorimeter, and an anticoincidence system, dedicated to the study of the non-thermal Universe in the photon energy range from 0.3 MeV to 3 GeV - the lower energy limit can be pushed to energies as low as 150 keV for the tracker, and to 30 keV for calorimetric detection. The mission is based on an advanced space-proven detector technology, with unprecedented sensitivity, angular and energy resolution, combined with polarimetric capability. Thanks to its performance in the MeV-GeV domain, substantially improving its predecessors, e-ASTROGAM will open a new window on the non-thermal Universe, making pioneering observations of the most powerful Galactic and extragalactic sources, elucidating the nature of their relativistic outflows and their effects on the surroundings. With a line sensitivity in the MeV energy range one to two orders of magnitude better than previous generation instruments, e-ASTROGAM will determine the origin of key isotopes fundamental for the understanding of supernova explosion and the chemical evolution of our Galaxy. The mission will provide unique data of significant interest to a broad astronomical community, complementary to powerful observatories such as LIGO-Virgo-GEO600-KAGRA, SKA, ALMA, E-ELT, TMT, LSST, JWST, Athena, CTA, IceCube, KM3NeT, and LISA.
FBQS J1644+2619: multiwavelength properties and its place in the class of gamma-ray emitting Narrow Line Seyfert 1s (1801.08750)
J. Larsson, F. D'Ammando, S. Falocco, M. Giroletti, M. Orienti, E. Piconcelli, S. Righini
Jan. 26, 2018 astro-ph.GA, astro-ph.HE
A small fraction of Narrow Line Seyfert 1s (NLSy1s) are observed to be gamma-ray emitters. Understanding the properties of these sources is of interest since the majority of NLSy1s are very different from typical blazars. Here, we present a multi-frequency analysis of FBQS J1644+2619, one of the most recently discovered gamma-ray emitting NLSy1s. We analyse an ~80 ks XMM-Newton observation obtained in 2017, as well as quasi-simultaneous multi-wavelength observations covering the radio - gamma-ray range. The spectral energy distribution of the source is similar to the other gamma-ray NLSy1s, confirming its blazar-like nature. The X-ray spectrum is characterised by a hard photon index (Gamma = 1.66) above 2 keV and a soft excess at lower energies.The hard photon index provides clear evidence that inverse Compton emission from the jet dominates the spectrum, while the soft excess can be explained by a contribution from the underlying Seyfert emission. This contribution can be fitted by reflection of emission from the base of the jet, as well as by Comptonisation in a warm, optically thick corona. We discuss our results in the context of the other gamma-ray NLSy1s and note that the majority of them have similar X-ray spectra, with properties intermediate between blazars and radio-quiet NLSy1s.
Science with the Cherenkov Telescope Array (1709.07997)
The Cherenkov Telescope Array Consortium: B.S. Acharya, I. Agudo, I. Al Samarai, R. Alfaro, J. Alfaro, C. Alispach, R. Alves Batista, J.-P. Amans, E. Amato, G. Ambrosi, E. Antolini, L.A. Antonelli, C. Aramo, M. Araya, T. Armstrong, F. Arqueros, L. Arrabito, K. Asano, M. Ashley, M. Backes, C. Balazs, M. Balbo, O. Ballester, J. Ballet, A. Bamba, M. Barkov, U. Barres de Almeida, J.A. Barrio, D. Bastieri, Y. Becherini, A. Belfiore, W. Benbow, D. Berge, E. Bernardini, M.G. Bernardini, M. Bernardos, K. Bernlöhr, B. Bertucci, B. Biasuzzi, C. Bigongiari, A. Biland, E. Bissaldi, J. Biteau, O. Blanch, J. Blazek, C. Boisson, J. Bolmont, G. Bonanno, A. Bonardi, C. Bonavolontà, G. Bonnoli, Z. Bosnjak, M. Böttcher, C. Braiding, J. Bregeon, A. Brill, A.M. Brown, P. Brun, G. Brunetti, T. Buanes, J. Buckley, V. Bugaev, R. Bühler, A. Bulgarelli, T. Bulik, M. Burton, A. Burtovoi, G. Busetto, R. Canestrari, M. Capalbi, F. Capitanio, A. Caproni, P. Caraveo, V. Cárdenas, C. Carlile, R. Carosi, E. Carquín, J. Carr, S. Casanova, E. Cascone, F. Catalani, O. Catalano, D. Cauz, M. Cerruti, P. Chadwick, S. Chaty, R.C.G. Chaves, A. Chen, X. Chen, M. Chernyakova, M. Chikawa, A. Christov, J. Chudoba, M. Cieślar, V. Coco, S. Colafrancesco, P. Colin, V. Conforti, V. Connaughton, J. Conrad, J.L. Contreras, J. Cortina, A. Costa, H. Costantini, G. Cotter, S. Covino, R. Crocker, J. Cuadra, O. Cuevas, P. Cumani, A. D'Aì, F. D'Ammando, P. D'Avanzo, D. D'Urso, M. Daniel, I. Davids, B. Dawson, F. Dazzi, A. De Angelis, R. de Cássia dos Anjos, G. De Cesare, A. De Franco, E.M. de Gouveia Dal Pino, I. de la Calle, R. de los Reyes Lopez, B. De Lotto, A. De Luca, M. De Lucia, M. de Naurois, E. de Oña Wilhelmi, F. De Palma, F. De Persio, V. de Souza, C. Deil, M. Del Santo, C. Delgado, D. della Volpe, T. Di Girolamo, F. Di Pierro, L. Di Venere, C. Díaz, C. Dib, S. Diebold, A. Djannati-Ataï, A. Domínguez, D. Dominis Prester, D. Dorner, M. Doro, H. Drass, D. Dravins, G. Dubus, V.V. Dwarkadas, J. Ebr, C. Eckner, K. Egberts, S. Einecke, T.R.N. Ekoume, D. Elsässer, J.-P. Ernenwein, C. Espinoza, C. Evoli, M. Fairbairn, D. Falceta-Goncalves, A. Falcone, C. Farnier, G. Fasola, E. Fedorova, S. Fegan, M. Fernandez-Alonso, A. Fernández-Barral, G. Ferrand, M. Fesquet, M. Filipovic, V. Fioretti, G. Fontaine, M. Fornasa, L. Fortson, L. Freixas Coromina, C. Fruck, Y. Fujita, Y. Fukazawa, S. Funk, M. Füßling, S. Gabici, A. Gadola, Y. Gallant, B. Garcia, R. Garcia López, M. Garczarczyk, J. Gaskins, T. Gasparetto, M. Gaug, L. Gerard, G. Giavitto, N. Giglietto, P. Giommi, F. Giordano, E. Giro, M. Giroletti, A. Giuliani, J.-F. Glicenstein, R. Gnatyk, N. Godinovic, P. Goldoni, G. Gómez-Vargas, M.M. González, J.M. González, D. Götz, J. Graham, P. Grandi, J. Granot, A.J. Green, T. Greenshaw, S. Griffiths, S. Gunji, D. Hadasch, S. Hara, M.J. Hardcastle, T. Hassan, K. Hayashi, M. Hayashida, M. Heller, J.C. Helo, G. Hermann, J. Hinton, B. Hnatyk, W. Hofmann, J. Holder, D. Horan, J. Hörandel, D. Horns, P. Horvath, T. Hovatta, M. Hrabovsky, D. Hrupec, T.B. Humensky, M. Hütten, M. Iarlori, T. Inada, Y. Inome, S. Inoue, T. Inoue, Y. Inoue, F. Iocco, K. Ioka, M. Iori, K. Ishio, Y. Iwamura, M. Jamrozy, P. Janecek, D. Jankowsky, P. Jean, I. Jung-Richardt, J. Jurysek, P. Kaaret, S. Karkar, H. Katagiri, U. Katz, N. Kawanaka, D. Kazanas, B. Khélifi, D.B. Kieda, S. Kimeswenger, S. Kimura, S. Kisaka, J. Knapp, J. Knödlseder, B. Koch, K. Kohri, N. Komin, K. Kosack, M. Kraus, M. Krause, F. Krauß, H. Kubo, G. Kukec Mezek, H. Kuroda, J. Kushida, N. La Palombara, G. Lamanna, R.G. Lang, J. Lapington, O. Le Blanc, S. Leach, J.-P. Lees, J. Lefaucheur, M.A. Leigui de Oliveira, J.-P. Lenain, R. Lico, M. Limon, E. Lindfors, T. Lohse, S. Lombardi, F. Longo, M. López, R. López-Coto, C.-C. Lu, F. Lucarelli, P.L. Luque-Escamilla, E. Lyard, M.C. Maccarone, G. Maier, P. Majumdar, G. Malaguti, D. Mandat, G. Maneva, M. Manganaro, S. Mangano, A. Marcowith, J. Marín, S. Markoff, J. Martí, P. Martin, M. Martínez, G. Martínez, N. Masetti, S. Masuda, G. Maurin, N. Maxted, D. Mazin, C. Medina, A. Melandri, S. Mereghetti, M. Meyer, I.A. Minaya, N. Mirabal, R. Mirzoyan, A. Mitchell, T. Mizuno, R. Moderski, M. Mohammed, L. Mohrmann, T. Montaruli, A. Moralejo, D. Morcuende-Parrilla, K. Mori, G. Morlino, P. Morris, A. Morselli, E. Moulin, R. Mukherjee, C. Mundell, T. Murach, H. Muraishi, K. Murase, A. Nagai, S. Nagataki, T. Nagayoshi, T. Naito, T. Nakamori, Y. Nakamura, J. Niemiec, D. Nieto, M. Nikołajuk, K. Nishijima, K. Noda, D. Nosek, B. Novosyadlyj, S. Nozaki, P. O'Brien, L. Oakes, Y. Ohira, M. Ohishi, S. Ohm, N. Okazaki, A. Okumura, R.A. Ong, M. Orienti, R. Orito, J.P. Osborne, M. Ostrowski, N. Otte, I. Oya, M. Padovani, A. Paizis, M. Palatiello, M. Palatka, R. Paoletti, J.M. Paredes, G. Pareschi, R.D. Parsons, A. Pe'er, M. Pech, G. Pedaletti, M. Perri, M. Persic, A. Petrashyk, P. Petrucci, O. Petruk, B. Peyaud, M. Pfeifer, G. Piano, A. Pisarski, S. Pita, M. Pohl, M. Polo, D. Pozo, E. Prandini, J. Prast, G. Principe, D. Prokhorov, H. Prokoph, M. Prouza, G. Pühlhofer, M. Punch, S. Pürckhauer, F. Queiroz, A. Quirrenbach, S. Rainò, S. Razzaque, O. Reimer, A. Reimer, A. Reisenegger, M. Renaud, A.H. Rezaeian, W. Rhode, D. Ribeiro, M. Ribó, T. Richtler, J. Rico, F. Rieger, M. Riquelme, S. Rivoire, V. Rizi, J. Rodriguez, G. Rodriguez Fernandez, J.J. Rodríguez Vázquez, G. Rojas, P. Romano, G. Romeo, J. Rosado, A.C. Rovero, G. Rowell, B. Rudak, A. Rugliancich, C. Rulten, I. Sadeh, S. Safi-Harb, T. Saito, N. Sakaki, S. Sakurai, G. Salina, M. Sánchez-Conde, H. Sandaker, A. Sandoval, P. Sangiorgi, M. Sanguillon, H. Sano, M. Santander, S. Sarkar, K. Satalecka, F.G. Saturni, E.J. Schioppa, S. Schlenstedt, M. Schneider, H. Schoorlemmer, P. Schovanek, A. Schulz, F. Schussler, U. Schwanke, E. Sciacca, S. Scuderi, I. Seitenzahl, D. Semikoz, O. Sergijenko, M. Servillat, A. Shalchi, R.C. Shellard, L. Sidoli, H. Siejkowski, A. Sillanpää, G. Sironi, J. Sitarek, V. Sliusar, A. Slowikowska, H. Sol, A. Stamerra, S. Stanič, R. Starling, Ł. Stawarz, S. Stefanik, M. Stephan, T. Stolarczyk, G. Stratta, U. Straumann, T. Suomijarvi, A.D. Supanitsky, G. Tagliaferri, H. Tajima, M. Tavani, F. Tavecchio, J.-P. Tavernet, K. Tayabaly, L.A. Tejedor, P. Temnikov, Y. Terada, R. Terrier, T. Terzic, M. Teshima, V. Testa, S. Thoudam, W. Tian, L. Tibaldo, M. Tluczykont, C.J. Todero Peixoto, F. Tokanai, J. Tomastik, D. Tonev, M. Tornikoski, D.F. Torres, E. Torresi, G. Tosti, N. Tothill, G. Tovmassian, P. Travnicek, C. Trichard, M. Trifoglio, I. Troyano Pujadas, S. Tsujimoto, G. Umana, V. Vagelli, F. Vagnetti, M. Valentino, P. Vallania, L. Valore, C. van Eldik, J. Vandenbroucke, G.S. Varner, G. Vasileiadis, V. Vassiliev, M. Vázquez Acosta, M. Vecchi, A. Vega, S. Vercellone, P. Veres, S. Vergani, V. Verzi, G.P. Vettolani, A. Viana, C. Vigorito, J. Villanueva, H. Voelk, A. Vollhardt, S. Vorobiov, M. Vrastil, T. Vuillaume, S.J. Wagner, R. Wagner, R. Walter, J.E. Ward, D. Warren, J.J. Watson, F. Werner, M. White, R. White, A. Wierzcholska, P. Wilcox, M. Will, D.A. Williams, R. Wischnewski, M. Wood, T. Yamamoto, R. Yamazaki, S. Yanagita, L. Yang, T. Yoshida, S. Yoshiike, T. Yoshikoshi, M. Zacharias, G. Zaharijas, L. Zampieri, F. Zandanel, R. Zanin, M. Zavrtanik, D. Zavrtanik, A.A. Zdziarski, A. Zech, H. Zechlin, V.I. Zhdanov, A. Ziegler, J. Zorn
Jan. 22, 2018 hep-ex, astro-ph.IM, astro-ph.HE
The Cherenkov Telescope Array, CTA, will be the major global observatory for very high energy gamma-ray astronomy over the next decade and beyond. The scientific potential of CTA is extremely broad: from understanding the role of relativistic cosmic particles to the search for dark matter. CTA is an explorer of the extreme universe, probing environments from the immediate neighbourhood of black holes to cosmic voids on the largest scales. Covering a huge range in photon energy from 20 GeV to 300 TeV, CTA will improve on all aspects of performance with respect to current instruments. The observatory will operate arrays on sites in both hemispheres to provide full sky coverage and will hence maximize the potential for the rarest phenomena such as very nearby supernovae, gamma-ray bursts or gravitational wave transients. With 99 telescopes on the southern site and 19 telescopes on the northern site, flexible operation will be possible, with sub-arrays available for specific tasks. CTA will have important synergies with many of the new generation of major astronomical and astroparticle observatories. Multi-wavelength and multi-messenger approaches combining CTA data with those from other instruments will lead to a deeper understanding of the broad-band non-thermal properties of target sources. The CTA Observatory will be operated as an open, proposal-driven observatory, with all data available on a public archive after a pre-defined proprietary period. Scientists from institutions worldwide have combined together to form the CTA Consortium. This Consortium has prepared a proposal for a Core Programme of highly motivated observations. The programme, encompassing approximately 40% of the available observing time over the first ten years of CTA operation, is made up of individual Key Science Projects (KSPs), which are presented in this document.
Cherenkov Telescope Array Contributions to the 35th International Cosmic Ray Conference (ICRC2017) (1709.03483)
F. Acero, B.S. Acharya, V. Acín Portella, C. Adams, I. Agudo, F. Aharonian, I. Al Samarai, A. Alberdi, M. Alcubierre, R. Alfaro, J. Alfaro, C. Alispach, R. Aloisio, R. Alves Batista, J.-P. Amans, E. Amato, L. Ambrogi, G. Ambrosi, M. Ambrosio, J. Anderson, M. Anduze, E.O. Angüner, E. Antolini, L.A. Antonelli, V. Antonuccio, P. Antoranz, C. Aramo, M. Araya, C. Arcaro, T. Armstrong, F. Arqueros, L. Arrabito, M. Arrieta, K. Asano, A. Asano, M. Ashley, P. Aubert, C. B. Singh, A. Babic, M. Backes, S. Bajtlik, C. Balazs, M. Balbo, O. Ballester, J. Ballet, L. Ballo, A. Balzer, A. Bamba, R. Bandiera, P. Barai, C. Barbier, M. Barcelo, M. Barkov, U. Barres de Almeida, J.A. Barrio, D. Bastieri, C. Bauer, U. Becciani, Y. Becherini, J. Becker Tjus, W. Bednarek, A. Belfiore, W. Benbow, M. Benito, D. Berge, E. Bernardini, M.G. Bernardini, M. Bernardos, S. Bernhard, K. Bernlöhr, C. Bertinelli Salucci, B. Bertucci, M.-A. Besel, V. Beshley, J. Bettane, N. Bhatt, W. Bhattacharyya, S. Bhattachryya, B. Biasuzzi, G. Bicknell, C. Bigongiari, A. Biland, A. Bilinsky, R. Bird, E. Bissaldi, J. Biteau, M. Bitossi, O. Blanch, P. Blasi, J. Blazek, C. Boccato, C. Bockermann, C. Boehm, M. Bohacova, C. Boisson, J. Bolmont, G. Bonanno, A. Bonardi, C. Bonavolontà, G. Bonnoli, J. Borkowski, R. Bose, Z. Bosnjak, M. Böttcher, C. Boutonnet, F. Bouyjou, L. Bowman, V. Bozhilov, C. Braiding, S. Brau-Nogué, J. Bregeon, M. Briggs, A. Brill, W. Brisken, D. Bristow, R. Britto, E. Brocato, A.M. Brown, S. Brown, K. Brügge, P. Brun, P. Brun, F. Brun, L. Brunetti, G. Brunetti, P. Bruno, M. Bryan, J. Buckley, V. Bugaev, R. Bühler, A. Bulgarelli, T. Bulik, M. Burton, A. Burtovoi, G. Busetto, S. Buson, J. Buss, K. Byrum, A. Caccianiga, R. Cameron, F. Canelli, R. Canestrari, M. Capalbi, M. Capasso, F. Capitanio, A. Caproni, R. Capuzzo-Dolcetta, P. Caraveo, V. Cárdenas, J. Cardenzana, M. Cardillo, C. Carlile, S. Caroff, R. Carosi, A. Carosi, E. Carquín, J. Carr, J.-M. Casandjian, S. Casanova, E. Cascone, A.J. Castro-Tirado, J. Castroviejo Mora, F. Catalani, O. Catalano, D. Cauz, C. Celestino Silva, S. Celli, M. Cerruti, E. Chabanne, P. Chadwick, N. Chakraborty, C. Champion, A. Chatterjee, S. Chaty, R. Chaves, A. Chen, X. Chen, K. Cheng, M. Chernyakova, M. Chikawa, V.R. Chitnis, A. Christov, J. Chudoba, M. Cieślar, P. Clark, V. Coco, S. Colafrancesco, P. Colin, E. Colombo, J. Colome, S. Colonges, V. Conforti, V. Connaughton, J. Conrad, J.L. Contreras, R. Cornat, J. Cortina, A. Costa, H. Costantini, G. Cotter, B. Courty, S. Covino, G. Covone, P. Cristofari, S.J. Criswell, R. Crocker, J. Croston, C. Crovari, J. Cuadra, O. Cuevas, X. Cui, P. Cumani, G. Cusumano, A. D'Aì, F. D'Ammando, P. D'Avanzo, D. D'Urso, P. Da Vela, Ø. Dale, V.T. Dang, L. Dangeon, M. Daniel, I. Davids, B. Dawson, F. Dazzi, A. De Angelis, V. De Caprio, R. de Cássia dos Anjos, G. De Cesare, A. De Franco, F. De Frondat, E.M. de Gouveia Dal Pino, I. de la Calle, C. De Lisio, R. de los Reyes Lopez, B. De Lotto, A. De Luca, M. De Lucia, J.R.T. de Mello Neto, M. de Naurois, E. de Oña Wilhelmi, F. De Palma, F. De Persio, V. de Souza, J. Decock, C. Deil, P. Deiml, M. Del Santo, E. Delagnes, G. Deleglise, M. Delfino Reznicek, C. Delgado, J. Delgado Mengual, R. Della Ceca, D. della Volpe, M. Detournay, J. Devin, T. Di Girolamo, C. Di Giulio, F. Di Pierro, L. Di Venere, L. Diaz, C. Díaz, C. Dib, H. Dickinson, S. Diebold, S. Digel, A. Djannati-Ataï, M. Doert, A. Domínguez, D. Dominis Prester, I. Donnarumma, D. Dorner, M. Doro, J.-L. Dournaux, T. Downes, G. Drake, S. Drappeau, H. Drass, D. Dravins, L. Drury, G. Dubus, K. Dundas Morå, A. Durkalec, V. Dwarkadas, J. Ebr, C. Eckner, E. Edy, K. Egberts, S. Einecke, J. Eisch, F. Eisenkolb, T.R.N. Ekoume, C. Eleftheriadis, D. Elsässer, D. Emmanoulopoulos, J.-P. Ernenwein, P. Escarate, S. Eschbach, C. Espinoza, P. Evans, C. Evoli, M. Fairbairn, D. Falceta-Goncalves, A. Falcone, V. Fallah Ramazani, K. Farakos, E. Farrell, G. Fasola, Y. Favre, E. Fede, R. Fedora, E. Fedorova, S. Fegan, M. Fernandez-Alonso, A. Fernández-Barral, G. Ferrand, O. Ferreira, M. Fesquet, E. Fiandrini, A. Fiasson, M. Filipovic, D. Fink, J.P. Finley, C. Finley, A. Finoguenov, V. Fioretti, M. Fiorini, H. Flores, L. Foffano, C. Föhr, M.V. Fonseca, L. Font, G. Fontaine, M. Fornasa, P. Fortin, L. Fortson, N. Fouque, B. Fraga, F.J. Franco, L. Freixas Coromina, C. Fruck, D. Fugazza, Y. Fujita, S. Fukami, Y. Fukazawa, Y. Fukui, S. Funk, A. Furniss, M. Füßling, S. Gabici, A. Gadola, Y. Gallant, D. Galloway, S. Gallozzi, B. Garcia, A. Garcia, R. García Gil, R. Garcia López, M. Garczarczyk, D. Gardiol, F. Gargano, C. Gargano, S. Garozzo, M. Garrido-Ruiz, D. Gascon, T. Gasparetto, F. Gaté, M. Gaug, B. Gebhardt, M. Gebyehu, N. Geffroy, B. Genolini, A. Ghalumyan, A. Ghedina, G. Ghirlanda, P. Giammaria, F. Gianotti, B. Giebels, N. Giglietto, V. Gika, R. Gimenes, P. Giommi, F. Giordano, G. Giovannini, E. Giro, M. Giroletti, J. Gironnet, A. Giuliani, J.-F. Glicenstein, R. Gnatyk, N. Godinovic, P. Goldoni, J.L. Gómez, G. Gómez-Vargas, M.M. González, J.M. González, K.S. Gothe, D. Gotz, J. Goullon, T. Grabarczyk, R. Graciani, J. Graham, P. Grandi, J. Granot, G. Grasseau, R. Gredig, A.J. Green, T. Greenshaw, I. Grenier, S. Griffiths, A. Grillo, M.-H. Grondin, J. Grube, V. Guarino, B. Guest, O. Gueta, S. Gunji, G. Gyuk, D. Hadasch, L. Hagge, J. Hahn, A. Hahn, H. Hakobyan, S. Hara, M.J. Hardcastle, T. Hassan, T. Haubold, A. Haupt, K. Hayashi, M. Hayashida, H. He, M. Heller, J.C. Helo, F. Henault, G. Henri, G. Hermann, R. Hermel, J. Herrera Llorente, A. Herrero, O. Hervet, N. Hidaka, J. Hinton, N. Hiroshima, K. Hirotani, B. Hnatyk, J.K. Hoang, D. Hoffmann, W. Hofmann, J. Holder, D. Horan, J. Hörandel, M. Hörbe, D. Horns, P. Horvath, J. Houles, T. Hovatta, M. Hrabovsky, D. Hrupec, J.-M. Huet, G. Hughes, D. Hui, G. Hull, T.B. Humensky, M. Hussein, M. Hütten, M. Iarlori, Y. Ikeno, J.M. Illa, D. Impiombato, T. Inada, A. Ingallinera, Y. Inome, S. Inoue, T. Inoue, Y. Inoue, F. Iocco, K. Ioka, M. Ionica, M. Iori, A. Iriarte, K. Ishio, G.L. Israel, Y. Iwamura, C. Jablonski, A. Jacholkowska, J. Jacquemier, M. Jamrozy, P. Janecek, F. Jankowsky, D. Jankowsky, P. Jansweijer, C. Jarnot, P. Jean, C.A. Johnson, M. Josselin, I. Jung-Richardt, J. Jurysek, P. Kaaret, P. Kachru, M. Kagaya, J. Kakuwa, O. Kalekin, R. Kankanyan, A. Karastergiou, M. Karczewski, S. Karkar, H. Katagiri, J. Kataoka, K. Katarzyński, U. Katz, N. Kawanaka, L. Kaye, D. Kazanas, N. Kelley-Hoskins, B. Khélifi, D.B. Kieda, T. Kihm, S. Kimeswenger, S. Kimura, S. Kisaka, S. Kishida, R. Kissmann, W. Kluźniak, J. Knapen, J. Knapp, J. Knödlseder, B. Koch, J. Kocot, K. Kohri, N. Komin, A. Kong, Y. Konno, K. Kosack, G. Kowal, S. Koyama, M. Kraus, M. Krause, F. Krauß, F. Krennrich, P. Kruger, H. Kubo, V. Kudryavtsev, G. Kukec Mezek, S. Kumar, H. Kuroda, J. Kushida, P. Kushwaha, N. La Palombara, V. La Parola, G. La Rosa, R. Lahmann, K. Lalik, G. Lamanna, M. Landoni, D. Landriu, H. Landt, R.G. Lang, J. Lapington, P. Laporte, O. Le Blanc, T. Le Flour, P. Le Sidaner, S. Leach, A. Leckngam, S.-H. Lee, W.H. Lee, J.-P. Lees, J. Lefaucheur, M.A. Leigui de Oliveira, M. Lemoine-Goumard, J.-P. Lenain, G. Leto, R. Lico, M. Limon, R. Lindemann, E. Lindfors, L. Linhoff, A. Lipniacka, S. Lloyd, T. Lohse, S. Lombardi, F. Longo, M. Lopez, R. Lopez-Coto, T. Louge, F. Louis, M. Louys, F. Lucarelli, D. Lucchesi, P.L. Luque-Escamilla, E. Lyard, M.C. Maccarone, T. Maccarone, E. Mach, G.M. Madejski, G. Maier, A. Majczyna, P. Majumdar, M. Makariev, G. Malaguti, A. Malouf, S. Maltezos, D. Malyshev, D. Malyshev, D. Mandat, G. Maneva, M. Manganaro, S. Mangano, P. Manigot, K. Mannheim, N. Maragos, D. Marano, A. Marcowith, J. Marín, M. Mariotti, M. Marisaldi, S. Markoff, J. Martí, J.-M. Martin, P. Martin, L. Martin, M. Martínez, G. Martínez, O. Martínez, R. Marx, N. Masetti, P. Massimino, A. Mastichiadis, M. Mastropietro, S. Masuda, H. Matsumoto, N. Matthews, S. Mattiazzo, G. Maurin, N. Maxted, M. Mayer, D. Mazin, M.N. Mazziotta, L. Mc Comb, I. McHardy, C. Medina, A. Melandri, C. Melioli, D. Melkumyan, S. Mereghetti, J.-L. Meunier, T. Meures, M. Meyer, S. Micanovic, T. Michael, J. Michałowski, I. Mievre, J. Miller, I.A. Minaya, T. Mineo, F. Mirabel, J.M. Miranda, R. Mirzoyan, A. Mitchell, T. Mizuno, R. Moderski, M. Mohammed, L. Mohrmann, C. Molijn, E. Molinari, R. Moncada, T. Montaruli, I. Monteiro, D. Mooney, P. Moore, A. Moralejo, D. Morcuende-Parrilla, E. Moretti, K. Mori, G. Morlino, P. Morris, A. Morselli, F. Moscato, D. Motohashi, E. Moulin, S. Mueller, R. Mukherjee, P. Munar, C. Mundell, J. Mundet, T. Murach, H. Muraishi, K. Murase, A. Murphy, A. Nagai, N. Nagar, S. Nagataki, T. Nagayoshi, B.K. Nagesh, T. Naito, D. Nakajima, T. Nakamori, Y. Nakamura, K. Nakayama, D. Naumann, P. Nayman, D. Neise, L. Nellen, R. Nemmen, A. Neronov, N. Neyroud, T. Nguyen, T.T. Nguyen, T. Nguyen Trung, L. Nicastro, J. Nicolau-Kukliński, J. Niemiec, D. Nieto, M. Nievas-Rosillo, M. Nikołajuk, K. Nishijima, K.-I. Nishikawa, G. Nishiyama, K. Noda, L. Nogues, S. Nolan, D. Nosek, M. Nöthe, B. Novosyadlyj, S. Nozaki, F. Nunio, P. O'Brien, L. Oakes, C. Ocampo, J.P. Ochoa, R. Oger, Y. Ohira, M. Ohishi, S. Ohm, N. Okazaki, A. Okumura, J.-F. Olive, R.A. Ong, M. Orienti, R. Orito, A. Orlati, J.P. Osborne, M. Ostrowski, N. Otte, Z. Ou, E. Ovcharov, I. Oya, A. Ozieblo, M. Padovani, S. Paiano, A. Paizis, J. Palacio, M. Palatiello, M. Palatka, J. Pallotta, J.-L. Panazol, D. Paneque, M. Panter, R. Paoletti, M. Paolillo, A. Papitto, A. Paravac, J.M. Paredes, G. Pareschi, R.D. Parsons, P. Paśko, S. Pavy, A. Pe'er, M. Pech, G. Pedaletti, P. Peñil Del Campo, A. Perez, M.A. Pérez-Torres, L. Perri, M. Perri, M. Persic, A. Petrashyk, S. Petrera, P.-O. Petrucci, O. Petruk, B. Peyaud, M. Pfeifer, G. Piano, Q. Piel, D. Pieloth, F. Pintore, C. Pio García, A. Pisarski, S. Pita, L. Pizarro, Ł. Platos, M. Pohl, V. Poireau, A. Pollo, J. Porthault, J. Poutanen, D. Pozo, E. Prandini, P. Prasit, J. Prast, K. Pressard, G. Principe, D. Prokhorov, H. Prokoph, M. Prouza, G. Pruteanu, E. Pueschel, G. Pühlhofer, I. Puljak, M. Punch, S. Pürckhauer, F. Queiroz, J. Quinn, A. Quirrenbach, I. Rafighi, S. Rainò, P.J. Rajda, R. Rando, R.C. Rannot, S. Razzaque, I. Reichardt, O. Reimer, A. Reimer, A. Reisenegger, M. Renaud, T. Reposeur, B. Reville, A.H. Rezaeian, W. Rhode, D. Ribeiro, M. Ribó, M.G. Richer, T. Richtler, J. Rico, F. Rieger, M. Riquelme, P.R. Ristori, S. Rivoire, V. Rizi, J. Rodriguez, G. Rodriguez Fernandez, J.J. Rodríguez Vázquez, G. Rojas, P. Romano, G. Romeo, M. Roncadelli, J. Rosado, S. Rosen, S. Rosier Lees, J. Rousselle, A.C. Rovero, G. Rowell, B. Rudak, A. Rugliancich, J.E. Ruíz del Mazo, W. Rujopakarn, C. Rulten, F. Russo, O. Saavedra, S. Sabatini, B. Sacco, I. Sadeh, E. Sæther Hatlen, S. Safi-Harb, V. Sahakian, S. Sailer, T. Saito, N. Sakaki, S. Sakurai, D. Salek, F. Salesa Greus, G. Salina, D. Sanchez, M. Sánchez-Conde, H. Sandaker, A. Sandoval, P. Sangiorgi, M. Sanguillon, H. Sano, M. Santander, A. Santangelo, E.M. Santos, A. Sanuy, L. Sapozhnikov, S. Sarkar, K. Satalecka, Y. Sato, F.G. Saturni, R. Savalle, M. Sawada, S. Schanne, E.J. Schioppa, S. Schlenstedt, T. Schmidt, J. Schmoll, M. Schneider, H. Schoorlemmer, P. Schovanek, A. Schulz, F. Schussler, U. Schwanke, J. Schwarz, T. Schweizer, S. Schwemmer, E. Sciacca, S. Scuderi, M. Seglar-Arroyo, A. Segreto, I. Seitenzahl, D. Semikoz, O. Sergijenko, N. Serre, M. Servillat, K. Seweryn, K. Shah, A. Shalchi, M. Sharma, R.C. Shellard, I. Shilon, L. Sidoli, M. Sidz, H. Siejkowski, J. Silk, A. Sillanpää, D. Simone, B.B. Singh, G. Sironi, J. Sitarek, P. Sizun, V. Sliusar, A. Slowikowska, A. Smith, D. Sobczyńska, A. Sokolenko, H. Sol, G. Sottile, W. Springer, O. Stahl, A. Stamerra, S. Stanič, R. Starling, D. Staszak, Ł. Stawarz, R. Steenkamp, S. Stefanik, C. Stegmann, S. Steiner, C. Stella, M. Stephan, R. Sternberger, M. Sterzel, B. Stevenson, M. Stodulska, M. Stodulski, T. Stolarczyk, G. Stratta, U. Straumann, R. Stuik, M. Suchenek, T. Suomijarvi, A.D. Supanitsky, T. Suric, I. Sushch, P. Sutcliffe, J. Sykes, M. Szanecki, T. Szepieniec, G. Tagliaferri, H. Tajima, K. Takahashi, H. Takahashi, M. Takahashi, L. Takalo, S. Takami, J. Takata, J. Takeda, T. Tam, M. Tanaka, T. Tanaka, Y. Tanaka, S. Tanaka, C. Tanci, M. Tavani, F. Tavecchio, J.-P. Tavernet, K. Tayabaly, L.A. Tejedor, F. Temme, P. Temnikov, Y. Terada, J.C. Terrazas, R. Terrier, D. Terront, T. Terzic, D. Tescaro, M. Teshima, V. Testa, S. Thoudam, W. Tian, L. Tibaldo, A. Tiengo, D. Tiziani, M. Tluczykont, C.J. Todero Peixoto, F. Tokanai, M. Tokarz, K. Toma, J. Tomastik, A. Tonachini, D. Tonev, M. Tornikoski, D.F. Torres, E. Torresi, G. Tosti, T. Totani, N. Tothill, F. Toussenel, G. Tovmassian, N. Trakarnsirinont, P. Travnicek, C. Trichard, M. Trifoglio, I. Troyano Pujadas, M. Tsirou, S. Tsujimoto, T. Tsuru, Y. Uchiyama, G. Umana, M. Uslenghi, V. Vagelli, F. Vagnetti, M. Valentino, P. Vallania, L. Valore, A.M. Van den Berg, W. van Driel, C. van Eldik, B. van Soelen, J. Vandenbroucke, J. Vanderwalt, G.S. Varner, G. Vasileiadis, V. Vassiliev, J.R. Vázquez, M. Vázquez Acosta, M. Vecchi, A. Vega, P. Veitch, P. Venault, C. Venter, S. Vercellone, P. Veres, S. Vergani, V. Verzi, G.P. Vettolani, C. Veyssiere, A. Viana, J. Vicha, C. Vigorito, J. Villanueva, P. Vincent, J. Vink, F. Visconti, V. Vittorini, H. Voelk, V. Voisin, A. Vollhardt, S. Vorobiov, I. Vovk, M. Vrastil, T. Vuillaume, S.J. Wagner, R. Wagner, P. Wagner, S.P. Wakely, T. Walstra, R. Walter, M. Ward, J.E. Ward, D. Warren, J.J. Watson, N. Webb, P. Wegner, O. Weiner, A. Weinstein, C. Weniger, F. Werner, H. Wetteskind, M. White, R. White, A. Wierzcholska, S. Wiesand, R. Wijers, P. Wilcox, A. Wilhelm, M. Wilkinson, M. Will, D.A. Williams, M. Winter, P. Wojcik, D. Wolf, M. Wood, A. Wörnlein, T. Wu, K.K. Yadav, C. Yaguna, T. Yamamoto, H. Yamamoto, N. Yamane, R. Yamazaki, S. Yanagita, L. Yang, D. Yelos, T. Yoshida, M. Yoshida, S. Yoshiike, T. Yoshikoshi, P. Yu, D. Zaborov, M. Zacharias, G. Zaharijas, A. Zajczyk, L. Zampieri, F. Zandanel, R. Zanin, R. Zanmar Sanchez, D. Zaric, M. Zavrtanik, D. Zavrtanik, A.A. Zdziarski, A. Zech, H. Zechlin, V.I. Zhdanov, A. Ziegler, J. Ziemann, K. Ziętara, A. Zink, J. Ziółkowski, V. Zitelli, A. Zoli, J. Zorn
Oct. 3, 2017 astro-ph.HE
List of contributions from the Cherenkov Telescope Array Consortium presented at the 35th International Cosmic Ray Conference, July 12-20 2017, Busan, Korea.
Enhanced Polarized Emission from the One-Parsec-Scale Hotspot of 3C 84 as a Result of the Interaction with Clumpy Ambient Medium (1709.06708)
H. Nagai, Y. Fujita, M. Nakamura, M. Orienti, M. Kino, K. Asada, G. Giovannini
Sept. 20, 2017 astro-ph.HE
We present Very Long Baseline Array polarimetric observations of the innermost jet of 3C$\sim$84 (NGC$\sim$1275) at 43$\sim$GHz. A significant polarized emission is detected at the hotspot of the innermost re-started jet, which is located $\sim$1 pc south from the radio core. While the previous report presented a hotspot at the southern end of the western limb, the hotspot location has been moved to the southern end of the eastern limb. Faraday rotation is detected within an entire bandwidth of the 43-GHz band. The measured rotation measure (RM) is at most (6.3$\pm$1.9)$\times10^{5}$$\sim$rad$\sim$m$^{-2}$ and might be slightly time variable on the timescale of a month by a factor of a few. Our measured RM and the RM previously reported by the CARMA and SMA observations cannot be consistently explained by the spherical accretion flow with a power-law profile. We propose that a clumpy/inhomogeneous ambient medium is responsible for the observed rotation measure. Using equipartition magnetic field, we derive the electron density of $2\times10^{4}$$\sim$cm$^{-3}$. Such an electron density is consistent with the cloud of narrow line emission region around the central engine. We also discuss the magnetic field configuration from black hole scale to pc scale and the origin of low polarization.
The second catalog of flaring gamma-ray sources from the Fermi All-sky Variability Analysis (1612.03165)
S. Abdollahi, M. Ackermann, M. Ajello, A. Albert, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, J. Becerra Gonzalez, R. Bellazzini, E. Bissaldi, R. D. Blandford, E. D. Bloom, R. Bonino, E. Bottacini, J. Bregeon, P. Bruel, R. Buehler, S. Buson, R. A. Cameron, M. Caragiulo, P. A. Caraveo, E. Cavazzuti, C. Cecchi, A. Chekhtman, C. C. Cheung, G. Chiaro, S. Ciprini, J. Conrad, D. Costantin, F. Costanza, S. Cutini, F. D'Ammando, F. de Palma, A. Desai, R. Desiante, S. W. Digel, N. Di Lalla, M. Di Mauro, L. Di Venere, B. Donaggio, P. S. Drell, C. Favuzzi, S. J. Fegan, E. C. Ferrara, W. B. Focke, A. Franckowiak, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, M. Giomi, F. Giordano, M. Giroletti, T. Glanzman, D. Green, I. A. Grenier, J. E. Grove, L. Guillemot, S. Guiriec, E. Hays, D. Horan, T. Jogler, G. Jóhannesson, A. S. Johnson, D. Kocevski, M. Kuss, G. La Mura, S. Larsson, L. Latronico, J. Li, F. Longo, F. Loparco, M. N. Lovellette, P. Lubrano, J. D. Magill, S. Maldera, A. Manfreda, M. Mayer, M. N. Mazziotta, P. F. Michelson, W. Mitthumsiri, T. Mizuno, M. E. Monzani, A. Morselli, I. V. Moskalenko, M. Negro, E. Nuss, T. Ohsugi, N. Omodei, M. Orienti, E. Orlando, V. S. Paliya, D. Paneque, J. S. Perkins, M. Persic, M. Pesce-Rollins, V. Petrosian, F. Piron, T. A. Porter, G. Principe, S. Rainò, R. Rando, M. Razzano, S. Razzaque, A. Reimer, O. Reimer, C. Sgrò, D. Simone, E. J. Siskind, F. Spada, G. Spandre, P. Spinelli, L. Stawarz, D. J. Suson, M. Takahashi, K. Tanaka, J. B. Thayer, D. J. Thompson, D. F. Torres, E. Torresi, G. Tosti, E. Troja, G. Vianello, K. S. Wood
We present the second catalog of flaring gamma-ray sources (2FAV) detected with the Fermi All-sky Variability Analysis (FAVA), a tool that blindly searches for transients over the entire sky observed by the Large Area Telescope (LAT) on board the \textit{Fermi} Gamma-ray Space Telescope. With respect to the first FAVA catalog, this catalog benefits from a larger data set, the latest LAT data release (Pass 8), as well as from an improved analysis that includes likelihood techniques for a more precise localization of the transients. Applying this analysis on the first 7.4 years of \textit{Fermi} observations, and in two separate energy bands 0.1$-$0.8 GeV and 0.8$-$300 GeV, a total of 4547 flares has been detected with a significance greater than $6\sigma$ (before trials), on the time scale of one week. Through spatial clustering of these flares, 518 variable gamma-ray sources are identified. Likely counterparts, based on positional coincidence, have been found for 441 sources, mostly among the blazar class of active galactic nuclei. For 77 2FAV sources, no likely gamma-ray counterpart has been found. For each source in the catalog, we provide the time, location, and spectrum of each flaring episode. Studying the spectra of the flares, we observe a harder-when-brighter behavior for flares associated with blazars, with the exception of BL Lac flares detected in the low-energy band. The photon indexes of the flares are never significantly smaller than 1.5. For a leptonic model, and under the assumption of isotropy, this limit suggests that the spectrum of the freshly accelerated electrons is never harder than $p\sim$2.
Exploring the connection between radio and GeV-TeV gamma-ray emission in the 1FHL and 2FHL AGN samples (1708.06201)
R. Lico, M. Giroletti, M. Orienti, L. Costamante, V. Pavlidou, F. D'Ammando, F. Tavecchio
Aug. 23, 2017 astro-ph.HE
The Fermi Large Area Telescope (LAT) revealed that blazars, representing the most extreme radio-loud active galactic nuclei (AGN) population, dominate the census of the gamma-ray sky, and a significant correlation was found between radio and gamma-ray emission in the 0.1-100 GeV energy range. However, the possible connection between radio and very high energy (VHE, E>0.1 TeV) emission still remains elusive, owing to the lack of a homogeneous coverage of the VHE sky. The main goal of this work is to quantify and assess the significance of a possible connection between the radio emission on parsec scale measured by the very long baseline interferometry (VLBI) and GeV-TeV gamma-ray emission in blazars, which is a central issue for understanding the blazar physics and the emission processes. We investigate the radio VLBI and high energy gamma-ray emission by using two large and unbiased AGN samples extracted from the first and second Fermi-LAT catalogs of hard gamma-ray sources detected above 10 GeV (1FHL) and 50 GeV (2FHL). For comparison, we perform the same correlation analysis by using the 0.1-300 GeV gamma-ray energy flux provided by the third Fermi-LAT source catalog. We find that the correlation strength and significance depend on the gamma-ray energy range with a different behavior among the blazar sub-classes. Overall, the radio and gamma-ray emission above 10 GeV turns out to be uncorrelated for the full samples and for all of the blazar sub-classes with the exception of high synchrotron peaked (HSP) objects, which show a strong and significant correlation. On the contrary, when 0.1-300 GeV gamma-ray energies are considered, a strong and significant correlation is found for the full blazar sample as well as for all of the blazar sub-classes. We interpret and explain this correlation behavior within the framework of the blazar spectral energy distribution properties.
ALMA polarization observations of the particle accelerators in the hot spot of the radio galaxy 3C 445 (1705.06465)
M. Orienti, G. Brunetti, H. Nagai, R. Paladino, K.-H. Mack, M.A. Prieto
May 18, 2017 astro-ph.HE
We present Atacama Large Millimeter Array (ALMA) polarization observations at 97.5 GHz of the southern hot spot of the radio galaxy 3C 445. The hot spot structure is dominated by two bright components enshrouded by diffuse emission. Both components show fractional polarization between 30 and 40 per cent, suggesting the presence of shocks. The polarized emission of the western component has a displacement of about 0.5 kpc outward with respect to the total intensity emission, and may trace the surface of a front shock. Strong polarization is observed in a thin strip marking the ridge of the hot spot structure visible from radio to optical. No significant polarization is detected in the diffuse emission between the main components, suggesting a highly disordered magnetic field likely produced by turbulence and instabilities in the downstream region that may be at the origin of the extended optical emission observed in this hot spot. The polarization properties support a scenario in which a combination of both multiple and intermittent shock fronts due to jet dithering, and spatially distributed stochastic second-order Fermi acceleration processes are present in the hot spot complex.
Gamma-ray blazar spectra with H.E.S.S. II mono analysis: the case of PKS 2155-304 and PG 1553+113 (1612.01843)
H.E.S.S. Collaboration: H. Abdalla, A. Abramowski, F. Aharonian, F. Ait Benkhali, A.G. Akhperjanian, T. Andersson, E.O. Angüner, M. Arrieta, P. Aubert, M. Backes, A. Balzer, M. Barnard, Y. Becherini, J. Becker Tjus, D. Berge, S. Bernhard, K. Bernlöhr, R. Blackwell, M. Böttcher, C. Boisson, J. Bolmont, P. Bordas, F. Brun, P. Brun, M. Bryan, T. Bulik, M. Capasso, J. Carr, S. Casanova, M. Cerruti, N. Chakraborty, R. Chalme-Calvet, R.C.G. Chaves, A. Chen, J. Chevalier, M. Chrétien, S. Colafrancesco, G. Cologna, B. Condon, J. Conrad, C. Couturier, Y. Cui, I.D. Davids, B. Degrange, C. Deil, J. Devin, P. deWilt, L. Dirson, A. Djannati-Ataï, W. Domainko, A. Donath, L.O'C. Drury, G. Dubus, K. Dutson, J. Dyks, T. Edwards, K. Egberts, P. Eger, J.-P. Ernenwein, S. Eschbach, C. Farnier, S. Fegan, M.V. Fern, A. Fiasson, G. Fontaine, A. Förster, S. Funk, M. Füßling, S. Gabici, M. Gajdus, Y.A. Gallant, T. Garrigoux, G. Giavitto, B. Giebels, J.F. Glicenstein, D. Gottschall, A. Goyal, M.-H. Grondin, D. Hadasch, J. Hahn, M. Haupt, J. Hawkes, G. Heinzelmann, G. Henri, G. Hermann, O. Hervet, A. Hillert, J.A. Hinton, W. Hofmann, C. Hoischen, M. Holler, D. Horns, A. Ivascenko, A. Jacholkowska, M. Jamrozy, M. Janiak, D. Jankowsky, F. Jankowsky, M. Jingo, T. Jogler, L. Jouvin, I. Jung-Richardt, M.A. Kastendieck, K. Katarzyński, U. Katz, D. Kerszberg, B. Khélifi, M. Kieffer, J. King, S. Klepser, D. Klochkov, W. Kluźniak, D. Kolitzus, Nu. Komin, K. Kosack, S. Krakau, M. Kraus, F. Krayzel, P.P. Krüger, H. Laffon, G. Lamanna, J. Lau, J.-P. Lees, J. Lefaucheur, V. Lefranc, A. Lemière, M. Lemoine-Goumard, J.-P. Lenain, E. Leser, T. Lohse, M. Lorentz, R. Liu, R. López-Coto, I. Lypova, V. Mar, A. Marcowith, C. Mariaud, R. Marx, G. Maurin, N. Maxted, M. Mayer, P.J. Meintjes, M. Meyer, A.M.W. Mitchell, R. Moderski, M. Mohamed, L. Mohrmann, K. Morå, E. Moulin, T. Murach, M. de Naurois, F. Niederwanger, J. Niemiec, L. Oakes, P. O'Brien, H. Odaka, S. Öttl, S. Ohm, M. Ostrowski, I. Oya, M. Padovani, M. Panter, R.D. Parsons, M. Paz Arribas, N.W. Pekeur, G. Pelletier, C. Perennes, P.-O. Petrucci, B. Peyaud, S. Pita, H. Poon, D. Prokhorov, H. Prokoph, G. Pühlhofer, M. Punch, A. Quirrenbach, S. Raab, A. Reimer, O. Reimer, M. Renaud, R. de los Reyes, F. Rieger, C. Romoli, S. Rosier-Lees, G. Rowell, B. Rudak, C.B. Rulten, V. Sahakian, D. Salek, D.A. Sanchez, A. Santangelo, M. Sasaki, R. Schlickeiser, F. Schüssler, A. Schulz, U. Schwanke, S. Schwemmer, M. Settimo, A.S. Seyffert, N. Shafi, I. Shilon, R. Simoni, H. Sol, F. Spanier, G. Spengler, F. Spies, Ł. Stawarz, R. Steenkamp, C. Stegmann, F. Stinzing, K. Stycz, I. Sushch, J.-P. Tavernet, T. Tavernier, A.M. Taylor, R. Terrier, L. Tibaldo, D. Tiziani, M. Tluczykont, C. Trichard, R. Tuffs, Y. Uchiyama, D.J. van der Walt, C. van Eldik, B. van Soelen, G. Vasileiadis, J. Veh, C. Venter, A. Viana, P. Vincent, J. Vink, F. Voisin, H.J. Völk, T. Vuillaume, Z. Wadiasingh, S.J. Wagner, P. Wagner, R.M. Wagner, R. White, A. Wierzcholska, P. Willmann, A. Wörnlein, D. Wouters, R. Yang, V. Zabalza, D. Zaborov, M. Zacharias, A.A. Zdziarski, A. Zech, F. Zefi, A. Ziegler, N. Żywucka, LAT Collaboration: M. Ackermann, M. Ajello, L. Baldini, G. Barbiellini, R. Bellazzini, R. D. Bl, R. Bonino, J. Bregeon, P. Bruel, R. Buehler, G. A. Cali, R. A. Cameron, M. Caragiulo, P. A. Caraveo, E. Cavazzuti, C. Cecchi, J. Chiang, G. Chiaro, S. Ciprini, J. Cohen-Tanugi, F. Costanza, S. Cutini, F. D'Amm, F. de Palma, R. Desiante, N. Di Lalla, M. Di Mauro, L. Di Venere, B. Donaggio, C. Favuzzi, W. B. Focke, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, F. Giordano, M. Giroletti, L. Guillemot, S. Guiriec, D. Horan, G. Jóhannesson, T. Kamae, S. Kensei, D. Kocevski, S. Larsson, J. Li, F. Longo, F. Loparco, M. N. Lovellette, P. Lubrano, S. Maldera, A. Manfreda, M. N. Mazziotta, P. F. Michelson, T. Mizuno, M. E. Monzani, A. Morselli, M. Negro, E. Nuss, M. Orienti, E. Orl, D. Paneque, J. S. Perkins, M. Pesce-Rollins, F. Piron, G. Pivato, T. A. Porter, G. Principe, S. Rainò, M. Razzano, D. Simone, E. J. Siskind, F. Spada, P. Spinelli, J. B. Thayer, D. F. Torres, E. Torresi, E. Troja, G. Vianello, K. S. Wood
Dec. 6, 2016 astro-ph.IM, astro-ph.HE
The addition of a 28 m Cherenkov telescope (CT5) to the H.E.S.S. array extended the experiment's sensitivity to lower energies. The lowest energy threshold is obtained using monoscopic analysis of data taken with CT5, providing access to gamma-ray energies below 100 GeV. Such an extension of the instrument's energy range is particularly beneficial for studies of Active Galactic Nuclei with soft spectra, as expected for those at a redshift > 0.5. The high-frequency peaked BL Lac objects PKS 2155-304 (z = 0.116) and PG 1553+113 (0.43 < z < 0.58) are among the brightest objects in the gamma-ray sky, both showing clear signatures of gamma-ray absorption at E > 100 GeV interpreted as being due to interactions with the extragalactic background light (EBL). Multiple observational campaigns of PKS 2155-304 and PG 1553+113 were conducted during 2013 and 2014 using the full H.E.S.S. II instrument. A monoscopic analysis of the data taken with the new CT5 telescope was developed along with an investigation into the systematic uncertainties on the spectral parameters. The energy spectra of PKS 2155-304 and PG 1553+113 were reconstructed down to energies of 80 GeV for PKS 2155-304, which transits near zenith, and 110 GeV for the more northern PG 1553+113. The measured spectra, well fitted in both cases by a log-parabola spectral model (with a 5.0 sigma statistical preference for non-zero curvature for PKS 2155-304 and 4.5 sigma for PG 1553+113), were found consistent with spectra derived from contemporaneous Fermi-LAT data, indicating a sharp break in the observed spectra of both sources at E ~ 100 GeV. When corrected for EBL absorption, the intrinsic H.E.S.S. II mono and Fermi-LAT spectrum of PKS 2155-304 was found to show significant curvature. For PG 1553+113, however, no significant detection of curvature in the intrinsic spectrum could be found within statistical and systematic uncertainties.
Contributions of the Cherenkov Telescope Array (CTA) to the 6th International Symposium on High-Energy Gamma-Ray Astronomy (Gamma 2016) (1610.05151)
The CTA Consortium: A. Abchiche, U. Abeysekara, Ó. Abril, F. Acero, B. S. Acharya, C. Adams, G. Agnetta, F. Aharonian, A. Akhperjanian, A. Albert, M. Alcubierre, J. Alfaro, R. Alfaro, A. J. Allafort, R. Aloisio, J.-P. Amans, E. Amato, L. Ambrogi, G. Ambrosi, M. Ambrosio, J. Anderson, M. Anduze, E. O. Angüner, E. Antolini, L. A. Antonelli, M. Antonucci, V. Antonuccio, P. Antoranz, C. Aramo, A. Aravantinos, M. Araya, C. Arcaro, B. Arezki, A. Argan, T. Armstrong, F. Arqueros, L. Arrabito, M. Arrieta, K. Asano, M. Ashley, P. Aubert, C. B. Singh, A. Babic, M. Backes, A. Bais, S. Bajtlik, C. Balazs, M. Balbo, D. Balis, C. Balkowski, O. Ballester, J. Ballet, A. Balzer, A. Bamba, R. Bandiera, A. Barber, C. Barbier, M. Barcelo, M. Barkov, A. Barnacka, U. Barres de Almeida, J. A. Barrio, S. Basso, D. Bastieri, C. Bauer, U. Becciani, Y. Becherini, J. Becker Tjus, V. Beckmann, W. Bednarek, W. Benbow, D. Benedico Ventura, J. Berdugo, D. Berge, E. Bernardini, M. G. Bernardini, S. Bernhard, K. Bernlöhr, B. Bertucci, M.-A. Besel, V. Beshley, N. Bhatt, P. Bhattacharjee, W. Bhattacharyya, S. Bhattachryya, B. Biasuzzi, G. Bicknell, C. Bigongiari, A. Biland, A. Bilinsky, W. Bilnik, B. Biondo, R. Bird, T. Bird, E. Bissaldi, M. Bitossi, O. Blanch, P. Blasi, J. Blazek, C. Bockermann, C. Boehm, L. Bogacz, M. Bogdan, M. Bohacova, C. Boisson, J. Boix, J. Bolmont, G. Bonanno, A. Bonardi, C. Bonavolontà, P. Bonifacio, F. Bonnarel, G. Bonnoli, J. Borkowski, R. Bose, Z. Bosnjak, M. Böttcher, J.-J. Bousquet, C. Boutonnet, F. Bouyjou, L. Bowman, C. Braiding, T. Brantseg, S. Brau-Nogué, J. Bregeon, M. Briggs, M. Brigida, T. Bringmann, W. Brisken, D. Bristow, R. Britto, E. Brocato, S. Bron, P. Brook, W. Brooks, A. M. Brown, K. Brügge, F. Brun, P. Brun, P. Brun, G. Brunetti, L. Brunetti, P. Bruno, T. Buanes, N. Bucciantini, G. Buchholtz, J. Buckley, V. Bugaev, R. Bühler, A. Bulgarelli, T. Bulik, M. Burton, A. Burtovoi, G. Busetto, S. Buson, J. Buss, K. Byrum, F. Cadoux, J. Calvo Tovar, R. Cameron, F. Canelli, R. Canestrari, M. Capalbi, M. Capasso, G. Capobianco, A. Caproni, P. Caraveo, J. Cardenzana, M. Cardillo, S. Carius, C. Carlile, A. Carosi, R. Carosi, E. Carquín, J. Carr, M. Carroll, J. Carter, P.-H. Carton, J.-M. Casandjian, S. Casanova, S. Casanova, E. Cascone, M. Casiraghi, A. Castellina, J. Castroviejo Mora, F. Catalani, O. Catalano, S. Catalanotti, D. Cauz, S. Cavazzani, P. Cerchiara, E. Chabanne, P. Chadwick, T. Chaleil, C. Champion, A. Chatterjee, S. Chaty, R. Chaves, A. Chen, X. Chen, X. Chen, K. Cheng, M. Chernyakova, L. Chiappetti, M. Chikawa, D. Chinn, V. R. Chitnis, N. Cho, A. Christov, J. Chudoba, M. Cieślar, M. A. Ciocci, R. Clay, S. Colafrancesco, P. Colin, J.-M. Colley, E. Colombo, J. Colome, S. Colonges, V. Conforti, V. Connaughton, S. Connell, J. Conrad, J. L. Contreras, P. Coppi, S. Corbel, J. Coridian, R. Cornat, P. Corona, D. Corti, J. Cortina, L. Cossio, A. Costa, H. Costantini, G. Cotter, B. Courty, S. Covino, G. Covone, G. Crimi, S. J. Criswell, R. Crocker, J. Croston, J. Cuadra, P. Cumani, G. Cusumano, P. Da Vela, Ø. Dale, F. D'Ammando, D. Dang, V. T. Dang, L. Dangeon, M. Daniel, I. Davids, I. Davids, B. Dawson, F. Dazzi, B. de Aguiar Costa, A. De Angelis, R. F. de Araujo Cardoso, V. De Caprio, R. de Cássia dos Anjos, G. De Cesare, A. De Franco, F. De Frondat, E. M. de Gouveia Dal Pino, I. de la Calle, C. De Lisio, R. de los Reyes Lopez, B. De Lotto, A. De Luca, J. R. T. de Mello Neto, M. de Naurois, E. de Oña Wilhelmi, F. De Palma, F. De Persio, V. de Souza, G. Decock, J. Decock, C. Deil, M. Del Santo, E. Delagnes, G. Deleglise, C. Delgado, J. Delgado, D. della Volpe, P. Deloye, M. Detournay, A. Dettlaff, J. Devin, T. Di Girolamo, C. Di Giulio, A. Di Paola, F. Di Pierro, M. A. Diaz, C. Díaz, C. Dib, J. Dick, H. Dickinson, S. Diebold, S. Digel, J. Dipold, G. Disset, A. Distefano, A. Djannati-Ataï, M. Doert, M. Dohmke, A. Domínguez, N. Dominik, J.-L. Dominique, D. Dominis Prester, A. Donat, I. Donnarumma, D. Dorner, M. Doro, J.-L. Dournaux, T. Downes, K. Doyle, G. Drake, S. Drappeau, H. Drass, D. Dravins, L. Drury, G. Dubus, L. Ducci, D. Dumas, K. Dundas Morå, D. Durand, D. D'Urso, V. Dwarkadas, J. Dyks, M. Dyrda, J. Ebr, E. Edy, K. Egberts, P. Eger, A. Egorov, S. Einecke, J. Eisch, F. Eisenkolb, C. Eleftheriadis, D. Elsaesser, D. Elsässer, D. Emmanoulopoulos, C. Engelbrecht, D. Engelhaupt, J.-P. Ernenwein, P. Escarate, S. Eschbach, C. Espinoza, P. Evans, M. Fairbairn, D. Falceta-Goncalves, A. Falcone, V. Fallah Ramazani, D. Fantinel, K. Farakos, C. Farnier, E. Farrell, G. Fasola, Y. Favre, E. Fede, R. Fedora, E. Fedorova, S. Fegan, D. Ferenc, M. Fernandez-Alonso, A. Fernández-Barral, G. Ferrand, O. Ferreira, M. Fesquet, P. Fetfatzis, E. Fiandrini, A. Fiasson, A. Filipčič, M. Filipovic, D. Fink, C. Finley, J. P. Finley, A. Finoguenov, V. Fioretti, M. Fiorini, H. Fleischhack, H. Flores, D. Florin, C. Föhr, E. Fokitis, M. V. Fonseca, L. Font, G. Fontaine, B. Fontes, M. Fornasa, M. Fornasa, A. Förster, P. Fortin, L. Fortson, N. Fouque, A. Franckowiak, A. Franckowiak, F. J. Franco, I. Freire Mota Albuquerque, L. Freixas Coromina, L. Fresnillo, C. Fruck, M. Fuessling, D. Fugazza, Y. Fujita, S. Fukami, Y. Fukazawa, T. Fukuda, Y. Fukui, S. Funk, A. Furniss, W. Gäbele, S. Gabici, A. Gadola, D. Galindo, D. D. Gall, Y. Gallant, D. Galloway, S. Gallozzi, J. A. Galvez, S. Gao, A. Garcia, B. Garcia, R. García Gil, R. Garcia López, M. Garczarczyk, D. Gardiol, C. Gargano, F. Gargano, S. Garozzo, F. Garrecht, L. Garrido, M. Garrido-Ruiz, D. Gascon, J. Gaskins, J. Gaudemard, M. Gaug, J. Gaweda, B. Gebhardt, M. Gebyehu, N. Geffroy, B. Genolini, L. Gerard, A. Ghalumyan, A. Ghedina, P. Ghislain, P. Giammaria, E. Giannakaki, F. Gianotti, S. Giarrusso, G. Giavitto, B. Giebels, T. Gieras, N. Giglietto, V. Gika, R. Gimenes, M. Giomi, P. Giommi, F. Giordano, G. Giovannini, P. Girardot, E. Giro, M. Giroletti, J. Gironnet, A. Giuliani, J.-F. Glicenstein, R. Gnatyk, N. Godinovic, P. Goldoni, G. Gomez, M. M. Gonzalez, A. González, D. Gora, K. S. Gothe, D. Gotz, J. Goullon, T. Grabarczyk, R. Graciani, J. Graham, P. Grandi, J. Granot, G. Grasseau, R. Gredig, A. J. Green, A. M. Green, T. Greenshaw, I. Grenier, S. Griffiths, A. Grillo, M.-H. Grondin, J. Grube, M. Grudzinska, J. Grygorczuk, V. Guarino, D. Guberman, S. Gunji, G. Gyuk, D. Hadasch, A. Hagedorn, L. Hagge, J. Hahn, H. Hakobyan, S. Hara, M. J. Hardcastle, T. Hassan, K. Hatanaka, T. Haubold, A. Haupt, T. Hayakawa, M. Hayashida, M. Heller, R. Heller, J. C. Helo, F. Henault, G. Henri, G. Hermann, R. Hermel, J. Herrera Llorente, J. Herrera Llorente, A. Herrero, O. Hervet, N. Hidaka, J. Hinton, W. Hirai, K. Hirotani, B. Hnatyk, J. Hoang, D. Hoffmann, W. Hofmann, T. Holch, J. Holder, S. Hooper, D. Horan, J. Hörandel, M. Hörbe, D. Horns, P. Horvath, J. Hose, J. Houles, T. Hovatta, M. Hrabovsky, D. Hrupec, J.-M. Huet, M. Huetten, G. Hughes, D. Hui, T. B. Humensky, M. Hussein, M. Iacovacci, A. Ibarra, Y. Ikeno, J. M. Illa, D. Impiombato, T. Inada, S. Incorvaia, L. Infante, Y. Inome, S. Inoue, T. Inoue, Y. Inoue, F. Iocco, K. Ioka, M. Iori, K. Ishio, K. Ishio, G. L. Israel, Y. Iwamura, C. Jablonski, A. Jacholkowska, J. Jacquemier, M. Jamrozy, P. Janecek, M. Janiak, D. Jankowsky, F. Jankowsky, P. Jean, I. Jegouzo, P. Jenke, J. J. Jimenez, M. Jingo, M. Jingo, L. Jocou, T. Jogler, C. A. Johnson, M. Jones, M. Josselin, L. Journet, I. Jung, P. Kaaret, M. Kagaya, J. Kakuwa, O. Kalekin, C. Kalkuhl, H. Kamon, R. Kankanyan, A. Karastergiou, K. Kärcher, M. Karczewski, S. Karkar, P. Karn, J. Kasperek, H. Katagiri, J. Kataoka, K. Katarzyński, S. Kato, U. Katz, N. Kawanaka, L. Kaye, D. Kazanas, N. Kelley-Hoskins, J. Kersten, B. Khélifi, D. B. Kieda, T. Kihm, S. Kimeswenger, S. Kisaka, S. Kishida, R. Kissmann, S. Klepser, W. Kluźniak, J. Knapen, J. Knapp, J. Knödlseder, B. Koch, F. Köck, J. Kocot, K. Kohri, K. Kokkotas, K. Kokkotas, D. Kolitzus, N. Komin, I. Kominis, A. Kong, Y. Konno, K. Kosack, G. Koss, M. Kossatz, G. Kowal, S. Koyama, J. Kozioł, M. Kraus, J. Krause, M. Krause, H. Krawzcynski, F. Krennrich, A. Kretzschmann, P. Kruger, H. Kubo, V. Kudryavtsev, G. Kukec Mezek, M. Kuklis, H. Kuroda, J. Kushida, A. La Barbera, N. La Palombara, V. La Parola, G. La Rosa, H. Laffon, R. Lahmann, M. Lakicevic, K. Lalik, G. Lamanna, D. Landriu, H. Landt, R. G. Lang, J. Lapington, P. Laporte, J.-P. Le Fèvre, T. Le Flour, P. Le Sidaner, S.-H. Lee, W. H. Lee, J.-P. Lees, J. Lefaucheur, K. Leffhalm, H. Leich, M. A. Leigui de Oliveira, D. Lelas, A. Lemière, M. Lemoine-Goumard, J.-P. Lenain, R. Leonard, R. Leoni, L. Lessio, G. Leto, A. Leveque, B. Lieunard, M. Limon, R. Lindemann, E. Lindfors, L. Linhoff, A. Liolios, A. Lipniacka, H. Lockart, T. Lohse, E. Łokas, S. Lombardi, F. Longo, A. Lopatin, M. Lopez, D. Loreggia, T. Louge, F. Louis, M. Louys, F. Lucarelli, D. Lucchesi, H. Lüdecke, T. Luigi, P. L. Luque-Escamilla, E. Lyard, M. C. Maccarone, T. Maccarone, T. J. Maccarone, E. Mach, G. M. Madejski, A. Madonna, F. Magniette, A. Magniez, M. Mahabir, G. Maier, P. Majumdar, P. Majumdar, M. Makariev, G. Malaguti, G. Malaspina, A. K. Mallot, A. Malouf, S. Maltezos, D. Malyshev, A. Mancilla, D. Mandat, G. Maneva, M. Manganaro, S. Mangano, P. Manigot, N. Mankushiyil, K. Mannheim, N. Maragos, D. Marano, P. Marchegiani, J. A. Marcomini, A. Marcowith, M. Mariotti, M. Marisaldi, S. Markoff, C. Martens, J. Martí, J.-M. Martin, L. Martin, P. Martin, G. Martínez, M. Martínez, O. Martínez, K. Martynyuk-Lototskyy, R. Marx, N. Masetti, P. Massimino, A. Mastichiadis, S. Mastroianni, M. Mastropietro, S. Masuda, H. Matsumoto, S. Matsuoka, N. Matthews, S. Mattiazzo, G. Maurin, N. Maxted, N. Maxted, J. Maya, M. Mayer, D. Mazin, M. N. Mazziotta, L. Mc Comb, N. McCubbin, I. McHardy, C. Medina, F. Mehrez, C. Melioli, D. Melkumyan, T. Melse, S. Mereghetti, M. Merk, P. Mertsch, J.-L. Meunier, T. Meures, M. Meyer, J. L. Meyrelles jr, A. Miccichè, T. Michael, J. Michałowski, P. Mientjes, I. Mievre, A. Mihailidis, J. Miller, T. Mineo, M. Minuti, N. Mirabal, F. Mirabel, J. M. Miranda, R. Mirzoyan, A. Mitchell, T. Mizuno, R. Moderski, I. Mognet, M. Mohammed, R. Moharana, L. Mohrmann, E. Molinari, P. Molyneux, E. Monmarthe, G. Monnier, T. Montaruli, C. Monte, I. Monteiro, D. Mooney, P. Moore, A. Moralejo, C. Morello, E. Moretti, K. Mori, P. Morris, A. Morselli, F. Moscato, D. Motohashi, F. Mottez, Y. Moudden, E. Moulin, S. Mueller, R. Mukherjee, P. Munar, M. Munari, C. Mundell, J. Mundet, H. Muraishi, K. Murase, A. Muronga, A. Murphy, N. Nagar, S. Nagataki, T. Nagayoshi, B. K. Nagesh, T. Naito, D. Nakajima, D. Nakajima, T. Nakamori, K. Nakayama, J. Nanni, D. Naumann, P. Nayman, L. Nellen, R. Nemmen, A. Neronov, N. Neyroud, T. Nguyen, T. T. Nguyen, T. Nguyen Trung, L. Nicastro, J. Nicolau-Kukliński, F. Niederwanger, A. Niedźwiecki, J. Niemiec, D. Nieto, M. Nievas-Rosillo, A. Nikolaidis, M. Nikołajuk, K. Nishijima, K.-I. Nishikawa, G. Nishiyama, K. Noda, K. Noda, L. Nogues, S. Nolan, R. Northrop, D. Nosek, M. Nöthe, B. Novosyadlyj, L. Nozka, F. Nunio, L. Oakes, P. O'Brien, C. Ocampo, G. Occhipinti, J. P. Ochoa, A. OFaolain de Bhroithe, R. Oger, Y. Ohira, M. Ohishi, S. Ohm, H. Ohoka, N. Okazaki, A. Okumura, J.-F. Olive, D. Olszowski, R. A. Ong, S. Ono, M. Orienti, R. Orito, A. Orlati, J. Osborne, M. Ostrowski, D. Ottaway, N. Otte, S. Öttl, E. Ovcharov, I. Oya, A. Ozieblo, M. Padovani, I. Pagano, S. Paiano, A. Paizis, J. Palacio, M. Palatka, J. Pallotta, K. Panagiotidis, J.-L. Panazol, D. Paneque, M. Panter, M. R. Panzera, R. Paoletti, M. Paolillo, A. Papayannis, G. Papyan, A. Paravac, J. M. Paredes, G. Pareschi, N. Park, D. Parsons, P. Paśko, S. Pavy, M. Pech, A. Peck, G. Pedaletti, A. Pe'er, S. Peet, D. Pelat, A. Pepato, M. d. C. Perez, L. Perri, M. Perri, M. Persic, M. Persic, A. Petrashyk, P.-O. Petrucci, O. Petruk, B. Peyaud, M. Pfeifer, G. Pfeiffer, G. Piano, D. Pieloth, E. Pierre, F. Pinto de Pinho, C. Pio García, Y. Piret, A. Pisarski, S. Pita, Ł. Platos, R. Platzer, S. Podkladkin, L. Pogosyan, M. Pohl, P. Poinsignon, A. Pollo, A. Porcelli, J. Porthault, W. Potter, S. Poulios, J. Poutanen, E. Prandini, E. Prandini, J. Prast, K. Pressard, G. Principe, F. Profeti, D. Prokhorov, H. Prokoph, M. Prouza, R. Pruchniewicz, G. Pruteanu, E. Pueschel, G. Pühlhofer, I. Puljak, M. Punch, S. Pürckhauer, R. Pyzioł, F. Queiroz, E. J. Quel, J. Quinn, A. Quirrenbach, I. Rafighi, S. Rainò, P. J. Rajda, M. Rameez, R. Rando, R. C. Rannot, M. Rataj, T. Ravel, S. Razzaque, P. Reardon, I. Reichardt, O. Reimann, A. Reimer, O. Reimer, A. Reisenegger, M. Renaud, S. Renner, T. Reposeur, B. Reville, A. Rezaeian, W. Rhode, D. Ribeiro, R. Ribeiro Prado, M. Ribó, G. Richards, M. G. Richer, T. Richtler, J. Rico, J. Ridky, F. Rieger, M. Riquelme, P. R. Ristori, S. Rivoire, V. Rizi, E. Roache, J. Rodriguez, G. Rodriguez Fernandez, J. J. Rodríguez Vázquez, G. Rojas, P. Romano, G. Romeo, M. Roncadelli, J. Rosado, J. Rose, S. Rosen, S. Rosier Lees, D. Ross, G. Rouaix, J. Rousselle, A. C. Rovero, G. Rowell, F. Roy, S. Royer, A. Rubini, B. Rudak, A. Rugliancich, W. Rujopakarn, C. Rulten, M. Rupiński, F. Russo, F. Russo, K. Rutkowski, O. Saavedra, S. Sabatini, B. Sacco, I. Sadeh, E. O. Saemann, S. Safi-Harb, A. Saggion, V. Sahakian, T. Saito, N. Sakaki, S. Sakurai, A. Salamon, M. Salega, D. Salek, F. Salesa Greus, J. Salgado, G. Salina, L. Salinas, A. Salini, D. Sanchez, M. Sanchez-Conde, H. Sandaker, A. Sandoval, P. Sangiorgi, M. Sanguillon, H. Sano, M. Santander, A. Santangelo, E. M. Santos, R. Santos-Lima, A. Sanuy, L. Sapozhnikov, S. Sarkar, K. Satalecka, K. Satalecka, Y. Sato, R. Savalle, M. Sawada, F. Sayède, S. Schanne, T. Schanz, E. J. Schioppa, S. Schlenstedt, J. Schmid, T. Schmidt, J. Schmoll, M. Schneider, H. Schoorlemmer, P. Schovanek, A. Schubert, E.-M. Schullian, J. Schultze, A. Schulz, S. Schulz, K. Schure, F. Schussler, T. Schwab, U. Schwanke, J. Schwarz, T. Schweizer, S. Schwemmer, U. Schwendicke, C. Schwerdt, E. Sciacca, S. Scuderi, A. Segreto, J.-H. Seiradakis, G. H. Sembroski, D. Semikoz, O. Sergijenko, N. Serre, M. Servillat, K. Seweryn, N. Shafi, A. Shalchi, M. Sharma, M. Shayduk, R. C. Shellard, T. Shibata, A. Shigenaka, I. Shilon, E. Shum, L. Sidoli, M. Sidz, J. Sieiro, H. Siejkowski, J. Silk, A. Sillanpää, D. Simone, H. Simpson, B. B. Singh, A. Sinha, G. Sironi, J. Sitarek, P. Sizun, V. Sliusar, V. Sliusar, A. Smith, D. Sobczyńska, H. Sol, G. Sottile, M. Sowiński, F. Spanier, G. Spengler, R. Spiga, R. Stadler, O. Stahl, A. Stamerra, S. Stanič, R. Starling, D. Staszak, Ł. Stawarz, R. Steenkamp, S. Stefanik, C. Stegmann, S. Steiner, C. Stella, M. Stephan, N. Stergioulas, R. Sternberger, M. Sterzel, B. Stevenson, F. Stinzing, M. Stodulska, M. Stodulski, T. Stolarczyk, G. Stratta, U. Straumann, L. Stringhetti, M. Strzys, R. Stuik, K.-H. Sulanke, T. Suomijärvi, A. D. Supanitsky, T. Suric, I. Sushch, P. Sutcliffe, J. Sykes, M. Szanecki, T. Szepieniec, P. Szwarnog, A. Tacchini, K. Tachihara, G. Tagliaferri, H. Tajima, H. Takahashi, K. Takahashi, M. Takahashi, L. Takalo, S. Takami, J. Takata, J. Takeda, G. Talbot, T. Tam, M. Tanaka, S. Tanaka, T. Tanaka, Y. Tanaka, C. Tanci, S. Tanigawa, M. Tavani, F. Tavecchio, J.-P. Tavernet, K. Tayabaly, A. Taylor, L. A. Tejedor, I. Telezhinsky, F. Temme, P. Temnikov, C. Tenzer, Y. Terada, J. C. Terrazas, R. Terrier, D. Terront, T. Terzic, D. Tescaro, M. Teshima, M. Teshima, V. Testa, D. Tezier, J. Thayer, J. Thornhill, S. Thoudam, D. Thuermann, L. Tibaldo, A. Tiengo, M. C. Timpanaro, D. Tiziani, M. Tluczykont, C. J. Todero Peixoto, F. Tokanai, M. Tokarz, K. Toma, J. Tomastik, Y. Tomono, A. Tonachini, D. Tonev, K. Torii, M. Tornikoski, D. F. Torres, M. Torres, E. Torresi, G. Toso, G. Tosti, T. Totani, N. Tothill, F. Toussenel, G. Tovmassian, T. Toyama, P. Travnicek, C. Trichard, M. Trifoglio, I. Troyano Pujadas, M. Trzeciak, K. Tsinganos, S. Tsujimoto, T. Tsuru, Y. Uchiyama, G. Umana, Y. Umetsu, S. S. Upadhya, M. Uslenghi, V. Vagelli, F. Vagnetti, J. Valdes-Galicia, M. Valentino, P. Vallania, L. Valore, W. van Driel, C. van Eldik, B. van Soelen, J. Vandenbroucke, J. Vanderwalt, G. Vasileiadis, V. Vassiliev, J. R. Vázquez, M. L. Vázquez Acosta, M. Vecchi, A. Vega, I. Vegas, P. Veitch, P. Venault, L. Venema, C. Venter, S. Vercellone, S. Vergani, K. Verma, V. Verzi, G. P. Vettolani, C. Veyssiere, A. Viana, N. Viaux, J. Vicha, C. Vigorito, P. Vincent, S. Vincent, J. Vink, V. Vittorini, N. Vlahakis, L. Vlahos, H. Voelk, V. Voisin, A. Vollhardt, A. Volpicelli, H. von Brand, S. Vorobiov, I. Vovk, M. Vrastil, L. V. Vu, T. Vuillaume, R. Wagner, R. Wagner, S. J. Wagner, S. P. Wakely, T. Walstra, R. Walter, T. Walther, J. E. Ward, M. Ward, K. Warda, D. Warren, S. Wassberg, J. J. Watson, P. Wawer, R. Wawrzaszek, N. Webb, P. Wegner, O. Weiner, A. Weinstein, R. Wells, F. Werner, H. Wetteskind, M. White, R. White, M. Więcek, A. Wierzcholska, S. Wiesand, R. Wijers, P. Wilcox, N. Wild, A. Wilhelm, M. Wilkinson, M. Will, M. Will, D. A. Williams, J. T. Williams, R. Willingale, N. Wilson, M. Winde, K. Winiarski, H. Winkler, M. Winter, R. Wischnewski, E. Witt, P. Wojcik, D. Wolf, M. Wood, A. Wörnlein, E. Wu, T. Wu, K. K. Yadav, H. Yamamoto, T. Yamamoto, N. Yamane, R. Yamazaki, S. Yanagita, L. Yang, D. Yelos, A. Yoshida, M. Yoshida, T. Yoshida, S. Yoshiike, T. Yoshikoshi, P. Yu, V. Zabalza, D. Zaborov, M. Zacharias, G. Zaharijas, A. Zajczyk, L. Zampieri, F. Zandanel, R. Zanmar Sanchez, D. Zaric, D. Zavrtanik, M. Zavrtanik, A. Zdziarski, A. Zech, H. Zechlin, A. Zhao, V. Zhdanov, A. Ziegler, J. Ziemann, K. Ziętara, A. Zink, J. Ziółkowski, V. Zitelli, A. Zoli, J. Zorn, P. Żychowski
Oct. 17, 2016 astro-ph.HE
List of contributions from the Cherenkov Telescope Array (CTA) Consortium presented at the 6th International Symposium on High-Energy Gamma-Ray Astronomy (Gamma 2016), July 11-15, 2016, in Heidelberg, Germany.
The Chandra Survey of Extragalactic Sources in the 3CR Catalog: X-ray Emission from Nuclei, Jets, and Hotspots in the Chandra Archival Observations (1609.07145)
F. Massaro, M. Orienti, G. R. Tremblay, S. A. Baum (Univ. Manitoba), C. P. O'Dea
As part of our program to build a complete radio and X-ray database of all the 3CR extragalactic radio sources, we present an analysis of 93 sources for which Chandra archival data are available. Most of these sources have been already published. Here we provide a uniform re-analysis and present nuclear X-ray fluxes and X-ray emission associated with radio jet knots and hotspots using both publicly available radio images and new radio images that have been constructed from data available in the VLA archive. For about 1/3 of the sources in the selected sample a comparison between the Chandra and the radio observations was not reported in the literature: we find X-ray detections of 2 new radio jet knots and 17 hotspots. We also report the X-ray detection of extended emission from the intergalactic medium of 15 galaxy clusters, two of which were most likely unknown previously.
The awakening of the gamma-ray narrow-Line Seyfert 1 galaxy PKS 1502+036 (1609.04845)
F. D'Ammando (Department of Physics, Astronomy of the University of Bologna, INAF-IRA Bologna), M. Orienti, J. Finke (U.S. Naval Research Laboratory), T. Hovatta (Aalto University Metsahovi Radio Observatory), M. Giroletti, W. Max-Moerbeck, T. J. Pearson (Cahill Center for Astronomy, Astrophysics), A. C. S. Readhead (Cahill Center for Astronomy, Astrophysics), R. A. Reeves (CePIA, Departamento de Astronomia, Universidad de Concepcion), J. L. Richards (Department of Physics, Purdue University)
After a long low-activity period, a gamma-ray flare from the narrow-line Seyfert 1 PKS 1502+036 (z=0.4089) was detected by the Large Area Telescope (LAT) on board Fermi in 2015. On 2015 December 20 the source reached a daily peak flux, in the 0.1-300 GeV band, of (93 $\pm$ 19) $\times$10$^{-8}$ ph cm$^{-2}$ s$^{-1}$, attaining a flux of (237 $\pm$ 71) $\times$10$^{-8}$ ph cm$^{-2}$ s$^{-1}$ on 3-hr time-scales, which corresponds to an isotropic luminosity of (7.3 $\pm$ 2.1) $\times$10$^{47}$ erg/s. The gamma-ray flare was not accompanied by significant spectral changes. We report on multi-wavelength radio-to-gamma-ray observations of PKS 1502+036 during 2008 August-2016 March by Fermi-LAT, Swift, XMM-Newton, Catalina Real-Time Transient Survey, and the Owens Valley Radio Observatory (OVRO). An increase in activity was observed on 2015 December 22 by Swift in optical, UV, and X-rays. The OVRO 15 GHz light curve reached the highest flux density observed from this source on 2016 January 12, indicating a delay of about three weeks between the gamma-ray and 15 GHz emission peaks. This suggests that the gamma-ray emitting region is located beyond the broad line region. We compared the spectral energy distribution (SED) of an average activity state with that of the flaring state. The two SED, with the high-energy bump modelled as an external Compton component with seed photons from a dust torus, could be fitted by changing the electron distribution parameters as well as the magnetic field. The fit of the disc emission during the average state constrains the black hole mass to values lower than 10$^8$ solar masses. The SED, high-energy emission mechanisms, and gamma-ray properties of the source resemble those of a flat spectrum radio quasar.
A Panchromatic View of Relativistic Jets in Narrow-Line Seyfert 1 Galaxies (1609.04434)
F. D'Ammando, M. Orienti, J. Larsson, M. Giroletti, C. M. Raiteri
The discovery by the Large Area Telescope on board Fermi of variable gamma-ray emission from radio-loud narrow-line Seyfert 1 (NLSy1) galaxies revealed the presence of a possible third class of Active Galactic Nuclei (AGN) with relativistic jets in addition to blazars and radio galaxies. Considering that NLSy1 are usually hosted in spiral galaxies, this finding poses intriguing questions about the nature of these objects and the formation of relativistic jets. We report on a systematic investigation of the gamma-ray properties of a sample of radio-loud NLSy1, including the detection of new objects, using 7 years of Fermi-LAT data with the new Pass 8 event-level analysis. In addition we discuss the radio-to-very-high-energy properties of the gamma-ray emitting NLSy1, their host galaxy, and black hole mass in the context of the blazar scenario and the unification of relativistic jets at different scales.
Supplement: Localization and broadband follow-up of the gravitational-wave transient GW150914 (1604.07864)
B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, C. Affeldt, M. Agathos, K. Agatsuma, N. Aggarwal, O. D. Aguiar, L. Aiello, A. Ain, P. Ajith, B. Allen, A. Allocca, P. A. Altin, S. B. Anderson, W. G. Anderson, K. Arai, M. C. Araya, C. C. Arceneaux, J. S. Areeda, N. Arnaud, K. G. Arun, S. Ascenzi, G. Ashton, M. Ast, S. M. Aston, P. Astone, P. Aufmuth, C. Aulbert, S. Babak, P. Bacon, M. K. M. Bader, P. T. Baker, F. Baldaccini, G. Ballardin, S. W. Ballmer, J. C. Barayoga, S. E. Barclay, B. C. Barish, D. Barker, F. Barone, B. Barr, L. Barsotti, M. Barsuglia, D. Barta, S. Barthelmy, J. Bartlett, I. Bartos, R. Bassiri, A. Basti, J. C. Batch, C. Baune, V. Bavigadda, M. Bazzan, B. Behnke, M. Bejger, A. S. Bell, C. J. Bell, B. K. Berger, J. Bergman, G. Bergmann, C. P. L. Berry, D. Bersanetti, A. Bertolini, J. Betzwieser, S. Bhagwat, R. Bhandare, I. A. Bilenko, G. Billingsley, J. Birch, R. Birney, S. Biscans, A. Bisht, M. Bitossi, C. Biwer, M. A. Bizouard, J. K. Blackburn, C. D. Blair, D. G. Blair, R. M. Blair, S. Bloemen, O. Bock, T. P. Bodiya, M. Boer, G. Bogaert, C. Bogan, A. Bohe, P. Bojtos, C. Bond, F. Bondu, R. Bonnand, B. A. Boom, R. Bork, V. Boschi, S. Bose, Y. Bouffanais, A. Bozzi, C. Bradaschia, P. R. Brady, V. B. Braginsky, M. Branchesi, J. E. Brau, T. Briant, A. Brillet, M. Brinkmann, V. Brisson, P. Brockill, A. F. Brooks, D. A. Brown, D. D. Brown, N. M. Brown, C. C. Buchanan, A. Buikema, T. Bulik, H. J. Bulten, A. Buonanno, D. Buskulic, C. Buy, R. L. Byer, L. Cadonati, G. Cagnoli, C. Cahillane, J. C. Bustillo, T. Callister, E. Calloni, J. B. Camp, K. C. Cannon, J. Cao, C. D. Capano, E. Capocasa, F. Carbognani, S. Caride, J. C. Diaz, C. Casentini, S. Caudill, M. Cavaglià, F. Cavalier, R. Cavalieri, G. Cella, C. B. Cepeda, L. C. Baiardi, G. Cerretani, E. Cesarini, R. Chakraborty, T. Chalermsongsak, S. J. Chamberlin, M. Chan, S. Chao, P. Charlton, E. Chassande-Mottin, H. Y. Chen, Y. Chen, C. Cheng, A. Chincarini, A. Chiummo, H. S. Cho, M. Cho, J. H. Chow, N. Christensen, Q. Chu, S. Chua, S. Chung, G. Ciani, F. Clara, J. A. Clark, F. Cleva, E. Coccia, P.-F. Cohadon, A. Colla, C. G. Collette, L. Cominsky, M. Constancio Jr., A. Conte, L. Conti, D. Cook, T. R. Corbitt, N. Cornish, A. Corsi, S. Cortese, C. A. Costa, M. W. Coughlin, S. B. Coughlin, J.-P. Coulon, S. T. Countryman, P. Couvares, E. E. Cowan, D. M. Coward, M. J. Cowart, D. C. Coyne, R. Coyne, K. Craig, J. D. E. Creighton, J. Cripe, S. G. Crowder, A. Cumming, L. Cunningham, E. Cuoco, T. Dal Canton, S. L. Danilishin, S. D'Antonio, K. Danzmann, N. S. Darman, V. Dattilo, I. Dave, H. P. Daveloza, M. Davier, G. S. Davies, E. J. Daw, R. Day, D. DeBra, G. Debreczeni, J. Degallaix, M. De Laurentis, S. Deléglise, W. Del Pozzo, T. Denker, T. Dent, H. Dereli, V. Dergachev, R. T. DeRosa, R. De Rosa, R. DeSalvo, S. Dhurandhar, M. C. Díaz, L. Di Fiore, M. Di Giovanni, A. Di Lieto, S. Di Pace, I. Di Palma, A. Di Virgilio, G. Dojcinoski, V. Dolique, F. Donovan, K. L. Dooley, S. Doravari, R. Douglas, T. P. Downes, M. Drago, R. W. P. Drever, J. C. Driggers, Z. Du, M. Ducrot, S. E. Dwyer, T. B. Edo, M. C. Edwards, A. Effler, H.-B. Eggenstein, P. Ehrens, J. Eichholz, S. S. Eikenberry, W. Engels, R. C. Essick, T. Etzel, M. Evans, T. M. Evans, R. Everett, M. Factourovich, V. Fafone, H. Fair, S. Fairhurst, X. Fan, Q. Fang, S. Farinon, B. Farr, W. M. Farr, M. Favata, M. Fays, H. Fehrmann, M. M. Fejer, I. Ferrante, E. C. Ferreira, F. Ferrini, F. Fidecaro, I. Fiori, D. Fiorucci, R. P. Fisher, R. Flaminio, M. Fletcher, J.-D. Fournier, S. Franco, S. Frasca, F. Frasconi, Z. Frei, A. Freise, R. Frey, V. Frey, T. T. Fricke, P. Fritschel, V. V. Frolov, P. Fulda, M. Fyffe, H. A. G. Gabbard, J. R. Gair, L. Gammaitoni, S. G. Gaonkar, F. Garufi, A. Gatto, G. Gaur, N. Gehrels, G. Gemme, B. Gendre, E. Genin, A. Gennai, J. George, L. Gergely, V. Germain, A. Ghosh, S. Ghosh, J. A. Giaime, K. D. Giardina, A. Giazotto, K. Gill, A. Glaefke, E. Goetz, R. Goetz, L. Gondan, G. González, J. M. G. Castro, A. Gopakumar, N. A. Gordon, M. L. Gorodetsky, S. E. Gossan, M. Gosselin, R. Gouaty, C. Graef, P. B. Graff, M. Granata, A. Grant, S. Gras, C. Gray, G. Greco, A. C. Green, P. Groot, H. Grote, S. Grunewald, G. M. Guidi, X. Guo, A. Gupta, M. K. Gupta, K. E. Gushwa, E. K. Gustafson, R. Gustafson, J. J. Hacker, B. R. Hall, E. D. Hall, G. Hammond, M. Haney, M. M. Hanke, J. Hanks, C. Hanna, M. D. Hannam, J. Hanson, T. Hardwick, K. Haris, J. Harms, G. M. Harry, I. W. Harry, M. J. Hart, M. T. Hartman, C.-J. Haster, K. Haughian, A. Heidmann, M. C. Heintze, H. Heitmann, P. Hello, G. Hemming, M. Hendry, I. S. Heng, J. Hennig, A. W. Heptonstall, M. Heurs, S. Hild, D. Hoak, K. A. Hodge, D. Hofman, S. E. Hollitt, K. Holt, D. E. Holz, P. Hopkins, D. J. Hosken, J. Hough, E. A. Houston, E. J. Howell, Y. M. Hu, S. Huang, E. A. Huerta, D. Huet, B. Hughey, S. Husa, S. H. Huttner, T. Huynh-Dinh, A. Idrisy, N. Indik, D. R. Ingram, R. Inta, H. N. Isa, J.-M. Isac, M. Isi, G. Islas, T. Isogai, B. R. Iyer, K. Izumi, T. Jacqmin, H. Jang, K. Jani, P. Jaranowski, S. Jawahar, F. Jiménez-Forteza, W. W. Johnson, D. I. Jones, R. Jones, R. J. G. Jonker, L. Ju, C. V. Kalaghatgi, V. Kalogera, S. Kandhasamy, G. Kang, J. B. Kanner, S. Karki, M. Kasprzack, E. Katsavounidis, W. Katzman, S. Kaufer, T. Kaur, K. Kawabe, F. Kawazoe, F. Kéfélian, M. S. Kehl, D. Keitel, D. B. Kelley, W. Kells, R. Kennedy, J. S. Key, A. Khalaidovski, F. Y. Khalili, I. Khan, S. Khan, Z. Khan, E. A. Khazanov, N. Kijbunchoo, C. Kim, J. Kim, K. Kim, N. Kim, N. Kim, Y.-M. Kim, E. J. King, P. J. King, D. L. Kinzel, J. S. Kissel, L. Kleybolte, S. Klimenko, S. M. Koehlenbeck, K. Kokeyama, S. Koley, V. Kondrashov, A. Kontos, M. Korobko, W. Z. Korth, I. Kowalska, D. B. Kozak, V. Kringel, A. Królak, C. Krueger, G. Kuehn, P. Kumar, L. Kuo, A. Kutynia, B. D. Lackey, M. Landry, J. Lange, B. Lantz, P. D. Lasky, A. Lazzarini, C. Lazzaro, P. Leaci, S. Leavey, E. O. Lebigot, C. H. Lee, H. K. Lee, H. M. Lee, K. Lee, A. Lenon, M. Leonardi, J. R. Leong, N. Leroy, N. Letendre, Y. Levin, B. M. Levine, T. G. F. Li, A. Libson, T. B. Littenberg, N. A. Lockerbie, J. Logue, A. L. Lombardi, J. E. Lord, M. Lorenzini, V. Loriette, M. Lormand, G. Losurdo, J. D. Lough, H. Lück, A. P. Lundgren, J. Luo, R. Lynch, Y. Ma, T. MacDonald, B. Machenschalk, M. MacInnis, D. M. Macleod, F. Magaña-Sandoval, R. M. Magee, M. Mageswaran, E. Majorana, I. Maksimovic, V. Malvezzi, N. Man, I. Mandel, V. Mandic, V. Mangano, G. L. Mansell, M. Manske, M. Mantovani, F. Marchesoni, F. Marion, S. Márka, Z. Márka, A. S. Markosyan, E. Maros, F. Martelli, L. Martellini, I. W. Martin, R. M. Martin, D. V. Martynov, J. N. Marx, K. Mason, A. Masserot, T. J. Massinger, M. Masso-Reid, F. Matichard, L. Matone, N. Mavalvala, N. Mazumder, G. Mazzolo, R. McCarthy, D. E. McClelland, S. McCormick, S. C. McGuire, G. McIntyre, J. McIver, D. J. McManus, S. T. McWilliams, D. Meacher, G. D. Meadors, J. Meidam, A. Melatos, G. Mendell, D. Mendoza-Gandara, R. A. Mercer, E. Merilh, M. Merzougui, S. Meshkov, C. Messenger, C. Messick, P. M. Meyers, F. Mezzani, H. Miao, C. Michel, H. Middleton, E. E. Mikhailov, L. Milano, J. Miller, M. Millhouse, Y. Minenkov, J. Ming, S. Mirshekari, C. Mishra, S. Mitra, V. P. Mitrofanov, G. Mitselmakher, R. Mittleman, A. Moggi, M. Mohan, S. R. P. Mohapatra, M. Montani, B. C. Moore, C. J. Moore, D. Moraru, G. Moreno, S. R. Morriss, K. Mossavi, B. Mours, C. M. Mow-Lowry, C. L. Mueller, G. Mueller, A. W. Muir, A. Mukherjee, D. Mukherjee, S. Mukherjee, N. Mukund, A. Mullavey, J. Munch, D. J. Murphy, P. G. Murray, A. Mytidis, I. Nardecchia, L. Naticchioni, R. K. Nayak, V. Necula, K. Nedkova, G. Nelemans, M. Neri, A. Neunzert, G. Newton, T. T. Nguyen, A. B. Nielsen, S. Nissanke, A. Nitz, F. Nocera, D. Nolting, M. E. N. Normandin, L. K. Nuttall, J. Oberling, E. Ochsner, J. O'Dell, E. Oelker, G. H. Ogin, J. J. Oh, S. H. Oh, F. Ohme, M. Oliver, P. Oppermann, R. J. Oram, B. O'Reilly, R. O'Shaughnessy, D. J. Ottaway, R. S. Ottens, H. Overmier, B. J. Owen, A. Pai, S. A. Pai, J. R. Palamos, O. Palashov, N. Palliyaguru, C. Palomba, A. Pal-Singh, H. Pan, C. Pankow, F. Pannarale, B. C. Pant, F. Paoletti, A. Paoli, M. A. Papa, H. R. Paris, W. Parker, D. Pascucci, A. Pasqualetti, R. Passaquieti, D. Passuello, B. Patricelli, Z. Patrick, B. L. Pearlstone, M. Pedraza, R. Pedurand, L. Pekowsky, A. Pele, S. Penn, A. Perreca, M. Phelps, O. Piccinni, M. Pichot, F. Piergiovanni, V. Pierro, G. Pillant, L. Pinard, I. M. Pinto, M. Pitkin, R. Poggiani, P. Popolizio, A. Post, J. Powell, J. Prasad, V. Predoi, S. S. Premachandra, T. Prestegard, L. R. Price, M. Prijatelj, M. Principe, S. Privitera, G. A. Prodi, L. Prokhorov, O. Puncken, M. Punturo, P. Puppo, M. Pürrer, H. Qi, J. Qin, V. Quetschke, E. A. Quintero, R. Quitzow-James, F. J. Raab, D. S. Rabeling, H. Radkins, P. Raffai, S. Raja, M. Rakhmanov, P. Rapagnani, V. Raymond, M. Razzano, V. Re, J. Read, C. M. Reed, T. Regimbau, L. Rei, S. Reid, D. H. Reitze, H. Rew, S. D. Reyes, F. Ricci, K. Riles, N. A. Robertson, R. Robie, F. Robinet, A. Rocchi, L. Rolland, J. G. Rollins, V. J. Roma, R. Romano, G. Romanov, J. H. Romie, D. Rosińska, S. Rowan, A. Rüdiger, P. Ruggi, K. Ryan, S. Sachdev, T. Sadecki, L. Sadeghian, L. Salconi, M. Saleem, F. Salemi, A. Samajdar, L. Sammut, E. J. Sanchez, V. Sandberg, B. Sandeen, J. R. Sanders, B. Sassolas, B. S. Sathyaprakash, P. R. Saulson, O. Sauter, R. L. Savage, A. Sawadsky, P. Schale, R. Schilling, J. Schmidt, P. Schmidt, R. Schnabel, R. M. S. Schofield, A. Schönbeck, E. Schreiber, D. Schuette, B. F. Schutz, J. Scott, S. M. Scott, D. Sellers, D. Sentenac, V. Sequino, A. Sergeev, G. Serna, Y. Setyawati, A. Sevigny, D. A. Shaddock, S. Shah, M. S. Shahriar, M. Shaltev, Z. Shao, B. Shapiro, P. Shawhan, A. Sheperd, D. H. Shoemaker, D. M. Shoemaker, K. Siellez, X. Siemens, D. Sigg, A. D. Silva, D. Simakov, A. Singer, A. Singh, R. Singh, A. Singhal, A. M. Sintes, B. J. J. Slagmolen, J. R. Smith, N. D. Smith, R. J. E. Smith, E. J. Son, B. Sorazu, F. Sorrentino, T. Souradeep, A. K. Srivastava, A. Staley, M. Steinke, J. Steinlechner, S. Steinlechner, D. Steinmeyer, B. C. Stephens, R. Stone, K. A. Strain, N. Straniero, G. Stratta, N. A. Strauss, S. Strigin, R. Sturani, A. L. Stuver, T. Z. Summerscales, L. Sun, P. J. Sutton, B. L. Swinkels, M. J. Szczepańczyk, M. Tacca, D. Talukder, D. B. Tanner, M. Tápai, S. P. Tarabrin, A. Taracchini, R. Taylor, T. Theeg, M. P. Thirugnanasambandam, E. G. Thomas, M. Thomas, P. Thomas, K. A. Thorne, K. S. Thorne, E. Thrane, S. Tiwari, V. Tiwari, K. V. Tokmakov, C. Tomlinson, M. Tonelli, C. V. Torres, C. I. Torrie, D. Töyrä, F. Travasso, G. Traylor, D. Trifirò, M. C. Tringali, L. Trozzo, M. Tse, M. Turconi, D. Tuyenbayev, D. Ugolini, C. S. Unnikrishnan, A. L. Urban, S. A. Usman, H. Vahlbruch, G. Vajente, G. Valdes, N. van Bakel, M. van Beuzekom, J. F. J. van den Brand, C. Van Den Broeck, D. C. Vander-Hyde, L. van der Schaaf, J. V. van Heijningen, A. A. van Veggel, M. Vardaro, S. Vass, M. Vasúth, R. Vaulin, A. Vecchio, G. Vedovato, J. Veitch, P. J. Veitch, K. Venkateswara, D. Verkindt, F. Vetrano, A. Viceré, S. Vinciguerra, D. J. Vine, J.-Y. Vinet, S. Vitale, T. Vo, H. Vocca, C. Vorvick, D. Voss, W. D. Vousden, S. P. Vyatchanin, A. R. Wade, L. E. Wade, M. Wade, M. Walker, L. Wallace, S. Walsh, G. Wang, H. Wang, M. Wang, X. Wang, Y. Wang, R. L. Ward, J. Warner, M. Was, B. Weaver, L.-W. Wei, M. Weinert, A. J. Weinstein, R. Weiss, T. Welborn, L. Wen, P. Weßels, T. Westphal, K. Wette, J. T. Whelan, D. J. White, B. F. Whiting, R. D. Williams, A. R. Williamson, J. L. Willis, B. Willke, M. H. Wimmer, W. Winkler, C. C. Wipf, H. Wittel, G. Woan, J. Worden, J. L. Wright, G. Wu, J. Yablon, W. Yam, H. Yamamoto, C. C. Yancey, M. J. Yap, H. Yu, M. Yvert, A. Zadrożny, L. Zangrando, M. Zanolin, J.-P. Zendri, M. Zevin, F. Zhang, L. Zhang, M. Zhang, Y. Zhang, C. Zhao, M. Zhou, Z. Zhou, X. J. Zhu, M. E. Zucker, S. E. Zuraw, J. Zweizig J. Allison, K. Bannister, M. E. Bell, S. Chatterjee, A. P. Chippendale, P. G. Edwards, L. Harvey-Smith, Ian Heywood, A. Hotan, B. Indermuehle, J. Marvil, D. McConnell, T. Murphy, A. Popping, J. Reynolds, R. J. Sault, M. A. Voronkov, M. T. Whiting (The Australian Square Kilometer Array Pathfinder A. J. Castro-Tirado, R. Cunniffe, M. Jelínek, J. C. Tello, S. R. Oates, Y.-D. Hu, P. Kubánek, S. Guziy, A. Castellón, A. García-Cerezo, V. F. Muñoz, C. Pérez del Pulgar, S. Castillo-Carrión, J. M. Castro Cerón, R. Hudec, M. D. Caballero-García, P. Páta, S. Vitek, J. A. Adame, S. Konig, F. Rendón, T. de J. Mateo Sanguino, R. Fernández-Muñoz, P. C. Yock, N. Rattenbury, W. H. Allen, R. Querel, S. Jeong, I. H. Park, J. Bai, Ch. Cui, Y. Fan, Ch. Wang, D. Hiriart, W. H. Lee, A. Claret, R. Sánchez-Ramírez, S. B. Pandey, T. Mediavilla, L. Sabau-Graziati T. M. C. Abbott, F. B. Abdalla, S. Allam, J. Annis, R. Armstrong, A. Benoit-Lévy, E. Berger, R. A. Bernstein, E. Bertin, D. Brout, E. Buckley-Geer, D. L. Burke, D. Capozzi, J. Carretero, F. J. Castander, R. Chornock, P. S. Cowperthwaite, M. Crocce, C. E. Cunha, C. B. D'Andrea, L. N. da Costa, S. Desai, H. T. Diehl, J. P. Dietrich, Z. Doctor, A. Drlica-Wagner, M. R. Drout, T. F. Eifler, J. Estrada, A. E. Evrard, E. Fernandez, D. A. Finley, B. Flaugher, R. J. Foley, W.-F. Fong, P. Fosalba, D. B. Fox, J. Frieman, C. L. Fryer, E. Gaztanaga, D. W. Gerdes, D. A. Goldstein, D. Gruen, R. A. Gruendl, G. Gutierrez, K. Herner, K. Honscheid, D. J. James, M. D. Johnson, M. W. G. Johnson, I. Karliner, D. Kasen, S. Kent, R. Kessler, A. G. Kim, M. C. Kind, K. Kuehn, N. Kuropatkin, O. Lahav, T. S. Li, M. Lima, H. Lin, M. A. G. Maia, R. Margutti, J. Marriner, P. Martini, T. Matheson, P. Melchior, B. D. Metzger, C. J. Miller, R. Miquel, E. Neilsen, R. C. Nichol, B. Nord, P. Nugent, R. Ogando, D. Petravick, A. A. Plazas, E. Quataert, N. Roe, A. K. Romer, A. Roodman, A. C. Rosell, E. S. Rykoff, M. Sako, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, D. Scolnic, I. Sevilla-Noarbe, E. Sheldon, N. Smith, R. C. Smith, M. Soares-Santos, F. Sobreira, A. Stebbins, E. Suchyta, M. E. C. Swanson, G. Tarle, J. Thaler, D. Thomas, R. C. Thomas, D. L. Tucker, V. Vikram, A. R. Walker, R. H. Wechsler, W. Wester, B. Yanny, Y. Zhang, J. Zuntz (The Dark Energy Survey, the Dark Energy Camera GW-EM Collaborations) V. Connaughton, E. Burns, A. Goldstein, M. S. Briggs, B.-B. Zhang, C. M. Hui, P. Jenke, C. A. Wilson-Hodge, P. N. Bhat, E. Bissaldi, W. Cleveland, G. Fitzpatrick, M. M. Giles, M. H. Gibby, J. Greiner, A. von Kienlin, R. M. Kippen, S. McBreen, B. Mailyan, C. A. Meegan, W. S. Paciesas, R. D. Preece, O. Roberts, L. Sparke, M. Stanbro, K. Toelge, P. Veres, H.-F. Yu, L. Blackburn M. Ackermann, M. Ajello, A. Albert, B. Anderson, W. B. Atwood, M. Axelsson, L. Baldini, G. Barbiellini, D. Bastieri, R. Bellazzini, E. Bissaldi, R. D. Blandford, E. D. Bloom, R. Bonino, E. Bottacini, T. J. Brandt, P. Bruel, S. Buson, G. A. Caliandro, R. A. Cameron, M. Caragiulo, P. A. Caraveo, E. Cavazzuti, E. Charles, A. Chekhtman, J. Chiang, G. Chiaro, S. Ciprini, J. Cohen-Tanugi, L. R. Cominsky, F. Costanza, A. Cuoco, F. D'Ammando, F. de Palma, R. Desiante, S. W. Digel, N. Di Lalla, M. Di Mauro, L. Di Venere, A. Domínguez, P. S. Drell, R. Dubois, C. Favuzzi, E. C. Ferrara, A. Franckowiak, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, P. Giommi, F. Giordano, M. Giroletti, T. Glanzman, G. Godfrey, G. A. Gomez-Vargas, D. Green, I. A. Grenier, J. E. Grove, S. Guiriec, D. Hadasch, A. K. Harding, E. Hays, J.W. Hewitt, A. B. Hill, D. Horan, T. Jogler, G. Jóhannesson, A. S. Johnson, S. Kensei, D. Kocevski, M. Kuss, G. La Mura, S. Larsson, L. Latronico, J. Li, L. Li, F. Longo, F. Loparco, M. N. Lovellette, P. Lubrano, J. Magill, S. Maldera, A. Manfreda, M. Marelli, M. Mayer, M. N. Mazziotta, J. E. McEnery, M. Meyer, P. F. Michelson, N. Mirabal, T. Mizuno, A. A. Moiseev, M. E. Monzani, E. Moretti, A. Morselli, I. V. Moskalenko, M. Negro, E. Nuss, T. Ohsugi, N. Omodei, M. Orienti, E. Orlando, J. F. Ormes, D. Paneque, J. S. Perkins, M. Pesce-Rollins, F. Piron, G. Pivato, T. A. Porter, J. L. Racusin, S. Rainò, R. Rando, S. Razzaque, A. Reimer, O. Reimer, D. Salvetti, P. M. Saz Parkinson, C. Sgrò, D. Simone, E. J. Siskind, F. Spada, G. Spandre, P. Spinelli, D. J. Suson, H. Tajima, J. B. Thayer, D. J. Thompson, L. Tibaldo, D. F. Torres, E. Troja, Y. Uchiyama, T. M. Venters, G. Vianello, K. S. Wood, M. Wood, S. Zhu, S. Zimmer E. Brocato, E. Cappellaro, S. Covino, A. Grado, L. Nicastro, E. Palazzi, E. Pian, L. Amati, L. A. Antonelli, M. Capaccioli, P. D'Avanzo, V. D'Elia, F. Getman, G. Giuffrida, G. Iannicola, L. Limatola, M. Lisi, S. Marinoni, P. Marrese, A. Melandri, S. Piranomonte, A. Possenti, L. Pulone, A. Rossi, A. Stamerra, L. Stella, V. Testa, L. Tomasella, S. Yang (The GRAvitational Wave Inaf TeAm A. Bazzano, E. Bozzo, S. Brandt, T. J.-L. Courvoisier, C. Ferrigno, L. Hanlon, E. Kuulkers, P. Laurent, S. Mereghetti, J. P. Roques, V. Savchenko, P. Ubertini M. M. Kasliwal, L. P. Singer, Y. Cao, G. Duggan, S. R. Kulkarni, V. Bhalerao, A. A. Miller, T. Barlow, E. Bellm, I. Manulis, J. Rana, R. Laher, F. Masci, J. Surace, U. Rebbapragada, D. Cook, A. Van Sistine, B. Sesar, D. Perley, R. Ferreti, T. Prince, R. Kendrick, A. Horesh (The Intermediate Palomar Transient Factory K. Hurley, S. V. Golenetskii, R. L. Aptekar, D. D. Frederiks, D. S. Svinkin, A. Rau, A. von Kienlin, X. Zhang, D. M. Smith, T. Cline, H. Krimm F. Abe, M. Doi, K. Fujisawa, K. S. Kawabata, T. Morokuma, K. Motohara, M. Tanaka, K. Ohta, K. Yanagisawa, M. Yoshida C. Baltay, D. Rabinowitz, N. Ellman, S. Rostami D. F. Bersier, M. F. Bode, C. A. Collins, C. M. Copperwheat, M. J. Darnley, D. K. Galloway, A. Gomboc, S. Kobayashi, P. Mazzali, C. G. Mundell, A. S. Piascik, Don Pollacco, I. A. Steele, K. Ulaczyk J.W. Broderick, R.P. Fender, P.G. Jonker, A. Rowlinson, B.W. Stappers, R.A.M.J. Wijers (The Low Frequency Array V. Lipunov, E. Gorbovskoy, N. Tyurina, V. Kornilov, P. Balanutsa, A. Kuznetsov, D. Buckley, R. Rebolo, M. Serra-Ricart, G. Israelian, N. M. Budnev, O. Gress, K. Ivanov, V. Poleshuk, A. Tlatov, V. Yurkov N. Kawai, M. Serino, H. Negoro, S. Nakahira, T. Mihara, H. Tomida, S. Ueno, H. Tsunemi, M. Matsuoka S. Croft, L. Feng, T. M. O. Franzen, B. M. Gaensler, M. Johnston-Hollitt, D. L. Kaplan, M. F. Morales, S. J. Tingay, R. B. Wayth, A. Williams S. J. Smartt, K. C. Chambers, K. W. Smith, M. E. Huber, D. R. Young, D. E. Wright, A. Schultz, L. Denneau, H. Flewelling, E. A. Magnier, N. Primak, A. Rest, A. Sherstyuk, B. Stalder, C. W. Stubbs, J. Tonry, C. Waters, M. Willman (The Pan-STARRS Collaboration) F. Olivares E., H. Campbell, R. Kotak, J. Sollerman, M. Smith, M. Dennefeld, J. P. Anderson, M. T. Botticella, T.-W. Chen, M. D. Valle, N. Elias-Rosa, M. Fraser, C. Inserra, E. Kankare, T. Kupfer, J. Harmanen, L. Galbany, L. Le Guillou, J. D. Lyman, K. Maguire, A. Mitra, M. Nicholl, A. Razza, G. Terreran, S. Valenti, A. Gal-Yam (The PESSTO Collaboration) A. Ćwiek, M. Ćwiok, L. Mankiewicz, R. Opiela, M. Zaremba, A. F. Żarnecki C. A. Onken, R. A. Scalzo, B. P. Schmidt, C. Wolf, F. Yuan P. A. Evans, J. A. Kennea, D. N. Burrows, S. Campana, S. B. Cenko, P. Giommi, F. E. Marshall, J. Nousek, P. O'Brien, J. P. Osborne, D. Palmer, M. Perri, M. Siegel, G. Tagliaferri A. Klotz, D. Turpin, R. Laugier (The TAROT, Zadko, Algerian National Observatory, C2PU Collaboration) M. Beroiz, T. Peñuela, L. M. Macri, R. J. Oelkers, D. G. Lambas, R. Vrech, J. Cabral, C. Colazo, M. Dominguez, B. Sanchez, S. Gurovich, M. Lares, J. L. Marshall, D. L. DePoy, N. Padilla, N. A. Pereyra, M. Benacquista (The TOROS Collaboration) N. R. Tanvir, K. Wiersema, A. J. Levan, D. Steeghs, J. Hjorth, J. P. U. Fynbo, D. Malesani, B. Milvang-Jensen, D. Watson, M. Irwin, C. G. Fernandez, R. G. McMahon, M. Banerji, E. Gonzalez-Solares, S. Schulze, A. de U. Postigo, C. C. Thoene, Z. Cano, S. Rosswog
July 21, 2016 gr-qc, astro-ph.HE
This Supplement provides supporting material for arXiv:1602.08492 . We briefly summarize past electromagnetic (EM) follow-up efforts as well as the organization and policy of the current EM follow-up program. We compare the four probability sky maps produced for the gravitational-wave transient GW150914, and provide additional details of the EM follow-up observations that were performed in the different bands.
Localization and broadband follow-up of the gravitational-wave transient GW150914 (1602.08492)
A gravitational-wave (GW) transient was identified in data recorded by the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO) detectors on 2015 September 14. The event, initially designated G184098 and later given the name GW150914, is described in detail elsewhere. By prior arrangement, preliminary estimates of the time, significance, and sky location of the event were shared with 63 teams of observers covering radio, optical, near-infrared, X-ray, and gamma-ray wavelengths with ground- and space-based facilities. In this Letter we describe the low-latency analysis of the GW data and present the sky localization of the first observed compact binary merger. We summarize the follow-up observations reported by 25 teams via private Gamma-ray Coordinates Network circulars, giving an overview of the participating facilities, the GW sky localization coverage, the timeline and depth of the observations. As this event turned out to be a binary black hole merger, there is little expectation of a detectable electromagnetic (EM) signature. Nevertheless, this first broadband campaign to search for a counterpart of an Advanced LIGO source represents a milestone and highlights the broad capabilities of the transient astronomy community and the observing strategies that have been developed to pursue neutron star binary merger events. Detailed investigations of the EM data and results of the EM follow-up campaign are being disseminated in papers by the individual teams.
VLBA observations of radio faint Fermi-LAT sources above 10 GeV (1607.03415)
R. Lico, M. Giroletti, M. Orienti, F. D'Ammando
July 12, 2016 astro-ph.HE
The first Fermi-LAT High-energy source catalog (1FHL), containing gamma-ray sources detected above 10 GeV, is an ideal sample to characterize the physical properties of the most extreme gamma-ray sources. We investigate the pc scale properties of a sub-sample of radio faint 1FHL sources with the aim to confirm the proposed blazar associations, by revealing a compact high brightness temperature radio core, and we propose new low-frequency counterparts for the unassociated gamma-ray sources (UGS). Moreover, we increase the number of 1FHL sources with high resolution observations to explore the possible connection between radio and gamma rays at E >10 GeV. We observed 84 1FHL sources, mostly blazars of High Synchrotron Peaked (HSP) type, in the northern sky with the Very Long Baseline Array (VLBA) at 5 GHz. These sources lack high resolution radio observations and have at least one NVSS counterpart within the 95% confidence radius. For those sources without a well identified radio counterpart we exploit the VLBA multiple phase-center correlation capability to discern among the possible low-frequency candidates. For about 93% of the sources of our sample we reveal a compact high brightness temperature radio core, supporting their proposed blazar association. The vast majority of the detected sources are radio weak, with a median VLBI flux density value of 16.3 mJy. For the detected sources we obtain an average brightness temperature of the order of $2\times10^{10} \, \rm{K}$. We find a compact component for 16 UGS, for which we propose a new low-frequency association. We find brightness temperature values which do not require high Doppler factors, and are in agreement with the expected values for the equipartition of energy between particles and magnetic field. We find strong indications about the blazar nature of all of the detected UGS, for which we propose new low-frequency associations.
Searching the Gamma-ray Sky for Counterparts to Gravitational Wave Sources: Fermi GBM and LAT Observations of LVT151012 and GW151226 (1606.04901)
J. L. Racusin, E. Burns, A. Goldstein, V. Connaughton, C. A. Wilson-Hodge, P. Jenke, L. Blackburn, M. S. Briggs, J. Broida, J. Camp, N. Christensen, C. M. Hui, T. Littenberg, P. Shawhan, L. Singer, J. Veitch, P. N. Bhat, W. Cleveland, G. Fitzpatrick, M. H. Gibby, A. von Kienlin, S. McBreen, B. Mailyan, C. A. Meegan, W. S. Paciesas, R. D. Preece, O. J. Roberts, M. Stanbro, P. Veres, B.-B. Zhang, M. Ackermann, A. Albert, W. B. Atwood, M. Axelsson, L. Baldini, J. Ballet, G. Barbiellini, M. G. Baring, D. Bastieri, R. Bellazzini, E. Bissaldi, R. D. Blandford, E. D. Bloom, R. Bonino, J. Bregeon, P. Bruel, S. Buson, G. A. Caliandro, R. A. Cameron, R. Caputo, M. Caragiulo, P. A. Caraveo, E. Cavazzuti, E. Charles, J. Chiang, S. Ciprini, F. Costanza, A. Cuoco, S. Cutini, F. D'Ammando, F. de Palma, R. Desiante, S. W. Digel, N. Di Lalla, M. Di Mauro, L. Di Venere, P. S. Drell, C. Favuzzi, E. C. Ferrara, W. B. Focke, Y. Fukazawa, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, R. Gill, M. Giroletti, T. Glanzman, J. Granot, D. Green, J. E. Grove, L. Guillemot, S. Guiriec, A. K. Harding, T. Jogler, G. Johannesson, T. Kamae, S. Kensei, D. Kocevski, M. Kuss, S. Larsson, L. Latronico, J. Li, F. Longo, F. Loparco, P. Lubrano, J. D. Magill, S. Maldera, D. Malyshev, J. E. McEnery, P. F. Michelson, T. Mizuno, A. Morselli, I. V. Moskalenko, M. Negro, E. Nuss, N. Omodei, M. Orienti, E. Orlando, J. F. Ormes, D. Paneque, J. S. Perkins, M. Pesce-Rollins, F. Piron, G. Pivato, T. A. Porter, G. Principe, S. Raino, R. Rando, M. Razzano, S. Razzaque, A. Reimer, O. Reimer, P. M. Saz Parkinson, J. D. Scargle, C. Sgro, D. Simone, E. J. Siskind, D. A. Smith, F. Spada, P. Spinelli, D. J. Suson, H. Tajima, J. B. Thayer, D. F. Torres, E. Troja, Y. Uchiyama, G. Vianello, K. S. Wood, M. Wood
June 15, 2016 astro-ph.HE
We present the Fermi Gamma-ray Burst Monitor (GBM) and Large Area Telescope (LAT) observations of the LIGO binary black hole merger event GW151226 and candi- date LVT151012. No candidate electromagnetic counterparts were detected by either the GBM or LAT. We present a detailed analysis of the GBM and LAT data over a range of timescales from seconds to years, using automated pipelines and new techniques for char- acterizing the upper limits across a large area of the sky. Due to the partial GBM and LAT coverage of the large LIGO localization regions at the trigger times for both events, dif- ferences in source distances and masses, as well as the uncertain degree to which emission from these sources could be beamed, these non-detections cannot be used to constrain the variety of theoretical models recently applied to explain the candidate GBM counterpart to GW150914.
Development of the Model of Galactic Interstellar Emission for Standard Point-Source Analysis of Fermi Large Area Telescope Data (1602.07246)
F. Acero, M. Ackermann, M. Ajello, A. Albert, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, R. Bellazzini, E. Bissaldi, E. D. Bloom, R. Bonino, E. Bottacini, T. J. Brandt, J. Bregeon, P. Bruel, R. Buehler, S. Buson, G. A. Caliandro, R. A. Cameron, M. Caragiulo, P. A. Caraveo, J. M. Casandjian, E. Cavazzuti, C. Cecchi, E. Charles, A. Chekhtman, J. Chiang, G. Chiaro, S. Ciprini, R. Claus, J. Cohen-Tanugi, J. Conrad, A. Cuoco, S. Cutini, F. D'Ammando, A. de Angelis, F. de Palma, R. Desiante, S. W. Digel, L. Di Venere, P. S. Drell, C. Favuzzi, S. J. Fegan, E. C. Ferrara, W. B. Focke, A. Franckowiak, S. Funk, P. Fusco, F. Gargano, D. Gasparrini, N. Giglietto, F. Giordano, M. Giroletti, T. Glanzman, G. Godfrey, I. A. Grenier, S. Guiriec, D. Hadasch, A. K. Harding, K. Hayashi, E. Hays, J.W. Hewitt, A. B. Hill, D. Horan, X. Hou, T. Jogler, G. Jóhannesson, T. Kamae, M. Kuss, D. Landriu, S. Larsson, L. Latronico, J. Li, L. Li, F. Longo, F. Loparco, M. N. Lovellette, P. Lubrano, S. Maldera, D. Malyshev, A. Manfreda, P. Martin, M. Mayer, M. N. Mazziotta, J. E. McEnery, P. F. Michelson, N. Mirabal, T. Mizuno, M. E. Monzani, A. Morselli, E. Nuss, T. Ohsugi, N. Omodei, M. Orienti, E. Orlando, J. F. Ormes, D. Paneque, M. Pesce-Rollins, F. Piron, G. Pivato, S. Rainò, R. Rando, M. Razzano, S. Razzaque, A. Reimer, O. Reimer, Q. Remy, N. Renault, M. Sánchez-Conde, M. Schaal, A. Schulz, C. Sgrò, E. J. Siskind, F. Spada, G. Spandre, P. Spinelli, A. W. Strong, D. J. Suson, H. Tajima, H. Takahashi, J. B. Thayer, D. J. Thompson, L. Tibaldo, M. Tinivella, D. F. Torres, G. Tosti, E. Troja, G. Vianello, M. Werner, K. S. Wood, M. Wood, G. Zaharijas, S. Zimmer
Feb. 23, 2016 astro-ph.HE
Most of the celestial gamma rays detected by the Large Area Telescope (LAT) aboard the Fermi Gamma-ray Space Telescope originate from the interstellar medium when energetic cosmic rays interact with interstellar nucleons and photons. Conventional point and extended source studies rely on the modeling of this diffuse emission for accurate characterization. We describe here the development of the Galactic Interstellar Emission Model (GIEM) that is the standard adopted by the LAT Collaboration and is publicly available. The model is based on a linear combination of maps for interstellar gas column density in Galactocentric annuli and for the inverse Compton emission produced in the Galaxy. We also include in the GIEM large-scale structures like Loop I and the Fermi bubbles. The measured gas emissivity spectra confirm that the cosmic-ray proton density decreases with Galactocentric distance beyond 5 kpc from the Galactic Center. The measurements also suggest a softening of the proton spectrum with Galactocentric distance. We observe that the Fermi bubbles have boundaries with a shape similar to a catenary at latitudes below 20 degrees and we observe an enhanced emission toward their base extending in the North and South Galactic direction and located within 4 degrees of the Galactic Center.
Radio follow-up of the gamma-ray flaring gravitational lens JVAS B0218+357 (1601.03591)
Cristiana Spingola, D. Dallacasa, M. Orienti, M. Giroletti, J. P. McKean, C. C. Cheung, T. Hovatta, S. Ciprini, F. D'Ammando, E. Falco, S. Larsson, W. Max-Moerbeck, R. Ojha, A. C. S. Readhead, J. L. Richards, J. Scargle
We present results on multifrequency Very Long Baseline Array (VLBA) monitoring observations of the double-image gravitationally lensed blazar JVAS B0218+357. Multi-epoch observations started less than one month after the gamma-ray flare detected in 2012 by the Large Area Telescope on board Fermi, and spanned a 2-month interval. The radio light curves did not reveal any significant flux density variability, suggesting that no clear correlation between the high energy and low-energy emission is present. This behaviour was confirmed also by the long-term Owens Valley Radio Observatory monitoring data at 15 GHz. The milliarcsecond-scale resolution provided by the VLBA observations allowed us to resolve the two images of the lensed blazar, which have a core-jet structure. No significant morphological variation is found by the analysis of the multi-epoch data, suggesting that the region responsible for the gamma-ray variability is located in the core of the AGN, which is opaque up to the highest observing frequency of 22 GHz.
Discovery of off-axis jet structure of TeV blazar Mrk 501 with mm-VLBI (1601.02497)
S. Koyama, M. Kino, M. Giroletti, A. Doi, G. Giovannini, M. Orienti, K. Hada, E. Ros, K. Niinuma, H. Nagai, T. Savolainen, T. P. Krichbaum, M. Á. Pérez-Torres
Jan. 6, 2016 astro-ph.GA, astro-ph.HE
High-resolution millimeter wave very-long-baseline interferometry (mm-VLBI) is an ideal tool for probing the structure at the base of extragalactic jets in detail. The TeV blazar Mrk 501 is one of the best targets among BL Lac objects for studying the nature of off-axis jet structures because it shows different jet position angles at different scales. The aim of this study is to investigate the properties of the off-axis jet structure through high-resolution mm-VLBI images at the jet base and physical parameters such as kinematics, flux densities, and spectral indices. We performed Very Long Baseline Array (VLBA) observations over six epochs from 2012 February to 2013 February at 43 GHz. Quasi-simultaneous Global Millimeter VLBI Array (GMVA) observations at 86 GHz were performed in May 2012. We discover a new jet component at the northeast direction from the core in all the images at 43 and 86 GHz. The new component shows the off-axis location from the persistent jet extending to the southeast. The 43 GHz images reveal that the scattering of the positions of the NE component is within ~0.2 mas. The 86 GHz data reveals a jet component located 0.75 mas southeast of the radio core. We also discuss the spectral indices between 43 and 86 GHz, where the northeast component has steeper spectral index and the southeast component has comparable or flatter index than the radio core does.
Fossil shell emission in dying radio loud AGNs (1511.03090)
M. Kino, H. Ito, N. Kawakatu, M. Orienti, H. Nagai, K. Wajima, R. Itoh
Nov. 10, 2015 astro-ph.HE
We investigate shell emission associated with dying radio loud AGNs. First, based on our recent work by Ito et al. (2015), we describe the dynamical and spectral evolutions of shells after stopping the jet energy injection. We find that the shell emission overwhelms that of the radio lobes soon after stopping the jet energy injection because fresh electrons are continuously supplied into the shell via the forward shock while the radio lobes rapidly fade out without jet energy injection. We find that such fossil shells can be a new class of target sources for SKA telescope. Next, we apply the model to the nearby radio source 3C84. Then, we find that the fossil shell emission in 3C84 is less luminous in radio band while it is bright in TeV gamma-ray band and it can be detectable by CTA.
Radio properties of Compact Steep Spectrum and GHz-Peaked Spectrum radio sources (1511.00436)
M. Orienti
Nov. 2, 2015 astro-ph.GA
Compact steep spectrum (CSS) and GHz-peaked spectrum (GPS) radio sources represent a large fraction of the extragalactic objects in flux density-limited samples. They are compact, powerful radio sources whose synchrotron peak frequency ranges between a few hundred MHz to several GHz. CSS and GPS radio sources are currently interpreted as objects in which the radio emission is in an early evolutionary stage. In this contribution I review the radio properties and the physical characteristics of this class of radio sources, and the interplay between their radio emission and the ambient medium of the host galaxy.
High-energy properties of the high-redshift flat spectrum radio quasar PKS 2149-306 (1510.06416)
F. D'Ammando, M. Orienti
We investigate the gamma-ray and X-ray properties of the flat spectrum radio quasar PKS 2149-306 at redshift z = 2.345. A strong gamma-ray flare from this source was detected by the Large Area Telescope on board the Fermi Gamma-ray Space Telescope satellite in 2013 January, reaching on January 20 a daily peak flux of (301$\pm$36)$\times$10$^{-8}$ ph/cm$^2$/s in the 0.1-100 GeV energy range. This flux corresponds to an apparent isotropic luminosity of (1.5$\pm$0.2)$\times$10$^{50}$ erg/s, comparable to the highest values observed by a blazar so far. During the flare the increase of flux was accompanied by a significant change of the spectral properties. Moreover significant flux variations on a 6-h time-scale were observed, compatible with the light crossing time of the event horizon of the central black hole. The broad band X-ray spectra of PKS 2149-306 observed by Swift-XRT and NuSTAR are well described by a broken power-law model, with a very hard spectrum ($\Gamma$$_1$ $\sim$ 1) below the break energy, at E$_{\rm\,break}$ = 2.5-3.0 keV, and $\Gamma$$_2$ $\sim$ 1.4-1.5 above the break energy. The steepening of the spectrum below $\sim$ 3 keV may indicate that the soft X-ray emission is produced by the low-energy relativistic electrons. This is in agreement with the small variability amplitude and the lack of spectral changes in that part of the X-ray spectrum observed between the two NuSTAR and Swift joint observations. As for the other high-redshift FSRQ detected by both Fermi-LAT and Swift-BAT, the photon index of PKS 2149-306 in hard X-ray is 1.6 or lower and the average gamma-ray luminosity higher than 2$\times$10$^{48}$ erg/s. | CommonCrawl |
let's get lost g eazy
• Because all the 2p orbitals are degenerate, it doesn't matter which one has the pair of electrons.. )The electronic configuration for barium is [Xe] 6s^2 2, 8, 18, 18, 8, 2. Since the nuclear charge +Ze increases from +12e to +13e on going from Mg to Al one might expect that IE1(Al) > IE1(Mg). All Rights Reserved. In its simplest form, we could write the electronic configuration of chlorine as 2,8,7 In terms of subshells, the electronic configuration would be represented as 1s 2 2s 2 2p 6 3s 2 3p 5. How old was Ralph macchio in the first Karate Kid? What is the rhythm tempo of the song sa ugoy ng duyan? Electron configurations are based primarily on three principles: the Aufbau principle, the Pauli exclusion principle, and the Heisenberg uncertainty principle. antimony. 1 decade ago. I checked my answer against the answer given in the back of my chemistry textbook, and the two didn't correspond. This electron is located on the second energy level, in the 2s-orbital. [Kr] 5s2 4d10 5p3. The answer my book gives is [Ar] 3d^2. It has, according to its atomic number, the same number of protons and electrons. V3+ I got 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2. [Kr] 4d^5 5s^1 5p^6 What is the correct electronic configuration for a ground-state divalent Barium cation (Ba2+)? How did Rizal overcome frustration in his romance? [Rn] 2. Every atom in its ground state is uncharged. 1 O2 3 QUESTION 30 What Is The Electron Configuration Of F-? In order to become an ion with a minus two charge, it must acquire two electrons — in this case another two 2 p. Choose the diamagnetic species from below. The shell number is followed by the letter of the sub-shell, with the number of electrons in the shell indicated by a superscript number. What happens is 4s, and 3d are pretty close in energy to each other. Why don't libraries smell like bookstores? So when writing electron configuration for #Ni^(2+)# we exclude last two electrones from the last shell of Ni electron confgiruation.. Get an answer to your question "What is the shorthand electron configuration for Ba2+ ..." in Chemistry if there is no answer or all answers are wrong, use a search bar and … The driving force for such gain or loss of electrons is the energetically optimal state of having a full valence (outermost) shell of electrons. Answer Save. You can determine whether the net effect in a sample is diamagnetic or paramagnetic by examining the electron configuration of each element. In this video we will write the electron configuration for Ca2+, the Calcium ion. Thus, the electron configuration for a Br-ion is 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 10 4 s 2 4 p 6. Your IP: 15.223.25.113 Or if you need more The Electron Configuration practice, you can also practice The Electron Configuration practice problems. Who is the longest reigning WWE Champion of all time? Based on our data, we think this problem is relevant for Professor Ferrence's class at ISU. So before we talked about orbital diagrams and up here we actually have a bunch of orbital diagrams, but instead of having to write all of this out to describe where electrons go and how they fit into the atom we're actually going to make it shorthand and that way we're going to do electronic configuration. Solution The electron element lead . [Kr] 4d^10 5s^1 5p^1 4. Question: 1) Give The Electron Configurations For The Following Ions: Al3+, Ba2+, Br- And O2-2) Compute The Percent Ionic Character Of The Interatomic Bonds For The Following Compounds: TiO2, ZnTe, CsCl, InSb And MgCl23) Show That The Atomic Packing Factor For HCP Is 0.74. Electron configuration is shorthand for the arrangement of electrons in atomic orbitals. This is described by the occupied sub-shells. An electron configuration is a description of the relative locations of electrons in an atom or ion. What textbook is this problem found in? Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Physical constants: h=6.626 x 10-34 Joule-sec c=3.00 x 10 8 m/s mass of electron = 9.1 x 10-31 kg. 8. The electron configuration of barium is [Xe]6s2.Barium has six electron shells with 2, 8, 18, 18, 8, 2 electrons. This means that a neutral lithium atom will have a total of 3 electrons surrounding its nucleus. Electronic configuration is the arrangement of electrons in an atom. How do you put grass into a personification? 2.7 Give the electron configurations for the following ions: Fe 2+, Al3+, Cu+, Ba2+, Br-, and O2-. The full step-by-step solution to problem: 27P from chapter: 8 was answered by , our top Chemistry solution expert on 11/08/17, 03:59AM. Electronic configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 6s 2 >> Back to key information about the elementBack to key information about the element At oxygen, with Z = 8 and eight electrons, we have no choice. This is described by the occupied sub-shells. Correct Electron Configuration for Copper (Cu) Half-filled and fully filled subshell have got extra stability. Copyright © 2021 Multiply Media, LLC. You are correct on both items. What professor is this problem relevant for? The electronic configuration of Ba2+ is the configuration of Ba excluding the last two 6s2 electrons. Full electron configuration of barium: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 6s2. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Like all Group 17 (halogen) elements, atoms of chlorine can gain an electron to form an anion (negatively charged ion) with a … In simplified form, I got [Ar]4s^2. Removal of this one electron leaves sodium stable: Its outermost shell now contains eight electrons, giving sodium the electron configuration of neon. You can determine the ground-state electron configuration of Cobalt (Co) by locating the position Co in the periodic table. Example: 1s 2 For writing ground state electron configurations, a few main steps should be followed. Chemistry was written by and is associated to the ISBN: 9780073402680. Question: 1) Give The Electron Configurations For The Following Ions: Al3+, Ba2+, Br- And O2-2) Compute The Percent Ionic Character Of The Interatomic Bonds For The Following Compounds: TiO2, ZnTe, CsCl, InSb And MgCl23) Show That The Atomic Packing Factor For HCP Is 0.74. So if we have these transition metals, basically, we would write out the electronic configuration for each of these. Write the electron configurations for the following species. Solution The electron configurations for the ions are determined using Table 2.2 (and Figure 2.8). Sn^2+ Choose the ground state electron configuration for Hf^2+ None. For example, Group 1 element sodium (Na) has a single electron in its valence shell, with full shells of 2 and 8 electrons beneath. Think […] Barium's atomic number is 56; this means that it has 56 protons in its nucleus and also puts it as a period 6 element. The electron configuration states where electrons are likely to be in an atom. Is green skull in the pirate bay is good? Neutral barium means it has no charge, meaning no electrons are removed or added in the atom. In order Identify the species that has the smallest radius. Ba2+ [Xe] Fe [Ar]4s23d6 Se2- [Kr] Os [Xe] 6s2 4f14 5d6 Page 3 8. The shell number is followed by the letter of the sub-shell, with the number of electrons in the shell indicated by a superscript number. I … How long will the footprints on the moon last? 21) The shorthand electron configuration of chlorine, element 17, is. 23) Write the shorthand electron configuration for the Ba2+ ion: www.njctl.org Chemistry Quantum Num. Therefore, one of the 4s2 electrons jumps to the 3d9. But Ba2+ ion and the noble gas xenon (Xe) will have the same number of electrons (54 electrons each). The electron configuration of barium (atomic number 56) can be written three different ways: - Full configuration: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2, - Compressed (without all standard script numbers): 1s2 2s2p6 3s2p6d10 4s2p6d10 5s2p6 6s2. The subshells have a distinct shape and configuration, in which the electrons move freely. This problem has been solved! Fe2+: From Table 2.2, the electron configuration for an atom of iron is 1s22s22p63s23p63d64s2. N^+3. [Rn] 6s^2 5d^2 5. When two different elements have the same electron configuration, they are called isoelectronic. [Xe] 6s^2 4f ^2 4. 1. See the answer. For Copper, remember Copper is one of those exceptions to the Electron Configurations, because normally you would write it 1s2, 2s2, 2p6, 3s2, 3p6, 4s2, and 3d9. In this video we will write the electron configuration for Br-, the Bromide ion. This give us the (correct) configuration of: 1s2 2s2 2p6 3s2 3p6 3d10 4s1. What is the electron configuration of barium. Post by Chem_Mod » Sat Oct 19, 2013 11:10 pm . 2.7 Give the electron configurations for the following ions: Fe2+, Al3+, Cu+, Ba2+, Br-, and O2-. It has a role as a cofactor. 3 - Which of the following ions have noble gas... Ch. This full solution covers the following key subjects: configurations, electron… One may also ask, what is KR 5s24d105p3? Pb and Br e. Sn , Sr, and Br Based on our data, we think this question is relevant for Professor Thompson's class at TTU. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. 1. In many cases, multiple configurations are within a small range of energies and the irregularities shown above are quite irrelevant chemically. How much money does The Great American Ball Park make during one game? It is written out, as opposed to orbital diagrams which are depicted pictorially. Chem 1110 - Chapter 8: Electron Configurations and Periodicity Practice Quiz 4. The maximum electrons that can be carried by the sub-shell S is 2, by P is 6, by D is 10, and the F sub-shell can carry 14. • 1. 3 - What noble gas has the same electron configuration... Ch. Sc3+ Cl− Ba2+ Se. In simplified form, I got [Ar]4s^2. Write the electron configuration for each of the following ions. Chlorine, with seven valence electrons, can gain one electron to attain the configuration of argon. Note that these electron configurations are given for neutral atoms in the gas phase, which are not the same as the electron configurations for the same atoms in chemical environments. Recall that, similar to copper, silver has a ground state electron configuration that violates the typical Aufbau (Building-Up) guidelines. Presenting the electron configuration of Br- as [Kr] would be correct, but keep in mind that they may be expecting you to give the full electron configuration rather than just the shorthand. 4) Vanadium Has An Atomic Radius Of 1.32 Angstroms And A Density Of 6.1 G/cm3 Isoelectronic refers to two atoms, ions, or molecules that have the same electronic structure and the same number of valence electrons.The term means "equal electric" or "equal charge". Atoms or ions with the same electronic configurations are said to be isoelectronic to each other or to have the same isoelectronicity. This means that the electron … Cloudflare Ray ID: 611f1f7548484bd7 The electron configuration describes the distribution of electrons in the shell of an atom at various energy states. Chem_Mod Posts: 19112 Joined: Thu Aug 04, 2011 8:53 pm Has upvoted: 801 times. For example, sodium (Na), which has a single electron in its outer 3s orbital, can lose that electron to attain the electron configuration of neon. Favorite Answer. copper. If your impeached can you run for president again? Ground-state means that the element is in its lowest energy form (not in excited state). Fluorine, for example, with the electron configuration [He]2s 2 2p 5, has seven valence electrons, so its Lewis dot symbol is constructed as follows: The number of dots in the Lewis dot symbol is the same as the number of valence electrons, which is the same as the last digit of … 01 00wN O O QUESTION 29 How Many Valence Electrons Does P Have? O 2-: From Table 2.2, the electron configuration for an atom of oxygen is 1 s 2 2 s 2 2 p 4. Barium Electronic configuration. Having gained a positive charge, the sodium ion is called a cation. Write the electron configurations for the following species. 16. 1) Write the ground-state electron configuration of Sb3+. Each shell and subshell have a limitation on the amount of electrons that it can carry. It is - 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6. Relevance. [Kr] 4d^10 5p^2 5. [Kr] 4d^10 5s^2 3. What is the denotative and connotative meaning of clouds? How many orbitals are there in the fourth shell? Chem_Mod Posts: 19112 Joined: Thu Aug 04, 2011 8:53 pm Has upvoted: 801 times. Ch. Here we have a challenging problem about Electron Configurations for Transition Metals. a. Al3+ b. Se2− c. Sc3+ d. As3− A: a) [He] 2s 2 2p 6 b) [Ar] 3d 10 4s 2 4p 6 c) [Ne] 3s 2 3p 6 d) [Ar] 3d 10 4s 2 4p 6 Another way to prevent getting this page in the future is to use Privacy Pass. What is the electron configuration of barium? What is the analysis of the poem song by nvm gonzalez? [Rn] 6s^1 3. 22) The 3p subshell in the ground state of atomic radon contains _____ electrons. Predicting Electron Configurations of Ions What is the electron configuration and orbital diagram of: (a) Na + (b) P 3 – (c) Al 2+ (d) Fe 2+ (e) Sm 3+ Solution First, write out the electron configuration for each parent atom. When Cobalt loses 2 electrons to become $\ce{Co^{2+}}$ it loses the electrons which are in $4s^2$, not the ones in $3d^7$ because the electrons in $4s^2$ have a high reactivity. 1. When did organ music become associated with baseball? cesium ← barium → lanthanum. Electrons are rather labile, however, and an atom will often gain or lose them depending on its electronegativity. Herein, which element has the electron configuration Xe 6s2 4f14 5d6? Find the amount of electrons in the atom. [Ar]452 Ô (Xe]452 O (Xe|4s23p2 [Kr]4s23p5 [Kr] [Xe] QUESTION 28 How Many Valence Electrons Does Mg Have? Who was the lady with the trophy in roll bounce movie? I … Isoelectronic chemical species typically display similar chemical properties. Identify the number of valence electrons in Cl^-1. Electron configurations are written using the principal quantum number n, followed by the orbital (s, p, d, or f) with the total number of electrons written as a superscript. You may need to download version 2.0 now from the Chrome Web Store. There is no noble gas with the same electronic configuration as the element barium (Ba). If the electron subshells are completely filled with electrons, the material will be diamagnetic because the magnetic fields cancel each other out. - Noble Gas form : [Xe] 6s2 (this works because [Xe] symbolizes the electronic configuration of Xenon, which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6. 6 Answers. [Rn] 2. Solution The electron configurations for the ions are determined using Table 2.2 (and Figure 2.6). The answer my book gives is [Ar] 3d^2. Therefore the Iron electron configuration will be 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 6. Each element has a unique atomic structure that is influenced by its electronic configuration, which is the distribution of electrons across different orbitals of an atom. Sr and Brd. (longhand) electron configuration of fluorine, ( atomic # 9) is . An uncharged barium atom would have … Barium. [Xe] 6s^2 4f ^2 4. Top. Charles M. Lv 6. For elements with many electrons, noble gas configuration is a useful way to abbreviate the electron configuration. The electronic configuration for $\ce{Br-}$ is: $$\mathrm{1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6}$$ Because it have one more electron than bromine, which ends its electronic configuration with $\mathrm{4p^5}$. Write the electron configuration for the following ion. [Kr] 4d^5 5s^1 5p^6 What is the correct electronic configuration for a ground-state divalent Barium cation (Ba2+)? 1) Write the ground-state electron configuration of Sb3+. [Kr] 4d^10 5s^2 3. [Ar]451 [Ar]451 O [Ne) [Kr)4523p5 [Kr] [Ar]4523p5. For the Cu+ ion we remove one electron from 4s1 leaving us with: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10. Note that when writing the electron configuration for an atom like Fe, the 3d is usually written before the 4s. Its electron configuration will be "Li: " 1s^2 color(red)(2)s^1 Now, the lithium cation, "Li"^(+), is formed when lithium loses the electron located on its outermost shell -> its valence electron. There are 118 elements in the periodic table. One electron must be paired with another in one of the 2p orbitals, which gives us two unpaired electrons and a 1s 2 2s 2 2p 4 electron configuration. The most important thing to remember is that electrons fill orbitals from lowest energy to highest energy. Performance & security by Cloudflare, Please complete the security check to access. [Kr] 4d^8 5s^1 5p^3 2. In this video we will write the electron configuration for O 2-, the Oxide ion. The K shell contains a 1s subshell hence it can carry 2 electrons, the L … The electron configurations of ions are isoelectronic with a noble gas, meaning they have the same electron configuration as a noble gas. Fe2+: From Table 2.2, the electron configuration for an atom of iron is 1s22s22p63s23p63d64s2. Besides, what element is Xe 6s2 4f14 5d10? Electron Configuration: The electron configuration of an atom describes its total number of electrons or atomic number or number of protons. The ground state electron configuration of ground state gaseous neutral xenon is [Kr]. Barium(2+) is a barium cation, a divalent metal cation and a monoatomic dication. [Kr] 4d^10 5s^1 5p^1 4. Write the electron configuration for the following ion. Re: Electron configuration fo Ag+. I checked my answer against the answer given in the back of my chemistry textbook, and the two didn't correspond. Keeping this in consideration, which element has the electron configuration Xe 6s2 4f14 5d6? Electronic configuration for Mg: [Ne] 3s2: Electronic configuration for Al: [Ne] 3s2 3p1 Thus, in Mg the most loosely bound electron is removed from a doubly occupied 3s orbital while in Al it is removed from a singly occupied 3p orbital. So what happens is one of the 4s-electrons goes in, helps fill up the d … Periodic table » Lanthanum » Electron configuration Lanthanum Full electron configuration of lanthanum: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 5d 1 6s 2 The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Top. [Kr] 4d^10 5p^2 5. Electronic configuration of $\ce{Co}$ is as follows: $1s^2\ 2s^2\ 2p^6\ 3s^2\ 3p^6\ 4s^2\ 3d^7$. Which is the following is an INCORRECT electron configuration for the ground state of these species? So the electron configuration for Ag + becomes [Kr] 4d 10. V3+ I got 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2. I suggest to watch Tyler DeWitt's YouTube video on electron configuration. 1. 2.7 Give the electron configurations for the following ions: Fe2+, Al3+, Cu+, Ba2+, Br-, O2-, Fe3+ and S2-. Please enable Cookies and reload the page. [Rn] 6s^2 5d^2 5. [Kr] 4d^8 5s^1 5p^3 2. View Notes - HW 1 Solutions from EE E501 at Universidad de Guadalajara. Electron Configuration Chart for All Elements in the Periodic Table. Ba2+ has the same electron configuration as Xenon, Xe. Electron configuration for Ni is #1s^2 2s^2 2p^6 4s^2 3d^8#.. #Ni^(2+)# has two electrons less than Ni ( that is why #Ni^(2+)# is positively charged). Does harry styles have a private Instagram account? & Periodic Table This decides the electron capacity of the shells. The electron configuration describes the distribution of electrons in the shell of an atom at various energy states. The configuration is just like Xenon. Use the element blocks of the periodic table to find the highest electron orbital. I believe the ground state electron configuration of Na is: 1s2 2s2 2p6 3s1. [Rn] 6s^1 3. Question: What Is The Electron Configuration Of Ba2+? In such a state, the resulting charged atom has the electr… [Xe] 6. If you don't have a chart, you can still find the electron configuration. When added to 6s2, it is equivalent to the full electron configuration of Barium, when neutral. Periodic table » Lanthanum » Electron configuration Lanthanum Full electron configuration of lanthanum: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 5d 1 6s 2 Na+ means that the atom has lost an electron would the config then be: 1s2 2s2 2p6 the same as Ne? 1. 4) Vanadium Has An Atomic Radius Of 1.32 Angstroms And A Density Of 6.1 G/cm3 What floral parts are represented by eyes of pineapple? So for Titanium, it would be pretty easy. Ba2+ [Xe] Fe [Ar]4s23d6 Se2- [Kr] Os [Xe] 6s2 4f14 5d6 Page 3 8. What is the best way to fold a fitted sheet? 3 - Write electron configurations for a. the cations... Ch. Problem: Which of the following ions have the same ground state electron configuration: Sn 4+ , Pb 4+ , Sr 2+ , Br - a. Sn and Pb b. Pb and Src. 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Gas xenon ( Xe ) will have a limitation on the moon last, 18, 8, 2 give. Use the element blocks of the following ions: fe2+, Al3+, Cu+,,... Electrons ( 54 electrons each ) 4f14 5d6 Page 3 8 did n't.... What element is in its ground state electron configurations for the ions are determined using Table 2.2 and! In roll bounce movie or lose them depending on its electronegativity that fill... Or if you don ' t have a challenging problem about electron configurations for a. the cations... Ch electron. For an atom x 10-34 Joule-sec c=3.00 x 10 8 m/s mass of electron = 9.1 x kg... A noble gas with the same isoelectronicity write the electron configuration for a ground-state divalent barium cation Ba2+! Thu Aug 04, 2011 8:53 pm has upvoted: 801 times no charge meaning... Solutions from EE E501 at Universidad de Guadalajara m/s mass of electron 9.1! 4F14 5d6 Building-Up ) guidelines other out, meaning no electrons are likely to be in an atom and. The longest reigning WWE Champion of all time, it doesn ' t matter which has... Electrons that it can carry paramagnetic by examining the electron configurations of ions are determined Table. I … the electron configurations for Transition Metals, basically, we would write out the electronic configuration for None... For barium is [ Ar ] 4s^2 3 8, multiple configurations are based primarily three! Each of the song sa ugoy ng duyan to prevent getting this in. Neutral lithium atom will have a total of 3 electrons surrounding its nucleus is 5s24d105p3. Electron configuration for Ca2+, the Pauli exclusion principle, and the two n't. Mass of electron = 9.1 x 10-31 kg is Kr 5s24d105p3 multiple are. Electron = 9.1 x 10-31 kg located on the amount of electrons that it can carry x. Back of my chemistry textbook, and the irregularities shown above are quite irrelevant chemically ID 611f1f7548484bd7!: from Table 2.2 ( and Figure 2.6 ) when added to 6s2, doesn! O QUESTION 29 how many orbitals are degenerate, it is equivalent the. Is no noble gas... Ch 8 m/s mass of electron = 9.1 x 10-31 kg pair of (. The electron configuration for barium is [ Kr ] Os [ Xe ] Fe [ Ar ].. A few main steps should be followed ion: www.njctl.org chemistry Quantum Num the denotative and connotative of. Determined using Table 2.2, the same number of protons and electrons the config then be 1s2! Form, i got [ Ar ] 4s23d6 Se2- [ Kr ] 4d^5 5s^1 5p^6 is... Gas with the same electron configuration of $ \ce { Co } is... Subshells are completely filled with electrons, we would write out the electronic configuration for a ground-state divalent cation. Number, the Pauli exclusion principle, the Pauli exclusion principle, and the Heisenberg uncertainty.. Who was the lady with the same electron configuration of neon of $ \ce { Co } $ as... Of ground state gaseous neutral xenon is [ Kr ] 4d^5 5s^1 5p^6 is! ( Xe ) will have a limitation on the second energy level, the. 30 what is the arrangement of electrons or atomic number, the configuration. The answer my book gives is [ Ar ] 4523p5 Co in the shell of an at!: 15.223.25.113 • Performance & security by cloudflare, Please complete the security check to access 3p6 4s1. 4F14 5d6 Page 3 8 ID: 611f1f7548484bd7 • Your IP: 15.223.25.113 • Performance security... Them depending on its electronegativity called a cation $ 1s^2\ 2s^2\ 2p^6\ 3s^2\ 4s^2\... Is green skull in the shell of an atom # 9 ) is a barium cation, a main... Determine whether the net effect in a sample is diamagnetic or paramagnetic by examining the electron for... Write electron configurations for the Ba2+ ion and the noble ba2+ electron configuration has the electr… Here we have a limitation the. Electrons are likely to be in an atom of iron is 1s22s22p63s23p63d64s2 that violates the typical Aufbau ( Building-Up guidelines. The element barium ( Ba ) above are quite irrelevant chemically however, and 3d are pretty close energy. Shell and subshell have got extra stability for O 2-, the resulting charged atom has the same isoelectronicity electrons... Cobalt ( Co ) by locating the position Co in the fourth shell ng duyan is the analysis of following... Aufbau ( Building-Up ) guidelines highest electron orbital = 9.1 x 10-31 kg fitted sheet are pretty in! From Table 2.2 ( and Figure 2.6 ba2+ electron configuration each element ( Xe will! Also practice the electron configuration for copper ( Cu ) Half-filled and fully filled subshell have got extra.. Correct electronic configuration of chlorine, element 17, is Cu ) Half-filled and filled. Configuration of barium: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 ] 4s23d6 [! The Heisenberg uncertainty principle when neutral Table 2.2, the sodium ion is called a cation 2s2. 1 O2 3 QUESTION 30 what is Kr 5s24d105p3 view Notes - HW ba2+ electron configuration Solutions from EE E501 Universidad! The arrangement of electrons or atomic number or number of protons filled subshell have total! 01 00wN O O QUESTION 29 how many Valence electrons Does P have Oxide ion,. Electrons that it can carry Thu Aug 04, 2011 8:53 pm has upvoted: times. Neutral xenon is [ Ar ] 4s^2, is same electron configuration of Sb3+ ) guidelines this consideration! Electrons are removed or added in the shell of an atom at various energy states the two did correspond. Of F- ( and Figure 2.8 ) barium, when neutral leaves sodium stable: its outermost shell now eight. The highest electron orbital 5s^1 5p^6 what is Kr 5s24d105p3 i checked my against. Human and gives you temporary access to the web property energy level, in future! In roll bounce movie: electron configurations for the ground state electron configuration a! Our data, we think this problem is relevant for Professor Ferrence 's class at.... Highest energy for president again various energy states Co ) by locating the position Co in the of! ] 4s23d6 Se2- [ Kr ] [ Ar ] 3d^2 electrons ( 54 each! Writing the electron configuration chart for all elements in the future is to Privacy! 1S^2\ 2s^2\ 2p^6\ 3s^2\ 3p^6\ 4s^2\ 3d^7 $ the Pauli exclusion principle, the Calcium ion have a,. Is as follows: $ 1s^2\ 2s^2\ 2p^6\ 3s^2\ 3p^6\ 4s^2\ 3d^7 $ solution the electron configuration Cobalt! Of my chemistry textbook, and the Heisenberg uncertainty principle one of the periodic Table to find the highest orbital... Are degenerate, it would be pretty easy which one has the electron configuration for Ca2+ the! States where electrons are likely to be in an atom Al3+, Cu+ Ba2+. Would write out the electronic configuration for an atom Valence electrons Does P have correct electron configuration for each these... All the 2p orbitals are degenerate, it is - 1s2 2s2 2p6 3s2 3p6 3d10 4p6! Recall that, similar to copper, silver has a ground state electron configuration of Cobalt Co! The amount of electrons or atomic number, the Oxide ion, 18,,... Mass of electron = 9.1 x 10-31 kg what is the best way to fold a sheet! Temporary access to the 3d9 for a. the cations... Ch moon last 2.2 and... To copper, silver has a ground state electron configuration, they are called isoelectronic of and. To fold a fitted sheet the irregularities shown above are quite irrelevant chemically ] Os [ Xe ] [! Roll bounce movie giving sodium the electron configuration for each of these species video on configuration... Sn^2+ Choose the ground state electron configuration chart for all elements in the future is to use Privacy..
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calculate the mass of 1 atom of nitrogen
Anonymous. (Relative atomic masses: H = 1.0, N = 14.0) Reveal answer Calculate the mass of {eq}2.50 \times 10^4 {/eq} molecules of nitrogen gas. Antimony has two naturally occurring isotopes. The molar mass of water is 18.015 g/mol. The four lead isotopes have atomic masses and relative abundances of 203.973 amu (1.4%), 205.974 amu (24.1%), 206.976 amu (22.1%) and 207.977 amu (52.4%). wt. This means that a single atom of sodium weighs 23 atomic mass units (AMUs), but it also means that one mole of sodium atoms weighs 23 grams. (a) Mass of nitrogen atom = number of moles x atomic mass = 1 x 14 = 14 g (b) Mass of aluminium atom = 4 x atomic mass = 4 x 27 = 108 g Question: 1)Calculate The Mass Defect Of The Nitrogennucleus . Calculating the mass of 1 molecule of ammonia: ... ( Mass of nitrogen atom / Mass of ammonia molecule ) … Problem: The ratio of the mass of a nitrogen atom to the mass of an atom of 12C is 7:6 and the ratio of the mass of nitrogen to oxygen in N2O is 7:4. The molar mass of nitrogen is 14.0067 g/mol. is 3.0 × 10 –25 J, calculate its wavelength. 1) Calculate the molar mass. So mass of 1 mole of N2 is 28g.Therefore mass of 0.5 mole is 28/2 =14g 1-30. Mass of each atom of calcium = 6.642 x 10-26 /1 = 6.642 x 10-26 kg. A compound which contains one atom of X and two atoms of Y for each three atoms of Z is made by mixing 5. They are equal to 11 and 23, respectively. Do the same for a hydrogen chloride molecule in which the chlorine atom has an atomic mass of 34.97. 714 1561 1189 You can determine the mass percent, or how much of the total mass of ammonium carbonate is nitrogen, by first determining the mass of nitrogen and the mass of the total compound. Calculate the percentage atom economy for the reaction in Stage 7. Add the mass of Hydrogen, Nitrogen, and two Oxygens. 1. 7. Unit Conversion: In a unit conversion, the mathematical method of calculation is dimensional analysis. Find the atomic number (Z) and mass number (A). Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00. Choose your element. Calculate the atomic mass of lead. Calculate the total mass of the 7 protons and 7 neutrons in one nucleus of ""_7^14"N". Multiply the relative atomic mass by the molar mass constant. This converts atomic units to grams per mole, making the molar mass of hydrogen 1.007 grams per mole, of carbon 12.0107 grams per mole, of oxygen 15.9994 grams per mole, and of chlorine 35.453 grams per mole. Calculate the mass of the following: (i) 0.5 mole of gas (mass from mole of molecule) (ii) 0.5 mole of N atoms (mass from mole of atom) 6 moles Nitrogen to grams = 84.0402 grams. Step 2: Determine the molar mass of the element. Gram atomic mass of nitrogen = 14 g Therefore, 1 mole of nitrogen atoms contains 14 g. Calculate the mass defect for this atom in atomic mass units given the following data: mass of a proton = 1.007276 u mass of a neutron = 1.008665 u mass of an electron = 0.000549 u … Q:-Determine the empirical formula of an oxide of iron which has 69.9% iron and 30.1% dioxygen by mass. The starting point here will be the molar mass of nitrogen, which is listed as . 8 moles Nitrogen to grams = 112.0536 grams ... (iii)€€€€ The percentage yield of ammonia is the percentage, by mass, of the nitrogen and hydrogen which has been converted to ammonia. How do you calculate the mass of a single atom or molecule? 1 u = 1/12 the mass of carbon 12 by definition. Favorite Answer. The charge is 0. Calculate the maximum mass of ammonia that can be made from an excess of nitrogen and 12.0 g of hydrogen. FREE Expert Solution Show answer. This was calculated by multiplying the atomic weight of hydrogen (1.008) by two and adding the result to the weight for one oxygen (15.999). > The mass defect is the difference between the calculated mass (the sum of the masses of the protons and neutrons) and the actual mass of the nucleus. Relevance. Calculate numbers of protons, neutrons, and electrons by using mathematical expressions (1-3): p = 11. n = 23 - 11 = 12. e = 11 - 0 = 11 4. 1 moles Nitrogen to grams = 14.0067 grams. The question also states that the molar mass of nitrogen (N2) is 28.0 g/mole. So let's think : if there are 14 grams in 6,02x10^23 atoms , we just need to find this value in one atom. Let's assume that it is the atom of sodium (Na). 7 moles Nitrogen to grams = 98.0469 grams. Each mole of nitrogen contains 6.0221415 × 10^23 molecules - known as Avogadro's number. The molar mass of an element is the mass in g of one mole of the element. Q:- Atomic number is the total number of the protons that are present in the nucleus of an element's atom. (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. We must give here N2 not N because N is only atom and N2 is the molecule Molar mass : The mass of the one mole of the substance is called molar mass. Answer: Molar mass of N2 is 28(since atomic weight of nitrogen is 14and forN2=14*2). 1 mole of nitrogen atoms is equivalent to the gram atomic mass of nitrogen. Question 1: The mass of an atom of uranium-235 is observed to be 235.044 u. Answer Save. To solve this we just need a little relation , if an element has 14 U it means it has 14gram every mole of this element . If its K.E. Finding molar mass starts with units of grams per mole (g/mol). Hence 6.02 x … Percent of nitrogen: 21.2% To get the percent composition of nitrogen is ammonium sulfate, (NH_4)_2SO_4, you need to know the molar mass of the compound and that of elemental nitrogen. mass of one nitrogen atom and nitrogen molecule in kg. (the kh value for nitrogen in water is 6.1 x 10-4 m/atm.) 3 moles Nitrogen to grams = 42.0201 grams. The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. What is the mass of a single nitrogen atom in grams? Weights of atoms and isotopes are from NIST article. Use the equation to calculate the maximum mass of magnesium oxide produced. Therefore, mass of one mole of nitrogen = 14g 100% (369 ratings) Problem Details. The mass of an electron is 9.1 × 10 –31 kg. The ratio of the mass of a nitrogen atom to the mass of an atom of {eq}_{12} {/eq}C is 7:6, and the ratio of the mass of nitrogen to oxygen in N{eq}_2 {/eq}O is 7:4. assume a total pressure of 1.0 atm and the mole fraction for nitrogen of 0.78. Weights of atoms and isotopes are from NIST article. I know that relative atomic mass of $\ce{^{12}C}$ is $12~\mathrm{u}$. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, Fe, Mg, Mn, S, O, H, C, N, Na, K, Cl, Al. 1-31. You're adding the masses of uncombined protons and neutrons, 1.0073 u and 1.0087 u respectively. The atomic mass of Nitrogen is 14.00674. 1 … Gram atomic mass of an element is the amount of that element in grams whose quantity is equal to atomic mass of that element. Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). Q:-Calculate the wavelength of an electron moving with a velocity of 2.05 × 10 7 ms –1. (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. Hydrogen has an atomic mass of 1.008. Get an answer to your question Calculate the mass of nitrogen dissolved at room temperature in a 80.0 l home aquarium. 2 moles Nitrogen to grams = 28.0134 grams. But when those particles fuse together to form an atom, some of the mass is converted into energy according to E=mc^2. 2 Answers. 6. 10 years ago. The important thing to notice is that ammonium sulfate contains 2 nitrogen atoms, each belonging to one ammonium ion, NH_4^(+). Find the mass of 1 mol of oxygen atoms. 5 moles Nitrogen to grams = 70.0335 grams. Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). Please remember that you need the molar mass first when trying to find the average mass of one molecule. The atomic mass of Nitrogen is 14.0067the atomic mass of nitrogen is 14.007, so if you round off you will get 14 Therefore, gram atomic mass of nitrogen atom = 14 g ( since, atomic mass of nitrogen = 14u) Mass of one mole of an atom equal to gram atomic mass of atom . Step 1: Calculate the number of moles from the number of atoms. Take the relative atomic masses of hydrogen and nitrogen to be 1 and 14 respectively. So, the molar mass of ammonium sulfate is 132.14 g/mol and the molar mass … The mass defect is 0.108 506 u. Divide the atomic mass by the avogadro's number = 2.3258736 x 10^ -23 ~= 2.326 x 10 ^ -23 Molecular mass of nitrogen (N 2) = 14 x 2 = 28 g. 1 mole of nitrogen is 28 g = 28 x 10-3 kg. This is defined as 0.001 kilogram per mole, or 1 gram per mole. 4 moles Nitrogen to grams = 56.0268 grams. 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Jakob Schwichtenberg in Lie Group Theory | 7. August 2014
Short Introduction to and Motivation for Representation Theory
What may seem at a first glance like just another mathematical gimmick of group theory, is of incredible importance in physics. One can consider the Poincaré group (the set of all transformations that leave the speed of light constant) and use the framework of representation theory to construct the irreducible representations of this group. (The irreducible representations are the basic building blocks all other representations can be built of.) The straight-forward examination of the irreducible representations of the Poincaré group gives us physicists the appropriate mathematical tools needed to describe nature at the most fundamental level.
The lowest dimensional representation is trivial and called scalar or spin $0$ representation, because the objects (scalars) the group acts on in this representation are used to describe elementary particles of spin $0$. (In this representation the group doesn't changes the objects in question at all.)
The next higher-dimensional representation is called spin $\frac{1}{2}$ or spinor representation, because the objects (spinors) the group acts on in this representation are used to describe elementary particles of spin $\frac{1}{2}$.
The third representation is called spin $1$ or vector representation, because the objects (vectors) the group acts on in this representation are used to describe elementary particles of spin $1$.
But what exactly is a representation?
For theoretical considerations its often useful to regard any group as an abstract group. This means defining the group by its manifold structure and the group operation. For example $SU(2)$ is the three sphere $S^3$, the elements of the group are points of the manifold and the rule associating a product point $ab$ with any two points $b$ and $a$ satisfies the usual group axioms. In physical applications one is more interested in what the group actually does, i.e. the group action.
An important idea is that one group can act on many different kinds of objects (this will make much more sense in a moment). This idea motivates the definition of a representation: A representation is a map between any group element $g$ of a group $G$ and a linear transformation $T(g)$ of some vector-space $V$ in such a way that the group properties are preserved:
$T(e)=I$ (The identity element of the group transforms nothing at all)
$T(g^{-1})=\big ( T(g) \big )^{-1} $ (Every inverse element is mapped to the corresponding inverse transformation
$T(g)\circ T(h) = T(gh)$ (The combination of transformations corresponding to $g$ and $h$ is the same as the transformation corresponding to the point $gh$)
This concept can be formulated more general if one accepts arbitrary (not linear) transformations of an arbitrary (not vector) space. Such a map is called a realization.
In physics one is concerned most of the time with linear transformations of objects living in some vector space (for example, Hilbert space in Quantum Mechanics or Minkowski space for Special Relativity), therefore the concept of a representation is more relevant to physics than the general concept called realization.
A representation identifies with each point (abstract group element) of the group manifold (the abstract group) a linear transformation of a vector space. The framework of representation theory enables one to examine the group action on very different vector spaces.
One of the most important examples in physics is $SU(2)$. For example one can examine how $SU(2)$ acts on the complex vector space of dimension two: $C^2$ (the action on $C^1$ is trivial). The objects living in this space are complex vectors of dimension two. Therefore $SU(2)$ acts on these objects as $2\times2$ matrices. The matrices (=linear transformations) acting on $C^2$ are just the usual matrices one identifies with $SU(2)$. Nevertheless we can examine how $SU(2)$ acts on $C^3$. There is a well defined framework for constructing such representations and as a result $SU(2)$ acts on complex vectors of dimension three as $3\times 3$ matrices for which a basis is given by
\begin{equation} J_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 0& 1 & 0 \\ 1&0 & 1 \\ 0 & 1 & 0 \end{pmatrix} , \qquad J_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 0& -i & 0 \\ i&0 & -i \\ 0 & i & 0 \end{pmatrix} , \qquad J_3 = \begin{pmatrix} 1& 0 & 0 \\ 0&1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{equation}
One can go on and inspect how $SU(2)$ acts on higher dimensional vectors. This can be quite confusing and maybe its better to call the group $S^3$ instead of $SU(2)$, because usually $SU(2)$ is defined as the set of complex $2\times 2$ (!!) matrices satisfying
$U^\dagger U = 1$ and $\det(U)=1$
and now we write $SU(2)$ as $3 \times 3$ matrices. Therefore one must always keep in mind that one means the abstract group, instead of the $2 \times 2 $ definition, when one talks about higher dimensional representation of $SU(2)$ or any other group.
Typically a group is defined in the first place by a representation. This enables one to study the group properties concretely. After this initial study its often more helpful to regard the group as an abstract group, because its possible to find other, useful representations of the group.
It seems to me that there exist a typo when you describe the properties of the representation, concretely in the second one you forgot the -1 in the exponent of the right hand of the equality.
Reply to Francisco
Yes, thanks a lot! I've corrected it now.
Reply to Jakob
A complete guide to the adjoint representation of a Lie group? 21. October 2017
[…] maps each element of the set of abstract groups elemento to a matrix that acts on a vector space (see this post). The problem here is that at the beginning this can be quite confusing: If we can study the […] | CommonCrawl |
Cytoprotective effect of palm kernel cake phenolics against aflatoxin B1-induced cell damage and its underlying mechanism of action
Ehsan Oskoueian1, 2View ORCID ID profile,
Norhani Abdullah1, 3Email author,
Idrus Zulkifli1,
Mahdi Ebrahimi4,
Ehsan Karimi5, 6,
Yong Meng Goh1, 4,
Armin Oskoueian1, 7 and
Majid Shakeri1
© Oskoueian et al. 2015
Palm kernel cake (PKC), a by-product of the palm oil industry is abundantly available in many tropical and subtropical countries. The product is known to contain high levels of phenolic compounds that may impede the deleterious effects of fungal mycotoxins. This study focused on the evaluation of PKC phenolics as a potential cytoprotective agent towards aflatoxin B1 (AFB1)-induced cell damage.
The phenolic compounds of PKC were obtained by solvent extraction and the product rich in phenolic compounds was labeled as phenolic-enriched fraction (PEF). This fraction was evaluated for its phenolic compounds composition. The antioxidant activity of PEF was determined by using 1,1-diphenyl-2-picryl-hydrazil scavenging activity, ferric reducing antioxidant power, inhibition of ß-carotene bleaching, and thiobarbituric acid reactive substances assays. The cytotoxicity assay and molecular biomarkers analyses were performed to evaluate the cytoprotective effects of PEF towards aflatoxin B1 (AFB1)-induced cell damage.
The results showed that PEF contained gallic acid, pyrogallol, vanillic acid, caffeic acid, syringic acid, epicatechin, catechin and ferulic acid. The PEF exhibited free radical scavenging activity, ferric reducing antioxidant power, ß-carotene bleaching inhibition and thiobarbituric acid reactive substances inhibition. The PEF demonstrated cytoprotective effects in AFB1-treated chicken hepatocytes by reducing the cellular lipid peroxidation and enhancing antioxidant enzymes production. The viability of AFB1-treated hepatocytes was improved by PEF through up-regulation of oxidative stress tolerance genes and down-regulation of pro-inflammatory and apoptosis associated genes.
The present findings supported the proposition that the phenolic compounds present in PKC could be a potential cytoprotective agent towards AFB1 cytotoxicity.
Aflatoxin B1
Cytoprotection
Antioxidant enzyme
Molecular mechanism
Aflatoxins produced by Aspergillus species, can be ubiquitously found in many foodstuffs. The aflatoxin B1 (AFB1) produced by both Aspergillus flavus and Aspergillus parasiticus is considered the most toxic among the mycotoxins [1]. Upon ingestion, this mycotoxin causes hepatotoxicity and alters the blood and immunological parameters [2]. The AFB1 triggers the generation of reactive oxygen species (ROS) in different organs, impairs the antioxidant/pro-oxidant imbalance, elevates lipid peroxidation and damages biological molecules including lipids, proteins and DNA. The combination of these manifestations leads to oxidative stress and initiates the malfunction of the liver, which is the main detoxifying organ in the body [3, 4].
Recent studies suggested that plant phenolics and flavonoids are capable of adsorbing mycotoxins and alleviating their side effects in animals. The adsorption of toxic metabolites including aflatoxins, ochratoxins and fumonisins, boosts liver function and consequently enhances animal health and production [3, 5–7]. In this respect, phenolics, including sylimarin [8], rosmarinic acid [9], carnosic acid [10], catechins [11, 12] hesperidin [13], thymol [3] and quercetin [6] have been found to posses cytoprotective effects. However, none of these compounds have been commercialised as AFB1 cytoprotective agents due to their limited supply. Consequently, easily available sources of phenolic compounds such as agro-industrial by-products should be considered as an alternative source of these metabolites. The palm kernel cake (PKC), the residue from the kernel during oil extraction, would offer a sustainable source of phenolic compounds as the by-product is abundantly produced in countries like Indonesia, Malaysia, Philippines, Thailand, India, Nigeria, Colombia and Ivory Coast [14, 15].
Some reports are available on the phenolic compounds present in the palm oil and leaf [16], but information concerning the characteristics and function of phenolic compounds present in the PKC is rather limited and inconclusive. Therefore, we hypotised that the phenolics present in PKC may have antioxidant potential to impede the AFB1 cytotoxixity effects. In this regard, chicken hepatocytes, as one of the most sensitive cells to AFB1 manifestations, were used to evaluate the cytoprotective effects of PKC phenolics and to determine the underlying mechanisms of protection against AFB1 cytotoxicity.
Agriculture by-product
The expeller type PKC was obtained from Oil Mill Sdn Bhd., Dengkil, Selangor, Malaysia. The samples were freeze dried (Labconco, Kansas City, USA) and ground (mesh 100) using a laboratory grinder and stored at –20 °C before used.
Extraction of phenolic compounds
The reflux extraction technique was applied to extract the phenolic compounds in PKC following the method as described by Crozier et al. [17]. Briefly, 5 g of dried PKC powder were transferred into a 500 ml round-bottom flask and 160 ml of methanol were added, followed by 40 ml of 6 M HCL solution. The flask was then heated for two hours at 90 °C and the mixture was filtered (No. 1, Whatman, England). The methanol was removed under vacuum using a Rotary Evaporator (Rotavapour Buchii, Flawil, Switzerland) and the aqueous phase was then washed using n-hexane (20 ml) and subjected to liquid-liquid extraction with diethyl ether (3 × 20 ml) and ethyl acetate (3 × 20 ml). The organic solvents were then removed by using a Rotary Evaporator (Buchii, Switzerland) at 50 °C. The dried fraction obtained was weighed and reconstituted in dimethyl sulfoxide and labeled as PKC phenolic-enriched fraction (PEF).
Total phenolic contents
The total phenolic content (TPC) of PEF was determined according to the method described by Ismail et al. [18]. The PEF solution (0.5 ml) was mixed with 2.5 ml Folin-Ciocalteu reagent (previously diluted with water 1:10, v/v) and 2 ml sodium carbonate solution (7.5 %, w/v) and subsequently incubated for 90 min in the dark. The absorbance of the mixture was determined using a spectrophotometer (Molecular Devices, Sunnyvale, CA, USA) at 765 nm and the result was expressed as milligram of gallic acid equivalent (GAE) per gram of dried PEF.
Analyses of phenolic compounds by HPLC
To determine the quantity and types of phenolic compounds, the PEF was analysed by a high performance liquid chromatograph (Waters, Milford, MA, USA) equipped with an analytical column (Intersil ODS-3, 5 μm 4.6x150 mm, Gl Science Inc) as described by Karimi et al. [19].
The mobile phase consisted of deionized water (solvent A) and acetonitrile (solvent B). The pH of deionized water was adjusted to 2.5 with trifluoroacetic acid. The column was equilibrated by 85 % solvent A and 15 % solvent B. The elution was established by increasing the ratio of solvent B from 15 % to 85 % in 50 min. Then, the solvent B was decreased to 15 % in the next 5 min and this ratio was maintained for an extra 10 min for re-equilibration. The flow rate was 0.6 ml/min and the phenolic compounds were detected at 280 nm. For quantification of phenolic compounds, a calibration curve was prepared by injection of different standard compounds. The results were expressed as milligram of each phenolic compound per gram of dried PEF.
Radicals scavenging activity
The 1,1-diphenyl-2-picryl-hydrazil (DPPH) free radicals were used to evaluate the radical scavenging activity of PEF following the method described by Gulcin et al. [20]. The PEF solution was diluted in methanol and 1 ml of the solution was mixed with DPPH methanolic solution (0.1 mM). The mixture was incubated in a dark condition, at room temperature for 30 min. The absorbance of the mixture was read using a spectrophotometer (Molecular Devices, Sunnyvale, CA, USA) at 517 nm. The gallic acid was used as the standard antioxidant and the free radical scavenging activity of the PEF was calculated as follows:
$$ \mathrm{Radical}\kern0.5em \mathrm{scavenging}\ \mathrm{activity}\left(\%\right)=\left[\left(a\hbox{-} b\right)/(a)\right]\times 100 $$
a= Absorbance of negative control; b= Absorbance of sample
Ferric reducing antioxidant power (FRAP)
The ferric reducing antioxidant power of PEF fraction was evaluated according to the method described by Yen and Chen [21]. Briefly, 1 ml of PEF solution, 2.5 ml of potassium phosphate buffer (0.2 M, pH 6.6) and 2.5 ml of potassium ferricyanide (1 %, w/v) were added to the test tube, and the mixture was incubated at 50 °C for 20 min. In order to stop the reaction, 2.5 ml trichloroacetic acid (10 %, w/v) were added and the mixture was centrifuged at 3000 × g for 15 min. The upper layer of solution (2.5 ml) was transferred to the test tube containing 2.5 ml of distilled water and 0.5 ml FeCl3 (0.1 %, w/v). The solutions were mixed properly, and the absorbance of the reaction was determined at 700 nm (Molecular Devices, Sunnyvale, CA, USA) . Gallic acid was used as the reference antioxidant and the ferric reducing antioxidant power of samples was calculated using the following formula.
$$ \mathrm{Antioxidant}\ \mathrm{activity}\ \left(\%\right) = \left[\left(\mathsf{a}\hbox{-} \mathsf{b}\right)\ /\ \left(\mathsf{a}\right)\right] \times 100 $$
Inhibition of ß-carotene bleaching
The ß-carotene bleaching assay was used to determine the antioxidant activity of PEF according to the method described by Ismail et al. [18]. Three milliliter of ß-carotene solution (5 mg ß-carotene/50 ml chloroform), linoleic acid (40 mg) and Tween 20 (400 mg) were mixed thoroughly and then a stream of nitrogen gas was passed to dry the mixture. The ß-carotene-linoleic acid emulsion was prepared by adding 100 ml of ultra-pure water. Then, to 1.5 ml of ß-carotene-linoleic acid emulsion, 20 μl of PEF solution were added and the mixture was incubated in a water bath (50 °C, 60 min). At the end of the incubation, the reacting mixtures were cooled and, the absorbance was read at 470 nm using a spectrophotometer (Molecular Devices, Sunnyvale, CA, USA). The gallic acid was used as the standard in this assay. The following formula was applied to calculate the antioxidant activity of PEF.
$$ \mathrm{Antioxidant}\ \mathrm{activity}\ \left(\%\right) = \left[\left(\mathsf{R}\mathsf{D}\mathsf{c}\ \hbox{--}\ \mathsf{R}\mathsf{D}\mathsf{s}\right)\ /\left(\mathsf{R}\mathsf{D}\mathsf{c}\right)\right] \times \mathsf{100} $$
RDc= Rate of degradation in the control: [(a/b) /60]; RDs= Rate of degradation in the sample: [(a/b) /60]; a = Initial absorbance of the sample; b = Absorbance after 60 min of incubation
Thiobarbituric acid reactive substances assay (TBARS)
The TBARS assay was used to evaluate the potential of PEF in preventing the oxidation of linoleic acid under oxidative condition, according to the method described by Hendra et al. [22]. The PEF (4 mg) was dissolved in 4 ml of absolute ethanol and then 4.1 ml of 2.5 % linoleic acid in 99 % ethanol, 8 ml of phosphate buffer (0.05 M, pH 7) and 3.9 ml of ultra-pure water were added. The mixture was transferred to the 15 ml screw cap test tube, capped tightly and incubated in a 40 °C oven for 6 days. At the end of the incubation period, 2 ml of sample solutions, aqueous solution of trichloroacetic acid [1 ml of 20 % (w/v)] and aqueous thiobarbituric acid [2 ml, 0.67 % (w/v)] were mixed in a screw cap test glass tube and incubated in boiling water bath for 10 min. The tube was cooled to room temperature and centrifuged at 3000 × g for 20 min. The absorbance of the supernatant was determined using a spectrophotometer (Molecular Devices, Sunnyvale, CA, USA) at 532 nm.
The antioxidant activity was reported as:
$$ \mathrm{Percent}\ \mathrm{inhibition} = \left[\left(\mathsf{a}\hbox{-} \mathsf{b}\right)\ /\ \left(\mathsf{a}\right)\right] \times \mathsf{100} $$
a = Absorbance of the control reaction; b: Absorbance of the sample reaction
Isolation and culture of primary chicken hepatocytes
The chicken hepatocytes were isolated using the 2-step collagenase method as described by Wang et al. [23]. Five-week-old male chickens were treated by intra-peritoneal injection of natrium thiopenthal (45 mg/kg) and heparin (1500 U/kg). The abdominal cavity was opened after full anaesthesia. The liver was perfused with different buffers as described by Wang et al. [23] and then the liver was excised and digested using 0.5 mg/ml of collagenase type IV for 25 min at 37 °C. The William's E medium (Gibco, Grand Island, NY) containing 5 % chicken serum and 2 mg/ml bovine serum albumin (BSA) was used to stop the digestion. The cells were passed through 100, 60 and 30 μm sieves and subsequently incubated with red blood cell lysis buffer (Sigma–Aldrich, St. Louis, MO, USA) and rewashed using William's E medium containing chicken serum to eliminate the red blood cells. The accuracy of the isolated hepatocytes were confirmed according to morphological characteristics. Cells were cultured in William's E medium supplemented with 100 U/ml of penicillin-streptomycin, 10 μg/ml insulin and 5 % chicken serum. The cells were incubated at 37 °C with 5 % CO2 in a humidified incubator. The approval of Animal Use and Care Committee (ACUC), Faulty of Medicine and Health Sciences, University Putra Malaysia for this procedure was obtained.
The cytotoxicity effect of PEF on chicken hepatocytes was determined using MTT assay [24]. The cells were grown in each well of 96-well plates with the density of 5 × 103 cells/ 100 μl of the medium. The cells were pre-treated with the serial concentrations of PEF (0, 5, 10, 20, 40 μg/ml) and gallic acid (10 μM or 1.7 μg/ml) as positive control. The cells incubated in a medium devoid of PEF (0 μg/ml) was considered as a negative control. The cells were incubated for 24 h, and then the media were replaced with the fresh media containing 5 μM of AFB1 (Cayman Chemical Company, Ann Arbor, MI, USA) and incubated for another 48 h. Finally, the viability of the cells was determined by using 3–(4,5–Dimethylthiazol-2-yl)-2,5-Diphenyltetrazolium Bromide (MTT) assay. The experiment was conducted in triplicates.
Antioxidant enzyme assay
The cells were treated as mentioned earlier in the cytotoxicity assay. Upon treatments, cells were rinsed with ice-cold phosphate-buffered saline (PBS, 0.1 M, pH 7.4) for three times. Then, 6 ml of PBS were added, cells were scraped and transferred into 15 ml centrifuge tube. The cells were centrifuged at 250 × g for 20 min at 4 °C and the supernatant was discarded. The cells were lysed immediately at 4 °C using 150 μl of lysis buffer (0.5 % Triton x-100, 2 mM ETDA in 20 mM Tris–HCl pH 7.5) and then sonicated for 10 s using a sonicator (Hielscher, Teltow, Germany). Then lysates were centrifuged at 2800 × g for 10 min at 4 °C. The supernatant was collected to determine the antioxidant enzymes activity. The activities of superoxide dismutase (SOD), catalase (CAT) and glutathione reductase (GR) were determined by using enzyme kits from Nanjing Jiancheng Bioengineering Institute (Nanjing, China) according to the instructions provided by the kits. The results were expressed as enzyme activity/g protein (U/ g protein) of the cells.
The lipid peroxidation in the chicken hepatocytes was determined by measuring the malondialdehyde (MDA) using thiobarbituric acid reactive substances (TBARS) [25]. The treatments were similar to the cytotoxicity test. Treated cells were rinsed with phosphate-buffered saline (PBS, 0.1 M) for three times and scraped. The scraped cells were suspended in 4 ml of potassium chloride (1 %) and homogenised using an Ultra-Turrax homogeniser (Wilmington, NC, USA) at 20,000 rpm for 25 s while kept on ice. Then, 300 μl distilled water, 200 μl of homogenised cells, 35 μl of BHT, 165 μl sodium dodecyl sulphate (SDS) and 2 ml TBA were added into the screw cap test tube. The solution was mixed and heated at 90 °C for 60 min. The solution was cooled immediately and 3 ml of n-butanol were added, shaken for 30 s and centrifuged at 2800 × g for 10 min. The absorbance of n-butanol fraction was recorded at 532 nm by a spectrophotometer (Molecular Devices, Sunnyvale, CA). The 1,1,3,3-tetraethoxypropane was used to construct the standard curve.
Gene expression analyses
The hepatocytes were cultured and treated as mentioned in the cytotoxicity assay. Treated cells were rinsed with phosphate-buffered saline (PBS, 0.1 M, pH 7.2) for two times and scraped. Total RNA was extracted from cells using a RNasey mini kit (Qiagen, Valencia, CA, USA). The total RNA was converted to cDNA through reverse transcript PCR technique using Maxime RT Permix kit (iNtRON Biotechnology, Sungnam, Korea). The expression of nuclear factor kappa-light-chain-enhancer of activated B cells (NF-kB), nitric oxide synthase (iNOS), tumor necrosis factor alpha (TNF-α), interleukin-1 beta (IL1ß), interleukin-6 (IL6), bax, bcl2, glyceraldehyde 3-phosphate dehydrogenase (GAPDH) and β-Actin genes (Table 1) were analysed by Real time PCR thermocycler (Bio-Rad, CA, USA) using iQ SYBR Green Supermix (Bio-Rad, CA, USA). The amplification conditions were optimised for all genes as follows: 95 °C for 5 min (1X), then 95 °C for 20 s, then 58 °C for 20 s and 72 °C for 25 s (35X). The expressions of genes were normalised to GAPDH and ß-actin as housekeeping genes according to Vandesompele et al. [26] method using CFX manager software version 2 (Bio-Rad Laboratories). All the real time PCR amplifications were conducted in triplicates.
The primer characteristics used for the gene expression analysis
Sequences (5′ to 3′)
NF-kB
gaaggaatcgtaccgggaaca
ctcagagggccttgtgacagtaa
iNOS
gaacagccagctcatccgata
cccaagctcaatgcacaactt
TNF-α
tgtgtatgtgcagcaacccgtagt
ggcattgcaatttggacagaagt
IL1ß
tgggcatcaagggctaca
tcgggttggttggtgatg
caaggtgacggaggaggac
tggcgaggagggatttct
tcctcatcgccatgctcat
ccttggtctggaagcagaaga
gatgaccgagtacctgaacc
caggagaaatcgaacaaaggc
GAPDH
gtcagcaatgcatcgtgca
ggcatggacagtggtcataaga
acacggtattgtcaccaact
taacaccatcaccagagtcc
The expression of 70 kilodalton heat shock protein (Hsp70), caspase-3 and GAPDH proteins were determined by Western blot analysis. The chicken hepatocytes were cultured and treated as described in the cytotoxicity assay. The cells were trypsinised and washed with ice-cold PBS (0.1 M, pH 7.2) and the cells were immediately lysed at 4 °C using 150 μl of lysis buffer (0.5 % Triton x-100, 2 mM ETDA in 20 mM Tris–HCl pH 7.5) containing 15 μl/ml of protease inhibitor (ProteoBlock Protease Inhibitor Cocktail, Fermentas, MD, USA). In order to facilitate the extraction of proteins, the cells were sonicated for 15 s using a sonicator (Hielscher, Teltow, Germany) and incubated on ice for 20 min. In order to collect the supernatant, cell lysates were centrifuged at 15,000 × g for 25 min and the protein concentration of the supernatant was determined using a Protein Assay kit (Bio-Rad, CA, USA). The protein (25 μg) was denatured at 95 °C for 5 min and subjected to electrophoresis using Tris-glycine polyacrylamide gel. The Hoefer Semi-Dry Transfer Unit was used to transfer the protein to a PVDF membrane, and the membrane was washed using Odyssey Blocking Buffer (LI-COR, Lincoln, NE, USA). The membrane was incubated in heat shock protein 70 (Hsp70) (Biorbyt orb10848), nuclear factor (erythroid-derived 2)-like 2 (Nrf2) (Biorbyt orb11165), caspase-3 (Biorbyt orb10237) and GAPDH (Thermo Scientific MA1-4711) primary antibodies with dilution rates ranging from 1:500 up to 1:1000 overnight at 4 °C. The PBST (phosphate buffer saline and Tween 20, 0.05 %) was used to wash the membrane for three times. The IRDye 680 Goat Anti-Mouse or IRDye 800 CW Goat Anti-Rabbit secondary antibodies were applied to detect the target proteins by using the Odyssey Infrared Imaging System (LI-COR, Lincoln, NE, USA). The intensity of the bands were analysed by the Odyssey software.
Apoptosis analysis by flow cytometry
The chicken hepatocytes were cultured at the density of 1 × 106 cells per 75 cm2 flask and treated as mentioned in the cytotoxicity test. The cells were trypsinised and washed with ice-cold PBS (0.1 M, pH 7.2). Then, cells were stained using FITC Annexin V Apoptosis Detection Kit I (BD Biosciences Pharmingen, San Diego, CA, USA) according to the manufacturer's instruction. The cell apoptosis was evaluated by flow cytometry (FACS-Canto II BD Biosciences) and the data were analysed using Diva software (BD Biosciences, Franklin Lakes, NJ, USA). The characteristics of viable cells were FITC Annexin V and PI negative, whereas early apoptotic cells to be FITC Annexin V positive and PI negative. The cells in late apoptosis or already dead were both FITC Annexin V and PI positive.
All the data obtained from this study were analysed in a completely randomised design using the GLM procedure of SAS [27]. The differences between means were determined by Duncan's Multiple Range Test and considered significant at p < 0.05. All measurements were performed in triplicate samples and carried out independently at least three times.
Extraction yield and total phenolic contents
The yield of PEF was 9.0 ± 0.86 g/100 g dry PKC, and the amount of total phenolics was 658.3 ± 26.32 mg gallic acid equivalents (GAE) /g dried PEF (Table 2).
The extraction yield and total phenolic content of phenolic-enriched fraction (PEF)
Extraction yield (g/100 g DM a)
9.0 ± 0.86
Total phenolic content b (mg/g DM)
658.3 ± 26.32
All data are presented as means (± S.E.M) of at least three replicates (n = 3)
S.E.M Standard error of the means
aDM: Dry matter
bmg gallic acid equivalents (GAE)/g dry matter PEF
As shown in Fig. 1, the PEF contained phenolic acids, including gallic acid, pyrogallol, vanillic acid, caffeic acid, syringic acid, epicatechin, catechin and ferulic acid with the concentrations ranging from 6.9 to 13.2 mg/g dry fraction (Table 3).
The HPLC chromatogram of phenolic acids present in phenolic-enriched fraction (PEF) obtained from PKC detected at 280 nm
The types of phenolic acids detected in phenolic-enriched fraction (PEF)
Phenolic compounds (mg/g dried PEF)
Gallic acid
Syringic acid
Pyrogallol
Epicatechin
Vanillic acid
Catechin
Caffeic acid
Ferulic acid
The IC50 values presented in Table 4 indicated the antioxidant activity of PEF and gallic acid (positive control). The DPPH scavenging activity, reducing power activity, ß-carotene bleaching inhibition and TBARS inhibition values for PEF were 24.6, 31.2, 37.1 and 42.9 μg/ml and these values were significantly (p < 0.05) higher than that of gallic acid with the values of 4.6, 7.4, 11.6 and 14.5 μg/ml, respectively.
The IC50 values indicating antioxidant activity of phenolic-enriched fraction (PEF) and positive control (gallic acid)
DPPH scavenging activity
Reducing power activity
ß-carotene bleaching inhibition
TBARS inhibition
S.E.M
All data are presented as means (± SEM) of at least three replicates (n = 3)
Means (n = 3) with different superscripts (a,b) within a column are significantly different (p < 0.05)
Cytotoxic assay
The cytoprotective activities of PEF and gallic acid against AFB1-cell damage are shown in Fig. 2. The AFB1 at the concentration of 5 μM decreased the cell viability to 57.1 % upon 48 h incubation. Treatment of cells with 20 and 40 μg/ml of PEF enhanced cell viability significantly (p < 0.01). Similarly, gallic acid with the concentration of 10 μM or 1.7 μg/ml significantly (p < 0.01) improved the cell survival.
The cytoprotective activity of PEF and gallic acid against AFB1-cell damage. All values are means ± S.E.M of three independent experiments. The cells were pre-treated with different concentrations of PEF and gallic acid as positive control (10 μM or 1.7 μg/ml) for 24 h and then the media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h. The experiment was performed in triplicate. ***p < 0.001 and **p < 0.01 indicated significant difference compared to the untreated control (0)
Antioxidant enzymes and lipid peroxidation
Table 5 shows the results of total protein, lipid peroxidation and antioxidant enzymes activity in chicken hepatocytes. It was observed that with the increase in the concentration of PEF, the total cellular protein increased from 0.6 mg/ml to 0.8 mg/ml. The lipid peroxidation values were reduced gradually from 7.5 to 3.8 nM MDA/mg protein. The activities of SOD, CAT and GR were 4.2, 3.8 and 0.2 U/mg protein, respectively, without the addition of PEF, but increased to 7.8, 6.9 and 0.5 U/mg protein, respectively, when cells were treated with 40 μg/ml of PEF. In general, the concentrations of 20 and 40 μg/ml of PEF significantly (p < 0.05) improved the oxidative activities of biomarkers, indicating the potential of the PEF to alleviate the negative impacts of AFB1 on hepatocytes functions. Furthermore, these concentrations of PEF exhibited comparable results to that of gallic acid as a reference antioxidant used in this study.
The total protein, lipid peroxidation and antioxidant enzyme activity of chicken hepatocytes pretreated with phenolic-enriched fraction (PEF) and exposed to AFB1
AFB1 (5 μM)
PEF (μg/ml)
Control+
Total protein (mg ml−1)
Lipid peroxidation (nM MDA mg−1 protein)
6.9ab
3. 8de
SOD activity (U mg−1 protein)
4.8cd
5.7bc
CAT activity (U mg−1 protein)
3.8de
4.9bcd
GR activity (U mg−1 protein)
All cells were pre-treated with different concentrations of PEF for 24 h, then the media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h
MDA Malondialdehyde as lipid peroxidation biomarker
Control+: 10 μM or 1.7 μg/ml gallic acid
SOD Superoxide dismutase
CAT Catalase
GR Glutathione reductase
Means (n = 3) with different superscripts (a,b,c,d,e) within a row are significantly different (p < 0.05)
Changes in molecular biomarkers of oxidative stress
The changes in the expression of various genes in chicken hepatocytes are presented in Table 6. The gradual increase in PEF concentration up-regulated the anti-apoptosis gene (bcl2) and down-regulated the expressions of NF-kB, proinflammatory mediators (iNOS, TNF-α, IL1ß and IL6) and pro-apoptotic gene (bax). The PEF at 40 μg/ml produced comparable results to gallic acid as a positive control.
The changes in the expression of different genes in chicken hepatocytes pretreated with phenolic-enriched fraction (PEF) and exposed to AFB1
Gene expression (Fold changes)
PEF concentration (μg/ml)
Up-regulated genes
+2.1cd
+3.2c
+4.3b
+5.9a
Down-regulated genes
−1.5d
−2.6c
−3.6b
−4.4a
−1.2cd
−2.4bc
The cells were pre-treated for 24 h with different concentrations of PEF ranging from 0 to 40 μg/ml and positive control (10 μM or 1.7 μg/ml gallic acid). The media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h
The expression of each gene was normalised to the GAPDH and ß-actin expressions as housekeeping genes and then the result normalised to the expression of that gene in the negative control (PEF 0 μg/ml)
Means (n = 3) with different superscripts (a,b,c,d) within a row are significantly different (p < 0.05)
The western blot analysis confirmed the changes in the expression of Hsp70 (Fig. 3), nrf2 (Fig. 4) and caspase-3 (Fig. 5) proteins. It was observed that the high concentrations of PEF (20 and 40 μg/ml) significantly (p < 0.05) down-regulated the Hsp70 and caspase-3 proteins while up-regulated the nrf2 protein as compared to the cells without PEF pretreatment (0 μg/ml). The gallic acid has also been found to down-regulate the Hsp70 and caspase-3 proteins and up-regulate the nrf2 protein.
Expression of Hsp70 protein in chicken hepatocytes. The cells were pre-treated for 24 h with different concentrations of PEF ranging from 0 to 40 μg/ml and positive control (gallic acid, 10 μM or 1.7 μg/ml). The media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h. All values represent means ± SEM from three independent experiments. *** p < 0.001 and ** p < 0.01 indicate significant difference compared to the untreated control (0)
Expression of nrf2 protein in chicken hepatocytes. The cells were pre-treated for 24 h with different concentrations of PEF ranging from 0 to 40 μg/ml and positive control (gallic acid, 10 μM or 1.7 μg/ml). The media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h. All values represent means ± SEM from three independent experiments. *** p < 0.001 and ** p < 0.01 indicate significant difference compared to the untreated control (0)
Expression of caspase-3 protein in chicken hepatocytes. The cells were pre-treated for 24 h with different concentrations of PEF ranging from 0 to 40 μg/ml and positive control (gallic acid, 10 μM or 1.7 μg/ml). The media were replaced with a medium containing 5 μM of AFB1 and the cells incubated for another 48 h. All values represent mean ± standard error from three independent experiments. *** p < 0.001 and ** p < 0.01 indicate significant difference compared to the untreated control
The flow cytometry analysis plots are presented in Fig. 6. As observed in Fig. 6-a, the majority of cells showed apoptosis. However, it was evident that the increase in the concentration of PEF resulted in the decrease of the population of apoptotic cells (Fig. 6–b to -e). Similarly, Fig. 6–f indicates that pretreatment of cells with the gallic acid decreased the apoptotic cells. The percentage values for the viable, apoptotic and dead cells obtained from flow cytometry analysis are presented in Table 7. In the medium without PEF (0 μg/ml), the majority of the cell mass (52.8 %) were recognised as apoptotic cells. The increasing concentrations of PEF reduced the percentage of apoptotic cells and enhanced the cell viability significantly (p < 0.05). The percentage of apoptotic and viable cells in hepatocytes treated with gallic acid as a positive control were 41.1 and 40.2 %, respectively, which were comparable to that of hepatocytes treated with 20 and 40 μg/ml of PEF.
Flow cytometry analyses of chicken hepatocytes pretreated with phenolic-enriched fraction (PEF) and exposed to 5 μM of AFB1. The a, b, c, d and e showed cells pretreated with 0, 5, 10, 20 and 40 μg/ml of PEF, respectively. The f indicated that the cells were pretreated with gallic acid at the concentration of 10 μM or 1.7 μg/ml
Percentage of viable, apoptotic and dead cells analysed by flow cytometry
Cells (%)
Viable
40.2ab
Apoptotic
43.4bc
41.1cd
The cells were pre-treated with different concentrations of PEF for 24 h, then the media were replaced with a medium containing 5 μM of AFB1 and the cells were incubated for another 48 h
A minimum of 15,000 cells per sample was analysed by flow cytometry
Means with different superscripts (a,b,c,d) within a row are significantly different (p < 0.05)
PEF Phenolic-enriched fraction
The types of phenolic compounds detected in PEF were slightly different from that of the oil palm fruit extract which showed the presence of protocatechuic, p-hydroxybenzoic and p-coumaric acids, besides gallic, vanillic, caffeic, syringic and ferulic acids [28]. Tan et al. [29] also reported the presence of p-hydroxybenzoic acid, cinnamic acid, ferulic acid and coumaric acid in the oil, while, Jaffri et al. [30] observed the presence of catechin derivatives in the oil palm leaf extract. It seemed that the PEF contained additional types of phenolic compounds as compared to those detected in the oil and leaf extracts. However, the types of phenolic compounds present are subjected to numerous factors including extraction, detection and identification procedures, besides other agronomic factors. Phenolic compounds have been reported as a potential antioxidant [31, 32], thereby PEF can be considered a reliable source of natural antioxidants for protecting cells against xenobiotics toxicity.
The results presented in Table 5 and Fig. 2 revealed that, pretreatment of hepatocytes with different concentrations of PEF improved the lipid peroxidation and cellular antioxidant enzymes, concomitant with the increase in cell viability when cells were exposed to AFB1. These findings reflected the cytoprotective properties of PEF, which could be attributed to its antioxidant activity. Previous studies have confirmed the ability of mycotoxins to impair the balance between pro-oxidants and antioxidants in the cells, which resulted in the production of reactive oxygen species (ROS), that led to lipoperoxidation and oxidative stress [3–7, 9, 12]. Consequently, it was emphasized that antioxidant enzymes would function as the main defense mechanism against the ROS in the cells. In the present study, the PEF exerted cytoprotective effect which reduced the toxic symptoms of AFB1 probably through activation of antioxidant enzymes and inhibition of lipid peroxidation chain reaction. Similar studies have reported the role of natural antioxidants in enhancing antioxidant enzymes, inhibiting lipid peroxidation and alleviating oxidative stress in cells exposed to mycotoxins [6, 33–35]. The PEF at 40 μg/ml showed cytoprotection activity and alleviation of oxidative biomarkers reaching up to that of gallic acid at 1.7 μg/ml. Although the concentrations of phenolic compounds used in the assay differ, the PEF should be considered effective in alleviating the AFB1 effects. These findings should be of interest in lieu of PEF source and availability.
Table 6 and Figs. 3, 4 and 5 show the expression of molecular biomarkers involved in oxidative stress, inflammatory response and apoptosis in cells pretreated with various concentrations of PEF and exposed to AFB1. The relationship between expressions of genes and proteins provides a better understanding of the mechanism of action of PEF against AFB1-cell damage. As shown in Fig. 7, the AFB1 was converted to the Aflatoxin B1–8,9 epoxide through epoxidation and this active form initiated the production of ROS in the cells. Thereafter, the excessive amount of ROS elicited the NF-kB up-regulation and nrf2 down-regulation. The NF-kB upon up-regulation induced the expression of various pro-inflammatory mediators, including iNOS, TNF-α, IL1ß and IL6 (Fig. 7). The nrf2 as the redox-sensitive transcription factor could activate the cellular antioxidant defense mechanism and enhance the production of antioxidant enzymes and heat shock proteins [36]. The down-regulation of nrf2 reduced the antioxidant enzymes production and increased the up-regulation of Hsp70 (Fig. 7). The Hsp70 plays a role as a sensor of cellular redox changes acting like antioxidant enzymes. For instance, under oxidative stress the intracellular components, particularly proteins could undergo oxidation. The Hsp70 restores and maintains the redox homeostasis of the cells even under oxidative stress [36].
This diagram illustrates the events taking place in un-treated chicken hepatocytes exposed to AFB1
The up-regulation of pro-inflammatory mediators accompanied by suppression of anti-inflammatory proteins disrupted the cell homeostasis leading to an imbalance between anti-apoptotic (bcl2) and pro-apoptotic (bax and caspase-3) effectors. The imbalance between anti-apoptotic and pro-apoptotic effectors induced apoptosis and cell death (Fig. 7). The pretreatment of cells with PEF induced the cells to attenuate the expression of NF-kB and subsequently pro-inflammatory mediators (iNOS, TNF-α, IL1ß and IL6) (Fig. 8). This is possibly due to the direct inhibition of ROS together with modulation of NF-kB expression. On the other hand, the suppression of pro-inflammatory mediators through direct action of PEF could be another plausible reason (Fig. 8).
This diagram illustrates the probable protective mechanisms of PEF in chicken hepatocytes exposed to AFB1
The PEF up-regulated the expression of nrf2 and its activation enhanced the antioxidant enzymes production (Table 5), while inhibiting the Hsp70 expression. The inverse relationship between antioxidant enzymes and Hsp70 protein suggests that the antioxidant enzymes regulate the cellular redox homeostasis in lieu of Hsp70 protein, thereby high expression of Hsp70 is no longer required. In line with this result, Costa et al. [37] also reported the critical role of phenolic compounds in up-regulation of nrf2 protein and enhancement of antioxidant enzymes production.
The results of the present study showed that, the PEF not only affected the NF-kB and nrf2 expressions, but also regulated the imbalance between anti-apoptotic and pro-apoptotic effectors resulting in the enhancement of cell survival. The anti-apoptotic activity of PEF was probably due to the presence of phenolic compounds (gallic acid, pyrogallol, vanillic acid, caffeic acid, syringic acid, epicatechin, catechin and ferulic acid) which up-regulated the bcl2 gene and down-regulated the bax gene and caspase-3 protein (Fig. 8, Table 6). Similarly, several observations demonstrated the close relationship between antioxidant activity of phenolic compounds with cellular antioxidant enzymes activity and reduction of apoptosis cell death in various animal and human cell lines [5, 11, 38].
The present study showed that PEF could be considered as a cytoprotective agent by up-regulating the antioxidant-related genes and down-regulating the pro-inflammatory and apoptosis associated genes in hepatocytes exposed to AFB1. The findings valorised the phenolic compounds of PKC and paved the way for production of an alternative cytoprotective agent against AFB1 cytotoxicity.
PKC:
Palm kernel cake
AFB1:
PEF:
Phenolic-enriched fraction
TPC:
Total phenolic content
GAE:
Gallic acid equivalent
DPPH:
1,1-diphenyl-2-picryl-hydrazil
ACUC:
Animal Use and Care Committee
MTT:
3-(4,5-Dimethylthiazol-2-yl)-2,5-Diphenyltetrazolium Bromide
SOD:
Superoxide dismutase
Catalase
Glutathione reductase
MDA:
Malondialdehyde
TBARS:
Thiobarbituric acid reactive substance
Sodium dodecyl sulphate
NF-kB:
Nuclear factor kappa-light-chain-enhancer of activated B cells
iNOS:
Tumor necrosis factor alpha
IL1ß:
Interleukin-1 beta
IL6:
GAPDH:
Glyceraldehyde 3-phosphate dehydrogenase
Hsp70:
70 kilodalton heat shock protein
Nrf2:
Nuclear factor (erythroid-derived 2)-like 2
The research funds provided by the Ministry of Education, Malaysia, under the Long-Term Research Grant Scheme (LRGS), Project number: UPM/700-1/3/LRGS is gratefully acknowledged.
The authors declared that they have no competing interests.
The contributions of authors are as follow: Conception and design of experiment: EO, NA, IZ and YMG. Conduction of experiments: EO, ME, MS, AO and EK. Data acquisition and statistical analysis: EO, ME, MS, AO and EK. Manuscript preparation: EO, ME and NA. All authors read and approved the final manuscript.
Institute of Tropical Agriculture, Univeristi Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Agricultural Biotechnology Research Institute of Iran (ABRII), East and North-East Branch, P.O.B. 91735/844, Mashhad, Iran
Department of Biochemistry, Faculty of Biotechnology and Biomolecular Sciences, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Department of Veterinary Preclinical Sciences, Faculty of Veterinary Medicine, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Department of Biochemistry and Biophysics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Department of Crop Science, Faculty of Agriculture, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Ferdowsi University of Mashhad, International Branch, Mashhad, Iran
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Distribution of extremal values
If an item follows normal distribution, average also follows normal distribution. What about minimum and maximum?
normal-distribution order-statistics
$\begingroup$ You might want to look into this book. $\endgroup$ – mpiktas Apr 18 '11 at 6:57
$\begingroup$ @user4211, do you ask about distribution of minimum and maximum of any sample distribution, or only normal? $\endgroup$ – mpiktas Apr 18 '11 at 6:58
You should have a look at the order statistics. Here is a very brief overview.
Let $X_{1}, \ldots X_{n}$ be an i.i.d. sample of size $n$ drawn from a population with distribution function $F$ and probability density function $f$. Define $Y_{1}=X_{(1)}, \ldots, Y_{r} = X_{(r)}, \ldots, Y_{n}=X_{(n)}$, where $X_{(r)}$ denotes the $r$th order statistic of the sample $X_{1}, \ldots X_{n}$, i.e., its $r$th smallest value.
It can be shown that the joint probability density function of $Y_{1}, \ldots, Y_{n}$ is
$f_{X_{(1)}, \ldots, X_{(n)}}(y_{1}, \ldots, y_{n}) = n! \prod_{i=1}^{n} f(y_{i})$ if $y_{1} < y_{2} < \ldots < y_{n}$ and $0$ otherwise.
By integrating the previous equation we get
$f_{X_{(r)}}(x) = \frac{n!}{(r - 1)! (n - r)!} f(x) (F(x))^{r-1} (1 - F(x))^{n - r}$
In particular, for the minimum and maximum, we respectively have
$f_{X_{(1)}}(x) = n f(x) (1 - F(x))^{n-1}$
$f_{X_{(n)}}(x) = n f(x) (F(x))^{n - 1}$
mpiktas
ocramocram
$\begingroup$ +1, I've edited a small mistake in the second last formula. $\endgroup$ – mpiktas Apr 18 '11 at 6:59
$\begingroup$ Thanks ocram, the answer is impressive so I checked as good answer but now can you make it in plain english thanks :) By the way how do you put equation in stackexchnage ? $\endgroup$ – user4211 Mar 5 '12 at 8:07
$\begingroup$ What do you mean exactly? You asked for the pdf's of the minimum and of the maximum, and these two are given by $f_{X_{(1)}}$ and $f_{X_{(n)}}$, respectively. So, if you draw many many samples and compute the min for each, then you end up with a random variable with pdf $f_{X_{(1)}}$. Is it ok? $\endgroup$ – ocram Mar 6 '12 at 9:21
You might also want to read up on the generalized extreme value (GEV) distribution. It turns out that as $n\rightarrow\infty$, the (shifted and scaled) distribution of the maximal value of the sample converges to one of the three special cases of the GEV distribution.
AnikoAniko
$\begingroup$ Great link will read it $\endgroup$ – user4211 Mar 5 '12 at 8:08
The sum of Gaussians is Gaussian. That is why the average is normal. The distribution of any non-linear function of (finitely many) Gaussians need not be Gaussian, and it usually isn't. Such is the case of the maximum function. To approximate the maximum of a multivariate Gaussian, Hothorn is a good place to start.
JohnRosJohnRos
$\begingroup$ very interesting will read hothorn $\endgroup$ – user4211 Mar 5 '12 at 8:08
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Probabilistic Combinatorics Online 2020
About MIPT
See the schedule for each day of the conference below.
All of the talks will be held in Zoom:
Meeting ID: 886-3882-5996
Password: first 6 decimal places of $\pi$ after the decimal point
Alternatively, you can watch a live stream on Twitch. No registration necessary.
The videos of the talks are available on the labs' YouTube channel and you may also find the links on this page in the schedule below.
Sept 23Sept 24Sept 25
11.50 - 12.00 MSK (UTC +3)
Alexey Pokrovskiy University College London
Rota's Basis Conjecture holds asymptotically
Video Slides
Rota's Basis Conjecture is a well known problem, that states that for any collection of $n$ bases in a rank $n$ matroid, it is possible to decompose all the elements into $n$ disjoint rainbow bases. Here an asymptotic version of this is will be discussed - that it is possible to find $n − o(n)$ disjoint rainbow independent sets of size $n − o(n)$.
Brendan McKay Australian National University
Some remarks on the method of switchings
The method of switchings has become a standard tool for combinatorial enumeration and probability problems. We will discuss some old and new applications and techniques. First, we discuss how switchings make a very general tool for bounding upper tails of distributions. Second, we illustrate how switchings can be used to study subgraphs of random graphs. Finally, we enumerate linear hypergraphs with a given number of edges.
Nicholas Wormald Monash University
Fast uniform generation of regular graphs and contingency tables
joint work with Andrii Arman and Jane Gao
We present a new technique for use in switching-based random generation of graphs with given degrees. For graphs with m edges and maximum degree $D=O(m^4)$, the "best" existing uniform sampler, given by McKay and Wormald in 1990, runs in time $O(m^2D^2)$. Our new one runs in time $O(m)$, which is effectively optimal. For $d$-regular graphs with $d =o(\sqrt n)$, the best existing ones run in time $O(nd^3)$. This is now improved to $O(nd+d^4)$. Similar results are obtained for generating random contingency tables with given marginals (equivalently, bipartite multigraphs with given degree sequence) in the sparse case.
Benjamin Sudakov ETH, Zurich
Large independent sets from local considerations
joint work with Matija Bucic
How well can global properties of a graph be inferred from observations that are purely local? This general question gives rise to numerous interesting problems. One such very natural question was raised independently by Erdos-Hajnal and Linial-Rabinovich in the early 90's. How large must the independence number $\alpha(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of size $r$? In this talk we discuss new methods to attack this problem which improve many previous results.
Pawel Pralat Ryerson University
Localization Game for Random Graphs
We consider the localization game played on graphs in which a cop tries to determine the exact location of an invisible robber by exploiting distance probes. The corresponding graph parameter is called the localization number. The localization number is related to the metric dimension of a graph, in a way that is analogous to how the cop number is related to the domination number. Indeed, the metric dimension of a graph is the minimum number of probes needed in the localization game so that the cop can win in one round. We investigate both graph parameters for binomial random graphs.
Jane Gao University of Waterloo
Random graphs with specified degree sequences
Random graphs with specified degree sequences are popular random graph models not yet well understood, especially when the degree sequences are not "nice". Most problems in random graph theory eventually boil down to estimating subgraph probabilities. We will discuss the configuration model and the switching method that are tools commonly used to study such random graphs. Then we will discuss some recent results on subgraph probabilities and distributions, the chromatic number and the connectivity. Finally we discuss the relation between these random graphs and the well-known Erdős-Renyi graphs, and show how well the former can be approximated by the latter.
David Gamarnik MIT Sloan School of Management
Low-Degree Hardness of Random Optimization Problems
joint work with Aukosh Jagannath and Alex Wein
We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model (to be introduced in the talk), and (b) finding a large independent set in a sparse Erdos-Renyi graph. We consider the family of algorithms based on low-degree polynomials of the input. This is a general framework that captures methods such as approximate message passing and local algorithms on sparse graphs, among others. We show this class of algorithms cannot produce nearly optimal solutions with high probability. Our proof uses two ingredients. On the one hand both models exhibit the Overlap Gap Property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions close to optimality are either close or far from each other. The second proof ingredient is the stability of the algorithms based on low-degree polynomials: a small perturbation of the input induces a small perturbation of the output. By an interpolation argument, such a stable algorithm cannot overcome the OGP barrier, thus leading to the inapproximability. The stability property is established using concepts from Gaussian and Boolean Fourier analysis, including noise sensitivity, hypercontractivity, and total influence.
Michael Krivelevich Tel Aviv University
Color-biased Hamilton cycles in random graphs
joint work with Lior Gishboliner and Peleg Michaeli
We show that a random graph $G(n,p)$ with the edge probability $p(n)$ above the Hamiltonicity threshold is typically such that for any $r$-coloring of its edges, for a fixed $r\geq2$, there is a Hamilton cycle with at least $(2/(r+1)-o(1))n$ edges of the same color. This estimate is asymptotically optimal.
Lior Gishboliner ETH Zürich
Very fast construction of bounded-degree spanning graphs via the semi-random graph process
joint work with Omri Ben-Eliezer, Danny Hefetz and Michael Krivelevich
Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. In this paper, we consider a recently proposed semi-random graph process, defined as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u$, adding the edge $\{u,v\}$ to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property. There are also natural offline and non-adaptive variants of this setting.
It was asked by $N$. Alon whether for every bounded-degree (spanning) graph $H$, Builder can construct a copy of $H$ with high probability in $O(n)$ rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree $\Delta$ can be constructed with high probability in $(3\Delta/2+o(\Delta))n$ rounds, where the $o(\Delta)$ term tends to zero as $\Delta$ tends to infinity. This is tight (even for the offline case) up to a multiplicative factor of $3+o_{\Delta}(1)$. Furthermore, for the special case where $H$ is a forest of maximum degree $\Delta$, we show that $H$ can be constructed with high probability in $O(\log\Delta)n$ rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of $\Omega(n\sqrt{\log n})$ on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact there are non-adaptive strategies for constructing a Hamilton cycle or a $K_r$-factor, which are successful w.h.p. within $O(n\sqrt{\log n})$ rounds.
Sergei Kiselev MIPT
Rainbow matchings in $k$-partite hypergraphs
joint work with Andrey Kupavskii
Let $[n]:=\{1,\ldots,n\}$. The following conjecture was made by Aharoni and Howard [1]:
Conjecture. Let $n\ge s$ and $k$ be positive integers. If $\mathcal F_1,\ldots,\mathcal F_s\subset [n]^k$ satisfy $\min_{i}|\mathcal F_i|>(s-1)n^{k-1}$ then there exist $F_1\in\mathcal F_1,\ldots, F_s\in \mathcal F_s,$ such that $F_i\cap F_j = \emptyset$ for any $1\le i < j\le s$.
In their paper, Aharoni and Howard proved this conjecture for $k=2,3$. Then, Lu and Yu [3] proved it for $n>3(s-1)(k-1).$
Our main result is the proof of the conjecture for all $s>s_0.$ The proof relies on the idea that intersection of any family with a random matching is highly concentrated around its expectation. This idea was introduced by the second author in the paper [2] in the context of the Erdős Matching Conjecture.
[1] R. Aharoni and D. Howard, A Rainbow $r$-Partite Version of the Erdős—Ko—Rado Theorem, Comb. Probab. Comput. 26 (2017), N3, 321—337.
[2] P. Frankl, A. Kupavskii, The Erdős os Matching Conjecture and Concentration Inequalities, arXiv:1806.08855.
[3] H. Lu and X. Yu, On rainbow matchings for hypergraphs, SIAM J. Disrete Math. 32 (2018), N1, 382—393.
Mihyun Kang Graz University of Technology
Topological aspects of random graphs
In this talk we will discuss various topological aspects of random graphs. How does the genus of a uniform random graph change as the number of edges increases? How does a topological constraint (such as imposing an upper bound on the genus) influence the structure of a random graph (such as the order of the largest component, the length of the shortest and longest cycles)?
Will Perkins University of Illinois at Chicago
Finite-size scaling for the random cluster model on random graphs
joint work with Tyler Helmuth and Matthew Jenssen
The random cluster model is a probability measure on edge sets of a graph given by exponentially tilting edge percolation by the number of connected components an edge set induces. It generalizes the ferromagnetic Potts model, and like the Potts model it exhibits a phase transition as the temperature changes on many classes of graphs. Here we study the large q behavior of the random cluster model on random regular graphs and give detailed information about the phase transition, including the distribution of the log partition function, correlation decay, and local weak convergence. Our technique involves approximating the model by a mixture of abstract polymer models with convergent cluster expansions.
Nikolaos Fountoulakis University of Birmingham
On the spectral gap and the expansion of random simplicial complexes
joint work with Michał Przykucki
In this talk, we will discuss the expansion properties and the spectrum of the combinatorial Laplace operator of a d-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. The focus of our discussion will be the spectral gap of the Laplace operator and the Cheeger constant.
Furthermore, we will discuss a notion of a random walk on such a complex, which generalises the standard random walk on a graph, and consider its mixing time.
Lutz Warnke Georgia Institute of Technology
Prague Dimension of Random Graphs
joint work with He Guo and Kalen Patton
Various notions of dimension are important in many area of mathematics, and for graphs the Prague dimension was introduced in the late 1970s Nesetril, Pultr and Rodl.
We show that the Prague dimension of the binomial random graph $G(n,p)$ is typically of order $n/\log n$ for constant edge-probabilities $p$; this proves a conjecture of Furedi and Kantor.
One key ingredient of our proof is a randomized greedy edge-coloring algorithm, that allows us to bound the chromatic index of random subhypergraphs with large edge-uniformities.
Matthew Kwan Stanford University
Perfect matchings in random hypergraphs
joint work with Asaf Ferber
For positive integers $d < k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_{1}(2,n)=\lceil n/2\rceil$. However, in general, our understanding of the values of $m_{d}(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In the first part of this talk, we discuss a new "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and "not too small" edge probability $p$ typically has the property that every spanning subgraph with minimum $d$-degree at least $(1+\varepsilon)m_{d}(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_{d}(k,n)$ without actually knowing its value.
The ideas in our work are quite powerful and can be applied to other problems: in the second part of this talk we highlight a recent application of these ideas to random designs, proving that a random Steiner triple system typically admits a decomposition of almost all its triples into perfect matchings (that is to say, it is almost resolvable).
Persi Diaconis Stanford University
A course on probabilistic combinatorics
Video Slides 1 Slides 2
This past April-June (2020) I gave a graduate course on probabilistic combinatorics at Stanford's departments of mathematics and statistics. This covered the usual topics: Let $X$ be a finite set, pick $x$ in $X$ uniformly, 'what does it look like?' With $X$ permutations, graphs, partitions, set partitions, matrices over a finite field. It also covered 'who cares?' That is, applications in statistics, computer science and physics/chemistry. Two features were lectures on 'from algorithm to theorem' and 'graph limit theory and its extensions'. The first emphasizes the place and usefulness of algorithms to actually pick $x$ uniformly (for example, there are many different ways to sample permutations, how does one efficiently sample partitions or set partitions? say when $n=10^6$?) Each such algorithm has an associated set of limit theorems that it 'makes transparent'. The second topic featured the work of Lovasz and Razborov and their coworkers. I will review the topics (stopping to be specific) and the course projects, some of which are turning into publishable papers.
Alexander Semenov MIPT
Probability thresholds estimates for coloring properties of random hypergraphs
joint work with Dmitry Shabanov
Recall that for an integer $j$, a $j$-independent set in a hypergraph $H=(V,E)$ is a subset $W\subset V$ such that for every edge $e\in E: |e\cap W| \leqslant j$. A $j$-proper coloring of $H=(V,E)$ is a partition of the vertex set $V$ of $H$ into disjoint union of $j$-independent sets, so called colors. The $j$-chromatic number $\chi_j(H)$ of $H$ is the minimal number of colors needed for a $j$-proper coloring of $H$.
We will talk about our latest results on colorings of the k-uniform random hypergraph $H(n,k,p)$. We are interested in asymptotic properties of $H(n,k,p)$ to have its $j$-chromatic number equal to some fixed number $r$. By asymptotic properties of $H(n,k,p)$ we consider $n$ as tending to infinity, while $k$ and $r$ are kept constant.
It can be shown that the previously mentioned property of random hypergraph has a sharp threshold. For the classic case of $(k-1)$-chromatic number, upper and lower bounds for that threshold were investigated by different researchers. It should be also mentioned that the gap between these bounds in terms of the parameter $c=p{n\choose k}/n$ has a bounded order $O_k(1)$.
We are going to present the results from our last series of works where we found very tight bounds for the case of arbitrary $r$ and $1 < k-j=o(k^{1/4})$.
Jakub Kozik Jagiellonian University
Bi-uniform Property B
How many edges do we need in order to construct a $k$-uniform hypergraph which is not two-colorable? This number, denoted by $m(k)$, has been intensively studied since its introduction by Erd\H{o}s and Hajnal in 1961. As a result, the lower bounds have been improved a number of times and nowadays we know that $m(k)=\Omega(\sqrt{k/\log(k)}\;2^k)$ (Radhakrishnan and Srinivasan 2000). The story of the upper bounds is much shorter. Bound $m(k)= O(k^2 \;2^k)$, obtained by Erdős in 1964, has not been improved since.
We are going to discuss what insights can be gained from considering analogous problem for non-uniform hypergraphs. We focus on an interesting class of bi-uniform hypergraphs, i.e. hypergraphs in which there are only two allowed sizes of edges. On the other hand, having only two sizes of edges eliminates a lot of technical difficulties.
Peleg Michaeli Tel-Aviv University
Greedy maximal independent sets via local limits
joint work with Michael Krivelevich, Tamás Mészáros and Clara Shikhelman
The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science — and even in chemistry. The algorithm builds a maximal independent set by inspecting the vertices of the graph one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.
In this talk, I will present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a useful notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to give new results for other models such as random trees.
Felix Joos Heidelberg University
Dirac-type results for hypergraph decompositions into cycles
I will discuss recent joint work with Marcus Kühn and Bjarne Schülke on decompositions of hypergraphs into cycles. One of the results answers a question of Glock, Kühn and Osthus and another one is an extension of the well-known result due to Rödl and Rucinski on the minimum degree threshold for Hamilton cycles in k-uniform hypergraphs.
Bhargav Narayanan Rutgers University
The threshold of the square of the Hamilton cycle
joint work with J. Kahn and J. Park
We show that the threshold for the appearance of the square of the Hamilton cycle in $G_{n,p}$ is $p = 1/\sqrt{n}$
József Balogh University of Illinois at Urbana-Champaign
Extensions of Mantel's Theorem
Mantel's theorem is a basic classical theorem of extremal graph theory. There are many different extensions and generalizations investigated and many open questions remained.
I will talk about four recent results, including `stability' and `supersaturation' properties.
The results are partly joined with Clemen, Katona, Lavrov, Lidicky, Linz, Pfender and Tuza.
Allan Sly Princeton University
Local functions for the Ising model on the tree
joint work with Danny Nam and Lingfu Zhang
This talk will look at the question of what processes can or cannot be constructed using local randomness. Work of Gamarnik and Sudan and later Rahman and Virag showed that local algorithms on random $d$-regular graphs can only construct independent sets of size approximately half the maximal size when $d$ is large. Like the optimization problem, a closely related question arising in ergodic theory asks can a particular distribution such as a uniformly random colouring on the tree be constructed as a factor of IID, a type of local functions. I'll survey results in this area and describe new work constructing a factor of IID for the Ising model on the tree in its intermediate regime.
Xavier Pérez-Giménez University of Nebraska-Lincoln
The chromatic number of a random lift of $K_d$
joint work with JD Nir
An $n$-lift of a graph $G$ is a graph from which there is an $n$-to-$1$ covering map onto $G$. Amit, Linial, and Matou\v sek (2002) raised the question of whether the chromatic number of a random $n$-lift of $K_5$ is concentrated on a single value. We consider a more general problem, and show that for fixed $d\ge 3$ the chromatic number of a random lift of $K_d$ is (asymptotically almost surely) either $k$ or $k+1$, where $k$ is the smallest integer satisfying $d < 2k \log k$. Moreover, we show that, for roughly half of the values of $d$, the chromatic number is concentrated on $k$. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random $n$-lifts of $G$, for any fixed $d$-regular graph $G$.
© Probabilistic Combinatorics Online 2020 | CommonCrawl |
LMS Journal of Computation and Mathematics
Explicit application of Waldspu...
Explicit application of Waldspurger's theorem
Soma Purkait (a1)
Mathematics Institute,Zeeman Building,University of Warwick,Coventry, CV4 7AL,United Kingdom email [email protected]
Published online by Cambridge University Press: 01 August 2013
For a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger's theorem relates the critical value of the $\mathrm{L} $ -function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger's recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our 'simplified Waldspurger' by giving several examples.
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URL: /core/journals/lms-journal-of-computation-and-mathematics
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MSC classification
11F37: Forms of half-integer weight; nonholomorphic modular forms
11F70: Representation-theoretic methods; automorphic representations over local and global fields
11F11: Holomorphic modular forms of integral weight | CommonCrawl |
Effects of treadmill with different intensities on bone quality and muscle properties in adult rats
Zhehao Liu1,
Jiazi Gao1 &
He Gong1
BioMedical Engineering OnLine volume 18, Article number: 107 (2019) Cite this article
Bone is a dynamically hierarchical material that can be divided into length scales of several orders of magnitude. Exercise can cause bone deformation, which in turn affects bone mass and structure. This study aimed to study the effects of treadmill running with different intensities on the long bone integrity and muscle biomechanical properties of adult male rats.
Forty-eight 5-month-old male SD rats were randomly divided into 4 groups: i.e., sedentary group (SED), exercise with speed of 12 m/min group (EX12), 16 m/min group (EX16), and 20 m/min group (EX20). The exercise was carried out for 30 min every day, 5 days a week for 4 weeks. The femurs were examined using three-point bending test, microcomputer tomography scanning and nanoindentation test; the soleus muscle was dissected for tensile test; ALP and TRACP concentrations were measured by serum analysis.
The failure load was significantly increased by the EX12 group, whereas the elastic modulus was not significantly changed. The microstructure and mineral densities of the trabecular and cortical bone were significantly improved by the EX12 group. The mechanical properties of the soleus muscle were significantly increased by treadmill exercise. Bone formation showed significant increase by the EX12 group. Statistically higher nanomechanical properties of cortical bone were detected in the EX12 group.
The speed of 12 m/min resulted in significant changes in the microstructure and biomechanical properties of bone; besides, it significantly increased the ultimate load of the soleus muscle. The different intensities of treadmill running in this study provide an experimental basis for the selection of exercise intensity for adult male rats.
Osteoporosis is a type of bone disease characterized by decrease in bone quality and deterioration of bone microstructure, which leads to an increase in bone fragility and susceptibility to fractures [1, 2]. More than 200 million people suffer from osteoporosis worldwide [3]. As a major public health problem all over the world, osteoporosis imposes severe health threats and financial burdens on the patients and their families.
Currently, a wide variety of drugs, such as alendronate sodium, raloxifene, and teriparatide, are used for the treatment of osteoporosis. However, these drugs are associated with serious adverse reactions, e.g., high blood pressure, hectic fever, and thrombus formation [4]. Alternatively, non-drug therapy has become a popular method for treating osteoporosis due to its low price, applicability, and good treatment outcomes. According to the previous studies, exercises such as jumping, jogging, and swimming can improve bone quality, density, and strength [5,6,7]. Physical exercise, in particular treadmill running, has been widely used in animal models associated with osteoporosis, as its mechanism is basically similar to human running on a treadmill [8, 9].
The data obtained from animal experiments do not fully represent the condition of the human body. However, animal experiments can replicate the development of the disease and can be studied for specific situations, and can be combined with a variety of experimental methods to further study the mechanism of disease development. In addition, animal experiments can explore the effectiveness of different therapeutic options and provide a theoretical basis for their clinical application. Due to the similarities of bone morphology and structure between rat and human, rats have become an excellent animal model for studying bone diseases. Rats are also the most commonly used experimental animals for studying the effects of exercise on the musculoskeletal system. Compared to female rats, male rats can reduce estrogen intervention and are more suitable for treadmill running.
Bone strength is related to not only its structure and geometry, but also the external load and the bone material properties [10, 11]. Studies have demonstrated that there is a "feedback system" in bone tissue, which is able to sense strain changes inside the bone [12]. Furthermore, In addition to age, gender, nutritional status and energy intake, the strain environment around the bone is also an important factor affecting bone mass, which decreased in the position where strain is reduced and added in the position where strain is increased, affecting the size, shape, and structure of the bone. Bone has a multi-scale hierarchical structure at macroscopic, microscopic, and nanoscopic levels with various characteristics at different scales [13]. Thus, studies at different scales are required to better understand the relationship between bone strength and structure.
Since the mechanical stimulus generated by treadmill running is mostly generated by impact with the ground or muscle contraction, it is essential to investigate the muscle force. The soleus and the gastrocnemius muscles are both posterior muscles of the lower limbs, and belong to the antigravity muscles that maintain body posture or exercise against gravity. Moreover, the soleus and gastrocnemius muscles are dominated by slow twitch (ST) and fast twitch (FT) fibers, respectively [14]. Contraction of the slow twitch fibers is dominant during endurance training; while the fast twitch fibers also participate in contraction during prolonged exercises. Therefore, treadmill running affects the soleus muscle first, and then the gastrocnemius muscle is affected. There are many reports about the effects of exercise on the soleus muscle under micro-gravity [15,16,17,18], which revealed that treadmill running could alleviate muscle atrophy to a certain extent in astronauts caused by long-term exposure to micro-gravity. Moreover, high-intensity treadmill running led to 59% less muscle loss in astronauts compared to low-intensity exercise [18]. As one of the main muscles providing calf tension and directly participating in the eccentric contraction, the soleus muscle has important significance in terms of its biomechanical properties.
A recent study of a 1-month-old female rat backpack with weight-bearing and moderate intensity treadmill found that when rats ran at moderate intensity with an additional 12% weight-bearing level, bone formation was promoted and cancellous bone microstructure was significantly improved [19]. The choice of age and gender has an important impact on the experimental results. Previous study showed that treadmill running had different adaptability to bone structure, biomechanical properties and molecular signals between 12-week-old male and female rats [20]. Compared to female rats, male rats reduced estrogen intervention and were more suitable for treadmill running. Adult male rat reaches peak bone mass, and it can reduce the effect of age on bone mass in the experiment. Therefore, 5-month-old male rats were used to study the effects of different intensities treadmill running on bone mass and muscle biomechanical properties in adult male rats. The study of bones and muscles can provide a more comprehensive experimental basis for the selection of exercise intensity in adult male rats.
In this study, treadmill running at different intensity levels was carried out, in which multi-level test and analyses, including serum analysis, mechanical tensile test of soleus muscle, three-point bending test, micro-CT scanning, and nanoindentation test, were performed to investigate the effects of different levels of mechanical stimuli on the long bone quality and muscle performance of adult male rats, thereby providing feasible suggestions for selecting a proper exercise intensity for adult males.
Mechanical properties and weight of lower soleus muscle
The soleus muscle is one of the most important lower limb muscles. Table 1 shows the ultimate load and ultimate displacement values obtained from the soleus muscle tensile tests. It can be seen that the ultimate load and ultimate displacement values of the exercise groups were significantly higher than those of the SED group (p < 0.05). Although there were no significant differences between the EX12 group and the other exercise groups, the EX12 group had the maximum ultimate load and the minimum ultimate displacement. The EX12 group had the maximum soleus muscle weight, which was 18.18% and 13.64% greater than those of the SED group and the EX16 group, respectively (p < 0.05) (Fig. 1).
Table 1 The ultimate load and ultimate displacement values of soleus muscle obtained from tensile test, mean ± SD
Soleus muscle weight. The error line represents SD. B Statistically different from EX12 group (p < 0.05)
Activity evaluation of osteoblasts and osteoclasts by the serum analysis
The concentrations of alkaline phosphatase (ALP) and tartrate-resistant acid phosphatase (TRACP) obtained by serum analysis are shown in Fig. 2. As can be seen from Fig. 2a, higher ALP concentrations were found in the exercise groups compared with the SED group, and the EX12 group had statistically higher ALP concentration than the SED group (p < 0.05). No statistical difference was found in TRACP concentration for all groups as shown in Fig. 2b; however, TRACP concentration in the EX12 and EX20 groups were smaller than that in the SED group (p > 0.05), while TRACP concentration in the EX16 group was similar with the SED group.
The concentrations of ALP and TRACP obtained from serum analysis. The error line represents SD. a ALP Alkaline phosphatase. b TRACP Tartrate-resistant acid phosphatase. A Statistically different from SED group (p < 0.05)
The failure load and elastic modulus of the femur evaluated by the three-point bending mechanical test
The failure load and elastic modulus values of the femurs were obtained by the three-point bending mechanical test as shown in Table 2. It was found that the failure loads of the SED and EX20 groups were significantly different from that of the EX12 group (p < 0.05). The EX12 group had the maximum failure load and the EX20 group had the minimum failure load. No statistical difference in the elastic modulus was found for all the groups (p > 0.05). The elastic modulus in the EX16 groups was smaller than those in the EX12 and EX20 groups (p > 0.05). On the one hand, it may be related to the porosity and the ET/HT ratio. The E/H ratio describes the material deformation during the indentation process, which is a crucial indicator for the toughness of materials [25]. On the other hand, it may be related to the chemical composition of the cortical one and the arrangement of the collagen fibers [26, 27].
Table 2 The failure load and elastic modulus values of femurs obtained from three-point bending mechanical test, mean ± SD
Microarchitectural evaluation of the left femur by micro-CT scanning
The 3D microstructure parameters of femoral distal trabecular bone were obtained by CTAn software as shown in Table 3. The EX12 group had the maximum bone volume fraction (BV/TV), trabecular bone mineral density (Tb.BMD), trabecular thickness (Tb.Th), trabecular number (Tb.N), structure model index (SMI), while the trabecular separation (Tb.Sp) was the minimum; BV/TV, Tb.BMD and Tb.Th in the EX12 group were statistically higher than those in the SED, EX16 and EX20 groups (p < 0.05); significantly lower Tb.Sp was found in the EX16 group compared with the SED group (p < 0.05); The EX12 group had the maximum structure model index (SMI) among all the components. Slightly lower SMI was found in the EX16 and EX20 groups than that of the SED group. SMI in the EX16 group was significantly lower than that in the EX12 group (p < 0.05).
Table 3 Microarchitecture parameters of trabecular bone in the distal femur evaluated by micro-CT, mean ± SD
The 3D microstructure parameters of cortical bone in the femoral diaphysis were obtained by CTAn software as shown in Table 4. Statistically lower cortical bone mineral density (Ct.BMD) was found in the SED and EX16 groups compared with the EX12 group (p < 0.05). Cortical bone porosity (Ct.P) in the EX12 and SED groups was significantly lower than that in the EX16 group (p < 0.05). Significantly higher cortical bone thickness (Ct.Th) was detected in the EX12 group compared with the SED and EX20 groups (p < 0.05).
Table 4 Microarchitecture parameters of cortical bone in the femur diaphysis evaluated by micro-CT scanning, mean ± SD
Elastic modulus and hardness measured by nanoindentation test
Modulus of indentation (E), indentation hardness (H) and the ratio of indentation modulus to indentation hardness (E/H) of cortical bone in the femoral diaphysis were obtained by nanoindentation tests as shown in Table 5. It can be found that statistically greater longitudinal indentation hardness (HL), transverse modulus of indentation (ET), ratio of transverse indentation modulus to indentation hardness (ET/HT) were detected in the EX12 group compared with the SED group; ratio of longitudinal indentation modulus to indentation hardness (EL/HL) was statistically smaller in the SED group (p < 0.05). No statistical differences were found in longitudinal modulus of indentation (EL), HL and EL/HL ratio between the EX16 and SED groups (p > 0.05); ET and transverse indentation hardness (HT) were significantly greater compared with the SED group (p < 0.05). EL, HL, HT and ET/HT ratio in the EX20 group were significantly smaller than those in the SED group (p < 0.05).
Table 5 E, H and E/H ratio of cortical bone obtained by nanoindentation tests, mean ± SD
In this study, forty-eight 5-month-old male SD rats were divided into three intensity exercise groups and one control group. Macroscopic, microscopic, and nanoscopic perspectives were used to investigate the effects of different treadmill exercise intensities on rat femur. Three-point bending test, micro-CT scanning, and nanoindentation test were performed to study bone quality from multiple perspectives. Moreover, since the mechanical stimulus generated during treadmill exercise is derived not only from gravity, but also from muscle contraction, the mechanical properties of the soleus muscle were also investigated. It was found that exercise at a speed of 12 m/min could significantly increase the long bone quality and increase the ultimate load carried by the soleus muscle of rats.
In recent years, female rats were used in most of the research on the effect of treadmill running on bone mass and strength, which were in the growing period or elderly; while there were relatively few studies on the effect of treadmill running on bone mass in male rats, 4- to 10-week-old growing period [28, 29] or 15- to 23-month-old [30, 31] rats were predominant. From the growth phase to the adult phase and the elderly phase, bone mass changes significantly, so the study for each phase is essential. Previous studies have only observed the effects of treadmill running on bone [6] or muscle [32], and rarely combine together. Adult male rats were used in this study, and the best form of exercise to improve bone quality in adulthood was explored by combining bone and muscle investigations. In addition, previous studies on skeletal muscles in the lower limb were usually focused on the gastrocnemius muscle, and relatively less attention was paid to the soleus muscle. These research protocols focused on the muscle fiber mass [33] and functional characteristics [18]. Studying the mechanical properties of soleus muscle is important for the treatment of muscle atrophy.
Mechanical stimuli play an important role in bone strength [34]. Exercise with different intensity, frequency, and duration produces different mechanical stimuli, which has different effects [8]. In this study, the EX12 group had the maximum failure load, which was significantly greater than those of the SED and the EX20 groups (p < 0.05); while the failure load in the EX20 group was smaller than that of the SED group. Bone is a living organ with functional adaptability. It can adjust itself according to the surrounding mechanical environment, developing the optimal structure able to bear the applied loads. Bone can achieve the maximum structural strength with the least materials [35]. Appropriate mechanical stimuli can increase bone quality and improve bone structure; while excessive mechanical stimuli may cause bone tissue deterioration in quality and microstructure, and decrease the mechanical properties [36]. Previous studies on 5-week-old male rats also revealed that, after high-intensity treadmill training, both tibial strength and proximal bone mineral density (BMD) reduced, while the trabeculae in the epiphysis thinned [37]. The treadmill running with 20 m/min speed in this study caused excessive mechanical stimulus to rat femur, which resulted in the decrease in failure load.
The mechanical properties of bone are closely related to its microstructure. Bone strength can be effectively predicted by combining three-point bending test and micro-CT scanning [38, 39]. Compared to the microstructure parameters of the trabecular bone in other groups, the EX12 group demonstrated higher BV/TV, Tb.BMD, and Tb.Th (p < 0.05). The EX12 group significantly improved the trabecular structure, increased bone density, and tightened trabecular arrangement as compared to the SED group. These results are consistent with the results from the previous studies, which indicated that exercise can improve the trabecular bone structure in rats [40]. For cortical bone, the EX12 group had higher Ct.BMD and Ct.Th compared to the SED group (p < 0.05); while Ct.P was the minimum among all groups. This indicated that applying a mechanical stimulus with a velocity of 12 m/min can increase bone density and cortical bone thickness.
It has been reported in the literature that exercise can increase the levels of growth hormone (GH), prostaglandin E2 (PGE2), parathyroid hormone (PTH), and thyroid hormone (TH) [41]. Among them, PGE2 can stimulate mesenchymal stem cells to differentiate into osteoblasts [42]. ALP is an indicator of bone formation, and is closely related to the activity of osteoblasts. Analysis of ALP in serum can effectively reveal the effect of treadmill running on the active of osteoblasts [43]. Recent studies showed that physical exercise has impacts on the amount and osteogenic differentiation potential of mesenchymal stem cells. After running on the treadmill for 5 weeks, the mesenchymal stem cells of 4-week-old male C57BI/6 mice significantly increased, and the ALP activity also increased [44]. In the present study, no significant difference was found in the serum TRACP concentration among groups. The ALP concentrations in the exercise groups were higher than that in the SED group; while the ALP concentration of the EX12 group was significantly higher than that of the SED group (p < 0.05). The results suggested that treadmill exercise with a 12 m/min speed contributed to increasing bone formation, but did not remarkably suppress bone resorption, which was consistent with the increase in bone density in the EX12 group, as observed by micro-CT evaluation. Previous studies have shown different results on whether treadmill running can reduce TRACP concentration. A study on the treadmill running in rats also found that rats increased bone formation after 8 weeks of running without affecting bone resorption [19]. This is consistent with our current conclusion. Other researchers found that treadmill running not only increases bone formation but also inhibits bone resorption [45]. This may be related to different animal species, age and experimental design. In this study, however, the EX16 and SED groups demonstrated similar TRACP concentrations, which should be further investigated.
Nanoindentation is an effective method for measuring nanomechanical properties, and has been widely used in studying the mechanical properties of bone tissue [46]. In the present study, the longitudinal hardness (HL) and the transverse elastic modulus (ET) of the EX12 group were both significantly improved compared to the SED group (p < 0.05), indicating that the nanomechanical properties in the EX12 group were superior to those in other groups. The E/H ratio describes the material deformation during the indentation process, which is proportional to the fracture toughness; thus, it is a crucial indicator for the toughness of materials [25]. The ET/HT ratio of the EX12 group was significantly greater than that of the SED group (p < 0.05), suggesting that the EX12 group had a high transverse fracture resistance. In addition, we also observed that the variation tendencies in elastic modulus and hardness were consistent.
In human activities, most of the loads acting on bones are produced by externally exerted forces and skeletal muscle contraction [47]. In this study, the ultimate load and the ultimate displacement values of the soleus muscle in the exercise groups were significantly higher than those in the SED group (p < 0.05). The EX12 group demonstrated the maximum ultimate load and the soleus muscle weight. Mechanical environment is a key factor in maintaining musculoskeletal functions [48]. Exercise increases the muscle myosin and actin contents [49]; and during exercise, muscle contraction increases the number of cells in the muscle tendon, thus improving the load-bearing capacity of the muscle [50]. The increase in the soleus muscle weight in the exercise groups may be due to the apparent "demand" for soleus muscle in rats from the treadmill running. With long-term running, muscle contraction increases the number and percentage of muscle fibers, as well as muscle mass. This is in accordance with the results of a previous study, in which the effects of treadmill training on the soleus muscles were explored in rats after complete spinal cord transection at T8–T9. It was found that 9 weeks of treadmill running increased soleus muscle mass and cross-sectional area of muscle fibers in rats [51].
There were some limitations in the present study. Firstly, findings obtained from 5-month-old male SD rats may not be applicable to female rats. A recent study showed that loading of a 1-month-old female rat with a weight-bearing and moderate-intensity treadmill can promote bone formation and improve trabecular microstructure [19]. Future work will include the effects of treadmill exercise with different intensities on bone quality of female SD rats with the same age to obtain more comprehensive results and conclusions. Secondly, three exercise groups with different intensities (EX12, EX16 and EX20 groups) were used, and the total distance per day for each group was not equal, which may possibly affect the changes in bone quality. However, the exercise time for different groups was kept the same. Thus, it is meaningful to study the effects of different exercise intensities on bone quality. Future work can specify the total distance traveled and explore the effects of treadmill running on the bone of the small mammalian. Finally, the effects of different exercise intensities on nearly one single intact bone (femur) were investigated, which may not be applicable to other bones. However, considering that femur is load-bearing bone prone to fracture, this study is still meaningful.
The results of this study demonstrated that after 4 weeks of treadmill running, among the three exercise intensities, the speed of 12 m/min could significantly affect the microstructure and mechanical properties of bone and improve the ultimate load of the soleus muscle. The treadmill running established in this study is effective and provides a reference for the selection of proper exercise intensity levels for adult males.
This study was approved by the Medical Ethics Committee of the First Hospital of Jilin University (No. 2018-238). This study was in strict accordance with the requirement of the Laboratory Animal Standardization Committee. And all efforts were made to alleviate suffering of animals.
Sixty 5-month-old male Sprague–Dawley rats were procured from the Animal Experimental Center of Jilin University. Animals were housed in groups of 4 (57 × 39 × 20 cm3), in laboratory cages under controlled laboratory conditions, a temperature of 23 ± 2 °C, and a relative humidity of 55 ± 5%, under a 12-h dark/light cycle. Rats were given freedom of movement in their cages, and provided ad libitum access to standard rodent food pellets (autoclaved diet National Institutes of Health-31 with 6% fat; 18% protein; Ca:P = 1:1; and supplemented vitamins and fortified minerals) and tap water. All the rats were healthy during the experimental period.
Experimental design
The temporal schematic is shown in Fig. 3a. The exercise groups underwent 1 week of adaption treadmill running. During the adaptive period, the rats were subjected to run on a flat treadmill (Fig. 3b) by gradually increasing the running speed from 10 m/min to 16 m/min, and 30 min/day, five times a week. In this stage, the rats were acclimated to treadmill running and those animals refused to run were eliminated. 48 rats with similar motor ability were selected and randomly assigned to sedentary control group (SED, n = 12) and exercise group (EX, n = 36). The rats in the exercise group were then randomly divided into 3 groups based on the average exercise speed: (1) Exercise group with a speed of 12 m/min (EX12); (2) Exercise group with a speed of 16 m/min (EX16); (3) Exercise group with a speed of 20 m/min (EX20). A 5-min "warming-up" at a speed of 10 m/min prior to each formal running was performed. Throughout the experimental period, the rats in the exercise groups ran on the treadmill for 30 min/day, 5 days a week, for 4 weeks. The rats in the sedentary control group were allowed to move freely in the cages, and were evaluated every day to ensure their health.
Temporal schematic of experiment and equipment for treadmill exercise. a The temporal schematic. b The equipment for treadmill exercise, which was composed of a treadmill platform and a controller
Specimens preparation
48 h after the final exercise session, the rats were anesthetized with isoflurane, and then the blood sample with no less than 5 ml was obtained through the abdominal aorta of each rat. Serum was separated by centrifugation at 3000 rpm/min for 15 min. The obtained serum samples were then immediately stored in a refrigerator at − 20 °C until further analysis.
The animals were then euthanized by cutting the abdominal aorta. The skin, muscles and tendons were peeled out carefully from both sides of the femur. The integrity of the left tibiofibula and muscles was maintained. All the dissected samples were wrapped in parafilm with saline solution and stored at − 20 °C until further analysis. Figure 4 represented the multi-scale test of the femur. For all the tests, the surveyors had no prior knowledge of the grouping of specimens.
The multi-scale test of the femurs
Soleus tensile test
The left tibia was thawed at room temperature for 6 h. The skin, surface muscle and gastrocnemius muscle at the back of the leg were cut away to expose the soleus muscle. During the sample preparation, the tibiofibula was dissected from the middle of the leg with the retained connection between the tendon and bone. This procedure was performed to ensure that the clamp can hold the sample appropriately, avoid direct contact with the muscle, and prevent stress concentration at the clamped position. The muscle specimen was prepared within 5 min, during which saline solution was titrated every 30 s to keep the sample moist. Immediately after the sample preparation, a tensile test was conducted using an electronic universal testing machine (AG-X plus, Shimadzu, Kyoto, Japan) (Fig. 5). The sample was loaded at a speed of 2 mm/min until the muscle fracture, and the load–displacement curve was recorded during the test. During the tensile phase, the soleus muscle was kept under moisture with dripping saline every 30 s. There were a total of 48 samples in this experiment, of which 46 samples were fractured at the position of muscle abdomen during the tensile test. One sample was injured by an operation error during peeling, and another sample was broken at the tendon position. These two samples were not counted in the final results. Immediately after muscle fracture, the soleus muscle was removed from the clamp and weighed on an electronic analytical balance. The intermediate processes were completed within 2 min.
Serum analysis
Alkaline phosphatase (ALP) and tartrate-resistant acid phosphatase (TRACP) were quantitatively analyzed by enzyme-linked immunosorbent assay (enzyme-linked immunosorbent assay assay kit, NanJing, China) using ELIASA (CLARIOstar, BMG LABTECH, Germany), and bone formation and absorption rates were measured.
Three-point bending mechanical test
Next, the right femur was thawed at room temperature and washed with saline solution to keep moist. As shown in Fig. 6, a three-point bending mechanical test was performed at room temperature of 22 °C using an electronic universal testing machine (AG-X plus, Shimadzu, Kyoto, Japan). The span of the fulcrum was adjusted to 20 mm, and the speed of the actuator was set at 1 mm/min. All samples were loaded in the same position with the middle of the femoral shaft as the loading point. The load–displacement curve was obtained with the bundled software (TRAPEZIUMX, Shimadzu, Kyoto, Japan), and the failure load was recorded. In this study, an elliptic ring was synthesized from a cross section of bone. The fractured samples were measured by a vernier caliper with the precision of 0.02 mm, and two surveyors who did not know the group information of the specimens were selected for testing. Each sample was measured for three times by each surveyor. The moment of inertia (I) and elastic modulus (Em) of the cross section were calculated using the following equations [21]:
$$I = \frac{{\pi \left( {BH^{3} - bh^{3} } \right)}}{64},$$
$$E_{m} = \frac{{L^{3} }}{48I}\left( {\frac{\Delta F}{\Delta f}} \right),$$
where B and H are the long and short axes of the outer ellipse of the cross section; b and h are the long and short axes of the inner ellipse of the cross section; L is the span of the two support point, and \(\left( {\frac{\Delta F}{\Delta f}} \right)\) is the slope of the load–displacement curve.
A right femur under the three-point bending mechanical test
Micro-CT scanning
Left femurs were initially fixed with 80% ethanol (EtOH). Subsequently, femur samples were scanned using a benchtop micro-CT system (Skyscan 1076, Bruker-MicroCT, Belgium) at 18-μm voxel image resolution with 70 kV, 100 μA, and a 1.0-mm aluminum filter, to obtain three-dimensional (3D) microstructure parameters of trabecular bone and cortical bone. The projection data were then reconstructed with NRecon (Skyscan, Belgium) to create 3D images. The region of interest (ROI) was manually selected for the analysis of micro-CT images. The entire trabecular bone of the femoral lateral condyle was selected as trabecular ROI, and 5-mm cortical bone from femoral shaft to the proximal was selected as cortical ROI. Image analysis was performed with CTAn analysis software (CTAn, Skyscan, Belgium). The 3D morphometrical parameters of femoral head trabecular bone were measured, including bone mineral density (BMD), bone volume fraction (BV/TV), trabecular thickness (Tb.Th), trabecular number (Tb.N) and trabecular separation (Tb.Sp). The 3D microstructure parameters of femoral shaft cortical bone were also measured, including cortical bone mineral density (Ct.BMD), cortical bone thickness (Ct.Th) and cortical bone porosity (Ct.P).
Nanoindentation test
Following the three-point bending test, all right femur samples were cleaned with deionized water. Two longitudinal cortical bone samples with equal size were cut along the axis of the femoral shaft. One of the samples was dissected through the medullary cavity along the axis of the femoral shaft to obtain the transverse cortical bone sample, and the other was used as the longitudinal cortical bone sample (Fig. 7). The specimens were dehydrated by a serial gradation ethyl alcohol (70%, 80%, 90%, and 100%) for 24–48 h per stage. After dehydration, all specimens were embedded in epoxy resin at room temperature [22]. All the embedded samples were metallographically polished using silicon carbide papers of decreasing grit sizes (600, 800, 1500, and 2000 grit), and subsequently on the microcloths with 0.05-µm grit of diamond suspension to obtain the smooth surfaces required for nanoindentation test. Finally, the samples were washed with deionized water to remove debris.
Preliminary preparation of the transverse and longitudinal samples of cortical bone. The sample on the left is a longitudinal cortical bone sample, and that on the right is a transverse cortical bone sample
A Berkovich diamond indenter was used for the measurements. Nanoindentation tests were performed with the indenter speed of 750 μN/s and the indentation depth of 1000 nm, holding this load for 10 s, and finally unloading to 15% of the peak load at a rate equal to half that used during loading [23]. E and H were calculated using the method described by Oliver and Pharr [24].
H is calculated as the peak load (Pmax) divided by the projected contact area of the Berkovich tip (A):
$$H = \frac{{P_{ \text{max} } }}{A}.$$
Eef is the effective indentation modulus, and S is the linear slope of the unloading curve, and their relationship is:
$$S = \frac{2}{\sqrt \pi }\beta E_{\text{ef}} \sqrt A .$$
The Eb (E) is calculated as:
$$\frac{1}{{E_{\text{ef}} }} = \frac{{1 - v_{b}^{2} }}{{E_{b} }} + \frac{{1 - v_{i}^{2} }}{{E_{i} }},$$
where ν is the Poisson's ratio, and E is the elastic modulus. The subscripts b and i refer to bone sample and the indenter, respectively. For the Berkovich indenter, vi = 0.07, Ei = 1140 GPa, β = 1.034. For the bone indenter, νb = 0.3.
All statistical analyses were performed using SPSS 19.0 software. The mean value of each parameter of each group was calculated. All data obtained were analyzed using one-way analysis of variance (ANOVA) for differences among all groups. If significant difference was observed, the least significant difference test was used for post hoc comparison to compare the difference between every two groups. The significance level of p was chosen to be 0.05.
ALP:
TRACP:
tartrate-resistant acid phosphatase
BMD:
BV/TV:
bone volume fraction
Tb.Th:
trabecular thickness
Tb.N:
trabecular number
Tb.Sp:
trabecular separation
SMI:
structure model index
Ct.BMD:
cortical bone density
Ct.Th:
cortical bone thickness
Ct.P:
cortical bone porosity
Ct.L:
longitudinal cortical bone
Ct.T:
transverse cortical bone
modulus of indentation
H :
indentation hardness
E/H :
ratio of indentation modulus to indentation hardness
EL :
longitudinal modulus of indentation
transverse modulus of indentation
HL :
longitudinal indentation hardness
HT :
transverse indentation hardness
EL/HL :
ratio of longitudinal indentation modulus to indentation hardness
ET/HT :
ratio of transverse indentation modulus to indentation hardness
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The work was supported by the National Natural Science Foundation of China (Nos. 11702110, 11872095 and 11432016) and the Natural Science Foundation of Jilin Province (Nos. 20170519008JH and 20170520093JH).
Department of Engineering Mechanics, Jilin University, Changchun, 130022, People's Republic of China
Zhehao Liu, Jiazi Gao & He Gong
Zhehao Liu
Jiazi Gao
He Gong
ZHL, JZG and HG conceived and designed the experiment, analyzed and interpreted the data; ZHL performed treadmill running and the statistical analysis; ZHL, JZG and HG performed the micro-CT scanning, three point bending mechanical test, serum analysis, soleus tensile test and nanoindentation test. ZHL, JZG and HG wrote the paper. All authors participated in the trial design, provided feedback on drafts of the manuscript. All authors read and approved the final manuscript.
Correspondence to He Gong.
This study was approved by the Medical Ethics Committee of the First Hospital of Jilin University (No. 2018-238). This study was in strict accordance with the requirement of the Laboratory Animal Standardization Committee.
Liu, Z., Gao, J. & Gong, H. Effects of treadmill with different intensities on bone quality and muscle properties in adult rats. BioMed Eng OnLine 18, 107 (2019). https://doi.org/10.1186/s12938-019-0728-0
Treadmill exercise
Exercise intensity
Femurs
Biomechanical properties
Multiscale analysis | CommonCrawl |
Inter-domain dynamics in the chaperone SurA and multi-site binding to its outer membrane protein clients
Antonio N. Calabrese ORCID: orcid.org/0000-0003-2437-77611 na1,
Bob Schiffrin1 na1,
Matthew Watson1 na1,
Theodoros K. Karamanos1 nAff5,
Martin Walko ORCID: orcid.org/0000-0002-7160-61361,2,
Julia R. Humes1,
Jim E. Horne1,
Paul White1,
Andrew J. Wilson ORCID: orcid.org/0000-0001-9852-63661,2,
Antreas C. Kalli ORCID: orcid.org/0000-0001-7156-94033,
Roman Tuma ORCID: orcid.org/0000-0003-0047-00131,4,
Alison E. Ashcroft ORCID: orcid.org/0000-0002-1676-123X1,
David J. Brockwell ORCID: orcid.org/0000-0002-0802-59371 &
Sheena E. Radford ORCID: orcid.org/0000-0002-3079-80391
Nature Communications volume 11, Article number: 2155 (2020) Cite this article
Bacterial structural biology
The periplasmic chaperone SurA plays a key role in outer membrane protein (OMP) biogenesis. E. coli SurA comprises a core domain and two peptidylprolyl isomerase domains (P1 and P2), but its mechanisms of client binding and chaperone function have remained unclear. Here, we use chemical cross-linking, hydrogen-deuterium exchange mass spectrometry, single-molecule FRET and molecular dynamics simulations to map the client binding site(s) on SurA and interrogate the role of conformational dynamics in OMP recognition. We demonstrate that SurA samples an array of conformations in solution in which P2 primarily lies closer to the core/P1 domains than suggested in the SurA crystal structure. OMP binding sites are located primarily in the core domain, and OMP binding results in conformational changes between the core/P1 domains. Together, the results suggest that unfolded OMP substrates bind in a cradle formed between the SurA domains, with structural flexibility between domains assisting OMP recognition, binding and release.
Chaperones play vital roles in multicomponent proteostasis networks, ensuring that proteins fold and avoid aggregation in the crowded cellular milieu, and that misfolded proteins which cannot be rescued by chaperones are targeted for degradation1,2. It is now established that many chaperones are in rapid dynamic exchange between co-populated conformations, and that this conformational plasticity is key to their functional mechanisms3. In the case of ATP-dependent chaperones, e.g. the Hsp60 chaperonins GroEL and TRiC, and the Hsp90 and Hsp70 families, ATP binding and/or hydrolysis promotes conformational changes that facilitate the folding and/or release of their clients1,2,4,5,6,7,8. However, some chaperones are not dependent on energy from nucleotide binding/hydrolysis, and instead their intrinsic structural flexibility is proposed to be key to their function3,9,10,11,12. The functional mechanisms of these ATP-independent chaperones, including how they bind and release their substrates in a controlled manner, are generally not well understood.
SurA is an ATP-independent chaperone involved in the biogenesis of outer membrane proteins (OMPs) in the periplasm of Gram-negative bacteria13,14,15,16,17,18. This protein is thought to be the major chaperone responsible for protecting OMPs from aggregation in the periplasm13,14,15,16,17,18 and facilitating OMP delivery to the β-barrel assembly machinery (BAM) for folding and insertion into the outer membrane (OM)13,14,19,20,21. Deletion of SurA leads to OMP assembly defects, the induction of stress responses, and increased sensitivity to antibiotics and detergents15,17,22,23,24. Further, ΔsurA strains show reduced assembly of virulence factors, such as pili and adhesins, and exhibit reduced pathogenicity in a number of species22,25,26. E. coli SurA has a three domain architecture, consisting of a core domain which is composed of its N- and C-terminal regions, and two parvulin-like peptidylprolyl isomerase (PPIase) domains (P1 and P2) (Fig. 1a)27. However, despite the availability of its crystal structure27, how SurA binds its unfolded OMP clients and the molecular mechanism(s) of SurA function remain unknown. A substrate binding crevice was proposed based on examination of molecular packing interactions in crystals of SurA (Fig. 1b), but the location of OMP binding regions and the roles of the PPIase domains (which are not essential for in vivo or in vitro function, at least for some clients24,28,29) in folding and binding its varied OMP clients remained unknown.
Fig. 1: SurA structure and domain architecture.
a Domain architecture of E. coli SurA. Regions are coloured grey (N-terminal region of the core domain), green (P1), yellow (P2) and orange (C-terminal region of the core domain). The signal sequence is not shown, and was not present in any of the constructs used in this study, but the numbering used throughout reflects the gene numbering (including the signal peptide). b Crystal structure of E. coli SurA WT (PDB 1M5Y27), with missing residues added using MODELLER106. A client binding crevice in the core domain was proposed based on crystal contacts with a neighbouring SurA molecule, as indicated27. P1 contacts the core domain in this crystal structure (Supplementary Fig. 1a-c). Regions are coloured as in (a).
In the crystal structure of full-length SurA27, an extended conformation is observed in which the core and P1 domains are in contact (Fig. 1b, Supplementary Fig. 1a-c), while P2 is separated from this globular region via a linker (Fig. 1b, Supplementary Fig. 1d). Examination of the molecular packing interactions in the crystal lattice revealed multiple contacts between all three domains and neighbouring molecules, which may stabilise the elongated architecture observed (Supplementary Fig. 2). SurA homologues and domain deletion variants have been crystallised in conformations with a variety of domain orientations (Supplementary Fig. 3) suggesting that SurA may have a dynamic structure. Further, tethering of the P1 and core domains via a disulfide bond resulted in impaired OMP assembly in vivo29. However, the precise nature of these conformational dynamics and how they are linked to OMP binding have remained elusive.
Here, we sought to determine the conformational properties of full-length E. coli SurA in solution in an effort to better understand its conformational dynamics and how inter-domain motions may be exploited or modified by client binding. Combining mass spectrometric (MS) methods (chemical cross-linking (XL) and hydrogen-deuterium exchange (HDX)), with single-molecule FRET (smFRET) and molecular dynamics (MD) simulations we show that SurA adopts conformations in solution that differ substantially from its crystal structure27. Specifically, the P1 domain samples open and closed states relative to the core domain, and P2 is primarily located closer to the core/P1 domains than observed in the crystal structure. We also show that multiple sites on the SurA surface, predominantly located in the core domain, are involved in client binding. OMPs bind to these specific sites in different orientations, consistent with a dynamically bound state, and the conformations adopted by the chaperone alter in response to OMP binding. Combined, our results portray a model in which the three domains of SurA form a cradle around its OMP clients, protecting them from misfolding and aggregation on their journey through the periplasm, with the conformational dynamics of the domains presumably facilitating their delivery to BAM for folding into the outer membrane.
Inter-domain conformational flexibility in SurA
We first investigated the structure and dynamics of apo-SurA in solution using XL-MS, which provides distance information in the form of spatial restraints, and enables comparison of the solution conformation(s) of the protein with structural data30. For this purpose, we used the bifunctional reagent disuccinimidyl dibutyric urea (DSBU), which primarily cross-links Lys residues31, and a SurA concentration at which the protein is monomeric in solution (5 μM)32. The concentration of SurA in the periplasm has been estimated to be ~7 μM suggesting it is a functional monomer33. DSBU has been shown to cross-link residues within a straight line distance (SLD) between their Cα atoms of ca. 27–30 Å34. More recently it has been shown that considering the solvent accessible surface distance (SASD) between residues may more reliably predict structural models35 (a Cα–Cα distance between cross-linked residues of up to ca. 35 Å is considered feasible for DSBU). For monomeric SurA, a total of 13 intra-domain (core-core, P1-P1 and P2-P2) (Supplementary Fig. 4, Supplementary Table 1) and 19 inter-domain (core-P1, core-P2 and P1-P2) cross-links were detected (Fig. 2a, Supplementary Table 1). Most of the intra-domain cross-links identified (8 of 13, based on the SASD) are consistent with the domain structures observed in the crystal structure of full-length E. coli SurA (Supplementary Fig. 4, Supplementary Table 1), but there is some suggestion of local fluctuations within the domain structures, with five unsatisfied cross-links (K90-K134, K134-K394 and K134-K405 in the core domain, and two, K278-K293 and K362-K388, in the P2 domain). Interestingly, two of the five unsatisfied intra-domain cross-links (K134-K394 in the core domain, and K362-K388 in the P2 domain) involve a residue that was not present/disordered in the crystal structure (and was built in using MODELLER), while the remaining three cross-links (K90-K134 and K134-K405 in the core domain, and K278-K293 in the P2 domain) involve Lys residues in regions of defined secondary structure (Supplementary Fig. 1d). By contrast, only four (K105-K278, K278-K394, K251-K278, K252-278) of the 19 inter-domain cross-links (core-P1, core-P2 or P1-P2) are compatible with SASD values derived from the SurA crystal structure (Fig. 2b–d, Supplementary Table 1). Interestingly, the three core-P1 cross-links detected (K251-K405, K252-K394 and K269-K394) involve residues that have SLDs of ~26 Å36. However, the vectors of the SLDs for these cross-links pass directly through the P1 and core domains, highlighting the importance of considering SASDs in judging cross-link violation35,37. Taken together, the data show that SurA populates structures in solution in which P2 is closer to both the core and P1 domains than portrayed by its crystal structure, as well as conformations in which the orientation and/or distance of P1 relative to the core is distinct from that observed in the crystal structure of the protein27.
Fig. 2: XL-MS suggests the P2 domain is closer to the core/P1 than implied by the crystal structure.
a Locations of the 19 identified SurA inter-domain cross-links (red and blue lines). b–d Crystal structure of SurA showing the identified inter-domain cross-links between b core-P1 (3 in total), c core-P2 (9 in total), and d P1-P2 (7 in total). Only four of the inter-domain cross-links identified (coloured in blue) are consistent with the crystal structure of full-length SurA (PDB 1M5Y27), based on a maximum SASD of 35 Å35. Other cross-links are inconsistent with this distance cut-off (red lines). Details of cross-linked residues are given in Supplementary Table 1. A representative mass spectrum for each cross-link can be found in Supplementary Data 1. Reactions contained 5 µM SurA, 50 µM DSBU, in 10 mM potassium phosphate buffer, pH 8.0, for 45 min, 25 °C. Note that for clarity cross-links are shown as straight lines between residues (rather than as SASDs which provide a more reliable measurement for comparison with protein structures (see text)). The N-terminal region of the core domain, P1, P2 and the C-terminal region of the core domain are shown in grey, green, yellow and orange, respectively.
As an independent validation of the cross-linking results, we investigated the inter-domain distances in monomeric SurA by smFRET38. We selected non-conserved residues in the core, P1 and P2 domains (Q85, N193 and E301, respectively) to substitute with Cys, and constructed three variants containing Cys substitutions at two positions (core-P1, core-P2 and P1-P2) (Fig. 3a–c, Supplementary Table 2). Each SurA variant was stochastically labelled with Alexa 488 and Alexa 594 dyes (R0 = 60 Å), enabling inter-domain distance distributions to be monitored in a pairwise fashion. Samples containing ~50 pM of labelled monomeric SurA were interrogated using confocal fluorescence detection and alternating laser excitation (ALEX) (see Methods). Fluorescence intensities in the donor and acceptor channels yielded FRET efficiencies (EFRET) for the passage of each single molecule through the confocal volume (fluorescence burst). These were collated into FRET efficiency histograms, fitted to two Gaussian components and compared with distributions predicted for each labelled SurA double Cys variant calculated from the SurA crystal structure (see Methods) (Fig. 3d–f)39,40.
Fig. 3: Comparison of smFRET data to values predicted from crystal structure of SurA.
The crystal structure of SurA indicating the available volumes calculated for each dye label at residues a 85 and 193 (core-P1 distance probes); b 85 and 301 (core-P2 distance probes) and c 193 and 301 (P1-P2 distance probes). d–f Predicted EFRET between each pair of dyes for the crystal structure of SurA (black line) and the observed EFRET distributions (green, red or blue lines) independently fitted to two Gaussians (pink/yellow). g–i BVA reveals high variance for all three dye pairs. The average values of the measured variance for the EFRET values (white filled circles) lie above the expected shot-noise limited standard deviation (black arc) indicating dynamics on a timescale faster than the duration of the bursts (here sub-ms). Predicted EFRET distributions d–f were calculated from the SurA crystal structure (PDB 1M5Y27) using distance distributions generated by the MtsslWizard plugin for PyMOL103, which takes into account both the location of the dyes and the flexibility of the dye linkers39. Note that all dyes can be considered as freely rotating, as manifested by their low anisotropy (Supplementary Table 2), hence changes in EFRET can be translated to distance variations. Samples contained ~50 pM labelled SurA variant in 50 mM Tris-HCl, pH 8.0, 25 °C.
The predicted EFRET distribution for the core-P1 SurA variant calculated from the crystal structure of E. coli SurA27 has a single maximum at ~0.6 (Fig. 3d). While the experimentally observed distribution recapitulates this predicted peak maximum, it also shows a second, smaller, population centred on EFRET ~ 0.2 (Fig. 3d). This suggests that SurA populates at least two distinct conformational ensembles in solution, one in which P1 is located close to the core domain with an inter-domain distance similar to that in the crystal structure (core-P1closed ensemble), and one in which the P1 and core domains are further apart (core-P1open ensemble). The observed EFRET distributions for the labelled core-P2 and P1-P2 variants were also obtained and fitted to two Gaussians with maxima of the most intense peak at ~0.3 and ~0.2, respectively, and smaller peaks at ~0.6 and ~0.5, respectively (Fig. 3e, f), both in marked contrast with the very low predicted values for the crystal structure (~0.1 and ~0.02, a spatial separation of ~85 and ~115 Å, for core-P2 and P1-P2, respectively). This indicates that, in the vast majority of molecules, P2 is located closer to the core and P1 domains than suggested by the SurA crystal structure27, consistent with the XL-MS data (Fig. 2b–d), and that there are at least two discernible conformational ensembles. Note that the concentrations of SurA used here are lower than those found in vivo (50 pM vs 7 μM in the periplasm33). At higher concentrations, and indeed in the crowded, dynamic periplasm, the conformational ensembles of SurA may be influenced by a number of factors, including excluded volume effects, higher viscosity, perturbed diffusion, and protein–protein interactions. For example, excluded volume effects may favour more compact conformations while higher viscosity generally slows down segmental motions.
Burst variance analysis (BVA)41 was used to determine if dynamic interconversion between states was occurring on a sub-millisecond timescale. BVA compares the FRET efficiency variance derived from the experimental data within each burst with the theoretical shot-noise limited variance expected for static FRET values. Dynamic interconversion between FRET states on a timescale faster than the duration of the burst (i.e. the average time it takes a molecule to diffuse through the confocal volume, in this case less than a millisecond) will add to the experimental variance which in turn will be larger than the shot-noise limited value. For each FRET pair (Fig. 3a–c), the experimental FRET variance was larger than that predicted for a static molecule, demonstrating that inter-domain motions involving each pair of domains occurs on the sub-millisecond timescale (Fig. 3g–i). Together, these data indicate a dynamic chaperone in which sub-ms motions involving all three domains are occurring, in particular at the core-P1 interface which interconverts between core-P1closed and core-P1open states. In addition, the data show that P2 is also mobile, consistent with dynamic interconversion between conformational states on the sub-ms timescale as suggested by BVA analysis, but spends most of its time closer to the core and P1 domains than suggested by the SurA crystal structure.
To help visualise the possible conformational excursions of the different domains of apo-SurA in solution we performed unrestrained all-atom molecular dynamics (MD) simulations. Initially three 1-μs simulations were performed starting from the crystal structure of full-length SurA (PDB 1M5Y27). However, in these simulations the individual domains of SurA were unstable, with both P1 and P2 unfolding. Therefore, we built an alternative starting model of full-length SurA in which the core and P1 domains are spatially separated, consistent with the smFRET data (SurAcore-P1-open) (Supplementary Fig. 5a, see Methods). In the three 1-μs simulations performed using this model as a starting structure, each domain remained folded and a wide variety of conformations was observed in which the distances between the domains differed markedly (Supplementary Fig. 5b-g & Supplementary Movies 1–3). While the three endpoint structures of these simulations satisfy more of the detected inter-domain cross-links than the crystal structure (an additional 5, Supplementary Table 3), they do not satisfy all of the cross-links observed. However, 18 of the 19 inter-domain cross-links are compatible with conformations of SurA that were sampled during the three 1-μs simulations (Supplementary Table 4), consistent with SurA adopting a broad array of conformations in solution that are in rapid exchange.
We also performed simulated annealing MD simulations of SurA using the detected inter-domain cross-links as distance restraints (see Methods) in order to visualise possible conformations of the chaperone in which P1 and/or P2 are docked onto the core domain (each domain was treated as a rigid body). All 19 inter-domain cross-links were used as restraints in these simulations. In the lowest energy structure obtained by this approach (structure 1 in Supplementary Fig. 6) all of the 19 inter-domain cross-links were satisfied (Supplementary Fig. 6, Supplementary Table 5). However, this does not suggest that SurA adopts a unique structure in solution, and indeed other structures obtained in the simulated annealing calculations with different domain orientations explain the observed cross-links almost equally well (the 10 lowest energy structures are shown in Supplementary Fig. 6, at least 15/19 inter-domain cross-links were satisfied in each structure, Supplementary Table 5). In these structures a range of SurA domain orientations are observed (Supplementary Fig. 6b), with P2 docking against the core in all 10 structures, whereas P1 adopts a range of conformations (Supplementary Fig. 6).
Given that the simulated annealing approach will drive SurA to adopt compact states that satisfy the maximum number of cross-link restraints within a single structure, more extended states of SurA that are significantly populated in solution, as shown by the smFRET data (Fig. 3), will not be captured by this approach. Indeed, as shown by the smFRET and unrestrained MD simulations, the dynamic nature of SurA makes it challenging to define its precise conformational landscape, wherein a repertoire of conformations in dynamic exchange on a sub-millisecond timescale are formed. Consistent with this, no single structure can satisfy the broad distributions observed by smFRET (Supplementary Table 6), providing further evidence that the cross-links observed cannot all result from a single SurA conformation, but result from different rapidly interconverting states. Together, the unrestrained all-atom MD and simulated annealing simulations demonstrate that the three domains of SurA are able to move independently of each other as rigid bodies, facilitated by the flexible linker regions between them (Supplementary Fig. 1d). This results in chaperone structures with a broad range of inter-domain distances and orientations in rapid exchange as confirmed by smFRET.
SurA binds its OMP substrates at multiple interaction sites
We next investigated how SurA binds its OMP clients, and how this affects conformations adopted by the chaperone. While NMR studies have shown that OMP substrates bound to SurA remain in a dynamic, unfolded state42,43,44, their binding site(s) on SurA remained unexplored. To map the OMP interaction surface on SurA we used E. coli OmpX (16 kDa) as a model substrate (OmpX forms an 8-stranded β-barrel in its native state). SurA binds unfolded OmpX with a Kd,app of ~800 nM, as measured by microscale thermophoresis (MST) (Supplementary Fig. 7), similar to the affinity of SurA for other OMPs28,45,46. SurA-OmpX complexes were assembled by rapid dilution of urea-denatured OmpX into a solution of SurA (final concentrations: 5 μM OmpX, 5 μM SurA, 0.24 M urea) (see Methods) immediately prior to cross-linking with DSBU. A band corresponding to cross-linked SurA-OmpX complexes could be observed by SDS-PAGE (Fig. 4a), and following in-gel digestion a total of 26 unique inter-molecular cross-linked peptides were detected (Fig. 4b, Supplementary Table 7, Supplementary Data 1). Sixteen of the 26 unique crosslinks between SurA and OmpX are in the core domain (Fig. 4c, Supplementary Table 7). Four Lys residues in P2, one in P1, and 7 in the core cross-linked to OmpX, but no cross-links were detected for the remaining 11 Lys residues in P1 and P2 indicating either that these regions are not involved in the interaction or that the orientations of residue side-chains hampered the XL reaction at these sites (Fig. 4b, c). Importantly, several cross-links were detected from the same residue in OmpX to several residues on SurA (e.g. residue 82 of OmpX cross-links to 13 different residues in SurA spanning all four regions of the chain, Supplementary Table 7). Similarly, the same site on SurA cross-linked to multiple sites on OmpX (e.g. residues 135, 294, 389 and 395 in SurA each cross-link to three residues (50, 71 and 82) in OmpX, Supplementary Table 7) consistent with a flexible and dynamic OmpX in the bound state (Fig. 4b).
Fig. 4: Multi-site binding of OmpX to SurA.
a SDS-PAGE analysis of DSBU cross-linked SurA-OmpX. Note that the species indicated with an asterisk (*) are higher order cross-linked species of mass corresponding to multiple SurA molecules bound to OmpX, consistent with multivalent binding observed previously46. These were not analysed further here. Source data are provided as a Source Data file. b Inter-molecular cross-links detected in the SurA-OmpX complex. The location of all Lys residues are indicated with orange arrows. c Crystal structure of SurA (PDB 1M5Y27). Purple spheres indicate identified cross-link sites (Supplementary Table 7). Samples contained 5 µM SurA, 5 µM OmpX, 0.24 M urea, 50 µM–2 mM DSBU, in 10 mM potassium phosphate buffer, pH 8.0, 25 °C. A representative mass spectrum for each cross-link can be found in Supplementary Data 1.
The extended spacer arm length of DSBU and its limited reactivity (primarily with Lys residues), makes it challenging to precisely define the interaction interface for OmpX on SurA. To probe the organisation of the SurA-OmpX complex in more detail we thus exploited the ability of the photoactivatable cross-linker MTS-diazirine (Supplementary Fig. 8a) to react rapidly (within ns47), and non-specifically with any residue within ~15 Å of the diazirine moiety (Cα-Cα Euclidean distance)48. This "tag transfer" method was developed specifically to enable detection of weak and transient protein–protein interactions48. We created four MTS-diazirine-labelled single-Cys variants of OmpX (M41C, I102C, K122C, V167C), formed complexes of each with SurA, and following rapid UV irradiation (for only 30 s)48, identified the cross-linked products by LC-MS/MS (Fig. 5a–c, Supplementary Fig. 8a-c and Supplementary Table 8, Supplementary Data 1)48. In this experimental design (Supplementary Fig. 8a), all amino acid substitutions were performed on the OMP, which remains dynamically unfolded upon binding SurA44. Despite the location of the labelled Cys residues in distant regions of the OmpX sequence and the lack of specificity of the diazirine cross-linker48, all four Cys-OmpX variants cross-linked to the N-terminal domain of SurA (11 cross-linked sites were identified, Supplementary Table 8) and P1 (two cross-linked sites), indicating that these regions form the heart of the binding epitope (Fig. 5b, c, Supplementary Fig. 8c). Notably, no cross-links were detected between OmpX and the SurA P2 domain or C-terminal region, despite the highly promiscuous photoactivatable cross-linker employed. This differs from the SurA-OmpX cross-links detected with DSBU, probably because (1) the much longer cross-linking time (45 min) required for cross-linking with DSBU, compared with 30 s for the diazirine, permits conformational excursions during the cross-linking reaction; and/or (2) the increased spacer arm length and cross-linking distance (ca. 27–30 Å) of DSBU, compared with the tag-transfer XL (ca. 15 Å), enables DSBU to capture longer-range interactions. Overall, therefore, the results suggest that OmpX adopts a range of likely interconverting conformations upon binding SurA, in which multiple specific interactions are formed predominantly with the N-terminal region of the chaperone core domain.
Fig. 5: Multiple locations across the OmpX sequence interact with similar sites on SurA.
a Tag-transfer photo-cross-linking48 of SurA-OmpX complexes using OmpX Cys variants labelled with MTS-diazirine analysed by SDS-PAGE. A band corresponding to the SurA-OmpX complex is observed for all OmpX variants following UV irradiation. These bands were not observed when analysed using reducing SDS-PAGE (Supplementary Fig. 8b). Source data are provided as a Source Data file. b Inter-molecular cross-links detected in the SurA-OmpX complex (Supplementary Table 8). c Structure of SurA with residues which were photo-cross-linked to labelled OmpX Cys variants shown in purple. Where the data quality did not permit residue level assignment, the cross-linked peptide is shown in light purple. Samples contained 10 μM SurA, 5 µM MTS-diazirine-labelled OmpX, 0.24 M urea, in 10 mM potassium phosphate buffer, pH 8.0, 25 °C and cross-linking was initiated by UV LED irradiation of the sample for 30 s (see Methods). A representative mass spectrum for each cross-link can be found in Supplementary Data 1.
Conformational changes in SurA upon OMP binding
Next, we examined the conformational changes induced by OMP binding to SurA using differential HDX-MS analysis (Fig. 6a–f and Supplementary Fig. 9). We first compared the uptake of deuterium by different regions of SurA in the presence or absence of OmpX under conditions which minimise OMP aggregation (10 mM potassium phosphate, pH 8.0, 0.24 M urea, 4 ˚C28,49,50) (Fig. 6a, b, Supplementary Fig. 9a). In the presence of OmpX, regions in SurA that are protected from deuterium uptake upon substrate binding all cluster to the core domain. No change in protection in P2 was detected in the presence of OmpX, consistent with the tag transfer XL-MS results and with previous results which have shown that P2 is not required to prevent the aggregation of the small (8-stranded) tOmpA28. Intriguingly, two regions of SurA (residues 46–72 in the N-terminal region and 212–239 in P1) (Fig. 6a, b, Supplementary Fig. 9a), that are located at the core-P1 interface (Supplementary Fig. 1d), were deprotected upon OmpX binding, demonstrating a structural reorganisation of this interface in response to substrate binding.
Fig. 6: SurA binding to different substrates leads to varying patterns of protection and deprotection by differential HDX-MS analysis.
Wood's plots showing the summed differences in deuterium uptake in SurA over all four HDX timepoints, comparing SurA alone with SurA in the presence of a OmpX, c OmpF or e WEYIPNV. Wood's plots were generated using Deuteros115. Peptides coloured in blue or red, respectively, are protected or deprotected from exchange in the presence of OmpX/OmpF/WEYIPNV. Peptides with no significant difference between conditions, determined using a 99% confidence interval (dotted line), are shown in grey. Regions of SurA protected or deprotected in the presence of b OmpX, d OmpF and f WEYIPNV coloured in blue or red, respectively. Example deuterium uptake curves are shown in Supplementary Fig. 9. See Methods for experimental details.
To determine whether deprotection at the core-P1 interface occurs in the presence of other OMPs, the effects of binding the larger substrate, OmpF (16-stranded), on the HDX properties of SurA was examined. In the presence of OmpF, residues in the N- and C-terminal regions of the core domain were also protected from exchange, consistent with shared OmpX and OmpF binding sites. However, in marked contrast with the results for OmpX in which residues 46–72 of SurA were deprotected from exchange upon substrate binding, these residues were instead protected from exchange in the presence of OmpF, suggesting that the larger OMP binds to, or occludes, a greater surface area on the core (Fig. 6c, d, Supplementary Fig. 9b). Importantly, as for OmpX, deprotection was observed in the P1 domain at the core-P1 interface, suggesting that structural reorganisation of this interface also occurs upon OmpF binding. Notably, the hinge region between P1 and P2 (residues 266–286) was also deprotected in the presence of OmpF, suggesting that binding of the larger substrate may also alter the conformational dynamics at locations more distal to the core.
To decouple the phenomena of protection arising as a result of OmpX/OmpF binding and deprotection as a result of conformational changes in SurA, we also compared the levels of deuterium uptake of SurA in the presence of a 7-residue peptide known to bind to the P1 domain (WEYIPNV, Kd 1–14 μM)51,52 (Fig. 6e, f, Supplementary Fig. 9c,d). Interestingly, extensive deprotection at the core-P1 interface (residues 39–74, 142–160 and 381–422) was observed in the presence of WEYIPNV, while protection was only observed in P1 at the known peptide binding site (residues 212–24352) (Supplementary Fig. 9c,d). Combined, these results demonstrate that the OMP substrate binding surface is more extensive in OmpX/OmpF compared with WEYIPNV, but in all three cases binding triggers a structural reorganisation between the core and P1 domains.
To further study the effects of substrate binding on the conformations of SurA adopted in solution we used smFRET to examine the inter-domain distances of SurA-bound to OmpX, OmpF or WEYIPNV (Fig. 7). Consistent with the HDX data (Fig. 6), binding of OmpX to SurA resulted in changes at the core-P1 interface. Instead of the ca. bimodal EFRET distribution observed for the apo-SurA (EFRET centred on ~0.2 and ~0.6, Fig. 3d), a single broad EFRET distribution with a maximum at an EFRET value (~0.5) between that of the open and closed states was observed for the SurA-OmpX complex (Fig. 7a). By contrast, the EFRET distributions for the core-P2 (Fig. 7b) and P1-P2 (Fig. 7c) SurA variants bound to OmpX were similar to those of apo-SurA. Control experiments in which EFRET was determined for apo-SurA in the presence of 0.24 M urea (used to aid solubilisation of the OMP50) showed broadening of the core-P1 distribution (relative to apo-SurA in the absence of urea) (Supplementary Fig. 10a), and little change in the core-P2 and P1-P2 distributions (Supplementary Fig. 10b-c). BVA showed increased dynamics on a sub-ms timescale for all three FRET pairs, suggesting more frequent opening/closing transitions in the presence of denaturant (Supplementary Fig. 10d–f). Effects similar to OmpX binding on the EFRET distributions were observed when OmpF was added to SurA (Fig. 7d–f). In marked contrast with the effects on the EFRET distributions on OmpX/OmpF binding, the addition of the P1-binding peptide WEYIPNV had a more profound effect, changing both the core-P1 and core-P2 inter-domain EFRET distributions, inverting the populations of the core-P1open and core-P1closed distributions to favour core-P1open (Fig. 7g), and shifting the maximum of the core-P2 EFRET distribution to a lower EFRET value (Fig. 7h), with only a subtle change to the P1-P2 EFRET distribution (Fig. 7i). Consistent with the HDX data, these results suggest that binding of WEYIPNV promotes the release of the P1 domain from the core. BVA on the SurA-substrate complexes indicated that all complexes remained dynamic on the sub-millisecond timescale (Supplementary Fig. 11a-i), although the dynamics of the larger OmpF-SurA complex were dampened relative to those of SurA-OmpX (Supplementary Fig. 11d–f).
Fig. 7: Response of SurA inter-domain distances to substrate binding measured by smFRET.
Experimentally measured EFRET distributions (grey) at equilibrium for the three pairwise combinations of fluorescently-labelled SurA double mutants (core-P1, core-P2, and P1-P2) in the presence of a–c OmpX, d–f OmpF, or g–i WEYIPNV. Kernel density estimations (KDEs) of the probability density function of the measured EFRET values are shown as green, red and blue solid lines for a, d, g SurA core-P1, b, e, h core-P2 and c, f, i P1-P2 pairwise measurements in the presence of OmpX, OmpF or peptide WEYIPNV, respectively. Each were fitted to a maximum of two Gaussians to appraise the ensemble heterogeneity (note that these do not necessarily represent true discrete states). Only a single Gaussian was used in (a) as this distribution is approximately unimodal. The corresponding apo-SurA distributions (black dashed lines, taken from Fig. 3d–f) are shown for reference to allow comparison between apo and holo SurA. Samples contained ~50 pM labelled SurA variant, 1.5 μM OmpX/OmpF/WEYIPNV, in 50 mM Tris-HCl, pH 8.0, 25 °C, with a final urea concentration of 0.24 M in the OMP-containing samples.
Despite its key role in OMP biogenesis and bacterial virulence53,54, how SurA binds its OMP substrates specifically, but weakly28,46,51, and how it is able to protect its clients from aggregation and deliver them to BAM for folding into the OM, remain poorly understood in molecular detail. Previous NMR studies have shown that OmpX, tOmpA and FhuA are dynamically disordered when bound to SurA42,43,44. However, precisely how SurA binds its OMP clients and how OMP binding alters the conformation(s) adopted by SurA in solution have remained unknown. Here, we have exploited XL, HDX-MS, MD and smFRET, to analyse the conformational dynamics of apo-SurA and to investigate how this is modulated by substrate binding. Further, we have identified the regions of SurA involved in substrate binding for both small (OmpX) and larger (OmpF) clients. The combined data presented are consistent with a model in which specific, yet multi-site, binding by a dynamically disordered substrate is accomplished within a cradle-like conformation of SurA that is very different to that observed in its crystal structure (Fig. 8).
Fig. 8: Summary of SurA conformational dynamics and a proposed mechanism of substrate binding.
a The crystal structure of SurA (PDB 1M5Y27). Note that this conformation is not significantly populated in solution, as demonstrated here. Instead, in solution the P2 domain is mostly found close to the core/P1 domains (b, c). In these conformations the P1 domain can adopt b core-P1closed and c core-P1open states. d Substrate binding results in the P1 domain adopting a structure intermediate between core-P1open and core-P1closed. Our data are consistent with the OMP client being captured as a dynamically unfolded state43 within a cradle formed by the three domains of SurA. Whether SurA-bound OMP is in a collapsed globule (represented here as a sphere) or a more extended state remains unclear. However, the XL data are consistent with the presence of multiple, specific OMP recognition sites on SurA, suggesting a dynamic ensemble of bound structures. Note that the images presented are schematic and aim to portray the dynamics of the P1, P2 and core domains relative to each other, rather than atomic-level detail.
Flexibility and/or inter-domain dynamics are key features of the mechanisms of several ATP-independent chaperones, such as Tim9/1055, Trigger Factor (TF)5,56, Spy9, SecB10, and the periplasmic OMP chaperones Skp57,58,59 and FkpA60. Like SurA, TF and FkpA are multi-domain proteins containing PPIase domains that exhibit inter-domain dynamics60,61. The ATP-dependent chaperones GroEL/TRiC, Hsp70 and Hsp90 also utilise inter-domain dynamics for their functional cycles7,8,62. For the ATP-independent chaperones HdeA63 and Hsp3364, which are activated by acidic and oxidative stress, respectively, conformational switching has been shown to trigger their chaperone function. The results presented here suggest that SurA populates a broad ensemble of structures in solution involving different inter-domain distances and orientations. Interconversion between these conformations occurs on a sub-ms timescale, as demonstrated here using smFRET BVA (Fig. 3g–i). We propose that such rapid conformational changes are likely to be important for SurA to be able to bind its clients in the periplasm and to release substrates to BAM for folding into the OM. Notably, a similarly broad range of inter-domain motions on comparable timescales to those observed here for SurA was observed previously in MD simulations for the homologous chaperone TF5.
Our combined XL-MS, HDX-MS and smFRET data show that the P1 domain of SurA is not statically bound to the core domain in solution. Instead, these domains are in a dynamic equilibrium between core-P1open and core-P1closed ensembles, suggesting a role for core-P1 dynamics in regulating access of the client OMP to the core chaperone domain and perhaps for access to P1 itself, providing client specificity and enhancing binding affinity52. Previous studies revealed that tethering the core and P1 domains by creation of a disulfide bond impairs OMP assembly in vivo, and that destabilising SurA can rescue OMP assembly defects in BAM-compromised strains29,65. These results can now be explained by the opening and closing motions between the core and P1 domains revealed here by smFRET and HDX-MS. We also show here that the P2 domain commonly populates conformations in close proximity to the core and P1 domains in solution (Figs. 1, 2) in marked contrast with the orientation of P2 in the SurA crystal structure in which it is distal to P1/core, most likely as a consequence of crystal packing (Supplementary Fig. 2)27. This positions the P2 domain such that it could readily provide additional binding/chaperoning capacity for specific substrates28. Thus, our data are consistent with SurA frequently adopting a compact, possibly cradle-like, structure (Fig. 8) that may act as the acceptor state for client binding. However, an alternative model in which more extended SurA structures initially capture the unfolded OMP, followed by compaction of the complex, is also possible. In either scenario, the result of binding is a compact SurA in which the client is bound to the core domain, protecting the OMP from aggregation within the dynamic complex. Such a dynamic structure could provide a mechanism for release of bound OMPs to BAM for folding into the OM without the requirement for ATP binding/hydrolysis to drive client release.
Previous reports based on crystal contacts proposed that SurA may bind its substrates via a binding crevice in the N-terminal domain (Fig. 1)27. By contrast, the results presented here show that SurA binds its OMP clients at multiple sites32,66, predominantly involving the core domain as indicated by tag-transfer XL and HDX. The additional binding capacity in the P1 and P2 domains may be employed in a substrate-specific manner, as suggested by the finding that P2 is required to suppress the aggregation of the 10-stranded OmpT, but not the 8-strand tOmpA28. Previous results have also shown that deletion of P2 can perturb SurA function in vivo, as measured by a decrease in the amount of assembled LamB in a BAM-compromised strain29. These assembly defects are rescued by further deletion of the P1 domain, suggesting that P2 may also play a role in regulating interactions between the P1 and core domains that inhibit SurA chaperone function29. Here we have shown that the three domains of SurA can come into close proximity in solution, consistent with the view that inter-domain communication between all three domains may a play a role in SurA activity. P2 has also been suggested to interact with the BAM complex67, to either localise OMPs to this folding catalyst, prime BAM for OMP insertion, promote substrate release, or create supercomplexes linking SecYEG and BAM across the periplasm68. Differential proteomics experiments have identified reduced levels of eight different OMPs upon SurA deletion17, including OmpX and OmpF which were studied here. Further work will be needed to understand the relay of interactions between SurA, its clients, other chaperones, and BAM, and to discern whether/how the mechanism of OMP delivery to the OM is dependent on the identity of the OMP client.
The XL-MS data presented show that OmpX is able to adopt multiple conformations and orientations when bound to SurA42. The cross-link locations identified on SurA by tag-transfer experiments show no clear correlation with areas with a particular electrostatic potential, hydrophobic patches, or regions of high sequence conservation (Supplementary Fig. 12). Instead, they cluster to regions within the cradle which forms by docking of the three domains of SurA and in which the OMP is sequestered and binds predominantly to the core domain (Fig. 8). Such a model is consistent with in vivo data showing that the SurA core domain alone can largely (but not wholly) complement deletion of wild-type SurA24,29,69. Specific client interaction sites have also been identified in TF using the substrate PhoA, which are also located in a cradle formed between its domains70. In the presence of substrate, the EFRET distribution observed by smFRET shifts to an intermediate state between core-P1open and core-P1closed, suggesting that substrate binding either alters the conformations of SurA, or that the interconversion between the open and closed states occurs more rapidly than in the unbound state, leading to signal averaging. In either case, this is reminiscent of the lid motions in Hsp70 that entrap its substrates71, the sequestration of OMPs within the cavity of the chaperone Skp57,58, and the conformational flexibility that has been suggested to be important for substrate capture/release by Spy9. The conformational properties of OmpX when bound to SurA cannot be resolved in detail using the data presented here. Interestingly, however, analysis of the locations of cross-linking sites on OmpX in the OmpX-SurA complex using DSBU does not show any obvious correlation with regions that form β-strands, loops or turns in the folded OMP, consistent with the tag-transfer results which showed cross-linking to SurA when the diazirine was positioned in regions that either form β-strands or loops in native OmpX (Fig. 5a, Supplementary Fig. 13). Combined, these results suggest that while specific SurA recognition motifs in OMP sequences may be present19,28,45,51, the interaction interface on SurA may be less well-defined, consistent with recent reports from Marx et al. who also used XL as well as neutron diffraction to examine SurA-client interactions32.
A number of sequence motifs in OMPs have been implicated in facilitating OMP biogenesis. In the case of OmpX, such motifs have been shown to nucleate assembly72, mediate hydrophobic collapse and membrane interactions73, and stabilise the folded state74. Aromatic-containing motifs in OMPs (Ar-Ar and Ar-X-Ar, where Ar is an aromatic residue and X is any amino acid) have also been implicated in OMP-SurA interactions19,32,45,51. Many of the cross-links we detect are close to these motifs (Supplementary Fig. 13). Available evidence suggests that when bound to SurA, OMPs adopt extended structures32,33, and that binding of multiple SurA molecules to different sites in the OMP may be essential to prevent aggregation from these extended states28,32,46.
The in vitro folding pathways of several OMPs have been explored75,76, however, how OMP folding is modulated by chaperone binding28,46,50,58,75 and interactions with BAM remain unresolved77,78. Once translocated into the periplasm by the SecYEG complex, OMP folding, comprising delivery to BAM and insertion into the OM, is achieved without an external energy source. Indeed, this process is thought to be driven thermodynamically by favourable free energies of folding of OMPs (ΔG°F as high as ~130 kJ/mol79) compared with the weak binding of OMPs to SurA (ΔG° ~ 35-44 kJ/mol80) or their self-association (ΔG° ~ 38 kJ/mol80,81,82). The weak binding events between OMPs and SurA are likely key to enable efficient transfer of OMPs between chaperone molecules and for handover to the BAM complex78. It has been proposed that unfolded OMPs are recognised by BAM via the so-called C-terminal "β-signal"83,84. Our XL-MS experiments suggest that this region of OmpX forms contacts with SurA (Supplementary Fig. 13). Hence, the transient nature of the SurA-OMP interactions may be essential both to facilitate presentation of the β-signal to BAM and, subsequently, to enable rapid handover of OMPs to BAM for folding and insertion into the OM. Recent studies using neutron diffraction and cross-linking support this model32.
SurA plays multiple roles in OMP biogenesis, including sequestration of OMPs in the periplasm to prevent their toxic aggregation85 and delivery of OMPs to the BAM complex to enable folding into the OM78. The results presented here demonstrate a role for SurA inter-domain dynamics in OMP binding, notably the reorganisation of the core and P1 domains, and dynamic localisation of P2 close to these domains. Such structural plasticity may also be important for facilitating binding of SurA:OMP complexes to BAM, assisting BAM catalytic activity, or priming the OMP for membrane insertion by pre-selecting favourable conformations for folding, thereby smoothing the energy landscape of folding33,44,78. Understanding the interplay between the conformational dynamics of SurA and those of BAM, in particular the communication and coordination between SurA and different BAM subunits, will be essential in unravelling the molecular mechanism of OMP biogenesis. The model of SurA action presented here, whereby a compact, dynamic and responsive chaperone structure is responsible for client binding, represents a first key step in this endeavour. This adds to the growing body of data suggesting that all components of the OMP assembly line, including SurA, Skp57,58 and BAM86,87,88,89,90 have intrinsic conformational dynamics which, in combination, may be key to achieving efficient OMP biogenesis in the absence of ATP.
Cloning, expression and purification of SurA
A pET28b plasmid containing the mature SurA sequence preceded by an N-terminal 6x His-tag and thrombin-cleavage site (pSK257) was a kind gift from Daniel Kahne (Harvard University, USA)91. The thrombin-cleavage site was mutated to a TEV-cleavage site using Q5 site-directed mutagenesis (NEB), and the resulting plasmid was transformed into BL21(DE3) cells (Stratagene). Cells were grown in LB medium supplemented with 30 µg/mL kanamycin at 37 °C with shaking (200 rpm) until an OD600 of ~0.6 was reached. The temperature was subsequently lowered to 20 °C, and expression induced with 0.4 mM IPTG. After ~18 h, cells were harvested by centrifugation, resuspended in 25 mM Tris-HCl, pH 7.2, 150 mM NaCl, 20 mM imidazole, containing EDTA-free protease inhibitor tablets (Roche), and lysed using a cell disrupter (Constant Cell Disruption Systems). The cell debris was removed by centrifugation (20 min, 4 °C, 39,000 × g), and the lysate was applied to 5 mL HisTrap columns (GE Healthcare). The columns were washed with 25 mM Tris-HCl, pH 7.2, 150 mM NaCl and 20 mM imidazole, followed by 25 mM Tris-HCl, 6 M Gdn-HCl, pH 7.2 (to denature the SurA on-column). After washing with 25 mM Tris-HCl, 150 mM NaCl, pH 7.2, SurA was eluted with 25 mM Tris-HCl, 150 mM NaCl, 500 mM imidazole, pH 7.2. The eluate was dialysed against 25 mM Tris-HCl, 150 mM NaCl, pH 8.0 overnight, and the following day TEV protease46 (ca. 0.5 mg) and 0.1% (v/v) β-mercaptoethanol were added. The cleavage reaction was left to proceed overnight at 4 °C on a tube roller. The cleavage reaction was again applied to the 5-mL HisTrap columns (GE Healthcare) to remove the cleaved His-tag and His-tagged TEV protease. The unbound, cleaved SurA product was dialysed extensively against 25 mM Tris-HCl, 150 mM NaCl, pH 8.0, before being concentrated to ~200 µM with Vivaspin 20 concentrators (Sartorius; 5-kDa MWCO), aliquoted, snap-frozen in liquid nitrogen and stored at −80 °C. Protein concentrations were determined spectrophotometrically using an extinction coefficient at 280 nm of 29450 M−1 cm−1.
Cys-containing variants (Q85C, N193C, E301C, Q85C-N193C, N193C-E301C and Q85C-E301C) were generated by Q5 site-directed mutagenesis (NEB) and were purified as detailed above, except for the addition of 10 mM DTT to all buffers in the purification procedure, up until the elution step.
Expression and purification of TEV protease
Vector pMHTDelta238 containing His-tagged TEV fused with MBP which is removed in vivo by autocleavage, was obtained from DNASU (Clone TvCD00084286). The vector was transformed into BL21-CodonPlus[DE3]-RIPL cells (Stratagene, UK). Cells were grown in LB medium containing 50 µg/mL kanamycin at 37 °C with shaking (200 rpm) until the culture reached an OD600 of ~0.6. The temperature was then lowered to 30 °C and expression induced with 0.5 mM IPTG. After ~4 h the cells were harvested by centrifugation, resuspended in 25 mM sodium phosphate buffer, pH 8.0, 200 mM NaCl, 10% (v/v) glycerol, 25 mM imidazole, 1 mM PMSF, 2 mM benzamidine, ~0.02 mg/ml DNase (Sigma, UK), and lysed by sonication (6 × 30 s bursts with 1 min cooling on ice between each sonication). The lysate was centrifuged to remove cell debris (20 min, 4 °C, 39,000 × g), applied to Ni2+ Sepharose beads (GE Healthcare) and washed twice with 25 mM sodium phosphate buffer, pH 8.0, 200 mM NaCl, 10% (v/v) glycerol, 25 mM imidazole. His-tagged TEV was eluted with 25 mM sodium phosphate buffer, pH 8.0, 200 mM NaCl, 10% (v/v) glycerol, 500 mM imidazole. The eluate was filtered (0.2 μM syringe filter, Sartorius, UK) and gel filtered on a HiLoad Superdex 75 26/60 column (GE Healthcare) equilibrated with 25 mM sodium phosphate buffer, pH 8.0, 200 mM NaCl, 25 mM imidazole, 10% (v/v) glycerol, 5 mM β-mercaptoethanol. Peak fractions were concentrated to ~1 mg/mL using Vivaspin 20 (5 kDa MWCO) concentrators (Sartorius), aliquoted, snap-frozen in liquid nitrogen and stored at −80 °C.
Cloning of OmpX, OmpF and Cys-OmpX
Codon-optimised synthetic genes (Eurofins) of the mature sequences of OmpX (residues 24–171) and OmpF (residues 23–362) were cloned into pET11a (Novagen) between the NdeI (5′) and BamHI (3′) restriction sites. To create the Cys-OmpX construct, the residues Gly-Ser-Cys were added immediately after the N-terminal Met residue using Q5 site-directed mutagenesis (NEB).
Expression and purification of OMPs
OMPs were purified using a method adapted from ref. 50. Briefly, E. coli BL21[DE3] cells (Stratagene) were transformed with a pET11a plasmid containing the gene sequence of the mature OMP. Overnight cultures were subcultured and grown in LB medium (500 mL) supplemented with carbenicillin (100 μg/mL), at 37 °C with shaking (200 rpm). Protein expression was induced with IPTG (1 mM) once an OD600 of 0.6 was reached. After 4 h the cells were harvested by centrifugation (5000 × g, 15 min, 4 °C). The cell pellet was resuspended in 50 mM Tris-HCl pH 8.0, 5 mM EDTA, 1 mM phenylmethylsulfonyl fluoride, 2 mM benzamidine, and the cells were subsequently lysed by sonication. The lysate was centrifuged (25,000 × g, 30 min, 4 °C) and the insoluble material was resuspended in 50 mM Tris-HCl pH 8.0, 2% (v/v) Triton X-100, before being incubated for 1 h at room temperature, with gentle agitation. The insoluble material was pelleted (25,000 × g, 30 min, 4 °C) and the inclusion bodies washed twice by resuspending in 50 mM Tris-HCl pH 8.0 followed by incubation for 1 h at room temperature with gentle agitation, and then collected by centrifugation (25,000 × g, 30 min, 4 °C). For the OmpX and OmpF constructs, the inclusion bodies were solubilised in 25 mM Tris-HCl, 6 M Gdn-HCl, pH 8.0 and centrifuged (20,000 × g, 20 min, 4 °C). The supernatant was filtered (0.2 µM syringe filter, Sartorius) and the protein was purified using a Superdex 75 HiLoad 26/60 gel filtration column (GE Healthcare) equilibrated with 25 mM Tris-HCl, 6 M Gdn-HCl, pH 8.0. Peak fractions were concentrated to ∼500 μM using Vivaspin 20 (5 kDa MWCO) concentrators (Sartorius), and the protein solution was snap-frozen in liquid nitrogen and stored at −80 °C.
Chemical cross-linking-mass spectrometry (XL-MS)
OmpX was buffer exchanged from storage buffer (25 mM Tris-HCl, 6 M Gdn-HCl, pH 8.0) into 10 mM potassium phosphate, pH 8.0, 8 M urea using Zeba spin desalting columns (Thermo Fisher Scientific). For cross-linking, apo-SurA was prepared at a concentration of 5 µM in 10 mM potassium phosphate, pH 8.0, while SurA-OmpX complexes were assembled by mixing SurA and OmpX such that the final concentration of each was 5 µM, in 10 mM potassium phosphate, pH 8.0, 0.24 M urea. DSBU (Thermo Fisher Scientific) was added at 10× (apo-SurA) or 10–400× (SurA:OmpX) molar equivalents relative to the concentration of SurA, and the cross-linking reaction was left to proceed for 45 min at room temperature before quenching by adding 0.2 M Tris-HCl, pH 8.0. The cross-linked material was separated by SDS-PAGE and gel bands corresponding to either SurA alone or the SurA:OmpX complex were excised and the proteins trypsinised in-gel48. Briefly, gel bands were cut into ca. 1-mm3 pieces, and the pieces were destained in 30% (v/v) ethanol at 60 °C for 30 min, dehydrated with 100% acetonitrile, and dried in a laminar flow hood for 60 min. The gel pieces were rehydrated with 20 µL of 0.02 μg.μL−1 trypsin solution (Promega) in 25 mM ammonium bicarbonate pH 8, and incubated at 37 °C for 18 h with shaking (500 rpm). Peptides were recovered by incubating gel pieces with 50 μL of 60% (v/v) acetonitrile/5% (v/v) formic acid (×3) for 10 min. The peptides were then evaporated to dryness and resuspended in 20 μL with 5% (v/v) acetonitrile/0.1% (v/v) formic acid prior to MS analysis. Peptides (5 µL) were injected onto a reverse-phase Acquity M-Class C18, 75 µm × 150 mm column (Waters) and separated by gradient elution of 1–50% (v/v) solvent B (0.1% (v/v) formic acid in acetonitrile) in solvent A (0.1% (v/v) formic acid in water) over 60 min at 300 nL.min−1. The eluate was infused into an Orbitrap Q Exactive (Thermo Fisher Scientific) mass spectrometer operating in positive ion mode. Orbitrap calibration was performed using Ultramark solution (Thermo Fisher Scientific). Data acquisition was performed in DDA mode and fragmentation was performed using HCD. Each high-resolution full scan (m/z range 500–2000, R = 120,000) was followed by high-resolution product ion scans (R = 15,000), with a normalised collision energy of 30%. The 15 most intense ions in the MS spectrum were selected for MS/MS. Dynamic exclusion of 60 s was used. RAW data files were converted to MGF format using PEAKS Studio 8.5 (Bioinformatics Solutions). Cross-link identification was performed using MeroX v1.6.692. Data were processed using RISE mode. Automated cross-link assignment was performed considering Lys-Lys cross-links and allowing a maximum of 2 of the four 26-u doublets, corresponding to fragmentation of the cross-linker, to be missing. Up to two oxidised Met residues per peptide were considered as variable modifications. Mass deviation tolerances of up to 10 ppm in both MS and MS/MS were used. Only results with scores corresponding to a false discovery rate (FDR) of <5% were taken forward. All spectra were manually verified to ensure they comprised fragment ions with significant coverage of each cross-linked peptide and for the presence of diagnostic fragments from cross-linker fragmentation. A representative mass spectrum for each cross-link can be found in Supplementary Data 1. The Raw DSBU XL-MS data have been deposited to the ProteomeXchange Consortium via the PRIDE93 partner repository with the dataset identifier PXD016993. A reporting summary (based on community guidelines94) can be found (Supplementary Data 2).
Preparation of SurA variants for smFRET
For each SurA variant, the protein was diluted to a concentration of 50 μM in 25 mM Tris-HCl, pH 7.2, 150 mM NaCl, 5 mM DTT. The protein solution was incubated for 30 min at room temperature before being buffer exchanged into 25 mM Tris-HCl, pH 7.2, 150 mM NaCl, 1 mM EDTA using 7 kDa MWCO Zeba spin desalting columns (Thermo Fisher Scientific). A ten-fold molar excess of Alexa Fluor 488 C5 maleimide/Alexa Fluor C5 594 maleimide (Thermo Fisher Scientific) was then added and the samples incubated for 2 h at room temperature with gentle rocking. The reaction was quenched with a 10-fold molar excess (over Alexa Fluor 488 C5 maleimide and Alexa Fluor C5 594 maleimide) of β-mercaptoethanol. Protein was separated from unbound dye by size exclusion chromatography on a Superdex 200 10/300 GL column (GE Healthcare, UK) equilibrated with 50 mM Tris-HCl, 150 mM NaCl, pH 8.0. Fractions containing labelled protein were combined, snap-frozen in liquid nitrogen and stored at −80 °C.
Single-molecule Förster resonance energy transfer (smFRET)
smFRET experiments were performed using a custom-built experimental set-up for μs ALEX95. Laser wavelengths and powers used were 488 nm, 140 μW and 594 nm, 120 μW, respectively before the objective and losses from the objective were on the order of 50%. The laser alternation period was set to 40 μs (duty cycle of 40%). Samples of labelled SurA were prepared on the day of use from concentrated stocks that had been stored at −80 °C and were kept on ice and in the dark while in use. A sample (100 µL, 50 mM Tris-HCl pH 8.0, 50 pM of labelled SurA) selectively supplemented with 1.5 μM OmpX/OmpF and/or 0.24 M urea or 1.5 μM peptide was added atop a coverslip set on the objective. A camera was used to monitor the distance of the focal plane from the coverslip and the objective height adjusted using a piezo-controller (piezo system jena) to 20 μm above the surface of the coverslip. Data acquisition was performed in 3 × 10 min runs with fresh sample prepared after every third collection to counteract the issues of protein aggregation and adherence to the coverslip as well as changes in solution osmolarity resulting from evaporation. Evaporation over the course of 30 min was minimised by employing a plastic lid that fitted over the coverslip. Data were collected using Labview graphical environment (LabView 7.1 Professional Development System for Windows, National Instruments)96. Separate photon streams were then converted and stored in an open file format for timestamp-based single-molecule fluorescence experiments (Photon-HDF5), which is compatible with many recent data processing environments97. Fluorescence bursts were analysed using customised Python 2.7 scripts98, and made use of FRETBursts, an open source toolkit for analysis of freely-diffusing single-molecule FRET bursts99. Functions from the FRETBursts package were used to recover single-molecule 'bursts' of fluorescence containing a minimum of 20 photons and being 1.7 times higher than the background signal of the measured time traces. Artefacts due to photophysical effects such as blinking were also removed. Apparent E and S values were calculated by relating the measured intensities in each of the four photon streams within each burst, i.e.,
donor emission during periods of donor excitation \((I_{Dem|Dex})\),
donor emission during periods of accpetor excitation \((I_{Dem|Aex})\),
acceptor emission during periods of donor excitation \((I_{Aem|Dex})\),
acceptor emission during periods of acceptor excitation \((I_{Aem|Aex})\),
according to Eqs. (1) and (2), respectively.
$$^iE_{app} = \frac{{I_{Aem|Dex}}}{{I_{Aem|Dex} + I_{Dem|Dex}}}$$
$$^iS_{app} = \frac{{I_{Aem|Dex} + I_{Dem|Dex}}}{{I_{Aem|Dex} + I_{Dem|Dex} + I_{Aem|Aex}}}$$
Four correction parameters described by relations between the photon streams were determined and applied to the data during the burst search algorithm of FRETBursts:
Donor leakage into the acceptor channel
$$\alpha = \frac{{g_{R|D}}}{{g_{G|D}}} = \frac{{\left( {{\,}^{ii}E_{app}^{({\mathrm{DO}})}} \right)}}{{1 - \left( {{\,}^{ii}E_{app}^{({\mathrm{DO}})}} \right)}}$$
Excitation of the acceptor dye by the donor excitation laser
$$\delta = \frac{{\sigma _{A|G}}}{{\sigma _{A|R}}}\frac{{I_{Dex}}}{{I_{Aex}}} = \frac{\left({{\,}^{ii}S_{app}^{({\mathrm{AO}})}}\right)}{{1 - \left( {{\,}^{ii}ES_{app}^{({\mathrm{AO}})}} \right)}}$$
Normalization of effective fluorescence quantum yields, \({}^{eff}{\mathrm{\Phi }}_F = a_b.{\mathrm{\Phi }}_F\) and detection efficiencies,\(g\), of \(A\) and \(D.\)
$$\gamma = \frac{{g_{R|A}}}{{g_{G|D}}}\frac{{{\,}^{eff}_{\Phi _{F,A}}}}{{{\,}^{eff}_{\Phi _{F,D}}}}$$
ab is the fraction of molecules in the bright state and \({\mathrm{\Phi }}_F\) is the fluorescence quantum yield without photophysical (saturation) effects.
Normalization of excitation intensities,\(I\), and cross-sections,\(\sigma\), of A and D.
$$\beta = \frac{{\sigma _{A|R}}}{{\sigma _{D|G}}}\frac{{I_{Aex}}}{{I_{Dex}}}$$
according to the standard FRET workflow developed by Hellenkamp et al.100, employing the following definitions:
FRET efficiency:
$$E\,{\mathrm{or}}\,E_{{\mathrm{FRET}}} = \frac{{F_{A|D}}}{{F_{D|D} + F_{A|D}}}$$
Stoichiometry:
$$S\,{\mathrm{or}}\,S_{{\mathrm{FRET}}} = \frac{{F_{D|D} + F_{A|D}}}{{F_{D|D} + F_{A|D} + F_{A|A}}}$$
Subscripts:
\(D\,{\mathrm{or}}\,A\) – Concerning donor or acceptor
\(A|D\) – Acceptor fluorescence given donor excitation
\(A_{em}|D_{ex}\), \(D_{em}|D_{ex}\), \(A_{em}|A_{ex}\) – Intensity in the acceptor channel given donor excitation, accordingly
Superscripts:
\({\mathrm{BG}}\) – Background
\({\mathrm{DO}}/{\mathrm{AO}}\) – Donor-only/acceptor-only species
\(i - iii\) – Indicates (i) the uncorrected intensity; (ii) intensity after BG correction; (iii) intensity after BG, alpha and delta corrections.
The data from each 10-min acquisition was merged prior to subsequent analysis. In order to remove bursts arising from singly labelled proteins, the data were also further filtered using ALEX-2CDE, yielding bursts with a Gaussian distribution of S values in a narrow range of dye stoichiometry (S within 0.25–0.75)101. Typically, ~5000 bursts were collected for each condition examined after all filters had been applied. The result of this procedure of correction and filtering is demonstrated graphically in the form of 2D plots of E versus S for the example of the core-P1 labelled SurA which also served as our etalon (Supplementary Fig. 14). Filtered bursts were then assembled into 1D histograms and kernel density estimation was used to approximate 1D probability density functions of the EFRET values in each condition which were then fitted to up to two Gaussians. Burst variance analysis (BVA)41 was performed using FRETBursts98, and plots were made using the Seaborn and Matplotlib102 packages in the Spyder IDE on python 3.7. Visualisations of the available volumes for FRET dyes attached at different positions in SurA were generated using the FRET Positioning and Screening (FPS) software with dye linker lengths and radii parameters suggested in the FPS manual for the FRET dyes used39. Predicted EFRET value distributions from the crystal structure of full-length SurA for each dye pair were calculated from distance distributions generated using the MtsslWizard PyMOL plugin103. Raw smFRET data are available at the University of Leeds data repository (https://doi.org/10.5518/701). Burst variance analysis (BVA)41 was performed using FRETBursts98, and plots were made using the Seaborn and Matplotlib102 packages. Visualisations of the available volumes for FRET dyes attached at different positions in SurA were generated using the FRET Positioning and Screening (FPS) software with dye linker lengths and radii parameters suggested in the FPS manual for the FRET dyes used39. Predicted EFRET value distributions from the crystal structure of full-length SurA for each dye pair were calculated from distance distributions generated using the MtsslWizard PyMOL plugin103. Raw smFRET data are available at the University of Leeds data repository (https://doi.org/10.5518/701).
Fluorescence anisotropy
Fluorescence anisotropy decay measurements were performed on single Cys variants of SurA with/without OmpX (600 nM SurA with/without 3 µM OmpX), with SurA labelled with Alexa Fluor 488 C5 maleimide or Alexa Fluor C5 594 using a Quantamaster 8000 (Horiba) equipped with a Whitelase supercontinuum pulsed laser (NKT) for excitation with a repetition rate of 10 MHz and TCSPC detection. Three pairs of scans were taken with VV and VH polarisation for each sample (500 µL, 600 nM in 50 mM Tris-HCl pH 8), and a peak height of 10,000 photons was collected for each scan. Normalisation and global fitting of each pair of polarised decay curves along with the IRF and HV and HH polarised decays that defined the G factor was performed using FelixGX v4.9.0.10243 (Horiba). The steady state and time-resolved anisotropy are related by the following expression (9):
$$r = \frac{{r_0}}{{1 + {t /{T_r}}}}$$
where r is the steady state anisotropy, r0 is the initial anisotropy, τ is the fluorescence lifetime and Tr is the rotational correlation timecalculated from the measured decay,
All-atom molecular dynamics simulations of the mature sequence of SurA (residues 21–428) in explicit solvent were performed with GROMACS 5.0.2104 using the CHARMM36 force field105. For simulations starting from the crystal structure of full-length SurA (PDB 1M5Y27), loop residues which are unresolved in the structure were modelled using MODELLER106, and the four missing N-terminal residues were added in Chimera107. The system was minimised (5000 steps) followed by equilibration for 25 ps, with backbone and sidechain position restraints of 400 and 40 kJ mol−1 nm−2, respectively, in the x, y and z directions. The temperature reached its target value (300 K) within the first 10 ps and remained stable for the rest of the equilibration. The system contained 202 sodium ions and 198 chloride ions (150 mM NaCl), and 70,091 TIP3P water molecules. The total number of atoms was 217,001 in a periodic box size of 13.2 nm × 13.2 nm × 13.2 nm. For simulations starting from a SurAcore-P1-open conformation a model was first built using the crystal structures of full-length SurA (PDB 1M5Y27) and SurA-ΔP2 (PDB 2PV352), in which the P1 domain is extended away from the core. The two structures were aligned on the core domain and the P1 and core domains were removed from the full-length SurA structure. Linker residues between domains were added using MODELLER106, and the four missing N-terminal residues were added in Chimera. The system was minimised (5000 steps) followed by equilibration for 25 ps with backbone and sidechain position restraints of 400 and 40 kJ mol−1 nm−2, respectively, in the x, y and z directions. The temperature reached its target value (300 K) within the first 10 ps and remained stable for the rest of the equilibration. The system contained 189 sodium ions and 185 chloride ions (150 mM NaCl), and 64,809 TIP3P water molecules. The total number of atoms was 201,129 in a periodic box size of 12.9 nm × 12.9 nm × 12.9 nm. Simulation systems were built using CHARMM-GUI108. In all simulations the pressure was maintained using a Parrinello-Rahman barostat109 and the temperature was maintained using a Nose-Hoover thermostat110. The temperature of the systems was 300 K and the timestep was 2 fs. Analysis of Cα–Cα distances between residue pairs identified in cross-linking experiments was performed using the 'gmx distance' GROMACS command. Calculations of solvent accessible surface distances (SASDs) made use of JWalk35. MD simulation data, including those starting from the SurA crystal structure, are available at the University of Leeds data repository (https://doi.org/10.5518/701). Included are GROMACS input files, starting structures, reduced MD trajectories and the final structures after 1 μs of simulation.
Simulated annealing calculations were carried out in XPLOR-NIH111. Cross-links were treated as distance restraints with a flat-well energy potential using noePot. A rigid-body calculation (100 calculations in total) was performed, where each domain was treated as a rigid body and residues in the linker regions were given torsion angle degrees of freedom. To ensure that the initial parameters used do not bias the outcome, the starting structures used in each of the 100 simulations were all different, generated by randomly orienting the domains relative to each other. Pseudo-potential energy terms describing covalent geometry restraints were applied to restrict deviation from bond lengths, angles and improper torsion angles. All cross-links were utilised as distance restraints in the 100 calculations. The first step in the structure calculation consisted of 10,000 steps of energy minimization, followed by simulated annealing dynamics with all the potential terms active, where the temperature is slowly decreased (3000–25 K) over 4 fs and a final energy minimization in torsion angle space. During the hot phase (T = 3000 K) the cross-link terms were underweighted to allow the domain to sample a large conformational space and they were geometrically increased during the cooling phase. For each calculation the coordinates of P1 and P2 were randomized by applying a random translation within 20 Å and a random rotation within 90° of their initial positions. The linkers were re-built using torsionDB112 to enforce correct geometry before the first step in the structure calculation protocol. The 100 generated structures were ranked based on their energies, taking into account how well the distance restraints are satisfied (this is the main contributor to the final energy) and covalent geometry/VDW terms to ensure that the selected models do not have any geometry violations. The 10 lowest energy structures were visualised and analysed in further detail. Given this simulated annealing approach will drive the structure to compact states, more extended states of SurA that smFRET data show are populated in solution, will not be captured by this method. The structures of the 10 lowest energy conformations of SurA are available at the University of Leeds data repository (https://doi.org/10.5518/701).
Labelling of Cys-OmpX with Alexa Fluor 488
Purified Cys-OmpX was covalently labelled with Alexa Fluor 488 dye via maleimide chemistry. A sample containing 200 μM Cys-OmpX in 25 mM Tris-HCl, 6 M Gdn-HCl, pH 7.2, was incubated with 10 mM DTT for 30 min. This sample was subsequently buffer exchanged into 25 mM Tris-HCl, 6 M Gdn-HCl, pH 7.2 (that had been sparged for 15 min with nitrogen gas) using Zeba spin desalting columns (Thermo Fisher Scientific). Alexa Fluor 488 C5 maleimide (Thermo Fisher Scientific) (10 mg/mL dissolved in DMSO) was immediately added to the OmpX sample at a final concentration of 2 mM. The total sample volume was 480 µL. The labelling reaction was kept at 25 °C for 1 h then left overnight at 4 °C. The reaction was then loaded onto a Superdex Peptide 10/300 column (GE Healthcare) equilibrated with 6 M Gdn-HCl, 25 mM Tris-HCl, pH 7.2 to remove the excess free dye. Samples were collected every 1 mL and peak protein fractions tested for dye labelling using a Nanodrop 2000 (Thermo Fisher Scientific). Samples containing labelled OmpX were snap-frozen using liquid nitrogen and stored at −80 °C until required.
Microscale thermophoresis (MST)
From a 200 μM SurA stock solution in 50 mM Tris-HCl, pH 8.0, a series of twofold serial dilutions was performed to obtain 16 15-µL samples. Labelled Cys-OmpX was buffer exchanged into 8 M urea, 50 mM Tris-HCl, pH 8.0, to a concentration of 1.7 µM. This stock was diluted 16.6-fold to a concentration of 100 nM with 50 mM Tris-HCl, pH 8.0, then immediately added to the sixteen SurA-containing samples in 15 µL aliquots (30 µL total sample volume). The final sample concentrations were 50 nM Cys-OmpX, 100 µM–3 nM SurA, 0.24 M urea, 50 mM Tris-HCl, pH 8.0. Samples were immediately added to capillaries by capillary action then read using a Monolith NT.115 MST instrument (NanoTemper, Germany). To obtain the dissociation constant, Kd, data were fitted to the Hill Eq. (10):
$$S_{{\mathrm{obs}}} = S_{\mathrm{U}} + \left( {S_{\mathrm{B}} - S_{\mathrm{U}}} \right).\left( {\frac{{[{\mathrm{SurA}}]^n}}{{K_{\mathrm{D}} + [{\mathrm{SurA}}]^n}}} \right)$$
where Sobs is the observed signal, SU is the signal from unbound OmpX, SB is the signal from bound OmpX, and n is the Hill coefficient. Data fitting was carried out using IgorPro 6.3.4.1 (Wavemetrics, Oregon, USA).
Tag-transfer photo-cross-linking
Single Cys variants of OmpX (M41C, I102C, K122C, V167C) were conjugated with MTS-diazirine48. Briefly, Each OmpX variant was buffer exchanged into 6 M guanidine-HCl, 50 mM Tris-HCl, 10 mM DTT, 1 mM EDTA, pH 8.0 that had been sparged with N2. After incubation at room temperature for 15 min, the OmpX variant was buffer exchanged (Zeba Spin Desalting Columns, 7 K MWCO, Thermo Scientific) into the same buffer without DTT. MTS-diazirine (from a stock solution in DMSO) was added in 20-times molar excess over the OmpX variant. Final concentrations were 200 µM OmpX, 4 mM MTS-diazirine, 20% (v/v) DMSO, 4.8 M guanidine-HCl, 40 mM Tris-HCl and 0.8 mM EDTA. This solution was incubated at room temperature for 1 h before being buffer exchanged into 6 M guanidine-HCl, 50 mM Tris-HCl, 1 mM EDTA, pH 8.0. The labelled protein was aliquoted, snap-frozen in liquid nitrogen and stored at −80 °C.
SurA-OmpX complexes were assembled by mixing SurA with each OmpX variant such that the final concentrations of each was 10 µM and 5 µM, respectively, in 10 mM potassium phosphate buffer, pH 8.0, 0.24 M urea. Photo-cross-linking was performed for 30 s using a UV LED irradiation platform48. The cross-linked material was separated by SDS-PAGE. The gel band corresponding to the cross-linked complex was excised and the proteins were trypsinised in-gel (see XL-MS methods above)48. To detect additional modified peptides, reduction of the cross-linker and thiol capture was performed to enrich cross-linked peptides. Peptides (5 µL) were injected onto a reverse-phase Acquity M-Class C18, 75 µm × 150 mm column (Waters) and separated by gradient elution of 1–50% (v/v) solvent B (0.1% (v/v) formic acid in acetonitrile) in solvent A (0.1% (v/v) formic acid in water) over 60 min at 300 nL.min−1. The eluate was infused into a Xevo G2-XS (Waters) mass spectrometer operating in positive ion mode. Mass calibration was performed by infusion of aqueous NaI (2 µg/µL). [Glu1]-Fibrinopeptide B (GluFib) was used for the lock mass spray, with a 0.5 s lock spray scan taken every 30 s. The lock mass correction factor was determined by averaging 10 scans. Data acquisition was performed in DDA mode with a 1 s MS scan over m/z 350–2000. The four most intense ions in the MS spectrum were selected for MS/MS by CID, each with a 0.5 s scan over m/z 50–2000. The collision energy applied was dependent upon the charge and mass of the selected ion. Dynamic exclusion of 60 s was used. Data processing and modification localization was performed using PEAKS Studio 8.5 (Bioinformatics Solutions). Search parameters were as follows: parent mass error tolerance = 10 ppm; fragment mass error tolerance = 0.05 Da, maximum number of missed cleavages = 3; fixed modification = carbamidomethylation (57.02 Da); variable modifications = deamidation (0.98 Da), oxidation of Met (15.99 Da), MTS tag (145.06 Da). A FDR cut-off of 1% was used. A representative mass spectrum for each cross-link can be found in Supplementary Data 1. The raw tag-transfer XL data have been deposited to the ProteomeXchange Consortium via the PRIDE93 partner repository with the dataset identifier PXD016993.
Hydrogen-deuterium exchange mass spectrometry
An automated HDX robot (LEAP Technologies, Ft Lauderdale, FL, USA) coupled to a Acquity M-Class LC and HDX manager (Waters, UK) was used for all HDX-MS experiments. For differential HDX-MS of SurA in the absence and presence of OmpX/OmpF, samples contained 8 µM of SurA or 8 µM of SurA with 8 µM OmpX/OmpF (in 10 mM potassium phosphate, pH 8.0, 0.24 M urea). For differential experiments with addition of WEYIPNV peptide, the samples contained 8 µM SurA with 110 µM peptide (in 10 mM potassium phosphate, pH 8.0). Note that the addition of 0.24 M urea does not dramatically alter the intrinsic rate of exchange113.
Thirty microlitres of protein-containing solution was added to 135 μL deuterated buffer (10 mM potassium phosphate buffer pD 8.0, 0.24 M d4-urea or 10 mM potassium phosphate buffer pD 8.0, 82% D2O) and incubated at 4 °C for 0.5, 2, 30 or 120 min. Four replicate measurements were performed for each condition and each time point. After labelling, HDX was quenched by adding 100 μL of quench buffer (10 mM potassium phosphate, 2 M Gdn-HCl, pH 2.2) to 50 μL of the labelling reaction. Fifty microlitres of the quenched sample was passed through immobilised pepsin and aspergillopepsin columns (Affipro, Mratín, Czech Republic) connected in series (20 °C) and the peptides were trapped on a VanGuard Pre-column [Acquity UPLC BEH C18 (1.7 μm, 2.1 mm × 5 mm, Waters, UK)] for 3 min. The peptides were separated using a C18 column (75 μm × 150 mm, Waters, UK) by gradient elution of 0–40% (v/v) acetonitrile (0.1% v/v formic acid) in H2O (0.3% v/v formic acid) over 7 min at 40 μL min−1. Peptides were detected using a Synapt G2Si mass spectrometer (Waters, UK). The mass spectrometer was operated in HDMSE mode, with dynamic range extension enabled (data independent analysis (DIA) coupled with IMS separation) were used to separate peptides prior to CID fragmentation in the transfer cell114. CID data were used for peptide identification, and uptake quantification was performed at the peptide level (as CID results in deuterium scrambling). Data were analysed using PLGS (v3.0.2) and DynamX (v3.0.0) software (Waters, UK). Search parameters in DynamX were as follows: peptide and fragment tolerances = automatic, min fragment ion matches = 1, digest reagent = non-specific, false discovery rate = 4. Restrictions for peptides in DynamX were as follows: minimum intensity = 1000, minimum products per amino acid = 0.3, max sequence length = 25, max ppm error = 5, file threshold = 3. The software Deuteros115 was used to identify peptides with statistically significant increases/decreases in deuterium uptake (applying a 99% confidence interval) and to prepare Wood's plots. The raw HDX-MS data, have been deposited to the ProteomeXchange Consortium via the PRIDE93 partner repository with the dataset identifier PXD017010. A summary of the HDX-MS data, as recommended by reported guidelines116, is shown in Supplementary Table 9.
Electrostatic surface potential and conservation analyses
Calculation of the surface electrostatic potential of SurA was performed using the APBS plugin for PyMOL117. Amino acid conservation analysis was carried out using the ConSurf webserver using default parameters118.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
MS data, have been deposited to the ProteomeXchange Consortium via the PRIDE93 partner repository with the dataset identifiers PXD016993 (XL-MS) and PXD017010(HDX-MS). Source data for Figs. 4a and 5a are provided with the paper. MD trajectories and the final structures after 1 μs of simulation) and the structures of the 10 lowest energy Raw smFRET data, MD simulation data (including GROMACS input files, starting structures, and conformations of SurA from simulated annealing are freely available at the University of Leeds data repository (https://doi.org/10.5518/701). The source data underlying Fig. 4a, Fig. 5a, Supplementary Fig. 8b, Supplementary Fig. 7 and Supplementary Table 2 are provided as a Source Data file. All other data are available from the corresponding author on reasonable request.
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We thank members of the Radford and Brockwell laboratories for helpful discussions, along with Nasir Khan and James Ault for excellent technical support and Tomas Fessl for advice on FRET data analysis. The plasmid containing the mature sequence of SurA was kindly provided by D. Kahne (Harvard). A.N.C. (BB/P000037/1), B.S. (BB/N007603/1, BB/T000635/1), M.W. (BB/N017307/1) and J.E.H. (BB/M011151/1) acknowledge funding from the BBSRC. M.W. was funded by the EPSRC (EP/N035267/1), J.R.H. was funded by the US National Institutes of Health (GM102829) and PW acknowledges funding from the MRC (MR/P018491/1). The Monolith NT.115 MST instrument was purchased with funding from the Wellcome Trust (105615/Z/14/Z). Funding from the Wellcome Trust (208385/Z/17/Z) and BBSRC (BB/M012573/1) enabled the purchase of mass spectrometry equipment.
Theodoros K. Karamanos
Present address: National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD, 20892, USA
These authors contributed equally: Antonio N. Calabrese, Bob Schiffrin, Matthew Watson.
Astbury Centre for Structural Molecular Biology, School of Molecular and Cellular Biology, Faculty of Biological Sciences, University of Leeds, Leeds, LS2 9JT, UK
Antonio N. Calabrese, Bob Schiffrin, Matthew Watson, Theodoros K. Karamanos, Martin Walko, Julia R. Humes, Jim E. Horne, Paul White, Andrew J. Wilson, Roman Tuma, Alison E. Ashcroft, David J. Brockwell & Sheena E. Radford
Astbury Centre for Structural Molecular Biology, School of Chemistry, University of Leeds, Leeds, LS2 9JT, UK
Martin Walko & Andrew J. Wilson
Astbury Centre for Structural Molecular Biology and School of Medicine, University of Leeds, Leeds, LS2 9JT, UK
Antreas C. Kalli
Faculty of Science, University of South Bohemia, Ceske Budejovice, Czech Republic
Roman Tuma
Antonio N. Calabrese
Bob Schiffrin
Matthew Watson
Martin Walko
Julia R. Humes
Jim E. Horne
Andrew J. Wilson
Alison E. Ashcroft
David J. Brockwell
Sheena E. Radford
A.N.C., B.S., M. Watson and T.K.K. designed and performed the experiments and analysed the data. M.Walko and A.J.W. designed and synthesised the tag-transfer cross-linker. J.R.H., J.E.H. and P.W. designed and purified protein variants. A.C.K., R.T., A.E.A., D.J.B. and S.E.R. supervised the research. The paper was written by A.N.C., B.S., M. Watson and S.E.R. with input and comments from all authors.
Correspondence to Sheena E. Radford.
Peer review information Nature Communications thanks Dirk Linke, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Calabrese, A.N., Schiffrin, B., Watson, M. et al. Inter-domain dynamics in the chaperone SurA and multi-site binding to its outer membrane protein clients. Nat Commun 11, 2155 (2020). https://doi.org/10.1038/s41467-020-15702-1
The role of membrane destabilisation and protein dynamics in BAM catalysed OMP folding
Samuel F. Haysom
Nature Communications (2021)
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Talk:Interval of Ordered Set is Convex
How do we extend the notion of interval to tosets without confusion? --Dfeuer (talk) 09:17, 13 February 2013 (UTC)
I don't know. Admittedly I never use that notion of interval. Also, at the end of the day it's just an order-convex set, no? --Lord_Farin (talk) 09:20, 13 February 2013 (UTC)
It depends on exact definitions. The analogy to the reals gives a bounded order-convex set. I don't know what's the most common definition. --Dfeuer (talk) 09:24, 13 February 2013 (UTC)
The reals surely also admit unbounded intervals... --Lord_Farin (talk) 09:25, 13 February 2013 (UTC)
These beasts be slippery. $\mathsf{Pr} \infty \mathsf{fWiki}$ currently defines Definition:Real Interval/Open to be bounded, defines Definition:Real Interval/Unbounded Open to be what I would call an open ray, and defines a Definition:Real Interval, before going into all these things, to be something that excludes the possibility that it's unbounded. --Dfeuer (talk) 09:31, 13 February 2013 (UTC)
That is indeed the case - because when "real interval" is usually encountered, it is supposed that it is bounded at either end. Thus the default position. The case when one of the endpoints is $\pm \infty$ is less commonly encountered, at least in basic analysis. Hence the more specific definition first, the more general one following - and the most general one of all, where $\R$ itself is considered as a degenerate case of a real interval, added at the end to ensure completeness. As it stands, it works. I would counsel against structural amendments. --prime mover (talk) 12:39, 13 February 2013 (UTC)
I have taken the libery of amending Definition:Real Interval slightly to emphasize that the informal definition has multiple interpretations. This takes weight off the reading that the informal definition imposes any interval to be bounded. --Lord_Farin (talk) 12:59, 13 February 2013 (UTC)
I don't particularly have an opinion, mind you, except I like using similar terminology for similar things. If interval = convex set, then I won't throw a fit..... --Dfeuer (talk) 09:33, 13 February 2013 (UTC)
Interval Defined by Betweenness says it all. --Lord_Farin (talk) 12:59, 13 February 2013 (UTC)
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International Conference on Document Analysis and Recognition
ICDAR 2021: Document Analysis and Recognition – ICDAR 2021 Workshops pp 147–162Cite as
Accurate Graphic Symbol Detection in Ancient Document Digital Reproductions
Zahra Ziran ORCID: orcid.org/0000-0002-3529-738010,
Eleonora Bernasconi ORCID: orcid.org/0000-0003-3142-308410,
Antonella Ghignoli ORCID: orcid.org/0000-0001-7399-055X10,
Francesco Leotta ORCID: orcid.org/0000-0001-9216-850210 &
Massimo Mecella ORCID: orcid.org/0000-0002-9730-888210
First Online: 04 September 2021
Part of the Lecture Notes in Computer Science book series (LNIP,volume 12916)
The original version of this chapter was revised: chapter 12 made as open access. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-86198-8_34
Digital reproductions of historical documents from Late Antiquity to early medieval Europe contain annotations in handwritten graphic symbols or signs. The study of such symbols may potentially reveal essential insights into the social and historical context. However, finding such symbols in handwritten documents is not an easy task, requiring the knowledge and skills of expert users, i.e., paleographers. An AI-based system can be designed, highlighting potential symbols to be validated and enriched by the experts, whose decisions are used to improve the detection performance. This paper shows how this task can benefit from feature auto-encoding, showing how detection performance improves with respect to trivial template matching.
Graphic symbol detection
This research is part of the project NOTAE: NOT A writtEn word but graphic symbols, which has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Advanced Grant 2017, GA n. 786572, PI Antonella Ghignoli). See also http://www.notae-project.eu. Copyright 2021 for this paper by its authors.
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A huge number of historical documents from Late Antiquity to early medieval Europe do exist in public databases. The NOTAE project (NOT A writtEn word but graphic symbols) is meant to study graphic symbols, which were added by authors of these documents with several different meanings. This task is very different though from processing words and letters in natural language as the symbols that we look for can be orthogonal to the content, making contextual analysis useless.
Labeling document pictures with positions of graphic symbols even in an unsupervised manner requires the knowledge of domain experts, paleographers in particular. Unfortunately, this task does not scale up well considering the high number of documents. This paper proposes a system that helps curators to identify potential candidates for different categories of symbols. Researchers are then allowed to revise the annotations in order to improve the performance of the tool in the long run.
A method for symbol detection has already been proposed in the context of the NOTAE project [1]. Here, the authors have created a graphic symbols database and an identification pipeline to assist the curators. The symbol engine takes images as input, then uses the database objects as queries. It detects symbols and reduces noise in the output by clustering the identified symbols. Before this operation, the user is required to decide the binarization threshold from a prepared selection.
The approach proposed in this paper has several differences with the original one. First of all, in this new version of the tool, we rely on OPTICS [2], instead of DBSCAN [3], for clustering purposes. OPTICS uses the hyper-parameters MaxEps, and MinPts almost the same way as DBSCAN, but it distinguishes cluster densities on a more continuous basis. In contrast, DBSCAN considers only a floor for cluster density and filters noise by identifying those objects that are not contained in any cluster. In addition, our proposed pipeline implements an algorithm that sorts the objects of a cluster by confidence scores and selects the top match. So, the pipeline can control the number of predictions over different types of graphic symbols.
Moreover, the first tool has shown an high number of false positive, which are difficult to filter out. Here, we show how automatic detection of symbols can benefit from feature auto-encoding, showing how detection performance improves with respect to trivial template matching.
The paper is organized as it follows. Section 2 summarizes prior work on document analysis and digital paleography tools, metric learning, and graphic symbols spotting. Section 3 present our data cleaning process and image pre-processing tailored to the specific data domain, i.e., ancient documents. Section 4 covers the inner details of the proposed method. Section 5 show experimental results. Finally, Sect. 6 concludes the paper with a final discussion.
2 Related Work
In [7], the authors discuss the recent availability of large-scale digital libraries, where historians and other scientists can find the information they need to help with answering their research questions. However, as they state, researchers are still left with their traditional tools and limitations, and that is why they propose two new tools designed to address the need for document analysis at scale. Firstly, they consider a tool to match handwritings which is applied to documents that are fragmented and collected across tens of libraries. They also note the shortcomings of computer software in recommending matching scores without providing persuasive and satisfactory reasoning for researchers, as the ground truth is itself the subject of study and active research. Secondly, they mention a paleographic classification tool that recommends matching styles and dates with a given writing fragment. According to them, it seems like paleography researchers are interested in the why of recommender systems outputs as much as they value their accuracy.
Variational auto-encoders (VAE) [8] train a model to generate feature embeddings under the assumption that samples of a class should have the same embeddings. In [9], given the intra-class variance such as illumination and pose, the authors challenge that assumption. They believe that minimizing a loss function risks over-fitting on training data by ignoring each class's essential features. Moreover, by minimizing the loss function, the model could learn discriminative features based on intra-class variances. Also, they illustrate how the model struggles to generalize as samples from different classes but with the same set of intra-class variances cluster at the central part of the latent space. In addition to the KL-divergence [10] and reconstruction loss terms as in prior work, they add two metric-based loss terms. One of the new terms helps minimize the distance between samples of the same class in the presence of intra-class variations. The other new loss term prevents intra-class variances, which different classes might share, overpower essential features representing representational value.
Their framework, deep variational metric learning (DVML), disentangles class-specific discriminative features from intra-class variations. Furthermore, per their claim, it significantly improves the performance of deep metric learning methods by experimenting on the following datasets: CUB-200-2011, Cars196, and Stanford Online Products. In this work, we sample from the latent space by calculating the Kaiming-normal function, also known as He initialization [11], and we use that as epsilon to relate the mean and variance of the data distribution.
In [12], the authors focus on the problem of symbol spotting in graphical documents, architectural drawings in particular. They state the problem in the form of a paradox, as recognizing symbols requires segmenting the input image. The segmentation task should be done on a recognized region of interest. Furthermore, they want a model that works on digital schematics and scanned documents where distortions and blurriness are natural. Moreover, they also aim to build an indexing system for engineers and designers who might want to access an old document in an extensive database with a given symbol drawing that could only partially describe the design. Having those considerations in mind, the authors then propose a vectorization technique that builds symbols as a hierarchy of more basic geometric shapes. Then, they introduce a method for tiling the input document picture in a flexible and input-dependent manner. Their approach approximates arcs with low poly segments [13, 14], and puts constraints on subsets of line segments such as distance ratios, angles, scales, etc. This way, they can model slight variations in the way that symbol queries build full representational graphs.
Related tables in the NOTAE database
3 Data Preprocessing
3.1 Scraping Public Databases
With their expertise and knowledge of the domain, the NOTAE curators have gathered a database to find documents, hands, symbols, and digital reproductions, among much other useful information. The database tables are connected as a knowledge graph [15] (see Fig. 1). For example, a symbol present inside a script has an associated hand. The hand, in turn, comes from a document with an identification number. Then, we can get the digital reproduction of where the symbol comes from using the document ID.
3.2 Cleaning Duplicates
One of the implicit assumptions in dataset design is that sample images are unique. Scraped data is not clean, and it is likely to have duplicates. Web pictures come from different sources with different sizes and compression algorithms for encoding/decoding. So, comparing an image against the rest of the dataset to determine if it is a duplicate will not work most of the time. Using coarser features instead of raw pixels can destroy many trivial details that are not noticeable to the human eyes. Moreover, it is observed to be the case that some digital reproductions of the same picture have been ever so slightly cropped across public databases. So, the features need to be invariant under minor variations in data distribution but different enough between two unique pictures. The difference hash (dHash) algorithm [16] processes images and generates fixed-length hashes based on visual features. dHash has worked outstanding for our use case. In particular, generating 256-bit hashes and then using a relative Hamming distance threshold of 0.25 detects all duplicates. Among duplicate versions of scraped data, we chose the one with a higher resolution. By comparing image hashes against each other, we managed to clean the scraped data and thus create two datasets of unique samples such as graphic symbols and digital reproductions.
The effect of the proposed solution. On the left the result without cleaning duplicates, on the right after the cleaning operation.
Figure 2 in particular shows the difference with respect to the quality of obtained results.
3.3 Binarization
Digital reproductions contain various supports such as papyrus, wooden tablets, slate, and parchment. In addition, due to preservation conditions and the passage of time, parts of the documents have been lost, and we deal with partial observations of ancient texts and symbols. Accordingly, a pre-processing step seems necessary to foreground the handwritten parts and clear the background of harmful features and noise. Then, the issue of what threshold works best for such a diverse set of documents surfaces. In that regard, we follow the prior work [1] and hand-pick one value out of the five prepared threshold values that are input dependent. We find the first two of the threshold values by performing K-means clustering on the input image and then choosing the red channel, which is the most indicative value. Next, we calculate the other three thresholds as linear functions of the first two (taking the average, for example).
Template matching works on each color channel (RGB) separately, and so it returns three normalized correlation values. Consequently, the proper peak-finding function should take the average of them in order to find the location of the most probable box (see further in Sect. 4.2 for more on peak-finding in template matching). However, since document pictures have a wide range of supports with various colors and materials, using color images is optimal, whereas binary images work the best. First, we remove the background using the selected threshold value. Next, we apply the erosion operator to remove noise and the marginal parts further. Finally, we fill the foreground with the black color to get the binary image. In our experiments, the binarization step has proven to be at least an order of magnitude more effective in reducing false positives, compared to when we tried color images.
3.4 Dataset Design
The simple baseline begins with the binarization of document pictures and template matching using the NOTAE graphic symbols database. These two steps make for an end-to-end pipeline already and identify graphic symbols with a given picture (see Fig. 3.) Next, we split our preferred set of unique binarized digital reproductions into three different subsets: train, validation, and test. The partitioning ratio is 80% training data and 10% for each test and validation subsets.
3.5 Initial Symbol Clustering
As discussed in the introduction, in this new version of the annotation tool we moved from DBSCAN to OPTICS for symbol clustering. A description of how OPTICS forms denser clusters follows. First, it defines core-distance as the minimum distance within the Eps-neighborhood of an object such that it satisfies the MaxEps condition. In general, core-distance is less than MaxEps, and that is why there is a MaxEps rather than a fixed Eps in OPTICS. Then, it uses core-distance to define the reachability score as a function of one object concerning another. The reachability of object o with respect to a different object p is defined to be the maximum between either of two values: the core-distance of o or the distance between o and p. Reachability is instead not defined if objects o and p are not connected. Using a cluster expansion loop with the given core distances, OPTICS can reorder database objects between subclusters and superclusters where cluster cores come earlier and noise later. Object ordering plus reachability values prove to be much more flexible than a naive cluster-density condition in the way DBSCAN works.
Identification pipeline
4 Modeling Approach
Our target is to determine very particular graphic symbols in a digital reproduction and find their positions as smaller rectangles inside the picture frame. The NOTAE database supplies the templates we look for, so the simplest possible model can be a template matching algorithm. It takes a picture and a set of templates as inputs and returns a set of bounding boxes and the confidence scores assigned to each one of them as outputs. Then, one could select the final predictions from the top of the boxes sorted based on their scores. However, in practice, we observed that the simple model also returns too many false positives, bounding boxes with relatively high confidence scores but contain non-symbols. Moreover, the rate of false positives increases as a linear function of the database size. This inefficiency in naive template matching poses a problem since the NOTAE system design relies on the growth of the database for making its suggestions brighter. So, template matching is a simple and fast model for identifying graphic symbols in a document picture, but it has relatively limited precision.
In the identification pipeline, the template matching step is done for every graphic symbols database object. For one object (template), the algorithm returns a field of correlation densities over the input document picture, as many as the number of pixels in the given picture. So we select the one with the maximum score as the final match. Also, template matching uses five different sizes of each object. The scales range from 5% of the picture width up to 20% because that is about the size of the symbols in documents. Hence, the first step of the pipeline produces five bounding boxes per database object.
After template matching is over, we can recover some precision by way of updating confidence scores. Fast template matching is possible by transforming visual data from the spatial dimension to the frequency dimension. One can ignore some high-frequency features to speed up the process and then transform the results back to the spatial dimension. In [17] Fourier transform does so to reduce computation complexity. However, once the algorithm has queried the database and is done with its prediction process, then we can afford to update the confidence scores using a more computationally complex approach that would be quite infeasible right from the beginning. We engineer visual features for both database objects and regions of interest (ROI) for that purpose, as template matching predicts. Suppose u and v are two such features extracted using a method of our choice (u represents a template while v represents an identification ROI, for example.) If we find the lengths of these feature vectors then it becomes easy to see how similar they are:
$$correlation = \frac{<u, v>}{|u|\cdot |v|},$$
\(<\cdot ,\cdot>\) denotes the inner product on the vector space of features, \(|\cdot |\) denotes the length of a vector and the correlation is in the closed interval \([-1,1]\).
Due to reasons that will be discussed later in this section, we can build features in a particular latent space to preserve the save the same metric from the previous step. In fact, we propose to build a discriminator that uses the correlation between features to update the prediction probabilities and prune away false positives.
We already identified potential candidates for graphic symbols in a document picture, then discriminated against some of them based on engineered features, and finally, filtered outliers based on size. However, all those steps pertain to more individual and local symmetries rather than considering what an ensemble of identifications has in common. That is where clustering and unsupervised classification comes into play and further reduce false positives. Using the same engineered features, be it histogram of oriented gradients (HOG) or learned embeddings, a clustering algorithm can group the identified symbols into one cluster and label the rest of the identifications as noise. In this last major step to improve the results, global symmetries are the main deciding factor as to whether a bounding box should be in the graphic symbols group or not. In the clustering step, individual boxes relate to each other via a distance function. Setting a minimum neighborhood threshold, clusters of specific densities can form, as discussed in the previous section. At the end of every promising identification pipeline, they apply a non-maximum suppression algorithm. In overlapping bounding boxes, those with lower confidence scores are removed in favor of the top match. See Fig. 3 for a representation of our identification pipeline.
4.1 Updating Identification Probabilities
Suppose T, F, M and D be events: T as the event that a box is true positive, F as the event that a box is false positive, M as the event that the template Matching model labels a box as true positive, and D is the event that the Discriminator model labels a box as true positive. Also, suppose MD be the event that both the template Matching and Discriminator models label a box as true positive.
Please note that the sum of prior probabilities should be equal to one.
$$P(T) + P(F) = 1.$$
Next, let's appeal to the Bayes theorem. In the symbol identification task, write down the posterior probabilities of such events occurring:
$$P(T | MD) = \frac{P(MD | T) \cdot P(T)}{P(MD | T) \cdot P(T) + P(MD | F) \cdot P(F)},$$
or, in an equivalent way:
$$P(T | MD) = \frac{P(MD | T) \cdot P(T)}{P(MD)}.$$
First, the template matching model acts on the graphics symbols database. The input document picture is implicit here as it stays constant throughout the pipeline. Then, the template matching model returns one match per pixel in the document picture. A suitable cut-off threshold as a hyper-parameter will reduce the number of symbols based on the confidence scores. So, we only select the top match for each database object (template). Next, the discriminator model acts on the top matches. Furthermore, thus the template-matching model and the discriminator model participate in a function composition at two different levels of abstraction. In this composition, template matching works with raw pixels, whereas discrimination works with high-level embedding vectors.
$$updated \ scores = Discriminate \circ Match(database),$$
where \(\circ \) denotes the function composition by first applying Match and then Discriminate on the database, object by object.
If we assume that events M and D are independent (or slightly correlated), then we can say that they are conditionally independent given T.
$$P(MD | T) = P(M | T) \cdot P(D | T)$$
Therefore the updated probability will be:
$$P(T | MD) = \frac{P(M | T) \cdot P(D | T) \cdot P(T)}{P(MD)}$$
Performing some computation to simplify the posterior probability:
$$P(T | MD) = \frac{P(M | T) \cdot P(DT)}{P(MD)}$$
$$P(T | MD) = \frac{P(M | T) \cdot P(T | D) \cdot P(D)}{P(MD)}$$
$$P(T | MD) = \frac{P(M | T) \cdot P(T | D)}{Q(1, 2)},$$
where \(Q(1, 2) = \frac{P(MD)}{P(D)}\).
Since \(1 = P(T | MD) + P(F | MD)\), therefore:
$$1 = \frac{P(M | T) \cdot P(T | D) + P(M | F) \cdot P(F | D)}{Q(1, 2)}.$$
Now, it is obvious that
$$Q(1, 2) = P(M | T) \cdot P(T | D) + P(M | F) \cdot P(F | D)$$
And that conclusion implies that the updated probability is as follows:
$$\begin{aligned} P(T | MD) = \frac{P(M | T) \cdot P(T | D)}{P(M | T) \cdot P(T | D) + P(M | F) \cdot P(F | D)} \end{aligned}$$
Q: Where do we get the value P(M|T) from? A: The Average Recall (AR) of the template matching function gives the value for P(M|T). It is the probability that the fast template matching algorithm identifies a symbol given that it is a true symbol. Q: Where do we get the value P(T|D) from? A: The Average Precision (AP) of the discriminator function gives the value for P(T|D). It is the probability that a symbol is true given that the discriminator model has labeled it positive. Q: What does P(M|F) mean? A: It is the probability that the template-matching model identifies a symbol given that it is negative. Q: What does P(F|D) mean? A: It is the probability that a symbol is false given that the discriminator model has labeled it positive.
The template matching model produces potential bounding boxes in a digital reproduction with the graphic symbols database. Next in the pipeline, we use an attention mechanism to discriminate for the boxes that are more likely to be true with the given digital reproduction. The discriminator is indifferent to the location of the query symbol and only cares about whether the matching box is similar to it or not. Therefore, the discrimination step is an image classification task in essence. Figure 4 shows how the two steps, symbol matching, and classification, share the same database objects. The discrimination model, step 5 in Fig. 3, introduces a posterior probability function P(T|D) and assigns to every box a value from −1 to 1. The sequential update of information now changes first to consider event M and then update with event D.
Filtering noise with low overhead as the inference has lower latency.
Finally, we can normalize the discrimination confidence score by adding one unit and dividing it by 2 to get a correct probability value in [0, 1], formally known as an affine transformation. Next, we use it to replace the score from the template matching step. The posterior probability P(T|MD) is correlated to the scores coming from both steps: template matching and discrimination. The rest of the pipeline will work the same (see Fig. 5). In the following subsection, we are going to use this result to focus on the feature engineering that maximizes P(T|D), that is, the true positive rate has given the second event, discrimination.
Identifications on the left side and ground truth on the right.
4.2 Latent Clustering
By now, we have established the probability that a graphic symbol is true given that the discriminator model has labeled it positive works based on a correlation between the source symbol and the target identification. As indicated, we need to look more closely at the choice of metric and distance functions. Because the more accurate we are in determining the actual distance between two objects, the better we can reason about if the two objects in question are related and why.
Suppose the distribution of the graphic symbols database is described by manifold M. Here, we do not assume any structure beyond that there is a probability p(x) that we discover a given object x in it. Except for maybe a smooth frame at x for applying convolutional filters. Since it is a complex manifold, as is the case with most objects in the real world, it could be intractable to explain with a reasonable amount of information. Therefore, we defer to a latent manifold \(\tilde{M}\) which is finite-dimensional and could potentially explain the most important aspects that we care about in objects from M. What we need here is a map, such as \(\phi \), from manifold M into manifold \(\tilde{M}\) such that our choice of metric in the latent manifold \(\tilde{M}\) results in a predictable corresponding metric in the original manifold M.
Accordingly, we could reason unseen objects knowing that for every input in the domain of graphic symbols manifold, there will be a predictable output in the co-domain of the latent manifold. Predictable in the sense that our metric in the latent space would work as expected. In this context, the encoder model plays the part of the inverse of a smooth map. It maps objects from the pixel space onto the latent space.
$$Encode : pixel \ space \mapsto latent \ space$$
Suppose that p and v are vector representations of an ROI (inside a document picture) and a graphic symbol, respectively. Next, we define a few smooth maps for computing the probabilities of our modeling approach.
$$P(M | T) := \mathop {\mathrm {arg}\,\mathrm {max}}\limits _{i,j} Match(p_{i,j}, v),$$
given by
$$Match(p_{i,j}, v) = p_{i,j} *v\,=\,{<}\hat{p_{i,j}}, \hat{v}{>},$$
The inner product between normalized elements from the template matching sliding window at (i, j) of the input picture and normalized database elements makes sense if both vector spaces are of the same actual dimension. Here i and j are the maximum arguments of the term on the right, which reflect our process of selecting the top match based on confidence scores. We take the maximum value among the inner products so that it corresponds to the most probable location in the document picture.
$$Discriminate(p_{i, j}, v) := P(T | D),$$
$$Discriminate(p_{i,j},v) =\,{<}Encode(p_{i,j}), Encode(v){>}.$$
For taking symbols from the pixel space to the latent space (embeddings), we can use the encoder part of a variational auto-encoder (VAE) model. We trained a VAE model on the graphic symbols database in a self-supervised manner to get the embeddings of unseen symbols. The model uses a deep residual architecture (ResNet18 in Fig. 6) [18] and the bottleneck in this neural network would be the latent layer where the features are sampled from.
The VAE encodes graphic symbols, upper row, and decodes them, lower row.
4.3 Optimization Objective
The loss function should look like the following equation since according to equation (1) from earlier in this section, we want the training objective to minimize P(M|F) and P(F|D) while maximizing P(M|T) and P(T|D) (up to a proxy function.)
$$\mathcal {L} = \alpha \cdot reconstruction + \beta \cdot [KL \ divergence],$$
where, \(\alpha \) and \(\beta \) are hyper-parameters in \(\mathbb {R}\). The reconstruction loss term above is the mean square error of the input image and its decoded counterpart. A point in the latent space should be similar to a sample from the normal distribution if we want the model to learn a smooth manifold. When the latent distribution and the normal distribution are the most alike, the KL-divergence loss term should be approximately equal to zero. Adding the relative entropy loss term to the loss function justifies our assumption on the learned manifold being a smooth one.
5 Quantifying Model Performance
In order to perform evaluation, it is helpful to imagine the annotation tool as a generic function that maps elements from an input domain to the output. In our case, in particular, we want to map tuples of the form (document_picture, symbol) to a bounding box array. As evaluation method, we employed mean Average Precision (mAP) [5], which outputs the ratio of true symbols over all of the identified symbols.
Additionally, we annotated the dataset using the Pascal VOC [6] format in order to evaluate the system using well-established tools.
We used an object detection model by the moniker CenterNet ResNet50 V2 512 \(\times \) 512 [19], which was pre-trained on the MS COCO 17 dataset [20]. It is a single-stage detector that has achieved 29.5% mAP with COCO evaluation tools. In order to repurpose it for our work, we generated annotations for 183 unique digital reproductions using our pipeline and then fine-tuned the object detection model on the annotated data. It is not so easy to measure how helpful our approach is using offline training as the model outputs have to be first justified by the model and then interpreted and validated by domain experts. Therefore, the evaluation protocol in this section merely focuses on the coherency and accuracy of the results. The different variants of the identified symbols datasets are partitioned with different ratios and random seeds, so they also serve as a multi-fold testing apparatus. This section considers improvements in precision since it is normal for symbol spotting methods to perform well in terms of recall.
Improvements in mAP validate the pipeline. The horizontal line is the baseline.
In the spirit of an iterative pipeline design, we generated seven different identified symbols datasets. Using roughly the identical digital reproductions and graphic symbols validated our data and modeling approach. For the baseline, we bypassed steps 3 and 5 in the pipeline (Fig. 3) and also used HOG features to have a model as close as possible to prior work [1]. Next, we used the encoder with a binary classifier and generated mark 3. This modification puts steps 3 and 5 of the pipeline into effect. We have compared the evaluation results of MK3 with that of the baseline model, which is about double the precision, suggesting the effectiveness of the discrimination step in improving the true positive rate. Mark 5 follows the same architecture as mark 3. However, it adds discrimination based on bounding box area and foreground density after discrimination with posterior probability, which further improved the results (compare the third and the fourth columns in Table 1).
Then, we modified the pipeline by training the encoder and hard-wiring a discriminator function to calculate posterior probabilities using cosine similarity. The object detection model trained on the identified symbols mark 6 dataset yielded new evaluation results. MK6 annotations look much better than their predecessors in a qualitative way. Interestingly, MK6 annotations seem to generalize well over different scales (see the bottom image in Fig. 5), as it is the first dataset among the series to identify small symbols as well.
The evaluation of MK3 was when we picked up on the trend that we could gain model performance by focusing more on the data rather than the model. By manually labeling the binarized version of the graphic symbols database, we excluded almost half of the objects as non-symbols to get to a dataset of 722 graphic symbols. So, we should attribute some of the improvements over the baseline model to the data cleaning process. That process called for training the auto-encoder model again with the clean data. Table 2 brings the final improvement rates over the baseline with MK3, MK5, and MK6. We added the validation set to Table 2 and Fig. 7 in order to show that our approach is not sensitive to the choice of hyper-parameters. Because test results are strongly correlated with validation. MK5 performs at least twice better than the baseline, and so it is a good candidate to replace it as a new baseline. So, we expect it to perform as well on unseen data. The following relation allows us to calculate the relative change in mAP:
$$relative \ change \ in \ mAP = \frac{proposed \ mAP - baseline \ mAP}{baseline \ mAP} \cdot 100\%.$$
Table 1. Symbol identification performance results related to the identified symbols datasets: the baseline, mark 3, mark 5 and mark 6 (all evaluated on their respective test sets at training step 2000.)
Table 2. Guiding the identification pipeline design by measuring the relative change in mAP, dataset to dataset.
Table 2 presents the relative change in mAP while Table 1 puts the main challenge metric into its proper context. As an illustration, mark 5 outperforms the baseline by \(102\%\) and \(119\%\) in the validation and test subsets, respectively.
In this paper, we have shown how the detection scores provided by fast template matching can be the key to annotate extensive databases in an efficient way. In previous work, the idea is that the bigger the database grows, the more brilliant the symbol engine gets. However, more significant databases also cause more false positives due to inefficiencies in template matching. In this work, we first removed duplicates and then hand-picked binarized versions of the scraped images. Then, a series of identified graphic symbols datasets to validate our hypotheses on data and modeling was designed. The confidence scores of symbol matching using a binary classifier where the discriminative features sampled from the latent space as an approximation of the original space updated. Next, we justified our assumptions about the effectiveness of our distance function in providing a metric for filtering false positives. Not only we managed to recover results from the baseline model, but also there was a significant improvement in model performance across validation and test subsets. Even though many false positives make it through the final stage of the pipeline, we illustrated how a trained detection model generalizes well on the annotated data and why it solves the paradox of segmenting for spotting or spotting for segmentation. Our approach applies to intelligent assistants for database curators and researchers. In a domain where labeled data is scarce, we have adopted evaluation metrics that enable researchers to quantify model performance with weakly labeled data.
The fact that modifications to the pipeline have a clear impact on model performance regarding the relative change in mAP helps define a reward function. Based on the behavior of model performance, we believe that the relative change in mAP could introduce a new term to the loss function. In future work, we would like to see agents that can use this metric to fill in the gaps between sparse learning signals from domain experts during interactive training sessions.
Chapter "Accurate Graphic Symbol Detection in Ancient Document Digital Reproductions" was previously published non-open access. It has now been changed to open access under a CC BY 4.0 license and the copyright holder updated to 'The Author(s)'.
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Sapienza Università di Roma, Rome, Italy
Zahra Ziran, Eleonora Bernasconi, Antonella Ghignoli, Francesco Leotta & Massimo Mecella
Zahra Ziran
Eleonora Bernasconi
Antonella Ghignoli
Francesco Leotta
Massimo Mecella
Correspondence to Zahra Ziran .
Editor information
Editors and Affiliations
Boise State University, Boise, ID, USA
Dr. Elisa H. Barney Smith
Indian Statistical Institute, Kolkata, India
Umapada Pal
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Ziran, Z., Bernasconi, E., Ghignoli, A., Leotta, F., Mecella, M. (2021). Accurate Graphic Symbol Detection in Ancient Document Digital Reproductions. In: Barney Smith, E.H., Pal, U. (eds) Document Analysis and Recognition – ICDAR 2021 Workshops. ICDAR 2021. Lecture Notes in Computer Science(), vol 12916. Springer, Cham. https://doi.org/10.1007/978-3-030-86198-8_12
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Quantum Decoherence of Gaussian Steering and Entanglement in Hawking Radiation and Thermal Bath
Shu-Min Wu ORCID: orcid.org/0000-0002-9175-58681,2,3 &
Hao-Sheng Zeng1,2,3
International Journal of Theoretical Physics (2020)Cite this article
We study the effects of Hawking radiation and bath temperature on quantum steering and entanglement for a two-mode Gaussian state exposed in the background of a black hole and immersed in the two independent thermal baths. We find that both the effects can destroy the quantum steering and entanglement. Quantum steering always exists sudden death for any Hawking temperature and any bath temperature, but entanglement does not in zero-temperature thermal bath. Both the Hawking radiation and the asymmetry of thermal baths can induce the asymmetry of quantum steering, but the latter effect is much weaker than the former. An unintuitive result is that the observer who stays in the Hawking radiation or in the thermal bath with higher temperature has more stronger steerability than the other one. We also find that Hawking radiation and thermal noise can change the asymptotic behavior of steering and entanglement versus the squeezing parameter.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11275064), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20124306110003), and the Construct Program of the National Key Discipline.
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, 410081, China
Shu-Min Wu
& Hao-Sheng Zeng
Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha, 410081, China
Department of Physics, Hunan Normal University, Changsha, 410081, China
Search for Shu-Min Wu in:
Search for Hao-Sheng Zeng in:
Correspondence to Hao-Sheng Zeng.
Here we present the solving process of master equation (16). We first consider the case of single mode bosonic field. Defining the covariance matrix
$$ \sigma(t) = \left( {\begin{array}{ll} \sigma_{xx}(t) & \sigma_{xp}(t) \\ \sigma_{px}(t) & \sigma_{pp}(t) \end{array}} \right), $$
for the quadrature operators \(\hat {x}\) and \(\hat {p}\) of the bosonic mode, we obtain from (16) the system of equations [51, 53, 54]
$$ \frac{\mathrm{d}\sigma (t)}{\mathrm{d}t}=W_{1}\sigma(t)+\sigma(t)W^{\mathrm{T}}_{1}+2D_{1}, $$
$$ W_{1} = \left( {\begin{array}{cc} -(\lambda_{1}-\mu_{1}) & 1 \\ -1 & -(\lambda_{1}+\mu_{1}) \end{array}} \right) $$
with dissipative constant λ1. The diffusion matrix D1 may be determined by the asymptotic condition, which leads to
$$ D_{1}=-\frac{1}{2}[W_{1}\sigma(\infty)+\sigma(\infty)W^{\mathrm{T}}_{1}]. $$
The solution of (23) can be written as
$$ \sigma(t)=X_{1}(t)\sigma(0)X^{\mathrm{T}}_{1}+Y_{1}(t), $$
$$ Y_{1}(t)=-X_{1}(t)\sigma(\infty)X^{\mathrm{T}}_{1}+\sigma(\infty). $$
Where the matrix \(X_{1}(t)=\exp (W_{1}t)\) fulfils the asymptotic condition \(\lim _{t\rightarrow \infty }X_{1}(t)=0\). In the underdamped case with ω1 > μ1, X1(t) is given by (19).
If the bosonic mode is finally in the thermal equilibrium at temperature T1, then the asymptotic state is described by the Gibbs state,
$$ \sigma(\infty) = \frac{1}{2}\coth\frac{1}{2T_{1}}\left( {\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}} \right). $$
For the two-mode Gaussian state (15) considered in the text, with each mode coupled to its own thermal bath, the evolution of the compound system is then given by (18), and the compound Gaibbs state is given by (20).
In our model, T1 denotes the temperature of the thermal bath in the flat spacetime, and T2 denotes the temperature of the thermal bath outside the horizon of the black hole. The bath temperature T2 would be enhanced by the Hawking radiation compared with the flat spacetime. Formally, T2 is determined by the total number of photons \(\bar {n}_{tot}\) through the expression \(\bar {n}_{tot}=[e^{h\varOmega /kT_{2}}-1]^{-1}\), where Ω is the frequency of the bosonic mode, and \(\bar {n}_{tot}=\bar {n}_{the}+\bar {n}_{rad}\), with \(\bar {n}_{the}\) being the thermal photon number irrelevant to Hawking radiation and \(\bar {n}_{rad}\) the photon number of Hawking radiation.
Wu, S., Zeng, H. Quantum Decoherence of Gaussian Steering and Entanglement in Hawking Radiation and Thermal Bath. Int J Theor Phys (2020) doi:10.1007/s10773-019-04372-5
Gaussian steering
Gaussian entanglement
Hawking radiation | CommonCrawl |
On a complete set of operations for factorizing codes
Clelia De Felice
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2006)
Volume: 40, Issue: 1, page 29-52
It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set 𝒪 of operations exists such that each factorizing code can be obtained by using the operations in 𝒪 and starting with prefix or suffix codes. 𝒪 is named here a complete set of operations (for factorizing codes). We show that composition and substitution are not enough in order to obtain a complete set. Indeed, we exhibit a factorizing code over a two-letter alphabet A = { a , b } , precisely a 3 - code, which cannot be obtained by decomposition or substitution.
Felice, Clelia De. "On a complete set of operations for factorizing codes." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 40.1 (2006): 29-52. <http://eudml.org/doc/245420>.
@article{Felice2006,
abstract = {It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set $\{\mathcal \{O\}\}$ of operations exists such that each factorizing code can be obtained by using the operations in $\{\mathcal \{O\}\}$ and starting with prefix or suffix codes. $\{\mathcal \{O\}\}$ is named here a complete set of operations (for factorizing codes). We show that composition and substitution are not enough in order to obtain a complete set. Indeed, we exhibit a factorizing code over a two-letter alphabet $A = \lbrace a,b\rbrace $, precisely a $3-$code, which cannot be obtained by decomposition or substitution.},
author = {Felice, Clelia De},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {variable length codes; formal languages; factorizations of cyclic groups; free monoid generated by a finite alphabet},
publisher = {EDP-Sciences},
title = {On a complete set of operations for factorizing codes},
url = {http://eudml.org/doc/245420},
AU - Felice, Clelia De
TI - On a complete set of operations for factorizing codes
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PB - EDP-Sciences
AB - It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set ${\mathcal {O}}$ of operations exists such that each factorizing code can be obtained by using the operations in ${\mathcal {O}}$ and starting with prefix or suffix codes. ${\mathcal {O}}$ is named here a complete set of operations (for factorizing codes). We show that composition and substitution are not enough in order to obtain a complete set. Indeed, we exhibit a factorizing code over a two-letter alphabet $A = \lbrace a,b\rbrace $, precisely a $3-$code, which cannot be obtained by decomposition or substitution.
KW - variable length codes; formal languages; factorizations of cyclic groups; free monoid generated by a finite alphabet
UR - http://eudml.org/doc/245420
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variable length codes, formal languages, factorizations of cyclic groups, free monoid generated by a finite alphabet
Theory of computing
68Q45
Formal languages and automata
Abelian groups
Finite abelian groups [For sumsets, see and ]
Communication, information
Prefix, length-variable, comma-free codes
Articles by Clelia De Felice | CommonCrawl |
Word contexts enhance the neural representation of individual letters in early visual cortex
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Representational similarity analysis reveals task-dependent semantic influence of the visual word form area
Xiaosha Wang, Yangwen Xu, … Yanchao Bi
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Fang Wang, Blair Kaneshiro, … Bruce D. McCandliss
Deaf readers benefit from lexical feedback during orthographic processing
Eva Gutierrez-Sigut, Marta Vergara-Martínez & Manuel Perea
Micha Heilbron1,2,
David Richter ORCID: orcid.org/0000-0002-3404-83741,
Matthias Ekman ORCID: orcid.org/0000-0003-1254-13921,
Peter Hagoort ORCID: orcid.org/0000-0001-7280-75491,2 &
Floris P. de Lange1
Nature Communications volume 11, Article number: 321 (2020) Cite this article
Visual context facilitates perception, but how this is neurally implemented remains unclear. One example of contextual facilitation is found in reading, where letters are more easily identified when embedded in a word. Bottom-up models explain this word advantage as a post-perceptual decision bias, while top-down models propose that word contexts enhance perception itself. Here, we arbitrate between these accounts by presenting words and nonwords and probing the representational fidelity of individual letters using functional magnetic resonance imaging. In line with top-down models, we find that word contexts enhance letter representations in early visual cortex. Moreover, we observe increased coupling between letter information in visual cortex and brain activity in key areas of the reading network, suggesting these areas may be the source of the enhancement. Our results provide evidence for top-down representational enhancement in word recognition, demonstrating that word contexts can modulate perceptual processing already at the earliest visual regions.
Context-based expectations can strongly facilitate perception, but how this is neurally implemented remains a topic of debate1,2. One famous and striking example of contextual facilitation is found in reading, where letters are more easily identified when embedded in a linguistic context such as a word or name (e.g., a road sign) than in a random string (e.g., a license plate)3.
Historically, two opposing accounts have been proposed to explain this so called 'word superiority effect'. Under the guessing-based account, letter identification occurs in a bottom-up fashion and the advantage offered by words constitutes only a post-perceptual advantage in 'guessing' the correct letter4,5. Alternatively, the perceptual account explains word superiority as a top–down effect, proposing that higher-order linguistic knowledge can enhance perceptual processing of the individual letters6,7. A rich behavioural literature, dating back several decades8,9, has documented that even when the ability to guess the correct letter is experimentally controlled, the word advantage persists10. This has been interpreted as evidence that the effect must (at least in part) reflect top–down perceptual enhancement—a view that remains dominant until today11.
However, some lingering doubts have persisted. For instance, ideal observer analysis has shown that the efficiency of letter recognition is much lower than that of a fully holistic (word-based) observer, and lies within the theoretical limits of a strictly letter-based (feedforward) observer—even when considering word superiority12. Moreover, advances in deep learning have shown that letters and other complex objects can be accurately recognised in context by bottom-up architectures, further questioning the need to invoke top–down explanations13. Beyond these theoretical arguments, neural evidence for the perceptual locus of this supposedly top–down effect is lacking. This is remarkable, since the top–down interpretation of word superiority makes a clear neural prediction: if the behavioural word advantage is due to a perceptual enhancement of letter stimuli, then it should be accompanied by an enhancement of sensory information in the early visual areas that process the individual letters already.
Here, we test this prediction using a simple paradigm involving streams of words and nonwords. We use neural network simulations of the paradigm to confirm that top–down models would uniquely predict the enhancement of letter representations by word contexts. When we then perform the same experiment in human observers while recording brain responses using functional magnetic resonance imaging (fMRI), we find that word contexts robustly enhance letter representations in early visual cortex. Moreover, compared to nonwords, words are associated with increased information-activation coupling between letter information in early visual cortex on the one hand, and blood-oxygen-level-dependent (BOLD) activity in key areas of the reading network on the other. These results suggest word superiority is (at a least in part) a perceptual effect, supporting prominent top–down models of word recognition.
Word contexts facilitate orthographic decisions
Participants (n = 34) were presented with streams of words or nonwords consisting of five letters (see Fig. 1a), while maintaining fixation. We used a blocked design in which word and nonword (i.e. unpronounceable letter string) stimuli were presented in long trials of ten items of which the middle letter (U or N) was kept fixed while the outer letters varied, creating a word or nonword context (each 10-s trial containing only stimuli of one condition). To make reading visually challenging, stimuli were embedded in Gaussian noise (see Methods). To keep participants engaged, they performed a spelling discrimination task on specific target stimuli that occurred occasionally (1–2 times) per trial. Target stimuli were learned during a prior training session. Targets were presented either in their regular form or with one letter permuted, and participants had to categorise targets as 'spelled' correctly or incorrectly (i.e. presented in the learned form or permuted). Participants were faster (median RT difference: −29.2 ms; Wilcoxon signed rank, T34 = 40, p = 1.07 × 10−5, r = 0.87) but not significantly more accurate (mean accuracy difference: 1.62%; t-test, t34 = 1.70, p = 0.098, d = 0.29) for word compared to nonword targets. This observation is in line with the word superiority effect, but from the behaviour alone it is unclear whether the word advantage was perceptual or post-perceptual.
Fig. 1: Experimental paradigm.
a Example stimuli for each condition. Participants observed words or nonwords (i.e. orthographically illegal, unpronounceable strings) with a U or N as middle letter, resulting in four conditions. b Functional localiser. During the functional localiser, the key letters (U and N) were presented in isolation and without visual noise, while participants performed an irrelevant task at fixation. c Trial structure. We used a blocked design, in which each 14-s trial consisted of ten words or nonwords with a fixed middle letter. Participants performed an orthographic discrimination task on specific, prelearned targets that occurred once or occasionally twice per trial. Participants were trained in a separate session to perform the task while maintaining fixation at the centre of the screen.
Representational enhancement is a hallmark of top–down models
Because our paradigm is different from the traditional paradigms in the (behavioural) word superiority literature, we performed simulations of our experiment to confirm that the top–down account indeed predicts the representational enhancement we set out to detect. We used a predictive coding implementation14 of the influential Interactive Activation architecture proposed by McClelland and Rumelhart6 (see Methods).
In the simulation, we ran artificial 'runs' in which we presented sets of word and nonword stimuli used in the experiment to the network (Fig. 2a). To simulate experimental viewing conditions, we added Gaussian noise and ran the network until convergence so as to mimic long stimulus duration (see Methods) resulting in stimuli that were presented well-above recognition threshold (Supplementary Fig. 2). Representational strength was quantified by dividing the activity level for the correct letter unit by the sum of activity levels of all letters—a fraction that asymptotically goes to 1 as representational strength increases. After running 34 simulated runs with the top–down model, the relative evidence for the middle letter was confirmed to be much higher in words than nonwords (paired t-test, t34 = 50.5, p = 7.72 × 10−33), despite the signal-to-noise ratio of the simulated stimuli being identical (Fig. 3a). Importantly, when the same stimuli were presented to a network lacking word-to-letter feedback connections, no such difference was found (paired t-test, t34 = −0.24, p = 0.81), resulting in a significant interaction (two-sample t-test t34 = 31.1, p = 3.5 × 10−41). This confirmed that despite the differences between our and the classic paradigm, representational enhancement of letters by word contexts is a hallmark of top–down models of letter perception.
Fig. 2: Probing representational enhancement in neural network models and the brain.
a Modelling representational enhancement in a hierarchical neural network model6,14. Stimuli used in the experiment were encoded into vectors of visual features and overlaid with Gaussian noise (bottom rows). Inputs were presented to a network with or without word-to-letter feedback connections. For both networks, representational strength was quantified from the distribution of activity levels of letter units for the third position (principle illustrated for the fifth letter, E). Solid circles indicate units (representing features, letters or words); lines indicate feedforward connections, and dotted lines with arrows indicate feedback connections. Note that we used a predictive coding formulation of the network14 but for simplicity only the state estimator (prediction) units are shown in the schematic (see Methods for details). b Quantifying representational letter enhancement using multivariate pattern analysis (MVPA). To probe letter representations in the brain, we used two MVPA techniques: classification (upper panel) and pattern correlation (lower panel).
Fig. 3: Word contexts enhance letter representations.
a Theoretical predictions. We simulated 34 artificial 'runs' in which we exposed a network with (top–down model) and without (feedforward model) word-to-letter feedback connections to the experimental stimuli, and computed the average representational strength of the middle letter in word and nonword contexts. Note that the strong dissociation was observed despite the fact that the middle letter was well-above threshold in all conditions for both models (Supplementary Fig. 2). b Letter representation in early visual cortex of 34 human observers. Two multivariate pattern analysis methods (see Methods) revealed that neural representations of letters were enhanced in word compared to nonword contexts, supporting the top–down model. In both panels, grey dots represent individual simulated 'runs' (a) or individual participants (b). Lines represent paired differences. White dots, boxes and whiskers represent between-subject medians, quartiles and 1.5 interquartile ranges, respectively. Significance levels correspond to p < 0.01 (**) or p < 0.001 (***) in a paired, two-tailed Student's t or Wilcoxon sign rank test.
Word contexts enhance letter representations
Next, we tested whether we could find a similar enhancement effect in early visual cortex in our participants. To do so, we first trained a classifier for each participant on an independent dataset from a functional localiser run, during which the two middle letters (U or N) were presented in isolation and without Gaussian noise (see Fig. 2b). We then tested the classifier's ability to identify the middle letter of the words and non-words presented in the main experiment, in a trial-based fashion (each trial lasting 14 s and consisting of ten stimuli). We reasoned that if word context enhances the sensory representations of letters (e.g. enhancing the letter features in noise), this should be apparent in early visual areas, which we defined as the union of V1 and V2 (see Methods). To focus on voxels sensitive to the relevant part of the visual field, we selected the 200 voxels (the same number we used in a previous study15) most responsive during the localiser run. We were able to classify letter identity well above chance level (one sample t-test, t34 = 18.84, p = 3.13 × 10−19, d = 3.23) reaching a mean overall decoding accuracy of 81.4% averaged over both conditions (see Fig. 3).
Having established that letter identity can be extracted with high fidelity from early visual cortex, we went on to test if representational content was enhanced by word context. Strikingly, we found that classification accuracy was indeed higher for words compared to nonwords (Wilcoxon sign rank test, T34 = 141.5, p = 7.55 × 10−3, r = 0.52; Fig. 3b). To further examine this enhancement effect, we quantified representational content using an (arguably simpler) supplementary multi-voxel pattern analysis (MVPA) technique: pattern correlation analysis—the difference in voxel response pattern correlation that could be attributed to letter identity ('Pearson ρ within-letter' minus 'Pearson ρ between-letter'; see Methods). Reassuringly, the results aligned with those of the classification analysis: the correlation difference score being significantly higher for words than nonwords (Wilcoxon sign rank, T34 = 103, p = 8.83 × 10−4, r = 0.67).
To confirm that the differences revealed by the classification and pattern correlation analysis were related to differences in representations of stimulus information and not to unrelated confounding factors, we performed a number of controls. First, we tested the stability of the results over different region of interest (ROI) definitions. Since both representational analyses used the 200 voxels that were most responsive during an independent functional localiser, we wished to ensure that the results were not unique to this a priori specified (but arbitrary) number. We therefore re-ran the same analyses for ROIs ranging from 50 to 1000 voxels with steps of 10. This revealed that the same pattern of effects was found over practically the entire range of ROI sizes (Supplementary Fig. 3).
Another possibility is that the increased estimates of representational content could be explained by a simple difference in signal amplitude, potentially related to participants being more attentive to words than nonwords. To address this, we quantified BOLD amplitude per condition using a standard generalised linear model (GLM) based approach (see Methods), but found no significant difference between conditions in the amplitude estimates for the corresponding voxels (paired t-test, t34 = −0.57, p = 0.57, d = 0.10; Bayesian paired t-test, BF10 = 0.21; see Supplementary Fig. 4). Importantly, we found no significant differences in eye-movement deviation from fixation between words and nonwords (Wilcoxon T32 = 197, p = 0.21, r = 0.25; Bayesian paired t-test, BF10 = 0.48; see Supplementary Fig. 6 and Methods), confirming participants' ability to maintain fixation during the task did not differ significantly between conditions.
As a final control analysis, we wanted to confirm that the MVPA results relied on retinotopically specific information. This would be an important indication that both the letter information extracted from visual cortex, and its enhancement by word contexts, indeed originate from sensory representations. To this end, we performed a searchlight variant of the classification and pattern correlation analysis (see Supplement for details). This revealed (see Supplementary Figs. 7, 8) that letter identity information was only visible in neural activity patterns in visual cortex, ruling out that decoding relied on a brain-wide signal. We further tested for retinotopic specificity within visual cortex by comparing the functionally defined central ROI (described above), to a functionally defined peripheral ROI (see Methods and Supplementary Note 1 for more details). This revealed (Supplementary Fig. 9) that overall letter decoding was greatly reduced for the peripheral ROI compared to the central ROI, both for classification (paired t-test, t34 = 15.59, p = 8.86 × 10−17, d = 2.67) and pattern correlation analysis (paired t-test, t34 = 8.06, p = 2.65 × 10−9, d = 1.38). Importantly, we found a similar reduction in the peripheral ROI for the enhancement effect (the difference in decoding between conditions), again both for the classification (paired t-test, t34 = 2.56, p = 0.015, d = 0.44) and pattern correlation analysis (paired t-test, t34 = 2.92, p = 6.31 × 10−3, d = 0.50).
In sum, these analyses show that sensory letter information in early visual cortex, as estimated by classification and pattern correlation analysis, was increased in words compared to nonwords. This enhancement was present over a range of ROI definitions, but was reduced for peripheral compared to central ROIs, and could not be explained by confounding factors such as BOLD amplitude or eye movements.
Representational enhancement across the visual hierarchy
Having established a perceptual enhancement effect by word context in early visual cortex, we then asked how this enhancement effect was distributed among specific visual areas. To this end, we further investigated five ROIs, four of which were defined anatomically (V1–V4) and one (visual wordform area (VWFA)) functionally; in each ROI, voxels were selected using the procedure described earlier (see Methods for details).
The results show consistent evidence for word enhancement in V1, V2 and V4 (all p's < 0.025; see Supplementary Fig. 10 for details), with both analyses. In contrast, V3 and VWFA showed no consistent evidence for word enhancement (see Supplementary Fig. 10). However, in these regions, the overall classification accuracy and pattern information scores were also close to chance, making the absence of differences between conditions difficult to interpret. For regions V1–V4, we also tested for univariate amplitude differences between word and nonword conditions. Interestingly, in all four regions, the sign of the univariate difference was negative (indicating weaker amplitude of responses to word stimuli), but note that only in V4 this difference was marginally significant (paired t-test, t34 = −2.11, p = 0.04, d = −0.36, uncorrected; Supplementary Fig. 5). In sum, we observed word enhancement across multiple regions in the visual hierarchy. Critically, none of the regions showed BOLD amplitude differences, ruling out the possibility that word enhancement was confounded by low-level attentional differences between conditions.
Information-activation coupling reveals putative neural sources
Having observed a hallmark of top–down perceptual enhancement by word contexts, we then asked what the potential neural source of this top–down effect could be. We reasoned that if a candidate brain region was involved in the observed enhancement, then activity levels in this region would be expected to covary with the amount of letter information represented in early visual cortex. Moreover, this relationship should not be driven by a categorical difference between conditions (e.g. that both BOLD amplitude in a candidate region and informational content in visual cortex are higher for words than nonwords, while the two are not related within conditions). Taking the two requirements together, we expected regions implicated in the top–down effect to show increased functional coupling between local BOLD activity and representational information in early visual cortex, for words compared to nonwords.
To test for this increased information-activation coupling, we used a GLM-based approach to model regional BOLD amplitude in both conditions as a function of early visual cortex classification evidence, and tested for an increased slope for words compared to nonwords (see Fig. 4b). This is analogous to the well-established psychophysiological interaction (PPI) analysis16, but uses classifier evidence instead of BOLD activity as seed timecourse. Classifier evidence here corresponds to the predicted probability of the correct (presented) letter stimulus for each brain volume (see Methods).
Fig. 4: Information-activation coupling analysis.
a ROI-based coupling analysis. For two ROIs, GLMs were fitted to estimate coupling between early visual cortex classification evidence and regional BOLD amplitude for words and nonwords separately. We then tested for increased coupling (higher coefficients) in words compared to nonwords. b Illustration for example participant. A single (averaged) timecourse was extracted from each ROI and regressed against visual cortex classification evidence to test for increased slopes in words compared to nonwords. For illustration purposes, only the predicted slopes based on the regressor of interest are shown. Note that classification evidence was defined as probability, but here expressed in arbitrary units due to the whitening operation. c Whole-brain results. Same analysis as in a and visualised in b, but performed for each voxel independently. Resulting contrast images (word-nonword) were tested at the group level for increases in coupling in words compared to nonwords. This revealed statistically significant clusters (p < 0.05 FWE corrected), in the left pMTG and in the left IFG. Glass brain plot rendered with nilearn63. For a non-thresholded slice-by-slice rendering of the whole-brain results in c, see Supplementary Fig. 11. Grey dots indicate coefficients of individual participants, and lines the within-subject differences; white dots, boxes and whiskers are between-subject medians, quartiles and interquartile ranges, respectively. Significance stars correspond to p < 0.01 (**) or p < 0.001 (***) in a paired two-tailed t-test or Wilcoxon sign rank test. pMTG posterior medial temporal gyrus, IFG inferior frontal gyrus.
We first tested for increased coupling in a hypothesis-driven, ROI-based fashion. We tested two candidate regions: the VWFA and the left posterior middle temporal gyrus (pMTG), associated with orthographic/visual17,18 and lexical/semantic processing19,20, respectively. Activity of all voxels was averaged to obtain a single BOLD timecourse per ROI. This BOLD timecourse was then modelled as a function of visual cortex classification strength to obtain separate coupling parameters for word and nonword conditions. We indeed observed a significantly increased coupling in both VWFA (Wilcoxon sign rank, T34 = 80, p = 2.00 × 10−4, r = 0.73) and pMTG (paired t-test t34 = 2.83, p = 8.2 × 10−3, d = 0.48; see Fig. 4a) for words. The increase in coupling appeared stronger in VWFA, but the difference in effects between regions was not statistically significant (paired t-test, t34 = 0.62, p = 0.54, d = 0.11).
Finally, we carried out an exploratory analysis by testing for increased functional coupling across the entire brain. In essence, the GLM procedure was identical to the one above, but carried out at the individual voxel level. This yielded, for each participant, a map of estimated differences in functional coupling for every voxel. These functional maps were then registered to a standard space after which we tested whether there were clusters of voxels that showed an increase in functional coupling for words compared to nonwords. We found two significant (FWE-corrected, cluster-forming p < 0.001, cluster-level p < 0.05) left-lateralised clusters at key nodes of the language network: one in pMTG and one in inferior frontal gyrus (IFG) (Fig. 4c; Supplementary Fig. 11). No significant cluster was found at VWFA, possibly due to individual neuro-anatomical variability in VWFA size and location21.
Altogether, these results demonstrate increased functional coupling between visual cortical classification evidence and neural activity in VWFA, pMTG and IFG. In all of these regions, we found a significant increase in functional coupling (here, meaning that classifier evidence increased when the regions became more active, and vice versa) for words compared to nonwords, which is consistent with the idea that these regions might constitute the neural source of the top–down effect.
Visual context facilitates perception1. Letter perception offers a striking example of such facilitation, as letters are more easily recognised when embedded in a word. Dominant, 'interactive' models of word recognition assume this facilitation occurs in the visual system already, proposing that linguistic knowledge can enhance perception in a top–down fashion6. Here we tested this perceptual enhancement hypothesis at the neural level. We presented streams of words or nonwords with a fixed middle letter while recording fMRI. Simulations of this paradigm confirmed that top–down models of word recognition uniquely predict that perceptual representations of the middle letter should be enhanced when embedded in a word. In line with the top–down account, information about the middle letter, probed using MVPA in early visual cortex, was enhanced when the letter was embedded in words compared to nonwords. Further, we found increased functional coupling between the informational pattern in early visual cortex, and regional BOLD amplitude in three key regions of the left-lateralized language network, i.e. VWFA, left pMTG and IFG. This points to these regions as potential neural sources of the representational enhancement effect. Together, these results constitute the first neural evidence for representational enhancement of letters by word contexts, as hypothesised by top–down accounts of word recognition6. The results also fit naturally with theoretical frameworks of top–down perceptual inference, such as hierarchical predictive coding22,23,24 (see refs. 2,25 for review) and with the broader literature on top–down, predictive effects in language processing26,27,28.
Our results are in line with a large behavioural literature on context effects in letter perception that support interactive activation (top–down) models. These works have demonstrated, for instance, that the word advantage persists when the guessing advantage afforded by words is constrained experimentally8,9, see ref. 10 for review; that readers subjectively perceive letters embedded in real words as sharper29; and that readers are better at detecting subtle perceptual changes in real words than in nonwords29. While the behavioural literature has extensively investigated top–down effects, work on the neural basis of visual word recognition has focussed almost exclusively on its bottom-up component, most notably by probing the bottom-up selectivity profile of VWFA (or ventral occipitotemporal cortex more broadly) to various visual and orthographic properties30,31,32. One study tried to disentangle word and letter encoding at a neural level33, but did not probe individual letter representations and their enhancement by word contexts. A recent study did investigate top–down processing, but was limited to attention-based response modulations in decision contexts34.
Beyond the domain of language, but converging with the results presented here, are results from object perception, where it was recently found that the facilitation of object recognition by familiar contexts was accompanied by enhancement of object representations in object selective cortex35. The similarity to the current findings speaks to the idea that representational enhancement reflects a more general principle of contextual effects in perception. These contextual effects have been extensively studied, and range from neurons in macaque early visual cortex responding differently to identical lines when presented as parts of different figures36,37 to neurons in mouse V1 encoding contextually expected but omitted stimuli (see ref. 38 for review). As such, it may be that although here the context is linguistic, the contextual effect in visual cortex reflects a more general mechanism that is not specific to reading or unique to humans. Interestingly, the idea that contextual enhancement reflects a more general perceptual mechanism was a key motivation to develop models of word recognition in the first place6,7.
When viewed as a more general principle of perception, contextual enhancement touches on an even broader question: are objects recognised by their parts or as wholes? On the one hand, word superiority has historically been taken as an example of 'holistic' perception8,10 and the enhancement we observed (in which 'wholes' enhance representations of 'parts') indeed seems to contradict a strictly letter-based (part-based) account. But on the other hand, it has been convincingly demonstrated, both for word and face recognition12,39, that the identifiability of parts poses a bottleneck on the identification of wholes, and hence, even for the most common words, recognition cannot be truly holistic12. Moreover, effects of wholes on the identification of parts are not always faciliatory: facial arrangements, for instance, have been both reported to have positive and negative effects on search performance40,41. Developing a theoretical framework that naturally accounts for top–down, contextual enhancement as reported here (see e.g. 23), while being properly constrained so as to incorporate feature-based bottlenecks, and the occasional detrimental effects of context, provides an important challenge for future research.
A limitation of the current study is that we cannot access sensory representations directly, and instead have to infer them by estimating sensory information from measured neural activity patterns. By itself, the fact that letter identity could be more readily decoded from words than nonwords could in principle merely reflect confounding differences in the BOLD signal between conditions, or in the our ability to extract information from that signal42. Importantly, however, we obtained converging evidence using two complementary techniques of probing representational content. One of these (classification) used an independent dataset for training purposes in which only single letters were presented without noise, which suggests that our MVPA techniques were picking up relevant information about the middle letters, rather than irrelevant signals that only covaried with reading (non)words with the respective middle letter. Moreover, enhancement was consistently found in multiple visual areas, was retinotopically specific, but not contingent on exact ROI definitions, and could not be explained by other confounds such as signal amplitude or eye movements. As such, we believe that the most parsimonious explanation of the observed effect is as reflecting an enhancement in the underlying sensory information available to visual cortex itself—in other words, a representational enhancement.
We interpret this representational enhancement as a neural signature of the perceptual enhancement of letters—a process formalised by top–down models of word recognition, and widely characterised in behavioural literature10,29. However, a limitation of fMRI is that it cannot differentiate between earlier and later activity. Hence, it is possible that the observed effects arise late, perhaps even much later than what is typically considered 'perceptual' (e.g. > 400 ms); instead perhaps reflecting what one might call iconic memory encoding. Although distinguishing between perceptual and post-perceptual effects is notoriously difficult, future studies might address this by probing perception more directly using an objective measure of perceptual sensitivity, or by using a high temporal resolution method (e.g. ECoG or MEG) in combination with a temporal criterion to arbitrate between perceptual and post-perceptual enhancement of sensory representations.
An apparent disconnect between our study and the existing literature concerns the level of representation at which enhancement occurs. We probed enhancement in early visual cortex (representing visual features such as edges and simple line conjunctions) while in theoretical models6,14 enhancement is probed at the level of letters, not features. However, perceptual enhancement is a generic mechanism and should not be unique to a specific level of representation. In fact, the main reason6 that in the classic models enhancement occurs only at the letter level is simplicity: because features comprise the input to the network and are hence not recognised, the possibility of enhancement occurring at the feature level is excluded by design.
Apart from model simplicity, one might argue there are more substantive, cognitive reasons that enhancement is primarily described at the letter level and not at the level of simple features. Specifically, word superiority has been reported with stimuli consisting of mixed case and font43, implying that sometimes the word superiority effect can be independent of the visual features that define exact letter shape, and may instead act at a more abstract level of letter identity. However, when the exact letter shape is well known and especially under visually noisy conditions (like in our experiment), enhancement of simple low-level visual features appears useful for letter recognition, thereby incentivising top–down enhancement to reach the (functionally well-localised) visual feature level. As such, we do not claim that our experiment shows that word superiority always acts at the level of sensory features. Rather, it demonstrates that in principle these enhancement effects can extend even to the earliest sensory cortical regions, contradicting purely bottom-up accounts in which such top–down enhancement is ruled out by design.
What kind of information is driving the observed representational enhancement effect? A possible source of information is lexical knowledge, although sublexical (orthographic/phonological) knowledge may be an equally plausible candidate. Indeed, behaviourally letters are also more easily recognised when embedded in pronounceable nonwords (pseudowords) than in unpronounceable, orthographically illegal nonwords44. Theoretically, such a pseudoword superiority effect can be either understood as originating via top–down connections in a dedicated sublexical route45 or as arising as a by-product from co-activations of lexical items with overlapping letter combinations6,46. The fact that we found increased functional coupling in both VWFA (associated with sublexical orthography) and pMTG (associated with lexical access) makes our data consistent with both types of feedback, originating from the sublexical and the lexical route. Future studies could examine the relative contributions of these forms of feedback, by examining the neural activity patterns to pronounceable pseudowords.
A final point of discussion concerns the interpretation of the information-activation coupling analysis. We interpret the results as pointing to putative candidate sources, because the observed increases in functional coupling match the expected pattern of results if the associated regions were indeed a neural source of the enhancement. However, we acknowledge that since the functional coupling analysis is correlational in nature, the direction of causality implied by this interpretation remains speculative. To get a better understanding of the sources involved this effect, future studies could either directly perturb candidate sources, use a more indirect method for inferring directionality such as laminar fMRI47 or a directional connectivity analysis48.
In conclusion, we have observed that word contexts can enhance sensory letter representations in early visual cortex. These results provide the first neural evidence for top–down enhancement of sensory letter representations by word contexts, and suggest that readers can better identify letters in context because they might, quite literally, see them better.
Thirty-six participants were recruited from the participant pool at the Donders Centre for Cognitive Neuroimaging. Sample size was chosen to detect a within-subject effect of at least medium size (d > 0.5) with 80% power using a two-tailed one-sample or paired t-test. The study was in accordance with the institutional guidelines of the local ethical committee (CMO region Arnhem-Nijmegen, The Netherlands, Protocol CMO2014/288), all participants gave informed consent and received monetary compensation. Participants were invited for an fMRI session and a prior behavioural training session, that took place no more than 24 h before the fMRI session. For one participant, who moved excessively between runs, decoding accuracy was never above chance; this participant was excluded from all fMRI analyses. One additional participant had their eyes closed for an extended duration during more than 20 trials, and was excluded from both the behavioural and fMRI analyses. All remaining participants (n = 34, 12 male, mean age = 23 ± 3.32) were included in all analyses. Due to technical problems, one participant only completed four instead of six blocks, all of which were analysed.
Stimuli were generated using Psychtoolbox-3 (ref. 49) running on MATLAB (MathWorks, MA, USA). Stimuli were rear-projected using a calibrated EIKI (EIKI, Rancho Santa Margarita, CA) LC XL 100 projector (1024 × 768, 60 Hz). Each stimulus was a five-letter word or nonword presented in a custom-made monospaced typeface. To prevent that the multivariate analyses would pick up on global low-level features (such as overall luminance or contrast) to discriminate between middle letter identity, the middle characters (U or N) were chosen to be identical in shape and size, but flipped vertically with respect to each other. Words were presented in a large font size, each letter 3.6° wide and with 0.6° spacing between letters. This size was chosen to make the middle letter as large as possible while retaining readability of all letters when fixating at the centre. In addition to the words and nonwords, a fixation dot of 0.8° in diameter was presented at the centre of the screen. To make reading visually challenging and incentivize top–down enhancement of low-level visual features, words were embedded in visual noise. The noise consisted of pixelated squares, each 1.2° wide, offset so that the pixels were misaligned with the letter strokes. Letters were presented on top of the noise with 80% opacity. We chose this type of noise after finding it impacted readability strongly even when the letters were presented at high physical luminance. Brightness values (in the range 0–255) of the noise 'pixels' were randomly sampled from a Gaussian distribution with a mean of 128 and an SD of 50. To make sure that the local brightness was on average identical for each trial and across the screen, the noise patches were generated using a pseudo-random procedure. In each trial, ten noise patches were presented, five of which were independent and randomly generated, while the other five were copies of the random patches, but polarity-inverted in terms of their relative brightness with respect to the mean. This way the brightness of each noise pixel was always 128 (grey) on average in each trial. The order of noise patches was pseudo-random, with the constraint that copied patches were never presented directly before or after their original noise patch. This way the re-use of noise patches was not noticeable and all seemed random.
In the main experiment, we used a blocked design, in which we presented blocks of four long trials (one of each of the four conditions), followed by a null-trial. Each trial was 14-s long, during which ten stimuli were presented. Of those stimuli, nine or occasionally (in 25% of trials) eight were (non-)word items and one or two were (learned) targets. A single presentation consisted of 900 ms of (non)word item plus noise background, and 500 ms of blank screen plus fixation dot (Fig. 1c). Targets were either presented in their regular (learned) form or with one of the non-middle letters permuted, and participants had to discriminate whether the target was regular or permuted. Target correctness and occurrence within the trial were counterbalanced and randomised, with the constraint that targets were never presented directly after each other. The order of word items was shuffled pseudo-randomly, with the constraint that the same letter never repeated twice at the same position (except for the middle letter).
In the functional localiser run, only the middle letters (U and N) plus fixation bulls' eye were presented. We again used a blocked design, with long trials that had a duration of 14 s during which one of letters was repeated at 1 Hz (500 ms on, 500 ms off; see Fig. 1b). During the localiser, each trial was followed by a null-trial in which only the fixation dot was presented for 9.8 s. This was repeated 18 times for each letter.
Two different sets of words and nonwords were used for the training and experimental session. For the experimental session, we used 100 five-letter words with a U or N as third character in Dutch (see Supplementary Table 1), plus equally many nonword items. This particular subset was chosen because they were the 100 most frequent five-letter words with a U or N in Dutch, according to the subtlex database50. Each item occurred at least four times and maximally five times (4.2 on average) during the entire experimental session; to ensure repetitions were roughly equally spaced, items were only repeated once all other items were presented equally often. Because we wanted to familiarise participants with the task and the custom-font, but not with the (non)word stimuli themselves (especially because there was considerable variation in the amount of training between participants), we used different (non)words for the training session. For the training session, we used the remaining 50 less frequent five-letter Dutch words with a U and N. For the nonwords, letters were randomly sampled according to the natural frequency of letters in written Dutch51, with the constraint that adjacent letters were never identical. The resulting nonwords were then hand-selected to ensure all created strings were unpronounceable, orthographically illegal nonwords. The four learned target stimuli were CLUBS and ERNST for the words, and KBUOT and AONKL for the nonwords. These were learned during the prior training session.
Each participant performed one behavioural training and one experimental fMRI session. The goal of the training was for participants to learn the four target items and learn how to perform the task while maintaining fixation at the centre of the screen. The fMRI session consisted of a brief practice of ~5 min during which the anatomical scan was acquired. This was followed by six experimental runs of 9−10 min, which were followed by a localiser run of ~15 min. We used a blocked design, in which we presented blocks of four long trials (one of each of the four conditions), followed by a null-trial experimental run consisted of 40 trials of 14 s. Trials were presented in blocks consisting of five trials: one of each condition (U-word, U-nonword, N-word, N-nonword), plus a null-trial during which only the fixation dot was present. The order of trial types within blocks was randomised and equalised: over the entire experiment, each order was presented twice, resulting in a total number of 240 trials (192 excluding nulls). In the functional localiser, single letters were presented blockwise: one letter was presented for 14 s, followed by a null-trial (9.8 s), followed by a trial of the other letter. Which letter came first was randomised and counterbalanced across participants.
Statistical testing
For each (paired/one-sample) statistical comparison, we first verified that the distribution of the data did not violate normality and was outlier free, determined by the D'Agostino and Pearson's test implemented in SciPy and the 1.5 IQR criterion, respectively. If both criteria were met, we used a parametric test (e.g. paired t-test); otherwise, we resorted to a non-parametric alternative (e.g. Wilcoxon sign rank). All statistical tests were two-tailed and used an alpha of 0.05. For effect sizes, we report Cohen's d for the parametric and biserial correlations for the non-parametric tests.
fMRI acquisition
Functional and anatomical images were collected with a 3T Skyra MRI system (Siemens), using a 32-channel headcoil. Functional images were acquired using a whole-brain T2*-weighted multiband-4 sequence (TR/TE = 1400/33.03 ms, voxel size = 2 mm isotropic, 75° flip angle, A/P phase encoding direction). Anatomical images were acquired with a T1-weighted MP-RAGE (GRAPPA acceleration factor = 2, TR/TE = 2300/3.03 ms, voxel size 1 mm isotropic, 8° flip angle).
fMRI preprocessing
fMRI data pre-processing was performed using FSL 5.0.11 (FMRIB Software Library; Oxford, UK52). The pre-processing pipeline included brain extraction (BET), motion correction (MCFLIRT), temporal high-pass filtering (128 s). For the univariate and univariate-multivariate coupling analyses, data were spatially smoothed with a Gaussian kernel (4 mm FWHM). For the multivariate analysis, no spatial smoothing was applied. Functional images were registered to the anatomical image using boundary-based registration as implemented in FLIRT and subsequently to the MNI152 T1 2-mm template brain using linear registration with 12 degrees of freedom. For each run, the first four volumes were discarded to allow for signal stabilisation. Most FSL routines were accessed using the nipype framework53. Using simple linear registration to align between participants can result in decreased sensitivity compared to more sophisticated methods like cortex-based alignment54. However, note that using a different inter-subject alignment method would not affect any of the main analyses, which were all performed in native EPI space. The only analysis that could be affected is the whole-brain version of the information-activation coupling analysis (Fig. 4c; Supplementary Fig. 11). However, this was only an exploratory follow-up on the pre-defined ROI-based coupling analysis, intended to identify potential other regions displaying the signature increase in coupling. For this purpose, the simple linear method was deemed appropriate.
Univariate data analysis
To test for differences in univariate signal amplitude between conditions, voxelwise GLMs were fit to each run's data using FSL FEAT. For the experimental runs, GLMs included four regressors of interest, one for each condition (U-word, U-nonword, etc). For the functional localiser runs, GLMs included two regressors of interest (U, N). Regressors of interest were modelled as binary factors and convolved with a double-gamma HRF. In addition, (nuisance) regressors were added for the first-order temporal derivatives of the regressors of interest, and 24 motion regressors (six motion parameters plus their Volterra expansion, following Friston et al.55). Data were combined across runs using FSL's fixed-effects analysis. All reported univariate analyses were performed on an ROI basis by averaging all parameter estimates within a region of interest, and then comparing conditions within participants (see Supplementary Figs. 4, 5).
Multivariate data analysis
For the multivariate analyses, spatially non-smoothed, motion-corrected, high-pass filtered (128 s) data were obtained for each ROI (see below for ROI definitions). Data were temporally filtered using a third-order Savitzky-Golay low-pass filter (window length 21) and z-scored for each run separately. Resulting timecourses were shifted by three TRs (i.e. 4.2 s) to compensate for HRF lag, averaged over trials, and null-trials discarded. For each participant, this resulted in 18 samples per class for the localiser (i.e. training data) and 96 samples per condition (word/nonword) for the main runs (i.e. testing data).
For the classification analysis, we used a logistic regression classifier implemented in sklearn 0.2 (ref. 56) with all default settings. The model was trained on the time-averaged data from the functional localiser run and tested on the time-averaged data from the experimental runs. Because we had the same number of samples for each class, binary classification performance was evaluated using accuracy (%).
For the pattern correlation analysis, only the time-averaged data from the main experiment were used. Data were randomly grouped into two arbitrary splits that both contained an equal number of trials of all four conditions (U-word, U-nonword, N-word, N-nonword). Within each split, the time-averaged data of each trial were again averaged to obtain a single average response for each condition per split. For both word/nonword conditions separately, these average responses were then correlated across splits. This resulted, for both word and nonword conditions, in two (Pearson) correlation coefficients: ρwithin and ρbetween, obtained by correlating the average response to stimuli with the same or different middle letter, respectively. This process was repeated 12 times, each time using a different random split of the data, and all correlation coefficients were averaged to obtain a single coefficient per comparison, per condition, per participant. Finally, pattern letter information for each condition was quantified by subtracting the two average correlation coefficients (ρwithin − ρbetween).
For the searchlight variant of the multivariate analyses, we performed exactly the same procedure as described in the manuscript. However, instead of using a limited number of a priori defined ROIs, we used a spherical searchlight ROI that slid across the brain. A searchlight radius of 6 mm was used, yielding an ROI size of about 170 voxels on average, similar to the 200 voxels in our main ROI. For both analyses, this resulted in a map for each outcome metric for each condition for each subject, defined in native EPI space. These maps were then used for subsequent analyses (see Supplementary Note 1).
Information-activation coupling analysis
For the information-activation coupling analysis, we used a GLM-based approach to predict regional BOLD amplitude as a function of early visual cortex classification evidence, and tested for an increase in coupling (slope) for words compared to nonwords (see Fig. 4b). The GLM had one variable of interest, visual cortex classification evidence (see below for definition) that was defined on a TR-by-TR basis, and split over two regressors, corresponding to both conditions (word/nonword). In addition, first-order temporal derivatives of the two regressors of interest and the full set of motion regressors (from the FSL FEAT GLM) were included to capture variability in HRF response onset and motion-related nuisance signals, respectively. Because the classification evidence was undefined for null-trials, these were omitted. To compensate for temporal autocorrelation in the data, pre-whitening of the data was applied using the AR(1) noise model as implemented in nistats56. The resulting GLM yielded two regression coefficients (one per condition) for each participant, which were then compared at the group level to test for an increase in coupling in word contexts. Conceptually, this way of testing for condition-dependent changes in functional coupling is analogous to PPI16 but using a multivariate time-course as a 'seed'. This timecourse, classification evidence, was defined as the probability assigned by the logistic regression model to the correct outcome—or \(\widehat p\left( {A|y = A} \right)\). This probabilistic definition combines aspects of both prediction accuracy and confidence into a single quantity. Mathematically it is defined, as in any binomial logistic regression classifier, via the logistic sigmoidal function:
$${\hat{p}}\left( {{{A}}|{{y}} = {{A}}} \right) = \left\{ {\begin{array}{*{20}{ll}}{\frac{1}{{1 \, + \, {\mathbf{e}}^{ - \theta\, ^{\mathrm{T}}{\mathbf{X}}}}}}\,& {\mathrm{if}}\;{y} = 1 \\ {1 - \frac{1}{{1 \, + \, {\mathrm{e}}^{ - \theta\, ^{\mathrm{T}}{\mathbf{X}}}}}}& {\mathrm{if}}\,y = 0 \end{array}} \right.,$$
where θ are the model weights, y is the binary stimulus category, X are the voxel response patterns for all trials, and the letter 'U' is coded as 1 and 'N' as 0. Note that while the value of \(\widehat p\left( {A|y = A} \right)\) itself is bounded between 0 and 1, the respective regressors were not after applying prewhitening to the design matrix (see Fig. 4b).
Two variants of the GLM analysis were performed: one on timecourses extracted from two candidate ROIs and one on each voxel independently. For the ROI-based approach, timecourses were extracted by taking the average timecourse of all amplitude-normalised (z-scored) data from two ROIs: left pMTG and VWFA (see 'ROI definition' for details). For the brain-wide variant, the same GLM was estimated voxelwise for each voxel independently. This resulted in a map with the difference in coupling parameters for each voxel, for each participant (βword − βnonword) defined in native MRI space. These maps were then transformed to MNI space, after which a right-tailed one-sample t-test was preformed to test for voxels showing an increase in coupling in word conditions. The resulting p-map was converted into a z-map and thresholded using FSL's Gaussian random-field-based cluster thresholding, using the default cluster-forming threshold of z > 3.1 (i.e., p < 0.001) and a cluster significance threshold of p < 0.05.
ROI definition
For the ROIs of V1–V4, fusiform cortex and inferior temporal cortex, Freesurfer 6.0 (ref. 57) was used to extract labels (left and right) per subject based on their anatomical image, which were transformed to native space and combined into a bilateral mask. Labels for V1–V2 were obtained from the default atlas58, whereas V3 and V4 were obtained from Freesurfer's visuotopic atlas59. Early visual cortex (EVC) was defined as the union of V1 and V2.
The VWFA was functionally defined following a procedure based on earlier work34. Briefly, first we took the union of left fusiform cortex and left inferior temporal cortex that were defined via individual cortical parcellations obtained from freesurfer, and trimmed the anterior parts of the resulting mask. Within this broad, left-lateralised ROI, we then selected the 200 voxels that were most selective to words over nonwords (i.e. words over orthographically illegal, unpronounceable letter strings) as defined by the highest Z-statistics in the respective word–nonword contrast in the univariate GLM. Similarly to Kay and Yeatman34, we found that for most participants this resulted in a single, contiguous mask and in other participants in multiple word-selective patches. There are two main reasons we used the simple contrast word–nonword from the main experiment, rather than running a separate, dedicated VWFA localiser. First, using the main task strongly increased statistical power per subject as we could use a full hour of data per participant to localise VWFA. Second, the comparison of words and unpronounceable letter strings (with matched unigram letter frequency) solely targets regions that are selective to lexical and orthographic information (i.e. the more anterior parts of VWFA, according to the VWFA hierarchy reported in ref. 32). As such, the localiser only targets regions selective to the type of linguistic (lexical or orthographic) knowledge that could underlie the observed effect. This stands in contrast to other, less-restrictive VWFA definitions (such as words > phase scrambled words, or words > false fonts).
For the multivariate stimulus representation analyses, we did not use the entire anatomical ROIs defined above, but performed a selectivity-selection to ensure we probed voxels that were selective to the relevant part of the visual field. In this procedure, we defined the most selective voxels as those with the k highest Z-statistics when we contrasted any letter (U or N) versus baseline in the functional localiser GLM. Following ref. 15, we took 200 voxels as our predefined value for k. To verify that our results were not contingent on this specific (but arbitrary) value, we also made a large range of masks for early visual cortex by varying k between 50 and 1000 with steps of 10. Repeating the classification and pattern correlation analyses over all these masks revealed that the same pattern of effects was obtained over almost the full range of mask definitions, and that the best classification performance was in fact at our predefined value of k = 200 (Supplementary Fig. 3).
For the peripheral visual ROI, voxels were selected based on the functional criterion that they showed a strong response to stimuli in the main experiment (which spanned a large part of the visual field), but a weak or no response to stimuli in the localiser (which were presented near fixation). Specifically, voxels were selected if they were both in the top 50% of Z-stats for the contrast visual stimulation > baseline in the main experiment, and in the bottom 50% of Z-scores for visuals stimulation > baseline in the localiser. This resulted in masks that contained on average 183 voxels, similar to the 200 voxels in the central ROI. In our initial analysis, we focussed on V1 (see Supplementary Fig. 9) because it has the strongest retinotopy. However, the same was also applied to early visual cortex with similar results (see Supplementary Note 1).
To define pMTG, we performed an automated meta-analysis using Neurosynth60. Because we were interested in pMTG as a hub for lexical access, we searched for the keyword 'semantic'. This resulted in a contrast map based on 1031 studies which we thresholded at an arbitrarily high Z-value of Z > 9. The resulting map was mainly restricted to two hubs, in the IFG and pMTG. We selected left pMTG by overlaying the map with an anatomical mask of medial temporal gyrus from FSL's Harvard-Oxford Atlas. The resulting map was brought to native space by applying the registration matrix for each participant.
Behavioural data analysis
Participants had 1.5 s after target onset to respond. Reaction times under 100 ms were considered spurious and discarded. If two non-spurious responses were given, only the first response was considered and evaluated. Median reaction times and mean accuracies were computed for both (word and nonword) conditions and compared within participants.
Eye movements were recorded using an SMI iView X eye monitor with a sampling rate of 50 Hz. Data were pre-processed and submitted to two analyses: number of trials during which eyes were closed for extended periods, and comparison of horizontal (reading-related) eye movements between conditions.
During pre-processing, all data points during which there was no signal (i.e. values were 0) were omitted. After omitting periods with no signal, data points with spurious, extreme values (which sometimes occurred just before or after signal loss) were omitted. To determine which values were spurious or extreme, we computed the z-score for each points, over the entire run and ignoring the periods where signal was 0, and considered all values higher than 4 extreme and spurious. Similar to the periods with no signal, these timepoints were also omitted in following analysis. The resulting 'cleaned' timecourses were then visually inspected to evaluate their quality. For two participants, the data were of insufficient quality to include in any analysis. For six participants, there were enough data of sufficient quality to perform the overall amount of reading-related eye movements between conditions, but signal quality was insufficient to quantify the number of trials during which the eyes were shut for an extended period. This is because in these participants there were various periods of intermittent signal loss that were related to signal quality, not to the eyes being closed. To compare eye movements between conditions, we took the standard deviance of the gaze position over the reading (horizontal) direction, and averaged this over each trial. Because the resulting data contained outliers (i.e. trials during which the participants failed to maintain fixation), we took the median over trials in each condition (word/nonword), and compared them within participants (Supplementary Fig. 6). For the participants where the data were consistently of sufficient quality, periods of signal loss longer than 1.2 s were considered 'eyes closed for extended period'. As an inclusion criterion, we allowed no more than 25 trials during which eyes were closed for an extended period. This led to the exclusion of one participant, who had 33 trials during which the eyes were closed for an extended period. This participant was a clear outlier: of all participants with sufficient quality eye tracking data to be included in this analysis, 14 had no trials during which eyes were closed for an extended period, and in the remaining 12 with at least one such trial the median number of trials was 3.5.
Neural network model
Simulations were performed using a predictive coding formulation of the classic interactive activation model6,7. We begin by explaining the model at an abstract level, then outline the algorithmic and mathematical details in generic terms, and then specify the exact settings we used for our model architecture, and how we used them in our simulations.
The interactive activation model is a hierarchical neural network model which takes visual features as inputs, integrates these features to recognise letters, and then integrates letters to recognise words. Critically, activity in word-units is propagated back to the letter-level, making the letter detectors sensitive not only to the presence of features (such as the vertical bar in the letter E), but also to neighbouring letters (such as the orthographic context HOUS_ preceding the letter E). This provides a top–down explanation for context effects in letter perception, such as (pseudo)word superiority. The predictive coding formulation of this model was first described by Spratling14. It uses a particular implementation of predictive coding—the PC/BC-DIM algorithm—that reformulates predictive coding (PC) to make it compatible with Biased Competition (BC) and uses Divisive Input Modulation (DIM) as the method for updating error and prediction activations. The goal of the network is to infer the hidden cause of a given pattern of inputs (e.g. the 'hidden' letter underlying a pattern of visual features) and create an internal reconstruction of the input. Note that the reconstruction is model-driven and not a copy of the input. Indeed, when the input is noisy or incomplete, the reconstruction will ideally be a denoised or pattern-completed version of the input pattern. Inference can be done hierarchically: at the letter-level, predictions represent latent letters given patterns of features, whilst at the word-level predictions represent latent words given patterns of letters (and reconstructions, inversely, represent reconstructed patterns of letters given the predicted word).
Mathematically, the network can be conveniently described as consisting of three components: prediction units (y), reconstruction units (r) and error units (e) that can be captured in only three equations. First, at each level, error units combine the input pattern (x) and the reconstruction of the input (r) to compute the prediction error (e):
$${\mathbf{e}} = {\mathbf{x}} \oslash \left[ {\mathrm{r}} \right]_{\varepsilon _2}.$$
Here, x is a (m by 1) input vector; r is a (m by 1) vector of reconstructed input activations, ∅ denotes pointwise division and the square brackets denote a max operator: [v]∈=max(∈, v). This max-operator prevents division-by-zero errors when all prediction units are silent and there is no reconstruction. Following Spratling14, we set ∈2 at 1 × 10−3. Division sets the algorithm apart from other versions of predictive coding that use subtraction to calculate the error (see Spratling61 for review). The prediction is computed from the error via pointwise and matrix multiplication:
$${\mathbf{y}} \leftarrow \left[ {\mathbf{y}} \right]_{{\it{\epsilon }}_1} \otimes {\mathrm{We}}.$$
Here, W is a (n by m) matrix of feedforward weights that map inputs onto latent causes (e.g. letters), ⊗ denotes pointwise multiplication, square brackets represents a max operator and ∈1 is set at 1 × 10−6. Each row of W maps the pattern of inputs to a specific prediction unit representing a specific latent cause (such as the letter) and can hence be thought of as the 'preferred stimulus' or basis vector for that prediction unit. The entire W matrix is then best thought of as comprising the layer's model of its environment. Finally, from the distribution of activities of the prediction units (y), the reconstruction of expected input features (r) is calculated as a simple linear generative model:
$${\mathbf{r}}\, {\mathrm{ = Vy}},$$
where V is a (m by n) matrix of feedback weights that map predicted latent causes (e.g. letters) back to their elementary features (e.g. strokes) to create an internal reconstruction of the predicted input, given the current state estimate. As in many multilayer networks, the model adheres to a form of weight symmetry: V is almost identical to WT, but its values are values normalised so that each column sums to one. To perform inference, prediction units can be initialised at zero (or with random values) and the Eqs. (2–4) are updated iteratively. To perform top–down hierarchical inference, reconstructions from a higher-order stage (e.g. recognising words) can be sent back to the lower-order stage (e.g. recognising letters) as additional input. To accommodate these recurrent inputs, additional weights have to be defined that are added to W and V as extra columns and rows, respectively. The strength of these weights is scaled to control the reliance on top–down predictions.
Architecture specification
The interactive activation architecture we used was a modification of the network described and implemented by Spratling14, extended to recognise five-letter words, trained on the Dutch subtlex vocabulary, and with a slight change in letter composition. Letters are presented to the network using a simulated font adapted from the one described by Rumelhart and Siple62 that composes any character using 14 strokes (Supplementary Fig. 12). For our five-letter network, the input layer comprises five 14-dimensional vectors (one per character) that each represent the presence of 14 line segments for one letter position. Note that conceptually it is easier to partition the input into five 14-dimensional vectors, in reality these were concatenated into a single 70-dimensional vector x.
At the first level, weight matrix W has 180 rows 250 columns: rows comprise five slots of 36 alphanumeric units (5 × 36 = 180); the first columns comprise five slots of 14 input features (5 × 14 = 70) and the last 180 columns route the top–down reconstruction from the word level. To define the weights of 70 (feedforward) columns, we used encoding function ϕ(c) that takes an alphanumeric character and maps it into a binary visual feature vector. For each alphanumeric character, the resulting feature vector was concatenated five times and the resulting 70 dimensional vector comprised the first row. This was repeated for all 36 alphanumeric characters and concatenated five times. The resulting numbers were then normalised so that the columns summed to one. Then we added the weights of the second 180 columns (inter-regional feedback coming from 5 × 36 letter reconstructions) were simply a 180 by 180 identity matrix multiplied by a scaling factor to control top–down strength. For our 'top–down model' (Fig. 3b), we set the scaling factor at 0.4; in the 'bottom-up model', we set it to 10−6 to effectively cancel the influence of feedback, resulting in a 'bottom-up' model. At the second level, weight matrix W had 6778 rows and 180 columns, representing 6776 Dutch five-letter words from the subtlex corpus, plus the two learned nonword targets (that we included in the vocabulary as participants learned these during training) and five times 36 alphanumeric characters. The orthographic frequency of letters as specified by the corpus was hard coded into the weights and then normalised to sum to one.
Although there are substantial implementational differences between this model and the classic connectionist version of the interactive activation model6,7, the version described here has been shown to capture all key experimental phenomena of the original model (see ref. 14 for details). Since our simulations only tried to validate and demonstrate a qualitative principle, not subtle quantitative effects, the exact numerical differences related to the differences in implementation should not matter for the effect we demonstrate here.
Because our paradigm is different from classical paradigms, we performed simulations to confirm that the top–down account indeed predicts the representational enhancement we set out to detect. Although the main simulation result (Fig. 3a) is not novel, our simulation, by mirroring our paradigm, departs from earlier simulations in some aspects, which we will clarify before going into the implementation details. First, most word superiority studies present stimuli near-threshold: words are presented briefly, followed by a mask, and average identification accuracies typically lie between 60 and 80%. This is mirrored in most classic simulations, where stimuli are presented to the network for a limited number of iterations and followed by a mask, leading to similar predicted response accuracies7,14. In our task, stimuli are presented for almost a second, and at least the critical middle letter is always clearly visible. This is mirrored in our simulations, where stimuli are presented to the network until convergence and predicted response accuracies of the network are virtually 100% in all conditions (see Supplementary Fig. 2). As such, an important aspect to verify was that enhancement of a critical letter can still occur when it is well-above threshold and response accuracy would be virtually at 100% already. Second, our simulations used the same Dutch word and nonword materials used in the experiment. This includes the occurrence of learned targets in the nonword condition, which we added to the vocabulary of the network and were hence a source of contamination as 12% of the items in the nonword condition were in fact in the vocabulary. Finally, unlike classical simulations, stimuli were corrupted by visual noise.
For Fig. 3a, we simulated 34 artificial 'runs'. In each run, 48 words and 48 nonwords were presented to a network with feedback connections (feedback weight strength 0.4) and without word-to-letter feedback (feedback weight strength 10−6). The same Dutch, five-letter (non)words were used as in the main experiment, and like in the experiment 12% of the (non)word items were replaced by target items. Critically, the nonword targets were learned and hence were part of the vocabulary of the network. To present a (non)word to the network, each character c has to be first encoded into a set of visual features and then corrupted by visual noise to produce an input vector x:
$${\mathbf{x}} = {\upvarphi}\left( c \right) + {\cal{N}}\left( {\mu ,\,\sigma ^2} \right).$$
For μ we used 0, σ was set to 0.125, and any values of x that became negative after adding white noise were zeroed. The network then tried to recognise the word by iteratively updating its activations using Eqs. (2–4), for 60 iterations. To compute the 'relative evidence' metric we used in Fig. 3a to quantify representational quality q(y), we simply take the fraction of activation for the correct letter (yi) of the sum of letter activations for all characters at the third slot:
$$q\left( {\mathbf{y}} \right) = \frac{{{\mathbf{y}}_i}}{{\mathop {\sum }\nolimits_{j \, = \, 37}^{73} {\mathbf{y}}_j}}.$$
Finally, to compute predicted response probabilities as in Supplementary Fig. 2, we followed McClelland and Rumelhart to use Luce's rule to compute responses probabilistically:
$$p\left( {R_i} \right) = \frac{{{\mathrm{e}}^{\beta {\mathrm{y}}_{\mathrm{i}}}}}{{\mathop {\sum }\nolimits_{j \, = \, 37}^{73} {\mathrm{e}}^{\beta {\mathrm{y}}_j}}}.$$
The β parameter (or inverse softmax temperature) determines how rapidly the response probability grows as yi increases (i.e. the 'hardness' of the argmax operation) and was set at 10, following McClelland and Rumelhart6,7, but results are similar for any typical beta value that is approximately in the same order of magnitude.
All simulations were performed using custom MATLAB code, which was an adaptation and extension of the MATLAB implementation published by Spratling14.
All raw data required to reproduce all analyses and figures are uploaded onto the Donders Data Repository and can be found at http://hdl.handle.net/11633/aacjymw7. A reporting summary for this Article is available as a Supplementary Information file.
Code availability
All code required to reproduce all analyses and figures are uploaded onto the Donders Data Repository and can be found at http://hdl.handle.net/11633/aacjymw7.
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This work was supported by The Netherlands Organisation for Scientific Research (NWO Research Talent grant to M.H.; NWO Vidi grant to F.P.d.L.; 016.Veni.195.435 to M.E.; Gravitation Program Grant Language in Interaction no. 024.001.006 to P.H.) and the European Union Horizon 2020 Program (ERC Starting Grant 678286, "Contextvision" to F.P.d.L). We thank Ashley Lewis for helpful comments on and discussions of an earlier version of this manuscript.
Donders Institute for Brain, Cognition and Behaviour, Radboud University, NL-6500, HB, Nijmegen, The Netherlands
Micha Heilbron, David Richter, Matthias Ekman, Peter Hagoort & Floris P. de Lange
Max Planck Institute for Psycholinguistics, 6525 XD, Nijmegen, The Netherlands
Micha Heilbron & Peter Hagoort
Micha Heilbron
David Richter
Matthias Ekman
Peter Hagoort
Floris P. de Lange
M.H., F.P.d.L., P.H., D.R. and M.E. designed the study. M.H. and D.R. collected the data. M.H., D.R., M.E. and F.P.d.L. conceived of the analysis plan. M.H. analysed the data. M.H. performed simulations. M.H. wrote the initial draft. All authors contributed to the final manuscript.
Correspondence to Micha Heilbron.
The authors declare no competing interests.
Peer review information Nature Communications thanks Paul Downing, Jason Yeatman and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
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Heilbron, M., Richter, D., Ekman, M. et al. Word contexts enhance the neural representation of individual letters in early visual cortex. Nat Commun 11, 321 (2020). https://doi.org/10.1038/s41467-019-13996-4
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Nature Communications (Nat Commun) ISSN 2041-1723 (online) | CommonCrawl |
Methodology Article
Dynamics of animal joint space use: a novel application of a time series approach
Justin T. French ORCID: orcid.org/0000-0002-0726-82241,
Hsiao-Hsuan Wang1,
William E. Grant1 &
John M. Tomeček1
Movement Ecology volume 7, Article number: 38 (2019) Cite this article
Animal use is a dynamic phenomenon, emerging from the movements of animals responding to a changing environment. Interactions between animals are reflected in patterns of joint space use, which are also dynamic. High frequency sampling associated with GPS telemetry provides detailed data that capture space use through time. However, common analyses treat joint space use as static over relatively long periods, masking potentially important changes. Furthermore, linking temporal variation in interactions to covariates remains cumbersome. We propose a novel method for analyzing the dynamics of joint space use that permits straightforward incorporation of covariates. This method builds upon tools commonly used by researchers, including kernel density estimators, utilization distribution intersection metrics, and extensions of linear models.
We treat the intersection of the utilization distributions of two individuals as a time series. The series is linked to covariates using copula-based marginal beta regression, an alternative to generalized linear models. This approach accommodates temporal autocorrelation and the bounded nature of the response variable. Parameters are easily estimated with maximum likelihood and trend and error structures can be modeled separately. We demonstrate the approach by analyzing simulated data from two hypothetical individuals with known utilization distributions, as well as field data from two coyotes (Canis latrans) responding to appearance of a carrion resource in southern Texas.
Our analysis of simulated data indicated reasonably precise estimates of joint space use can be achieved with commonly used GPS sampling rates (s.e.=0.029 at 150 locations per interval). Our analysis of field data identified an increase in spatial interactions between the coyotes that persisted for the duration of the study, beyond the expected duration of the carrion resource. Our analysis also identified a period of increased spatial interactions before appearance of the resource, which would not have been identified by previous methods.
We present a new approach to the analysis of joint space use through time, building upon tools commonly used by ecologists, that permits a new level of detail in the analysis of animal interactions. The results are easily interpretable and account for the nuances of bounded serial data in an elegant way.
Quantifying spatial overlap, or joint space, use between individual animals is of interest in many branches of ecology. How animals utilize space is a function of many factors, including resource availability [1], risk [2], and competition [3]. How these factors affect interactions between individuals is of key importance for many ecological issues. For example, joint space use has been linked to animal contact rates, and thus disease transmission [4, 5], animal social behavior [6, 7], as well as population genetics [8]. Though a common procedure, the analysis of joint space use remains problematic [9].
Ecologists commonly analyze space use in terms of an animal's utilization distribution (hereafter UD), the 2-dimensional relative frequency (probability) distribution of animal locations in space [10]. UDs provide a continuous representation of the relative amount of time an animal spent at a given location, or the intensity of space use, facilitating easy interpretation. The probabilistic nature of UDs provides attractive properties that make them useful for home range estimation. For example, taking the isopleth contour at a given probability density can provide a demarcation of where an animal spent an arbitrary proportion of its time [10]. However, utilizing the parent distribution in further analyses permits deeper inference into the spatial interactions between individuals.
Quantifying the degree of joint space use between 2 individuals permits the testing of a variety of hypotheses about inter-individual interactions [11]. The 3-dimensional intersection of 2 UDs provides an estimate of spatial overlap that incorporates information about the relative intensity of space use by each individual. This provides a more robust estimate of joint space use compared to 2-dimensional approaches that use the shared area of UD isopleths. This joint volume can be measured using several indices, however Bhattacharyya's Affinity (BA; [12]) has been shown to be minimally biased and has attractive properties that lend interpretability [11]. BA scales from 0 to 1, where 0 represents no spatial overlap and 1 represents identical space use. Theoretical bounds on behavioral metrics greatly facilitate ecological interpretation [13]. Several authors have utilized these pairwise comparisons to examine changes in joint space use between blocks of time (sensu [4, 14, 15]).
Though a common procedure in ecological literature, such an analysis oversimplifies temporal variation in joint space use. These interactions are dynamic in both time and space, making analysis of interactions inherently high-dimensional. Comparisons between few, relatively long time blocks provide limited insight into these processes, and overlook considerable temporal detail. Furthermore, they implicitly assume that animal space use patterns are stationary, or unchanging within the time period over which UDs are estimated [16]. This is unlikely to be the case for long periods of time, but such an assumption is much more reasonable over shorter intervals. Comparing UDs over finer, regular intervals (e.g. week or month) would reveal considerably more detail in patterns of spatial interactions, and permit statistical analysis of interaction dynamics, which was previously elusive [17].
We achieve such an analysis with a novel approach that synthesizes tools already familiar to ecologists and applies an appropriate regression framework. Abrahms et al. [18] derived a UD-based index of space use stability by measuring the intersections of successive monthly UD estimates for an individual. Though they did not consider trends in the sequence of measurements, their approach is readily extendable to examine dynamic interactions using a time series framework [17, 19], a logical avenue for the analysis of space use dynamics. When coupled, existing UD intersection metrics and time series analyses provide a simple, interpretable, and rigorously testable summary of complex dynamics of joint space use. This reduces a 5-dimensional problem (latitude, longitude, use intensity of 2 individuals, and time) to 2 manageable dimensions (spatial overlap and time). However, the bounded nature of BA precludes the use of standard regression procedures, such as normal linear regression or generalized linear models (GLMs). This is because GLMs are strictly suited to distributions with orthogonal (independent) parameters. The orthogonality assumption is violated when dispersion depends on the mean, which is a key property of bounded variables [20]. Other, analogous methods are needed to link the index to covariates.
Copula regression methods are a commonly used alternative to traditional GLMs in the financial and actuarial sectors [21] though, to our knowledge, their use in ecology is limited to one example [22]. They accommodate any response distribution, and are used to model complex correlation structures [23]. Recent work extends these methods to bounded time series [24], providing a link between the intersection index and explanatory variables.
Extending UD intersection metrics to a time series framework provides a flexible and interpretable approach to the analysis of space use interactions between individuals. Modeling joint space use in this way shows how the proportion of time 2 individuals use the same places changes through time, which is not only mathematically tractable, but intuitively understandable. This makes the results of our approach simple to communicate to both peers and non-scientists alike.
The success of this framework depends on the precision with which BA can be estimated with current GPS technology, which will affect both the sampling distribution of BA itself and the estimates of the effect of covariates on BA. Therefore, the goals of this work are: 1) To determine the precision with which BA could be estimated over reasonable sampling intensities; 2) to evaluate the accuracy and precision of effect size estimates of a covariate; and 3) to demonstrate the application of our methodology to a real data set. We simulated GPS data sets arising from known UDs at varying sampling intensities, then examined the precision of BA estimates from these simulations at high and low true values. We then evaluated the accuracy and precision of effect size estimates as sampling intensity increases. Finally, as an example, we examined the change in spatial interaction of 2 coyotes (Canis latrans) in southern Texas in response to a carrion deposition event.
Simulation study
We expanded simulation methods previously developed to evaluate kernel density estimator (KDE; [25]) performance as home range estimators [26, 27]. We used these simulations to a produce a known series of BA values with which we could compare estimates (Fig. 2). Each series consisted of 100 time windows (t). The true UD of each individual was held constant for the first half of the series, shifted to produce a known change in BA at t=50, and then held constant throughout the remainder of the series. We drew a specified number of locations randomly from the true UD of each individual at each time window, representing artificial GPS location data, to examine bias and precision as sampling intensity increases. By defining time periods a priori, we separate this analysis from home range estimation [27, 28]. In this context, an autocorrelated movement model would lead to an observed movement pattern that did not reflect the true UD on which we based our BA calculation. We sampled randomly from the true UD in order to ensure consistency between the within-window range and the location samples. We used simple bivariate normal (BVN) UDs with equal, unit variances with means separated by a fixed distance. We induced a 0.60 change in BA, from 0.20 to 0.80, at t=50 by changing the distances between means from 3.580 to 1.319.
We used a fixed KDE to fit a UD estimate for each individual at each time window. We used a bivariate normal kernel according to
$$ \widehat{UD_{it}} = \frac{1}{nh^{2}}\sum^{n}_{i-1}\frac{1}{2\pi} exp\left(\frac{-(\mathbf{x} - \mathbf{X}_{i})'(\mathbf{x} - \mathbf{X}_{i})}{2h^{2}}\right) $$
where \(\widehat {UD_{it}}\) is the estimated UD surface of animal i at time t, x is any location in 2-d space, Xi is the ith observation of the animal's location, n is the number of observations, and h is a smoothing parameter [25]. We used the reference smoothing parameter for computational simplicity, calculated as
$$ h = \sqrt{\frac{s^{2}_{x} + s^{2}_{y}}{2}}\cdot n^{-1/6} $$
where \(s^{2}_{x}\) and \(s^{2}_{y}\) are the variances of the x and y coordinates, respectively [29].
We then calculated BA between the 2 simulated individuals at each time window to obtain a series of BA estimates,
$$ BA_{t} = \iint{\sqrt{\widehat{UD_{1t}}(x,y)}*\sqrt{\widehat{UD_{2t}} (x,y)} dx dy} $$
where \(\widehat {UD_{1t}}\) and \(\widehat {UD_{2t}}\) are the UD estimates of individuals 1 and 2, respectively, at time t. We evaluated the bias and precision of BA estimates for sampling intensities of 50–1000 locations per temporal window, at increments of 50. We fit KDEs and calculated BA using the adehabitatHR package [30] in R [31].
We then evaluated how well we could estimate the effect size (magnitude of change) in BA due to our simulated disturbance at t=50. We used a marginal beta regression with a Gaussian copula [24] of the form
$$ \begin{aligned} Y_{t}|X \sim Beta(\mu_{t},\kappa_{t})\\ logit(\mu_{t}) = X^{\top}_{t}\beta \end{aligned} $$
where Yt|X is the value of the BA series at time t, given covariates X, μt and κt are the mean and precision of the beta distribution at time t, respectively, and β is the vector of regression coefficients. Copula methods exploit the probability integral transformation to relate the beta distributed response Yt to covariates Xt,
$$ Y_{t} = F^{-1}_{t}\{\Phi(\epsilon_{t});\beta\} $$
where Yt is assumed to be marginally beta distributed, \(F^{-1}_{t}\{\cdot ;\beta \}\) represents the appropriate cumulative density function linking the density to covariates (see [24]), and Φ(εt) is the cumulative distribution function of the normal distribution with mean 0 and variance εt. This allows the use of autoreggresive and moving average (ARMA(p,q)) terms, which are a special case of a multivariate normal covariance matrix [32], to model serial dependence in a non-Gaussian context [24]. The ARMA(p,q) term is defined as
$$ \epsilon_{t} = \sum^{p}_{i = 1}\psi_{i}\epsilon_{t-i} + \sum^{q}_{j = 1}\lambda_{j}\eta_{t-j} + \eta_{t} $$
where εt−i is the error of the previous observation, ψi is an autoregressive parameter vector, λj is a moving average parameter vector, and ηt are independent zero-mean normal variables [24]. Parameters are estimated with maximum likelihood. The copula-based approach separates the linear predictor from the correlated error structure, meaning the regression coefficients are interpreted in the same manner as a GLM and not confounded by the ARMA(p,q) term. We refer interested readers to [24] for a detailed treatment on the role and advantages of copulas in the analysis of bounded time series.
We fit marginal beta regression models using a binary covariate corresponding to the known change in UDs at t=50 using the gcmr package [33] in R [31]. In ecological terms, this is analogous to estimating the effect of the presence of a resource, the implementation of some disturbance, a hypothesized season, or some other relevant binary variable, on the degree of spatial interaction between two individuals. We replicated the entire process 100 times for each level of sampling intensity to obtain the sampling distribution of our effect size as a function of sampling intensity.
Application to empirical data
We then used field data representing 2 coyotes to demonstrate the practical utility of our approach in describing the dynamics of animal space use (Fig. 1). We collected these data on the East Foundation's 61,000 ha San Antonio Viejo Ranch (SAVR) in Jim Hogg and Starr counties in southern Texas. The East Foundation's ranches are managed as a living laboratory to promote the advancement of land stewardship through ranching, science, and education. The area is dominated by shrub savannas, primarily composed of honey mesquite (Prosopis glandulosa), prickly pear (Opuntia spp.), cat-claw acacia (Acacia greggii), blackbrush (Acacia rigidula), whitebrush (Alloysia gratissima), and granjeño (Celtis palida), with early to mid-successional grasses, including three-awns (Aristida spp.), little bluestem (Schizachyrium scoparium) and windmill grasses (Chloris spp.).
Territories of the 2 GPS-collared, coyotes M09 and F13, used in our example from the East Foundation's San Antonio Viejo Ranch. Territories were delineated using the 75% isopleth of a fixed kernel density estimate of all locations for each individual. Note the location of the carrion resource near, but outside, both territories
Distribution of estimated Bhattacharyya's Affinity (BA) values as sampling intensity increases. Blue lines represent the true BA values of the parent utilization distributions
We captured individuals by helicopter using a net gun [34], fitted them with a Vertex Plus or Vertex Lite GPS collar (Vectronic Aerospace GmbH, Berlin), and released them at the site of capture on 10 December 2016 (n = 1) and 1 April 2017 (n = 1) as part of an ongoing study of coyote space use. These collars collected location data every 2 hours until 31 December 2017, when they automatically released from the animal. While our collars collected location data on identical schedules, this is not strictly necessary, as long as collars collect comparable numbers of locations over the same time windows. To standardize across collars, we omitted data prior to 1 April 2017 from the analyses presented below. Both coyotes were considered territorial [35], and occupied distinct, non-overlapping territories. A domestic cow (Bos taurus x B. indicus) died of unknown causes in an area well outside both territories (Fig. 1) during the week of 23 September 2017. Coyotes alter their patterns of space use to utilize carrion resources [36], so this event afforded us the opportunity to evaluate whether our methods would detect a change in spatial overlap between the coyotes in response to the presence of carrion.
We included time relative to death of the cow (before or after) as a dummy coded variable
$$ \begin{aligned} x_{t} \in \{0,1\} \\ x_{t} = \left\{\begin{array}{ll} 0,& \text{if}\,\, t < t_{carrion} \\ 1,& \text{if} \,\,t \geq t_{carrion} \end{array}\right\} \end{aligned} $$
where tcarrion is the week of carrion deposition, to test whether that event had a persistent effect on the mean BA. Autocorrelation was modeled with ARMA(1,1) terms. This model is consistent with an interrupted time series design [37] and is analogous to an ANOVA for a beta-distributed variable with serial dependence. The resulting regression form consists of the marginal model
$$ \begin{aligned} BA_{t}|x_{t} \sim Beta(\mu_{t},\kappa_{t}) \\ logit(\mu_{t}) = x_{t}\beta_{1} + \beta_{0} \end{aligned} $$
and copula
$$ \begin{aligned} \Phi(\epsilon_{t})\\ \epsilon_{t} \sim ARMA(1, 1) \end{aligned} $$
Succinctly, this model tests for a persistent change in spatial interaction between 2 coyotes following the carrion deposition event, and estimates its magnitude.
Our simulation showed that reasonably precise estimates of BA can be achieved with 150 sampled locations per time window at both high and low values of BA (s.e.=0.029; Fig. 2). Estimates based on as few as 50 relocations per window could be useful if the hypothesized effect of some covariate is sufficiently large. These results also suggest a slight positive bias at low BA values, which decreases with sampling intensity. At 50 locations per window, the average bias at a true BA of 0.20 was 0.0311 (SE= 0.00919), while at a true BA of 0.80 the average bias was -0.00077 (SE= 0.00641). The bias at low BA declined with increasing sampling intensity to 0.0155 (SE= 0.00253). The average bias at high true BA values never exceeded 0.0105 (SE= 0.00342).
Parameter estimates from regression models stabilized quickly at 150 relocations, while error around the prediction slowly contracts beyond that point (Fig. 3). These estimates were slightly negatively biased, with an average bias of -0.0427 (se= 0.00106) at 50 locations/window, decreasing to a minimum of -0.00508 (se= 0.00106) as sampling intensity increased. This is likely due to the slight positive bias of low-valued BA estimates, which was strongly correlated with effect size bias across simulations (r= -0.784).
Estimated effect size of binary covariate on Bhattacharyya's Affinity (BA) as a function of sampling intensity (sampled locations per time window). The blue line represents the true effect size
Application to coyote data
The time series of BA values between the two coyotes indicated an obvious change in behavior following the appearance of the carrion resource (Fig. 4) and the beta regression model showed a significant effect of the carrion event (P<0.001; Fig. 4). The average UD intersection increased by 0.246, meaning that, on average, the 2 coyotes spent approximately 25% more time in the same places following the carrion deposition event. Upper and lower 95% CIs of this effect were 0.437 and 0.092, respectively. The graphs of observed and fitted values (Fig. 4), and the residuals (Fig. 5a) showed unaccounted structural differences between weeks 0–9 and weeks 10–24. Weeks 20, 27, 29, and 36 were identified as potential outliers (Fig. 5b), but overall the distributional form was appropriate. The ARMA(1,1) terms were significant (P<0.001 for both). Autocorrelation diagnostic plots supported the appropriateness of the assumed autocorrelation structure (Fig. 5c-d).
Time series of joint space use between the 2 GPS-collared coyotes from the East Foundation's San Antonio Viejo Ranch, measured by Bhattacharyya's Affinity (BA; blue line) and fitted values of the copula regression model (black, dashed line)
Residual diagnostics of beta regression model of two coyotes from the East Foundation's San Antonio Viejo Ranch. a The plot of residuals through time shows an unaccounted for structural difference between weeks 0-9 and subsequent weeks, as well as potential outliers at weeks 20, 27, 29, and 36. b The Q-Q plot shows reasonable model performance, again suggesting possible outliers at weeks 20 and 36. c-d Autocorrelation and partial autocorrelation plots show no significant residual autocorrelation, meaning the ARMA(1,1) term adequately captured the dependence structure
Our results are a proof of concept for the use of sequential measurements of UD intersections in a time series framework to capture dynamics of spatial interactions between 2 individuals. Results with simulated data reveal slight positive biases in low-valued BA estimates leading to slight negative biases in effect size estimates. However, the effect of such small biases on the ecological interpretation of results likely would be negligible in most cases. Further, sampling error is reasonable at achievable sample sizes with current GPS technology. Our framework is based on familiar analytic tools and results are readily interpretable. The framework also provides a much more detailed view of interactions through time compared to existing methods, as we demonstrated with the coyote example.
Practical application and performance
Our methodology is applicable to a wide variety of ecological questions where there is an a priori hypothesis about the drivers of joint space use. Our coyote example focuses on the presence of a resource, however the imposition of some disturbance, management action, or life history events (e.g. breeding associated behavior) are equally well treated with our approach. Because our approach is couched in a regression context, continuous covariates are also valid, though beyond the scope of our simulations. These could include such variables as available forage, precipitation, or temperature extremes within time windows, or the researcher could include cosine transformations of time to evaluate seasonal effects, to name but a few. This allows considerable flexibility to address questions of joint space use.
The length of the temporal window over which UDs are estimated is a key consideration in applying this analysis. The appropriate choice will depend on the temporal scale of the motivating question and the ecology of the species. The length of time window must be matched to the scale of the phenomenon of interest. Specifically, the window must be fine enough to capture variation in joint space use attributable to the phenomenon [38]. Highly mobile animals, that change their patterns of space use often, may require shorter windows in order to capture relevant variation in joint space use than sedentary species. For example, cougars (Puma concolor) are known to exhibit frequent, recursive space use patterns [39], which would require short time windows relative to their return frequency to capture. The analysis may also be conducted with multiple window lengths to examine how overlap varies with temporal scale, allowing the researcher to identify when individuals partition space at fine temporal scales but overlap at larger ones. However, the finest temporal scale that can be considered is limited by the number of locations required to adequately estimate a UD.
Various authors have reported minimum numbers of locations required to obtain a reliable UD estimate with the methods we used [26, 29, 40]. Our simulations show acceptable results using a first-generation estimator with 150 samples per UD window and 100 windows, approximating hourly collection intervals over a 2-year period. This sampling regime is common for larger species [41–43], yielding 168 locations per week. This sampling intensity is sufficient to generate reliable UDs, given the inherently unbiased design of sampling at regular time intervals [26, 29], and gave adequate performance in our simulations. This sampling intensity is relatively easy to achieve for large species, but presently unattainable for smaller species incapable of carrying large batteries. These constraints may be alleviated by improvements in battery technology and efficiency of GPS collar circuits, as well as more efficient UD estimators.
The precision of BA estimates is a function of the performance of the KDE method used. While we utilized a first-generation estimator for simplicity and computational speed, any KDE method is suitable for this approach and the appropriate estimator will depend on the particular research question [16, 44]. Given that the true UDs in our simulations were bivariate normal, our use of the reference parameter is justified in the literature [25, 26]. However, this procedure is known to overestimate the 95% isopleth area of more complex UDs [26, 45, 46], suggesting that the density in the tails of the UD is overestimated. This may also be the case in our simulations, which would explain the greater degree of bias when the UDs intersect mainly in their tails (at low true BA values). This greater positive bias at low values would compress effect size estimates in cases when BA increased following disturbance, as in our simulations. On the other hand, if the effect was negative following the disturbance, its magnitude would be slightly overestimated. The magnitude of the bias is small in either case, as indicated at our lowest sampling intensity. A bias of 3% (our largest average bias) is unlikely to affect ecological interpretation of results, and may be safely considered negligible in most cases. More sophisticated methods may be less biased in the tails of the UD, reducing bias in parameter estimates. The relative performance of various KDE procedures within this context is an open question that warrants further research.
Beyond technological improvements, there are analytical limitations to overcome to realize the full potential of our approach. Our techniques provide pair-level series, permitting analysis at the dyad level. Population level inference will require multivariate time series methodologies that accommodate potentially non-independent, beta-distributed response variables, which to our knowledge are currently unavailable. However, such methods do exist for short, non-stationary, Gaussian series that could serve as a conceptual basis for similar approaches with beta-distributed response variables [47]. Additionally, the approach we demonstrate here treats BA measurements as fixed values, though we show that they are estimated with error. Recent work provides a potential means to handle this source of error [9], and an appropriate hierarchical structure could be derived. Such development would be particularly important in sampling situations like our coyote example. Our simulation results suggest that sampling error of UDs at our bi-hourly schedule (84 locations/week) is appreciable at the lower BA values we observed between these individuals throughout the monitoring period (Figs. 2 and 4), thus the uncertainty of our parameter estimates may be particularly underestimated.
Advantages of this approach
The residual analysis of the beta regression model of coyote interactions reveals an important advantage of our approach; there is another period of interaction early in the series that we have captured, but failed to explain (Fig. 5). This early period of interaction would have been masked in a simple analysis of UD intersections before and after the death of the cow, as would be done using previous methods. Assuming space use itself to be stationary over these time blocks is unwarranted. The time series framework we propose captures the nonstationary dynamics of space use patterns and provides a means to explain them. Additionally, our methodology yields a statistical test of the effect that until now was not possible. Although [9] produced a method to test the significance of a single BA estimate, our framework permits modeling the influence of 1 or more variables on the dynamics of joint space use in an interpretable way.
Each stage of our framework was selected for straightforward interpretability (Fig. 6). The probabilistic nature of UDs, and their widespread use by ecologists make them an attractive starting point. The intuitive interpretation of BA as a symmetric index of how much 2 individuals use the same space makes it a natural choice. More subtly, the choice of marginal copula regression over other appropriate time series methods also aids interpretability. The separation of the regression component from the correlated error structure allows straightforward interpretation of model coefficients, which is not possible with other available methods [24]. Despite the substantially different mathematical architecture, this means that interpretation of model coefficients is done in the same manner as GLMs, which are common in ecological literature. This familiarity makes our approach easily accessible to ecologists.
Visualization of the quantification of joint space use by the 2 coyotes from the East Foundation's San Antonio Viejo Ranch during the week prior to the carrion deposition event (t23: carrion location marked with green dot) and during the week in which the event occurred (t24). Relocation data are analyzed to estimate the 2 individual space utilization distributions (UD; red dots and shading for the female, blue for the male), from which the joint UD volume is calculated (the integral of which is BA), which indicates the area of joint space use (green shading)
Fine scale dynamics, such as how movement trajectories change, or patterns in the distances between individuals could also be considered to examine inter-individual interactions [48, 49]. However, these approaches focus on fine-scale properties of movement, and answer related, but different questions [50]. Indeed, such analyses could serve as complimentary tools to our method. For example, joint space use may be used to examine similarity in habitat use, while information on the distances between individuals would provide information on how those individuals respond to each other at a finer scale (e.g. avoidance or attraction). Capturing these dynamics through time may elucidate mechanisms of resource partitioning between species.
The results of our approach are also readily visualized, which is of great heuristic value and lends intuitive context to the quantitative results. For example, we can visualize the change in joint space use by the 2 coyotes immediately before and after the carrion deposition event (Fig. 6). Mapping the UDs and the joint UD volume (the integral of which is BA) shows that joint space use before the event was concentrated along the boundary between the 2 territories. After the event, joint space use increased markedly as the female expanded her activity range toward the southeast, engulfing the activity range of the male, which also shifted slightly toward the southeast. Interestingly, both individuals moved synchronously away from the carrion initially, and did not converge on it until the following week, as confirmed by GPS locations converging at the carcass site (Fig. 1). The cause of these movements remains unknown, but their identification provides important contextual information that aids interpretation and the generation of ecologically-based hypotheses.
We argue that these properties also simplify communication of results to scientific peers and non-scientist stakeholders alike. The statement "on average, the 2 coyotes spent 25% more time in the same places each week after the carrion resource became available" is an accurate and meaningful interpretation of our results. An important caveat is that the individuals were not necessarily in those places at the same time within the week. Thus, the temporal grain and scale used in the analysis will affect interpretation. Nonetheless, such a statement carries implications for a variety of disciplines.
Finally, though we discuss linking joint space use to covariates selected for a priori hypotheses, other time series methods are applicable. For example, change detection methods allow researchers to segment time series into periods of similar behavior [51, 52]. These exploratory methods could be of great utility when periods of attraction or avoidance are expected, but when the time of their occurrence is not known. For example, some ungulates are known to partition space between sexes for most of the year, but aggregate during the breeding season [53]. Change detection methods could be used with a BA time series between sexes to objectively delineate when the breeding season occurs.
This work represents a marked advance towards informative, tenable analysis linking variables to the dynamics of joint space use that is also communicable to non-scientists. This methodology has applications in many areas of applied ecology where the dynamics of animal interactions are of interest. Given limited time, money, and material resources, successful management requires focused efforts. Our methodology provides needed information that is intuitively understood by stakeholders. This facilitates effective communication between scientists and decision makers, ideally leading to efficient, spatio-temporally targeted management actions supported by valid analyses.
The authors intend to archive the coyote data with MoveBank (https://www.movebank.org/)
ARMA:
Autoregressive moving average
Bhattacharya's affinity
Kernel density estimate
SAVR:
San Antonio Viejo Ranch
UD:
Utilization distribution
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The authors would like to acknowledge and thank the East Foundation for funding support, land access, and assistance with field work. We also extend our thanks to N.J. Silvy and X.B. Wu for their guidance and input. We thank Carlos Gonzalez, Dana Karelus, Tyler Campbell, and 2 anonymous reviewers for constructive commentary on previous drafts that greatly enhanced the quality of this work. Field data were collected under Texas A&M University AUP No. 2016-016A and Texas Parks and Wildlife scientific research permit No. SPR-0715-097. This is manuscript number 039 for the East Foundation….
This study was funded by the East Foundation. East Foundation personnel participated in coyote captures.
Department of Wildlife & Fisheries Science, Texas A&M University, 534 John Kimbrough Blvd., College Station, 77843, USA
Justin T. French, Hsiao-Hsuan Wang, William E. Grant & John M. Tomeček
Justin T. French
Hsiao-Hsuan Wang
William E. Grant
John M. Tomeček
JF conceived the ideas, designed methodology, performed analyses, and led the writing of the manuscript; JF and JT collected field data; HW and WG contributed greatly to simulations; HW contributed conceptual figures. All authors contributed critically to draft manuscripts and gave final approval for publication….
Correspondence to Justin T. French.
French, J.T., Wang, HH., Grant, W.E. et al. Dynamics of animal joint space use: a novel application of a time series approach. Mov Ecol 7, 38 (2019). https://doi.org/10.1186/s40462-019-0183-3
Bhattacharyya's affinity
Beta distribution
Copula marginal regression
Joint space use
GPS telemetry | CommonCrawl |
LaTeX is a blessing and a curse. It's a blessing because it provides so many nice style features for paper-writing. It's a curse because it is based on TeX, which was designed before PostScript revolutionized concepts in good printing and font design. TeX's font technology was brilliant for its time: but that time was twenty years ago, and it causes problems with current PDF documents and high-resolution printers. Modern LaTeX systems can overcome many of these difficulties, but by default aren't set up to. As such, about 90% of LaTeX authors compile their papers blissfully unaware that their papers are difficult to print nicely in very high resolution (>600 lpi) or display on various PDF viewers. In my experience, the large bulk of LaTeX papers out there using Computer Modern result in a PDF file that looks absolutely hideous when viewed with Adobe Acrobat, MacOS X's Preview.app, and other PDF viewers.
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Error 1: Using Bitmapped Computer Modern Fonts
This is an error that nearly every Computer Modern user does; as a result, nearly every computer modern paper on the web looks well nigh horrible on PDF viewers. It's so frustrating too, because it can be relatively easily fixed.
Computer Modern is by default a bitmapped font. That means that each symbol is rendered as a bunch of dots, rather than as an outline equation. PostScript fonts, TrueType fonts, and other modern fonts are rendered as outline equations. Outline equations scale indefinitely to arbitrarily high precision. Bitmaps look like bitmaps. Here's an example of the word "Conference", in Computer Modern Italic at 12pt bitmapped, blown up 400%:
You can see the bitmaps in action. That's what it looks like printed on your printer, if you look really close. Most Computer Modern users don't notice, because current LaTeX systems render their bitmaps at 600dpi, which is enough for many laser printers, though still bad quality for good webpresses. But bitmaps bulk up your paper considerably and slow down printing. But that's not the big problem. The big problem is that, as far as PostScript and PDF is concerned, your Computer Modern text is nothing but lots of tiny bitmaps. Lacking true information about what the font's really supposed to look like, many PDF viewers, like Acrobat Reader, will antialias the bitmaps, resulting in a nasty mess. Here's a snippet of an actual Computer Modern paper I reviewed, when viewed in Acrobat Reader at 100%:
That's pretty typical of Computer Modern papers. Again, this is because LaTeX here isn't actually using real fonts -- it's really constructing a PDF consisting of thousands of little bitmap pictures. Yuck!
Fortunately, you can fix this, but you'll have to figure out how to set it up on your system. Modern LaTeX distributions, particularly teTeX, come with PostScript (outline equation) versions of Computer Modern. These are known as the "Blue Sky" Computer Modern fonts, and they're free. Unfortunately, these distributions usually have the PostScript versions turned off by default. You'll need to figure out how to turn them on in your system. In tetex, you might be able to do it by editing the file /usr/local/teTeX/share/texmf/dvips/config/updmap, and turning on the type1_default=true switch, then executing the updmap script. This may or may not work. If you don't have the Blue Sky fonts on your system, you can download them for free from the American Mathematical Society. I can say that tetex/TeXShop.app for MacOS X has the fonts turned on by default now.
Here's Conference again, this time using the Blue Sky Computer Modern fonts, blown up to 400%:
Ahhhh... finally a professional look. And I retyped the above bibliographic entry, only using the Blue Sky Computer Modern font. Here's what it looks like in Acrobat Reader, shown at 100%:
As you can see, a huge difference in readability. It looks even better with ClearType turned on.
It's not a good thing to be publishing PDF documents with Computer Modern bitmaps. Even the lowliest information technology worker is pumping out Microsoft Word documents using good-quality outline fonts. As AI researchers, we're claiming to have a modicum of computer experience, aren't we? Bitmap fonts are computer technology that's twenty years out of date. We need to get with the program.
If your paper has no math in it, another option is to do \usepackage{times} etc. to use a modern outline PostScript font such as Times Roman, New Century Schoolbook, etc. Here's the same text in Times Roman:
However, if you use any math, the math will still be in the Computer Modern bitmaps. So regardless, it's well worth it to get the Blue Sky Computer Modern fonts working right on your system.
Error 2: Mixing Computer Modern and Times
\begin{rant}As much respect as I have for its creator, Computer Modern is, in my opinion, a bad font. It looks fine for mathematics equations, but in paragraphs of text, it is difficult to read because its letterforms are far too thin and light and the bowls are too wide and different from one another. The contrast is very poor. I think it pales in comparison to modern fonts by professional font designers.\end{rant}
Okay, that's over. But there are two real, serious problems with Computer Modern: first, it is a very very wide font, which translates directly into you not being able to write nearly as much text in your conference paper. By substituting another font you can often get well over 30% more room on a page to type more stuff. Second, lots of authors are (unfortunately) using Microsoft Word for conference papers. No system except for TeX has the Computer Modern font: thus if a conference wants to have a consistent look from paper to paper, something has to give. And usually that means abandoning Computer Modern and going with Times Roman.
Now I'm not going to claim that Times Roman is a great looking font either --- it's not, and I'd much prefer something else, though I do think that Computer Modern is worse. But my feelings aside, Times does have three things going for it: it's available on everyone's system, it's available on every laser printer (so doesn't have to be embedded inside your PDF document, saving lots of space), and it packs a lot of text in a small space while still remaining reasonably readable.
Switching to Times causes a problem for math users, though: Times Roman doesn't have the symbols necessary to render LaTeX's equations. Thus LaTeX must revert to Computer Modern to handle the math, even when it displays text in Times. Even after you've set LaTeX up to use Blue Sky Computer Modern, you're going to get text like this:
Notice that the cursive a's and f's look different from one another. This example doesn't really do the problem justice: you have to see the mix on a printed page. If you back up and look at a printed, mixed Computer Modern / Times document from a distance, you'll see that the Computer Modern just jumps right out. This is because the letterforms are spaced differently from Times, they're thinner and curvier, and at an odd slant. It's not a disaster to mix the two (I've done many such papers). But fixing this little bit of ickiness is really really easy, and you should consider doing it.
First, I should mention that you can purchase a Times-lookalike math symbol collection called mathtime from Y&Y. But if you're like me and don't want to cough up the $$$, there's another great option: use the \usepackage{mathptm} package in combination with the \usepackage{times} package. Mathptm substitutes as many PostScript Symbol and Times glyphs as possible in the equations, resorting to Computer Modern only for the unusual ones. This results in very consistent looking text:
There's a problem with this second approach, though. The Symbol PostScript font has some downright ugly glyphs. The worst one is the capital Sigma, used in summations. Hence the following math prettily rendered in Computer Modern:
...turns into the following monstrosity when using mathptm:
This is almost entirely a problem with the "big symbols" (Sigma, Integrate, etc.). Fortunately, you can fix this to look better and still have a consistent Times "look" throughout your paper. Edit your mathptm.sty file (mine is in /usr/local/teTeX/share/texmf/tex/latex/mathptm/mathptm.sty) and change the following line:
\DeclareSymbolFont{largesymbols}{OMX}{psycm}{m}{n}
...to...
\DeclareOption{bigsym}{\DeclareSymbolFont{largesymbols}{OMX}{psycm}{m}{n}}
\ProcessOptions
This by default sets up mathptm to use Times and Symbol for all the math symbols it can find, except for the big symbols like Big Sigma. Here's what the equation looks like now:
Try it! I think you'll like it, and it'll make your paper look much more professional with very little work. And with this modification in place, if for some reason you still want to use the big Times/Symbol symbols, you can always revert to "true" mathptm by saying \usepackage[bigsym]{mathptm}. | CommonCrawl |
Home > Proceedings > Proc. Centre Math. Appl. > National Research Symposium on Geometric Analysis and Applications
National Research Symposium on Geometric Analysis and Applications
June 26-30, 2000 | Centre for Mathematics and its Applications, Australian National University, Canberra
Editor(s) Andrew Hassell, Alexander Isaev, Adam Sikora
Proc. Centre Math. Appl., 39: 237pp. (2001).
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This volume contains the proceeding of the National Research Symposium on Geometric Analysis and Applications held at the Centre for Mathematics and its Applications, Australian National University, Canberra, from June 26-30, 2000. The Symposium celebrated the many significant contributions of Professor Derek W.
This volume contains the proceeding of the National Research Symposium on Geometric Analysis and Applications held at the Centre for Mathematics and its Applications, Australian National University, Canberra, from June 26-30, 2000. The Symposium celebrated the many significant contributions of Professor Derek W. Robinson to mathematics, on the occasion of his 65th birthday. he first day, Monday June 26, in particular was devoted to Derek; speakers with a particularly close connection to Derek, including A. Carey, M. Cowling, D. Evans, P. Jorgensen and T. ter Elst recalled and elaborated on important aspects of Derek's work, and the day ended with a banquet in Derek's honour.
The Symposium brought together researchers working in harmonic analysis, linear and nonlinear partial differential equations, quantum mechanics and mathematical physics, and included researchers from North America, Europe and Asia as well as Australasia.
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Proceedings of the Centre for Mathematics and its Applications, Volume 39
Rights: Copyright © 2001, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.
First available in Project Euclid: 17 November 2014
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Australian National University, Mathematical Sciences Institute
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Proceedings of the Centre for Mathematics and its Applications Vol. 39, i-iii (2001).
Derek W. Robinson: list of publications
Proceedings of the Centre for Mathematics and its Applications Vol. 39, iv-viii (2001).
Electrons with self-field as solutions to nonlinear PDE
Hilary Booth
Proceedings of the Centre for Mathematics and its Applications Vol. 39, 1-14 (2001).
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The Maxwell-Dirac equations give a model of an electron in an electromagnetic (e-m) field, in which neither the Dirac or the e-m fields are quantized. The two equations are coupled via the Dirac current which acts as a source in the Maxwell equa- tion, resulting in a nonlinear system of partial differential equations (PDE's). In this way the self-field of the electron is included.
We review our results to date and give the four real consistency conditions (one of which is conservation of charge) which apply to the components of the wavefunction and its first derivatives. These must be met by any solutions to the Dirac equation. These conditions prove to be invaluable in the analysis of the nonlinear system, and generalizable to higher dimensional supersymmetric matter.
In earlier papers, we have shown analytically that in an isolated stationary system, the surrounding electon field must be equal and opposite to the central (external) field. The nonlinearity forces electric neutrality, at least in the static case. We illustrate these properties with a numerical family of orbits which occur in the (static) spherical and cylindrical ODE cases. These solutions are highly localized and die off exponentially with increasing distance from the central charge.
Quantum mechanics as an intuitionistic form of classical mechanics
Murray Adelman , John V. Corbett
Proceedings of the Centre for Mathematics and its Applications Vol. 39, 15-29 (2001).
Intuitionistic real numbers are constructed as sheaves on the state space of the Schrodinger representation of a CCRalgebra with a finite number of degrees of freedom. These numbers are used as the values of position and momentum variables that obey Newton's equations of motion. Heisenberg's operator equations of motion are shown to give rise to numerical equations that, on a family of open subsets of state space, are local approximations to Newton's equations of motion for the intuitionistically valued variables.
Vilenkin bases in non-commutative $L_p$-spaces
P. G. Dodds , F. A. Sukochev
We study systems of eigenspaces arising from the representation of a Vilenkin group on a semifinite von Neumann algebra. In particular, such systems form a Schauder decomposition in the reflexive non-communative $L_p$-spaces of measurable operators affiliated with the underlying von Neumann algebra. Our results extend classical results of Paley concerning the familiar Walsh-Paley system to the non-commutative setting.
Orbital convolutions, wrapping maps and $e$-functions
A. H. Dooley
We survey the theory of wrapping maps as applied to compact groups and vector times compact semidirect products, and give an explicit description of the e-function for compact sym- metric spaces. The latter is globally defined.
Norms of $0$-$1$ matrices in $C_p$
Ian Doust
We announce a new result (proved in collaboration with T.A. Gillespie) on the boundedness of a class of Schur mul- tiplier projections on the von Neumann-Schatten ideals $C_p$. We also show that for $1 \leq p \leq 2$ the average Cp norm of a $0-1$ matrix grows just as quickly as the largest norm of such a matrix.
Spectral multipliers for self-adjoint operators
Xuan Thinh Duong , El Maati Ouhabaz , Adam Sikora
In this article, we give a survey of spectral multipliers and present (without proof) sharp Hörmander-type multiplier theorems for a self adjoint operator $A$ under the assumption that $A$ has Gaussian heat kernel bounds and satisfies appropriate estimates of the $L^2$ norm of the kernels of spectral multipliers. Our theorems imply several important, previously known results on spectral multipliers and give new results for sharp estimates for the critical exponent for the Riesz means summability.
Subelliptic operators and Lie groups
Derek W. Robinson , A. F. M. ter Elst
We review the theory of subelliptic operators on Lie groups. The principal themes are subcoercivity, weighting and Gaussian bounds.
From XY to ADE
David E. Evans
Proceedings of the Centre for Mathematics and its Applications Vol. 39, 85-105 (2001).
We survey the role of non-commutative operator algebras in statistical mechanics and the relation between the classification of modular invariant partition functions in conformal field theories and braided subfactors.
The heat-flow method in contact geometry
Robert Gulliver
Proceedings of the Centre for Mathematics and its Applications Vol. 39, 106-117 (2001).
Contact geometry treats such questions as the existence and classification of contact structures on manifolds of odd dimension and specified topological structure. See inequality (1) below. The geometric/analytic approach treated in this report introduces parabolic systems of partial differential equations (PDEs) in a way which complements the more algebraic methods, which until now are better known in contact geometry.
Manipulating the electron current through a splitting
M. Harmer , A. Mikhailova , B. S. Pavlov
The description of electron current through a splitting is a mathematical problem of electron transport in quantum networks [5, 1]. For quantum networks constructed on the interface of narrow-gap semiconductors [29, 2] the relevant scattering problem for the multi-dimensional Schrödinger equation may be substituted by the corresponding problem on a one-dimensional linear graph with proper selfadjoint boundary conditions at the nodes [11, 10, 25, 24, 16, 19, 4, 28, 20, 18, 6, 5, 1]. However, realistic boundary conditions for splittings have not yet been derived.
Here we consider some compact domain attached to a few semiinfinite lines as a model for a quantum network. An asymptotic formula for the scattering matrix for this object is derived in terms of the properties of the compact domain. This allows us to propose designs for devices for manipulating quantum current through a splitting [3, 15, 22, 9, 21].
What's new for the Beltrami equation?
Tadeusz Iwaniec , Gaven Martin
The existence theorem for quasiconformal mappings has found a central role in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers we give an account of the "state of the art" as it pertains to this theorem, that is to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation.
Some second-order partial differential equations associated with Lie groups
Palle E. T. Jorgensen
In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.
Principal series and wavelets
Christopher Meaney
Recently Antoine and Vandergheynst $[1, 2]$ have produced continuous wavelet transforms on the n-sphere based on a principal series representation of $SO(n; 1)$. We present some of their calculations in a more general setting, from the point of view of Fourier analysis on compact groups and spherical function expansions.
Singularities and the wave equation on conic spaces
Richard B. Melrose , Jared Wunsch
Some remarks on oscillatory integrals
Gerd Mockenhaupt
Yet another construction of the central extension of the loop group
Michael K. Murray , Daniel Stevenson
We give a characterisation of central extensions of a Lie group $G$ by $C^\times$, in terms of a differential two-form on G and a differential one-form on $G \times G$. This is applied to the case of the central extension of the loop group.
Spectrum of the Ruelle operator and zeta functions for broken geodesic flows
Luchezar Stoyanov
On the Banach-isomorphic classification of $L_p$ spaces of hyperfinite semifinite von Neumann algebras
F. A. Sukochev
We present a survey of recent results in the Banach space classification of non-commutative $L_p$-spaces.
Introducing quaternionic gerbes
Finlay Thompson
The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group $\mathbb{R}^+SO(4)$ appears naturally as a (non-commutative) monoidal structure on this groupoid. Using this monoidal structure we indicate the existence of a canonical quaternionic gerbe associated to a conformal structure on a four manifold. | CommonCrawl |
The ST7565 display controller
This page covers the theory of using a graphic LCD based on the ST7565 controller. These are widely available with popular sizes of 132x32 and 128x64, a number cost below £10.
(Image © Electronic Assembly, lcd-module.de)
The tutorial follows the same path I took whilst developing a simple library for a screen I had bought. I set myself the challenge of doing this from scratch, rather than using code from the internet.
Whilst this post uses a 128x64 screen as an example it is equally applicable to different sizes.
Easy to interface using fewer pins than other graphic LCDs
Onboard voltage booster for the liquid crystal display
Software controlled contrast and rotation
Works at faster clock speeds
Good selection of backlight colours
3.3v logic only (not such a problem any more)
In SPI mode a large buffer is required in local memory
The most important thing needed when developing a library is a good datasheet. The information provided by screen manufacturers is often lacking in detail or quality. The datasheet I used can be found here.
Basic operation (hardware)
The most basic pinout for the ST7565 is shown below. To drive it from a microcontroller in SPI mode 5 pins are required. All of these pins are outputs from the microcontroller and inputs to the screen. Because it is not possible to read from the screen there is no requirement for inputs.
It is necessary to connect RESET to the microcontroller as the initialisation procedure involves toggling this pin. This pin should not be tied high or low or it may not be possible to reset the screen correctly.
Depending on your GLCD module it may be necessary to connect a number of capacitors, this is discussed in the next section.
Voltage boosting and regulation
The ST7565 has an integrated voltage booster (charge pump) circuit that can provide the high voltages needed by the liquid crystal display. If the circuit only has a single supply such as 3.3v then the charge pump is the easiest way to power the screen. If the circuit contains a voltage source of between 9v and 12v then it is possible to reduce the number of external components needed by using this directly.
The internal booster can provide from 2x to 6x the input voltage depending on how capacitors are connected between the CAPxx terminals. Full details are provided on page 31 to 34 of the datasheet above. The only limitation is that the maximum voltage after boosting must not exceed Vout, which is 13.5v.
Many LCD specific datasheets will provide the exact capacitors required for a particular model. For some display modules these capacitors are included on the circuit board and do not need to be added. Check the datasheet carefully! For example, the image below is taken from the datasheet for an inexpensive model available from Mouser, the EA DOGM132-5:
(© Electronic Assembly, lcd-module.de)
In addition to the voltage booster the ST7565 contains an integrated voltage regulator. This has 8 steps and regulates the reference voltage used to drive the liquid crystal display. It is necessary to find a value between 0 and 7 which works well with the specific hardware as this is used with the dynamic contrast feature. Typically a "middle" value such as 3 will work with any screen.
Basic operation (software)
SPI mode provides an easy mechanism to drive the screen using few pins. A single bit is written to the data pin and the clock is pulsed high then low. The implementation of SPI by the ST7565 is fairly standard, more information can be found on the Wikipedia SPI Bus page.
One limitation of using the screen in SPI mode is that it only supports writing commands or data. It is not possible to read from the screen therefore it is necessary to keep a copy of the current screen data in memory on the microcontroller. The size of this software buffer can be calculated easily, there are 128 columns and 64 rows for a total of 8192 bits (1 bit per pixel for black and white). Therefore 1024 bytes in total are required to store a copy of the screen, which can be a large amount on many smaller microcontrollers.
With software ('bit banged') SPI it is possible to use the screen at very fast clock speeds. On an PIC 18F I have used my screen with a 64Mhz internal oscillator and it works absolutely fine. It is possible to calculate (or simulate/measure) whether the microcontroller can exceed the ST7565 minimum clock period.
Unlike displays using the KS0108 the ST7565 has a single controller for the whole screen. There is no need to swap between the left and right in order to send commands.
A 128x64 screen is made up of 8 pages, each of which is 8 pixels high and 132 wide. This maps nicely to the software buffer required for caching the screen contents because each page is a byte high.
It is easiest to imagine the top left of the screen as 0,0 and the bottom right as 127,63. This makes some of the maths required in software much easier. To write data at (0,0) it is necessary to select page 0, column 0 and then set the individual bit. In contrast the location (32,32) is in page 5. The simplest method of writing data to the screen is to send a whole page at a time, starting from the left in column 0 and continuining to the right at column 127.
Sending a command to the screen
Issuing a single command to the screen is the most basic building block of the interface code. The steps required are:
Set the A0 pin low to indicate that command data is being sent
Set the chip select (CS) pin low
Send each bit of the command, starting with the most significant bit and working down to the least significant
Set the chip select (CS) pin high, to free the bus
Therefore the "display on" command (which is 0xAF or 0b10101111) should be sent as: 1, 0, 1, 0, 1, 1, 1, 1.
As long as the SPI timings are not violated there is no need to introduce any delays as the screen will never be "busy".
Screen initialisation
Now that we can send a single command to we need to follow a number of steps to setup the screen.
Strobe the RESET pin low then high, to initiate a hardware reset
Setup the duty cycle (either 1⁄7 or 1⁄9) depending on the physical LCD
Set the horizontal and vertical orientation to a known state
Configure the internal resistor divider which is used by the voltage regulator
Turn on the internal voltage booster to provide power to the LCD glass
Initialise the dynamic contrast to a default value
Reset the current display position to the top left
The commands for these are a single byte except the dynamic contrast, which is explained later. Below is an image captured from a logic analyser while the initialisation code runs.
In this trace the following things can be seen:
Command 0xA2: set the LCD bias to 1/9th
Command 0xA0: horizontally "normal" (not flipped)
Command 0xC8: vertically "flipped" (complements the command above)
Command 0x23: the internal resistor divider set to 3 (from 0..7)
Command 0x2F: power control, all internal blocks ON
Command 0x81: enter dynamic contrast mode
Data for the dynamic contrast mode, set to 31 (from 0..63)
Command 0x40: go back to the top left of the display
When power is first applied to the screen it is also important to clear the onboard screen memory. As it is non-volatile RAM it will start in an unknown state.
Sending data to the screen
The only difference between sending a command and data is the A0 pin. When sending data the A0 pin must be high to differentiate the bus data from a command.
To start we send a command to reset the position to the top left corner (0,0). After this a quick trick is to write a single byte and see how it is displayed. It is important to know which way round the screen will display the data it receives, whether the "most significant bit" or "least significant bit" is displayed at the top. The illustration below shows the output if the byte 0x80 is sent four times to the screen:
From this it can be seen that the most significant bit displays at the bottom of the page. The first bit we send is at the bottom, then subsequent bits work upwards toward the top of the page. This will be important later when we try to set an individual pixel.
Displaying a basic test pattern
The easiest way to test if the driver code is working is to send a very basic test pattern. A quick check is to send constant bytes of 0xFF to fill the screen. If this works then variations can be tried, such as 0xF0 or 0xAA.
At the end of each row, after every 128 bytes, it is necessary to select the next page and move to the start of the row. The following image is the basic test card I used to demonstrate that my display was wired up correctly.
Writing to specific pixels
Writing specific pixels opens up a whole world of possibilities from lines, circles, 1 bit-per-pixel graphics and variable height fonts. As these graphics functions are not dependant on a specific controller or screen it is best to split them out into a separate library which is portable across any screen. This is perhaps the most difficult portion whilst developing a library to work with any screen.
In order to set a specific pixel it is necessary to break the process into a number of steps:
Find the corresponding byte in the software buffer
Calculate the bit within this byte
Use logical XOR (to set the bit) or AND (to clear the bit)
Finding the correct byte is straightforward. The first position in our array is at the top left of the screen. The 128th byte in the array (Array[127]) is on the first page at the right hand side of the screen. The next bit "wraps" to the next page, at page 1 column 1.
To calculate which byte we need to use we need to divide the Y position by 8 and multiple the result by 128. To this we add the X position, then subtract 1 (as C arrays start from 0). The equation $ X + ({ \frac{Y}{8} \times 128 }) - 1 $ works well in C. To verify this is correct:
(1,1) on the screen would give $ 1 + ( \frac{1}{8} \times 128 ) - 1 = 0 $
(61,52) on the screen would give $ 61 + ( \frac{52}{8} \times 128 ) - 1 = 828 $
To calculate the bit inside this byte we must remember that the 8th (most significant) bit appears at the bottom of the page. So to write to Y location 8 we need to set the 8th bit of a byte on the first page. To write to Y location 16 we need to write to the 8th bit of the corresponding byte on the second page. Therefore we need the remainder after the Y position is divided by 8, which can be found using the modulo operator. This is as simple as Y % 8:
(1,1) means setting 1 % 8 = the 1st bit
(61,49) means setting 49 % 8 = the 1st bit
(61,53) means setting 53 % 8 = the 5th bit
It is now possible to set the correct bit inside the byte, on PIC microcontrollers this can be achieved using very efficient bit operators such as bsf or bcf.
Displaying text
Like many other graphic LCDs the ST7565 has no built-in font. The quickest & easiest way of displaying text on this screen is to define an 8 pixel high font. This maps directly to one page (or 'line') of the display and therefore we can use very fast instructions to copy the data from a predefined font to the right area of the screen.
However this limits us to a fixed height and only 8 lines of the screen. Now that we know how to set individual pixels there are much nicer ways of displaying text. Consider the following image:
Larger fonts take up more room but can be used to provide a much nicer looking user interface for a project. With the knowledge that we can set a specific pixel it is easy to make advanced graphics functions. However these are separate to the ST7565 screen itself, so I will cover them in a future blog post. Impatient readers can check my graphics library (Doxygen documentation can be found here).
Using advanced features
So far all of the features are common to all standard graphic LCDs. However there are a number of additional tricks that ST7565 based displays can offer.
Inverting the screen
With a single command the ST7565 can be instructed to invert the display. This can be useful for emphasis or to make the screen easier to read in certain situations. The command "display normal or reverse" is 0b1010011X, where X is 1 for reversed or 0 for normal.
The picture below shows the same font demonstration as above, but with the inverse feature turned on.
Blacking the screen
The ST7565 is capable of setting all points on the display to black without affecting the screen RAM. Whilst it is hard to think of a situation where this might be useful the command "display all points" is a single byte and can be used to turn the display black if required.
Dynamic contrast control
Unlike other common graphic LCDs the contrast for the ST7565 is set by an internal resistor divider. This means that it can be set in software and altered dynamically. To change the contrast we need to send two commands. The first enters "volume mode set" and the second sets the contrast from 0..63.
The number of useful steps may be less than 64, on my test screen the visible steps are from 12 (very light) to 40 (very dark). Lower values are not visible as the voltage is too low, higher values make the screen completely black as there is no distinction between pixels which are off or on.
This feature could be used with a light sensor to dim the screen depending on ambient conditions. It can also be used to fade the screen in or out.
Screen rotation and mirroring
In the same way that the screen can be inverted without affecting the display data it is possible to flip the screen in either the horizontal or vertical direction. This means that the screen can be used either way up with no changes required to the software except sending extra two commands to the screen.
Flipping the screen horizontally (from left to right) is achieved with the confusingly named "ADC Select" command. Flipping the screen vertically (from top to bottom) is possible with the "COM Output Mode Select" command. Unless you want to view the screen in a mirror it is necessary to use both commands at once!
As mentioned earlier the memory on the screen is actually 132 columns wide, to support a maximum screen size of 132x64. On 128 pixel wide models there is an important consideration if rotating the display. If the screen is not rotated as the extra 4 rows are on the right hand side and not visible, as illustrated below:
When the screen is rotated the controller outputs these columns in the opposite order, starting with the last column. Therefore the unused 4 columns from the screen memory are output first, leaving an empty space on the left hand side.
In order fix this it is necessary to keep track in software of whether the screen is rotated or not. When it is rotated the column address should be incremented by 4 each time a new line is sent to the screen, skipping the extra columns which would otherwise be displayed.
Refreshing the screen involves sending the entire local buffer to the screen RAM. If only a few changes have been made then this is a very inefficient way to update the screen.
It would be possible to send each individual write to the screen, simply by copying the single byte that has changed each time a pixel is written. However this is inefficient if more than a single bit (or pixel) is set in the same byte in subsequent operations. Changes to the screen would also be very visible at slower speeds.
As the screen has 8 pages the most efficient method on a small microcontroller is to send only the pages which have changed. Each time a pixel is set or cleared the corresponding page in the local buffer is marked as 'dirty'. When it comes to refreshing the screen only the dirty pages are transferred.
In the example above only pages 2 and 5 have had changes made. Therefore these will be sent to the screen, which takes 1⁄4 of the time as sending all 8 pages.
The screen could be subdivided into even smaller regions, for example 16 half pages. The trade-off of extra code each time a pixel is set and the reduced bus time sending data should be considered carefully.
Verifying speeds
Above I commented that it was possible to run a PIC at 64Mhz and not violate the SPI timings. The important values are shown in Table 28 of the datasheet:
We are interested in Tscyc, Tshw and Tslw. These relate to the total clock cycle, the HIGH portion of the cycle and the LOW portion respectively. The overall clock pulse must be at least 50ns and the high and low portions must be at least 25ns each.
Crunching the numbers
Assume that the SPI code turns ON the clock pin and then OFF again immediately in 2 instructions. The pin will be HIGH at the end of the first instruction and LOW again after the second, therefore it is only HIGH for the period of 1 instruction.
On the PIC 12/16/18 family of microcontrollers each instruction takes 4 cycles. At 64Mhz this means that each instruction takes 1/(64000000⁄4) seconds, or 62.5ns. From this we can expect that the HIGH period of the clock pulse should be about 62.5ns, easily within the required timings.
Checking with a logic analyser
It is also possible to measure and see how long the pulse lasts on real hardware. The screenshot below shows the HIGH portion of the clock pulse being measured, this is the shortest part of the cycle. The PIC being used is an 18F25K22 running at 64Mhz. The code toggles the pin ON and OFF in two instructions.
Remember that it must be at least 25ns long in order to be acceptable. The measurement shows 41.7ns, which is shorter than calculated but still acceptable. A number of reasons could produce the difference between our calculation and the physical measurement:
The resolution of the logic analyser (24Mhz in this example) might not be sufficient for the measurement.
Setting or clearing bits in hardware might not take the entire 4 cycles.
The PIC might not be running at exactly 64Mhz due to instability of the oscillator or PLL.
If you are interested in using a ST7565 based screen in an Arduino project then you should check out Ladyada's tutorial on Adafruit. My C library is available which was designed for PIC 18F parts with HiTech C. It would be trivial to change for another compiler or chip.
David Cannings
Cyber Security Geek
My interests include computer security, digital electronics and writing tools to help analysis of cyber attacks.
© David Cannings · Powered by the Academic theme for Hugo. | CommonCrawl |
Ionic transport through a protein nanopore: a Coarse-Grained Molecular Dynamics Study
Nathalie Basdevant ORCID: orcid.org/0000-0002-6775-29071,
Delphine Dessaux ORCID: orcid.org/0000-0002-6541-11051 &
Rosa Ramirez1
Scientific Reports volume 9, Article number: 15740 (2019) Cite this article
Biological physics
Computational biophysics
The MARTINI coarse-grained (CG) force field is used to test the ability of CG models to simulate ionic transport through protein nanopores. The ionic conductivity of CG ions in solution was computed and compared with experimental results. Next, we studied the electrostatic behavior of a solvated CG lipid bilayer in salt solution under an external electric field. We showed this approach correctly describes the experimental conditions under a potential bias. Finally, we performed CG molecular dynamics simulations of the ionic transport through a protein nanopore (α-hemolysin) inserted in a lipid bilayer, under different electric fields, for 2–3 microseconds. The resulting I − V curve is qualitatively consistent with experiments, although the computed current is one order of magnitude smaller. Current saturation was observed for potential biases over ±350 mV. We also discuss the time to reach a stationary regime and the role of the protein flexibility in our CG simulations.
Single-molecule experimental techniques are widely used to understand the kinetics and energetics involved into biological phenomena. Among them, because of its simplicity and amenability to parallelization, nanopore force spectroscopy is an increasingly used technique for ultra-sensitive detection of macromolecules. The method consists in applying an electric potential difference to guide a charged molecule through a nanopore (artificial or biological) inserted in a solid or lipid membrane, in presence of a salt solution1,2,3,4,5. A macromolecule going through the pore, partially blocking it, creates a variation in the electrical current according to the nature of the molecule and the pore characteristics: size, conformation, and structure. Ionic current measurements are a probe for the detection. Amongst the various biological nanopore, α-hemolysin is the most widely employed in nanopore analytics. It has been commonly used as a nanopore for DNA or RNA translocation1,3, DNA sequencing or biosensing6, DNA unzipping of individual hairpins7,8,9, and the study of protein translocation4.
The understanding of the physical mechanisms governing the translocation of ions or molecules through a nanopore is not only helpful to reveal relevant biological phenomena but also for the design of new devices based on such nanopores. The microscopic details of these processes cannot be deduced straightforwardly from the experiments. In this sense, an increasing effort to provide theoretical and computational approaches in order to understand such processes is being done10. In general, these approaches require a choice between the amount of computational resources and the microscopic level of accuracy. Among continuum approaches, the Poisson–Nernst Planck theory (PNP)11,12,13,14,15,16 is useful to study ion transport at a relatively small computational cost, although not very accurate for narrow pores. At the same level, Brownian or Langevin dynamics methods (BD)12,17,18,19 can be an alternative. In this case, the main challenge is the representation of the protein-continuum solvent interface.
In All-Atom Molecular Dynamics (AA-MD) simulations, the individual motion of ions, water, protein and membrane atoms are fully computed based on a classical atomic force field. This approach will be limited by the classical force field approximation used, but AA-MD simulations are undoubtedly excellent to provide microscopic details of biological translocation processes "at short times". The huge computational cost is the main disadvantage of this method and therefore, there are several ways to accelerate the transport of molecules through the α-hemolysin pore: using very high electric potentials (1 V compared to hundreds of mV for experiments)8,20,21 or forcing the transport of molecules through the pore using velocity-pulling methods22. Although some results of these AA-MD are in accordance with experiments, long time phenomena cannot be observed in such simulations, generally limited to a few hundred nanoseconds. The high-potential/pulling methods can also induce artifacts in the results.
Coarse-Grained (CG) force fields, representing several atoms by one site, for lipid membrane and biomolecules, are a very promising alternative to All-Atoms models, as they enable simulations of time-lengths closer to experiments. In this sense, our article focuses on MD simulations where the AA representation is replaced by an "atomistic" CG force field, in our case the MARTINI force field. The CG MARTINI force field23, originally developed for lipids, is widely used to simulate membrane systems for time-lengths up to hundreds of microseconds24,25,26. Furthermore, a polarizable CG water model is needed to correctly describe the electrostatic behavior of electrolytes. Among the several CG polarizable solvent models available27,28,29, the Polarizable Water (PW) model29 has been parametrized to be used together with the standard MARTINI force field. Whereas classical AA simulations of ionic current through α-hemolysin have been done, to our knowledge, such a study using a CG model has not been yet performed.
In this work, our MARTINI CG MD simulations for determining ion currents through α-hemolysin are detailed and discussed. We begin by studying the electrostatic behavior of solvent, ions and membrane using this force field. The ionic conductivity of CG MARTINI ions in PW solution as a function of concentration is discussed. Secondly, we study the electrostatic behavior of a solvated DPPC (dipalmitoylphosphatidylcholine) lipid bilayer with excess ions in solution. We will confirm that applying an electric field in a direction perpendicular to the membrane plan is accurate to represent the experimentally applied potential difference. Finally, we will study the current through the protein nanopore at various electric potentials on microseconds-long MD.
All our MD simulations were performed using the Gromacs software package30 (version 5.0.2) and the MARTINI 2.2 Coarse-Grained (CG) model for ions, lipids23, proteins31,32 and the PW (Polarizable Water) model29 for water. The systems were simulated either in the NVT or the NPT ensemble at a temperature of T = 320 K, where PW water presents a density of 1020 kg/m3 29. MARTINI CG ions consist in particles of the same size and mass as a PW particle, carrying positive or negative charge. Although they are called "Na+" and "Cl−" in the force field parameters files, they do not represent a realistic sodium or chloride ion33. Since the MARTINI salt is perfectly symmetrical, we considered that it represents the KCl salt instead of NaCl. Methodological details of the simulations can be found in the Methods. The details of the ionic molar concentration setting for NVT simulations is discussed in the Supplementary information.
Ionic conductivity
First of all, we focused on the ionic conductivity of MARTINI ions in bulk PW water. The correct interplay between ions and water force fields is of critical importance to simulate conduction processes. In the case of MARTINI PW water, bulk properties such as density and dielectric constant have been optimized to construct the PW force field, while for ions the equilibrium bulk correlation function ION-PW water has been benchmarked to determine ions parameters29. We would like to address the ability of the combination of these ion and water force fields to yield reasonable results when used to simulate transport processes.
Our simulations consisted of a cubic box containing PW water particles, cations (K+) and anions (Cl−) with concentrations from 0.1 M to 2 M (see Methods for details). For each concentration, three different conditions were simulated: no external electric field, and with electric fields (E) of 0.02 V/nm and 0.04 V/nm in the z direction.
To calculate the ionic conductivity κ of CG ions as a function of concentration, we computed the average total ionic current <I> in the z direction as follows: the z-component of the velocity of the center of mass of all K+ ions, vz+, and the one of all Cl− ions, vz−, were calculated every 500 ps, and the instantaneous current in the z direction was then computed using the formula:
$$I(t)=\frac{1}{{L}_{z}}{N}_{ions}e({v}_{z}^{+}-{v}_{z}^{-}),$$
where Nions is the total number of ions of each category, Lz is the box dimension in z direction and e is the elementary charge. The cumulative sum of the current over the length of the simulation (1 μs) was then calculated and fitted with a linear regression, the resulting slope of this fit yielding the average current <I>. The conductivity κ is defined by \(\kappa =\frac{ < I > }{EA}\), where A is the cross-sectional area, which is here: A = LxLy (Lx and Ly are the boxes dimensions in x and y directions). Since two distinct electric fields were applied in the simulations at each concentration, we computed the average of the two calculated conductivities.
The concentration dependence of the conductivity computed at a fixed temperature of 320 K is shown on Fig. 1, as well as the experimental results and AA simulations using the CHARMM and AMBER force field performed by Pezeshki et al.34,35 for KCl (numerical values from the figure of ref.34 were kindly provided by Ulrich Kleinekathöfer). The CG conductivity increases with the ionic concentration for concentrations below 1.15 M and is slightly lower than the reference values. For concentrations above this value, the conductivity decreases with the concentration. This effect is due to the size of the CG particles and principally PW particles: at concentrations larger than 1.15 M, the CG model cannot correctly represent the water Coulomb screening, and non-linear effects appear dramatically. As an example, at 2.3 M, there are only twice more PW particles than CG ions (see Supplementary Information for further discussion on the concentration calculation for the ions CG model).
Ionic conductivity of KCl as a function of molarity. Experimental results (in red) and all-atom simulations using the CHARMM (blue) and AMBER (green) force field performed by Pezeshki et al.34 for KCl at 320 K, and CG results (black) computed with our CG MARTINI MD simulations.
Using AA models, a saturation of the conductivity was also observed above 1 M, compared to experiments, but there is no decrease. A saturation effect is also measured experimentally for conductivity in the bulk for certain types of ions with a large hydration shell36.
The CG ionic conductivities values with PW at 320 K are in very good agreement with experimental and all-atom theoretical data for concentrations of KCl below 1.15 M. This result is quite different from37 where the authors, using a Nernst-Einstein method, show that conductivity of CG MARTINI ions is close to experiments only if an effective concentration is used.
Charge distribution and electric potential of a DPPC bilayer
The electric potential across lipid membranes is essential for various biological processes in the cell, such as signaling, ion transport through membrane channels, or translocation of molecules through pores. In vivo, such potential difference across the membrane is caused by ionic concentration gradients.
In nanopore experiments, this difference is established and monitored using two electrodes placed across the membrane1, while there are several options to simulate the trans-membrane potential in silico. In this work, a uniform electric field E in the z direction perpendicular to the membrane surface is applied to every charged atom of the system38. This method has been previously used by Aksimentiev and Schulten21 on the α-hemolysin pore, and successfully applied to various biological problems8,10,34,39. An extensive MD study on several membrane systems showed that this method accurately represents an electric potential difference of ΔV = ELz, where Lz is the system size in the z direction, independently of the membrane thickness40.
In the original paper on the PW model29, the electric potential around a DPPC membrane was computed without any ions and external electric field. Therefore, a study of the electrostatic properties of a system composed of MARTINI {DPPC + PW + IONS} under an external electric field is needed to assure the validity of the model for describing the ionic transport. After equilibration of a system composed of DPPC lipids, PW water molecules and ions (see Methods), the MD production was performed in the NPT ensemble for two microseconds at 320 K and 1 bar. Several ionic concentrations were simulated: pure water, 0.2 M, 0.4 M and 0.8 M. Moreover, at each concentration, three different electric fields were applied: no external field, 0.01 V/nm and 0.02 V/nm in the z direction. These electric fields correspond to a theoretical transmembrane potential difference of 0 mV, 160 mV and 320 mV, respectively. It should be noted that, in the NPT ensemble, the box dimension in the z direction increased at the beginning of the dynamics, and the Lz length was around 16 nm on average during all the membrane MD (from 16.24 nm without ions to 16.41 nm at 0.8 M).
The averaged charge density, on the last 500 ns of the 2 μs-simulation along the z direction for each species, DPPC, water and ions, was computed for each simulation. Figure 2a shows the total linear charge density and its decomposition (b–d), without an electric field, as a function of the z coordinate, without ions and at 0.4 M and 0.8 M concentrations. The total charge density is globally similar to previous AA simulations41, showing four peaks from either side of the membrane. The decomposition of the charge density is also consistent with the AA one, so that water and ions, when present, rearrange in reaction of the membrane charge density. The DPPC charge density (Fig. 2b) is slightly different when ions are present in the system, showing that the lipids rearrange a little in the presence of salt. Compared to AA simulations41, the effect of the salt concentration on the membrane potential is weaker since it is limited to the charged head group. Moreover, the influence of the ionic concentration on the area per lipid is much smaller than for AA simulations, decreasing from 0.635 nm2 with no ion to 0.617 nm2 at 0.8 M, instead of 0.62 nm2 with no ions to 0.5 nm2 at 0.8 M. The solvent charge density is very different with and without the presence of ions (Fig. 2c), since the dipolar moments of water molecules react to the ionic charge distribution. Figure 2e,f show the particle density for each ions species (K+ (e) and Cl− (f)). From these figures, we can infer that there is a layer of cations close to the negative charges of the phosphate groups, corresponding to the two peaks around |z| = 2 nm on Fig. 2e. There is also an overlay of anions close to the positive beads of the lipid heads, corresponding to the two peaks around |z| = 2.5 nm on Fig. 2f. This ionic structure on the membrane surface is also consistent with AA simulations41.
(a) Total linear charge density along the z axis around a CG DPPC bilayer with (0.4 M in red and 0.8 M in green) and without ions (in black), and its decomposition: (b) DPPC charge density, (c) solvent, (d) ions. (e–f): Particle density for K+ (e) and Cl− (f) ions. The z axis is centered at the middle of the lipid bilayer.
The electrostatic potential along the z direction for each was then calculated by the double numerical integration of the charge density, according to42,43. In the absence of an external electric field, the system, centered on the bilayer, is theoretically symmetrical in the transformation (x, y, z) to (x, y, −z). Therefore, for the systems without electric field, the charge density ρ(z) was symmetrized before calculation of the potential. Figure 3a shows the decomposition of the electric potential without ions. When no ions are present, the PW particles are oriented to compensate the electric potential created by the lipids. However, the total electric potential inside the membrane is negative, contrary to AA41 for which the total electric potential is slightly positive inside the membrane. The PW solvent electric potential does not compensate enough the potential created by the DPPC membrane, as it was previously shown in the original PW paper29.
(a–c) Total electric potential along the z axis of a CG DPPC membrane (black) and its decomposition, without ions (a), with 0.8 M ionic concentration (b), with no external electric field, and in the presence of an external electric field of 0.02 V/nm (c). (d) Total electrostatic potentials with no electric field (black), electric fields of 0.01 V/nm (red) and 0.02 V/nm (green) without ions (solid line) and with 0.4 M and 0.8 M concentrations (dashed lines).
On Fig. 3b, the decomposition of the electric potential at 0.8 M ionic concentration is shown. The total electric potential inside the membrane is less negative than without ions. The potential created by the membrane increases with the ion concentration, as in AA simulations41, proving that the bilayer is sensitive to the ions presence. When ions are present, the solvent potential is much lower and the rearrangement of ions is compensating the lipids potential. This effect is consistent with the AA study by Rodriguez and Garcia41, that showed that in the presence of ions, water plays a significantly smaller contribution to the total electric potential. Therefore, although the total electric potential inside the membrane is negative instead of positive, CG ions and water adopt the same behavior around a lipid membrane as an AA force-field.
Figure 3c shows the same decomposition for simulations with an external electric field. The total potential thus also contains the applied external potential (in dashed line). The figure shows that the ionic polarization potential is strongly altered by the external potential. This is not the case for other species whose charge density distribution is much less modified by the external field. On Fig. 3d, the total electrostatic potentials for simulations with pure water and with 0.8 M ions are represented. In the bulk, the reaction polarization potential is opposed to the external imposed one, and the resulting total electric potential difference is therefore only effective over the membrane. This result is consistent with previous AA-MD40 studies which attest that the electric potential does not depend on the thickness of the membrane. The authors showed that the reaction potential of the solution, composed of solvent and ions, to the external field, combined to the external electric field, results in an electric potential difference across the membrane only, with a zero-potential difference within the two bulk regions, proving that using the relation ΔV = ELz is accurate. Our results show that this assumption is also accurate with the MARTINI CG model.
We showed that the MARTINI force field with the PW solvent reproduces accurately the ionic conductivity in the bulk and the electrostatics properties around a lipid membrane in the presence of an external electric field. We can therefore now try to simulate ionic transport through a nanopore using this methodology.
Ionic current through the nanopore
In this section, we present the results of the simulations for the system composed by the α-hemolysin (PDB entry: 7AHL44) inserted in a DPPC bilayer and surrounded by PW water, ions and counter-ions in the presence of an external electric field. A value of 1.1 M± 0.1 M ionic concentration was measured at the bulk region. Further details of these procedures and the parameters used can be found in Methods.
The initial condition for the ensemble α-hemolysin suspended in a lipid bilayer membrane was taken from final state of Prof. Sansom simulations45, where the lipid bilayer grows spontaneously around the protein. We observe a curvature of the membrane in the pore vicinity for our system (see Fig. 4). This distortion could already be measured in the original membrane arrangement45. Moreover, this kind of membrane curvature was also observed in AA simulations of α-hemolysin in other lipids bilayers and it appears to be characteristic of the lipids involved and of the way they interact with the protein46.
Representation of simulation setup. Water has been removed for clearness purposes. Section of the pore (orange) along the z axis. The DPPC membrane (blue) is curved. K+ (green) and Cl− (magenta) ions are going through the pore channel.
MD simulations of α-hemolysin in DMPC lipid membrane using MARTINI force field were recently studied47 and showed a considerable distortion of the protein stem. In our simulations, the ElNeDyn network48 was applied inside and between the different protomers, and, as a result, the protein stem remains stable during the whole simulation.
After equilibration of the system, production simulations were carried out in the NPT ensemble (T = 320 K and P = 1 atm), with an external electric field applied perpendicular to the lipid layer, ranging from −0.04 V/nm to +0.04 V/nm. The potential bias ΔV value for a given electric field is calculated using an average Lz = 18.5 nm, ΔV = ELz. This Lz value varies less than 5% during the simulation time. All simulations, including the no-field simulation, start from the same initial condition (see Methods). The complete list of simulations performed with their corresponding electric fields and simulation times are shown in Table 1.
Table 1 Simulation parameters and number of ion passages during simulations.
To provide reasonable statistics, we performed simulations of 2 and 3 μs, depending on the applied external electric field. We do not present data for an electric field E = −0.04 V/nm, since the membrane broke early during the simulation. This is also the case in experiments where the allowed potential differences are |ΔV| < 350 mV.
The instant current was calculated computing the number of cations and anions (nc(t) and na(t)) crossing a xy plane perpendicular to the z axis, with z abscissa equal to the position of the center of mass of the groups in the trans end of the pore, between t and t + Δt (Δt = 100 ps) (see Supplementary Information). The sign represents the direction of the velocity in the z axis. These values are normally 0 or 1, so we define the number of ions passing through the plane in the interval [0, t] as \({N}_{c,a}(t)={\int }_{0}^{t}\,{n}_{c,a}(t^{\prime} )dt^{\prime} \), and the total charge going through the pore, which is the cumulative function of the current, is: Q(t) = e(Nc(t) − Na(t)). The instant current is defined then as the slope of the function \(Q(t):I(t)=\frac{dQ(t)}{dt}\).
The number of ions crossing the pore during the whole simulation time, Na(ts) and Nc(ts) are represented in Table 1, together with the total charge variation during the whole simulation Q(ts) in e units. These values are in the limit of "reasonable" statistics although, as we will see below, they evolve with time. The rectification and selectivity of the pore can be also deduced from this global values. The instant cumulative current Q(t) is represented in Fig. 5 for potentials under (side a) and over (side b) 300 mV. Since the instant current I(t) is the slope of this function at each time t, the linear increase of the cumulative current with time would indicate a stationary current. This seems to be the case for potential bias under 300 mV, although it should be pointed out that the stationary current is established only after some hundreds of nanoseconds, especially in the case of positive potential bias. The fact that the slope of the function increases with increasing potential is the reflect of a linear-response regime.
Cumulative current as a function of time, for potentials (a) under or (b) over 300 mV computed during 3 or 2 μs, respectively. (a): ±92 mV (green), ±185 mV (red) and ±277 mV (blue). (b): ±370 mV (green), ±462 mV (red), ±555 mV (blue) and ±740 mV (violet).
This is not the case for potentials over 300 mV, as shown in Fig. 5b). The cumulative currents are not proportional to the potential bias at the end of the simulation, implying that the simulated systems are not in the linear response regime. Moreover, we cannot be confident that the stationary state has been reached at the end of some simulations as the slope of the cumulative current is still varying.
Figure 6 shows the evolution in time of the I − V curve of the system averaged at a fix number of crossings (N = 40 events). For every simulation, an initial time tini is defined as Q(tini) = 5 e, which is the time when five charges have already crossed the pore. From this origin tini, we computed the total time tend such as ΔQ = Q(tend) − Q(tini) = 40 e. In this interval, the current Δt = tend − tini, is estimated as \(\bar{I}=\frac{\Delta Q}{\Delta t}\). The error bars were calculated with the standard deviation \(\sigma =\frac{1}{N}{\sum }_{i\mathrm{=1..}N}\,|\Delta {Q}_{i}-I\Delta {t}_{i}|\) so that \(\delta I=\frac{2\,\sigma }{\Delta t}\).
Average current during different passages windows as a function of potential bias. Note the difference between the beginning of the simulation (5–45 crossings) and the end of the simulation (195–235 crossings). The black solid line corresponds to Iref/10.
In order to compare our results to experimental values, we have defined a reference current in the linear regime, as Iref = RrefΔV, where Rref = 1 GΩ is a typical resistance observed experimentally for these systems36. For negative electric fields, Iref value is multiplied by 0.7 to take into account the observed rectification of 30% in experimental setups. The black continuous line observed in the figure is the function Iref/10. Thus, in our simulations, we observe a Qs value ten times smaller than the expected one. This is a consequence of the CG model: the resistance of the pore increases since the mobility of the ions inside the channel is highly reduced because of the larger size of PW water molecules. This ratio increases for higher potential differences where the linear regime is not valid anymore.
The measured current is clearly evolving during our simulations. In spite of this, a linear response regime can be established for potential biases below ~350 mV. Details of currents for potential biases between −370 and +370 mV and their anionic and cationic contributions are detailed in Table 2. In this regime, the final current appears slightly smaller than the one measured at the beginning of the simulation and, as mentioned, it is one order of magnitude smaller than the experimental one. For potentials above this value, a limiting current is observed. We find some asymmetry between positive and negative biases: while, for negative biases, the limiting current seems to converge to a well defined value, for the positive ones, this value appears much more noisy.
Table 2 Currents in the linear response regime (±400 mV).
The origin of the non-linear ohmic behavior of the current-voltage curve has been explained as due to the crowding or depletion regions at the trans and cis ends of the pore in presence of an applied external potential49. Both limiting and over-limiting currents in nanopores have been discussed in van Oeffelen et al.50. In our simulations, we observe a limiting current at a quite low voltage ~350 mV. This value is probably higher in AA simulations, although it is difficult to observe as the simulated time is not enough to get to the steady state21. In ref.51, the authors compare linear-response theory with AA-MD simulations to determine the departure from the linear regime for two pore models, one of them being a modified α-hemolysin pore. Their conclusion, and also ours as regard of our results, is that α-hemolysin has a small linear response range and that currents at low voltages cannot be safely extrapolated from the values obtained at higher potential biases. The limiting current cannot be experimentally observed since, as mentioned, the membrane breaks over ~300 mV.
It is not clear from our results if an over-limiting current appears over +700 mV50. This could be the case but it cannot be excluded that the stationary state has not been reached, that the effective protein charge (protein plus decorating ions) evolved, or that the periodic boundary conditions are inducing such phenomenon.
As observed, the average current cannot be accurately computed before reaching the stationary state and therefore, an interesting issue is which simulation time is needed to get to this regime. In van Oeffelen et al.50, it is pointed out that, due to slow co-diffusion of ions, the time for the charge-current to decay to a stationary state value is determined by the diffusion of particles. This time is estimated as a typical diffusion time τ = L2/D, where L is a typical distance and D = 2(D+D−)/(D+ + D−) is an averaged diffusion coefficient for cations and anions. In our simulations, this time is estimated to ~140 ns, where we have used L the length of the system occupied by the solvent, D+ = 2.54 10−9 m2/s and D− = 2.45 10−9 m2/s measured during our conductivity calculations. We have computed the evolution of the number of ions inside the stem, the region of the pore going from the trans side to 4 nm above (see Supplementary Information). We observed that the time to get to a stationary number of ions goes from ~100 to 300 ns, being shorter for negative voltage biases. The asymmetry for positive and negative biases is in agreement with AA simulations, as well as the number of ions in the stem computed in previous AA and PNP studies12,52.
The time to get to a stationary regime is longer than this estimation according to our simulations. The relatively quick convergence of the number of ions inside the pore is not a guarantee for the stationary state, where anions and cations currents, or conversely charge-current and particle-current, should be, in average, time independent. This is not the case for all our simulations. We observe the convergence of the current for voltage biases under +350 mV as well as for negative higher biases. Positive high biases would require longer simulations to get confident results. We can conclude that a 1–1.5 μs simulation should be enough to get a stationary state in the linear regime.
A final pertinent question about CG-MD simulations of such complex systems is if the pore dynamics play a determining role on the current. If this was not the case, simulation time could be saved by tabulating the pore and membrane potential. To test this possibility, we have performed CG-MD simulations where the backbone of the α-hemolysin was constrained to a fixed configuration. The external electric fields imposed were ±0.01 V/nm, ±0.02 V/nm and ±0.03 V/nm. The measured current in this case is much smaller than the corresponding one for the flexible protein and we found clogging periods during the simulation, going up to 200 ns. The current saturates for small potential biases going to a value close to zero (see the Supplementary Information). This result proves that the flexibility of the protein is essential to simulate the translocation process.
The use of MARTINI force field to simulate the transport of ions through the α-hemolysin in a DPPC bilayer is studied for the first time, to our knowledge, in this article.
In first place, we have investigated the transport properties of the original Martini hydrated ions as a function of concentration. This leads us to the problem of comparing CG results to AA or experimental ones. Since the ratio ion/water valid for AA systems is no longer a suitable parameter to set a molar concentration for the CG simulations, we have decided to use the number of ions per volume as concentration. We are aware that this definition may not represent exactly the same concentration in NVT and NPT ensembles, and definitely cannot be directly compared to AA results. In spite of this incertitude, we can conclude that MARTINI ions in PW water present a conductivity that could represent a KCl solution below 1.1 M concentration.
Secondly, we have investigated the properties of the {DPPC + IONS + PW} system in the absence and under different applied electric fields. It is already known that the MARTINI force field leads to a unrealistic positive potential inside the membrane in the absence of an applied electric field29. In the presence of an external field, we found that the expected potential differences through the membrane are recovered, such as in previous AA simulations40. We have also pointed out the role of the polarization of water to describe the correct electrostatics, proving that it is necessary to use a polarizable model of water.
Finally, we have studied the whole system {α-hemolysin + DPPC + PW + IONS} under different applied external fields focusing on the ionic transport through the pore. Principally due to the size of MARTINI PW water, the pore presents a smaller conductance than the experimental system. We have also performed simulations with a fixed protein which shows that the channel flexibility plays also an important role on the reduced CG conductance since we observed almost no current. This deviation in conductance seems to be systematic for low applied external fields. The qualitative behavior of the system nevertheless appears to be correct. Our I − V curves show a small range ±350 mV where the linear response regime is established. For higher potential biases, a saturation of the current is observed. These limits cannot be compared to experiments, for practical reasons, or to AA-MD simulations, since the simulated time is not enough as to get to a stationary state.
PNP solutions have provided to be extremely useful to describe the stationary state of ionic currents in many systems, but a precise description of this particular system would require excluded volume effects, water-ion correlations, hydrodynamics, finite system size correlations and the possibility to take into account the flexibility of the protein. On the other side, AA-MD simulations provide good current statistics at reasonable simulation times for high voltages but a) reaching the steady state seems to take a long time, if it is possible due to the finite system size, and b) the system is at a non-linear regime. Although the non-linearity will appear at higher potentials in AA simulations than in CG simulations, in ref.51, they estimate this limit to be below 500 mV in a AA-MD study of a modified α-hemolysin, interpolating low voltage values from high voltage ones can be somehow risky. At low voltages, direct sampling of net ion fluxes with AA simulations would require very long simulation times to acquire sufficient statistics.
Our conclusion, as regard to our results, is that CG-MD simulations could result a good alternative to study the details, such as mutation effects, of translocation through protein nanopores, saving some of the simulation time required for the equivalent AA simulations for these kind of problems.
Simulation details
We performed Molecular Dynamics (MD) using the Gromacs software package30 (version 5.0.2) and the MARTINI 2.2 CG model for ions, lipids23, proteins31,32 and the PW (Polarizable Water) model29 for water. Conductivity simulations systems contained only water and ions at different concentrations, whereas membrane simulations systems contained a DPPC bilayer, water, and ions at different concentrations. In the nanopore simulations, the protein nanopore was inserted in the lipid membrane.
A timestep of 20 fs was used for all MD simulations, which is the recommended timestep for CG MD simulations using MARTINI force field. Periodic boundary conditions in the three dimensions were applied. All MD were performed using the PME method for electrostatic interactions, with a 2 Å spacing for the Fourier grid and a direct space cut-off radius of 13 Å. A relative dielectric constant of εr = 2.5 was used for PW as recommended29. Van der Waals interactions were shifted at 13 Å. A Nosé-Hoover thermostat53,54 was used to maintain the temperature at 320 K.
Conductivity simulations
For conductivity simulations, the system was originally built as a cubic box of 15.8 nm3, containing 32,756 PW water particles. The system was minimized and slightly equilibrated (2 ns at NVT) before the introduction of ions at different concentrations. The genion tool of Gromacs was used to replace random water particles by positive (the MARTINI Na+ ion) and negative (Cl−) ions in the same quantity, in order to keep the system neutral. The system containing water and ions was then again minimized and equilibrated with a short MD (10 ns) in the NVT ensemble, using the Berendsen thermostat. Then, a one-microsecond-long MD was performed in the NVT ensemble at 320 K, using the Nosé-Hoover thermostat, with a characteristic time interval τT = 1 ps for both groups (ions and water). For each concentration, ranging from 0.1 M to 2 M, three different simulations were performed: one with no external electric field, one with an electric field of 0.02 V/nm in the z direction, and one with an electric field of 0.04 V/nm in the same direction.
Membrane simulations
As for the membrane simulations, a box containing 128 DPPC CG lipids organized in a bilayer was downloaded from the MARTINI website55. 4,232 water molecules (PW) were added around the membrane in a box of originally 6.7 × 6.1 × 12.0 nm3, the z axis oriented perpendicularly to the membrane plan. The system was minimized and shortly equilibrated before addition of ions at different concentrations using the same procedure as the conductivity simulations. Simulations without ions were also performed using the same methodology. After another minimization of the system, a NVT MD simulation was performed for 2 ns using the Berendsen thermostat, followed by 120 ns of NPT MD using the Berendsen thermostat and Berendsen semi-isotropic barostat at 1 bar. The MD production was performed for two microseconds using the Nosé-Hoover thermostat at 320 K (with a characteristic time interval τT = 1 ps for each group) and the Parrinello-Rahman barostat to keep the pressure at 1 bar (with a characteristic time interval of τP = 5 ps and a compressibility of 4.5 × 10−5 bar−1). It should be noted that, in the NPT ensemble, the box dimension in the z direction increased at the beginning of the dynamics, and the Lz length was around 16 nm on average during all the membrane MD. In addition to simulations without ions, three different KCl concentrations were simulated: 0.2 M (60 ions of each species), 0.4 M (124 ions) and 0.8 M (248 ions). For each concentration, three different simulations were performed: one with no external electric field, one with an electric field of 0.01 V/nm in the z direction, and one with an electric field of 0.02 V/nm. The averaged charge density was calculated on the last 500 ns of the 2 μs-simulation, using the g_potential tool of Gromacs, decomposing the system into 100 slices in the z direction.
Nanopore simulations
The crystallographic structure of α-hemolysin was taken from the PDB (entry 7AHL44). The coordinates of the missing atoms were added using the pdb2pqr software and the crystallographic water molecules were removed. The protein was reduced in coarse grains using the martinize.py script. The ElNeDyn MARTINI force field was applied on the protein48. It is composed of an elastic network model that maintains the secondary structures during the dynamics using springs (500 kJ.mol−1.nm−2 elastic bond strength and a 0.9 nm cutoff). After a short energy minimization, the resulting CG protein structure was inserted in a DPPC bilayer using the following procedure: the PDB structure of a CG α-hemolysin inserted in a DPPC bilayer of 756 lipid molecules obtained by Prof. Sansom45 was downloaded from CGDB (coarse grain database, a database of inserted membrane proteins inside DPPC bilayers56). Since the protein CG model used for CGDB is different from the MARTINI one, a least-squared fit of the backbone of our CG protein on the CGDB one was performed, then the initial protein from CGDB was deleted to keep only our MARTINI CG protein inserted in the DPPC bilayer. The size of the simulation box was initially set at 15.85 × 15.85 × 20 nm. The system was solvated with 28,758 standard W MARTINI CG water molecules using the genbox Gromacs command. Another minimization was carried out followed by two equilibration steps: 1 ns in the NVT ensemble (using a 10 fs timestep) followed by 5 ns in the NPT ensemble (using a 20 fs timestep).
The ionic concentration was set around 1 M, corresponding to 2,054 ions of each species (calculated according to the number of initial water molecules and using the ions/PW molecules ratio from conductivity simulations concentrations). Under normal conditions, the nanopore carries a global charge of +7e. Therefore, in order to neutralize the system, 2,050 K+ ions and 2,057 Cl− ions were added to the system. A short dynamics of 5 ns was then performed at NPT. After transformation of MARTINI W water into polarizable PW water, a short minimization was carried out, followed by two additional equilibration dynamics of 1 ns at NVT and 10 ns at NPT, both using position restraints on the protein backbone (with a 20 fs integration timestep). A last dynamics of 1 ns at NPT without position restraints was performed using the PME method for electrostatic interactions. The resulting system was used as initial state for all the simulations of the pore as well as the dynamics with the pore under position restraints on the protein backbone.
Several electric fields (E) were applied in the z direction of the system, perpendicular to the membrane plane, both positive and negative: no electric field, +/−0.005 V/nm, +/−0.01 V/nm, +/−0.015 V/nm, +/−0.02 V/nm, +/−0.025 V/nm, +/−0.03 V/nm and +/−0.04 V/nm. Molecular dynamics were carried out for 2 to 3 μs (+/−0.005 V/nm, +/−0.01 V/nm, +/−0.015 V/nm). See Table 1 for further details.
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This work was performed using HPC resources from GENCI-CINES under the grants 2015 - c2015077139, 2016 - c2016077139 and 2017 - c2017077139. Part of this work was funded by CNRS, Défi InFIniTi for the DYNANO project (2016 and 2017).
LAMBE, Univ Evry, CNRS, CEA, Université Paris-Saclay, 91025, Evry, France
Nathalie Basdevant, Delphine Dessaux & Rosa Ramirez
Nathalie Basdevant
Delphine Dessaux
Rosa Ramirez
N.B., D.D. and R.R. conducted the simulations and analyzed the results. All authors participated in writing and reviewed the manuscript.
Correspondence to Rosa Ramirez.
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Basdevant, N., Dessaux, D. & Ramirez, R. Ionic transport through a protein nanopore: a Coarse-Grained Molecular Dynamics Study. Sci Rep 9, 15740 (2019). https://doi.org/10.1038/s41598-019-51942-y
Received: 07 November 2018
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I Notation for vectors in tensor product space
Thread starter hilbert2
composite system spin tensor product
hilbert2
About writing the state vector of a composite system
Suppose I have a system of two (possibly interacting) spins of 1/2. Then the state of each separate spin can be written as a ##\mathbb{C}^2## vector, and the spin operators are made from Pauli matrices, for instance the matrices
##\sigma_z \otimes \hat{1}## and ##\hat{1} \otimes \sigma_z##,
which are tensor products of the z-direction spin matrix acting on one spin, and a unit matrix acting on the other spin, correspond to the spin-z operators for individual spins.
Now, in the bra-ket notation it is easy to write the product form states of the composite system as something like ##|s_1 \rangle |s_2 \rangle##. Is it also a correct notation to use the tensor product symbol ##\otimes## for constructing these from the ##\mathbb{C}^2## vectors:
##|s_1 \rangle |s_2 \rangle = \begin{bmatrix}a_1 \\ a_2 \end{bmatrix}\otimes\begin{bmatrix}b_1 \\ b_2 \end{bmatrix}##,
with ##a_1 ,a_2 ,b_1## and ##b_2## being the complex number components of ##|s_1 \rangle## and ##|s_2 \rangle## ?
fresh_42
2018 Award
I don't really understand the Dirac notation, but I would say it is
##|s_1\rangle \langle s_2| = \begin{bmatrix}a_1\\a_2\end{bmatrix} \otimes \begin{bmatrix}b_1\\b_2\end{bmatrix}## since ##|s_1\rangle## is a column vector and ##\langle s_2|## a row vector as I understood it.
martinbn
fresh_42 said:
But on the right you have two column vectors. I'd say it is as written. This ##|s_1\rangle |s_2\rangle## is short for ##|s_1\rangle\otimes |s_2\rangle##.
Reactions: hilbert2
martinbn said:
But on the right you have two column vectors.
In the tensor notation it is irrelevant how you write the vectors because the tensor product determines completely the result. Btw. you made use of this fact by your suggested abbreviation. Written as matrix multiplication it is ##s_1s_2^\tau = s_1\otimes s_2## where the vectors are columns. A dyad is column times row. If ##|s_1\rangle |s_2\rangle## is short for ##|s_1\rangle \otimes |s_2\rangle##, what is ##|s_1\rangle \langle s_2|## then?
My first thought was that the operator ##|s_1 \rangle \langle s_2 |## would just be the product
##\begin{bmatrix}a_1 \\ a_2 \end{bmatrix}\begin{bmatrix}b_1 & b_2\end{bmatrix} = \begin{bmatrix}a_1 b_1 & a_1 b_2 \\ a_2 b_1 & a_2 b_2 \end{bmatrix}##
but then it wouldn't have the whole 4-dimensional state space as its domain... You could probably write it that way if the ##|s_1 \rangle## and ##|s_2 \rangle## are, for instance, the ##s_z = 1/2## and ##s_z = -1/2## eigenstates of a single spin. Or the ##s_x = 1/2## and ##s_y = 1/2## eigenstates.
Thanks for the comments, anyway.
hilbert2 said:
but then it wouldn't have the whole 4-dimensional state space as its domain...
But linear combinations of them have. You can never span a four dimensional space by two vectors however you define their binary operation.
I remember having used the "kronecker()" function of R language when constructing matrix representations of operators that act on a quantum system composed from two non-interacting subsystems. It did produce results that were the same as calculated on pen and paper, and I think it is the same product that is meant with the ##\otimes## symbol.
Sure. The question is about the notation, not that Kronecker product and tensor product are the same.
The standard physicist's notation (the way I understand it) is that things like ##|a\rangle, |b\rangle, |c\rangle## belong to some Hilbert space ##H##, while things like ##\langle a|, \langle b|, \langle c|## to the dual ##H^*##, and ##\langle a| (v) = \left(|a\rangle,v\right)_H##.
So ##|s_1\rangle |s_2\rangle\in H\otimes H##, while ##|s_1\rangle \langle s_2|\in H\otimes H^*##.
Reactions: Klystron, Cryo, hilbert2 and 1 other person
## |\rangle \langle|## is the outer product if my memory of Dirac notation is correct. (Hopefully I latex this correct) the inner product of two vectors for column and row should be
##\vec{a}\otimes\vec{b}=\dbinom{a}{b}(a^*b^*)##
I had to double check this but two state vectors will be in notation
##\langle\psi|\phi\rangle## with standard scalar product being ##\langle\psi|\phi\rangle=\langle \phi|\psi\rangle^*##
The tensor product of two Hilbert spaces oft denoted as
##\mathcal{H}=\mathcal{H_1}\otimes\mathcal{H_2}##
Summary: About writing the state vector of a composite system
The above notation is accurate for the tensor products as far as I am familiar with it
Though I am not familiar enough with the tensor products in Dirac notation to have confidence in your questions I would surmise that the answer to both as being yes with that in mind
Lol took too long a break from physics
Is it also a correct notation to use the tensor product symbol ##\otimes## for constructing these from the ##\mathbb{C}^2## vectors:
with ##a_1 ,a_2 ,b_1## and ##b_2## being the complex number components of ##|s_1 \rangle## and ##|s_2 \rangle##?
The idea is correct: it is a vector in the four-dimensional tensor product space of the Hilbert space with itself. But "the" complex number components of ##|s_1 \rangle## and ##|s_2 \rangle## don't exist. Components only come into play after a basis has been chosen. The LHS of your equation is an abstract vector, the RHS implies that a basis has been chosen.
Don't mix representations, i.e., the matrix-vector notation with components of the various objects of linear algebra with respect to a basis and the basis-independent objects themselves. This almost always leads to confusion. The great thing with Dirac's notation is that it usese the basis-independent objects only and easily lets you derive anything in terms of a representation (i.e., using some specific basis) if necessary.
Now let's look at the different products which have occurred in this thread so far. Let's start with one Hilber space ##\mathcal{H}##. By definition you have a scalar product, i.e., a sesquilinear form mapping two vectors to a complex number, the notation is ##\langle a|b \rangle## with ##|a \rangle## and ##b \rangle## arbitrary vectors in ##\mathcal{H}##.
Then there's the construct ##|b \rangle \langle a|##. As the notation suggests that's a special linear mapping of any vector ##|c \rangle \in \mathcal{H}## another vector, namely ##|b \rangle \langle a|c \rangle \in \mathcal{H}##.
Now there are also direct products of two Hilbert spaces ##\mathcal{H}_1## and ##\mathcal{H}_2## leading to a new Hilbert space ##\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2##. The vectors in the new Hilbert space are spanned by the direct product of vectors from the two spaces. These special vectors in ##\mathcal{H}## are written as ##|a \rangle \otimes |b \rangle## with ##|a \rangle \in \mathcal{H}_1## and ##|b \rangle \in \mathcal{H}_2##. This tensor product is by definition linear in both arguments. From this you can construct any vector in ##\mathcal{H}## by linear combinations. Further one defines the scalar product on ##\mathcal{H}## also through the product states and the usual sesquilinearity property of the scalar product by
$$(\langle a_1| \otimes \langle b_1|)(|a_2 \rangle \otimes |b_2 \rangle)=\langle a_1 | a_2 \rangle \langle b_1|b_2 \rangle.$$
Some authors simply write ##|a \rangle |b \rangle## or even ##|ab \rangle## for product states. The meaning is always the same.
Reactions: Mordred
Thanks Vanhees71. Even though this isn't my thread the reminder helped me with details I had forgotten. Been far too long since I last worked with the notation. ( Though it's extremely useful ).
Get gonna have to sit down and study thankfully I still have my textbook collection...
Hopefully it isn't considered a hijack to ask a quick related question on the notation if so I apologize in advance.
Is this correct ##|i\rangle\langle i|\psi\rangle##
The two i's in the ket- bra being the projection operators operating on state ##\psi## ?
Sure, it's a valid expression. If ##|i \rangle## is normalized to 1, i.e., ##\langle i|i \rangle=1##, then ##\hat{P}=|i \rangle \langle i|## is indeed the projection operator to the direction of ##|i \rangle##.
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Journal of Analytical Science and Technology
Investigation of DNA binding and molecular docking propensity of phthalimide derivatives: in vitro antibacterial and antioxidant assay
Rizwan Arif1,
Pattan Sirajuddin Nayab1,
Akrema1,
Mohammad Abid2,
Umesh Yadava3 &
Rahisuddin ORCID: orcid.org/0000-0002-7139-69761
Journal of Analytical Science and Technology volume 10, Article number: 19 (2019) Cite this article
A series of N-substituted tetrabromphthalimide derivatives was synthesized by condensation reaction using tetrabromophthalic anhydride with 3,5-diamino-1,2,4-triazole/ 2,6-diaminopyridine/ 2,6-diamino-4-hydroxy pyrimidine/ o-tolidine. All the synthesized phthalimide derivatives were characterized by elemental analysis, infrared, and NMR spectroscopy. In vitro antibacterial evaluation was carried out for the synthesized compounds. Results revealed that compound 1 showed potential activity against Escherichia coli (100 μg/mL) and Streptococcus mutans (150 μg/mL). On the basis of antibacterial activity, compound 1 was selected for DNA binding interaction, though DNA target most of the antibacterial drugs. The DNA binding modes of the compound 1 with Ct-DNA (calf thymus) were studied by absorption measurements, hydrodynamic measurements and cyclic voltammetry methods. Molecular docking also confirms that compound 1 recognizes both the strands of the DNA dodecamer d(CGCGAATTCGCG)2 within minor groove and showing the best binding capability with the duplex. Compound 1 also showed better antioxidant activity by 2,2-diphenyl-1-picryl-hydrazyl (DPPH) free radical and hydrogen peroxide.
Heterocyclic phthalimides are well explored derivatives among important class of pharmacophores for the preparation of variety of drugs because of its similar structural properties (hydrophobic aryl ring and an electron-donating group), they possess variety of important biological activities. Due to its wide range of applications in medicinal chemistry and pharmaceutics (Bhat and Al-Omar 2011; Mohamed et al. 2015) viz. antibacterial (Lohan et al. 2014; Silva et al. 2014; Zhang et al. 2015), antifungal, anti-inflammatory (Buddana et al. 2015; Al-Qaisi et al. 2014), antivirus (Shi et al. 2011), antagonistic (Lin et al. 2009), and anti-androgens (Roy et al. 2007), it is an important and tremendous subject for today's research. Apart from their pharmaceutical applications, phthalimide also serves as herbicides (Balachandran et al. 2012), used in production of pesticides (Wang et al. 2013) and dyes (Choi et al. 2010). They were also found with industrial applications as bleaching agent, heat-resistant polymer, and flame retardants (Krishnakumar et al. 2005). For the amine protection in organic synthesis, phthalimide moiety is an important constituent.
In the process of production, the knowledge of thermodynamic parameters or solvent crystallization is significant and process for purifying the phthalimide derivatives by virtue of crystallization method induces by the awareness of their solubility in appropriate solvent system. Li and his co-workers determine the solubility of various phthalimide derivatives experimentally by using the isothermal dissolution equilibrium method in mixed solvent systems (Li et al. 2017). Phthalimide derivatives exhibited various biological activities and proved as a noteworthy pharmacophore and can interact with the peripheral anionic site of the enzyme. A series of phthalimide derivatives was synthesized as multi-function inhibitors for the treatment of Alzheimer's disease (AD) and found to be a balanced multi-target active molecule which exhibited potent and balanced inhibitory activities against cholinesterase inhibitors (Sang et al. 2017). Due to industrial development and high requirement, phthalimide was prepared on a large scale by various methods in high yield. Due to very low solubility in water, separation of phthalimide is not easy. Hence to rectify these problems, many methods have been developed by the researchers for the separation and purification of phthalimides in high yield (Kushwaha and Kaushik 2016). Heterocyclic phthalimide derivatives are very much useful for the construction of macromolecules, in supramolecular chemistry, useful in catalytic reaction and photochemistry (Yoon et al. 1997; Cho et al. 2010). A series of polyester derived from phthalimide, which are thermally stable and highly soluble in polar solvents, has been also reported (Behniafar et al. 2015).
DNA plays major role in many physiological and biological processes. Various targeting drugs for DNA which was approved as significant antimicrobial agents currently available in the market but due to their side effects and highly clinical costs, many researchers are working on developing new antimicrobial drugs. The biological activity of the antibacterial drugs containing heterocyclic ring is due to inhibition of DNA replication (Nayab et al. 2015a, b, 2016). In view of these prospective, the study of DNA interaction with phthalimides has been considered a subject of great importance for the investigation of potential antimicrobial and anticancer drugs and molecular docking technique can be used for knowing the aspect of intermolecular interactions of proteins and ligands.
Considering these facts, in the current research work, we have synthesized tetrabromo phthalimide derivatives by the condensation reaction of tetrabromo phthalic anhydride and 3,5-diamino-1,2,4-triazole/ 2,6-diaminopyridine/ 2,6-diamino-4-hydroxy pyrimidine/ o-tolidine in order to prepare potent antibacterial and DNA binding agents.
1,2,4-Triazole-3,5-diamine and tetrabromophthalic anhydride purchased from Sigma Aldrich Chemicals Pvt. Ltd. and 2,6-diaminopyridine, o-tolidine, 2,6-diamino-4-hydroxypyrimidine, and 2,2-diphenyl-1-picryl-hydrazyl (DPPH) from Merck, India. All the reagents were used as received without further purification. For thin-layer chromatography, precoated aluminum sheets (silica gel 60 F254, Merck, Germany) were used to monitor the reaction by using methanol:dichloromethane (1:4) as a solvent system and for the spots visualization UV light cabinet was used.
To perform elemental analysis, Vario Micro Elementar Analyzer was used. Electronic spectra were recorded on Perkin Elmer Lamda 40 UV-visible spectrophotometer. IR spectra were recorded on Agilent Cary 630 FTIR spectrometer as neat sample. 1H NMR spectra were recorded on Bruker DPX-300 NMR spectrometer operating at 400 MHz using DMSO-d6 as solvent with TMS as internal standard. Chemical shift values are given in ppm. Cyclic voltammetric measurements were performed by using DY2312 potentiostat. The antibacterial experiment was performed against Streptococcus mutans (MTCC 3224), and Escherichia coli (ATCC 25922) bacterial strains. To perform molecular docking study Gaussian 03 software was used. The antioxidant potential of the phthalimide derivatives was also estimated using DPPH free radical and hydrogen peroxide assay.
Synthesis of phthalimide derivatives
Similar procedure was employed according to our earlier reported method (Arif et al. 2016). The color solids that obtained were filtered, washed with distilled water, and finally dried under vacuum on fused calcium chloride and recrystallized in chloroform.
4,5,6,7-tetrabromo-2-[3-(4,5,6,7-tetrabromo-1,3-dioxo-2,3-dihydro-1H-isoindol-2-yl)-1H-1,2,4-triazol-5-yl]-2,3-dihydro-1H-isoindole-1,3-dione (1):
White solid; mp > 300 °C; C18HN5O4Br8: yield 60%; IR (cm−1): νC=O(asym) 1786, νC=O(sym) 1736, νNH 3453. 1H NMR (300 MHz, DMSO-d6), δH 9.46 (s, 1H, NH). 13C NMR (100 MHz, DMSO-d6) δ in ppm 166.63, 145.02, 138.41, 133.29, 131.51, 129.19, 128.38, 126.22, 118.93, 116.30. Anal. Calcd. for C18HN5O4Br8: C, 21.82; H, 0.11; N, 7.05; found: C, 21.87; H, 0.11; N, 7.05.
4,5,6,7-tetrabromo-2-[6-(4,5,6,7-tetrabromo-1,3-dioxo-2,3-dihydro-1H-isoindol-2-yl) pyridin-2-yl]-2,3-dihydro-1H-isoindole-1,3-dione (2):
Brown solid; mp > 300 °C; yield: 62%; IR (cm−1): νC=O(asym) 1780, νC=O(sym) 1730. 1H NMR (300 MHz, DMSO-d6): δH 7.46 (d, 2H, Ar-H); δH 9.84 (s, 1H, Ar-H). 13C NMR (100 MHz, DMSO-d6) δ in ppm 169.33, 132.59, 128.29, 129.23, 128.67, 124.88, 122.50, 119.29, 112.79. MS (m/z): 1000.35 (M+1). Anal. Calcd. for C21H3N3O4Br8: C, 25.20; H, 0.30; N, 11.03; found: C, 25.24; H, 0.34; N, 11.22.
4,5,6,7-tetrabromo-2-[4-hydroxy-6-(4,5,6,7-tetrabromo-1,3-dioxo-2,3-dihydro-1H-isoindol-2-yl)pyrimidin-2-yl]-2,3-dihydro-1H-isoindole-1,3-dione (3):
Off-white solid; mp > 255 °C; yield: 69%; IR (cm−1): νC=O(asym) 1783, νC=O(sym) 1720, νOH 3190. 1H NMR (300 MHz, DMSO-d6): δH 9.54 (s, 1H, −OH); δH 5.02 (s, 1H, Ar-H). 13C NMR (100 MHz, DMSO-d6) δ in ppm 169.08, 162.06, 160.07, 131.80, 131.53, 131.11, 127.87, 127.65, 126.42. MS (m/z): 1017.75 (M+1). Anal. Calcd. for C20H2N4O5Br8: C, 23.60; H, 0.19; N, 5.50; found: C, 23.25; H, 0.34; N, 5.87.
4,5,6,7-tetrabromo-2-{2-methyl-4-[3-methyl-4-(4,5,6,7-tetrabromo-1,3-dioxo-2,3-dihydro-1H-isoindol-2-yl) phenyl] phenyl}-2,3-dihydro-1H-isoindole-1,3-dione (4):
Mud color solid; mp > 300 °C; yield 71%; IR (cm− 1): νC=O(asym) 1777, νC=O(sym) 1719, νCH 2985; 1H NMR ((δ, ppm in DMSO-d6): 2.18 (s, 6H, −CH3); 7.84–7.87 (3H, Ar-H), 8.06–8.09 (3H, Ar-H). 13C NMR (100 MHz, DMSO-d6) δ in ppm: 168.12, 167.26, 147.55, 136.87, 135.94, 131.36, 129.14, 126.97. 28.22. MS (m/z) 1104.5 (M+1). Anal. Calcd. for C30H12N2O4Br8: C, 32.64; H, 1.09; N, 2.53; found: C, 32.66; H, 1.11; N, 2.49.
DNA binding studies
Absorption measurements
UV-visible spectroscopy is very suitable and significant method to evaluate the binding nature of small molecules and DNA. All the experiments were performed in tris-HCl buffer. The absorption ratio of Ct-DNA in buffer was 1.9:1 at 260 nm which reveals that DNA was possibly absolutely free from protein contamination (Raja et al. 2016). Absorption intensity was determined at 260 nm for the investigation of DNA concentration of stock solution using a molar absorption coefficient ε260 = 6600 L mol−1 cm−1 by UV-visible spectrophotometer (Kumar et al. 2016). Test samples were allowed to equilibrate at room temperature for 10 min before recording the absorption spectrum, carried out in the range of 190–500 nm by varying the concentration of the DNA (1.2–3.0 × 10−5 M) and adapting the constant concentration of compound (1 × 10− 4 M).
Viscosity measurements
The viscosity was determined in the presence of increasing concentration of test compound (0.4–2.0 × 10−5 M) and fixing the DNA concentration (2.5 × 10−5 M) in the 5 mM Tris–buffer (pH = 7.2). All the experiments were carried out using an Ostwald capillary viscometer maintained at 25 ± 0.1 °C. The flow time of the test compounds through the viscometer was determined in triplicate to get the average and accurate value (Patel et al. 2014). The obtained data ploted as (η/ηo)1/3 versus [compound]/[DNA], where η is the viscosity of Ct-DNA in the presence of compound and ηo is the viscosity of Ct-DNA alone. Relative viscosity of the test compound 1 was determined from the observed flow time of DNA solution (t) corrected for the flow time of tris-buffer alone (t0), using the expression 1.
$$ {\eta}_{\mathrm{o}}=\left(t-{t}^0\right)/{t}^0 $$
Electrochemical study
Cyclic voltammetry (CV) was performed to investigate the redox behavior of test compound 1 using DY2312 potentiostat. The electrochemical experiments were carried out with and without DNA in tris-buffer (pH = 7.5). All the experiments were done at a scan rate 0.2 Vs−1 in the potential range + 1.2 to − 2.0 V at room temperature (Mazhabi and Arvand 2014). Experiment was performed in a electrochemical cell consisting three-electrode, platinum wire auxiliary electrode, glassy carbon working electrode, and silver/silver nitrate as a reference electrode. Alumina powder was used to polish the electrode surface, and for the deoxygenation of the test compounds, nitrogen gas had been used 20 min before the experiments.
Antibacterial activity
All the synthesized compounds were screened for antibacterial activity against gram-negative bacterium E. coli and gram-positive bacterium S. mutants using Kirby Bauer methods (1953) and broth dilution method (Boufas et al. 2014) which conformed to the recommended standards of the Clinical and Laboratory Standards Institute (CLSI). Ampicillin was used as a standard antibacterial drug. A diluted series with 10 mL nutrient broth medium containing 50–200 μg/mL of synthesized phthalimide derivatives were prepared. One hundred microliters of respective bacterial suspension (approximately 106 CFU/mL) was used to inoculate the each set. The bacteria were plated onto solid nutrient agar plates. MIC was defined as the lowest concentration for the inhibition of bacterial growth. All the experiments were done in triplicate and average was reported.
Molecular docking study
Phthalimide derivative 1 has been optimized using B3LYP method in conjunction with 6-31G** basis set utilizing Gaussian 03 (Yadava et al. 2015a, b). Absence of imaginary frequency modes for each molecule indicates that the true minima were achieved. Conformations of the molecule were generated and prepared through LIGPREP wizard of the SCHRODINGER suite (Yadava et al. 2013). The three-dimensional structure of the DNA duplex was retrieved from the protein data bank (PDB ID: 1BNA) which was prepared using protein preparation wizard. The electron affinity grid map was generated around the center of the DNA, and the docking of molecule was carried out using XP (extra precision) mode of GLIDE (Yadava et al. 2015a, b). During docking, ligand was treated as flexible while DNA was taken as rigid. Docking complex was considered for the calculation of glide energy and glide scores.
Antioxidant activity
DPPH free radical scavenging assay
Antioxidant activity of all the phthalimide derivatives was evaluated against DPPH free radical according to the method reported by the Miliauskas et al. which was the best method based on electron transfer (Miliauskas et al. 2004). Test compound (0.5–3.0 mg/mL) and ascorbic acid (0.2–1.4 mg/mL) in DMSO was added to 0.1 mM DPPH (3 mL) in ethanol. All test compounds were incubated at 60 °C for 2 h, and the decrease in absorbance was noted at 510 nm using UV-Vis spectrophotometer against a blank of ethanol and DMSO in 1:1. Absorbance of DPPH (control) solution was also recorded at same wavelength for comparative study. The IC50 in (mg/mL) was calculated from the graph between % antioxidant activity vs concentrations. For each of the tests, compound experiment was done in triplicate and antioxidant property of the compounds was measured by using Eq. 2:
$$ \%\kern0.5em \mathrm{Inhibition}\kern0.5em =\kern0.5em \frac{A_{\mathrm{Control}}-{A}_{\mathrm{Sample}}}{A_{\mathrm{Control}}}\kern0.5em \times \kern0.5em 100 $$
where Acontrol = absorbance of DPPH free radical in methanol without an antioxidant and Asample = absorbance of DPPH free radical in the presence of an antioxidant.
Hydrogen peroxide scavenging activity
The antioxidant ability of the phthalimide derivatives was also estimated by hydrogen peroxide using standard method (Ruch et al. 1989). To the different concentrations of test compounds, 1.8 mL of a 2 mM H2O2 solution prepared in phosphate buffer (50 mM, pH 7.4) was added and the samples were incubated for 10 min. We record the decrease in absorbance against phosphate buffer as a blank by UV-Vis spectrophotometer. The absorbance of test samples was noted at 240 nm and compared with hydrogen peroxide which was taken as a control. The antioxidant ability of hydrogen peroxide was calculated using following equation:
$$ \%\mathrm{Inhibition}=\frac{A_{\mathrm{B}}-{A}_{\mathrm{T}}}{A_{\mathrm{B}}}\times \kern0.5em 100 $$
where AB was the absorbance of blank (without compounds) and AT was the absorbance of tested samples.
Phthalimide derivatives were prepared in good yield by the condensation reaction of tetrabromophthalic anhydride with substituted diamines of triazole, pyridine, and pyrimidine (Scheme 1). The structure of the compounds (1–4) was proposed on the basis of spectral data and elemental analysis. All the compounds are stable in solid state and soluble in DMSO and DMF. The progress of the reaction was monitored by thin layer chromatography (TLC) in methanol: dichloromethane (1:4).
Synthesis of tetrabromo-bisphthalimide derivatives
IR spectra
IR spectra of synthesized phthalimide derivatives were recorded in the range 4000–400 cm−1. All the compounds were characterized by a vibrational band in the range of 1777–1793 cm−1. The stretching frequency due to υ(−NH2 group) (3,5-diamino-1,2,4-triazole, 2,6-diaminopyridine and 2,4-diamino-6-hydroxypyrimidine and o-tolidine) which disappear in the compounds (1–4) confirms the formation of bis-phthalimide derivatives. A characteristic peak appeared at 3454 cm−1 for the compounds (1) may be assigned due to the υ(−NH) group. However, a peak appeared at 3190 cm−1 for the compound (3) is assigned due to υ(−OH) (Sas et al. 2015; Collin et al. 2001).
1H NMR spectra
1H NMR spectra of compounds were recorded in DMSO-d6 at room temperature exhibited well resolved signals depicted in Additional file 1: Figure S1. The signal due to –NH2 group in substituted amines disappear which indicate the formation bis-phthalimide derivatives (1–4). A singlet due to −NH group of triazole appear at 9.46 ppm was assigned for the compound 1. Moreover, a singlet appeared at 9.54 ppm attributed to the –OH for the compound (3). The signals due to aromatic protons in the compounds (1–4) appear at 6.52–8.13 ppm (Pawluc et al. 2012; Antunes et al. 2003).
DNA binding study
The binding ability of DNA with compound 1 was studied efficiently through absorption spectroscopy. The interaction ability of the compounds and DNA was investigated by the increase or decrease in the absorbance and shift in wavelength. Hypochromism or decrease in absorbance with red shifts in wavelength involved a strong stacking interaction between the DNA base pairs and aromatic chromophore although hyperchromism is associated with electrostatic or groove binding which also responsible for the damage of secondary structure of DNA. The absorption spectra of the compound 1 in presence and absence of DNA is given in Fig. 1. On increasing the concentration of DNA, absorbance decrease or hypochromism was observed which show that compound 1 bind with Ct-DNA by strong intercalative mode (Nayab et al. 2016).
Absorption spectra of compound (1) (1 × 10−4 M) in the presence of increasing amounts of Ct-DNA (1.2–3.0 × 10−5 M). The inset is plot of [DNA]/(εa − εf) vs [DNA] for the titration of DNA to compound
Hypochromism was observed due to the stacking of the planar aromatic group of the compounds between adjacent base pairs of double helix DNA. Intrinsic binding constant (Kb) was calculated by the Eq. 4:
$$ \frac{\left[\mathrm{DNA}\right]}{\left({\varepsilon}_a-{\varepsilon}_f\right)}\kern0.5em =\kern0.5em \frac{\left[\mathrm{DNA}\right]}{\left({\varepsilon}_b-{\varepsilon}_f\right)}\kern0.5em +\kern0.5em \frac{1}{K_b}\left({\varepsilon}_b-{\varepsilon}_f\right) $$
where ɛa, ɛf, and ɛb refers to Aobsd/[Compound], the absorption extinction coefficient for free compound and the absorption extinction coefficient for the compound in fully bound form, respectively. Kb is slope to intercept ratio given by the plot, [DNA]/(ɛa−ɛf) vs [DNA]. The DNA binding constant calculated for compound 1 was 4.7 × 106 M−1.
Viscometry measurement is a very significant tool for further estimation of binding nature of Ct-DNA with small molecule which is very critical and sensitive to length change. Classical intercalation leads to the adjustment of molecules between the base pairs of DNA which in turn increase the separation between base pairs at intercalation site and increases it helix length; therefore, viscosity increases. However, due to partial or non classical intercalation, led to bend or kink of DNA helix, results in reduction of the effective length of Ct-DNA and thereby decrease in viscosity (Arif et al. 2016). The effect of addition of increasing amount of test compound 1 on relative specific viscosity of DNA is shown in Fig. 2. On the addition of increasing amount of test compound 1 and fixing the Ct-DNA concentration, viscosity of Ct-DNA increases remarkably which notify that compound can intercalate between the base pairs of DNA and results are very similar with classical intercalators as corroborated by UV-Vis spectroscopic study.
Effect of increasing amounts of test compound (1) on the relative viscosity of DNA at pH 7.4 and 25 °C, [DNA] = 2.5 × 10−5 M and [Compound] = (0.4–2.0 × 10−5 M)
Electrochemical method is another significant tool to clarify the mode of binding of compound with Ct-DNA. The positive shift in peak potential is due to the intercalation of the compound with Ct-DNA; however, negative shift is attributed to the electrostatic binding. The cyclic voltammograms of test compound 1 with and without DNA are shown in Fig. 3. It was inferred that after the addition of DNA, peak current was dropped by 19% for compound 1. Decrease in current may be due to the formation of compound-DNA complex and decrease in free ligand concentration. These results are in accordance with viscosity measurements and absorption study.
Cyclic voltammograms of 3.2 × 10−4 M of test compound 1 in 1 mM tris-buffer, pH 7.5 at 50 mV s−1 scan rate without DNA (red) and with DNA (black)
In vitro antimicrobial evaluation of synthesized phthalimide derivatives (1–4) was carried out against two bacterial strains Escherichia coli (E. coli) and Streptococcus mutans (S. mutans) and ampicillin used as a positive control (Anacona et al. 2015). Results obtained revealed that out of all synthesized compounds, only compound 1 inhibited bacterial growths. MIC (minimal inhibitory concentration) was defined at lowest concentration that completely inhibited visible bacterial growth. The MBC values are 100 μg/mL and 150 μg/mL for compound 1 against E. coli and S. mutants, respectively, and compound 1 exhibits better activity than other compounds.
The results of extra precision glide docking of the molecules with the DNA duplex 5′(CGCGAATTCGCG)3′ are presented in Fig. 4. Glide docking demonstrates that the best pose of compound 1 has the better glide score in comparison to other compounds. The glide scores in their best poses of compound 1 is − 5.772 kcal/mol. The best docking pose of the compound 1 showed that the Emodel and glide energies equal to − 59.905 and − 45.704 kcal/mol respectively. Figure 4 demonstrates that the compound 1 recognizes both strands of the DNA and binds within minor groove of the duplex. One hydrogen bonding has also been demonstrated by the compound 1 involving the thymine (DT7) base of the DNA.
Molecular docking study of the compound 1
Molecular docking results demonstrate that this compound recognizes both the strands of the DNA dodecamer d(CGCGAATTCGCG)2 within minor groove. Docking results reveal that compound 1 has the best binding capability with the duplex. It may further be concluded that the binding of drug molecule with DNA is sequence dependent and the specific sequence of the DNA may be playing a key role in the binding process. These results may enhance future prospects for the drug development targeting DNA.
For the estimation of antioxidant activity of the test compounds, DPPH free radical assay is a very suitable and easy method. It is a stable free radical that can accept hydrogen ion or electron on reaction from an antioxidant compound and become reduced. After the incubation of 2 h at 60 °C, the color of the compounds changed from violet to light yellow which clearly reveals that some antioxidant moiety was present in compound. The decrease in the absorbance was noted at 510 nm. The IC50 values of DPPH radical scavenging activity for the compounds 1, 2, 3, and 4 were found to be 0.27 ± 0.02, 0.74 ± 0.04, 1.43 ± 0.03, and 0.97 ± 0.02 mg/mL respectively (Additional file 1: Figure S2). Higher IC50 value refers to weaker capacity of compounds to scavenge DPPH free radical. The results obtained from antioxidant assay reveals that compound 1 shows better scavenging activity.
Hydrogen peroxide is another very reactive species among all the oxygen-containing compound. Therefore, the antioxidant potential of test compounds was also estimated by hydrogen peroxide scavenging assay. The capabilities of target compounds to scavenge the hydrogen peroxide radicals were monitored using UV-Vis spectrophotometer. The IC50 values of scavenging activity for the compounds 1, 2, 3, and 4 were found to be 0.41 ± 0.014, 0.79 ± 0.015, 1.30 ± 0.028, and 1.23 ± 0.021 mg/mL respectively (Additional file 1: Figure S3). The investigation of antioxidant assay demonstrates that compound 1 showed the greater rate of H2O2 scavenging activity than other compounds and the results are in accordance with the results obtained by DPPH free radical method.
As compared to our previous study (Arif et al. 2016), it has been found that the derivatives of tetrabromo-phthalimide could interact to calf thymus DNA via strong intercalative mode of binding. It also conluded that tetrabromo-phthalimide derivatives bind more strongly to Ct-DNA as compared to phthalimide and tetrachloro-phthalimide derivatives followed by absorption measurements. Antioxidant study demonstrates that compounds (1–4) showed better antioxidant activity against DPPH free radical and hydrogen peroxide.
In this work, we have synthesized a series of N-substituted tetrabromophthalimide derivatives (1–4) and characterized structurally by elemental analysis, FT-IR, and 1H NMR spectral analysis. Synthesized compounds were evaluated for their antibacterial potential against E. coli and S. mutans, and it has been found that better results were found in the case of compound 1. On the basis of antibacterial potency, binding interactions of compound 1 with DNA were investigated by using absorbance, hydrodynamics, and cyclic voltammetry measurement methods against Ct-DNA. Compound 1 showed strong intercalation with Ct-DNA, and it was concluded that compound 1 could interact with DNA via classical intercalative mode. Antioxidant assays demonstrated that the test compounds showed good scavenging activity against DPPH free radical and H2O2.
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The financial support from University Grants Commission, Govt. of India for Major Research Project, is gratefully acknowledged. Mr. Rizwan Arif grateful to UGC, New Delhi for the financial support through Non-NET fellowship.
University Grants Commission (F.No. 41-238/2012).
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Department of Chemistry, Jamia Millia Islamia, New Delhi, 110025, India
Rizwan Arif, Pattan Sirajuddin Nayab, Akrema & Rahisuddin
Department of Biosciences, Jamia Millia Islamia, New Delhi, 110025, India
Mohammad Abid
Department of Physics, DDU Gorakhpur University, Gorakhpur, 273009, India
Umesh Yadava
Rizwan Arif
Pattan Sirajuddin Nayab
Akrema
Rahisuddin
RA and PSN synthesized the phthalimide derivatives and investigate DNA binding interactions. RU and AA interpreted the data. MA and UY conducted the docking experiments. All authors read and approved the final manuscript.
Correspondence to Rahisuddin.
Additional file
Figure S1. 1H and 13C NMR spectra of the compound 1. Figures S2. and S3. Graphs for the antioxidant assay of the compounds 1,2,3 and 4 against DPPH free radical and hydrogen peroxide are given. (DOC 4623 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Arif, R., Nayab, P.S., Akrema et al. Investigation of DNA binding and molecular docking propensity of phthalimide derivatives: in vitro antibacterial and antioxidant assay. J Anal Sci Technol 10, 19 (2019). https://doi.org/10.1186/s40543-019-0177-1
Phthalimide
DPPH
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DOI:10.1103/PhysRevB.100.165136
Path-integral Monte Carlo study of electronic states in quantum dots in an external magnetic field
@article{THoke2019PathintegralMC,
title={Path-integral Monte Carlo study of electronic states in quantum dots in an external magnetic field},
author={Csaba THoke and Tam'as Haidekker Galambos},
C. THoke, T. Galambos
We explore correlated electron states in harmonically confined few-electron quantum dots in an external magnetic field by the path-integal Monte Carlo method for a wide range of the field and the Coulomb interaction strength. Using the phase structure of a preceding unrestricted Hartree-Fock calculation for phase fixing, we find a rich variety of correlated states, often completely different from the prediction of mean-field theory. These are finite temperature results, but sometimes the…
table I
table II
table III
View All 36 Figures & Tables
Abnormal quantum moment of inertia and structural properties of electrons in 2D and 3D quantum dots: an ab initio path-integral Monte Carlo study
T. Dornheim, Yangqian Yan
We present extensive new direct path-integral Monte Carlo results for electrons in quantum dots in two and three dimensions. This allows us to investigate the nonclassical rotational inertia (NCRI)…
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A. Güçlü, G. Jeon, C. Umrigar, J. Jain
Composite fermion wave functions projected onto the lowest Landau level, provide an accurate description of two-dimensional quantum dots in the limit of strong magnetic fields. We show that the range…
Composite fermion theory of correlated electrons in semiconductor quantum dots in high magnetic fields
G. Jeon, Chia-Chen Chang, J. Jain
Interacting electrons in a semiconductor quantum dot at strong magnetic fields exhibit a rich set of states, including correlated quantum fluids and crystallites of various symmetries. We develop in…
FOUR-ELECTRON QUANTUM DOT IN A MAGNETIC FIELD
M. Tavernier, E. Anisimovas, F. Peeters, B. Szafran, J. Adamowski, S. Bednarek
We present a theoretical treatment of four two-dimensional electrons in a harmonic confinement potential in the presence of an external magnetic field using the exact diagonalization approach. The…
Wigner molecules in quantum dots
B. Reusch, W. Hausler, Hermann Grabert University of Freiburg, U. Hamburg, H Germany
We perform unrestricted Hartree-Fock (HF) calculations for electrons in a parabolic quantum dot at zero magnetic field. The crossover from Fermi liquid to Wigner molecule behavior is studied for up…
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1. S. Yu. Orevkov, "Arrangements of a plane $M$-sextic with respect to a line", Algebra i Analiz, 34:1 (2022), 123–143
2. S. Yu. Orevkov, "Counting lattice triangulations: Fredholm equations in combinatorics", Mat. Sb.
3. N. G. Kruzhilin, S. Yu. Orevkov, "Plane algebraic curves in fancy balls", Izv. Math., 85:3 (2021), 407–420
4. S. Yu. Orevkov, "Multivariate Signatures of Iterated Torus Links", Funct. Anal. Appl., 55:1 (2021), 59–74
5. Grigory Mikhalkin, Stepan Orevkov, "Rigid isotopy of maximally writhed links", Algebr. Geom., 8:3 (2021), 268–285 (cited: 2) (cited: 2)
6. S. Yu. Orevkov, "Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves", Geom. Funct. Anal., 31 (2021), 930–947
7. Dominique Bakry, Stepan Orevkov, Marguerite Zani, "Orthogonal polynomials and diffusion operators", Ann. Fac. Sci. Toulouse; Math., Serie 6, 30:5 (2021), 985–1073
8. S. Fiedler-Le Touzé, S. Orevkov, E. Shustin, "Corrigendum to "A flexible affine $M$-sextic which is algebraically unrealizable"", J. Algebraic Geom., 29:1 (2020), 109–121 (cited: 1) (cited: 1)
9. S. Yu. Orevkov, "Quasipositive links and connected sums", Funct. Anal. Appl., 54:1 (2020), 64–67 (cited: 2) (cited: 2)
10. V. I. Zvonilov, S. Yu. Orevkov, "A cell structure of the space of branched coverings of the two-dimensional sphere", St. Petersburg Math. J., 32:5 (2021), 885–904
11. S. Yu. Orevkov, "On Alternating Quasipositive Links", Dokl. Math., 102:2 (2020), 403–405
12. S. Yu. Orevkov, "Separating semigroup of hyperelliptic curves and of genus 3 curves", St. Petersburg Math. J., 31:1 (2020), 81–84
13. Grigory Mikhalkin, Stepan Orevkov, "Maximally writhed real algebraic links", Invent. Math., 216:1 (2019), 125–152 (cited: 3) (cited: 6)
14. S. Yu. Orevkov, "Products of conjugacy classes in $\mathrm{SL}_2(\mathbb{R})$", Trans. Moscow Math. Soc., 80 (2019), 73–81
15. Grigory Mikhalkin, Stepan Orevkov, "On Osculating Framing of Real Algebraic Links", Arnold Math. J., 5 (2019), 393–399
16. S. Yu. Orevkov, "Irreducibility of lemniscates", Russian Math. Surveys, 73:3 (2018), 543–545 (cited: 1) (cited: 1)
17. G. B. Mikhalkin, S. Yu. Orevkov, "Topology of maximally writhed real algebraic knots", Dokl. Math., 97:1 (2018), 28–31 (cited: 2) (cited: 1)
18. S. Yu. Orevkov, "On the Hyberbolicity Locus of a Real Curve", Funct. Anal. Appl., 52:2 (2018), 151–153
19. Patrick M. Gilmer, Stepan Yu. Orevkov, "Signatures of real algebraic curves via plumbing diagrams", J. Knot. Theor. Ramif., 27:3 (2018), 1840003 , 33 pp. (cited: 1) (cited: 1)
20. V. I. Zvonilov, S. Yu. Orevkov, "Compactification of the Space of Branched Coverings of the Two-Dimensional Sphere", Proc. Steklov Inst. Math., 298 (2017), 118–128 (cited: 1)
21. Yu. Orevkov, "Automorphism group of the commutator subgroup of the braid group", Annales de la Faculté des Sciences de Toulouse. Mathématiques, Sér. 6, 26:5 (2017), 1137–1161
22. Stepan Yu. Orevkov, "Remark on Tono's theorem about cuspidal curves", Math. Nachr., 290:17 (2017), 2992–2994 (cited: 1)
23. S. Yu. Orevkov, E. I. Shustin, "Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of singularity $X_{21}$", St. Petersburg Math. J., 28:2 (2017), 225–257 (cited: 3) (cited: 2)
24. S. Yu. Orevkov, "Markov traces on the Funar algebra", Comm. Math. Phys., 344:2 (2016), 371–396
25. Grigory Mikhalkin, Stepan Orevkov, "Real algebraic knots and links of small degree", J. Knot. Theor. Ramif., 25:12 (2016), 1642010 , 34 pp. (cited: 4) (cited: 5)
26. S. Yu. Orevkov, "Algorithmic recognition of quasipositive braids of algebraic length two", J. Algebra, 423 (2015), 1080–1108 (cited: 2) (cited: 3) (cited: 3)
27. S. Yu. Orevkov, "On the Hurwitz action on quasipositive factorizations of 3-braids", Dokl. Math., 91:2 (2015), 173–177 (cited: 1) (cited: 1) (cited: 1) (cited: 1)
28. S. Yu. Orevkov, "Criterion of Hurwitz equivalence for quasipositive factorizations of 3-braids", Dokl. Math., 92:1 (2015), 443–447
29. Stepan Orevkov, "Parametric equations of plane sextic curves with a maximal set of double points", J. Algebra Appl., 14:9 (2015), 1540013 , 14 pp. (cited: 1) (cited: 2) (cited: 2)
30. Stepan Yu. Orevkov, "Algorithmic recognition of quasipositive 4-braids of algebraic length three", Groups Complex. Cryptol., 7:2 (2015), 157–173
31. S. Yu. Orevkov, "On modular computation of Gröbner bases with integer coefficients", J. Math. Sci. (N. Y.), 200:6 (2014), 722–724
32. S. Yu. Orevkov, "Cubic Hecke algebras and invariants of transversal links", Dokl. Math., 89:1 (2014), 115–118 (cited: 1) (cited: 1) (cited: 1) (cited: 1)
33. S. Yu. Orevkov, "On commutator subgroups of Artin groups", Dokl. Math., 85:1 (2012), 117–119 (cited: 1) (cited: 1)
34. S. Yu. Orevkov, "Complex orientation formulas for $M$-curves of degree $4d+1$ with 4 nests", Ann. Fac. Sci. Toulouse Math. (6), 19:1 (2010), 13–26
35. S. Yu. Orevkov, Yu. P. Orevkov, "The Agnihotri–Woodward–Belkale Polytope and Klyachko Cones", Math. Notes, 87:1 (2010), 96–101
36. A. I. Aptekarev, V. K. Beloshapka, V. I. Buslaev, V. V. Goryainov, V. N. Dubinin, V. A. Zorich, N. G. Kruzhilin, S. Yu. Nemirovski, S. Yu. Orevkov, P. V. Paramonov, S. I. Pinchuk, A. S. Sadullaev, A. G. Sergeev, S. P. Suetin, A. B. Sukhov, K. Yu. Fedorovskiy, A. K. Tsikh, "Evgenii Mikhailovich Chirka (on his 75th birthday)", Russian Math. Surveys, 73:6 (2018), 1137–1144
37. A. V. Parusnikova, O. S. Ogneva, V. A. Bykovskii, A. A. Vedenov, D. O. Golovin, Yu. O. Golovin, V. M. Zaitsev, B. K. Zuev, E. Yu. Zueva, K. M. Efimov, A. V. Ivanov, V. S. Isaev, A. S. Karyagina, A. L. Kolosov, S. P. Konovalov, P. A. Kornilov, V. V. Lavrov, V. A. Lavrova, A. A. Letunov, I. E. Letunova, V. D. Maiorov, A. B. Merkov, Yu. I. Ozhigov, S. Yu. Orevkov, O. A. Presnyakova, D. V. Reut, A. V. Skaraev, V. N. Sorokin, P. N. Sorokin, P. G. Sushilin, E. V. Fokina, N. N. Chentsova, T. I. Shvarts, A. S. Zibrov, S. V. Liskina, T. F. Nikiforova, E. G. Orevkova, M. L. Shvarts, "V. I. Parusnikov. Obituary", Chebyshevskii Sb., 17:1 (2016), 284–285 ; ; ; ;
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References of "Devillet, Jimmy 50009525"
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Reducibility of n-ary semigroups: from quasitriviality towards idempotency
Couceiro, Miguel; Devillet, Jimmy ; Marichal, Jean-Luc et al
in Beiträge zur Algebra und Geometrie (in press)
Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the ... [more ▼]
Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathcal{F}^n_{n-1}\subseteq\mathcal{F}^n_n$ and we give conditions on the set $X$ for the last inclusions to be strict. The class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet \cite{CouDev}, who showed that its elements are reducible to binary associative operations. However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper, we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$. [less ▲]
Detailed reference viewed: 146 (21 UL)
Invariance in a class of operations related to weighted quasi-geometric means
Devillet, Jimmy ; Matkowski, Janusz
in Fuzzy Sets and Systems (in press)
Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left ... [more ▼]
Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance\ question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed. [less ▲]
Detailed reference viewed: 77 (6 UL)
Decomposition schemes for symmetric n-ary bands
Devillet, Jimmy ; Mathonet, Pierre
Scientific Conference (2020, August 27)
We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an ... [more ▼]
We extend the classical (strong) semilattice decomposition scheme of certain classes of semigroups to the class of idempotent symmetric n-ary semigroups (i.e. symmetric n-ary bands) where n \geq 2 is an integer. More precisely, we show that these semigroups are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1. We then use this main result to obtain necessary and sufficient conditions for a symmetric n-ary band to be reducible to a semigroup. [less ▲]
On idempotent n-ary semigroups
Devillet, Jimmy
This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various ... [more ▼]
This thesis, which consists of two parts, focuses on characterizations and descriptions of classes of idempotent n-ary semigroups where n >= 2 is an integer. Part I is devoted to the study of various classes of idempotent semigroups and their link with certain concepts stemming from social choice theory. In Part II, we provide constructive descriptions of various classes of idempotent n-ary semigroups. More precisely, after recalling and studying the concepts of single-peakedness and rectangular semigroups in Chapters 1 and 2, respectively, in Chapter 3 we provide characterizations of the classes of idempotent semigroups and totally ordered idempotent semigroups, in which the latter two concepts play a central role. Then in Chapter 4 we particularize the latter characterizations to the classes of quasitrivial semigroups and totally ordered quasitrivial semigroups. We then generalize these results to the class of quasitrivial n-ary semigroups in Chapter 5. Chapter 6 is devoted to characterizations of several classes of idempotent n-ary semigroups satisfying quasitriviality on certain subsets of the domain. Finally, Chapter 7 focuses on characterizations of the class of symmetric idempotent n-ary semigroups. Throughout this thesis, we also provide several enumeration results which led to new integer sequences that are now recorded in The On-Line Encyclopedia of Integer Sequences (OEIS). For instance, one of these enumeration results led to a new definition of the Catalan numbers. [less ▲]
Associative, idempotent, symmetric, and order-preserving operations on chains
Devillet, Jimmy ; Teheux, Bruno
in Order: A Journal on the Theory of Ordered Sets and its Applications (2020), 37(1), 45-58
We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In ... [more ▼]
We characterize the associative, idempotent, symmetric, and order-preserving operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n-element chain is the nth Catalan number. [less ▲]
Classifications of quasitrivial semigroups
Devillet, Jimmy ; Marichal, Jean-Luc ; Teheux, Bruno
in Semigroup Forum (2020), 100(3), 743-764
We investigate classifications of quasitrivial semigroups defined by certain equivalence relations. The subclass of quasitrivial semigroups that preserve a given total ordering is also investigated. In ... [more ▼]
We investigate classifications of quasitrivial semigroups defined by certain equivalence relations. The subclass of quasitrivial semigroups that preserve a given total ordering is also investigated. In the special case of finite semigroups, we address and solve several related enumeration problems. [less ▲]
On the structure of symmetric $n$-ary bands
E-print/Working paper (2020)
We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\ldots,x)=x$ for all $x ... [more ▼]
We study the class of symmetric $n$-ary bands. These are $n$-ary semigroups $(X,F)$ such that $F$ is invariant under the action of permutations and idempotent, i.e., satisfies $F(x,\ldots,x)=x$ for all $x\in X$. We first provide a structure theorem for these symmetric $n$-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong $n$-ary semilattice of $n$-ary semigroups and we show that the symmetric $n$-ary bands are exactly the strong $n$-ary semilattices of $n$-ary extensions of Abelian groups whose exponents divide $n-1$. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric $n$-ary band to be reducible to a semigroup. [less ▲]
Every quasitrivial n-ary semigroup is reducible to a semigroup
Couceiro, Miguel; Devillet, Jimmy
in Algebra Universalis (2019), 80(4),
We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in ... [more ▼]
We show that every quasitrivial n-ary semigroup is reducible to a binary semigroup, and we provide necessary and sufficient conditions for such a reduction to be unique. These results are then refined in the case of symmetric n-ary semigroups. We also explicitly determine the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲]
On idempotent n-ary uninorms
Devillet, Jimmy ; Kiss, Gergely; Marichal, Jean-Luc
in Torra, Vicenç; Narukawa, Yasuo; Pasi, Gabriella (Eds.) et al Modeling Decisions for Artifical Intelligence (2019, July 24)
In this paper we describe the class of idempotent n-ary uninorms on a given chain.When the chain is finite, we axiomatize the latter class by means of the following conditions: associativity ... [more ▼]
In this paper we describe the class of idempotent n-ary uninorms on a given chain.When the chain is finite, we axiomatize the latter class by means of the following conditions: associativity, quasitriviality, symmetry, and nondecreasing monotonicity. Also, we show that associativity can be replaced with bisymmetry in this new axiomatization. [less ▲]
On the single-peakedness property
Scientific Conference (2019, June 28)
Characterizations and enumerations of classes of quasitrivial n-ary semigroups
Devillet, Jimmy ; Couceiro, Miguel
Detailed reference viewed: 103 (8 UL)
Bisymmetric and quasitrivial operations: characterizations and enumerations
in Aequationes Mathematicae (2019), 93(3), 501-526
We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and ... [more ▼]
We investigate the class of bisymmetric and quasitrivial binary operations on a given set and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set is finite. [less ▲]
Quasitrivial semigroups: characterizations and enumerations
Couceiro, Miguel; Devillet, Jimmy ; Marichal, Jean-Luc
in Semigroup Forum (2019), 98(3), 472498
We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order ... [more ▼]
We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences. [less ▲]
Detailed reference viewed: 353 (103 UL)
Single-peakedness in aggregation function theory
Devillet, Jimmy ; Couceiro, Miguel; Marichal, Jean-Luc
Presentation (2019, May 14)
Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role ... [more ▼]
Due to their great importance in data fusion, aggregation functions have been extensively investigated for a few decades. Among these functions, fuzzy connectives (such as uninorms) play an important role in fuzzy logic. We establish a remarkable connection between a family of associative aggregation functions, which includes the class of idempotent uninorms, and the concepts of single-peakedness and single-plateaudness, introduced in social choice theory by D. Black. Finally, we enumerate those orders when the underlying set is finite. [less ▲]
Characterizations of biselective operations
Devillet, Jimmy ; Kiss, Gergely
in Acta Mathematica Hungarica (2019), 157(2), 387-407
Let X be a nonempty set and let i,j in {1,2,3,4}. We say that a binary operation F:X^2 -> X is (i,j)-selective if F(F(x_1,x_2),F(x_3,x_4)) = F(x_i,x_j), for all x_1,x_2,x_3,x_4 in X. In this paper we ... [more ▼]
Let X be a nonempty set and let i,j in {1,2,3,4}. We say that a binary operation F:X^2 -> X is (i,j)-selective if F(F(x_1,x_2),F(x_3,x_4)) = F(x_i,x_j), for all x_1,x_2,x_3,x_4 in X. In this paper we provide characterizations of the class of (i,j)-selective operations. We also investigate some subclasses by adding algebraic properties such as associativity or bisymmetry. [less ▲]
On quasitrivial semigroups
Presentation (2019, March 27)
Characterizations and classifications of quasitrivial semigroups
Scientific Conference (2019, March 03) | CommonCrawl |
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TD #
TD(16)02001 Emergency Ad-Hoc Networks by Using Drone Mounted Base Stations for a Disaster Scenario Margot Deruyck, Jorg Wyckmans, Luc Martens, Wout Joseph In case of a large scale disaster, the wireless access network can become quickly saturated. This is of course undesirable because for this kind of situations we actually need a reliable wireless connectivity. In this study, the potential of mounting LTE femtocell base stations on drones to offer an alternative for the saturated existing wireless infrastructure is investigated. Our preliminary results show that this is a very promising approach although a high amount of drones are needed to cover all users in the city center of Ghent, Belgium during a 1h intervention. The number of drones can be significantly reduced (up to 64%) by using a more advanced type of drone, by decreasing the user coverage requirement (11% less drones when requiring 80% instead of 90%) or by increasing the fly height of the drones (about 10% less drones needed when increasing the fly height by 10m). This study shows that it is interesting to further investigate the use of drones to provide an emergency wireless access network. DWG3
TD(16)02002 Characteristics of the Polarised Off-Body Channel in Indoor Environments Kenan Turbic, Slawomir J. Ambroziak, Luis M. Correia This paper addresses the depolarisation effect in off-body Body Area Networks channels, based on measurements performed at 2.45 GHz in an indoor environment. Seven different scenarios, involving both static and dynamic users, were considered, taking a statistical perspective. The analysis of the cross-polarisation discrimination is performed, as well as the analysis of path loss in co- and cross-polarised channels. Results show a strong dependence of the cross-polarisation discrimination and of channel characteristics on the polarisation and propagation condition, i.e., line-of-sight (LoS), non-LoS or quasi-LoS. The variation in distance, from 1 m to 6 m, shows very little impact. In dynamic scenarios, the shadow fading is observed to exhibit a Lognormal distribution, whereas multipath fading is seen to follow the Nakagami one, with essentially different parameter values in the co- and cross-polarised channels, showing a trend towards Rice in the former and Rayleigh in the latter cases. Based on results, a model is proposed for a dynamic off-body channel. DWG1
TD(16)02003 LTE Delay Assessment for Real-Time Management of Future Smart Grids Ljupco Jorguseski, Haibin Zhang, Sylvie Dijkstra-Soudarissanane, Michal Golinski This study investigates the feasibility of using Long Term Evolution (LTE), for the real-time state estimation of the smart grids. This enables monitoring and control of future smart grids. The smart grid state estimation requires measurement reports from different nodes in the smart grid and therefore the uplink LTE radio delay performance is selected as key performance indicator. The LTE delay evaluation approach is via 'snap-shot' system level simulations of an LTE system where the physical resource allocation, modulation and coding scheme selection and retransmissions are modelled. The impact of the schedulers on the LTE delay performance is analyzed at different granularities of LTE resource allocation, for both urban and suburban environments. DWG3
TD(16)02004 Long term measurements over fixed links in the mm wave band A. Cheema, S. Salous, X. Raimundo, Y. Cao The millimeter wave band has been identified for future wireless links with various applications ranging from device to device for ranges over a few cm to backhaul applications with links varying in range from a few meters to km's. To study the impact of precipitation on mm wave fixed links, three links are being set up in Durham at two frequencies with single and dual polarisation configurations. The TD will give an overview of the experimentals set up and the performance of the system. DWG1
TD(16)02005 Wide Band Propagation in Train-to-Train Scenarios – Measurement Campaign and First Results Paul Unterhuber, Stephan Sand, Mohammad Soliman, Benjamin Siebler, Andreas Lehner, Thomas Strang Within the next decades the railway system will change to fully automotive High Speed Trains (HSTs). Along this way new applications and a change in the train control and management is necessary to increase the efficiency and the safety but reduce the costs at the same time. Two of these future applications are dynamic coupling or virtual coupling. For such maneuvers, the train control need to be changed from today's centralized railway management systems. Such changes demand new reliable control communication links between Train-to-Ground (T2G) as well as Train-to-Train (T2T). The T2G will be covered by Long Term Evolution-Railway (LTE-R) which replaces today's Global System for Mobile Communications-Railway (GSM-R). This publication focuses on the wide band propagation in T2T scenarios and describes a measurement campaign with two HSTs. Next to the measurement campaign first results are presented. DWG1
TD(16)02006 Base Station Over-the-Air Testing in Reverberation Chamber Christian Patané Lötbäck, Klas Arvidsson, Mats Högberg, Mattias Gustafsson Base station conformance and performance testing is traditionally carried out in a conducted manner, i.e. test instruments are connected via cables to physical ports on the base station. In this way, the quality of the output signal is assessed. Various metrics need to be measured and the levels comply with requirements in standard specifications. As the complexity of the base station transceiving circuitry increases and more and more antennas are added to the transceiving links, new measurement techniques are needed to capture the true performance. This is especially important for base stations for the new emerging 5G standard, where a very large number of antennas and new technologies such as massive MIMO and active antenna systems will be used for the signal transmission and reception. The conducted testing will not show the true radio frequency performance when several transceivers are used to combine the signal in the air interface. For these systems, it might not either be possible to incorporate physical ports on the base station to which test equipment can be connected. Thus, Over-the-Air testing will be needed, where the signals are transmitted and received over the antenna interface. It is important that such test methods are time and cost effective, in order not to significantly increase the test efforts compared to today. The reverberation chamber is a good candidate for these tests, given its low test time and test setup complexity. The reverberation chamber is already a frequently and well-proven tool in the wireless industry to assess performance of user equipment and the extension to base station testing is straightforward. This paper elaborates on the feasibility of the reverberation chamber for base station Over-the-Air testing. Several key parameters are measured and compared to results from conducted testing, showing that the metrics currently measured in conducted mode can be translated to Over-the-Air metrics with high accuracy. In addition, an analysis of major uncertainty contributions is provided. This analysis shows that there is insignificant impact on the measurement accuracy when measuring antennas with high gain. EWG-OTA
TD(16)02007 Practical Interference-Aware R-ML SIC Receiver for LTE SU-MIMO Spatial Multiplexing Elena LUKASHOVA, Florian KALTENBERGER, Raymond KNOPP In the last years, the receiver design has become crucial for the performance of MIMO systems. However, the high complexity of the optimum Maximum-Likelihood (ML) receivers raises the problem of complexity-performance trade-off. This paper presents a practical Reduced Complexity ML Interference-Aware Receiver with Successive Interference Canceling principle (R-ML IA SIC) for single user MIMO system. The low computational complexity is achieved by decoupling real and imaginary parts of the interference-aware soft log-likelihood metrics. The proposed receiver does not introduce loss of information compared to ML receivers thanks to employed SIC principle, meaning the complexity reduction comes without the performance penalty. Our receiver achieves up to 1 bit/dim throughput gain in moderate and high SNR regimes in flat-fading environments compared to R-ML IA receiver based on Parallel detection (R-ML IA PAI). The throughput achieved from link-level simulations of implemented receiver is close to theoretically predicted one by the means of probability of outage analysis. DWG2
TD(16)02008 Virtualization of Spatial Streams for Enhanced Spectrum Sharing Hamed Ahmadi, Irene Macalusoy, Ismael Gomezy, Luiz DaSilvay, Linda Doyle In this work we propose a virtualized network architecture for an infrastructure provider that shares the physical resources of a Massive MIMO cell among several virtual network operators (VNOs) using spatial multiplexing. In this architecture the infrastructure provider allocates spatial streams to the VNOs, which enables each VNO to select its own scheduling policy and user priority to differentiate its service from the other VNOs. To assign the spatial streams to the VNOs that value them the most, we propose an auction-based spatial stream allocation approach. We show that the proposed auction-based approach performs very close to the optimal (fixed) approach in the case of homogeneous static VNOs demand. In case of heterogeneous demands, the auction mechanism is able to dynamically allocate the resources according to the needs of different VNOs. DWG3
TD(16)02009 Tailor-Made Tissue Phantoms Based on Acetonitrile Solutions for Microwave Applications up to 18 GHz Sergio Castelló-Palacios, Concepcion Garcia-Pardo, Alejandro Fornes-Leal, Narcís Cardona, and Ana Vallés-Lluch Tissue-equivalent phantoms play a key role in the development of new wireless communications devices, which are tested on such phantoms prior to their commercialization. However, existing phantoms cover a small number of tissues and do not reproduce them accurately within wide frequency bands. This paper aims at enlarging the number of mimicked tissues as well as their working frequency band. Thus, a variety of potential compounds are scanned according to their relative permittivity from 0.5 to 18 GHz. Next, a combination of these compounds is characterized so the relation between their dielectric properties and composition is provided. Finally, taking advantage of the previous analysis, tailor-made phantoms are developed for different human tissues up to 18 GHz and particularized for the main current Body Area Network (BAN) operating bands. The tailor-made phantoms presented here exhibit such a high accuracy that would allow researchers and manufacturers to test microwave devices at high frequencies for large bandwidths as well as the use of heterogeneous phantoms in the near future. The key of these phantoms lies in the incorporation of acetonitrile to aqueous solutions. Such compound has a suitable behavior to achieve the relative permittivity values of body tissues within the studied frequency band. EWG-IoT
TD(16)02010 Geometry-Based Polarised Static Off-Body Channel Model Kenan Turbic, Luis M. Correia, Marko Beko The paper presents a theoretical polarised off-body channel model, based on geometrical optics and modified Uniform Theory of Diffraction (UTD). It takes into account Line-of-Sight (LoS) propagation, reflections and diffractions, and allows arbitrary antenna orientations, polarisations and radiation patterns. The model is used to simulate an indoor environment scenario, where the transmission between co- and cross-polarised transmitter (Tx) and receiver (Rx) is analysed. The obtained results show that significantly stronger signal is received by the co-polarised antenna, as compared to the case when the antennas are orthogonal, where different propagation mechanisms dominate the two channels. DWG1
TD(16)02011 Using the iMinds w-iLab.t testbed for IoT experiments Margot Deruyck, Wout Joseph Presentation of iMinds w-iLab.t that can be used for IoT experiments EWG-IoT
TD(16)02012 A Model for Virtual Radio Resource Management in C-RAN Behnam Rouzbehani, Luís M. Correia, Luísa Caeiro This paper proposes a model of Virtual Radio Resource Management (VRRM) to provide Quality of Service (QoS) guarantees for different classes of services in a heterogeneous Cloud-based Radio Access Network (C-RAN). In this model, a single Virtual Network Operator (VNO) asks for wireless capacity from a set of physical network providers to serve its subscribers, and not having to deal with the physical infrastructure. The algorithm estimates the available capacity of the network based on the accessible radio resources from different Radio Access Technologies (RATs), and then allocates to each service of VNO, a portion of the available data rate based on the VNO's Service Level Agreements (SLAs). This process is done by solving a constrained nonlinear optimisation problem, which tries to balance and prioritise the allocated data rate of different services according to their specific QoS needs. The performance of the proposed algorithm is evaluated through implementing practical heterogeneous network scenarios. Results show that the algorithm is capable of satisfying the predefined SLAs, while maximising the utilisation of resources. DWG3
TD(16)02013 The Delay, Angular and Polarization Characteristics of Geometry-based Clusters in an Indoor Environment at 11 GHz Band Panawit Hanpinitsak, Kentaro Saito, Junichi Takada, Minseok Kim, Lawrence Materum This paper presents indoor cluster spread and polarization characteristics taken from an estimated multiple-input multiple-output (MIMO) channel of an indoor hall environment at 11 GHz band. The clusters were estimated by utilizing the authors' proposed geometrybased clustering method in which the single bounce (SB) and double bounce (DB) multipath components (MPCs) can be distinguished. After that, cluster angular spread, delay spread and cross-polarization ratio (XPR) and copolarization ratio (CPR) for each type of clusters were calculated and investigated. The results showed that the spreads of DB clusters are larger compared to SB clusters. Moreover, the polarization characteristics comparison between them also showed significant difference due to high power loss in horizontally-polarized wave of the DB clusters. Furthermore, based on the statistical fitting results, it was found that the cluster delay and angular spreads both can be approximately described the lognormal distribution, as well as the cluster polarization ratio. This indicated that the channel modeling accuracy could be improved by modeling the SB and DB clusters separately. DWG1
TD(16)02014 Dense Multipath Component Characteristics in 11GHz-band Indoor Environments Kentaro Saito, Jun-ichi Takada, Minseok Kim In the next-generation mobile communication system, utilization of higher frequency bands above 10 GHz has become a hot research topic because it has the potential to improve network capacity drastically by utilizing the available wideband spectrum. However, in the higher frequency band, in addition to the higher propagation loss, diffuse scatterings of propagation waves affect the multiple-input multiple-output (MIMO) transmission performance more significantly. In this paper, 11 GHz band MIMO channel measurements were conducted in line-of-sight (LoS) indoor environments to clarify the characteristics of diffuse scatterings in the higher frequency band. The frequency, angular, and the polarization domain dense multipath component (DMC) propagation parameters were jointly estimated by using RiMAX-based estimator to deal with the DMC contribution to MIMO channels quantitatively. In the measurements, significant DMCs were observed in all areas. The DMCs had directional and the polarization dependencies as well as frequency dependency. The DMC characteristics were different in each area. The angular spreads of the DMCs tended to increase and their decay factor tended to decrease as the room size decreased owing to the contribution of reverberation waves. For the validation, MIMO channel matrices were reconstructed from the estimated propagation parameters, and the results show that the MIMO transmission performance tended to be underestimated without the DMC contribution. The result is expected to be utilized for the novel MIMO channel model proposal in the higher frequency band that includes the DMC contribution. DWG1
TD(16)02015 Massive MIMO Propagation Models Henry Brice, Mark Beach, Evangelos Mellios The demand for capacity within existing mobile networks continues to increase as more subscribers and more devices communicate and as data-rich applications become more popular. The evolving 5G telecommunications standards aim to respond to such demand. A promising approach to increasing capacity and reliability within the context of 5G is Massive Multiple-Input Multiple-Output (MIMO) where many transmit antennas are used relative to the number of users, thus providing a greater opportunity to use the spatial characteristics of the channel for spatial diversity and multiplexing. The deployment of Massive MIMO systems requires the development of new propagation models, which is the focus of this research, in order to simulate this new communication medium for multiple environments so that the design of the networks can be optimised in terms of reliability and efficiency. This project addresses the particular challenges of Massive MIMO propagation models through the use of simulation techniques. Ray-Tracing is used to create detailed deterministic models for urban environments, and data obtained through practical measurement campaigns in Bristol and with other universities is analysed to investigate, calibrate and identify the relevant statistical features of the channel. DWG1
TD(16)02016 On the Use of Serious Game Engineering for 5G System Performance Evaluation Carlos Herranz, David Martín-Sacristán, Saúl Inca, Jose F. Monserrat, Narcís Cardona This paper summarizes the current proposal of the METIS-II project on the use of serious game engineering approach for the evaluation and visualization of 5G technologies. Based on UNITY 3D, a realistic scenario has already been implemented, including a portion of a city with every level of detail. This article discusses the representation of results and interaction with a conventional simulation tool, showing clear cohesion between these two entities. (This TD corresponds to a paper presented in EUCNC'16 by the same authors) DWG1,DWG3r,EWG-RA
TD(16)02017 Bandwidth Dependence of the Ranging Error Variance in Dense Multipath Stefan Hinteregger, Erik Leitinger, Paul Meissner, Josef Kulmer, and Klaus Witrisal It is well known that the time-of-flight ranging performance is heavy influenced by multipath propagation within a radio environment. This holds in particular in \textit{dense }multipath channels as encountered in indoor scenarios. The signal bandwidth has a tremendous influence on this effect, as it determines whether the time resolution is sufficient to resolve the useful line-of-sight (LOS) signal component from interfering multipath. This paper employs a geometry-based stochastic channel model to analyze and characterize the ranging error variance as a function of the bandwidth, covering the narrowband up to the UWB regimes. The Cram\'er-Rao lower bound (CRLB) is derived for this purpose. It quantifies the impact of bandwidth, SNR, and parameters of the multipath radio channel and can thus be used as an effective and accurate \textit{channel model} (e.g.) for the cross-layer optimization of positioning systems. Experimental data are analyzed to validate our theoretical results. DWG1; EWG-LT
TD(16)02018 MIMO Gain and Bandwidth Scaling for RFID Positioning in Dense Multipath Channels Stefan Hinteregger, Erik Leitinger, Paul Meissner, and Klaus Witrisal This paper analyzes the achievable ranging and positioning performance for two design constraints in a radio frequency identification (RFID) system: (i) the bandwidth of the transmit signal and (ii) the use of multiple antennas at the readers. The ranging performance is developed for correlated and uncorrelated constituent channels by utilizing a geometry-based stochastic channel model for the downlink and the uplink. The ranging error bound is utilized to compute the precision gain for a ranging scenario with multiple collocated transmit and receive antennas. The position error bound is then split into a monostatic and bistatic component to analyze the positioning performance in a multiple input, multiple output (MIMO) RFID system. Simulation results indicate that the ranging variance is approximately halved when utilizing uncorrelated constituent channels in a monostatic setup. It is shown that both the bandwidth and the number of antennas decrease the error variance roughly quadratically. DWG1;EWG-LT
TD(16)02019 Consideration of directivity of antennas for high frequency wireless body area networks during human movements Takahiro Aoyagi As increased requirements for much high speed and capacity in wireless communications, frequency bands become higher, such as millimeter wave or terahertz wave. In these high frequency bands, beamforming is employed to gain stable connec- tivity. On-body body area network is one of fascinate application of these high capacity frequency bands. However, directions of on- body antennas largely varies and shadowing frequently occurs due to human movements. In this paper, variation of antenna directions and shadowing of on-body propagation during human walk movement is investigated. As a result, range of antenna rotation and shadowing rate, which can be used future system design of high frequency body area networks, is clarified. DWG1
TD(16)02020 Distributed Consensus Estimator of Hierarchical Network Transfer Function in WPNC Networks Jan Sykora Wireless Physical Layer Network Coding (WPNC) network nodes determine their front-end, back-end, and node processing operation depending on the knowledge of the global Hierarchical Network Transfer Function (H-NTF). The H-NTF can be derived from the network connectivity state and it is essential in determining how to perform hierarchical decoding, what hierarchical network coding function to use, and how to encode the Network Coded Modulation at each node. The data payload in WPNC network uses a common signal space of mutually interacting signals that superpose on the receive nodes. There are no orthogonalized communication links that would allow traditional signaling based approaches to determine the network state. Also, the procedure where each node establishes a complete picture of the network state must be performed before the payload phase and with superposing non-orthogonal pilot signals that fully respect the WPNC paradigm. This paper proposes a distributed consensus-based estimator of the network state which operates on a common signal space shared by all nodes in the network. The algorithm allows each node to find the global network state based only on local neighbor superposed pilot observation. DWG2
TD(16)02021 / / / /
TD(16)02022 Joint channel and carrier frequency offset estimation for UFMC Eric Simon and Florian Kaltenberger Many efforts are currently undertaken for the definition of enhanced multicarrier waveforms (post 4G LTE). These waveforms relax the time domain localiza- tion to impose well localized spectrum shape. UFMC is considered as one of the most serious candidate for the 5th Generation of wireless communication systems, which aims at replacing OFDM and enhances system robustness and performance in relaxed synchronization condition e.g. time-frequency misalignment. An extensive literature has shown the advantages of UFMC with respect to OFDM for these new requirements of 5G, however joint channel and CFO estimation for UFMC has not been investigated yet. This is the topic of this paper. We proposed an algorithm based on expectation-maximization to jointly estimate the channel impulse response and the CFO that occurs due to frequency mismatch between the oscillators at the receiver and at the transmitter DWG2
TD(16)02023 The Capacity of Cloud-RAN: Outer bound with Quantisation and Constrained Fronthaul Load Qinhui Huang and Alister Burr In this paper, we consider a distributed 'massive MIMO' system with multiple intermediate relays serving multiple sources jointly via a constrained fronthaul. Each relay employs uniform quantisation and the modulo operation to meet the bandwidth constraint. We derive the loss of the mutual informationdue to these processes. By adaptively optimising the interval of the lattice quantiser based on the channel, we evaluate the inevitable loss of mutual information under different modulo operations (fronthaul load per channel use). Numerical results reveal the outer bound of this scheme and the inevitable gap compared to the ideal C-RAN with infinite bandwidth. Implementation issues associated with this scheme are also investigated. DWG2
TD(16)02024 Iterative interference cancellation for FBMC and reduced-CP OFDM Yahya Harbi and Alister Burr Iterative decoding has been widely used to achieve reliable high data rate transmission for broadband multi-carriers communication systems. However, in Orthogonal Frequency Division Multiplexing (OFDM) systems with insufficient cyclic prefix (CP), there are significant challenges for efficient receiver design under the effect of the time-variant Long-Term Evolution (LTE) multipath channel. In this work, an iterative interference cancellation (IIC) with Wiener filter (WF) channel estimation is proposed using a Low-Density Parity-Check (LDPC) decoder with different patterns of scattered and preamble pilots for Filter Bank Multi-Carrier/Offset QAM (FBMC/OQAM) and OFDM systems. Pilots and data aided are considered. The bit error probability is compared with that of the conventional FFT- OFDM system with insufficient cyclic prefix (CP) under different environments. The results obtained show that the probability of error in the FBMC/OQAM scheme is improved in many scenarios. DWG2
TD(16)02026 Validation of SCME wave fields Moray Rumney, Ya Jing, Sergio Cobos, Manuel Salmeron The verification of UE MIMO performance over the air (MIMO OTA) is being standardized by CTIA and 3GPP using the SCME UMa and UMi channel models. Two test methods that can emulate these spatial channels are the multi-probe anechoic (MPAC) boundary array method and the radiated two-stage (RTS) method. Validation of the correctness of the emulated spatial channels is one of the procedures that is being standardized. These procedures use reference dipoles to measure the temporal, spatial, power and cross polarization parameters of the test signal. Results show close alignment between different channel emulators and MIMO OTA test methods however, unexplained differences between test systems when measuring some commercial devices has been found. This paper investigates the sufficiency of the channel model validation procedures and the robustness of the definition of the SCME channel models when implemented in different channel emulators with a view to developing traceable validation procedures that can explain the observed differences between systems when measuring real devices. EWG-OTA
TD(16)02028 Building testability into mmWave 5G Moray Rumney For most of the history of cellular radio we have relied on conducted measurements of base stations and devices where the impact of the antenna is discounted. This has been sufficient during the period when antenna performance has been reliable enough to extrapolate end-user experience in real networks. The introduction of smaller less efficient integral antennas in devices motivated the move towards over the air (OTA) testing to ensure antennas were of sufficient quality. SISO OTA test methods have been around since 2003 and will soon be augmented with MIMO OTA test methods. For the base station, SISO OTA is soon to be introduced for antenna arrays supporting elevation beamforming (full-dimension MIMO). With the plan to develop a new air interface for 5G at mmWave frequencies, all most base station and all device testing will need to be carried out OTA since cabled connections are not viable at mmWave frequencies involving antennas with multiple elements. This move to OTA represents an unprecedented step function in the complexity of testing cellular systems and will require the development of a variety of new test methods that provide a range of spatial and non-spatial test capability at different complexity/cost levels. Part of the development of viable OTA test methods will be the identification of base station and device special conformance test functions that enable control over the DUT in order to simplify or speed up the process of device testing. Due to the timescales of 5G, viable cost-effective test solutions need to be found very quickly as there is no cabled fallback like in earlier generations. The work to identify test methods and possible conformance test functions has started in 3GPP and this paper describes some early ideas with a view to motivating further timely contributions from industry and academia. EWG-OTA
TD(16)02029 A Study of the Energy Detection Threshold in the IEEE 802.15.6 CSMA/CA Martina Barbi, Kamran Sayrafian, Mehdi Alasti A Body Area Network (BAN) is a radio interface standard for wireless connectivity of wearable and implantable sensors located inside or in close proximity to the human body. Medical applications requirements impose stringent constraints on the reliability, and quality of service performance in these networks. Interference from other co-located BANs or nearby devices that share the same spectrum could greatly impact the data link reliability in these networks. Specifically, the CSMA/CA MAC protocol as outlined in the IEEE802.15.6 BAN standard involves the use of an energy detection threshold to determine the status of the transmission channel i.e. idle versus busy. In this technical document, we would like to show that the use of such static thresholds could negatively impact the performance of the system composed of multiple co-located BANs. It could also lead to starvation or unfair treatment of a node that is experiencing excessive interference due to its physical location relative to all other nodes in the system. A simulation platform is presented to highlight this problem and investigate the performance impact. EWG-IoT
TD(16)02030 SER analysis of QPSK modulated Physical Layer Network Coding for system-level simulation Cheng Chen and Alister Burr System-level simulation has been widely used recently to evaluate the performance of wireless networks. In this paper we consider the simulation of a dense, multi-hop wireless network which applies Physical Layer Network Coding (PNC), as developed in the DIWINE project. The DIWINE system-level simulator calculates the packet error probability (PER) at the relay against the signal and noise power. The above PER could be deduced from the relay's symbol error rate (SER). Here we analyse the SER of QPSK modulated Physical Layer Network Coding for the case where two source signals are received at one relay, since this forms a building block of many typical networks. DWG2
TD(16)02031 Massive MIMO Mobility Measurements in LOS with Power Control Paul Harris, Wael Boukley Hasan, Henry Brice, Mark Beach, Evangelos Mellios, Andrew Nix, Simon Armour, Angela Doufexi Massive Multiple-Input, Multiple-Output (MIMO) has shown great potential as a capacity enhancing technology for 5G wireless, but performance under mobility has been largely unexplored. Furthermore, it is under these circumstances that power control becomes critical to obtain acceptable performance. In this paper, we provide an overview of the first real-time massive MIMO mobility measurements conducted only weeks ago at Lund University, and present results for a spatial domain power control algorithm applied in one of these scenarios. DWG2,EWG-RA
TD(16)02032 Ground-to-X polarimetric radio channel characterization in forest scenarios Pierre Laly, Rose Mazari, Guy Grunfelder, Davy P. Gaillot, Shiqi Cheng, Jean-Marie Floch, Martine Liénard, Pierre Degauque, Emmeric Tanghe, Wout Joseph Coarse localization of an injured person in a forest environment can be made owing to his mobile cell phone and through the cellular network. However if this person is unable to give a call and if its phone is not equipped with a software automatically sending GPS information to rescue members, determining its accurate position can only be made by deploying a dedicated localization system in the search zone. This can be done owing to direction finding equipment onboard either a vehicle or a drone flying over the zone, this equipment forcing the mobile phone to transmit a signal. The localization accuracy being strongly dependent on the propagation channel characteristics, measurements have been carried out with a MIMO channel sounder at a center frequency of 1.35 GHz and with an 80 MHz bandwidth. Each array element is a dual-polarized patch antenna allowing a multidimensional polarimetric estimation of the channel. The transmitting array is always placed near the ground surface while the receiving array is below or over the canopy when onboard a vehicle or at different altitudes when onboard a drone. Path loss, delay spread and coherence bandwidth are studied for different relative orientations of the antennas, including co- and cross-polarization configurations. Directions of arrival of the rays are deduced from the MIMO matrix owing to a high resolution algorithm. DWG1,DWG2
TD(16)02033 Non-linear PNC mapping for hierarchical wireless network Alister Burr We discuss the requirements of a general (not necessarily linear) physical-layer network coding (PNC) mapping function for application in a network containing multiple sources and multiple relays. We consider the definition of linearity in such a mapping, then the requirements for unambiguous decoding of the output of the network, and finally the effect of singular fading in the links to a given relay. Hence we show how a mapping scheme can be designed for a specific topology and a specific set of channels. We note also however that such schemes are not likely to be readily compatible with a layered network coded modulation scheme, in which error control coding is separated from the network coding function. DWG2
TD(16)02034 Characterization and Modeling of the MIMO Radio channel in the W-band Davy P. Gaillot, Maria-Teresa Martinez-Ingles, Juan Pascual-Garcia, Martine Lienard, José-Víctor Rodríguez, and Jose-Maria Molina-Garcia-Pardo In this work, polarimetric MIMO matrices were measured in an indoor office at 94 GHz with a 3 GHz bandwidth using a VNA-based virtual radio channel sounder approach. Initial investigation of the polarimetric large-scale parameters such as the root mean square (rms) delay spread and path loss computed from the transfer functions is discussed for 15 Tx – Rx positions. In addition, the RiMAX framework, validated thanks to a ray-tracer tool developed at UPCT, was used to estimate the specular multipath components (MPC) and dense multipath components (DMC) from which the delay and angular RMS values, weighted mean delay and angles, number of MPC were computed. In addition, the reverberation time and DMC contribution to the radio channel total power is presented as a function of distance. This preliminary work will be used to build more advanced polarimetric path loss models and/or parametric/stochastic channel models which are missing in the literature for the W-band. DWG1,DWG2
TD(16)02035 Frequency Dependency of Measured Highly Resolved Directional Propagation Channel Characteristics Jonas Medbo, Nima Seifi, Henrik Asplund, Fredrik Harrysson An indoor measurement campaign has been conducted in order to determine any frequency dependent characteristics of the directional radio propagation channel over the frequency range 6-60 GHz. For the analysis a novel method, which previously has been proven to provide exceptional measurement accuracy at 58.7 GHz, is used. Herein it is shown that the measured channel power distributions over direction and delay are surprisingly similar over the full frequency range in both LOS and NLOS conditions. One exception is that the window transmission attenuation and reflectivity is substantially different at the two frequencies 5.8 GHz and 14.8 GHz. This difference results in that one of two dominant pathways at 14.8 GHz goes out of the building and is reflected off an adjacent building back in again to the receiver location. This does not occur at 5.8 GHz as the windows block penetration at this frequency. DWG1
TD(16)02036 Experimental Ultra Wideband Path Loss Models for Implant Communications C. Garcia-Pardo, R. Chávez-Santiago, A. Fornes-Lea), S. Castelló-Palacios, A. Vallés-Lluch, C. Andreu, I. Balasingham and N. Cardona Abstract—Ultra wideband (UWB) signals possess characteristics that may enable high data rate communications with deeply implanted medical sensors and actuators. Nevertheless, this application could be hindered in part by international spectrum regulations, which restrict UWB communications to 3.1-10.6 GHz where propagation conditions through the human body are rather unfavorable. Therefore, for the proper feasibility assessment and design of implant communications using UWB signals, accurate models of the radio channel are of utmost importance. Hence, we present UWB path loss models for the two most commonly used implant communication scenarios, i.e., in-body to on-body (IB2OB) and in-body to off-body (IB2OFF). These models were extracted from in vivo measurements in the abdominal cavity within 3.1-8.5 GHz using a living porcine subject. A thorough comparison between this modeling approach and channel measurements using a homogeneous phantom, which mimics the electromagnetic behaviour of muscle tissue, is presented too. Measurements in a homogeneous propagation medium are simpler to perform, but they fail to capture several physiological effects observed in a living subject. Thus, we measured the deviation between the phantom-based and in-vivo-based path loss models. In general, phantom measurements yielded a more pessimistic estimation of the path loss. We provide the correction factors to adjust easy-toperform phantom-based measurements to more realistic path loss values, which can assist the biomedical engineer in the early stages of design and testing of wireless implantable devices. EWG-IoT
TD(16)02037 How to optimally tune sparse network coding over wireless links Pablo Garrido, Ramón Agüero Despite their high computational complexity, Random Linear Network Coding (RLNC) techniques offer a notable robustness against packet erasure wireless links. Some novel approaches have been recently proposed to reduce such computational burden, for both encoder and decoder elements. One of those are the so-called Tunable Sparse Network Coding (TSNC) techniques, which advo- cate limiting the number of packets that are combined to build a coded packet. They also propose to dynamically adapt the corresponding sparsity level, as the transmission evolves, although an optimum tuning has not been proposed so far. In this paper we propose a TSNC implementation that exploits a novel analytical model to estimate the probability of generating an innovative packet (linearly independent combination), given the current status at the decoder. Taking advantage of the model's accuracy, the proposed scheme shows a gain of ≈ 2 times, compared to previous TSNC implementations. Furthermore, we broaden the analysis of TSNC techniques by thoroughly assessing their performance over wireless networks over the ns- 3 platform. The results yield a remarkable complexity reduction (≈ 70%), without jeopardizing the network performance. DWG3
TD(16)02038 Electrical Balance Duplexer performance in a High Speed Rail Applications Leo Laughlin, Chunqing Zhang, Mark Beach, Kevin Morris, John Haine Electrical Balance Duplexers (EBDs) can achieve high transmit-to-receive (Tx-Rx) isolation, but can be affected by interaction between the antenna and the environment. Dynamic antenna reflections coefficients measured on board a high speed train have been embedded into EBD circuit simulations to determine the Tx-Rx isolation, and requirements for circuit adaptation, in a high speed rail scenario. Results show that electromagnetic interaction between the antenna and the environmental outside the train is limited, and thus that high speed circuit adaptation is not required in this environment. However the results may have been affected by the metallized tinted window on board the train, and therefore the investigation should be repeated on older rolling stock without metallized windows to determine what effect this may have had. DWG2
TD(16)02039 CHANNEL PROPAGATION EXPERIMENTAL MEASUREMENTS AND SIMULATIONS AT 52 GHz B. Montenegro-Villacieros, J. Bishop, S. Salous, X. Raimundo This paper presents initial results on the comparison of channel propagation experimental measurements and ray-tracing simulations at 52 GHz in an outdoor scenario. The results show that long range reflections from metallic structures contribute to the received multipath components in both the simulation and the measurements. However, the diffuse multipath components arising from rough surfaces captured by the high resolution channel sounder are not reproduced in the simulator. Further calibration of such tools is therefore necessary prior to their application as channel prediction tools DWG1
TD(16)02040 Envisioning Spectrum Management In Virtualised C-RAN Imad Al-Samman, Matteo Artuso, Henrik Christiansen, Angela Doufexi, Mark Beach Cloud Radio Access Network (C-RAN) has attracted a worldwide attention in both academia and industry. This network architecture re-forming has been considered as a potential solution to meet the increasing capacity demands for future mobile data traffic. In addition, Network irtualisation is a promising technique for efficient resource utilisation. This paper proposes a customizable resource virtualization algorithm for multi user data scheduling in a Long Term Evolution (LTE) C-RAN deployment. The algorithm is based on the hypervisor's specific allocation assignment of air interface resources between the virtual operators (VOs) dynamically, based on either joint scheduling or per cell schemes. The objective is to improve the resource allocation mechanism based on traffic conditions and a database of pre-defined services priorities. Two distinctive scenarios are considered and evaluated against standard Round Robin (RR) C-RAN scheduling technique. Simulation results show improvements in the overall throughput traffic and reduction in end-to-end delay for delay sensitive applications. In addition, an assessment of fairness guarantee is considered across all users. DWG2,EWG-OTA,EWG-RA
TD(16)02041 Impulsive noises and dependence – preliminary considerations Emilie Soret, Laurent Clavier, Gareth Peters Following works presented in IC1004, we propose in this TD some further considerations about solutions for modeling dependence in impulsive noises. We use the copula framework that allows to reprensent, for instance, the upper and lower tail dependencies that can not be captured by classical correlation (which, besides, is not adapted to alpha-stable distributions). To illustrate the copula approach we consider a receiver architecture. The noise is modeled as a bivariate dependent Cauchy noise. If the copula represents the dependence structure we can derive the likelihood ratio that exhibits two components: one from the marginals and one from the copulas. We can then illustrate the impact of the dependence structure on the decision regions and we show that ignoring this dependence at the receiver side can importantly degrade the system performance. DWG2
TD(16)02042 Peer-Assisted Individual Assessment in a Multi-Agent System Li Wenjie, Francesca Bassi, Laura Galluccio, and Michel Kieffer Consider a community of agents, all collaborating to perform a predefined task (sensing, detection, classification…), but with different levels of ability (LoA). Initially, each agent does not know how well it performs in comparison with its peers and it is thus willing to assess its ability. This general scenario is relevant, e.g., in Wireless Sensor Networks (WSNs), or in the context of crowd sensing, where devices with embedded sensing capabilities collaboratively collect data to characterize the surrounding environment: the global performance is very sensitive to the quality of the gathered measurements and agents providing outliers or bad-quality measurements should themselves avoid transmitting data. This paper presents a distributed algorithm allowing each agent to assess its ability at doing some task. This assessment involves pairwise interactions with peers and a local comparison test, able to determine which, among two agents performs better. The dynamics of the proportions of agents with similar beliefs in their LoA are described using continuous-time state equations. The existence of an equilibrium is shown. Closed-form expressions for the various proportions of agents with similar beliefs in their LoA is provided at equilibrium. Simulation results match well theoretical results in the context of agents equipped with sensors and aiming at determining the performance of their equipment. EWG-IoT
TD(16)02043 Ground reflection modelling in millimeter wave channels Shangbin Wu, Stephan Jaeckel, Fabian Undi A typical deployment scenario in the millimeter-wave is that antennas at base stations (BSs) are elevated while user terminals (UTs) are placed in urban environment, e.g. on street level height. In this case, the ground reflection can produce a strong propagation path that superimposes with the direct path and induces severe fading effects. In this TD, the modelling of ground reflection in milllimeter-wave will be presented. DWG1
TD(16)02044 OpenAirInterface Massive MIMO Testbed: A 5G Innovation Platform Florian Kaltenberger, Xiwen Jiang Massive MIMO is one of the key technologies enabling the next generation of wireless communications. With its high potential in increasing network capacity, offering high spectrum efficiency, saving transceivers' energy and many other advantages,Massive MIMO has attracted great attention from the research and industrial community. The OpenAirInterface Massive MIMO testbed is built on the open source 5G platformOpenAirInterface. It is one of the world's' first LTE full protocol stack compatible base stations equipped with large antenna array, which can directly provide services to commercial user equipments (UEs). It shows the feasibility of using Massive MIMO in LTE standard, indicating the possibility of smoothly evolving the wireless network from 4G to5G. It provides an innovation platform in solving 5G challenges, by giving the possibility of advanced algorithm testing, concept validation, channel measurements, etc. EWG-RA
TD(16)02045 A miniaturized pattern reconfigurable antenna for automotive applications Jerzy Kowalewski, Tobias Mahler, Jonathan Mayer, Thomas Zwick This paper presents a realization approach of a pattern reconfigurable antenna. Based on the results of the previous research using wave propagation simulation, the radiation patterns optimized for automotive urban scenarios are chosen. The patterns are determined by a special antenna synthesis method. The antenna in this work generates two switchable patterns obtained from this synthesis. The first one is in and against the driving direction and the second is directed orthogonal to the driving direction to the left and right hand sides of the vehicle. The pattern switching is realized by switching the phase between the parallel fed radiating elements. An easy method of phase switching with means of a tapered line balun and p-i-n diodes is proposed. The antenna covers the 2.45 GHz ISM band, and therefore can be easily used for measurements in an unlicensed band. As a proof of concept a prototype of the antenna utilizing p-i-n diodes as switching elements has been fabricated and measured. The maximal gain achieved is about 6.5 dBi. The measurement results correspond well with the simulation results in terms of S-parameter and radiation. DWG1
TD(16)02046 Double-directional Dual-polarimetric Ultra-wideband Cluster-based Characterization of 70-77 GHz Indoor Propagation Channels Cen Ling, Xuefeng Yin, Robert Müller, Stephan Häfner, Diego Dupleich, Christian Schneider, Jian Luo, Hua Yan and Reiner S. Thomä Recently, a measurement campaign for characterizing delay, spatial and polarimetric radio channels ranging from 70 GHz to 77 GHz was carried out in the small office and entrance hall scenarios respectively. Composite channel behaviors and statistics are analyzed and compared for various measurement configurations. Based on the multi-dimensional power spectra of delay, direction (i.e. azimuth and co-elevation) of departure and arrival, the multipaths are further grouped into clusters via K- means, threshold-based and Gaussian Mixture Model (GMM) approaches. Besides, the spatial positions of the clusters are identified by using the measurement-based ray tracer (MBRT) method, and both first- and last-hop scatterers along propagation paths between the transceivers are localized. Those results manifest the merits of the threshold-based clustering algorithm in case of clusters' compactness, separation and exclusiveness, and the significance of the localization techniques for the propagation clusters. Additionally, it is noted that the deployment of the networks, such as the geometry size of the environment, both positions and heights of the Txs and Rxs, and the polarization combinations exerts tremendous influence on the statistical chan- nel models and characteristics of the indoor millimeter- (mm- )wave propagation. DWG1
TD(16)02048 Hybrid self-interference cancellation for small form factor realization of in-band full duplex Chunqing Zhang, Leo Laughlin, Mark A. Beach, Kevin A. Morris, and John Haine "In-band full-duplex (IBFD) which operates on the same frequency at the same time has the potential to double the spectrum efficiency. However, this proposal needs high isolation of the transceiver to mitigate the self-interference (SI) which results from co-located co-channel transmitter and receiver. This paper presents a hybrid SI cancellation architecture which consists of two stages of analog cancellation and one stage of digital base band cancellation. A prototype based on this hybrid architecture and corresponding achievable cancellation performance will be discussed. Simulation results show the overall isolation suppresses the SI 3dB higher than the noise floor of the receiver DWG1
TD(16)02049 Presenting the VISTA (Virtual Road) experimental facility for automotive research of TU Ilmenau F. Wollenschläger, W. Kotterman, C. Bornkessel, G. Del Galdo, R. Thomä, M. Hein The Virtual Road experimental facility in Ilmenau has been built by the Technische Universität Ilmenau for advanced automotive research in the context of the Thyringian Innovation Centre on Mobility, established at Ilmenau. VISTA with its large (semi-)anechoic chamber with its integrated antenna measurement range for automobiles and its dynamometer for experiments with cars with running engines is a unique facility that is presented in this TD to the IRACON EWG IoT EWG-IoT
TD(16)02050 Network Function Virtualization of Software-Defined Internet of Things Chiara Buratti, Franco Callegati, Simone Cerboni, Walter Cerroni, Slavica Tomovic, Roberto Verdone One of the challenges of 5G (5-th Generation) wireless networks will be the integration of mobile radio access with the Internet of Things (IoT) paradigm. Billions of objects equipped with sensors and radio interfaces will be connected to network entities managing the control plane in a centralized way, as postulated by many stake- holders. To pave the way towards this novel approach, we present an architecture for virtualization of IoT networks, including an intent-based north-bound interface and a virtualized infrastructure manager, allowing virtualization of IoT resources. The architecture exploits the Software- Defined paradigm, including an IoT Controller able to program different networks with the aim of providing to users the intended service at the requested QoS. A first prototype of the architecture is presented and some preliminary results, related to round trip time are provided. EWG-IoT
TD(16)02051 The Over-the-Air facilities at Fraunhofer's FORTE W. Kotterman, M. Landmann, F. Raschke, M. Hein, R. Thomä, and G. Del Galdo The Fraunhofer Institute for Integrated Systems (IIS) has established FORTE (Facility for Over-the-Airesearch and Testing) in Ilmenau, in cooperation with TU Ilmenau. In this paper, its two OTA facilities are presented to COST IRACON EWG IoT in connection with the stand-alone LTE-network. The OTA facilities comprise the virtual electromagnetic environment for test objects up to the size of small cars in the frequency band of about 800 to 6000 GHz and the virtual satellite test range in Ka and Ku band for SatCom-on-the-Move and LTE-over-Sat. EWG-IoT
TD(16)02056 Reproducing Standard SCME Channel Models for Massive MIMO Base Station Radiated Testing Wei Fan, Fengchun Zhang, Tommi Jämsä, Mattias Gustafsson, Pekka Kyösti and Gert F. Pedersen Massive MIMO is a multi-user technology, where radio base stations (BSs) are equipped with a large-scale antenna array to simultaneously serve many terminals in the same time-frequency resource. As a comparison, only up to eight antennas for BSs are specified in 4G LTE standards. Performance evaluation of such large-scale antenna systems is challenging. In this TD, we propose to evaluate massive MIMO BSs with a sectorized MPAC setup. A sectorized MAPC setup with 16 OTA antennas distributed uniformly within $[-60^o, 60^o]$ is utilized to evaluate performance of a $8\times8$ and $16\times16$ uniform planar array at 3.5 GHz, respectively. Radio channel emulation in terms of power anglular spectrum, spatial correlation and beamforming pattern is investigated for the proposed MPAC setup and desired channel models. EWG-OTA
TD(16)02057 GNSS-SDR: An open source software-defined GNSS receiver Carles Fernandez-Prades This paper presents a new platform for the experimentation with GNSS signals. It includes a set of commercial of-the-shelf hardware and an open source software, constituting a state-of-the-art platform for research and development of next-generation GNSS receivers. The core of the platform is GNSS-SDR, an open source software-defined GNSS receiver which has been extended to support multi-band and multi-system operations. As a relevant case of use to validate the research facility, we present a triple band GNSS-SDR customization capable of receiving four GNSS signals in real- time: GPS L1 C/A, GPS L2CM and Galileo E5a. In addition, we provided detailed descriptions of the receiver architecture, identifying the synchronization challenges of the multi-system satellite channels and providing practical and reproducible solutions. The source code developed to produce this paper has been released under the General Public License, and it is freely available on the Internet. EWG-LT
TD(16)02058 Device-To-Device graph-oriented Resource Allocation in LTE Uplink using SC-FDMA Johannes Baumgarten and Thomas Kürner Future cellular networks will support device-to-device (D2D) communication and thereby enable a plethora of new applications. Through the reuse of cellular resources in the currently less used uplink spectrum, D2D can increase the efficiency of resource use and help to cope with the ever-increasing data traffic. In this paper, we present an interference-graph based approach for LTE D2D resource allocation with multi-user sharing of Resource Blocks(RB) while respecting the SC-FDMA constraint of assigning continuous RBs to each UE. The numerical simulations show great potential gains with an increasing amount of D2D communication pairs per cell. DWG3
IRACON ACTION © 2016 - COST is supported by the EU Framework Programme Horizon 2020 - Privacy | CommonCrawl |
Nearly Tight Bounds for Testing Function Isomorphism
Sourav Chakraborty Chennai Mathematical Institute Plot No. H1 SIPCOT IT Park Padur PO Siruseri - 603103<
Wednesday, 2 February 2011 (All day)
AG-69
We study the problem of testing structural equivalence (isomorphism) between a pair of Boolean functions $f,g:\\{0,1\\}^n \\to \\{0,1\\}$. Our main focus is on the most studied case, where one of the functions is given (explicitly), and the other function can be queried.
We prove that for every $k \\leq n$, the query complexity of testing isomorphism to $k$-juntas is $\\Omega(k)$ and $O(k \\log k)$. In particular, the (worst-case) query complexity of testing isomorphism to a given function $f:\\{0,1\\}^n \\to \\{0,1\\}$ is $\\widetilde\\Theta (n)$.
Thus our bounds are nearly tight. Our lower bound and upper bound results improves the known bound obtained by Fischer et al. (2004), Blais and O'Donnell (2010), and recently by Alon and Blais (2010).
Our proof can also be extended to give polynomial query-complexity lower bounds for the problems of testing whether a function has a circuit of size $
One implication of our techniques is a query-efficient procedure that given oracle access to any $k$-junta $g:\\{0,1\\} \\to \\{0,1\\}$ can draw uniformly-random samples $(x,a) \\in \\{0,1\\}^k \\times \\{0,1\\}$ labelled by the core of $g$, each sample being correct with high
probability. Generating such samples is one of the main ingredients of the testers from Diakonikolas et al. (2007) while the procedure therein makes roughly $k$ queries to $g$ for obtaining each sample, our procedure requires only ONE query to $g$.
We also study the query complexity of testing isomorphism to $k$-juntas with one-sided error. We prove that for any $1
We also consider the problem of testing isomorphism between two unknown functions that can be queried. We prove that the (worst-case) query complexity in this setting is $\\Omega(\\sqrt{2^n})$ and $O(\\sqrt{ 2^n n \\log n})$ (this is a joint work with Arie Matsliah and David Garcia Soriano).
Siddharth Bhandari wins ACM India 2022 Doctoral Dissertation Award
Siddharth Sandipkumar Bhandari is the recipient of the ACM India 2022 Doctoral Dissertation Award for his dissertation titled "Exact Sampling and L
Kumar Saurav's Infocom 2021 paper presentation
Kumar Saurav, a graduate student at the School of Technology and Computer Science has been invited by ACM-India's Academic Research and Careers for
Google India Research Awards
Prahladh Harsha and Akshayaram Srinivasan are recipients of the Google India Research Award for their proposals titled "Super-efficient verificatio
STCS Seminar
An Approximate Generalization of the Okamura-Seymour Theorem
31 Jan 2023, 9:30 to 10:30 | CommonCrawl |
The increasing likelihood of temperatures above 30 to 40 °C in the United Kingdom
Increasing heat and rainfall extremes now far outside the historical climate
Alexander Robinson, Jascha Lehmann, … Dim Coumou
Setting and smashing extreme temperature records over the coming century
Scott B. Power & François P. D. Delage
Anthropogenic influence in observed regional warming trends and the implied social time of emergence
Francisco Estrada, Dukpa Kim & Pierre Perron
Wet-Bulb Globe Temperature, Universal Thermal Climate Index, and Other Heat Metrics for US Counties, 2000–2020
Keith R. Spangler, Shixin Liang & Gregory A. Wellenius
Quantifying the role of variability in future intensification of heat extremes
Claudia Simolo & Susanna Corti
Constraining the increased frequency of global precipitation extremes under warming
Chad W. Thackeray, Alex Hall, … Di Chen
Anthropogenically-driven increases in the risks of summertime compound hot extremes
Jun Wang, Yang Chen, … Jiangjiang Xia
Enhanced risk of concurrent regional droughts with increased ENSO variability and warming
Jitendra Singh, Moetasim Ashfaq, … Deepti Singh
State-of-the-art global models underestimate impacts from climate extremes
Jacob Schewe, Simon N. Gosling, … Lila Warszawski
Nikolaos Christidis ORCID: orcid.org/0000-0002-5661-711X1,
Mark McCarthy ORCID: orcid.org/0000-0003-2692-14171 &
Peter A. Stott1
Nature Communications volume 11, Article number: 3093 (2020) Cite this article
1584 Altmetric
Projection and prediction
As European heatwaves become more severe, summers in the United Kingdom (UK) are also getting warmer. The UK record temperature of 38.7 °C set in Cambridge in July 2019 prompts the question of whether exceeding 40 °C is now within reach. Here, we show how human influence is increasing the likelihood of exceeding 30, 35 and 40 °C locally. We utilise observations to relate local to UK mean extremes and apply the resulting relationships to climate model data in a risk-based attribution methodology. We find that temperatures above 35 °C are becoming increasingly common in the southeast, while by 2100 many areas in the north are likely to exceed 30 °C at least once per decade. Summers which see days above 40 °C somewhere in the UK have a return time of 100-300 years at present, but, without mitigating greenhouse gas emissions, this can decrease to 3.5 years by 2100.
Intensification of hot extremes has continued unabated in recent decades1, posing a threat to human health2,3 and bringing forth a raft of further socio-economic impacts4,5. Europe is gearing up for more frequent and intense heatwaves6 and while the UK has not yet borne the brunt of extreme continental heat, its summer temperatures are decidedly on the rise7,8. Attribution research provides strong evidence that hot extremes are becoming more frequent and intense9 under the influence of human-caused climate change10,11. The UK summer temperature of 2018 was a joint record, estimated to have become 30 times more likely due to anthropogenic causes12. A year later, during a severe heatwave in western Europe13, the warmest daily temperature averaged over the UK reached a new peak (Fig. 1a) and the highest temperature in the country ever recorded was registered in Cambridge. These consecutive summer extremes are exposing the UK's vulnerability to such weather with ensuing impacts highlighted in the media, including a mortality spike in tandem with the 2019 event14,15, and a sharp heatwave-driven fall in overseas holiday demand that might have contributed to the collapse of the Thomas Cook travel group16. Therefore, the need to understand how the likelihood of extremely hot temperatures is changing under the anthropogenic effect on the climate is pressing and essential to decision-makers planning the UK's adaptation strategy.
Fig. 1: Warmest daytime temperatures (tx01) in the UK.
a Timeseries of the UK mean tx01 from HadUK-Grid observations (black line), and simulations with 16 CMIP5 models with all climatic forcings (red lines) and natural forings only (blue lines). The observed value in 2019 is marked with a cross. Simulations of future years follow the RCP 4.5 scenario. The model data were bias-corrected to have the same mean during a reference period as the observations. b A map of the tx01 trends during 1960–2019 computed with HadUK-Grid data. Circles mark areas (of ~60 × 60 km) where most grid boxes have trends not significantly different from zero (tested at the 10% level), as determined by a Mann–Kendall test.
To respond to this need, we compute observed and modelled values of the warmest daily maximum temperature in individual years (tx01) and estimate how the likelihood of exceeding extreme thresholds has been changing since 1900 and how it may further change in the remaining of this century under different emission scenarios17. We find that the likelihood of extremely warm days in the UK has been increasing and will continue to do so during the course of the century with the most extreme temperatures expected to be observed in the southeast England. The likelihood of exceeding 40 °C anywhere in the UK in a given year has also been rapidly increasing, and, without curbing of greenhouse gas emissions, such extremes could be taking place every few years in the climate of 2100.
Observed changes in UK tx01 extremes
Limitations arising from the spatial resolution of climate models and the coverage of observation stations often prevent attribution studies on local scales and have kept the focus on extremes over larger, sub-continental areas18. Although downscaling of model output has occasionally been employed to investigate local events19, the lack of reliable observations makes it difficult to evaluate the models. Here, we take advantage of the recent upgrade of the HadUK-Grid dataset20 for daily maximum temperature, which now provides observational data on a high-resolution grid of 1 × 1 km. The resolution of the dataset enables us to model the relationship between local and UK mean tx01. This simple downscaling technique allows us to estimate changes in the likelihood of UK extreme temperatures locally from model experiments with and without anthropogenic forcings that provide data on a relatively coarse resolution.
The HadUK-Grid data cover the entire UK and are available for the period 1960–present. Annual values of tx01 are calculated for all the grid boxes, and a map of the trends over the observational period is illustrated in Fig. 1b. Warming trends dominate and are most prominent in the southeast, where they may locally reach 1 °C decade−1. Testing the significance of the trends with the Mann–Kendall test, indicates they are significantly different than zero in most regions, but not in parts of Scotland where there is a weaker warming and also areas of cooling. It should be noted that although this is a useful qualitative assessment, the trend estimates are sensitive to the start and end dates. The UK's warmest day in a year is also estimated (Fig. 1a) after averaging daily maximum temperatures of each day over the observational area. Year 2019 has the highest UK mean tx01 value on record, though climate models indicate that due to internal variability there is a current risk of even higher temperatures.
Transfer functions for the estimation of local tx01
Even though climate models can provide reliable estimates of the UK mean tx01, as discussed later, their spatial resolution is still too coarse to yield local estimates on a 1-km grid. We therefore derive observationally based transfer functions to obtain local tx01 values from the UK mean that we can later apply to model data. A simple linear fit is applied to all the grid boxes of HadUK-Grid to model the relationship between the grid-box tx01 and its UK-mean counterpart. An example for a grid-box in London is shown in Fig. 2a. We account for the range of values of the response variable (local tx01) by estimating its confidence bounds for each percentile21 (orange lines in Fig. 2b) and so end up with a set of 100 possible transfer functions at each grid-box ('Methods'). Moreover, given the limited sample of 60 years, our analysis also investigates the uncertainty in the transfer functions, by applying a Monte Carlo bootstrap procedure that resamples the observational data. This procedure offers alternative transfer functions (grey lines in Fig. 2c), each with an associated set of 100 variants, as previously explained. Finally, it is important to establish whether grid-box temperatures on spatial scales of 1 km can adequately represent local temperatures. To this end, we compare station observations across the UK with the HadUK-Grid temperature of the grid-box where each station is located and confirm a good agreement in all cases (Fig. 2d; Supplementary Note 1 and Supplementary Fig. 1).
Fig. 2: Transfer functions for the estimation of the local warmest daytime temperature (tx01).
An example for a grid-box in London. a Local observations of tx01 plotted against the UK mean observed values (crosses). A linear fit to the data (red line) represents the transfer function for the grid-box. b Inclusion of the confidence bounds for the response variable (orange lines) leads to a set of a 100 transfer functions in total. c A bootstrapping procedure applied to the observed data (crosses) provides alternative transfer functions (grey lines), used to assess the effect of sampling uncertainty. For each of the grey lines, a set of 100 transfer functions can be obtained as shown in panel b. d Observed tx01 data from a station within the reference grid-box agree well with the HadUK-Grid data.
The CMIP5 ensemble
Estimates of the UK mean tx01 are next obtained from simulations with 16 climate models that participated in the World Climate Research Programme's Coupled Model Intercomparison Project phase 5 (CMIP5)22. The models provide simulations of the actual climate (all forcings) under the influence of both natural and anthropogenic forcings, as well as of a hypothetical natural world without the effect of human influence. Anthropogenic forcings include historical changes in well-mixed greenhouse gases, aerosols, ozone and land-use. Natural forcings include only volcanic aerosol emissions and changes in the solar irradiance. Data from the following atmosphere–ocean coupled CMIP5 models are used in the analysis:
ACCESS1-3, bcc-csm1-1, CCSM4, CESM1-CAM5, CNRM-CM5, CSIRO-Mk3-6-0, CanESM2, GFDL-CM3, GFDL-ESM2M, HadGEM3-ES, IPSL-CM5A-LR, IPSL-CM5A-MR, MIROC-ESM, MIROC-ESM-CHEM, MRI-CGCM3, NorESM1-M.
The all-forcings experiment was also extended to the end of the twenty-first century with projections that follow the representative concentration pathway (RCP) scenarios17 RCP 4.5 and 8.5 scenarios. Given the substantial volume of the simulated daily data, we employ only one simulation per model and per experiment (r1i1p1), thus placing equal weight on all models. We apply a simple bias-correction to all the model data to make sure that the mean tx01 value in the all-forcing simulations during the base period 1961–1990 agrees with the observed mean value. For consistency, we re-grid all the models on a common 60-km resolution grid, on which the observations are also available, and then mask the model data with the observations on the same grid and compute the UK-mean value. For each model, we subtract the mean observed from the mean modelled tx01 over the base period and remove the resulting bias from the tx01 estimates with and without anthropogenic forcings derived from the same model.
Evaluation of the models against observations in attribution analyses is essential, in order to determine whether they are fit-for-purpose. Here we compare the data of the UK mean tx01 during 1960–2019 simulated by the all-forcings experiment with observationally based data from HadUK-Grid. We apply a set of standard evaluation tests to assess how well the models represent the trends, variability and distribution of tx01. Results are shown in Fig. 3. First, we estimate the trends in tx01 over the observational period and its associated ± 2 standard deviation range computed with least-square fits (Fig. 3a). The observations indicate a small positive trend, but its precise value is uncertain because of the effect of variability. Although some models produce weaker trends than the observations, the relatively short length of the observational period prevents a more detailed assessment, and since all models have a range that overlaps with the one from the observations, they are all included in the analysis. We next assess the simulated variability over different timescales with power spectra from detrended tx01 timeseries (Fig. 3b). The observed spectrum is found to be within the range of the modelled spectra, albeit towards the higher end, though again sampling limitations need to be taken into consideration. The observed and modelled distribution of the UK mean tx01 in the period 1960–2019 is illustrated in Fig. 3c. The modelled distribution is constructed with data from all 16 models and is found to be indistinguishable from the observed distribution, when a Kolmogorov–Smirnov (KS) test is applied (P-value greater than 0.1). The shape of the observed distribution (histogram) indicates that temperatures in upper tail are not well sampled because of the limited length of the record. This is also reflected in the quantile–quantile plot of Fig. 3d, which nonetheless still indicates that the models can realistically represent the distribution of tx01. In conclusion, the simple evaluation tests described here do not raise concerns about the ability of the multi-model ensemble to represent the UK tx01, but, on the contrary, indicate it is a sufficiently good dataset for the attribution analysis.
Fig. 3: Model evaluation.
a The ±2 standard deviation range of the 1960–2019 trend (°C decade−1) in the UK mean warmest daytime temperature (tx01) estimated with HadUK-Grid observations (grey band) and CMIP5 model simulations with all forcings (vertical bars). b Power spectra from detrended timeseries of the UK mean tx01 computed with observations (black line) and model simulations (orange lines). c Normalised distributions of the UK mean tx01 in period 1960–2019 from observations (blue histogram) and aggregated data from the all-forcing simulations (pink line). The P-value of a Kolmogorov–Smirnov test that assesses whether the two distributions are significantly different is also shown. d Quantile–quantile (Q–Q) plots for each of the 16 models, comparing the simulated and the observed UK mean tx01 distributions.
Testing the transfer functions with the CMIP5 models
A main assumption in our methodology is that the transfer functions we derive from the observations are not sensitive to the non-stationarity of the climate to an extent that would weaken our results. Local and regional temperatures are influenced by both internal variability and external forcings, and the interplay between these two factors would be different in different periods23. Using the CMIP5 models, we test the effect of non-stationarities and the interplay of variability and external forcings. First, we establish that the choice of the training period for which the transfer functions are derived does not compromise the analysis. We derive a set of three transfer functions from the models, corresponding to three different scenarios:
Strong forcing influence: The transfer functions are derived from simulations with all forcings for future years 2020–2100, characterised by a strong anthropogenic influence.
Variability influence only: The transfer functions are derived from simulations with natural forcings only, i.e., in the absence of any anthropogenic influence.
Mixed response: The transfer functions are derived from simulations with all forcings for the period 1960–2019, i.e., the same as the observational period used in our main analysis, for which anthropogenic influence increases with time.
The transfer functions obtained for four different grid boxes (in the Southeast London area, Scotland, Central England and Northern Ireland) are shown in Fig. 4. It is evident that different training periods yield similar transfer functions. The largest discrepancies are generally within a degree (in most cases much smaller) and are eclipsed by uncertainties due to sampling (Fig. 2c) and internal variability (Fig. 2b), which have been included in our approach. For example, the uncertainty range in Fig. 2c spans ~10 °C, and is accounted for by using 100 variants of the transfer function. We repeated the sensitivity tests over different grid boxes, training periods and with individual models, and found no indication of a large uncertainty due to the non-stationary climate that would adversely affect our results.
Fig. 4: Transfer functions derived from the 16 models.
Functions computed with simulations with all external forcings and training periods 2020–2100 (strong forcing) and 1960–2019 (mixed response) are shown in black and red, respectively. Functions from simulations with natural forcings only (variability only) are shown in blue. Each panel corresponds to a different grid-box.
We also test whether internal variability may significantly change in a non-stationary climate over the course of the century. We use the multi-model ensemble mean of the UK tx01 timeseries as an estimate of the forced response and subtract it from all the timeseries by individual models. We then compute the standard deviation in 5-year rolling windows, which provides the timeseries of the standard deviation shown in Fig. 5. The models indicate no major change in variability over the period 1900–2100. Finally, as a way of assessing the quality of the of the simulated UK mean tx01 samples used in the analysis, we examine how different model combinations might affect the local tx01 distributions. We find that different samples yield similar distributions (Supplementary Note 2 and Supplementary Fig. 2) and conclude that the sample choice does not introduce a considerable uncertainty.
Fig. 5: Timeseries of the standard deviation of the UK mean warmest daytime temperature (tx01) constructed with each of the 16 models.
The standard deviation was computed in 5-year rolling windows after subtracting the forced response from model simulations with all forcings.
Attribution on local scales
We adopt a popular risk-based event attribution framework24, whereby tx01 estimates from the two multi-model ensembles (with and without human influence) are used to generate probability distributions for the actual and natural climate. Local distributions are constructed on the observational (1 × 1 km) grid by applying the previously derived transfer functions to the simulated UK mean tx01 and the uncertainty in the transfer functions is accounted for by the bootstrapping procedure. Details on the construction of the local distributions are given in the 'Methods'.
Results from our analysis for a grid-box in London are illustrated in Fig. 6. For this example, the all-forcing simulations were extended to 2100 with the RCP 4.5 scenario. The cumulative distribution function (CDF) of tx01 shifts to higher values with time, increasing the likelihood of exceeding 40 °C, which is near-zero in the natural climate (Fig. 6a). Exceeding the lower threshold of 30 °C in London is common and occurs almost every year, even without the anthropogenic effect (Fig. 6b). However, temperatures above 35 °C are now 2–3 times more likely than in the natural climate (Fig. 6f), and model projections suggest they will occur at least twice a decade at the end of the century (Fig. 6c). The likelihood of exceeding 40 °C in the reference location is still extremely low, but is rapidly increasing, with the return time falling from thousands of years in the natural world to hundreds, or even tens of years by 2100 (Fig. 6d). For London, these likelihoods could increase even further as a consequence of increased urbanisation in future or from higher rates of local anthropogenic heat release25, for example from wider adoption of air conditioning during heatwaves.
Fig. 6: Increasing chance of high-threshold exceedance illustrated for a location in London.
a Cumulative distribution functions of the local warmest daytime temperature (tx01) for the natural climate (green line), the present-day climate (pink solid line) and the climate of the late twenty-first century (pink dashed line). The 30, 35 and 40 °C thresholds are marked by the vertical black lines. Panels b-d show timeseries of the return time (inverse probability) for the exceedance of the three thresholds with all forcings (in pink). The thickness of the timeseries illustrates the uncertainty in the transfer functions used in the analysis. The expected range in the natural climate is marked in green. Panels e–g show timeseries of the risk ratio (in blue) for the three thresholds, measuring the change in the likelihood of exceeding the threshold relative to the natural climate. The thickness of the timeseries represents the 5–95% uncertainty range. The vertical grey lines in panels b–g mark year 2020 (i.e., the present climate).
Repeating the analysis on all grid boxes, we also produce maps showing the return time for local exceedances of the three temperature thresholds (Fig. 7). Given the high spatial resolution of the plotted fields, certain topographic or coastal effects become evident on close inspection. Besides the noticeable contrast between warmer summers in the south and cooler in the north, the southeast England clearly stands out as the region where high temperature extremes are most likely to occur. Compared with the natural world, there are now more areas likely to see temperatures exceeding 30 or 35 °C, while the 40 °C threshold is still very rare, even in the southeast. By the end of the century, most areas in the north of the UK will also be regularly experiencing days with temperatures at least as warm as 30 °C, while crossing the 35 °C becomes common in the southeast under RCP 4.5 and over most of England under RCP 8.5. The highest threshold of 40 °C is to be exceeded at least once a century in the London area under RCP 4.5, and several times a century over most of southeast England under RCP 8.5. The effect of the uncertainty in the transfer functions has also been assessed (Supplementary Note 3 and Supplementary Figs. 3 and 4), and although it may change to some extent the intensity and spread of the map features in Fig. 7, the main conclusions still hold.
Fig. 7: The changing likelihood of locally exceeding high thresholds of the warmest daytime temperature (tx01) in the UK.
Maps of the return time for tx01 going above 30 °C (panels a–d), 35 °C (e–h) and 40 °C (i–l) in the natural climate (panels a, e, i), the present climate (b, f, j), and the climate of the late twenty-first century simulated with the RCP 4.5 (c, g, k) and RCP 8.5 scenarios (d, h, l).
Chances of exceeding extreme thresholds anywhere in the UK
We finally compute the likelihood of exceeding an extreme threshold in a given year, not at a specific location, but anywhere in the UK. The chance of rising above, for example, 40 °C, the most extreme threshold examined here, might still be very low for a given location, but has been increasing in most areas under the influence of warming trends (Fig. 1b). When all grid boxes are examined together, the likelihood of getting at least one grid-box that exceeds 40 °C in a specific year is expected to be higher than a local likelihood. The CMIP5 models provide a large number of alternative representations for every year. Each representation may yield a hit, i.e., at least one location where the reference tx01 threshold is exceeded, or not, and the likelihood of exceeding the threshold anywhere in the UK may thus be determined by the count of hits ('Methods'). Figure 8 depicts timeseries of the return time for different threshold exceedances and its expected range in the natural climate. Rising above 35 °C is estimated to occur once every 5 years at present and almost every year by the end of the century (Fig. 8b). Also, the probability of recording 40 °C, or above, in the UK is now rapidly accelerating and begins to rise clearly above the range of the natural climate (Fig. 8c). The return time for the 40 °C threshold is reduced from 100–1000s of years in the natural climate to 100–300 years in the present climate and to only about 15 years by 2100 under the medium-emissions scenario (RCP 4.5) and 3.5 years under the high-emissions scenario (RCP 8.5).
Fig. 8: The increasing likelihood of exceeding high temperature thresholds anywhere in the UK.
Timeseries of the return time for observing temperatures in the UK above a 30 °C, b 35 °C and c 40 °C with all forcings and future projections following the RCP 4.5 (in pink) and RCP 8.5 (in grey) scenarios. The thickness of the timeseries illustrates the uncertainty in the transfer functions used in the analysis. The expected range in the natural climate is marked in green. The vertical grey lines mark year 2020 (i.e., the present climate).
Our study demonstrates that human-caused climate change has set hot-day extremes in the UK on a course towards temperatures that would be too high to be observed in the natural climate. As the warming continues, new records are expected in coming decades, with the most severe extremes likely to occur in the southeast of the UK. Our attribution analysis derives local information from observations rather than regional models and investigates high-impact extremes that could break out anywhere in the UK rather than in prescribed locations. There are, of course, uncertainties in our analysis, some of which we have explored and tried to address, including uncertainties in the transfer functions and the limited number of years they are based on, or the limited number of models employed and their ability to represent the UK climate. Although the transfer functions make a distinction between urban and rural locations, large future changes in the UK's urban landscape could present a caveat in the analysis, though this is likely to affect only a small fraction of grid boxes. Future probability estimates are found to be sensitive to the choice of the RCP in the model simulations. Here, we estimate the future likelihood of extremes under both a mid- and high range RCP scenario. However, if emissions are reduced in line with the Paris climate agreement, the future probabilities are expected to be lower. Despite these uncertainties, our analysis still clearly establishes the nature of already realised and future changes in extreme temperatures including their spatial characteristics, information that can help the UK plan its resilience to heat extremes.
The local and UK mean tx01 is computed from the observations for every year in the period 1960–2019. For a given grid-box, we represent the dependence of the local tx01 on the UK mean with a simple linear model:
$$tx01\left( {{\mathrm{local}}} \right) = \alpha _0 + \alpha _1tx01\left( {{\mathrm{UK}}} \right).$$
The linear regression is fitted to the n = 60 observed annual values of tx01(local) and tx01(UK) and ordinary least squares are used to estimate the coefficients α0 and α1.
If yiobs and yifit denote the observed and fitted values of tx01(local) in year i, and SSE the sum of squared errors:
$${\it{{\mathrm{SSE}}}} = \mathop {\sum}\limits_{i = 1}^n {\left( {y_i^{{\mathrm{obs}}} - y_i^{{\mathrm{fit}}}} \right)^2} ,$$
then the confidence interval for the response variable and the (1 + p)/2 quantile of the t(n − 2) distribution is estimated19 as:
$${\mathrm{ \pm }}t_{{\mathrm{(1 + }}p{\mathrm{)/2}}}\sqrt {\frac{{{\mathrm{SSE}}}}{{n{\mathrm{ - 2}}}}} \sqrt {{\mathrm{1 + }}\frac{1}{n} + \frac{{\left( {x_i^{{\mathrm{obs}}} - \overline X } \right)^2}}{{{\it{{\mathrm{SXX}}}}}}} ,$$
where xiobs denotes the observed value of tx01(UK) in year i, \(\overline {\mathrm{X}}\) the mean of the observed tx01(UK) values and SXX is calculated as:
$${\mathrm{SXX}} = \mathop {\sum }\limits_{i = 1}^n \left( {x_i^{obs} - \overline X } \right)^2.$$
The confidence bounds are represented by very shallow hyperbolas that can be almost perfectly approximated by straight lines, as done in this study. Using the best fit to the observed data and the 1st to 99th percentiles for the estimation of the uncertainty range for the response variable, we end up with a set of 100 transfer functions per grid-box. Therefore, when we apply the transfer functions to a model simulated value of tx01(UK), we obtain 100 values that represent the possible range of the tx01 at the reference location.
Uncertainty in the transfer functions
Although the 60-year long observational dataset used to derive the transfer functions is deemed large enough to provide reliable estimates of the linear fits at every grid-box, sampling uncertainty will still have some effect on the analysis results. This kind of uncertainty is commonly accounted for by a simple a Monte Carlo bootstrap procedure26 that we also employ here. The procedure involves random resampling of the 60 annual pairs of tx01(UK) and tx01(local) and deriving a new set of transfer functions from the resampled data. Multiple resampling provides multiple sets of transfer functions. Each set, as explained next, can provide an estimate of the likelihood of the local tx01 exceeding a certain temperature threshold and by repeating the calculations with all the bootstrapped sets of transfer functions, we obtain multiple estimates of the likelihood, which enables us to estimate its uncertainty range.
Estimation of the local tx01 probabilities
For every year of each experiment we obtain samples of 1600 tx01 values for every grid-box (16 CMIP5 models that provide estimates of the UK mean tx01 × 100 transfer functions). We further increase the sample size in the all-forcings experiment to 32,000 by calculating the probabilities in 20-year rolling windows during the period 1900–2100 (i.e., in time segments 1900–1919, 1901–1920, …, 2081–2100). We select years 2011–2030 to represent the present-day climate and 2081–2100 to represent the late twenty-first century climate. Future probabilities are estimated with both RCP 4.5 and 8.5, whereas only simulations with RCP 4.5 were used to estimate present-day probabilities. For the natural world, we aggregate all simulated years (1900–2005), assuming that the natural climate is stationary in the long run, which yields samples of 169,600 tx01 values (16 models × 100 transfer functions × 106 years) for every grid-box. The resulting samples provide estimates of the likelihood of exceeding the pre-selected thresholds of 30, 35 and 40 °C. Given the large sample sizes, probabilities are computed by a simple count of threshold exceedances. Using alternative sets of the transfer functions from the bootstrapping procedure described earlier, we re-calculate the probabilities multiple times and estimate their 5–95% range to account for the uncertainty in the empirical relationships.
Probability of exceeding a threshold anywhere in the UK
The 16 CMIP5 models provide 320 simulated years in consecutive 20-year rolling windows. The sample increases to 32,000 when we apply the set of 100 transfer functions for each grid-box to obtain high-resolution annual maps of the tx01. Counting how many of these 32,000 maps include at least one location where the reference threshold is exceeded, allows us to the calculate the probability estimate. As before, the probability is recomputed with alternative sets of the transfer functions to assess their uncertainty. The natural probabilities are also aggregated to provide the 5–95% range for the natural climate.
The HadUK-Grid temperature data and station temperature data from the Met Office Integrated Data Archive System (MIDAS) that support the findings of this study are available from the CEDA Archive, http://archive.ceda.ac.uk. The CMIP5 simulated temperature data that support the findings of this study are available from the Earth System Grid Federation (ESGF) Archive, https://esgf.llnl.gov/.
IDL code used for the analysis is available upon request.
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This work was supported by the Met Office Hadley Centre Climate Programme funded by BEIS and Defra and the EUPHEME project, which is part of ERA4CS, an ERA-NET initiated JPI Climate and co-funded by the European Union (Grant 690462).
Met Office Hadley Centre, FitzRoy Road, Exeter, EX1 3PB, UK
Nikolaos Christidis, Mark McCarthy & Peter A. Stott
Nikolaos Christidis
Mark McCarthy
Peter A. Stott
N.C. organised the research and carried out the analysis. M.M. provided data and together with PAS helped design the study. N.C. wrote the paper with help from all co-authors.
Correspondence to Nikolaos Christidis.
Peer review information Nature Communications thanks Jakob Zscheischler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
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Clinical value of radiomics and machine learning in breast ultrasound: a multicenter study for differential diagnosis of benign and malignant lesions
Valeria Romeo1,
Renato Cuocolo2,3,
Roberta Apolito1,
Arnaldo Stanzione ORCID: orcid.org/0000-0002-7905-57891,
Antonio Ventimiglia1,
Annalisa Vitale1,
Francesco Verde1,
Antonello Accurso1,
Michele Amitrano1,
Luigi Insabato1,
Annarita Gencarelli1,
Roberta Buonocore4,
Maria Rosaria Argenzio4,
Anna Maria Cascone4,
Massimo Imbriaco1,
Simone Maurea1 &
Arturo Brunetti1
European Radiology volume 31, pages 9511–9519 (2021)Cite this article
We aimed to assess the performance of radiomics and machine learning (ML) for classification of non-cystic benign and malignant breast lesions on ultrasound images, compare ML's accuracy with that of a breast radiologist, and verify if the radiologist's performance is improved by using ML.
Our retrospective study included patients from two institutions. A total of 135 lesions from Institution 1 were used to train and test the ML model with cross-validation. Radiomic features were extracted from manually annotated images and underwent a multistep feature selection process. Not reproducible, low variance, and highly intercorrelated features were removed from the dataset. Then, 66 lesions from Institution 2 were used as an external test set for ML and to assess the performance of a radiologist without and with the aid of ML, using McNemar's test.
After feature selection, 10 of the 520 features extracted were employed to train a random forest algorithm. Its accuracy in the training set was 82% (standard deviation, SD, ± 6%), with an AUC of 0.90 (SD ± 0.06), while the performance on the test set was 82% (95% confidence intervals (CI) = 70–90%) with an AUC of 0.82 (95% CI = 0.70–0.93). It resulted in being significantly better than the baseline reference (p = 0.0098), but not different from the radiologist (79.4%, p = 0.815). The radiologist's performance improved when using ML (80.2%), but not significantly (p = 0.508).
A radiomic analysis combined with ML showed promising results to differentiate benign from malignant breast lesions on ultrasound images.
• Machine learning showed good accuracy in discriminating benign from malignant breast lesions
• The machine learning classifier's performance was comparable to that of a breast radiologist
• The radiologist's accuracy improved with machine learning, but not significantly
Ultrasound (US) has gained an established role in the assessment of breast lesions, showing several indications in female subjects including cases of palpable lumps, as first diagnostic tool in patients younger than 40, and for the evaluation of suspicious findings at mammography or magnetic resonance imaging [1]. According to the Breast Imaging-Reporting and Data System (BI-RADS) risk assessment and quality assurance tool, lesions are classified into different categories reflecting malignancy probability [2]. Several BI-RADS US descriptors are provided to aid in standardizing breast lesion characterization. According to a recent meta-analysis, the pooled sensitivity and specificity of US used as primary tool in detecting breast cancer lesions are 80.1% and 88.4%, respectively [3].
Radiomics is a complex multi-step process that allows extracting quantitative data from medical images, for example using texture analysis, to build clinically useful prediction models and decision support tools [4, 5]. In oncologic patients, radiomics features can be used to non-invasively assess intratumoral heterogeneity on routinely performed imaging exams [6]. When applied to medical images, artificial intelligence (AI) techniques employing machine learning (ML) algorithms have shown valuable results in image-recognition tasks, also being able to extract quantitative parameters reflecting image heterogeneity [7]. Mainly embraced for classification tasks, different ML approaches can be considered, with the most commonly applied in radiology being supervised learning (requiring labeled input data) and unsupervised learning (requiring unlabeled input data) [8]. Furthermore, several different algorithms are currently available, from those more easily interpretable (such as decision trees) to more complicated and harder to interpret ones (such as the convolutional neural networks used in deep learning) [9]. A large variety of possible clinical applications for AI in breast imaging has been described, applied to either US, digital breast tomosynthesis, or magnetic resonance imaging, ranging from differential diagnosis of breast lesions to breast cancer molecular subtype identification and prognosis prediction [10, 11]. According to previous experiences, radiomic analyses applied to breast US images have shown a good accuracy in the differential diagnosis of BI-RADS 4 and 5 lesions as well as to discriminate triple negative breast cancer from fibroadenomas [12, 13]. Often, these investigations lack external validation as analyzed data originates from a single institution. Similarly, the added clinical value of the proposed ML tools is not always assessed, leaving some doubts on the real-world benefits of AI.
Therefore, the scope of this study was threefold: (1) to assess the accuracy of a radiomic approach paired to ML applied to US images acquired in routine clinical practice to differentiate benign from malignant BIRADS 2–5 breast lesions, with internal and external testing; (2) to compare its diagnostic accuracy with that of a dedicated breast radiologist; (3) to verify whether the performance of the radiologist could be improved by the use of the proposed ML algorithm.
Patient population
The institutional review board approved this retrospective study and written informed consent was waived. All breast ultrasound examinations performed between November 2018 and June 2019 at the University Hospital "San Giovanni di Dio e Ruggi D'Aragona" in Salerno, Italy (Institution 1), and the Diagnostic Imaging Unit of the University of Naples "Federico II," Italy (Institution 2). Clinical indications for performing breast US were both routine check-up and assessment of palpable breast lesions or diagnostic in-depth analysis of breast lesions detected elsewhere. US examinations were performed by two radiologists in each institution, with 8 to 20 years of experience in breast imaging. Inclusion criteria were as follows: > 18-year-old patients with at least one BI-RADS 2, 3, 4, or 5 lesion. Exclusion criteria were as follows: unavailable follow-up for BI-RADS 3 lesions, or pathological confirmation for BI-RADS 4 and 5 lesions; US images not suitable for radiomic analysis due to artifacts; BI-RADS 2 cystic lesions. Breast lesions from Institution 1 were used as the training set, while those from Institution 2 were used as an external test set, to assess the radiologist's performance with and without the aid of AI. The standard of reference consisted of 6 months follow-up (US and/or mammography) for BI-RADS 3 lesions and pathology examination by means of Tru-Cut biopsy or surgical excision for BI-RADS 4 and 5 lesions.
Image acquisition
US examinations were performed using a LOGIQ S8, GE Healthcare (Institution 1) and a Logos HiVision E-Hitachi (Institution 2) US scanners, employing a high-frequency linear probe with radial, transverse, and longitudinal scans on both breasts. DICOM images were recorded and stored in the respective institutional digital archives.
Image conversion and segmentation
US examinations were evaluated by a dedicated breast radiologist (VR) with 8 years of experience in breast imaging who selected and retrieved the DICOM image of each breast lesion. 2D B-mode images in which the lesion was fully included, free from artifact and any measurements, were selected. As US images were originally encoded as three-channel RGB images, they were converted to grayscale applying an ITU-R 601-2 luma transform:
$$ \mathrm{L}=\mathrm{R}\ast 299/1000+\mathrm{G}\ast 587/1000+\mathrm{B}\ast 114/1000 $$
Where L is the luminance value, and R, G, and B the original values for the red, green, and blue channels for each pixel. This was performed with the Image module of the PILLOW Python package (v2.2.1).
Subsequently, the same radiologist performed manual lesion segmentation using a dedicated software (ITKSNAP, v3.8.0) obtaining 2D regions of interest (ROIs) (Fig. 1) [14]. To assess feature stability, manual segmentation was also performed independently by two senior radiology residents (A.V. and A.V.) on 30 randomly selected patients from the training set in order to calculate feature intraclass correlation coefficient (ICC) [15].
Examples of lesion annotation. The upper row (a, b) shows placement of a region of interest on a benign lesion, while c and d depict a malignant lesion before and after manual segmentation.
Image preprocessing and feature extraction
A dedicated open-source Python-based software (PyRadiomics, v2.2.0) was employed for image preprocessing and 2D radiomic feature extraction [16] [17].
As voxels were already isotropic in-plane, no resampling was necessary prior to feature extraction. On the other hand, gray-level whole-image normalization was performed to ensure comparability of images acquired on different scanners and with varying settings, with a resulting range of 0–600. As suggested by the developers, a fixed bin width (= 3) was used for discretization. Other than from the original images, features were also extracted from Laplacian of Gaussian (LoG, sigma = 1, 2, 3, 4, 5) and wavelet (all high- and low-pass filter combinations on x and y planes) filtered ones. The use of image filters can reduce image noise and highlight textural characteristics. In particular, the LoG filter performs an image smoothing operation, enhancing structural edges within the image of interest. In this setting, the sigma value specifies the desired fineness or coarseness of the resulting output (lower values produce finer images and vice versa) [18]. Wavelet decompositions represent an alternative approach to remove low signal areas from the images (i.e., image smoothing and edge detection). Using high- and low-pass filter combinations, the original image is decomposed in distinct components, expanding the original signal [19]. As the best practice for medical image analysis is not established, these alternative filtering approaches have been both included in the investigation.
In regard to feature classes, 2D shape, first order, gray level co-occurrence matrix (GLCM), gray level run length matrix, gray level size zone matrix, and gray level dependence matrix ones were extracted. In particular, as image acquisition could vary in terms of depth and zoom, only adimensional 2D shape features were included to avoid biases, in particular perimeter-surface ratio, sphericity, spherical disproportion, and elongation. For the remaining classes, all available features were calculated with the exception of GLCM sum average, as suggested by the PyRadiomics developers due to known redundancy with other GLCM parameters.
Formulas and definitions of the extracted features can be found on the official documentation (https://pyradiomics.readthedocs.io/en/latest/features.html).
Data analysis and feature selection
Feature stability testing was performed by calculating a two-way random effect, single rater, absolute agreement ICC for each. Only features with good reproducibility (ICC value ≥ 0.75) were considered stable and included in the following steps [15, 20]. ICC calculation was performed using the R "irr" package [21]. The numbers of patients (n = 30) and readers (n = 3) as well as the ICC cutoff value were based on suggestions by recent guidelines and previous ML studies [15, 20]. A MinMax scaler with 0–1 range was fitted on the training data alone, to avoid any information leakage, and used to transform both training and test sets. Successively, non-informative features showing low variance (≤ 0.01) were excluded. Similarly, highly intercorrelated features were discarded based on the pairwise correlation matrix (r ≥ 0.8). At this point, the Synthetic Minority Over-sampling Technique (SMOTE) was employed on the training data to balance the dataset [22, 23]. In detail, SMOTE creates new instances (i.e., synthetic patients) of the minority class by interpolation of data from k (= 5 in our study) nearest neighbors from the original population with the same label. The process is repeated until the two classes are perfectly balanced. Finally, stratified 10-fold cross-validated recursive feature elimination (RFECV) with a Logistic Regression (LBFGS solver) estimator identified the optimal number of parameters to train the ML algorithm. These data processing steps were conducted using the pandas and scikit-learn Python packages [21, 24].
Machine learning analysis
Given the tabular nature of the data, expected number of instances available, and previous experiences, also following the recommendations made by the scikit-learn developers, a Random Forest (RF) ensemble algorithm was selected for this classification task.
Algorithm performance during the random search tuning process was assessed on the training set through 5-fold stratified cross-validation. This approach is more robust than a single train-test split and can be expected to give a better estimation of generalizability [25]. In stratified cross-validation, each of the folds the data is split preserves the class balance and is used as a validation set for an algorithm trained on the remaining (n = 4) data folds. Then, the final model was fitted on the whole training set and tested on the data from Institution 2. Its accuracy was also compared to a baseline reference value (no information rate, NIR) corresponding to the accuracy obtainable by classifying all lesions as belonging to the most frequent class (i.e., the mode of the classes). A p value < 0.05 was considered statistically significant. The Brier score was calculated for the model on the test set, as well as a calibration curve, to assess prediction and calibration loss of predicted probability and lesion class.
The machine learning analysis was performed using the scikit-learn Python package. Accuracy metrics were computed with the same Python package and the caret R package [21, 24].
Radiological evaluation
A dedicated breast radiologist (R.B., 8 years of experience) from Institution 1 evaluated the same US images of the test set used for the ML analysis and classified each lesion as benign or malignant, according to the BI-RADS V edition. In detail, the radiologist assessed lesion shape, margins, orientation, echo-pattern, and posterior features assigning a score from 2 to 5. BI-RADS scores were then dichotomized as 2–3 = benign, and 4–5 = malignant for the subsequent analysis. The radiologist was blinded to patient clinical history and final diagnosis. After a 4-week washout period, the same radiologist performed a new evaluation, this time with the availability of ML predictions and probabilities for each lesion. The accuracy of the radiologist was calculated using the caret R package and also compared to the NIR baseline accuracy reference.
Kolmogorov-Smirnov test was first performed to assess whether data were normally distributed. Accordingly, t-test and Mann-Whitney U test were performed to assess differences in terms of age and lesion size (maximum diameter) of malignant and benign breast lesions between training and test sets. Accuracy, sensitivity, specificity, and positive and negative predictive values of both ML classifier and expert radiologist were calculated. McNemar's test was performed to assess differences in the performance between ML and the human reader and between the reader without and with the use of ML. A p value < 0.05 was considered statistically significant.
Based on inclusion criteria, 441 patients of which 309 were from Institution 1 and 132 from Institution 2 were reviewed. Applying our exclusion criteria, a final population of 117 patients from Institution 1 (mean age 48 years, range 15–94 years), with 135 lesions (91 benign and 44 malignant), and 57 patients from Institution 2 (mean age 52 years, range 12–85 years), with 66 lesions (21 benign and 45 malignant), was therefore included. The flowchart of patient selection process is illustrated in Fig. 2. Age was not statistically different between training and test sets at Student's t-test (p = 0.177).
Flowchart of the patient selection process. Pts, patients; BLs, breast lesions
Mean size of breast lesions from Institution 1 was 13 mm (range 4–44 mm), while mean size of breast lesions from Institution 2 was 16.37 mm (range 4–47 mm). Training and test sets did not differ in terms of lesion size between benign and malignant breast lesions at Mann-Whitney U test (p values 0.794 and 0.325, respectively).
The BI-RADS assessment of the included lesions can be found in the supplementary materials as Table S1.
Fourteen BI-RADS 4 lesions were revealed as benign, while the remaining BI-RADS 4 and 2 BI-RADS 3 lesions, who showed a significant increase of lesion size at follow-up examinations, resulted in malignant pathology. Final diagnoses of histologically proven BI-RADS 3, 4, and 5 breast lesions are reported in Table 1.
Table 1 Diagnosis of histologically proven BI-RADS 3, 4, and 5 lesions in both training and test sets
Feature extraction, data analysis, and feature selection
A total of 520 features were extracted. Of these, 198 resulted unstable after ICC assessment and were discarded, with 322 features left. Then, 10 low variance parameters were also excluded as well as 278 highly intercorrelated ones, as resulted from the pairwise correlation matrix shown in Figure S1. After class balancing with SMOTE, from the remaining 34 features, RFECV identified a subset of 10 (Figure S2), including "original shape2D PerimeterSurfaceRatio"; "original shape2D Elongation"; "original glcm Autocorrelation"; "original gldm DependenceNonUniformityNormalized"; "log-sigma-1-0-mm-3D glcm Imc2"; "log-sigma-2-0-mm-3D glcm Correlation"; "log-sigma-3-0-mm-3D glrlm GrayLevelNonUniformityNormalized"; "log-sigma-4-0-mm-3D glcm Imc1"; "wavelet-H glcm Imc2"; "wavelet-H glrlm GrayLevelNonUniformityNormalized." The feature selection process is summarized in Figure S3.
The RF hyperparameters were set as follows: bootstrap = true, class weight = none, criterion = Gini, maximum depth = none, maximum features = 5, maximum leaf nodes = none, minimum impurity decrease = 0.0, minimum impurity split = none, minimum samples leaf = 1, minimum samples split = 2, minimum weight fraction leaf = 0.0, number of estimators = 400.
In the training set, RF obtained an overall mean accuracy of 82% (standard deviation, SD, ± 6%) and positive predictive value (PPV), sensitivity, specificity, and AUC for malignant lesions respectively of 78% (SD ± 5%), 89% (SD ± 7%), 75% (SD ± 5%), and 0.90 (SD ± 0.06).
In the test set, ML accuracy also was 82% (95% confidence intervals (CI) = 70–90%) with a PPV and negative predictive value (NPV) of 82% (95% CI = 74 to 89%) and 80% (95% CI = 56–93%), sensitivity of 93% (95% CI = 82–99%), and specificity of 57% (95% CI = 34–78%) for malignant lesions. RF's accuracy results are significantly better than the NIR (p = 0.0098). The AUC was 0.82 (95% CI = 0.70–0.93) (Fig. 3). Regarding prediction and calibration loss on the test set, the Brier score was 0.17 and Fig. 4 presents the calibration curve plot.
Receiver operating characteristic curve of the machine learning classifier for distinguishing benign and malignant lesions in the test set
Calibration curve plot of the model in the test set. Average predicted probability is represented in the x-axis while the proportion of malignant lesions in the y-axis
The expert radiologist obtained an accuracy of 79.4% (95% CI = 67–91%) on the test set, not significantly different from ML reading at McNemar's test (p = 0.815). Sensitivity and specificity in identifying malignant breast lesions were 77.8% (95% CI = 62.9 to 88.8%) and 81% (95% CI = 58.1 to 94.6%), respectively, while PPV was 89.7% (95% CI = 78.1 to 95.5%) and NPV 63% (95% CI = 48.7 to 75.3%). With the availability of ML predictions, these metrics improved as follows: accuracy = 80.2% (95% CI = 67–93%), sensitivity = 88.9% (95% CI = 75.6 to 96.3%), specificity = 71.4% (95% CI = 47.8 to 88.7%), PPV = 87% (95% CI = 77.1 to 93%), and NPV = 75% (95% CI = 55.7 to 87.7%). The McNemar test comparing the ML and expert radiologist's readings was not significantly different (p = 0.508). Classification tables of the comparison between the performance of the expert radiologist and ML as well as between the expert radiologist without and with the support of ML are reported in Tables S2 and S3, respectively. Examples of cases in which the expert radiologist was aided by the ML algorithm in correctly classifying benign and malignant lesions are illustrated in Fig. 5. Accuracy metrics of ML and expert radiologist without and with the availability of ML prediction are summarized in Table 2.
B-mode US images of a benign (a) and malignant (b) breast lesion initially misclassified by the expert radiologist and correctly diagnosed with the availability of ML reading. a A case of a 13-year-old patient with a 4-cm oval breast lesion with circumscribed margins but heterogeneous echo-pattern, proved to be a sclerosing papilloma after surgical excision. b A case of a 59-year-old patient with a 5-mm oval, hypoechoic breast lesion with circumscribed margins, histologically proved as Luminal A, G1, ductal invasive carcinoma
Table 2 Accuracy metrics (95% confidence interval) of ML classifier and expert radiologist without and with the availability of ML reading
In this study, we built a radiomic-based ML model to differentiate benign from malignant breast lesions on US. The resulting RF algorithm obtained an accuracy of 82%, with high sensitivity (93%) but low specificity (57%) in classifying benign and malignant breast lesions. It performed significantly better than the baseline NIR (p = 0.0098) and showed higher accuracy compared to the expert radiologist (82% vs 79.4%), even if this difference did not reach statistical significance (p = 0.815). Even though the radiologist's accuracy increased to 80.2% with the aid of ML predictions, this difference also proved not statistically significant (p = 0.508).
As with all ML studies, our results should be interpreted by taking into account the data that was employed to build and test the model. We wished to focus on challenging lesions, excluding all cystic lesions which do not pose a real diagnostic challenge. This design choice is also reflected in the performance of the breast radiologist, slightly lower than what could be expected from the literature [3, 26]. It should be acknowledged that reviewing US images is not the same as performing a complete examination, but this could not be avoided due to the study's retrospective nature. In this setting, RF outperformed both the baseline reference and the radiologist, demonstrating promising performance. The improved performance of the radiologist with the aid of ML is also suggestive of its usefulness in clinical practice. Indeed, the sensitivity of the expert radiologist raised up to 88.9% vs 77.8%, to the detriment of the specificity that, instead, decreased from 81 to 71%. The increase in terms of sensitivity is advisable as it could reduce the possibility to miss malignant lesions. It is interesting to note that the radiologist with ML still performed worse than ML alone, probably due to lack of trust in the model's predictions.
While the model's accuracy was stable across the cross-validation and test set assessment, in the latter we observed a reduction in specificity, compensated by increased sensitivity. It must be considered that the test set had a different proportion of malignant cases (n = 45/66, 68%) compared to the training one (44/135, 33%), and overall challenging lesions, as demonstrated by the radiologist's performance.
Our findings support a possible clinical role for a US radiomic-ML tool in the characterization of benign and malignant breast lesions, in line with previous studies conducted using US radiomic features with [27,28,29] and without ML [12, 13, 30, 31]. For example, logistic regression models were developed using radiomic features extracted from US images to discriminate benign from malignant lesions [31], predict the presence of cancer in BIRADS 4 and 5 lesions [12], and differentiate fibroadenomas from triple negative breast cancer [13] with AUC values of 0.886, 0.928, and 0.834–0.864, respectively. Furthermore, images obtained from automated whole breast US were used to extract texture, shape, and ellipsoid features and also analyzed using a logistic regression model, reporting an accuracy of 85% in classifying breast masses [30]. Deep learning networks were also tested to characterize benign and malignant breast lesions, with AUC values ranging from 0.80 [28] to 0.95 [29]. ML algorithms using radiomic features based on US BIRADS lexicon were also assessed by Fleury et al [32]; the Support Vector Machine resulted as the best classifier, with an AUC value of 0.84 in characterizing breast lesions. To the best of our knowledge, this is the first study assessing the usefulness of a ML classifier using shape, first order, and texture features from both original and filtered US images to classify benign and malignant breast lesions in a challenging dataset. The possibility to implement such a tool in the routine clinical practice would have tremendous implications in the management of breast lesions, considering that US is the first-level imaging modality for their assessment. In a future perspective, it would be possible to non-invasively characterize breast lesions using a widespread imaging modality, thus reducing the recourse to breast biopsy, as well as to reserve the use of second level and more expensive imaging techniques, such as MRI, to selected cases.
Limitations of our study are represented by its retrospective nature and the relatively limited patient population. However, the multicentric study design allowed to validate the AI algorithm on an external test set from a different institution, thus determining its robustness and generalizability. As stated above, the retrospective nature of the study did not allow for a standard evaluation of patients by the radiologist, who was limited to reviewing US images. Also, the final populations from the two institutions showed different proportions of malignant lesions, also understandable in light of the retrospective design of the investigation.
In conclusion, a radiomic approach paired to ML was accurate to differentiate benign from malignant BIRADS 2–5 breast lesions on US, showing a performance comparable to that of an experienced radiologist. Further studies on a larger cohort of patients and with a prospective design are necessary to confirm our promising findings.
AI:
BI-RADS:
Breast Imaging-Reporting and Data System
GLCM:
Gray level co-occurrence matrix
ICC:
ML:
No information rate
NPV:
PPV:
RF:
Forest Ensemble Algorithm
RFECV:
Cross Validated Recursive Feature Elimination
ROIs:
Regions of interest
SMOTE:
Synthetic Minority Oversampling Technique
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Sivaramakrishna R, Powell KA, Lieber ML, Chilcote WA, Shekhar R (2002) Texture analysis of lesions in breast ultrasound images. Comput Med Imaging Graph 26:303–307. https://doi.org/10.1016/S0895-6111(02)00027-7
Fleury E, Marcomini K (2019) Performance of machine learning software to classify breast lesions using BI-RADS radiomic features on ultrasound images. Eur Radiol Exp 3:34. https://doi.org/10.1186/s41747-019-0112-7
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
The authors state that this work has not received any funding.
Valeria Romeo and Renato Cuocolo contributed equally to the study and should be considered co-first authors.
Department of Advanced Biomedical Sciences, University of Naples "Federico II", Via S. Pansini, 5, 80131, Naples, Italy
Valeria Romeo, Roberta Apolito, Arnaldo Stanzione, Antonio Ventimiglia, Annalisa Vitale, Francesco Verde, Antonello Accurso, Michele Amitrano, Luigi Insabato, Annarita Gencarelli, Massimo Imbriaco, Simone Maurea & Arturo Brunetti
Department of Clinical Medicine and Surgery, University of Naples "Federico II", Naples, Italy
Renato Cuocolo
Laboratory of Augmented Reality for Health Monitoring (ARHeMLab), Department of Electrical Engineering and Information Technology, University of Naples "Federico II", Naples, Italy
Department of Radiology, A.O.U. San Giovanni di Dio e Ruggi d'Aragona, Salerno, Italy
Roberta Buonocore, Maria Rosaria Argenzio & Anna Maria Cascone
Valeria Romeo
Roberta Apolito
Arnaldo Stanzione
Antonio Ventimiglia
Annalisa Vitale
Antonello Accurso
Michele Amitrano
Luigi Insabato
Annarita Gencarelli
Roberta Buonocore
Maria Rosaria Argenzio
Anna Maria Cascone
Massimo Imbriaco
Simone Maurea
Arturo Brunetti
Correspondence to Arnaldo Stanzione.
The scientific guarantor of this publication is Professor Arturo Brunetti.
The authors of this manuscript declare no relationships with any companies whose products or services may be related to the subject matter of the article.
Statistics and biometry
No complex statistical methods were necessary for this paper.
Written informed consent was waived by the Institutional Review Board.
Ethical approval
Institutional Review Board approval was obtained.
• Retrospective
• Observational
• Multicenter study
(DOCX 4037 kb)
Romeo, V., Cuocolo, R., Apolito, R. et al. Clinical value of radiomics and machine learning in breast ultrasound: a multicenter study for differential diagnosis of benign and malignant lesions. Eur Radiol 31, 9511–9519 (2021). https://doi.org/10.1007/s00330-021-08009-2
Revised: 06 April 2021 | CommonCrawl |
Category:Deceleration Parameter
Revision as of 20:34, 23 February 2015 by Cosmo All (Talk | contribs) (Created page with " <div id="1301_1"></div> <div style="border: 1px solid #AAA; padding:5px;"> '''Problem 1''' <p style= "color: #999;font-size: 11px">problem id: 1301_1</p> Show that for a spa...")
problem id: 1301_1
Show that for a spatially flat Universe consisting of one component with equation of state $p=w\rho$ the deceleration parameter is equal to $q=(1+3w)/2$.
\[\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho(1+3w).\] In the flat case \[\rho=\frac{3}{8\pi G}H^2\] and \[q=-\frac{\dot a}{aH^2}=\frac12(1+3w).\]
Find generalization of the relation $q=(1+3w)/2$ for the non-flat case.
\[\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho(1+3w).\] Substitution \[\rho=\frac{3}{8\pi G}\left(H^2+\frac k{a^2}\right)\] gives \[q=-\frac{\dot a}{aH^2}=\frac12(1+3w)\left(1+\frac k{a^2H^2}\right).\]
Find deceleration parameter for multi-component non-flat Universe.
In this case \[q=-\frac{\dot a}{aH^2}=\frac12(1+3\frac p\rho)\left(1+\frac k{a^2H^2}\right) =\] \[=\frac12\left[1+3\frac p\rho\left(1+\frac k{a^2H^2}\right)\right]+\frac k{2a^2H^2},\] \[1+\frac k{a^2H^2}=\sum\limits_i\Omega_i,\quad\frac p\rho=\frac{\sum\limits_iw_i\Omega_i}{\sum\limits_i\Omega_i},\] \[q=\frac12\left(1+3\sum\limits_i\Omega_i\right)+\frac k{2a^2H^2}.\]
Show that the expression for deceleration parameter obtained in the previous problem can be presented in the following form \[q=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]
Using results of the previous problem one obtains \[q=\frac12\left(1+3\sum\limits_iw_i\Omega_i\right)+\frac k{2a^2H^2}=\frac12\left(1+\frac k{a^2H^2}\right)+\frac32\sum\limits_iw_i\Omega_i=\] \[=\frac12\sum\limits_i\Omega_i+\frac32\sum\limits_iw_i\Omega_i=\frac{\Omega_{total}}2+\frac32\sum\limits_i w_i \Omega_i.\]
problem id:
Let us consider the model of two-component Universe [ J. Ponce de Leon, cosmological model with variable equations of state for matter and dark energy, arXiv:1204.0589]. Such approximation is sufficient to achieve good accuracy on each stage of its evolution. At the present time the two dark components - dark matter and dark energy - are considered to be dominating. We neglect meanwhile the interaction between the components and as a result they separately satisfy the conservation equation. Let us assume that the state equation parameter for each component depends on the scale factor \begin{align} \nonumber p_{de} & = W(a) \rho_{de};\\ \nonumber p_m & = w(a)\rho_{m}. \end{align} Express the relative densities $\Omega_m$ and $\Omega_{de}$ in terms of the deceleration parameter and the state equation parameters $w(a)$ and $W(a)$.
In the considered case \[\Omega_m +\Omega_{de}=1+\frac{k}{a^2H^2},\] \[q=\frac12+\frac32\left[W\Omega_{de}+w\Omega_m\right]+\frac{k}{2a^2H^2}.\] These can, formally, be regarded as two equations for $\Omega_{de}$ and $\Omega_{m}$. Solving them we get \begin{align} \nonumber \Omega_{m} & = \frac{2q-1-3W}{3(w-W)}-\frac{k(1+3W)}{3a^2H^2(w-W)};\\ \nonumber \Omega_{m} & = \frac{2q-1-3w}{3(w-W)}-\frac{k(1+3w)}{3a^2H^2(w-W)}. \end{align}
Find the upper and lower limits on the deceleration parameter using results of the previous problem.
We note that the denominator in these expressions is always positive because $W<0$ for dark energy. Thus, the fact that $\Omega_{m(de)}>0$ imposes an upper and lower limit on $q$, \[\frac{1+3W}{2}\left(1+\frac{k}{a^2H^2}\right)\le q\le \frac{1+3w}{2}\left(1+\frac{k}{a^2H^2}\right).\]
Find the relation between the total pressure and the deceleration parameter for the flat one-component Universe
It follows from the conservation equation that \[p=-\frac{\dot\rho}{3H}\frac{w}{1+w}.\] Using \[w = \frac{2q-1}3;\quad \dot\rho=\frac{3}{4\pi G}H\dot H,\quad \dot H=-H^2(1+q)\] one finds \[p=\frac{H^2}{8\pi G}(2q-1).\]
The expansion of pressure by the cosmic time is given by \[p(t)=\left.\sum\limits_{k=0}^\infty\frac1{k!}\frac{d^kp}{dt^k}\right|_{t=t_0}(t-t_0)^k.\] Using cosmography parameters , evaluate the derivatives up to the fourth order.
Show that for one-component flat Universe filled with ideal fluid of density $\rho$.
\[q=-1-\frac12\frac{d\ln\rho}{d\ln a}.\]
problem id: 1301_10
For what values of the state parameter $w$ the rate of expansion of a one-component flat Universe increases with time?
Express the age of the spatially flat Universe filled with a single component with equation of state $p=w\rho$ through the deceleration parameter.
For spatially flat one-component Universe with state equation $p=w\rho$ the scale factor is \[a\propto t^\frac{2}{3(1+w)}\] and therefore \[H=\frac{2}{3(1+w)}\frac1t.\] Using $q=(1+3w)/2$ one can find a simple relation between the current age of the Universe and the DP \[t_0=\frac{H_0^{-1}}{1+q}.\]
Suppose we know the current values of the Hubble constant $H_0$ and the deceleration parameter $q_0$ for a closed Universe filled with dust only. How many times larger will it ever become? Find lifetime of such a Universe.
In a closed Universe filled with non-relativistic matter the current values of the Hubble constant is $H_0$, the deceleration parameter is $q_0$. Find the current age of this Universe.
In a closed Universe filled with dust the current value of the Hubble constant is $H_0$ and of the deceleration parameter $q_0$.
a) What is the total proper volume of the Universe at present time?
b) What is the total current proper volume of space occupied by matter which we are presently observing?
c) What is the total proper volume of space which we are directly observing?
For the closed ($k=+1$) model of Universe, filled with non-relativistic matter, show that solutions of the Friedmann equations can be represented in terms of the two parameters $H_0$ and $q_0$. [Y.Shtanov, Lecture Notes on theoretical cosmology, 2010 ]
The Friedman equations can then be represented in the form \begin{align} \label{background_2_29} H^2 & +\frac1{a^2} =\frac{8\pi G}3\rho,\\ \nonumber 2\frac{\ddot a}{a} & +H^2+\frac1{a^2} = 0. \end{align} Using this equations one can find the relations between the current values of the Universe's parameters \begin{align} \label{background_2_30} H_0^2 & =\frac1{a_0^2(2q_0-1)},\\ \nonumber q_0 & = \frac{4\pi G}{3H_0^2}\rho_0. \end{align} Note that in general \[q_0=\frac{4\pi G}3\frac{\rho_0+3p_0}{H_0^2}=\frac{\rho_0+3p_0}{2\rho_{0,crit}},\quad \rho_{0,crit}=\frac{3H_0^2}{8\pi G}.\] For Universe filled only with non-relativistic matter one has $\Omega_m = 2q_0$. It is easy to see that $q_0>1/2$ and $\Omega_m>1$ as was expected in the closed model. Using (\ref{background_2_30}) one can rewrite the equation for scale factor in the form \begin{equation}\label{background_2_31}\dot a^2=\frac\alpha a-1,\quad \alpha\equiv\frac{2q_0}{H_0(2q_0-1)^{3/2}}.\end{equation} It is easy to see that the considered model includes both parameters $H_0$ and $q_0$. integration of (\ref{background_2_31}) gives \[t=\int \sqrt{\frac{a}{\alpha-a}}\,da.\] Substitution \[a=\frac\alpha2(1-\cos\tau)\] leads to \begin{equation}\label{background_2_33}t=\frac\alpha2(\tau-\sin\tau).\end{equation} Because of the relation $dt=ad\tau$ it is evident that the variable $\tau$ is the conformal time. Taking the constants of integration so that $a=0$ as $t=0$ (and $\tau=0$) we can see that $a=a_0$ at $\tau=\tau_0$. Consequently, \[\cos\tau_0=\frac{1-q_0}{q_0},\quad \sin\tau_0=\frac{\sqrt{2q_0-1}}{q_0}.\] From (\ref{background_2_33}) it follows that the age of Universe in close model is \[t_0=\frac\alpha2(\tau_0-\sin\tau_0)=\frac{q_0}{H_0(2q_0-1)^{3/2}}\left(\arccos{\frac{1-q_0}{q_0}} - \frac{\sqrt{2q_0-1}}{q_0}\right).\] The maximum of scale factor reaches at $\tau=\pi$, \[a_{\rm max}=\alpha=\frac{2q_0}{H_0(2q_0-1)^{3/2}}.\] As one can see, all parameters of the model can be expressed in terms of parameters $H_0$ and $q_0$.
Do the same as in the previous problem for the case of open ($k=-1$) model of Universe.
In the case of open Universe the formulae (\ref{background_2_29})-(\ref{background_2_31}) transform into \begin{align} \label{background_2_37} H^2 & -\frac1{a^2} =\frac{8\pi G}3\rho,\\ \nonumber 2\frac{\ddot a}{a} & +H^2-\frac1{a^2} = 0,\\ \label{background_2_38} H_0^2 & =\frac1{a_0^2(1-2q_0)},\\ \nonumber q_0 & = \frac{4\pi G}{3H_0^2}\rho_0, \end{align} \begin{equation}\label{background_2_39}\dot a^2=\frac\beta a+1,\quad \beta\equiv\frac{2q_0}{H_0(1-2q_0)^{3/2}}.\end{equation} Now $q_0<1/2$, $0\le\Omega_m< 1$. The solution of (\ref{background_2_39}) in the conformal time parametrization is \[a=\frac\beta2(\cosh\tau-1),\quad t=\frac\beta2(\sinh\tau-\tau).\] Current value $\tau_0$ of the conformal time is defined by the relation \[\cosh\tau_0=\frac{1-q_0}{q_0}.\] The age of the Universe is \[t_0=\frac\beta2(\sinh\tau_0-\tau_0) = \frac{q_0}{H_0(1-q_0)^{3/2}}\left(\frac{\sqrt{1-2q_0}}{q_0} - \ln\frac{1-q_0+\sqrt{1-2q_0}}{q_0}\right).\] In this case again all characteristics of the model are expressed in terms of the parameters $H_0$ and $q_0$.
1 Classification of models of Universe based on the deceleration parameter
2 Deceleration as a cosmographic parameter
3 Cosmological scalars and the Friedmann equation
4 Averaging deceleration parameter
5 Energy conditions in terms of the deceleration parameter
6 Distance-Deceleration Parameter Relations
7 Horizons
8 Power-Law Universes
9 The Effects of a Local Expansion of the Universe
Classification of models of Universe based on the deceleration parameter
When the rate of expansion never changes, and $\dot a$ is constant, the scaling factor is proportional to time $t$, and the deceleration term is zero. When the Hubble parameter is constant, the deceleration parameter $q$ is also constant and equal to $-1$, as in the de Sitter model. All models can be characterized by whether they expand or contract, and accelerate or decelerate. Build a classification of such type using the signature of the Hubble parameter and the deceleration parameter.
a) $H>0$, $q>0$: expanding and decelerating
b) $H>0$, $q<0$: expanding and accelerating
c) $H<0$, $q>0$: contracting and decelerating
d) $H<0$, $q<0$: contracting and accelerating
e) $H>0$, $q=0$: expanding, zero deceleration
f) $H<0$, $q=0$: contracting, zero deceleration
g) $H=0$, $q=0$: static
There is little doubt that we live in an expanding Universe, and hence only (a), (b), and (e) are possible. Evidences in favor of the fact that the expansion is presently accelerating continuously grows in number and therefore the current dynamics belongs to type (b).
Can the Universe variate its type of evolutions in frames of the classification given in the previous problem?
Of course, generally speaking, both the Hubble parameter and deceleration parameter can change their sign during the evolution. Therefore the evolving Universe can transit from one type to another. It is one of the basic tasks of cosmology to follow this evolution and clarify its causes.
Point out possible regime of expansion of Universe in the case of constant deceleration parameter.
In the case of constant deceleration parameter the Universe would exhibit decelerating expansion if $q>0$, an expansion with constant rate if $q=0$, accelerating power-law expansion if $-1<q<0$, exponential expansion (also known as de Sitter expansion) if $q=-1$ and super-exponential expansion if $q<-1$.
Having fixed the material content we can classify the model of Universe using the connection between the deceleration parameter and the spatial geometry. Perform this procedure in the case of Universe filled with non-relativistic matter.
\begin{align} \nonumber q>\frac12 & :\quad closed\ spherical\ space & k&=+1;\\ \nonumber q=\frac12 & :\quad open\ flat\ space & k&=0;\\ \nonumber q<\frac12 & :\quad open\ hyperbolic\ space & k&=-1. \end{align}
Models of the Universe can be classified basing on the relation between the deceleration parameter and age of the Universe. Build such a classification in the case of Universe filled with non-relativistic matter (the Einstein-de Sitter Universe)
For the Einstein-de Sitter Universe where the age is \[t^*=\frac23H^{-1}\] we have \begin{align} \nonumber q>\frac12 & :\quad age<t^*;\\ \nonumber q=\frac12 & :\quad age=t^*;\\ \nonumber q<\frac12 & :\quad age>t^*. \end{align}
Show that sign of the deceleration parameter determines the difference between the actual age of the Universe and the Hubble time.
In a decelerating Universe with $q>0$, the age of the Universe will be less than the Hubble time, because at earlier times it was expanding at a faster rate, whereas a Universe that has always been accelerating, that is, $q<0$ for all time, will have an age that is greater than the Hubble time. A Universe that expands at a constant rate, $q=0$, has an age equal to the Hubble time.
Suppose the flat Universe is filled with non-relativistic matter with density $\rho_m$ and some substance with equation of state $p_X=w\rho_X$. Express the deceleration parameter through the ratio $r\equiv\rho_m/\rho_X$.
\[q=-\frac{\ddot a/a}{H^2}=\frac{\frac{4\pi G}{3}(\rho+3p)}{\frac{8\pi G}{3}\rho}=\frac12+\frac32\frac p\rho= \frac12+\frac32\frac{w\rho_X}{\rho_m+\rho_X} =\frac12+\frac32\frac w{(1+r)}.\]
problem id: 1301_23_1
Obtain Friedmann equations for the case of spatially flat $n$-dimensional Universe. (see Shouxin Chen, Gary W. Gibbons, Friedmann's Equations in All Dimensions and Chebyshev's Theorem, arXiv: 1409.3352)
Consider an $(n+1)$-dimensional homogeneous and isotropic Lorentzian spatially flat spacetime with the metric \begin{equation} \label{1} ds^2=g_{\mu\nu} dx^\mu dx^\nu=-dt^2+a^2(t)g_{ij} dx^i dx^j,\quad i,j=1,\dots,n, \end{equation} where $t$ is the cosmological (or cosmic) time and $g_{ij}$ is the metric of an $n$-dimensional Riemannian manifold $M$ of constant scalar curvature characterized by an indicator, $k=-1,0,1$, so that $M$ is an $n$-hyperboloid, the flat space $\Bbb R^n$, or an $n$-sphere, with the respective metric \begin{equation} \label{2} g_{ij}dx^i dx^j=\frac1{1-kr^2}\,dr^2+r^2\,d\Omega^2_{n-1}, \end{equation} where $r>0$ is the radial variable and $d\Omega_{n-1}^2$ denotes the canonical metric of the unit sphere $S^{n-1}$ in $\Bbb R^n$. Inserting the metric (\ref{1})--(\ref{2}) into the Einstein equations \begin{equation} G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}, \end{equation} where $G_{\mu\nu}$ is the Einstein tensor, $G$ the universal gravitational constant, and $\Lambda$ the cosmological constant, the speed of light is set to unity, and $T_{\mu\nu}$ is the energy-momentum tensor of an ideal cosmological fluid given by \begin{equation} \label{4} T^{\mu\nu}=\mbox{\footnotesize diag}\{\rho_m,p_m,\dots,p_m\}, \end{equation} with $\rho_m$ and $p_m$ the $t$-dependent matter energy density and pressure, we arrive at the Friedmann equations \begin{align} H^2=&\frac{16\pi G}{n(n-1)}\rho-\frac k{a^2},\label{5}\\ \dot{H}=&-\frac{8\pi G}{n-1}(\rho+p)+\frac k{a^2},\label{6} \end{align} in which $\rho,p$ are the effective energy density and pressure related to $\rho_m,p_m$ through \begin{equation} \label{8} \rho=\rho_m+\frac{\Lambda}{8\pi G},\quad p=p_m-\frac{\Lambda}{8\pi G}. \end{equation} On the other hand, recall that, with (\ref{1}) and (\ref{4}) and (\ref{8}), the energy-conservation law, $\nabla_\nu T^{\mu\nu}=0$, takes the form \begin{equation} \label{9} \dot{\rho}_m+n(\rho_m+p_m)H=0. \end{equation}
Analyze exact solutions of the Friedmann equations obtained in the previous problem in the case of flat ($k=0$) $n$-dimensional Universe filled with a barotropic liquid with the state equation \begin{equation} \label{10} p_m=w \rho_m. \end{equation} and obtain corresponding explicit expressions for the deceleration parameter.
Inserting (\ref{10}) into (\ref{9}), we have \begin{equation}\label{11} \dot{\rho}_m+n(1+w)\rho_m \frac{\dot{a}}a=0, \end{equation} which can be integrated to yield \begin{equation}\label{12} \rho_m=\rho_0 a^{-n(1+w)}, \end{equation} where $\rho_0>0$ is an integration constant. Using (\ref{12}) in (\ref{8}), we arrive at the relation \begin{equation}\label{13} \rho=\rho_0 a^{-n(1+w)}+\frac{\Lambda}{8\pi G}. \end{equation} From (\ref{5}) and (\ref{13}), we get the following equation of motion for the scale factor $a$: \begin{equation} \label{14} \dot{a}^2=\frac{16\pi G\rho_0}{n(n-1)}a^{-n(1+w)+2}+\frac{2\Lambda}{n(n-1)} a^2-k. \end{equation} To integrate (\ref{14}), we recall Chebyshev's theorem: For rational numbers $p,q,r$ ($r\neq0$) and nonzero real numbers $\alpha,\beta$, the integral \[\int x^p(\alpha+\beta x^r)^q\,dx\] is elementary if and only if at least one of the quantities \begin{equation}\label{cd} \frac{p+1}r,\quad q,\quad \frac{p+1}r+q, \end{equation} is an integer. Another way to see the validity of the Chebyshev theorem is to represent the integral of concern by a hypergeometric function such that when a quantity in (\ref{cd}) is an integer the hypergeometric function is reduced into an elementary function. Consequently, when $k=0$ or $\Lambda=0$, and $w$ is rational, the Chebyshev theorem enables us to know that, for exactly what values of $n$ and $w$, the equation (\ref{14}) may be integrated. For the spatially flat situation $k=0$ we rewrite equation (\ref{14}) as \begin{equation} \label{15} \dot{a}=\pm\sqrt{c_0 a^{-n(1+w)+2}+\Lambda_0 a^2},\quad c_0=\frac{16\pi G\rho_0}{n(n-1)},\quad\Lambda_0=\frac{2\Lambda}{n(n-1)}. \end{equation} Then (\ref{15}) reads \begin{equation} \label{15a} \pm\int a^{-1}\left(c_0 a^{-n(1+w)}+\Lambda_0 \right)^{-\frac12}da=t+C. \end{equation} It is clear that the integral on the left-hand side of (\ref{15a}) satisfies the integrability condition stated in the Chebyshev theorem for any $n$ and any rational $w$. It turns out that (\ref{15}) might be integrated for any real $w$ as well, not necessarily rational. To do so we apply $a>0$ and get from (\ref{15}) the equation \begin{equation} \label{16} \frac{d}{dt}\ln a=\pm\sqrt{c_0 a^{-n(1+w)}+\Lambda_0}, \end{equation} or equivalently, \begin{equation}\label{17} \dot{u}=\pm\sqrt{c_0 e^{-n(1+w)u}+\Lambda_0},\quad u=\ln a. \end{equation} Set \begin{equation}\label{18} \sqrt{c_0 e^{-n(1+w)u}+\Lambda_0}=v. \end{equation} Then \begin{equation}\label{19} u=\frac{\ln c_0}{n(1+w)}-\frac 1{n(1+w)} \ln(v^2-\Lambda_0). \end{equation} Inserting (\ref{19}) into (\ref{17}), we find \begin{equation}\label{20} \dot{v}=\mp\frac12 n(1+w)(v^2-\Lambda_0), \end{equation} whose integration gives rise to the expressions \begin{equation} \label{21} v(t)=\left\{\begin{array}{rl} &v_0\left(1\pm \frac12 n(1+w)v_0t\right)^{-1},\quad \Lambda_0=0;\\ &\\ & \sqrt{\Lambda_0}(1+C_0 e^{\mp n(1+w)\sqrt{\Lambda_0} t})(1-C_0 e^{\mp n(1+w)\sqrt{\Lambda_0} t})^{-1},\\& C_0=(v_0-\sqrt{\Lambda_0})(v_0+\sqrt{\Lambda_0})^{-1},\quad \Lambda_0>0;\\ &\\ &\sqrt{-\Lambda_0}\tan\left(\mp\frac12 n(1+w)\sqrt{-\Lambda_0} t +\arctan\frac{v_0}{\sqrt{-\Lambda_0}}\right),\quad \Lambda_0<0, \end{array} \right. \end{equation} where $v_0=v(0)$. Hence, in terms of $v$, we obtain the time-dependence of the scale factor $a$: \begin{equation} \label{22} a^{n(1+w)}(t)=\frac{8\pi G \rho_0}{\frac12 n(n-1)v^2(t)-\Lambda}. \end{equation} We now assume \(w>-1\) in the equation of state in our subsequent discussion. We are interested in solutions satisfying \(a(0)=0.\) When $\Lambda=0$, we combine (\ref{21}) and (\ref{22}) to get \begin{equation} \label{x1} a^{n(1+w)}(t)=4\pi G\rho_0\left(\frac n{n-1}\right)(1+w)^2 t^2. \end{equation} When $\Lambda>0$, we similarly obtain \begin{equation}\label{x2} a^{n(1+w)}(t)=\frac{8\pi G\rho_0}{\Lambda}\sinh^2\left(\sqrt{\frac{n\Lambda}{2(n-1)}}(1+w) t\right). \end{equation} When $\Lambda<0$ we rewrite (\ref{22}) as \begin{equation} \label{23} a^{n(1+w)}(t)=\frac{8\pi G \rho_0}{(-\Lambda)}\cos^2\left(\sqrt{\frac{n(-\Lambda)}{2(n-1)} }(1+w)t \mp\arctan\sqrt{\frac{n(n-1)}{-2\Lambda}}\, v_0\right). \end{equation} If we require $a(0)=0$, then (\ref{23}) leads to the conclusion \begin{equation} \label{24} a^{n(1+w)}(t)=\frac{8\pi G \rho_0}{(-\Lambda)}\sin^2\sqrt{\frac{n(-\Lambda)}{2(n-1)} }(1+w)t, \end{equation} which gives rise to a periodic Universe so that the scale factor $a$ reaches its maximum $a_m$, \begin{equation} \label{25} a^{n(1+w)}_m=\frac{8\pi G \rho_0}{(-\Lambda)}, \end{equation} at the times \begin{equation}\label{26} t=t_{m,k}=\left(\frac\pi2+k\pi\right)\frac1{(1+w)}\sqrt{\frac{2(n-1)}{n(-\Lambda)}},\quad k\in\Bbb Z, \end{equation} and shrinks to zero at the times \begin{equation}\label{27} t=t_{0,k}=\frac{k\pi}{(1+w)}\sqrt{\frac{2(n-1)}{n(-\Lambda)}},\quad k\in\Bbb Z. \end{equation} Using the equations (\ref{x1},\ref{x2},\ref{24}), one easily obtains the expressions for the DP \begin{equation} q(t)=\left\{\begin{array}{cc} \frac n2 (1+w)-1, & \Lambda_0=0;\\ &\\ \frac{n(1+w)}{2\cosh^2\left(\sqrt{\frac{n\Lambda}{2(n-1)}}(1+w)t\right)}-1, & \Lambda_0>0;\\ &\\ \frac{n(1+w)}{2\cos^2\left(\sqrt{\frac{n\Lambda}{2(n-1)}}(1+w)t\right)}-1, & \Lambda_0<0. \end{array} \right. \end{equation}
Deceleration as a cosmographic parameter
Make transition from the derivatives w.r.t. cosmological time to that w.r.t. conformal time in definitions of the Hubble parameter and the deceleration parameter.
\[H=\frac{a'}{a^2},\quad q=-\left(\frac{aa''}{a'^2}-1\right),\] where prime denotes derivative w.r.t. the conformal time.
Make Taylor expansion of the scale factor in time using the cosmographic parameters.
\begin{align} \nonumber a(t)=&a_0\left[1+H_0(t-t_0)-\frac12q_0H_0^2(t-t_0)^2+ \frac1{3!}j_0H_0^3(t-t_0)^3\right.\\ \label{background_4_3} &\left.+\frac1{4!}s_0H_0^4(t-t_0)^4+\frac1{5!}l_0H_0^5(t-t_0)^5 + O\left((t-t_0)^6\right)\right]. \end{align}
Make Taylor expansion of the redshift in time using the cosmographic parameters.
\begin{align} \nonumber 1+z=&\left[1+H_0(t-t_0)-\frac12q_0H_0^2(t-t_0)^2+ \frac1{3!}j_0H_0^3(t-t_0)^3\right.\\ \nonumber &\left.+\frac1{4!}s_0H_0^4(t-t_0)^4+\frac1{5!}l_0H_0^5(t-t_0)^5 + O\left((t-t_0)^6\right)\right]^{-1}.\\ \nonumber z=&H_0(t-t_0)+\left(1+\frac{q_0}2\right)H_0^2(t-t_0)^2+\cdots. \end{align}
Show that \[q(t)=\frac{d}{dt}\left(\frac1H\right)-1.\]
\[q(t)=\frac{d}{dt}\left(\frac1H\right)-1=-\frac{\dot H}{H^2}-1=-\frac{\frac{\ddot a}{a}-H^2}{H^2}-1=q.\]
Show that the deceleration parameter as a function of the red shift satisfies the following relations \begin{align} \nonumber q(z)=&\frac{1+z}{H}\frac{d H}{dz}-1;\\ \nonumber q(z)=&\frac12(1+z)\frac1{H^2}\frac{d H^2}{dz}-1;\\ \nonumber q(z)=&\frac12\frac{d\ln H^2}{d\ln(1+z)}-1;\\ \nonumber q(z)=&\frac{d\ln H}{dz}(1+z)-1. \end{align}
Show that the deceleration parameter as a function of the scale factor satisfies the following relations \begin{align} \nonumber q(a)=&-\left(1+\frac{a\frac{dH}{da}}{H}\right);\\ \nonumber q(a)=&\frac{d\ln(aH)}{d\ln a}. \end{align}
Show that the derivatives $dH/dz$, $d^2H/dz^2$, $d^3H/dz^3$ and $d^4H/dz^4$ can be expressed through the deceleration parameter $q$ and other cosmological parameters.
\begin{align} \frac{dH}{dz}=&\frac{1+q}{1+z}H;\\ \nonumber \frac{d^2H}{dz^2}=&\frac{j-q^2}{(1+z)^2}H;\\ \nonumber \frac{d^3H}{dz^3}=&\frac{H}{(1+z)^3}\left(3q^2+3q^3-4qj-3j-s\right);\\ \nonumber \frac{d^4H}{dz^4}=&\frac{H}{(1+z)^4}\left(-12q^2-24q^3-15q^4+32qj+25q^2j+7qs+12j-4j^2+8s+l\right). \end{align}
Use results of the previous problem to make Taylor expansion of the Hubble parameter in redshift.
\[H(z)=H_0+\left.\frac{dH}{dz}\right|_{z=0}z +\frac12\left.\frac{d^2H}{dz^2}\right|_{z=0}z^2 +\frac16\left.\frac{d^3H}{dz^3}\right|_{z=0}z^3+\dots=\] \[=H_0(1+(1+q_0)z+\frac12(j_0-q_0^2)z^2+\frac16(3q_0^2+3q_0^3-4q_0j_0-3j_0-s)z^3).\] There is a decomposition for the inverse Hubble parameter \begin{align} \nonumber \frac{d}{dz}\left(\frac1H\right)=&-\frac1{H^2}\frac{dH}{dz}=-\frac{1+q}{1+z}\frac1H;\\ \nonumber \frac{d^2}{dz^2}\left(\frac1H\right)=& 2\left(\frac{1+q}{1+z}\right)^2\frac1H =\left(\frac{2+4q+3q^2-j}{(1+z)^2}\right)\frac1H ;\\ \nonumber \frac1{H(z)}=& \frac1{H_0}\left[1-(1+q_0)z+\frac{2+4q_0+3q_0^2-j_0}{6}z^2+\dots\right]. \end{align}
Express derivatives of the Hubble parameter squared w.r.t. the redshift $d^iH^2/dz^i$, $i=1,2,3,4$ in terms of the cosmographic parameters.
\begin{align} \nonumber \frac{dH^2}{dz}=&\frac{2H^2}{1+z}(1+q),\\ \nonumber \frac{d^2H^2}{dz^2}=&\frac{2H^2}{(1+z)^2}(1+2q+j),\\ \nonumber \frac{d^3H^2}{dz^3}=&\frac{2H^2}{(1+z)^3}(-qj-s),\\ \nonumber \frac{d^4H^2}{dz^4}=&\frac{2H^2}{(1+z)^4}(4qj+3qs+3q^2j-j^2+4s+l). \end{align}
Express the current values of deceleration and jerk parameters in terms of $N=-\ln(1+z)$.
\begin{align} \nonumber q_0=& \left.-\frac1{H^2}\left\{\frac12\frac{d\left(H^2\right)}{dN}+H^2\right\}\right|_{N=0},\\ \nonumber j_0=&\left.\left\{\frac1{2H^2}\frac{d^2\left(H^2\right)}{dN^2} +\frac3{2H^2}\frac{d\left(H^2\right)}{dN}+1\right\}\right|_{N=0}. \end{align}
Express time derivatives of the Hubble parameter in terms of the cosmographic parameters.
\begin{align} \dot H=& -H^2(1+q),\\ \nonumber \ddot H=& H^3(j+3q+2),\\ \nonumber \dddot H=& H^4[s-4j-3q(q+4)-6],\\ \nonumber \ddddot H=& H^5[l-5s+10(q+2)j+30(q+2)q+24]. \end{align}
Express the deceleration parameter as power series in redshift $z$ or $y$-redshift $z/(1+z)$.
\begin{align} \nonumber q(z)=&q_0 + \left(-q_0-2q_0^2+j_0\right)z+\frac12\left(2q_0+8q_0^2+8q_0^3-7q_0j_0-4j_0-s_0\right)z^2+O(z^3),\\ \nonumber q(y)=&q_0 +\left(-q_0-2q_0^2+j_0\right)y+\frac12\left(4q_0+8q_0^3-7q_0j_0-2j_0-s_0\right)y^2+O(y^3). \end{align}
Show that the Hubble parameter is connected to the deceleration parameter by the integral relation \[H=H_0\exp\left[\int\limits_0^z[q(z')+1]d\ln (1+z')\right].\]
Show that derivatives of the lower order parameters can be expressed through the higher ones, for instance \[\frac{dq}{d\ln(1+z)}=j-q(2q+1).\]
Differentiate the relation \[q(z)=\frac{1+z}{H}\frac{d H}{dz}-1\] to find \[\frac{dq}{dz}=\frac1H\frac{dH}{dz}-\frac{1+z}{H^2}\left(\frac{dH}{dz}\right)^2+\frac{1+z}{H}\frac{d^2H}{dz^2}.\] Using \[\frac{dH}{dz}=\frac{1+q}{1+z}H,\quad \frac{d^2H}{dz^2}=\frac{j-q^2}{(1+z)^2}H,\] one obtains \[\frac{dq}{d\ln(1+z)}=j-q(2q+1).\]
Cosmological scalars and the Friedmann equation
Dunajski and Gibbons [M. Dunajski, Gary Gibbons, Cosmic Jerk, Snap and Beyond, arXiv:0807.0207] proposed an original way to test the General Relativity and the cosmological models based on it. The procedure implies expressing the Friedmann equation in terms of directly measurable cosmological scalars constructed out of higher derivatives of the scale factor, i.e. cosmographic parameters $H,q,j,s,l$. In other words, the key idea is to treat the Friedmann equations as one algebraic constraint between the scalars. This links the measurement of the cosmological parameters to a test of General Relativity, or any of its modifications.
Express the curvature parameter $k$ terms of the cosmographic parameters for the case of Universe filled with non-interacting cosmological constant and non-relativistic matter.
Using the relation $\rho_m=M/a^3$ ($M=const$) we rewrite the first Friedmann equation in the form \begin{equation}\label{1301_38_e1} \dot a^2+k=\frac13\frac M a+\frac13\Lambda a,\quad 8\pi G=1.\end{equation} Then we differentiate the latter equation two times to find \begin{align} \label{1301_38_e2} \ddot a=&-\frac16\frac M{a^2}=\frac13\Lambda a,\\ \nonumber \dddot a=&-\frac13\frac{M\dot a}{a^3}=\frac13\Lambda\dot a. \end{align} Using the definitions of the cosmographic parameters \[H=\frac{\dot a}{a},\quad q=-a\frac{\ddot a}{\dot a^2},\quad j=a^2\frac{\dddot a}{\dot a^3},\] we represent (\ref{1301_38_e1}) in the form \begin{align} \nonumber q &= \frac12A-B;\\ \nonumber j &= A+B;\\ \nonumber A &\equiv\frac13 \frac{M}{a^3H^2};\\ \nonumber B &\equiv\frac13 \frac{\Lambda}{H^2}. \end{align} Then we find \begin{align} \nonumber A &=\frac23(j+q);\\ \nonumber B &=\frac23(\frac12j-q). \end{align} The Friedmann equation (\ref{1301_38_e1}) in terms of the above introduced variables $A$ and $B$ takes on the form \[\frac k{a^2}=(A+B-1)H^2\] or \[k=a^2H^2(j-1).\]
Do the same as in the previous problem for the case of Universe filled with non-interacting non-relativistic matter $\rho_m=M_m/a^3$ and radiation $\rho_r=M_r/a^4$.
We represent the first Friedmann equation in the form \begin{equation}\label{1301_39_e1} \frac{\dot a^2}{a^2}+\frac k{a^2}=\frac{M_m}{a^3}+\frac{M_r}{a^4},\quad \frac{8\pi G}{3}=1.\end{equation} We twice differentiate the latter equation to find \begin{align} \label{1301_39_e2} \ddot a=&-\frac12\frac{M_m}{a^2}-\frac{M_r}{a^3},\\ \nonumber \dddot a=&\frac{M_m}{a^3}\dot a+3\frac{M_r}{a^4}\dot a. \end{align} Using definitions of the cosmographic parameters $q$ and $j$, one obtains \begin{align} \nonumber q &= \frac12A+B;\\ \nonumber j &= A+3B;\\ \nonumber A &\equiv\frac{M_m}{a^3H^2};\\ \nonumber B &\equiv\frac{M_r}{a^4H^2}. \end{align} Then we find \begin{align} \nonumber A &=-2j+6q;\\ \nonumber B &=j-2q. \end{align} The Friedmann equation (\ref{1301_39_e1}) in terms of the above introduced variables $A$ and $B$ takes on the form \[\frac k{a^2}=(A+B-1)H^2\] or in terms of the cosmographic parameters \[k=a^2H^2(4q-j-1).\]
Check the expressions for the curvature $k$ obtained in the previous problem for two cases: a) a flat Universe solely filled with non-relativistic matter; b) a flat Universe solely filled with radiation.
In the first case $B=0$, $q=1/2$. Then $A=1$ and $k=0$. Note that in this case $j=1$. In the second case $A=0$, $q=1$. Then $B=1$, $k=0$. In that case $j=3$.
Find relation between the cosmographic parameters free of any cosmological parameter for the case of Universe considered in the problem #1301_38.
Using expression for $\dddot a$ obtained in the problem \ref{1301_38}, one finds \[\ddddot a=\frac M3\frac{\ddot a}{a^3}-M\frac{\dot a^2}{a^4}+\frac\Lambda3\ddot a.\] For the snap parameter \[s\equiv a^3\frac{\ddddot a}{\dot a^4}\] one obtains \[s=-3(A+B)q-3A.\] Substitution of the parameters \begin{align} \nonumber A &=\frac23(j+q);\\ \nonumber B &=\frac23(\frac12j-q). \end{align} introduced in the problem \ref{1301_38}, one finally finds \[s+2(q+j)+qj=0.\] This fourth order ODE is equivalent to the Friedmann equation and has an advantage that it appears as a constraint on directly measurable quantities.
Perform the same procedure for the Chaplygin gas with the equation of state $p=-A/\rho$.
For small values of $a(t)$ density and pressure of the Chaplygin gas reduces to that of dust $\rho\propto a^{-3}$, and for large $a$ one gets the de Sitter Universe: $\rho=const$, $p=-\rho$. In between these two regimes one can use the approximation \begin{equation}\label{scalar_5}\rho=\sqrt A+\frac{B}{\sqrt{2A}}a^{-6}.\end{equation} Thus $\sqrt A$ plays the role of a cosmological constant. We insert this to the Friedmann equation with $\Lambda$ and follow the procedure of eliminating the constants by differentiation. This leads to an approximate constraint \begin{equation}\label{scalar_6}s+5(q+j)+qj=0.\end{equation}
Perform the same procedure for the generalized Chaplygin gas with the equation of state $\rho=-A/\rho^\alpha$.
\begin{equation}\label{scalar_7}s+(3\alpha+2)(q+j)+qj=0.\end{equation} Note that for $\alpha=1$ we reproduce the above obtained result for the Chaplygin gas, while $\alpha=0$ we return to the results obtained in the problem (\ref{1301_41}). If we want to exclude the parameter $\alpha$ from the latter equation we must take one more derivative of the Friedmann equation and introduce an additional cosmological parameter \[l=\frac1a\frac{d^5a}{dt^5}\left(\frac1a\frac{da}{dt}\right)^{-5}.\] As a result one obtains \begin{equation}\label{scalar_8} -2qs-2jq^2-lq-2sj-3sq^2-j^2q-lj+s^2-3q^2j-qsj+j^3-2j^2q^2=0. \end{equation} This constraint is again approximate and is valid only in the regime where the higher order terms in the expansion of $\rho$ can be dropped.
Averaging deceleration parameter
Since the deceleration parameter $q$ is a slowly varying quantity (e.g. $q = 1/2$ for matter-dominated case and $q = -1$ in the Universe dominated by dark energy in form of cosmological constant), then the useful information is contained in its time average value, which is very interesting to obtain without integration of the equations of motions for the scale factor. Let us see how it is possible [J.Lima, Age of the Universe, Average Deceleration Parameter and Possible Implications for the End of Cosmology, arXiv:0708.3414]. For that purpose let us define average value $\bar q$ of this parameter on time interval $[0,t_0]$ with the expression \[\bar q(t_0)=\frac1{t_0}\int\limits_0^{t_0}q(t)dt.\]
Express current value of the average deceleration parameter in terms of the Hubble parameter.
Making use of the definition of the deceleration parameter \[q(t)=-\frac{\ddot a a}{\dot a^2}=\frac{d}{dt}\left(\frac1H\right)-1,\] it is easy to obtain \[\bar q(t_0)=-1+\frac1{t_0H_0}.\]
Show that current age of the Universe depends solely on average value of the deceleration parameter.
Using results of the previous problem one finds \begin{equation}\label{background_5_3}t_0=\frac{H_0^{-1}}{1+\bar q}.\end{equation} Naturally current age of the Universe is proportional to $H_0^{-1}$, but the proportionality coefficient is solely determined by the average value of the deceleration parameter. It is worth noting that this purely kinematic result depends on curvature of the Universe, nor on number of components filling it, nor on the type of gravity theory used.
Show that the Hubble time $H_0)^{-1}$ represents a characteristic time scale for age of the Universe at any stage of the evolution.
As the average value of the deceleration parameter $\bar q$ is of order of unity then from (see result of the previous problem) \[t_0=\frac{H_0^{-1}}{1+\bar q}\] it immediately follows that the Hubble time $H_0)^{-1}$ represents a characteristic time scale at any stage of the evolution of the Universe.
Energy conditions in terms of the deceleration parameter
Dynamic model-independent constraints on the kinematics of the Universe can further be obtained from the so-called energy conditions. These conditions, based on quite general physical principles, impose restrictions on the components of the energy-momentum tensor $T_{\mu\nu}$. In choosing a model for the medium (a model, but not the equation of state!), these conditions can be transformed into inequalities restricting the possible values of pressure and density of the medium. In terms of density and pressure the energy conditions take on the form \begin{align} \nonumber NEC&\Rightarrow & \rho+p\ge&0, & &\\ \nonumber WEC&\Rightarrow & \rho\ge&0,&\rho+p\ge&0,\\ \nonumber SEC&\Rightarrow & \rho+3p\ge&0,&\rho+p\ge&0,\\ \nonumber DEC&\Rightarrow & \rho\ge&0,&-\rho\le p\le&\rho. \end{align} Here, NEC, WEC, SEC, and DEC correspond to the zero, weak, strong, and dominant energy conditions. Because these conditions do not require any definite equation of state for the substance filling the Universe, they impose very simple and model-independent constraints on the behavior of the energy density and pressure. Hence, the energy conditions provide one of the possibilities for explaining the evolution of the Universe on the basis of quite general principles.
Express the energy conditions in terms of the scale factor and its derivatives.
\begin{align} \nonumber NEC&\Rightarrow & -\frac{\ddot a}{a}+\frac{\dot a^2}{a^2}+\frac{k}{a^2}\ge&0,\\ \nonumber WEC&\Rightarrow & \frac{\dot a^2}{a^2}+\frac{k}{a^2}\ge&0,\\ \nonumber SEC&\Rightarrow & \frac{\ddot a}{a}\le&0,\\ \nonumber DEC&\Rightarrow & \frac{\ddot a}{a}+2\left[\frac{\dot a^2}{a^2}+\frac{k}{a^2}\right]\ge&0. \end{align}
Transform the energy conditions to constraints on the deceleration parameter for the flat Universe.
\begin{align} \label{background_6_5} NEC&\Rightarrow q\ge1,\\ \nonumber SEC&\Rightarrow q\ge0,\\ \nonumber DEC&\Rightarrow q\le2. \end{align} There is no WEC among above conditions because it is always satisfied for arbitrary real $a(t)$.
Analyze the constraints on the regimes of accelerated and decelerated expansion of the Universe following from the energy conditions.
The conditions (\ref{background_6_5}) considered separately in principle allow a possibility for both decelerated ($q>0$) and accelerated ($q<0$) expansion of the Universe. The constraints by NEC in (\ref{background_6_5}) have clear sense: as follows from the second Friedmann equation, the inequality $\rho+3p\le0$ gives necessary condition for accelerated expansion of the Universe, i.e. the accelerated expansion of the Universe is possible only in presence of components with high negative pressure $p<-\rho/3$. The SEC excludes existence of such components. As a result, $q\ge0$ in this case. At the same time, NEC and DEC are compatible with the condition $p<-\rho/3$ and therefore they allow the regimes with $q<0$. It worth noting that even before the discovery of the accelerated expansion of the Universe in 1997, Visser [M. Visser, Science 276, 88 (1997), M. Visser, Phys. Rev. D 56, 7578(1997)] already concluded, basing on analysis of the energy conditions, that current observations suggest that SEC was violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of "normal" matter is capable of fitting the observational data.
Distance-Deceleration Parameter Relations
In cosmology there are many different and equally natural definitions of the notion of distance between two objects or events. In particular, the luminosity distance $d_L$ of an object at redshift $z$ is $d_L=(L/2\pi F)^{1/2}$, where $L$ is the bolometric luminosity for a given object and $F$ is the bolometric energy flux received from that object. The expression for the luminosity distance in a FLRW Universe is \begin{equation}\label{distance_7_1} d_L(z)=(1+z)\left\{ \begin{array}{lr} R\sinh\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & open\\ H_0^{-1}\int\limits_0^z\frac{dz'}{E(z')}, & flat\\ R\sin\left[\frac1{H_0R}\int\limits_0^z\frac{dz'}{E(z')}\right], & closed \end{array} \right.\end{equation} (Here and below we set $c=1$). Here $R$ is the (comoving) radius of curvature of the open or closed Universe, $E=H/H_0$.
Represent the luminosity distance in the flat Universe in terms of the deceleration parameter.
\[d_L(z)=(1+z)\int\limits_0^z\frac{dz'}{H(z')}=(1+z)H_0\int\limits_0^z du\exp\left\{-\int\limits_0^u [1+q(v)]d[\ln(1+v)]\right\}.\]
Find expression for the luminosity distance up to terms of order of $z^2$.
\begin{equation}\label{distance_7_3}d_L=\frac{z}{H_0}\left[1+\left(\frac{1-q_0}{2}\right)z+O(z^2)\right],\end{equation} where in the spatially flat case \[q_0=\frac12\sum\limits_i\Omega_{i0}(1+3w_i).\]
Find expression for the luminosity distance in the next order $O(z^3)$ in the redshift.
\begin{align} \nonumber d_L(z)= & \frac{cz}{H_0}\left[ 1+\frac12(1-q_0)z-\frac16(1-q_0-3q_0^2+j_0)z^2\right.\\ +&\left.\frac1{24}(2-2q_0-15q_0^2-15q_0^3+5j_0+10j_0q_0+s_0)z^3+O(z^4)\right]. \end{align} As was expected, the latter decomposition contains the cosmographic parameters defined in terms of the higher order time derivatives of the scale factor ($j\propto d^3a/dt^3$).
Calculate the luminosity distance in the Universe filled with non-relativistic matter (Einstein-de Sitter model ($k=0$)).
\begin{equation}\label{distance_7_5}d_L=a_0r_1(1+z),\quad r_1=\int\limits_{t_1}^{t_0}\frac{dt}{a(t)}.\end{equation} In the considered case \[r_1=\frac{3t_0}{a_0}\left[1-\left(\frac{t_1}{t_0}\right)^{1/3}\right],\quad 1+z=\frac{a_0}{a_1}=\left(\frac{t_0}{t_1}\right)^{2/3},\] thus \[r_1=\frac{3t_0}{a_0}\left[1-\left(1+z\right)^{-1/2}\right]=\frac{2}{a_0H_0} \left[1-\left(1+z\right)^{-1/2}\right].\] We took into account here that in the Einstein-de Sitter model $h_0t_0=2/3$. For the luminosity distance one ultimately obtains \[d_L=a_0r_1(1+z)=\frac2{H_0}\left(1+z-\sqrt{1+z}\right).\]
The expression (\ref{distance_7_1}) allows to find the luminosity distance given the function $H(z)$. Solve the inverse problem for the flat case: find the Hubble parameter as a function of the luminosity distance [S. Nesseris and J. Garcia-Bellido, Comparative analysis of model-independent methods for exploring the nature of dark energy, arXiv:1306.4885].
In the flat case, differentiating the expression \[d_L(z)=(1+z)H_0^{-1}\int\limits_0^z\frac{dz'}{E(z')},\] one obtains \[\frac{d(d_l)}{dz}=\frac{1+z}{H(z)}+\frac{d_L(z)}{1+z}\Rightarrow H(z)=\frac{(1+z)^2}{d'_L(1+z)-d_L},\] where prime denotes derivative with respect to redshift $z$.
Use result of the previous problem to express deceleration parameter a s a function of luminosity distance and its derivatives.
Using result of the previous problem, one finds \[q(z)=-1\frac{\dot H}{H^2}=-1+(1+z)\frac{H'}{H}=1-\frac{(1+z)^2d''_L(z)}{d'_L(1+z)-d_L},\] where prime denotes derivative with respect to redshift $z$. The latter formula can be represented in the form \[q(N)=-1-\frac{H'(N)}{H(N)}=1+\frac{d''_L(N)+d'_L(N)}{d'_L(N)+d_L(N)}.\] Here primes denote derivatives with respect to $N\equiv\ln a=-\ln(1+z)$. For arbitrary geometry \[q(z)=\frac{1+\omega_K d_L(z)d'_L(z)/(1+z)}{1+\omega_K d_L^2(z)/(1+z)^2}-\frac{(1+z)^2d''_L(z)}{d'_L(z)(1+z)-d_L}.\]
The Hubble sphere in general does not coincide with the horizons, except when it becomes degenerate with the particle horizon at $q=1$ and with event horizon at $q=-1$. Show that.
The component with $q=1$ represents the ultra-relativistic matter (radiation) for which $a\propto t^{1/2}$ and therefore $H=1/(2t)$. In this case the particle horizon is finite and equals to \[L_p=a(t)\int\limits_0^t\frac{dt'}{a(t')}=2t=\frac1{H(t)}=R_H.\] The component with $q=-1$ represents the cosmological constant for which $a\propto e^{Ht}$, $H=const$. In this case the particle horizon is absent because the origin of the Universe is moved to $t=-\infty$. The event horizon has the following size \[L_e=a(t)\int\limits_t^\infty\frac{dt'}{a(t')}=\frac1H=R_H.\]
Show that the Hubble sphere contracts when $q<-1$, remains stationary when $q=-1$ and expands when $q>-1$.
\[\frac{d}{dt}(R_H)=c\frac{d}{dt}\left(\frac1H\right)= -\frac{c}{H^2}\left(\frac{\ddot a}{a}-\frac{\dot a^2}{a^2}\right)=c(1+q).\] As one can see, the Hubble sphere contracts when $q<-1$, remains stationary when $q=-1$ and expands when $q>-1$.
Analyze correspondence between kinematics of the Hubble sphere and boundaries of the observable Universe for different expansion regimes.
In the Universe with decelerated expansion ($q>0$) the Hubble's sphere has velocity exceeding the light speed by the quantity $cq$ and thus it overtakes the galaxies situated on its surface. Therefore the galaxies initially situated outside the Hubble's sphere will initially enter inside. Galaxies at distance $R>R_H$ are later $R<R_H$, and their superluminal recession in the course of time becomes subluminal. The light emitted toward the observer by a galaxy outside the Hubble sphere recedes until it enters inside the sphere. Therefore it starts approaching us and becomes available to observations. Therefore, all decelerating Universes lack event horizons unless they terminate at some future time. In uniformly expanding Universes $q=0$ the Hubble surface and the galaxies situated on it have equal velocities. Thus number of galaxies in the observable Universe remains constant. Then both particle horizon and event horizon are absent in such Universes. In case of the accelerating expansion $q<0$ the Hubble sphere has velocity which is less than the light speed by the quantity $q$ and thus it falls behind the galaxies, and therefore number of them decreases inside the Hubble sphere. All accelerating Universes have the property that galaxies at distance $R<R_H$ are later $R>R_H$, and their subluminal recession in the course of time becomes superluminal. Light emitted outside the Hubble sphere recedes from the observer and can never approach the observer. There are events that can never be observed, and such Universe have event horizons.
Calculate the derivatives $dL_p/dt$ and $dL_e/dt$.
Using the definitions of the particle horizon \[L_p=a(t)\int\limits_0^t\frac{dt'}{a(t')}\] and the event horizon \[L_e=a(t)\int\limits_t^\infty\frac{dt'}{a(t')},\] one finds \begin{align} \nonumber \frac{dL_p}{dt}=&\frac{d}{dt}\left[a(t)\int\limits_0^t\frac{dt'}{a(t')}\right]=L_p(z)H(z)+1;\\ \label{distance_11_3} \frac{dL_e}{dt}=&\frac{d}{dt}\left[a(t)\int\limits_t^\infty\frac{dt'}{a(t')}\right]=L_e(z)H(z)-1. \end{align}
Calculate the derivatives $d^2L_p/dt^2$ and $d^2L_e/dt^2$.
Differentiating the relations obtained in the previous problem w.r.t. time, one obtains \begin{align} \nonumber \frac{d^2L_p}{dt^2}=&H(1-qHL_p);\\ \label{distance_11_5} \frac{dL_e}{dt}=&-H(1+qHL_e). \end{align}
Find the expressions for the current particle horizon in the single-component Universe filled with non-relativistic matter.
Current value of the proper distance $L_p$ is \[L_p=a_0\int\limits_0^{t_0}\frac{dt'}{a(t')}=a_0\eta|_0^{\eta_0}=a_0\eta_0,\] where $\eta$ is the conformal time. Using the expressions for $a_0$ and $\eta_0$ obtained in problems \ref{1301_38} and \ref{1301_39}, one finds \[L_p=a_0\int\limits_0^{r_0}\frac{dr}{\sqrt{1-kr^2}} =a_0\int\limits_0^{t_0}\frac{dt}{a(t)}=\frac1{H_0} \left\{\begin{array}{lcr} 2, & k=0, & q_0=1/2;\\ \frac{\arcsin{\frac{\sqrt{2q_0-1}}{q_0}}}{\sqrt{2q_0-1}}, & k=1, & q_0>1/2;\\ \frac{\mathrm{arcsinh}{\frac{\sqrt{1-2q_0}}{q_0}}}{\sqrt{1-2q_0}}, & k=-1, & q_0<1/2. \end{array}\right.\]
Power-Law Universes
Assume that the scale factor $a(t)$ varies as $t^n$, where $t$ is the age of Universe and $n=const$. Show that in decelerated expanding Universe $n<1$.
In these power-law Universes we have \[H\equiv\frac{\dot a}{a}=\frac n t, \quad q\equiv-\frac{\ddot a}{aH^2}=\frac{1-n}t.\] Hence $n<1$ in a decelerating ($q>0$) Universe.
Show that if the scale factor $a(t)$ varies as $t^n$, \[L_p=\frac{R_h}{q}.\]
In the considered case \[L_p=a(t)\int\limits_0^t\frac{dt'}{a(t')}=\frac{t}{1-n}=\frac t n \frac1q=\frac{R_h}{q}\]
How, in a Universe of age $t$ can causally connected distances of $L\gg ct$ exist?
We cite below an extremely bright discussion of this question in [E.Harrison, Hubble's spheres and particle horizons, The Astrophisical Journal, 383, 63, 1991] " In the study of causal connections, the Hubble sphere bounded by the Hubble surface is as important as the observable universe bounded by the particle horizon. Let two comoving bodies be separated by a distance $L$ sufficiently small that each lies in the observable Universe of the other. Each body remains thereafter permanently in the other's observable Universe, and the ratio $L/(ct)$ during expansion depands on the behavior of the Hubble sphere. In a decelerating Universe the Hubble sphere expands faster than Universe, and a body at distance $L$ either is inside or will soon be inside the Hubble sphere. Hence, any two bodies must eventually recede from each other at subluminal velocity, and the ratio $L/(ct)$ will then decrease in time. In an accelerating Universe the Hubble sphere expands slower then universe, and a body at distance $L$ either is outside or will soon be outside the Hubble sphere. Hence, any two bodies must eventually recede from each other at superluminal velocity and the ratio $L/(ct)$ will then increase in time. How can causally connected distances of $L\gg ct$ exist? The answeris that the universe passes through a period of accelerated expansion, and causal connections of $L<ct$, established before acceleration, expand superluminally outside the Hubble sphere... A period of accelerated expansion distend all previously established causal connections and increases the distance to the particle horizon."
The Effects of a Local Expansion of the Universe
Considering the radial motion of a test particle in a spatially-flat expanding Universe find the Newtonian limit the radial force $F$ per unit mass at a distance $R$ from a point mass $m$.
In order to describe the cosmological expansion one commonly uses two sets of coordinates: the physical (or Euler) coordinates ($R,\theta,\varphi$) and comoving (or fixed, Lagrangian) coordinates ($r,\theta,\varphi$). (The angular coordinates are the same for both sets as the cosmological expansion is assumed to be radial.) The two sets are related by the formula $R(t)=a(t)r$. Therefore a point which is fixed w.r.t. cosmological expansion, i.e. with constant coordinates ($r,\theta,\varphi$), has additional radial acceleration \begin{equation}\nonumber \left.\frac{d^2R}{dt^2}\right|_{expansion}=R\frac{\ddot a}{a}=-qH^2R.\end{equation} Consequently, in the Newtonian limit the radial force $F$ per unit mass at a distance $R$ from a point mass $m$ is given by \begin{equation}\nonumber F=-\frac{m}{R^2}-q(t)H^2(t)R.\end{equation} The force consists of the usual $1/R^2$ inwards component due to the central (point) mass $m$ and a cosmological component proportional to $R$ that is directed outwards (inwards) when the expansion of the Universe is accelerating (decelerating).
Find the Newtonian limit of the radial force $F$ per unit mass at a distance $R$ from a point mass $m$ in a Universe which contains no matter (or radiation), but only dark energy in the form of a non-zero cosmological constant $\Lambda$.
In this case, the Hubble parameter and, hence, the deceleration parameter become time-independent and are given by $H=\sqrt{\Lambda/3}$ and $q=-1$. Thus, the force, obtained in the previous problem, also becomes time-independent, \begin{equation}\nonumber F=-\frac{m}{R^2}+\frac13\Lambda R.\end{equation}
Solve the previous problem for the case finite (i.e. non-pointlike) spherically-symmetric massive objects.
\begin{equation}\nonumber F=-\frac{M(R)}{R^2}+\frac13\Lambda R.\end{equation} where $M(R)$ is the total mass of the object contained within the radius $R$. If the object has the radial density $\rho(R)$ then \[M(R)=\int\limits_0^R4\pi R'^2\rho(R')dR'.\]
Show that a non-zero $\Lambda$ should set a maximum size, dependent on mass, for galaxies and clusters.
Although the de Sitter background is not an accurate representation of our universe, the SCM is dominated by dark-energy in a form consistent with a simple cosmological constant. Even in the simple Newtonian case (previous problems), we see immediately that there is an obvious, but profound, difference between the cases $\Lambda=0$ and $\Lambda\ne0$. In the former, the force on a constituent particle of a galaxy or cluster (say) is attractive for all values of $R$ and tends gradually to zero as $R\to\infty$ (for any sensible radial density profile). In the latter case, however, the force on a constituent particle (or equivalently its radial acceleration) vanishes at the finite radius $R_F$ which satisfies $R_F=[3M(R_F)/\Lambda]^{1/3}$, beyond which the net force becomes repulsive. This suggests that a non-zero $\Lambda$ should set a maximum size, dependent on mass, for galaxies and clusters. In the Newtonian limit, the speed of a particle in a circular orbit of radius $r$ is given by \[V(R)=\sqrt{\frac{M(R)}{R}-\Lambda R^2},\quad \left(F=-\frac{M(R)}{R^2}+\frac13\Lambda R\right).\] from which it is clear that no circular orbit can exist beyond the radius $R_F$.
The radius $R=R_F$ (see previous problem) does not necessarily correspond to the maximum possible size of the galaxy or cluster: many of the gravitationally-bound particles inside $R_F$ may be in unstable circular orbits. Therefore the so-called "outer" radius becomes of great importance, which one may interpret as the maximum size of the object, that is corresponding to the largest stable circular orbit $R_S$. Find in the Newtonian limit radius of the largest stable circular orbit $R_S$ in the case of gravity field created by a point mass $m$.
The radius $R_S$ may be determined as the minimum of the (time-dependent) effective potential for a test particle in orbit about the central mass [R. Nandra, A. N. Lasenby and M. P. Hobson, The effect of an expanding universe on massive objects, arXiv:1104.4458] . Remaining within the frames of the Newtonian approximation (a weak gravitational field and low velocities), the equation of motion for the test particle is \begin{equation}\nonumber \ddot R\approx-\frac m{R^2}-q(t)H^2(t)R+\frac{L^2}{R^3}.\end{equation} Here $L$ is angular momentum per unit mass. This is simply the Newtonian radial force expression with the inclusion of a centrifugal term. The test particle moves in the effective one-dimensional potential $V(R)$ \begin{equation}\label{1301_69_e3} V(R)=+\frac{L^2}{2R^2}-\frac m{R}+\frac12q(t)H^2(t)R^2.\end{equation} Extrema of the effective potential in which the test particle moves occur at the $R$-values for which $d^2R/dt^2=0$, namely the solutions of \begin{equation}\nonumber -\frac m{R^2}-q(t)H^2(t)R+\frac{L^2}{R^3}=0.\end{equation} Consider the function \begin{equation}\nonumber y = -mR-q(t)H^2(t)R^4+L^2.\end{equation} The polynomial has single extremum --- a minimum in the point \[R^*=R_S=\left(-\frac m{4q(t)H^2(t)}\right)^{1/3}\] Condition for existence of real roots of the equation (\ref{1301_69_e3}) reads $y(R^*)\le0$. Critical value of the angular momentum for which the minimum of the effective potential disappears can be found from the condition $y(R^*)=0$ and it equals \[L_{crit}=\frac{\sqrt3}{2}m^{1/2}\left(-\frac m{4q(t)H^2(t)}\right)^{1/6}= \left(-\frac{27m^4}{256q(t)H^2(t)}\right)^{1/6}\]
If we consider $R_S(t)$ (see previous problem) as the maximum possible size of a massive object at cosmic time $t$, and assume that the object is spherically-symmetric and has constant density, then it follows that there exists a time-dependent minimum density (due to the maximum size). Determine the corresponding density.
\[\rho_{min}(t)=\frac{3m}{4\pi R_S^3(t)}=-\frac{3q(t)H^2(t)}{\pi}.\] The latter relation is valid only for $q(t)<0$ (accelerating expansion). For $q(t)>0$, $\rho_{min}(t)=0$.
Show that minimum relative density $\Omega_{min}\equiv\rho_{min}/\rho_{crit}$ is determined only by the deceleration parameter.
Using result of the previous problem one obtains \[\Omega_{min}\equiv\frac{\rho_{min}}{\rho_{crit}}=-8q(t).\]
An important characteristics -- the minimum fractional density contrast $\delta_{min}(t)\equiv[\rho_{min}(t)-\rho_m(t)]/\rho_m(t)$ is immediately related to the deceleration parameter. Estimate this quantity in the SCM.
Since $\rho_m(t)=\Omega_m(t)\rho_{crit}(t)$, then \[\delta_{min}(t)=-\left[1+\frac{8q(t)}{\Omega_m(t)}\right].\] For the present time ($\Omega_{m0}\approx0.3$, $q_0\approx-0.55$) one finds that $\delta_{min0}\approx14$.
Retrieved from "http://universeinproblems.com/index.php?title=Category:Deceleration_Parameter&oldid=2131" | CommonCrawl |
Speaker: Ivan Izmestiev (Freie Universität Berlin)
Title: Variational properties of the discrete Hilbert-Einstein functional
Abstract: The discrete Hilbert-Einstein functional (also known as Regge action) for a 3-manifold glued from euclidean simplices is the sum of edge lengths multiplied with angular defects at the edges. There is an analog for hyperbolic cone-manifolds; a discrete total mean curvature term appears if the manifold has a non-empty boundary. Variational properties of this functional are similar to those of its smooth counterpart. In particular, critical points correspond to vanishing angular defects, i.e. to metrics of constant curvature. We give a survey on isometric embeddings and rigidity results that can be obtained by studying the second derivative of the discrete Hilbert-Einstein and speak about possible future developments.
Speaker: James Dibble (Rutgers University)
Title: Totally geodesic maps into manifolds with no focal points
Abstract: A classical result of Eells-Sampson is that every homotopy class of maps between compact Riemannian manifolds, where the target has non-positive sectional curvature, contains an energy-minimizing harmonic representative. They proved this by inventing the harmonic map heat flow, the first geometric flow defined on manifolds. Their work was refined by Hartman, who proved the monotonicity of certain distance functions under the flow and used this to deduce that that the space of harmonic maps in each homotopy class is path-connected and that energy is constant on it. Applying an identity that dates to the work of Bochner, Eells-Sampson also proved that, when the domain has non-negative Ricci curvature, all harmonic maps are totally geodesic.
It will be shown that, for domains with non-negative Ricci curvature, the results of Eells-Sampson, along with certain qualitative consequences of Hartman's results, generalize to energy-minimizing maps into manifolds with no focal points. These are manifolds whose universal covers satisfy a simple synthetic condition: For each point and each maximal geodesic, there is a unique geodesic connecting them that intersects the latter perpendicularly. By contrast with previous approaches, the proof uses neither a geometric flow nor the Bochner identity for harmonic maps.
Colloquium at 4:30pm
Speaker: Jonathan Williams (University of Georgia)
Title: A new approach to general smooth 4-manifolds
Abstract: Some consider smooth 4-manifolds to be a mature field, which typically means its approachable yet nontrivial problems have become scarce. This is mainly due to a lack of tools. In this talk I will present a new way to depict any smooth, closed oriented 4-manifold that opens the doors to two of the most successful tools from 3-manifolds: pseudoholomorphic curves and discrete groups.
Speaker: Eric Swartz (Western Australia)
Title: Generalized quadrangles with symmetry
Abstract: A generalized quadrangle is a point-line incidence geometry Q such that (1) any two points lie on at most one line, and (2) given a line l and a point P not incident with l, P is collinear with a unique point of l. Generalized quadrangles are a specific type of generalized polygon, which were first introduced by Tit s in 1959 as geometries associated to classical groups. It is natural, then, to ask the question: if one starts with the abstract definition of a generalized quadrangle, which ones are highly symmetric? I will discuss the background of this question, leading to the following recent work:
An antiflag of a generalized quadrangle is a non-incident point-line pair (P, l), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations (automorphisms that send points to points and lines to lines) is transitive on the set of all antiflags. We prove that if a finite, thick generalized quadrangle Q is antiflag-transitive, then Q is one of the following: the unique generalized quadrangle of order (3,5), a classical generalized quadrangle, or a dual of one of these.
This is joint work with John Bamberg and Cai-Heng Li, and this talk will assume no prior knowledge of finite geometry.
Speaker: Niels Martin Moeller (Princeton University)
Title: Gluing of Geometric PDEs - Obstructions vs. Constructions for Minimal Surfaces & Mean Curvature Flow Solitons
Abstract: For geometric nonlinear PDEs, where no easy superposition principle holds, examples of (global, geometrically/topologically interesting) solutions can be hard to come about. In certain situations, for example for 2-surfaces satisfying an equation of mean curvature type, one can generally "fuse" two or more such surfaces satisfying the PDE, as long as certain global obstructions are respected - at the cost (or benefit) of increasing the genus significantly. The key to success in such a gluing procedure is to understand the obstructions from a more local perspective, and to allow sufficiently large geometric deformations to take place. In the talk I will introduce some of the basic ideas and techniques (and pictures) in the gluing of minimal 2-surfaces in a 3-manifold. Then I will explain two recent applications, one to the study of solitons with genus in the singularity theory for mean curvature flow (rigorous construction of Ilmanen's conjectured "planosphere" self-shrinkers), and another to the non-compactness of moduli spaces of finite total curvature minimal surfaces (a problem posed by Ros & Hoffman-Meeks). Some of this work is joint w/ Steve Kleene and/or Nicos Kapouleas.
Speaker: Ross Geoghegan (Binghamton University)
Title: A theorem about extensions of groups
Title:The $\Sigma$-invariants of Thompson's group $F$, via Morse theory
Abstract: In a paper published in 2010, Bieri, Geoghegan and Kochloukova computed the $\Sigma$-invariants (also called Bieri-Neumann-Strebel-Renz invariants) of Thompson's group $F$. In recent joint work with Stefan Witzel, we recomputed these using the action of $F$ on a certain CAT(0) cube complex called the Stein-Farley complex. The main tool is a version of discrete Morse theory. I will explain what all of these words mean over the course of the talk, and it should be accessible to non-experts.
Title: The $\Sigma$-invariants of the generalized Thompson's groups $F_n$
Abstract: Building off the first talk, I will shift from $F$ to a family of groups $F_n$, of which $F$ is $F_2$. Using the action of $F_n$ on a CAT(0) cube complex, I was recently able to compute all the $\Sigma$-invariants of all the $F_n$. In this talk I will focus on the aspects of the $\Sigma$-invariants that only come up when $n>2$, and will highlight a new technique, building off work of Belk and Forrest, for proving higher connectivity properties of certain complexes. This talk will still be accessible to non-experts, though it will help to have gone to the first talk.
Speaker: Jim Belk (Bard College)
Title: Rearrangement Groups of Fractals
Abstract: The definition of Thompson's group $F$ depends crucially on the self-similar structure of the unit interval. In this talk, I will describe a family of Thompson-like groups that act on a variety of self-similar structures. Each of these groups has an associated CAT(0) cubical complex, analogous to the Farley complexes for $F$, $T$, and $V$. By analyzing descending links on these complexes, I will show that some of these groups have type $F_\infty$. This is joint work with Bradley Forrest.
April 9 no seminar.
Speaker: Stefan Witzel (Bielefeld University)
Title: Arithmetic groups, finiteness properties, and homology
Abstract: The group $SL_2(F_p[t,t^{-1}])$ is finitely generated but not finitely presented. In fact, it has a finite-index subgroup $G$ with $H_2(G,F_p)$ infinite (this and more was shown by Stuhler). I will talk about results of the same kind for related groups.
April 22 at 3:30pm in WH 309
Speaker: Ralf Spatzier (University of Michigan)
Title: Higher Rank Rigidity and Positive Curvature
Abstract: I will review rigidity and non-rigidity results about "higher rank" in Riemannian geometry. Specifically we consider "higher rank" spaces in which subobjects of extremal curvature are plentiful. I will emphasize recent joint work with Schmidt and Shankar on Riemannian manifolds of higher spherical rank where every geodesic c has a perpendicular parallel field making sectional curvature 1 with c, and the sectional curvature is bounded below by 1.
Hilton Memorial Lecture at 3pm in Science II, Room 140.
Title: Higher Rank in Geometry and Dynamics - How isometric and hyperbolic behavior force rigidity
Abstract: Higher rank phenomena have led to surprising rigidity results in group theory, geometry and dynamics. Examples start with Margulis superrigidity theorem for lattices in higher rank semisimple Lie groups, followed by the classification of nonpositively curved Riemannian manifolds with lots of flats. In recent years similar phenomena have been found in dynamics, in particular in the classification of hyperbolic actions on tori and nilmanifolds of higher rank Abelian groups and their measure rigidity.
Dean's Lecture in Geometry and Topology.
Speaker: Karsten Grove (Notre Dame University)
Title: Symmetry and Positive Curvature
Abstract: Although constituting a vast extension of ancient Spherical Geometry, the beautiful class of positively curved (Riemannian) spaces is like the "Tip of the Iceberg" among all (Riemannian) spaces. Accordingly, non-symmetric positively curved spaces are known only in a few sporadic dimensions, and yet only a few obstructions to their existence are known.
In this talk, we will describe the current state of affair of the subject including tools and methods, with emphasis on the impact symmetries have had on the development during the last few decades.
Speaker: Boris Kalinin (Pennsylvania State University)
Title: Smooth rigidity and classification for hyperbolic systems and actions
Abstract: Hyperbolic actions of $\mathbb{Z}^k$ and $\mathbb{R}^k$ extend the classical notion of Anosov diffeomorphisms and flows, which are hyperbolic actions of $\mathbb{Z}$ and $\mathbb{R}$. In contrast to the rank one case, higher rank hyperbolic actions exhibit various rigidity properties. I will focus on the problem of smooth classification, that is finding a smooth conjugacy to an algebraic model. I will give an overview of this area and compare it with the rank one case, where the natural problem is either topological classification or smooth classification under extra assumptions.
seminars/topsem/spring2015.txt · Last modified: 2015/05/27 10:19 by qiao | CommonCrawl |
> gr-qc > arXiv:2206.00882v1
gr-qc
astro-ph.IM
General Relativity and Quantum Cosmology
Title: Assessing the impact of non-Gaussian noise on convolutional neural networks that search for continuous gravitational waves
Authors: Takahiro S. Yamamoto, Andrew L. Miller, Magdalena Sieniawska, Takahiro Tanaka
(Submitted on 2 Jun 2022 (this version), latest version 22 Jun 2022 (v2))
Abstract: We present a convolutional neural network that is capable of searching for continuous gravitational waves, quasi-monochromatic, persistent signals arising from asymmetrically rotating neutron stars, in $\sim 1$ year of simulated data that is plagued by non-stationary, narrow-band disturbances, i.e., lines. Our network has learned to classify the input strain data into four categories: (1) only Gaussian noise, (2) an astrophysical signal injected into Gaussian noise, (3) a line embedded in Gaussian noise, and (4) an astrophysical signal contaminated by both Gaussian noise and line noise. In our algorithm, different frequencies are treated independently; therefore, our network is robust against sets of evenly-spaced lines, i.e., combs, and we only need to consider perfectly sinusoidal line in this work. We find that our neural network can distinguish between astrophysical signals and lines with high accuracy. In a frequency band without line noise, the sensitivity depth of our network is about $\mathcal{D}^{95\%} \simeq 43.9$ with a false alarm probability of $\sim 0.5\%$, while in the presence of line noise, we can maintain a false alarm probability of $\sim 10\%$ and achieve $\mathcal{D}^\mathrm{95\%} \simeq 3.62$ when the line noise amplitude is $h_0^\mathrm{line}/\sqrt{S_\mathrm{n}(f_k)} = 1.0$. We evaluate the computational cost of our method to be $O(10^{19})$ floating point operations, and compare it to those from standard all-sky searches, putting aside differences between covered parameter spaces. Our results show that our method is more efficient by one or two orders of magnitude than standard searches. Although our neural network takes about $O(10^8)$ sec to employ using our current facilities (a single GPU of GTX1080Ti), we expect that it can be reduced to an acceptable level by utilizing a larger number of improved GPUs.
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE); Instrumentation and Methods for Astrophysics (astro-ph.IM)
Cite as: arXiv:2206.00882 [gr-qc]
(or arXiv:2206.00882v1 [gr-qc] for this version)
From: Takahiro Yamamoto S. [view email]
[v1] Thu, 2 Jun 2022 05:54:28 GMT (383kb,D)
[v2] Wed, 22 Jun 2022 03:43:02 GMT (383kb,D) | CommonCrawl |
Long first exons and epigenetic marks distinguish conserved pachytene piRNA clusters from other mammalian genes
Tianxiong Yu ORCID: orcid.org/0000-0003-1151-46241,2 na1,
Kaili Fan ORCID: orcid.org/0000-0002-8723-79021,2 na1,
Deniz M. Özata ORCID: orcid.org/0000-0001-5215-86843,
Gen Zhang4,
Yu Fu ORCID: orcid.org/0000-0003-1244-94732,5 nAff6,
William E. Theurkauf4,
Phillip D. Zamore ORCID: orcid.org/0000-0002-4505-96183 &
Zhiping Weng ORCID: orcid.org/0000-0002-3032-79661,2
Nature Communications volume 12, Article number: 73 (2021) Cite this article
Spermatogenesis
In the male germ cells of placental mammals, 26–30-nt-long PIWI-interacting RNAs (piRNAs) emerge when spermatocytes enter the pachytene phase of meiosis. In mice, pachytene piRNAs derive from ~100 discrete autosomal loci that produce canonical RNA polymerase II transcripts. These piRNA clusters bear 5′ caps and 3′ poly(A) tails, and often contain introns that are removed before nuclear export and processing into piRNAs. What marks pachytene piRNA clusters to produce piRNAs, and what confines their expression to the germline? We report that an unusually long first exon (≥ 10 kb) or a long, unspliced transcript correlates with germline-specific transcription and piRNA production. Our integrative analysis of transcriptome, piRNA, and epigenome datasets across multiple species reveals that a long first exon is an evolutionarily conserved feature of pachytene piRNA clusters. Furthermore, a highly methylated promoter, often containing a low or intermediate level of CG dinucleotides, correlates with germline expression and somatic silencing of pachytene piRNA clusters. Pachytene piRNA precursor transcripts bind THOC1 and THOC2, THO complex subunits known to promote transcriptional elongation and mRNA nuclear export. Together, these features may explain why the major sources of pachytene piRNA clusters specifically generate these unique small RNAs in the male germline of placental mammals.
In animal germ cells, 21–35 nt long PIWI-interacting RNAs (piRNAs) guide PIWI proteins to repress transposons and regulate gene expression1. Eutherian mammals produce three distinct types of germline piRNAs during spermatogenesis. In mice, piRNAs first appear in the fetal testis; these piRNAs initiate transposon silencing via DNA methylation2,3,4,5. A second set of piRNAs, pre-pachytene piRNAs, emerges in the neonatal testis2. The function of pre-pachytene piRNAs, which typically derive from mRNAs, remains unknown. Finally, when spermatocytes enter the pachytene phase of meiosis, male mice produce a third type of piRNAs: pachytene piRNAs6,7,8,9. Unique to placental mammals, the 26–30 nt long pachytene piRNAs derive from ~100 discrete autosomal loci that contain fewer transposon sequences than the genome as a whole7,10,11,12. Pachytene piRNAs are both highly diverse—a single spermatocyte produces hundreds of thousands of different piRNA species—and highly abundant, reaching a peak intracellular concentration (7.2 µM) that rivals the abundance of ribosomes13. Yet the biological functions and target RNAs of pachytene piRNAs remain to be defined. Some studies suggest that pachytene piRNAs silence mRNAs during spermiogenesis14,15,16,17,18, while others support the view that they have no sequence-specific function19.
The genomic regions that produce piRNAs, referred to as piRNA clusters or piRNA source loci, resemble canonical mRNA- and long noncoding RNA- (lncRNA) producing genes: they are transcribed by RNA polymerase II (RNA pol II); bear 5′ caps and 3′ poly(A) tails; their transcription start sites (TSS) are marked with histone H3 trimethylated on lysine 4 (H3K4me3); and their transcripts often contain introns that are removed before nuclear export and processing into piRNAs11. Mouse pachytene piRNAs derive from 100 well-defined loci. An additional 84 annotated loci generate pre-pachytene piRNAs, while 30 "hybrid" loci produce piRNAs with characteristics from both classes11. Human adult and juvenile testes produce piRNAs from at least 83 pre-pachytene, 10 hybrid, and 89 pachytene loci12. In both mice and humans, the pachytene piRNA clusters and the genes encoding the proteins required for piRNA production are coordinately activated by the transcription factor A-MYB, which is first expressed as spermatogonia enter meiosis I11,12. A-MYB also activates transcription of other genes required for meiosis, but not implicated in piRNA biogenesis or function20.
Like many genes transcribed in the germline during spermatogenesis, the promoters of pachytene piRNA clusters contain fewer CG dinucleotides than protein-coding genes expressed in the soma, and most of the CGs are cytosine methylated21, a DNA modification normally associated with silencing rather than active transcription. These atypical promoters tend to be inactive in the soma. High levels of 5-hydroxymethylcytosine (5hmC) and active histone marks, such as H3K4me3 and histone H3 acetylated on lysine 9 (H3K9ac) or lysine 27 (H3K27ac) have been proposed to facilitate expression of pachytene piRNA clusters in mouse and human testis21. In mice, roughly half of pachytene piRNA clusters bind the transcription elongation factor BTBD18, and Btbd18 mutant male mice are sterile and show decreased transcription and piRNA production from these loci22.
Among the 100 well-annotated mouse pachytene piRNA clusters11, 41 are divergently transcribed from a central promoter. These include 15 pairs that produce piRNAs from both arms, 7 that produce piRNAs from one arm and an mRNA from the other, and 4 that produce piRNAs from one arm and a lncRNA from the other. Similarly, among the 89 well-annotated human pachytene piRNA clusters12, 46 are divergently transcribed from a central promoter, including 18 pairs that produce piRNA precursors from both arms, 7 that produce an mRNA from the other arm, and 3 that generate a lncRNA from the other arm. What genic or epigenetic features direct the transcripts of pachytene piRNA clusters into the piRNA pathway, what determines which arm of these divergently transcribed loci makes piRNAs, and what prevents the expression of pachytene piRNAs outside the testis remain unknown.
Here, we report that an unusually long (≥10 kb) first exon or long unspliced transcript correlates with germline-specific production of piRNA precursor transcripts from mouse pachytene piRNA clusters. We further report that pachytene piRNA precursors from long-first-exon or long unspliced transcripts are preferentially bound by THOC1 and THOC2, subunits of the THO complex, which is required for transcription elongation and nuclear export of mRNAs23,24,25. In Drosophila, the THO complex is essential for piRNA biogenesis, and tho mutants are female sterile26,27,28. Our integrative analysis of transcriptome, piRNA, and epigenome datasets across multiple species reveals that a long first exon is an evolutionarily conserved feature of pachytene piRNA clusters. Finally, comparison of testis germ cells with a variety of somatic tissues and cell types suggests that a highly methylated promoter, often containing a low or intermediate level of CG dinucleotides, correlates with germline expression and somatic silencing of pachytene piRNA clusters.
Pachytene piRNA clusters are expressed in the testis but silent in somatic tissues
In placental mammals, mature piRNAs are primarily detected in gonads6,7,9,29,30. Our analysis of transcriptome data31 shows that the primary precursor transcripts of pachytene piRNAs, but not pre-pachytene or hybrid piRNAs, are also restricted to male gonads. Pachytene piRNA precursor transcripts were abundant in the adult mouse testis, but not in adult brain, colon, liver, skeletal muscle, heart, kidney, lung, or spleen. In contrast, the precursor transcripts of pre-pachytene and hybrid piRNAs, which often also function as mRNAs, were found in all tissues examined (Fig. 1a). Like human fetal ovaries30, mouse fetal ovaries (embryonic day 11.5–20.5) behave like somatic tissues: most pachytene piRNA clusters are not expressed, whereas hybrid and pre-pachytene piRNA clusters are actively transcribed (Fig. 1a). Among the 100 mouse pachytene piRNA clusters, transcripts from 69 were detected solely in the testis—in the other tissues we examined, these transcripts were present at <0.1 RPKM (reads per kilobase of transcript per million mapped reads). The steady-state transcript abundance of another 21 pachytene piRNA clusters was ≥4-fold greater in the testis than in any of the eight somatic tissues analyzed. Of the remaining ten pachytene piRNA clusters, eight—two encoding proteins and six producing lncRNAs—were more abundant in the testis than in other tissues, but by <4-fold, while two additional loci were more abundant in the soma than in the germline. Thus, 90% of annotated mouse pachytene piRNA clusters are testis specific. In contrast, just 13% of hybrid (4/30) piRNA clusters and only one of the 84 well-defined pre-pachytene piRNA clusters are testis specific.
Fig. 1: Mouse pachytene piRNA clusters show testis-specific expression and epigenetic signals.
a Each column represents a piRNA gene, grouped by type (pachytene, hybrid, and pre-pachytene) and sorted by tissue-specificity score ("Methods") within each group. Each piRNA gene is further annotated by its function as an mRNA (black/grey), low, intermediate, or high-CG level at the promoter (low-CG, intermediate-CG, or high-CG; in blue, green, and yellow), length of first exon or length of intronless gene (≥10 or <10 kb; in red and pink), piRNA abundance (RPM) in adult testis, and transcript expression level (RPKM) in the testis, eight somatic tissues, and fetal ovary. piRNA abundance and transcript expression levels are log2 and log10 transformed, respectively, with a pseudo-count of 1 and share the same color scale. b Each panel indicates the levels of RNA pol II binding or a histone mark; tissues are color coded. Pachytene piRNA clusters exhibit high levels of activating histone marks (H3K4me3, H3K4me2, H3K4me1, H3K27ac, and H3K36me3), low levels of the repressive histone mark H3K27me3, and high levels of RNA pol II binding specifically in adult testis. ChIP-seq signals are shown in a ±2 kb window around the TSS in 10 bp bins. H3K4me2 ChIP-seq data were publicly available only for testis, liver, and brain. We performed two-sided Wilcoxon signed-rank tests to compare epigenetic signals at pachytene piRNA clusters in the testis versus somatic tissues. Pol II, p < 2.2 × 10−16; H3K4me1, 2.3 × 10−16 < p < 6.9 × 10−16; H3K4me2, 1.0 × 10−11 < p < 3.0 × 10−10; H3K4me3, p < 2.2 × 10−16; H3K27ac, p < 2.2 × 10−16; H3K27me3, 2.3 × 10−11 < p < 7.9 × 10−6; H3K36me3, 1.7 × 10−16 < p < 2.4 × 10−16.
On average, piRNAs from the 90 testis-specific pachytene piRNA clusters were 4.2-fold more abundant than from the ten pachytene piRNA clusters with broader expression (Wilcoxon rank-sum test p value = 0.036). Consistent with their testis-specific expression, pachytene piRNA clusters showed higher RNA pol II occupancy in the testis than in somatic tissues. Active histone marks—the promoter mark H3K4me3; the enhancer marks H3K4me1, H3K4me2, and H3K27ac; and the transcriptional elongation mark H3K36me3—were also substantially higher for these 90 loci in the testis than the soma (Wilcoxon signed-rank test p value ≤ 3.0 × 10−10; Fig. 1b). The signal profiles at individual piRNA clusters (Supplementary Fig. 1 shows H3K27ac) reveal that the enrichment of these histone marks is across most piRNA clusters. We note that several mutually exclusive histone marks are enriched at piRNA clusters, such as H3K4me1, H3K4me2, and H3K4me3; acetylation and butylation; and 5mC and 5hmC (also see below). We presume that the marks are likely to exist in different subsets of cells. Conversely, pachytene piRNA clusters had 1.3–2.0-fold lower levels of the repressive histone mark H3K27me3 in the testis than in somatic tissues (Wilcoxon signed-rank test p value ≤ 7.9 × 10−6; Fig. 1b).
Consistent with their expression profiles (Fig. 1a), three hybrid pachytene piRNA clusters showed testis-specific profiles of histone marks, while the remaining hybrid pachytene piRNA clusters had high levels of active histone marks in multiple somatic tissues (Supplementary Fig. 2a). As expected from their broad germline and somatic expression, pre-pachytene piRNA clusters on average showed similar profiles of histone marks in testis and somatic tissues (Supplementary Fig. 2b).
For comparison with pachytene piRNA clusters, we identified a set of 1,171 testis-specific, protein-coding genes with a steady-state transcript abundance ≥10 RPKM in the testis and ≥4-fold greater expression in testis than any of the eight reference somatic tissues. Like pachytene piRNA clusters, testis-specific protein-coding genes had substantially greater RNA pol II binding, higher levels of active histone marks, and lower levels of the repressive histone mark H3K27me3 in adult testis than in somatic tissues (Wilcoxon signed-rank test p value < 2.2 × 10−16; Supplementary Fig. 2c).
In mice and other placental mammals, a single protein, A-MYB, activates transcription of pachytene piRNA clusters and a large fraction of genes encoding piRNA pathway proteins11,12. Mice have 17 such annotated piRNA pathway genes, and the steady-state mRNA abundance was higher than 10 RPKM in the testis for all 17 genes; furthermore, 15 of the 17 were expressed at levels ≥4-fold greater in testis than in somatic tissues (Supplementary Fig. 3a). Of the remaining two genes, Ddx39 (DExD-box helicase 39 A) was expressed in all tissues, while Mov10l1 (Mov10-like RISC complex RNA helicase 1) was also expressed in the heart, where it was originally discovered32. Consistent with their high expression in testis, the 15 genes had substantially higher levels of active histone marks and RNA pol II binding, and significantly lower levels of the repressive histone mark H3K27me3 in adult testis compared with the panel of somatic tissues (Supplementary Fig. 3b). Most piRNA pathway genes are expressed in mouse fetal ovaries (Supplementary Fig. 3a), consistent with the activities of the piRNA pathway in the female gonads in mouse and human3,29,30,33,34,35.
Sixty nine of the 100 annotated pachytene piRNA clusters, but only one of the 17 piRNA pathway genes, were exclusively expressed in the testis (<0.1 RPKM in all somatic tissues examined; Chi-square test p value = 3.9 × 10−5). Thus, most pachytene piRNA clusters are expressed exclusively or specifically in the mouse testis, and generally show greater tissue specificity than genes encoding piRNA pathway proteins.
Long first exon and low CG correlate with piRNA production
Pachytene piRNA clusters display two unusual features: long first exons (long is defined as ≥10 kb and short as <10 kb throughout) and low level of CG dinucleotides (defined in "Methods") at their promoters (Fig. 1a), features known to repress somatic expression of protein-coding genes36,37,38. We asked whether these two features had a similar relationship to germline expression of pachytene piRNA clusters. Because most precursor transcripts of piRNA clusters are processed into mature piRNAs, we used piRNA abundance as a surrogate for the expression levels of piRNA clusters. Our data show that for pachytene piRNA clusters, long first exons and low-CG promoters positively correlate with piRNA abundance (Fig. 2a).
Fig. 2: Low-CG promoter, long-first-exon or long intronless gene, and high histone acylation correlate with the production of pachytene piRNAs.
a Bar plot reports the Spearman correlation coefficient between piRNA abundance and first exon length and promoter O/E CG, H3K4me3, H3K27me3, H3K4me2, H3K27ac, H4K5ac, H4K5bu, H4K8ac, H4K8bu, pan-lysine acetylation (Kac), pan-lysine crotonylation (Kcro), chromatin accessibility measured by ATAC-seq, DNA methylation, and RNA pol II binding at the promoters (TSS ± 2 kb) of pachytene piRNA clusters in pachytene spermatocytes. Significant correlations (Benjamin-adjusted p values < 0.05; n = 100) are marked in black, while nonsignificant correlations are marked in grey. b Histograms of first exon length of intronless pachytene piRNA clusters and the first exons of intron-containing pachytene piRNA clusters, pre-pachytene piRNA clusters, hybrid piRNA clusters, all protein-coding genes, and all lncRNAs. c The percentages of low-CG, intermediate-CG, and high-CG promoters. Eight groups are shown: pachytene piRNA clusters, pre-pachytene piRNA clusters, hybrid piRNA clusters, piRNA pathway genes, A-MYB-regulated protein-coding genes, A-MYB-regulated lncRNA genes, testis-specific protein-coding genes, and testis-specific lncRNA genes. d. Boxplots and meta-gene plots show histone modifications, ATAC, and RNA pol II levels at 100 pachytene piRNA clusters, 84 pre-pachytene piRNA clusters, 30 hybrid piRNA clusters, 17 piRNA pathway genes, 789 A-MYB-regulated protein-coding genes, 46 A-MYB-regulated lncRNAs, 1171 testis-specific protein-coding genes, and 164 testis-specific lncRNAs in pachytene spermatocytes. The x-axis for the box and the y-axis for the meta-gene plots report log2 ChIP signal or ATAC-seq read coverage relative to input, using a pseudo-count of 1. Asterisks indicate statistical significance (two-sided Wilcoxon rank-sum test p value < 0.001) for pairwise comparisons between 100 pachytene piRNA clusters and each of the other gene types. Red (blue) indicates that the epigenetic level is significantly higher (lower) in pachytene piRNA clusters. For boxplots, whiskers show 95% confidence intervals, boxes represent the first and third quartiles, and the vertical midline is the median.
Most protein-coding and lncRNA genes have short exons, including first exons (median length of annotated first exons = 229 nt); a long first exon delays splicing and hinders transcriptional elongation36,37. In contrast, half of mouse pachytene piRNA clusters are either long and intronless (n = 39; median gene length = 31,826 nt) or possess a long first exon (n = 8; median first exon length = 39,772 nt). Moreover, for pachytene piRNA clusters, first-exon length (or gene length for unspliced genes) correlates with piRNA abundance (ρ = 0.63 in pachytene spermatocytes; Fig. 2a; p value < 2.2 × 10−16). Among the 21,956 annotated protein-coding and 3496 lncRNA-producing genes in mice, all have short first exons (Fig. 2b) except for three protein-coding genes that have both long- and short-first-exon isoforms and one lncRNA gene with multiple long-first-exon isoforms. However, none of these long-first-exon isoforms is expressed in the testis. No hybrid and only three pre-pachytene piRNA clusters have long-first-exon isoforms (i.e., supported by RNA-seq reads; Chi-square test between pachytene and other piRNA clusters p value = 6.8 × 10−14). Furthermore, for two of these pre-pachytene piRNA clusters, the long-first-exon isoforms produced more piRNAs than the short-first-exon isoforms (piRNA density was 15.8 vs. 2.6 RPKM for pi-Ccrn4l.1 and 12.2 vs. 0.1 RPKM for pi-Phf20.1 in postnatal day 10.5 testes). The third pre-pachytene piRNA cluster, 7-qD2-40.1, only makes a long-first-exon isoform, and it makes abundant piRNAs (242.6 RPKM in postnatal day 10.5 testes). Together, our data indicate that a long first exon (or a long unspliced transcript) is a specific feature of pachytene piRNA clusters, which may distinguish them from protein-coding and lncRNA genes.
Paradoxically, high levels of promoter DNA methylation, which would typically repress the expression of protein-coding genes, do not impede expression of pachytene piRNA clusters in the testis: pachytene piRNA clusters with high piRNA abundance tend to have low promoter CG and high promoter methylation. More than 80% of pachytene piRNA clusters have promoters with low CG (defined as the ratio of the observed count over the expected count of CG dinucleotides, or O/E CG < 0.25, see "Methods"; 49/100 loci) or intermediate CG (0.25 < O/E CG < 0.5; 33/100 loci), whereas the promoters of most pre-pachytene (78/84) or hybrid (26/30) piRNA clusters are high CG (O/E CG > 0.5; Fig. 2c; Chi-square test p value < 2.2 × 10−16 for high-CG vs. low-CG and intermediate-CG combined). In somatic cells, the promoter O/E CG of a protein-coding gene is anticorrelated with its methylation level and correlated with its expression level: low-CG promoters are typically methylated and repressed, while high-CG promoters are unmethylated and expressed39. Similarly, for protein-coding genes in the germline, promoter O/E CG correlates with expression level (Spearman correlation coefficient ρ = 0.68 in the forebrain and ρ = 0.54 in pachytene spermatocytes, p values < 2.2 × 10−16), while both promoter O/E CG and expression level anticorrelate with promoter methylation level (ρ = −0.70 and −0.56 in the forebrain; ρ = −0.83 and −0.49 in pachytene spermatocytes; all p values < 2.2 × 10−16). In pachytene spermatocytes, promoter O/E CG and methylation level of pachytene piRNA clusters are similarly anticorrelated (ρ = −0.81; p value < 2.2 × 10−16); however, expression anticorrelates with promoter O/E CG (ρ = −0.31; p value = 1.5 × 10−3) and correlates with promoter methylation level (ρ = 0.21; p value = 8.7 × 10−4; Fig. 2a). We note that the level of promoter methylation of most genes, including pachytene piRNA clusters, remains constant throughout mouse spermatogenesis21.
A broad domain of histone acylation decorates long-first-exon and long unspliced pachytene piRNA clusters
Despite their high levels of promoter DNA methylation, pachytene piRNA clusters are efficiently transcribed and generate abundant piRNAs in the testis. What additional features of the pachytene piRNA-producing loci allow them to be expressed in the testis, while other genes with highly methylated promoters are repressed? Among the various chromatin and RNA pol II signatures measured in mouse male germ cells, promoter lysine acylation—including acetylation, butyrylation, and crotonylation—correlated with the abundance of mature piRNAs (ρ = 0.49–0.66, p values ≤ 2.1 × 10−7; Fig. 2a). Acylation of histone lysine residues activate transcription by opening chromatin, and allowing transcription factors and the transcriptional machinery to access the DNA40,41,42,43,44,45. The promoters of pachytene piRNA clusters displayed higher acylation levels than other classes of piRNA clusters, genes encoding piRNA pathway proteins, A-MYB-bound protein-coding genes, or testis-specific protein-coding or lncRNA genes (Fig. 2d, left panels, median ratio = 1.43–3.04, Wilcoxon rank-sum p values < 2.2 × 10−16 with respect to each of the other gene sets). Moreover, many pachytene piRNA clusters were broadly marked with acylated histones, extending far into the gene body (Fig. 2d, right panels; we note that the apparent enrichment of signals upstream of pachytene piRNA cluster is due to the large number of bidirectionally transcribed piRNA clusters, and this apparent enrichment is not observed for the RIP signal described below that is specific to the transcribed strand). In contrast, other active genes showed a narrow peak of histone acylation at the transcriptional start site. These data suggest that such broad domains of acylated histones enable testis-specific transcription of pachytene piRNA clusters despite their high levels of promoter DNA methylation.
Supporting this view, only long unspliced or long-first-exon pachytene piRNA clusters displayed high levels of histone acylation across a broad domain. Among the 100 annotated mouse pachytene piRNA clusters, 47 make long (i.e., >10,000 nt) transcripts: 39 of the 52 unspliced genes have transcripts longer than 10,000 nt; 5 of the 45 intron-containing loci have long first exons; two genes produce both long unspliced and long-first-exon transcripts from the same locus; and a single locus, pi-1700016M24Rik.1, generates one long unspliced transcript and two short-first-exon transcripts. The histone acylation profiles of these 47 pachytene piRNA clusters differ markedly from the 53 that make short unspliced or short-first-exon transcripts. Long unspliced or long-first-exon pachytene piRNA clusters show a broad domain of histone acylation extending far downstream from the promoter, often continuing for more than 40,000 nucleotides. In contrast, histone acylation of short intronless or short-first-exon pachytene piRNA clusters is confined to the promoter region, a pattern typical for actively transcribed mRNA- and lncRNA-producing genes (Fig. 3a–d, and Supplementary Figs. 4 and 5). Unlike acylation, other histone marks (e.g., H3K4me3), as well as most regions of high chromatin accessibility (ATAC-seq), were confined to the promoter for pachytene piRNA-producing loci (Fig. 3c, d and Supplementary Fig. 5a, b). Notably, among the 47 long unspliced or long-first-exon pachytene piRNA clusters, the density of histone acylation across the first exon or the entire intronless gene, rather than the density across the promoter, best correlated with piRNA abundance (e.g., ρ = 0.78 and p value < 2.2 × 10−16 for lysine acetylation at any position of the histone tail; Supplementary Fig. 4c). Together, our data suggest that a broad domain of histone acylation across either the first exon or the entire body of intronless genes, explains the testis-specific transcription of nearly half the annotated pachytene piRNA clusters.
Fig. 3: Half of pachytene piRNA clusters have long first exons, or are long and intronless and display high histone acylation signals across their first exon or gene body.
a Heatmap reports the enrichment of the pan-lysine acetylation ChIP signal relative to input in pachytene spermatocytes for 100 pachytene piRNA clusters in the −10 kb to +80 kb window flanking the TSS. Black bar, first exon–intron boundary; blue bar, 3′ end of transcript. b A meta-gene plot shows average lysine acetylation signal in pachytene spermatocytes for long-first-exon (or long intronless) pachytene piRNA clusters, short-first-exon pachytene piRNA clusters, and other groups of genes in the −10 kb to +80 kb window flanking the TSS. c UCSC genome browser view of 4-qC5-17839.1, a long, intronless pachytene piRNA-producing gene with a low-CG promoter. All signal tracks are from pachytene spermatocytes except for BTBD18, THOC1, THOC2, and 5hmC, which are from adult testis. d UCSC genome browser view of 6-qF3-8009.1, an intron-containing, long first exon, piRNA-producing gene with a low-CG promoter. Vertical dashed line marks the end of the first exon. e Scatter plot compares the change in transcript and piRNA abundance for Btbd18 mutant compared to Btbd18 heterozygous testis. Green: BTBD18-dependent pachytene piRNA clusters; yellow: BTBD18-independent pachytene piRNA clusters. The transcripts of pachytene piRNA clusters are further classified as long and intronless (n = 39); short and intronless (n = 13); spliced with long first exons (n = 5); spliced with short first exons (n = 40); intronless transcripts isoforms are long and spliced transcript isoforms have long first exons (n = 2); intronless transcript isoforms are long and spliced transcript isoforms have short first exons (n = 1).
BTBD18 is essential for transcription of long unspliced or long-first-exon pachytene piRNA clusters
A subset of pachytene piRNA clusters require the nuclear protein BTBD18 to generate high levels of precursor transcripts and piRNAs22. BTBD18 binds the promoters of 51 pachytene piRNA clusters, and for 48 of these loci, the steady-state levels of both precursor transcripts and piRNAs decrease ≥2-fold in Btbd18Null homozygous testes, compared with heterozygous littermates. These 48 BTBD18-dependent pachytene piRNA-producing loci include 46 of the 47 pachytene piRNA clusters that produce long transcripts—39 long intronless genes, 5 long-first-exon genes, and 2 genes producing both long intronless and long-first-exon transcript isoforms (Fig. 3e).
Two additional BTBD18-dependent pachytene piRNA-producing loci have first exons shorter than 10,000 nt, but nonetheless longer than typical: the first exon of 7-qD2-11976.1 is 9436 bp and the first exon of 7-qD1-654.1 is 4620 bp. Another exception is pi-1700016M24Rik.1, the only pachytene piRNA gene that makes both a long unspliced transcript and two spliced transcripts with short first exons; BTBD18 binds the pi-1700016M24Rik.1 promotor (Supplementary Fig. 5c). This locus failed to qualify as BTBD18-dependent because in Btbd18Null homozygous testis, exonic reads increased 12% while intronic reads decreased 33%. Yet piRNAs mapping to the locus decrease 16-fold in Btbd18Null mutant testis. In Btbd18 heterozygotes, 85% of piRNAs from the locus map to introns, suggesting that the piRNAs derive from the long, unspliced isoform. Supporting this idea, intronic piRNAs decreased 17-fold in Btbd18Null testis. Just 4.6% of pi-1700016M24Rik.1 piRNAs mapped to sequences unique to the two short-first-exon transcript isoforms, and even these piRNAs decreased only 4.3-fold in Btbd18Null testes. Together, these data suggest that BTBD18 is essential for transcription of long unspliced and long-first-exon pachytene piRNA clusters.
Chromatin immunoprecipitation (ChIP)-seq data22 show that BTBD18 binds the promoter of all 47 unspliced or long-first-exon pachytene piRNA clusters and the two BTBD18-dependent pachytene piRNA clusters, whose long first exons were slightly shorter than the 10 kb cutoff we imposed (7-qD2-11976.1 and 7-qD1-654.1). For 33 of the 49 genes, BTBD18 binding extends at least to the midpoint of the first exon (Supplementary Fig. 4a; relative extension index ≥ 0.5, see "Methods"). Similarly, for 32 of the 49 genes, the ATAC-seq signal extends at least halfway across the first exon or gene body (Supplementary Fig. 4a). Both the BTBD18 and the ATAC-seq signal extends to or beyond the midpoint of the first exon or intronless gene body for more than half of the 49 pachytene piRNA clusters (25/49). Together, these data support the proposal that BTBD18 facilitates transcription of pachytene piRNA clusters by increasing chromatin accessibility22.
Twenty six of the 49 pachytene piRNA clusters form divergent pairs transcribed from a common, bidirectional promoter. Four additional BTBD18-dependent pachytene piRNA clusters are also transcribed from a central promoter, in which the other member of the pair does not produce piRNAs: three are paired with protein-coding genes, and one is paired with a lncRNA. For example, the BTBD18-dependent pachytene piRNA gene 10-qB4-6488.1 is transcribed from a 475-bp long, bidirectional promoter that generates the Ddx50 mRNA from the opposite arm (Supplementary Fig. 5d). Both BTBD18 and A-MYB bind this divergently transcribed promoter. The ChIP-seq signal of BTBD18 is broad and covers the entire promoter, while the ChIP-seq peak of A-MYB is equidistant from the two TSSs. Yet only the piRNA-producing arm requires BTBD18 and A-MYB for its transcription (Supplementary Fig. 5d). Notably, 10-qB4-6488.1 makes three piRNA-producing transcript isoforms: a long unspliced RNA and two long-first-exon RNAs. In contrast, Ddx50 has a short first exon and makes no piRNAs. Since these two genes share a promoter, their distinct transcriptional regulation and the contrasting fates of their transcripts likely reflects their different first-exon lengths and not BTBD18 binding per se.
THOC1 and THOC2 specifically bind transcripts from long unspliced and long-first-exon pachytene piRNA clusters
The THO complex, which acts in transcription elongation and mRNA nuclear export23,24,25,46,47,48, also participates in piRNA biogenesis in Drosophila26,27,28. In the fly ovary, THO complex components specifically bind piRNA precursor transcripts, and loss of THO complex subunits disrupts transposon silencing, reduces germline piRNA abundance, and leads to male and female sterility. Our data suggest that in adult mouse testis, the THO complex also binds pachytene piRNA precursor transcripts.
We immunoprecipitated two mouse THO subunits—THOC1 (also named HPR1) and THOC2—from whole adult testes and sequenced the co-immunoprecipitated RNA (RIP-seq); immunoprecipitation with IgG served as a control (Supplementary Data 1). THOC1 and THOC2 specifically associated with long unspliced and long-first-exon transcripts from pachytene piRNA clusters: on average, binding of pachytene piRNA precursor transcripts to THOC1 and THOC2 was 7.2 and 5.5 times greater than spliced mRNAs, respectively (Fig. 4a; Wilcoxon rank-sum test p value < 2.2 × 10−16). In contrast, THOC1 and THOC2 bound less RNA from short intronless or short-first-exon pachytene, pre-pachytene, or hybrid piRNA precursor transcripts (1.6 and 1.4 times more than protein-coding transcripts, respectively, Wilcoxon rank-sum test p value < 1.1 × 10−13). Remarkably, for all seven spliced piRNA clusters with long first exons, the RIP signals of THOC1 and THOC2 were largely confined to long first exons of the piRNA precursor transcripts, suggesting that the THO complex binds to transcripts from the 5′-end to the first 5′ splice site (Fig. 4b and Supplementary Fig. 6a).
Fig. 4: THOC1 and THOC2 specifically bind precursor transcripts from long intronless and long-first-exon pachytene piRNA clusters.
a RIP-seq was used to measure the abundance of RNA bound to THOC1 (n = 5) or THOC2 (n = 3) compared to IgG control (n = 4) in adult testis. Each dot represents the mean abundance normalized for transcript length (RPKM). b Heatmaps report the enrichment of THOC1 or THOC2 binding, measured by RIP-seq, relative to IgG control for pachytene piRNA clusters in adult testis. TSS, transcription start site. Each row corresponds to pachytene piRNA gene, in first-exon (spliced) or transcript (unspliced) length order. For each gene, the black bar denotes the first exon–intron boundary; the blue bar indicates the 3′ end of the transcript.
As described in the previous section, the pachytene piRNA gene 10-qB4-6488.1 and the protein-coding gene Ddx50 are divergently transcribed from a common A-MYB- and BTBD18-bound promoter. The first exon of 10-qB4-6488.1 is 50,297 bp long, whereas the first exon of Ddx50 spans just 187 bp. Supporting the hypothesis that the THO complex marks transcripts for entry into the piRNA biogenesis pathway, THOC1 and THOC2 bound across the 10-qB4-6488.1 first exon, but did not detectably bind the Ddx50 mRNA (Supplementary Fig. 5d). Another pachytene piRNA gene described in the previous section, pi-1700016M24Rik.1, produces three RNA isoforms: a 10,179 bp unspliced transcript that appears to be the source of piRNAs and two spliced mRNAs with short first exons. Normalized for transcript length, THOC1 and THOC2 bound six times more to the long intronless transcript than either of the short-first-exon transcripts (Supplementary Fig. 5c).
In the testis, low-CG pachytene piRNA clusters have high levels of histone acylation and 5-hydroxymethylcytosine
Nearly half (49/100) of pachytene piRNA clusters have low-CG promoters (Fig. 2c); these genes are moderately enriched in the long intronless or long-first-exon group (30/49; Chi-square test p value = 8.8 × 10−3) and BTBD18 dependence (31/49; Chi-square test p value = 4.8 × 10−3). Unlike protein-coding genes with low-CG promoters, the low-CG pachytene piRNA clusters maintain high histone acylation levels in adult testis. They also exhibit high 5hmC levels, which may aid their transcriptional activation49.
Consistent with the known correlation between the expression level of a gene and the O/E CG of its promoter, protein-coding genes with low-CG promoters had significantly lower steady-state mRNA abundance in the testis than those with intermediate-CG or high-CG promoters (Supplementary Fig. 6b). Accordingly, these low-CG promoters have higher levels of DNA methylation, lower levels of active histone marks, reduced chromatin accessibility, and less RNA pol II binding than intermediate- or high-CG promoters (Supplementary Fig. 6b). Pachytene piRNA clusters defy the rule that promoters with low O/E CG and high DNA methylation are poorly expressed in the testis; instead, the pachytene piRNA clusters with low-CG, highly methylated promoters were as highly expressed and made as many piRNAs as those loci with intermediate- or high-CG promoters (Fig. 5a). Furthermore, low-CG pachytene piRNA clusters had significantly higher levels of histone acylation, despite significantly lower levels of H3K4me3, chromatin accessibility, and RNA pol II binding, compared with loci having intermediate- or high-CG promoters (Fig. 5b). These paradoxical results support the view that, in the testis, histone acylation overrides other determinants of transcription, allowing the expression of pachytene piRNA clusters irrespective of the O/E CG or methylation level of their promoters.
Fig. 5: Activation of pachytene piRNA clusters in the male germline correlates with high 5hmC levels, whereas the extent of somatic suppression of these genes correlates with first-exon length and high levels of DNA methylation and H3K27me3.
a piRNA and transcript abundance for pachytene piRNA clusters in pachytene spermatocytes. Promoter O/E CG is indicate by color: Blue, low (n = 49); green, intermediate (n = 33); yellow, high (n = 18). Two-sided Wilcoxon rank-sum test was used to determine p value; n.s. not significant. For all boxplots in this figure, whiskers show 95% confidence intervals, boxes represent the first and third quartiles, and the vertical midline is the median. b Level of histone modifications, ATAC, and RNA pol II for pachytene piRNA clusters in pachytene spermatocytes. Promoter O/E CG is as in a. Asterisks indicate two-sided Wilcoxon rank-sum test p values < 0.05 between 49 low-CG and 33 intermediate- or 18 high-CG promoters. Red (blue) indicates that the epigenetic level is significantly higher (lower) in low-CG pachytene piRNA clusters. c 5hmC levels in pachytene spermatocytes. Asterisks indicate two-sided Wilcoxon rank-sum test p values < 0.05 between 49 pachytene piRNA clusters and 397 testis-specific mRNAs, 95 testis-specific lncRNAs, all 6177 mRNAs, and all 2050 lncRNAs with low-CG promoters. d Top, DNA methylation levels for pachytene piRNA clusters and other groups of genes. Bottom, DNA methylation levels of pachytene piRNA clusters grouped as in a. Vertical line indicates the 50% DNA methylation level. e DNA methylation levels of pachytene piRNA clusters between forebrain and pachytene spermatocytes. Colors indicate promoter CG level; genes with forebrain expression level higher than 0.1 RPKM are denoted as triangles. f H3K27me3 levels, relative to control, are shown in adult testis and forebrain. Only the 23 pachytene piRNA clusters in the bottom-left quadrant defined in e are included and further classified by their expression levels in the forebrain: RPKM ≥ 0.1 (n = 12) or RPKM < 0.1 (n = 11). g First exon length of the same 23 pachytene piRNA clusters as in f, classified into the same groups as in f.
How do low-CG pachytene piRNA clusters maintain high levels of histone acylation? In the testis, highly expressed genes with low promoter O/E CG, but high levels of H3K4me3 in their promoters also have high levels of 5hmC21, a DNA modification leading to transcriptional activation49,50,51. Supporting this idea, pachytene piRNA clusters with low-CG promoters had high 5hmC levels. Using published hMeDIP data which specifically measure 5hmC levels (see "Methods"), we found that the promoter 5hmC levels of low-CG pachytene piRNA clusters were significantly higher than those of control genes (i.e., low-CG genes that do not produce piRNAs) in cells where pachytene piRNAs are abundant—pachytene spermatocytes (Wilcoxon rank-sum test p value < 9.7 × 10−7) and round spermatids (p value < 5.0 × 10−6)—but not in hepatocytes, which lack piRNAs (Fig. 5c and Supplementary Fig. 6c, d). We conclude that despite their hypermethylation, pachytene piRNA clusters with low-CG promoters can be expressed to high levels in the testis because of compensating 5hmC DNA modification.
Genic and epigenomic features correlated with silencing of pachytene piRNA clusters in the soma
What prevents expression of pachytene piRNA clusters in somatic cells, which generally lack the machinery to process precursor transcripts into piRNAs? We found that two genic features associated with pachytene piRNA clusters—low-CG promoters and long first exons—correlated with their somatic silencing. Furthermore, we identified two epigenomic signals—DNA methylation and the repressive histone mark H3K27me3—likely underpin this silencing.
The promoters of both pachytene piRNA clusters and testis-specific protein-coding genes were highly methylated (>50%) in the neonatal forebrain, heart, and liver—representative somatic tissues. Yet a subset of these two classes of genes showed lower levels of promoter methylation in pachytene spermatocytes, representative of male germline cells (Fig. 5d). These subsets of genes with different methylation levels between germline and soma mostly had promoters with intermediate levels (0.25–0.5) of O/E CG (Supplementary Fig. 7). Specifically, the median methylation level for the pachytene piRNA clusters with intermediate-CG promoters was 16% in pachytene spermatocytes vs. 72–81% in somatic tissues (Fig. 5d; Wilcoxon rank-sum p values < 3.2 × 10−6). Similarly, the median methylation level for the testis-specific protein-coding genes with intermediate-CG promoters was 17% in pachytene spermatocytes vs. 71–81% in the soma (p values < 2.2 × 10−16). These results suggest that DNA methylation may be an essential mechanism for silencing these intermediate-CG genes in the soma.
Indeed, most low-CG promoters of pachytene piRNA clusters (40/49) were highly methylated in both pachytene spermatocytes and forebrain, and most high-CG promoters (17/18) showed low levels of methylation in both pachytene spermatocytes and the forebrain. In contrast, most intermediate-CG (23/33) and some low-CG (9/49) promoters had low methylation in pachytene spermatocytes, but high methylation in the forebrain (Fig. 5e and Supplementary Fig. 7). Consistent with the idea that genes with highly methylated, low-CG promoters are silenced in the soma, the steady-state forebrain transcription levels were <0.1 RPKM for all of the 77 pachytene piRNA clusters with high DNA methylation in the forebrain. Further supporting this idea, all of the 12 pachytene piRNA clusters with forebrain transcript abundance ≥0.1 RPKM had intermediate- or high-CG promoters with low levels of DNA methylation. Our analysis in two other somatic tissues, neonatal heart, and liver, revealed the same repressive effect of DNA methylation (Supplementary Fig. 8a). Thus, DNA methylation likely silences a majority of pachytene piRNA clusters in the soma, especially those with low- or intermediate-CG promoters.
Among the 23 pachytene piRNA clusters with low DNA methylation in the forebrain, 11 showed some degree of testis-specific expression—their forebrain transcript abundance was <0.1 RPKM. All of these 11 genes have high (n = 8) or intermediate- (n = 3) CG promoters. For these genes, H3K27me3 rather than DNA methylation may repress somatic expression: the median H3K27me3 level on the promoters of these genes was 1.9 times higher in the forebrain than in pachytene spermatocytes (Wilcoxon rank-sum p value = 9.8 × 10−4). For the remaining 12 pachytene piRNA clusters expressed in forebrain, which constituted all of the pachytene piRNA clusters with forebrain transcript abundance ≥0.1 RPKM, promoter H3K27me3 levels in forebrain and pachytene spermatocytes do not significantly differ (Fig. 5f). Similar results were observed in neonatal heart and liver (Supplementary Fig. 8b). Thus, high H3K27me3 may be another mechanism for silencing pachytene piRNA clusters in the soma, especially those with high-CG and intermediate-CG promoters that are not DNA methylated in the soma.
The presence of long first exons may be yet another mechanism for somatic silencing of pachytene piRNA clusters with high-CG and intermediate-CG promoters lacking somatic DNA methylation. The 11 genes in this subset that are silenced in the forebrain have substantially longer first exons than the 12 genes that are expressed in the forebrain (Fig. 5g; median first-exon length = 35,227 vs. 308; Wilcoxon rank-sum test p value = 1.3 × 10−3). Similar results were observed in neonatal heart and liver (Supplementary Fig. 8c).
In theory, the absence of A-MYB in somatic cells could explain why pachytene piRNA clusters are not transcribed outside the germline. However, most non-pachytene piRNA cluster loci that are regulated by A-MYB in testis (n = 930, defined in "Methods") are expressed in somatic tissues (696 of 930 with >1 RPKM), despite low levels of A-Myb transcript in somatic cells (0.54 RPKM). The A-Myb paralogs B-Myb and C-Myb bind similar sequence motifs to A-Myb, but they are expressed in only a subset of the somatic tissues (B-Myb in colon and spleen and C-Myb in colon, lung, and spleen among the tissues we examined); therefore, they are unlikely to be responsible for the expression of A-MYB target genes in somatic tissues. We conclude that the absence of A-MYB in the soma is not sufficient to silence pachytene piRNA clusters in somatic tissues. Thus, three mechanisms—low CG and the associated high hypermethylation, high H3K27me3, and long first exons—may explain the somatic silencing of pachytene piRNA clusters. We performed the same analysis for testis-specific protein-coding genes, and observed that high DNA methylation and high H3K27me3, but not long first exons (Fig. 5d and Supplementary Fig. 9), likely repress these genes in the soma: among 1171 testis-specific protein-coding genes, 452 and 157 were expressed in the forebrain at >0.1 and >1 RPKM, respectively. Thus, testis-specific protein-coding genes and pachytene piRNA-producing genes may be silenced in the soma by similar mechanisms.
Evolutionarily conserved features of pachytene piRNA clusters
Although individual piRNA sequences are rarely conserved across species, the genomic location (synteny), promoter elements, and exon–intron structure of the major pachytene clusters are preserved among placental mammals12. Notably, long, unspliced and long-first-exon mouse pachytene piRNA clusters are more likely to be syntenic in other species than those with short first exons. We did not detect significant conservation of low-CG promoters between the species examined.
We analyzed the 89 pachytene piRNA clusters in human, 133 in rhesus, 132 in marmoset, 114 in rat, 117 in cow, 194 in opossum, and 89 in platypus12 for their evolutionary conservation with the 100 mouse pachytene piRNA clusters. We determined whether the genes were syntenic and whether the syntenic genes also produced piRNAs. For 21 mouse pachytene piRNA clusters, the syntenic loci in at least three other eutherian species produced piRNAs at similar levels (<5-fold change from mouse). For 45 other mouse pachytene piRNA clusters, the syntenic loci in rat produced similar levels of piRNAs. These data define three groups of pachytene piRNA clusters: eutherian-conserved, murine-conserved, and mouse-specific (34 pachytene piRNA clusters; Fig. 6a). Among the 100 mouse pachytene piRNA clusters, most eutherian-conserved loci had long first exons, while most murine-conserved and mouse-specific loci had short first exons (Fig. 6a; Fisher's exact test p value = 6.2 × 10−5). In each of the six eutherian mammals examined, eutherian-conserved pachytene piRNA clusters had longer first exons than other pachytene piRNA clusters (Fig. 6b; Wilcoxon rank-sum test p values < 1.3 × 10−3), suggesting that a long first exon promotes pachytene piRNA production across eutherian mammals.
Fig. 6: Long first exons and broad histone acylation domains are conserved features of pachytene piRNA clusters in placental mammals.
a Heatmap shows piRNA abundance (RPM) for the 100 mouse pachytene piRNA clusters and the syntenic loci in several mammalian species with increasing evolutionary distance. Pachytene piRNA clusters are ranked first by conservation and then by average piRNA abundance. b First exon length for pachytene piRNA clusters in mouse, rat, human, rhesus, marmoset, and cow. Two-sided Wilcoxon rank-sum p values are provided for comparing eutherian-conserved with other pachytene piRNA clusters in mouse (n = 21 and 79), rat (n = 20 and 94), human (n = 19 and 70), rhesus (n = 18 and 115), marmoset (n = 17 and 115), and cow (n = 18 and 99). Vertical line marks 10 kbp. c Left, meta-gene plot shows average H3K27ac signal in rhesus adult testis across the −5 to +50 kb window flanking the TSS for six groups of genes: 54 long-first-exon pachytene piRNA clusters, 79 short-first-exon pachytene piRNA clusters, 56 piRNA clusters that overlap protein-coding genes, 17 piRNA pathway genes, 593 A-MYB-regulated protein-coding genes, and 745 testis-specific protein-coding genes. Right, boxplots show the H3K27ac density relative to input for the same six groups of genes. d Mosaic plots for pachytene piRNA clusters. Filled boxes, eutherian-conserved pachytene piRNA clusters; unfilled boxes, other pachytene piRNA clusters. Red, long-first-exon and long intronless pachytene piRNA clusters; pink, short-first-exon and short intronless pachytene piRNA clusters. The area of each patch indicates the number of pachytene piRNA clusters in the group; numbers report the percentage of total piRNAs in each group. e Schematic diagram shows our proposed model that suppression of splicing steers piRNA precursor transcripts into the piRNA biogenesis pathway across animal species. Credit: silhouettes in e are from http://phylopic.org. The cabbage looper silhouette was by Gareth Monger (https://creativecommons.org/licenses/by/3.0/). The silhouettes have not been altered in any way. For all boxplots, whiskers show 95% confidence intervals, boxes represent the first and third quartiles, and the vertical midline is the median. Two-sided Wilcoxon rank-sum tests were used to calculate p values.
The 47 long intronless or long-first-exon mouse pachytene piRNA clusters show high levels of histone acylation extending across their first exon or gene body. High acylation is evolutionarily conserved among pachytene piRNA-producing loci in Rhesus macaque, which diverged from mouse 90 million years ago. H3K27ac ChIP-seq in adult Rhesus testis revealed that H3K27 acetylation levels were higher at pachytene piRNA-producing genes than other gene types, and the H3K27ac signal spread broadly from the TSS as far as 40 kbp (Fig. 6c). Moreover, a long, unspliced transcript and shared synteny correlated with high levels of piRNA production across placental mammals. Just 12–18% of the pachytene piRNA clusters in each of the six eutherian mammals have long first exons and are related by synteny with pachytene piRNA-producing loci in at least three of the six other species examined; yet this small set of genes produce 45–76% pachytene piRNAs in adult testis (Fig. 6d).
Pachytene piRNA clusters, which are unique to placental mammals, are expressed almost exclusively in the male germline. Of the 100 mouse pachytene piRNA clusters, 47 are long and unspliced or have long first exons, 49 have low-CG promoters, and 30 display both features. In general, a lack of splicing or a low O/E CG promoter is anticipated to prevent somatic expression, and indeed, genes with these features are silenced in the soma. Yet pachytene piRNA clusters with these same features are highly expressed in the germline, and produce most of the pachytene piRNAs in the adult mouse and other eutherian mammals.
Low-CG promoters are hypermethylated and recognized by proteins with methyl-CG-binding domains (MBDs). MBD proteins, in turn, recruit histone deacetylases (HDACs), especially class III HDACs, such as sirtuins, which remove lysine acetyl, butyryl, and crotonyl modifications52. Decreased histone acylation correlates with transcriptional repression53,54. Although the promoters of low-CG pachytene piRNA clusters are hypermethylated in the testis, they are also highly 5-hydroxymethylated, a modification that can block binding of MDB proteins. Consequently, 5-hydroxymethylation prevents binding of HDACs50. We suggest that the high 5-hydroxymethylation of the low-CG promoters of pachytene piRNA clusters allows them to maintain high lysine acylation and active transcription in the male germline.
Three additional features likely act to repress pachytene piRNA clusters in the soma. First, both low-CG and intermediate-CG promoters show high levels of DNA methylation in the soma. Second, the subset of pachytene piRNA clusters with high- or immediate-CG promoters show a high density of the repressive histone mark H3K27me3. Third, pachytene piRNA clusters tend to have long first exons or long unspliced transcripts. Pachytene piRNA clusters with intermediate-CG promoters may particularly rely on DNA methylation and H3K27me3 for their somatic silencing: just one-third (11/33) of intermediate-CG pachytene piRNA clusters are unspliced with long transcripts or spliced with long first exons; nearly two-thirds (30/49) of low-CG pachytene piRNA clusters display these features.
Most mammalian promoters inherently drive bidirectional transcription. Co-transcriptional binding of the spliceosomal U1 snRNP enhances transcriptional elongation in the productive direction55,56. Long first exons delay the recruitment of the splicing machinery and hence hinder transcriptional elongation. Co-transcriptional splicing has been proposed to be required for the recruitment of the THO complex, which is essential for transcriptional elongation and nuclear export of spliced RNA57. Consistent with this model, deletion of the first intron of a protein-coding gene markedly decreases its expression58. How then do long unspliced and long-first-exon pachytene piRNA transcripts recruit the THO complex in the male germline? Our analyses identify several features that may promote THO complex binding in the absence of splicing: broad histone acylation and binding of the transcriptional elongation factor BTBD18. Perhaps BTBD18 recruits histone acyl transferases to open the chromatin of long-first-exon or unspliced pachytene piRNA clusters. We do not yet know whether the efficient transcriptional elongation that results from BTBD18 binding and histone acylation suffices to allow co-transcriptional loading of the THO complex or if BTBD18 also participates in the direct recruitment of THO. Once bound to the piRNA precursor transcript, the THO complex may enhance transcriptional elongation and facilitate export of the piRNA precursor transcripts to the perinuclear nuage, where piRNA processing initiates (Fig. 6e). Clearly a major question for future study is how BTBD18 or other proteins identify pachytene piRNA precursor transcripts.
Suppression of splicing is a unifying feature of piRNA-producing loci across the animal kingdom. In flies, transcription of the heterochromatic piRNA clusters relies on a noncanonical mechanism, in which binding of the protein Rhino to the chromatin mark H3K9me3 allows promoter-independent recruitment of RNA pol II59,60,61,62. The complex of proteins assembled around Rhino also serves to suppress splicing and to bypass cleavage at polyadenylation signal sequences61,62,63,64. Lepidoptera lack this noncanonical transcription machinery, and data from the Trichoplusia ni germ cell line Hi5 suggest that piRNA clusters are transcribed from standard RNA pol II promoters, yet the piRNA precursors from these clusters also undergo little splicing65. In koala males bearing a KoRV-A provirus, piRNAs are produced only from the unspliced proviral transcript66. Similarly, the long-first-exon or long unspliced pachytene piRNA precursor transcripts of mice and other placental mammals are inherently depleted of introns (Fig. 6e). The link between piRNA production and suppression of splicing calls to mind the antagonism between splicing and siRNA generation in Cryptococcus neoformans, where the two pathways compete for RNA substrates67. We suggest that a failure to engage the splicing machinery near the transcription start marks transcripts for processing into piRNAs. Because most protein-coding genes contain short first exons and are efficiently spliced, such a mechanism would protect mRNAs from inappropriately entering the piRNA pathway. Moreover, channeling unspliced transcripts into the piRNA pathway would allow the unspliced genomic transcripts of retroviruses and retrotransposons to serve as precursors for sense piRNAs. Conversion of the unspliced transcripts to sense piRNAs—which cannot target the viral or transposon transcripts directly via sequence complementarity—would serve to destroy the RNA transcript, preventing accumulation of viral or transposon genomic RNA that can catalyze retro-insertion into the host genome66. Such a mechanism would provide an initial defense against these pathogens until the establishment of an antisense piRNA response that can directly target the spliced, protein-coding transcripts of retroviruses and transposons.
All mice were maintained and used according to the guidelines of the Institutional Animal Care and Use Committee of the University of Massachusetts Medical School. C57BL/6 J mice (RRID: IMSR_JAX:000664) were used as wild-type controls.
H3K27ac ChIP-seq
To perform ChIP, small pieces of frozen testis tissue were cross-linked with 2% (w/v) formaldehyde at room temperature for 30 min using end-over-end tumbler. We next crushed fixed tissues in the presence of ChIP lysis buffer (1% (w/v) SDS, 10 mM EDTA, 50 mM Tris-HCl, pH 8.1) by 40 strokes with a B pestle in a Dounce homogenizer (Kimble-Chase, Vineland, USA). Thereafter, lysate was sonicated using Covaris ultrasonicator (Covaris, E220) to shear the chromatin to 150–200 bp. Lysate was then diluted 1:10 with ChIP dilution buffer (0.01% (w/v) SDS, 1.1% (w/v) Triton X-100, 1.2 mM EDTA, 16.7 mM Tris-HCl, pH 8.1, 167 mM NaCl). We performed immunoprecipitation using 6 µg rabbit polyclonal anti-H3K27Ac antibody (Abcam, ab4729). Following the immunoprecipitation, we extracted DNA with phenol:chloroform:isoamyl alcohol (25:24:1; pH 8) and prepared libraries for anti-H3K27ac, and input DNA as previously described68. ChIP-seq libraries were sequenced as 79-nt paired-end reads using NextSeq500 (Illumina).
THOC1 and THOC2 RIP-seq
Testis from 4-month-old C57BL/6 J mice were frozen in liquid nitrogen and stored at −80 °C until use. One testis was lysed in 1 ml lysis buffer (50 mM HEPES-KOH, pH 7.5, 150 mM KCl, 3.2 mM MgCl2, 0.5% (v/v) NP-40, 1 mM phenylmethylsulphonyl fluoride; 1× Proteinase Inhibitor cOmplete, EDTA-free Protease Inhibitor Cocktail (Roche), 0.4 U/µl RNAseOUT (Invitrogen)) in 10 strokes with pestle A and 20 strokes with pestle B in a 2 ml Dounce homogenizer (Kimble-Chase). Lysate was sonicated (Bioruptor, Diagenode) at medium strength, 15 s on and 1 min off for 10 min at 0 °C. Lysate was cleared by centrifugation for 15 min at 16,050 × g at 4 °C. Lysate (100 µl) was used for input RNA extraction using RNeasy Mini Kit (Qiagen). A total of 10 µl rabbit anti-THOC2 and anti-IgG, or mouse anti-THOC1 and anti-IgG antibody was conjugated to the 100 µl Dynabeads (Invitrogen) protein A or protein G, respectively, in citric phosphate (CP) buffer (7.10 g Na2HPO4, 11.5 g citric acid in 1 l water, pH 5.6) for 2 h at room temperature with rotation. Antibody-conjugated beads were washed three times with CP buffer containing 0.1% (v/v) Tween-20 and incubated with mouse testis lysate overnight at 4 °C with rotation. Beads were washed three times in cold (4 °C) lysis buffer. The immunoprecipitated RNA was recovered from the antibody beads using the RNeasy Mini kit (Qiagen). The extracted RNA was used to generated RNA-seq libraries as described68.
We reanalyzed previously published RNA-seq31,69, small RNA-seq70, whole-genome bisulfite sequencing or WGBS21, ChIP-seq of histone marks42,45,71, ChIP-seq of RNA pol II (ref. 72), and ATAC-seq73 data. We also reanalyzed ovary RNA-seq data (E11.5, E12.5, E14.5, E16.5, E18.5, and E20.5) from GEO (accession: GSE119411). Public ChIP-seq data of H3K4me1, H3K4me3, H3K27me3, H3K27ac, and RNA pol II in six adult mouse tissues (testis, liver, heart, kidney, lung, and spleen) were obtained from the mouseENCODE consortium74. We also included H3K4me2 ChIP-seq data for the adult mouse testis, liver, and brain69, but we did not find ChIP-seq data in other somatic tissues to fully complement the RNA-seq data. We analyzed the WGBS data generated by the ENCODE consortium on fetal mouse tissues at eight embryonic time points until birth (E10.5 to P0), along with published WGBS data on mouse round spermatids21.
Gene annotation
Gene annotations and ribosomal RNA (rRNA) sequences were from Ensembl (Release 82). To determine whether Ensembl Release 100 might yield more lncRNAs with long first exons, we reanalyzed the data using the latest Ensembl gene annotation. Indeed, Ensembl Release 100 contains 924 more lncRNAs than Release 82 (5,586 vs. 4,662), but just one of the 924 lncRNAs (AC160336.1) has long first exon. RNA-seq data indicate that this lncRNA is not expressed in adult mouse testis. Thus, our conclusions remain unchanged using Ensembl Release 100.
RNA-seq and RIP-seq data
To analyze RNA-seq and RIP-seq data, rRNA was first removed from RNA-seq and RIP-seq data by mapping to rRNAs using Bowtie2 (version 2.2.5) with default parameters75. Unmapped reads were then mapped to the mm10 genome from the UCSC genome browser using STAR (version 020201) with default parameters (allowing up to 2 mismatch and up to 100 mapping locations)76. SAMtools (version 1.8) was used to transform the alignment results from sam format to bam format throughout this study77. HTSeq (version 0.9.1) was used with default parameters to count uniquely mapping reads78. For each protein-coding or lncRNA gene (Ensembl Release 82) and piRNA-producing locus, we then calculated and normalized the number of uniquely mapped reads as Reads Per Million mapped reads (RPM) and the number of uniquely mapped reads normalized by the total transcript length of each gene as RPKM.
ChIP-seq and ATAC-seq data
ChIP-seq data and ATAC-seq data were mapped to the mouse reference genome mm10 using Bowtie2 with the parameter–very-sensitive. Enrichment of ChIP signal over input (.bedGraph) was computed from alignment files (.bam) across the mouse genome using the MACS2 callpeak and bdgcmp modules with default parameters79. The H3K27ac ChIP-seq data in primary spermatocytes and round spermatids (accession GSE107398) were not provided with input, so we calculated read density normalized by sequencing depth. Furthermore, we normalized read density by sequencing depth for all ATAC-seq data. To calculate the level of each epigenetic mark for the various groups of genes (piRNA clusters, protein-coding genes, lncRNA genes, and their subsets), we averaged the enrichment or read density at the promoter (TSS ± 2 kb) or gene body (TSS + 2 kb till the 3′-end) as indicated.
We used a control gene set to normalize ChIP-seq signal levels across different datasets. We first identified 1230 genes with similar expression levels (0.66 < change < 1.5 and maximal expression >0.1 RPKM) in three cell types (spermatogonia, primary spermatocytes, and round spermatids). We then used the median levels of each histone mark for these 1230 control genes in three cell types to normalize the levels of all histone marks, chromatin accessibility (ATAC signal), and RNA pol II binding in the corresponding cell types. We used the same strategy to normalize the H3K27me3 levels in testis and somatic tissues. We identified a set of control genes whose expression levels differed by <1.5-fold between testis and each somatic tissue: 1132, 913, and 817 control genes for comparing testis with forebrain, heart, and liver, respectively. This normalization method reduced the bias caused by the difference in antibody specificity across different batches or different tissue or cell types.
WGBS data
We applied quality control and adapter trimming to all WGBS data using Trim Galore! (version 0.4.1). Sequence alignment and methylation calls were performed using Bismark (version v0.15.0, no mismatch allowed; mapping statistics in Supplementary Data 3) with default parameters80. Only C in CG positions was used for computing methylation levels. Methylation level was computed as mCG/CG averaged over all detected CG sites in a region. Each WGBS dataset had four replicates, and we calculated the methylation level for each region in each replicate and then averaged the levels across the four replicates.
Small RNA-seq data
Small RNA sequencing data were mapped to the mouse genome (mm10) using Bowtie (version 1.1.0) after removing rRNA (from Ensembl) and miRNA (from miRBase) with parameters -a–best–strata81. Only 24–32 nt small RNAs were retained. We then calculated the piRNA abundance of each piRNA gene using BEDTools intersectBed, apportioning the reads that mapped to multiple locations82. piRNA abundance was reported normalized to the total number of reads mapping uniquely to one genomic location. As most of the mapped reads are uniquely mapped to the mouse genome (90.6% in adult testis, 89.7% in pachytene spermatocyte, and 90.0% in round spermatid), normalizing to uniquely mapping or uniquely mapping plus multiply mapping reads result in similar piRNA abundance and the same conclusions.
5hMe-DIP data
5hMe-DIP (5hmC) data were mapped to the mouse genome (mm10) using Bowtie2 with the parameter–very-sensitive. The number of aligned reads were processed into enrichment relative to input in the bedGraph format. The 5hmC level for each gene was calculated by averaging the enrichment in the TSS ± 500 bp window.
Definition of genes regulated by A-MYB
A-MYB-regulated genes were defined based on A-MYB ChIP-seq data on wild-type mice, and long RNA-seq data on A-Myb mutant and heterozygous mice. We first identified the genes whose TSSs were within 500 bp of a A-MYB ChIP-seq peak called by MACS2 with parameters -q 0.05–keep-dup all -B (ref. 79). We then filtered out the genes with less than threefold enrichment of A-MYB ChIP-seq signal over input in the TSS ± 500 bp window. Keeping only those genes with >2-fold decrease in A-Myb mutant testis compared to heterozygotes at both 14.5 and 17.5 dpp yielded 837 A-MYB-regulated genes, including 791 protein-coding and 46 lncRNA genes (Supplementary Data 2).
Definition of high-, intermediate- and low-CG promoter classes
Normalized CG dinucleotide content in a promoter was calculated as described previously83: the ratio of observed to the expected number of CG dinucleotides in the promoter (O/E CG), where the expected number = [(Fraction of C + Fraction of G)/2]2, i.e.,
$${\mathrm{O}}/{\mathrm{E}}\,{\mathrm{CG}} = \frac{{{\mathrm{Fraction}}\,{\mathrm{of}}\,{\mathrm{CpG}}}}{{\left[ {\left( {{\mathrm{Fraction}}\,{\mathrm{of}}\,{\mathrm{C}} +{\mathrm{Fraction}}\,{\mathrm{of}}\,{\mathrm{G}}} \right)/2} \right]^2}}.$$
Promoters were classified into three groups: low-CG, O/E CG < 0.25; intermediate-CG, 0.25 < O/E CG < 0.5; and high-CG, O/E CG > 0.5.
Definition of testis-specific genes
Testis-specific genes corresponded to those genes with expression levels in testis >4-fold higher than the maximal expression in any somatic tissues that we examined. We also calculated a tissue-specific score (ts-score),
$${\mathrm{ts}}-{\mathrm{score}} = \frac{{\mathop {\sum }\nolimits_{i = 1}^n \left( {1 - \frac{{{\mathrm{Exp}}_i}}{{{\mathrm{Exp}}_{{\mathrm{max}}}}}} \right)}}{{n - 1}},$$
where n denotes the total number of tissues examined, Expi denotes the expression levels in a particular tissue, and Expts denotes the expression level in testis (we set the ts-score to 0 if any Expi > Expts). We used ts-score to rank piRNAs in Fig. 1a.
Refining the annotations of piRNA clusters
Based on the RNA-seq and small RNA-seq data produced after 2013, we made small modifications to our previous annotations11 of several piRNA clusters (Supplementary Data 4): (1) we removed the intron in 12-qE-23911.1 because it is in a low mappability region and not supported by uniquely mapping reads. (2) We removed the intron in 5-qF-14224.1 which is 59 nt long with a noncanonical splice site motif CC-CT and supported by few reads. (3) We changed the TSS of 17-qC-59.1 from chr17: 50,237,659 to chr17:50,239,160 based on A-MYB ChIP-seq, RNA-seq, and small RNA-seq data. (4) We changed the main TSS of 4-qB3-639.1 from chr4: 62,230,936 to chr4: 62,228,511 based on A-MYB ChIP-seq, RNA-seq, small RNA-seq, and BTBD18 ChIP-seq data. (5) We added a long and intronless isoform (chr2: 92,529,805–92,540,950) for Gm13817, which is divergently transcribed from 2-qE1-35981.1. Gm13817 is an unannotated pachytene piRNA-producing gene and produces >100 piRNAs per million unique mapped reads. (6) We removed the long-first-exon isoform of pi-Zfp652.1 (a hybrid piRNA clusters), which is not supported by RNA-seq or small RNA-seq reads. (7) We changed the main TSS of 10-qC-875.1 from chr10: 86,617,011 to chr10: 86,591,510 based on A-MYB and H3K4me3 ChIP-seq, RNA-seq, small RNA-seq, and BTBD18 ChIP-seq data. We also separated 10-qC-875.1 from a spliced lncRNA Gm48485, which is primarily expressed in round spermatid.
Definition of BTBD18-dependent pachytene piRNA clusters
We defined BTBD18-dependent pachytene piRNA clusters as those whose expression level was ≥2-fold lower in pachytene spermatocytes of Btbd18 mutant mice than Btbd18 heterozygous mice, and whose piRNA abundance was also ≥2-fold lower in the 18-day postpartum testis tissue of Btbd18 mutant mice than that of Btbd18 heterozygous mice.
Annotation of piRNA clusters in human, rhesus, marmoset, rat, cow, rabbit, opossum, and platypus
Human piRNA clusters were annotated previously12. We refined the previously annotated piRNA clusters in rhesus, marmoset, rat, cow, opossum, and platypus12 to delineate exons and introns (Supplementary Data 4). We considered 24–32 nt small RNA reads that could map to each mammalian genome, after rRNA, miRNA, tRNA, snRNA, and snoRNA were removed, as piRNAs. piRNA abundance was then computed for 20 kb sliding windows (with a 1 kb step) in the genome, and windows with >100 piRNAs per million uniquely mapped piRNAs were deemed potential piRNA clusters. To remove false positives due to unannotated miRNA, rRNA, tRNA, snRNA, or snoRNA, which mostly produce reads with the same sequences, we also filtered out the 20-kb genomic windows with fewer than 200 distinct reads. We then calculated the first-nucleotide composition for each 20-kb window and discarded those windows with fewer than 50% of its piRNAs having a 1 U or 10 A (with the 10 A possibly resulting from ping-pong amplification). The remaining contiguous 20-kb windows were deemed putative piRNA clusters. To obtain the precise promoter position of each piRNA gene, we performed trimming from the 5′ and 3′ ends by examining adjacent 100-bp windows. The first and last two nearby windows (closer than 1000 bp) each with more than two piRNAs per million uniquely mapped piRNAs were considered as the 5′ and 3′ ends of the piRNA gene. We used the RNA-seq reads after rRNA removal for annotating introns of piRNA clusters. Finally, we performed manual curation for each piRNA gene using piRNA profile, long RNA profile, and exon–exon junctions detected using RNA-seq reads. The transcriptional direction of a piRNA gene was indicated by the direction of the main long RNA transcript. The final piRNA clusters were classified according to their genomic location, and those with >50% base pairs overlapping protein-coding genes were defined as genic and the rest as intergenic. We treated intergenic piRNA clusters as pachytene piRNA clusters in rhesus, marmoset, rat, cow, opossum, and platypus. We defined 17 piRNA pathway genes in rhesus, 747 testis-specific protein-coding genes, 593 A-MYB-regulated protein-coding genes using one-to-one orthology between mouse and rhesus.
Evolutionary conservation of pachytene piRNA clusters
We performed evolutionary conservation analysis on mouse pachytene piRNA clusters with a similar approach as described before for human piRNA clusters12. Briefly, we mapped the 100 mouse pachytene piRNA clusters to human, rhesus, marmoset, rat, cow, opossum, and platypus genomes, using UCSC chain files and liftOver with the parameter -minMatch = 0.1 and recorded the syntenic regions of pachytene piRNA clusters that could be lifted over to each of the other species12. The coordinates of the syntenic region in the other species which overlapped a mouse piRNA-producing gene on the same genomic strand were adjusted to be the piRNA gene coordinates we annotated in that species. To be inclusive of piRNAs that map to the boundaries of the syntenic regions in that other species that did not overlap piRNA clusters on the same genomic strand, we extended these syntenic regions by 10 kb in both ends and then calculated the piRNA abundance in the regions, using small RNA-seq data in that other species. Finally, eutherian-conserved pachytene piRNA clusters were defined as those for which three or more eutherian mammals (among human, rhesus, marmoset, rat, or cow) could produce similar amounts of piRNAs (change < 5) to mouse in the syntenic regions. The remaining pachytene piRNA clusters were defined as murine-conserved when the syntenic region in rat produced similar amount of piRNAs to mouse, mouse-specific if otherwise.
Calculation of extension index and relative extension index
In order to quantify how much the signals of histone modification, chromatin accessibility, or factor binding extend into the gene body, we first cut the genome into 200-bp bins and then computed the average enrichment of each type of signal over input or average read count per one million reads over these 200-bp bins. Bins far downstream pachytene piRNA clusters (TSS + 80 kb to TSS + 100 kb) were considered background bins and bins with signal higher than 95% quantile of the background bins were regarded as signal-enriched bins. Transposon copies reside in pachytene piRNA clusters can hinder the signal identification and lead to signal gaps. To avoid this, we identified the furthest continually enriched bins for each pachytene piRNA clusters allowing 3800-bp gaps (19 bins). The extension index was defined as the distance from the TSS to the furthest continually enriched bin. The relative extension index was defined as the extension index divided by the relative first-exon length, with the maximum set to 1.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
The THOC2 RIP-seq data in adult mouse testis and the H3K27ac ChIP-seq data in adult rhesus testis have been deposited in the GEO with the accession GSE147724. All public data used in this study were downloaded from GEO and ENCODE, and are shown in Supplementary Fig. 10 and Supplementary Data 3 with their accessions. The data supporting the findings of this study are available from the corresponding authors upon reasonable request.
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We thank the members of the Weng, Zamore, and Theurkauf laboratories for their critical comments. This work was supported in part by Chinese National Natural Science Foundation grants (31571362 and 31871296) to Z.W. and National Institutes of Health grant P01 HD078253 to W.E.T, Z.W., and P.D.Z.
Yu Fu
Present address: Oncology Drug Discovery Unit, Takeda Pharmaceuticals, Cambridge, MA, 02139, USA
These authors contributed equally: Tianxiong Yu, Kaili Fan.
Department of Thoracic Surgery, Clinical Translational Research Center, Shanghai Pulmonary Hospital, The School of Life Sciences and Technology, Tongji University, 200092, Shanghai, China
Tianxiong Yu, Kaili Fan & Zhiping Weng
Program in Bioinformatics and Integrative Biology, University of Massachusetts Medical School, Worcester, MA, 01605, USA
Tianxiong Yu, Kaili Fan, Yu Fu & Zhiping Weng
RNA Therapeutics Institute, University of Massachusetts Medical School, Worcester, MA, 01605, USA
Deniz M. Özata & Phillip D. Zamore
Program in Molecular Medicine, University of Massachusetts Medical School, Worcester, MA, 01605, USA
Gen Zhang & William E. Theurkauf
Bioinformatics Program, Boston University, 44 Cummington Mall, Boston, MA, 02215, USA
Tianxiong Yu
Kaili Fan
Deniz M. Özata
Gen Zhang
William E. Theurkauf
Phillip D. Zamore
Zhiping Weng
T.Y., K.F., P.D.Z., and Z.W. conceived the project. T.Y. and K.F. performed computational analyses. Y.F. aided computational analyses. D.M.O. performed the Rhesus H3K27ac ChIP-seq experiment. G.Z. and W.E.T. performed the THOC2 RIP-seq experiment. T.Y., K.F., P.D.Z., and Z.W. wrote the manuscript.
Correspondence to William E. Theurkauf or Phillip D. Zamore or Zhiping Weng.
Peer review information Nature Communications thanks Benjamin Czech, Molly Hammell and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Description of Additional Supplementary Files
Supplementary Data 1
Yu, T., Fan, K., Özata, D.M. et al. Long first exons and epigenetic marks distinguish conserved pachytene piRNA clusters from other mammalian genes. Nat Commun 12, 73 (2021). https://doi.org/10.1038/s41467-020-20345-3
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arXiv.org > astro-ph > arXiv:1908.06988
Astrophysics > Solar and Stellar Astrophysics
Title:Super-Earth ingestion can explain the anomalously high metal abundances of M67 Y2235
Authors:Ross P. Church, Alexander J. Mustill, Fan Liu
(Submitted on 19 Aug 2019)
Abstract: We investigate the hypothesis that ingestion of a terrestrial or super-Earth planet could cause the anomalously high metal abundances seen in a turn-off star in the open cluster M67, when compared to other turn-off stars in the same cluster. We show that the mass in convective envelope of the star is likely only $3.45\,\times 10^{-3}\,{\rm M}_\odot$, and hence $5.2\,{\rm M}_\oplus$ of rock is required to obtain the observed 0.128 dex metal enhancement. Rocky planets dissolve entirely in the convective envelope if they enter it with sufficiently tangential orbits: we find that the critical condition for dissolution is that the planet's radial speed must be less than 40% of its total velocity at the stellar surface; or, equivalently, the impact parameter must be greater than about 0.9. We model the delivery of rocky planets to the stellar surface both by planet-planet scattering in a realistic multi-planet system, and by Lidov-Kozai cycles driven by a more massive planetary or stellar companion. In both cases almost all planets that are ingested arrive at the star on grazing orbits and hence will dissolve in the surface convection zone. We conclude that super-Earth ingestion is a good explanation for the metal enhancement in M67 Y2235, and that a high-resolution spectroscopic survey of stellar abundances around the turn-off and main sequence of M67 has the potential to constrain the frequency of late-time dynamical instability in planetary systems.
Comments: 12 pages, 10 figures, submitted to MNRAS
Subjects: Solar and Stellar Astrophysics (astro-ph.SR); Earth and Planetary Astrophysics (astro-ph.EP)
DOI: 10.1093/mnras/stz3169
Cite as: arXiv:1908.06988 [astro-ph.SR]
(or arXiv:1908.06988v1 [astro-ph.SR] for this version)
From: Ross Church [view email]
[v1] Mon, 19 Aug 2019 18:00:02 UTC (733 KB)
astro-ph.SR
astro-ph.EP | CommonCrawl |
Bistable reaction equations with doubly nonlinear diffusion
Alessandro Audrito
Dipartimento di Matematica "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italia
Dedicated to Professor Juan Luis Vázquez
Received January 2018 Revised November 2018 Published February 2019
Figure(11)
Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $ 0\leq u(x, t)\leq 1 $ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [5]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries.
Finally, as a complement of [5], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").
Keywords: Bistable equations, doubly nonlinear diffusion, free boundary, long-time behaviour, travelling waves.
Mathematics Subject Classification: Primary: 35K57, 35K65; Secondary: 35C07.
Citation: Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124
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$ (m, p-1) $-plane. The yellow and orange area are called "fast diffusion" and "very fast diffusion" range, respectively, and they will not be studied in this paper">Figure 1. The "slow diffusion" area and the "pseudo-linear" line in $ (m, p-1) $-plane. The yellow and orange area are called "fast diffusion" and "very fast diffusion" range, respectively, and they will not be studied in this paper
Figure 2. Qualitative representation of the reactions of type C and type C', respectively.
Figure 3. Examples of admissible TWs: Finite and Positive types
Figure 4. Reactions of type C, range $ \gamma > 0 $, case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3, 0.7 $. The second case is excluded by the assumption $ \int_0^1u^{m-1}f(u)du > 0 $
Figure 6. Reactions of type C, range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first two pictures show the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 5. Reactions of type C, range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 7. Reactions of type C, range $ \gamma = 0$ , case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $.
Figure 8. Reactions of type C, range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first picture shows the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 9. Reactions of type C', range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 10. Reactions of type C', range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 11. Reactions of type C', range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373
A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185
Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775
Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15
Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469
H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121
Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873
Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273
Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357
Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265
Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102
Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete & Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2021209
Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360
Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088
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Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920 | CommonCrawl |
It is not because of the few thousand francs which would have to be spent to put a roof [!] over the third-class carriages or to upholster the third-class seats that some company or other has open carriages with wooden benches. What the company is trying to do is to prevent the passengers who can pay the second class fare from traveling third class; it hits the poor, not because it wants to hurt them, but to frighten the rich. And it is again for the same reason that the companies, having proved almost cruel to the third-class passengers and mean to the second-class ones, become lavish in dealing with first-class passengers. Having refused the poor what is necessary, they give the rich what is superfluous.
Like caffeine, nicotine tolerates rapidly and addiction can develop, after which the apparent performance boosts may only represent a return to baseline after withdrawal; so nicotine as a stimulant should be used judiciously, perhaps roughly as frequent as modafinil. Another problem is that nicotine has a half-life of merely 1-2 hours, making regular dosing a requirement. There is also some elevated heart-rate/blood-pressure often associated with nicotine, which may be a concern. (Possible alternatives to nicotine include cytisine, 2'-methylnicotine, GTS-21, galantamine, Varenicline, WAY-317,538, EVP-6124, and Wellbutrin, but none have emerged as clearly superior.)
Caffeine dose dependently decreased the 1,25(OH)(2)D(3) induced VDR expression and at concentrations of 1 and 10mM, VDR expression was decreased by about 50-70%, respectively. In addition, the 1,25(OH)(2)D(3) induced alkaline phosphatase activity was also reduced at similar doses thus affecting the osteoblastic function. The basal ALP activity was not affected with increasing doses of caffeine. Overall, our results suggest that caffeine affects 1,25(OH)(2)D(3) stimulated VDR protein expression and 1,25(OH)(2)D(3) mediated actions in human osteoblast cells.
Racetams, such as piracetam, oxiracetam, and aniracetam, which are often marketed as cognitive enhancers and sold over-the-counter. Racetams are often referred to as nootropics, but this property is not well established.[31] The racetams have poorly understood mechanisms, although piracetam and aniracetam are known to act as positive allosteric modulators of AMPA receptors and appear to modulate cholinergic systems.[32]
Table 4 lists the results of 27 tasks from 23 articles on the effects of d-AMP or MPH on working memory. The oldest and most commonly used type of working memory task in this literature is the Sternberg short-term memory scanning paradigm (Sternberg, 1966), in which subjects hold a set of items (typically letters or numbers) in working memory and are then presented with probe items, to which they must respond "yes" (in the set) or "no" (not in the set). The size of the set, and hence the working memory demand, is sometimes varied, and the set itself may be varied from trial to trial to maximize working memory demands or may remain fixed over a block of trials. Taken together, the studies that have used a version of this task to test the effects of MPH and d-AMP on working memory have found mixed and somewhat ambiguous results. No pattern is apparent concerning the specific version of the task or the specific drug. Four studies found no effect (Callaway, 1983; Kennedy, Odenheimer, Baltzley, Dunlap, & Wood, 1990; Mintzer & Griffiths, 2007; Tipper et al., 2005), three found faster responses with the drugs (Fitzpatrick, Klorman, Brumaghim, & Keefover, 1988; Ward et al., 1997; D. E. Wilson et al., 1971), and one found higher accuracy in some testing sessions at some dosages, but no main effect of drug (Makris et al., 2007). The meaningfulness of the increased speed of responding is uncertain, given that it could reflect speeding of general response processes rather than working memory–related processes. Aspects of the results of two studies suggest that the effects are likely due to processes other than working memory: D. E. Wilson et al. (1971) reported comparable speeding in a simple task without working memory demands, and Tipper et al. (2005) reported comparable speeding across set sizes.
MPH was developed more recently and marketed primarily for ADHD, although it is sometimes prescribed off label or used nonmedically to increase alertness, energy, or concentration in conditions other than ADHD. Both MPH and AMP are on the list of substances banned from sports competitions by the World Anti-Doping Agency (Docherty, 2008). Both also have the potential for abuse and dependence, which detracts from their usefulness and is the reason for their classification as Schedule II controlled substances. Although the risk of developing dependence on these drugs is believed to be low for individuals taking them for ADHD, the Schedule II classification indicates that these drugs have a high potential for abuse and that abuse may lead to severe dependence.
Sounds too good to be true? Welcome to the world of 'Nootropics' popularly known as 'Smart Drugs' that can help boost your brain's power. Do you recall the scene from the movie Limitless, where Bradley Cooper's character uses a smart drug that makes him brilliant? Yes! The effect of Nootropics on your brain is such that the results come as a no-brainer.
I can only talk from experience here, but I can remember being a teenager and just being a straight-up dick to any recruiters that came to my school. And I came from a military family. I'd ask douche-bag questions, I'd crack jokes like so... don't ask, don't tell only applies to everyone BUT the Navy, right? I never once considered enlisting because some 18 or 19 year old dickhead on hometown recruiting was hanging out in the cafeteria or hallways of my high school.Weirdly enough, however, what kinda put me over the line and made me enlist was the location of the recruiters' office. In the city I was living in at the time, the Armed Forces Recruitment Center was next door to an all-ages punk venue that I went to nearly every weekend. I spent many Saturday nights standing in a parking lot after a show, all bruised and bloody from a pit, smoking a joint, and staring at the windows of the closed recruiters' office. Propaganda posters of guys in full-battle-rattle obscured by a freshly scrawled Anarchy symbol or a collage of band stickers over the glass.I think trying to recruit kids from school has a child-molester-vibe to it. At least it did for me. But the recruiters defiantly being right next to a bunch of drunk and high punks, that somehow made it seem more like a truly bad-ass option. Like, sure, I'll totally join. After all, these guys don't run from the horde of skins and pins that descend every weekend like everyone else, they must be bad-ass.
That is, perhaps light of the right wavelength can indeed save the brain some energy by making it easier to generate ATP. Would 15 minutes of LLLT create enough ATP to make any meaningful difference, which could possibly cause the claimed benefits? The problem here is like that of the famous blood-glucose theory of willpower - while the brain does indeed use up more glucose while active, high activity uses up very small quantities of glucose/energy which doesn't seem like enough to justify a mental mechanism like weak willpower.↩
Flow diagram of cognitive neuroscience literature search completed July 2, 2010. Search terms were dextroamphetamine, Aderrall, methylphenidate, or Ritalin, and cognitive, cognition, learning, memory, or executive function, and healthy or normal. Stages of subsequent review used the information contained in the titles, abstracts, and articles to determine whether articles reported studies meeting the inclusion criteria stated in the text.
Nootropics are becoming increasingly popular as a tool for improving memory, information recall, and focus. Though research has not yet determined the mechanism for how nootropics work, it is clear that they provide significant cognitive benefits. Additionally, through a variety of hypothesized biological mechanisms, these compounds are thought to have the potential to improve vision.
Between midnight and 1:36 AM, I do four rounds of n-back: 50/39/30/55%. I then take 1/4th of the pill and have some tea. At roughly 1:30 AM, AngryParsley linked a SF anthology/novel, Fine Structure, which sucked me in for the next 3-4 hours until I finally finished the whole thing. At 5:20 AM, circumstances forced me to go to bed, still having only taken 1/4th of the pill and that determines this particular experiment of sleep; I quickly do some n-back: 29/20/20/54/42. I fall asleep in 13 minutes and sleep for 2:48, for a ZQ of 28 (a full night being ~100). I did not notice anything from that possible modafinil+caffeine interaction. Subjectively upon awakening: I don't feel great, but I don't feel like 2-3 hours of sleep either. N-back at 10 AM after breakfast: 25/54/44/38/33. These are not very impressive, but seem normal despite taking the last armodafinil ~9 hours ago; perhaps the 3 hours were enough. Later that day, at 11:30 PM (just before bed): 26/56/47.
There is a similar substance which can be purchased legally almost anywhere in the world called adrafinil. This is a prodrug for modafinil. You can take it, and then the body will metabolize it into modafinil, providing similar beneficial effects. Unfortunately, it takes longer for adrafinil to kick in—about an hour—rather than a matter of minutes. In addition, there are more potential side-effects to taking the prodrug as compared to the actual drug.
Fortunately, there are some performance-enhancing habits that have held up under rigorous scientific scrutiny. They are free, and easy to pronounce. Unfortunately, they are also the habits you were perhaps hoping to forego by using nootropics instead. "Of all the things that are supposed to be 'good for the brain,'" says Stanford neurology professor Sharon Sha, "there is more evidence for exercise than anything else." Next time you're facing a long day, you could take a pill and see what happens.
Price discrimination is aided by barriers such as ignorance and oligopolies. An example of the former would be when I went to a Food Lion grocery store in search of spices, and noticed that there was a second selection of spices in the Hispanic/Latino ethnic food aisle, with unit prices perhaps a fourth of the regular McCormick-brand spices; I rather doubt that regular cinnamon varies that much in quality. An example of the latter would be using veterinary drugs on humans - any doctor to do so would probably be guilty of medical malpractice even if the drugs were manufactured in the same factories (as well they might be, considering economies of scale). Similarly, we can predict that whenever there is a veterinary drug which is chemically identical to a human drug, the veterinary drug will be much cheaper, regardless of actual manufacturing cost, than the human drug because pet owners do not value their pets more than themselves. Human drugs are ostensibly held to a higher standard than veterinary drugs; so if veterinary prices are higher, then there will be an arbitrage incentive to simply buy the cheaper human version and downgrade them to veterinary drugs.
Speaking of addictive substances, some people might have considered cocaine a nootropic (think: the finance industry in Wall Street in the 1980s). The incredible damage this drug can do is clear, but the plant from which it comes has been used to make people feel more energetic and less hungry, and to counteract altitude sickness in Andean South American cultures for 5,000 years, according to an opinion piece that Bolivia's president, Evo Morales Ayma, wrote for the New York Times.
The next morning, four giant pills' worth of the popular piracetam-and-choline stack made me... a smidge more alert, maybe? (Or maybe that was just the fact that I had slept pretty well the night before. It was hard to tell.) Modafinil, which many militaries use as their "fatigue management" pill of choice, boasts glowing reviews from satisfied users. But in the United States, civilians need a prescription to get it; without one, they are stuck using adrafinil, a precursor substance that the body metabolizes into modafinil after ingestion. Taking adrafinil in lieu of coffee just made me keenly aware that I hadn't had coffee.
Maj. Jamie Schwandt, USAR, is a logistics officer and has served as an operations officer, planner and commander. He is certified as a Department of the Army Lean Six Sigma Master Black Belt, certified Red Team Member, and holds a doctorate from Kansas State University. This article represents his own personal views, which are not necessarily those of the Department of the Army.
The majority of studies seem to be done on types of people who are NOT buying nootropics. Like the elderly, people with blatant cognitive deficits, etc. This is analogous to some of the muscle-building research but more extreme. Like there are studies on some compound increasing muscle growth in elderly patients or patients with wasting, and supplement companies use some of those studies to back their supplements.
Neuroplasticity, or the brain's ability to change and reorganize itself in response to intrinsic and extrinsic factors, indicates great potential for us to enhance brain function by medical or other interventions. Psychotherapy has been shown to induce structural changes in the brain. Other interventions that positively influence neuroplasticity include meditation, mindfulness , and compassion.
NGF may sound intriguing, but the price is a dealbreaker: at suggested doses of 1-100μg (NGF dosing in humans for benefits is, shall we say, not an exact science), and a cost from sketchy suppliers of $1210/100μg/$470/500μg/$750/1000μg/$1000/1000μg/$1030/1000μg/$235/20μg. (Levi-Montalcini was presumably able to divert some of her lab's production.) A year's supply then would be comically expensive: at the lowest doses of 1-10μg using the cheapest sellers (for something one is dumping into one's eyes?), it could cost anywhere up to $10,000.
So what about the flip side: a drug to erase bad memories? It may have failed Jim Carrey in Eternal Sunshine of the Spotless Mind, but neuroscientists have now discovered an amnesia drug that can dull the pain of traumatic events. The drug, propranolol, was originally used to treat high blood pressure and heart disease. Doctors noticed that patients given the drug suffered fewer signs of stress when recalling a trauma.
"I have a bachelors degree in Nutrition Science. Cavin's Balaster's How to Feed a Brain is one the best written health nutrition books that I have ever read. It is evident that through his personal journey with a TBI and many years of research Cavin has gained a great depth of understanding on the biomechanics of nutrition has how it relates to the structure of the brain and nervous system, as well as how all of the body systems intercommunicate with one another. He then takes this complicated knowledge and breaks it down into a concise and comprehensive book. If you or your loved one is suffering from ANY neurological disorder or TBI please read this book."
"How to Feed a Brain is an important book. It's the book I've been looking for since sustaining multiple concussions in the fall of 2013. I've dabbled in and out of gluten, dairy, and (processed) sugar free diets the past few years, but I have never eaten enough nutritious foods. This book has a simple-to-follow guide on daily consumption of produce, meat, and water.
A fundamental aspect of human evolution has been the drive to augment our capabilities. The neocortex is the neural seat of abstract and higher order cognitive processes. As it grew, so did our ability to create. The invention of tools and weapons, writing, the steam engine, and the computer have exponentially increased our capacity to influence and understand the world around us. These advances are being driven by improved higher-order cognitive processing.1Fascinatingly, the practice of modulating our biology through naturally occurring flora predated all of the above discoveries. Indeed, Sumerian clay slabs as old as 5000 BC detail medicinal recipes which include over 250 plants2. The enhancement of human cognition through natural compounds followed, as people discovered plants containing caffeine, theanine, and other cognition-enhancing, or nootropic, agents.
Low-dose lithium orotate is extremely cheap, ~$10 a year. There is some research literature on it improving mood and impulse control in regular people, but some of it is epidemiological (which implies considerable unreliability); my current belief is that there is probably some effect size, but at just 5mg, it may be too tiny to matter. I have ~40% belief that there will be a large effect size, but I'm doing a long experiment and I should be able to detect a large effect size with >75% chance. So, the formula is NPV of the difference between taking and not taking, times quality of information, times expectation: \frac{10 - 0}{\ln 1.05} \times 0.75 \times 0.40 = 61.4, which justifies a time investment of less than 9 hours. As it happens, it took less than an hour to make the pills & placebos, and taking them is a matter of seconds per week, so the analysis will be the time-consuming part. This one may actually turn a profit.
Up to 20% of Ivy League college students have already tried "smart drugs," so we can expect these pills to feature prominently in organizations (if they don't already). After all, the pressure to perform is unlikely to disappear the moment students graduate. And senior employees with demanding jobs might find these drugs even more useful than a 19-year-old college kid does. Indeed, a 2012 Royal Society report emphasized that these "enhancements," along with other technologies for self-enhancement, are likely to have far-reaching implications for the business world.
The experiment then is straightforward: cut up a fresh piece of gum, randomly select from it and an equivalent dry piece of gum, and do 5 rounds of dual n-back to test attention/energy & WM. (If it turns out to be placebo, I'll immediately use the remaining active dose: no sense in wasting gum, and this will test whether nigh-daily use renders nicotine gum useless, similar to how caffeine may be useless if taken daily. If there's 3 pieces of active gum left, then I wrap it very tightly in Saran wrap which is sticky and air-tight.) The dose will be 1mg or 1/4 a gum. I cut up a dozen pieces into 4 pieces for 48 doses and set them out to dry. Per the previous power analyses, 48 groups of DNB rounds likely will be enough for detecting small-medium effects (partly since we will be only looking at one metric - average % right per 5 rounds - with no need for multiple correction). Analysis will be one-tailed, since we're looking for whether there is a clear performance improvement and hence a reason to keep using nicotine gum (rather than whether nicotine gum might be harmful).
As with any thesis, there are exceptions to this general practice. For example, theanine for dogs is sold under the brand Anxitane is sold at almost a dollar a pill, and apparently a month's supply costs $50+ vs $13 for human-branded theanine; on the other hand, this thesis predicts downgrading if the market priced pet versions higher than human versions, and that Reddit poster appears to be doing just that with her dog.↩
Autism Brain brain fuel brain health Brain Injury broth Cholesterol choline DAI DHA Diabetes digestion Exercise Fat Functional Medicine gastric Gluten gut-brain Gut Brain Axis gut health Health intestinal permeability keto Ketogenic leaky Gut Learning Medicine Metabolism Music Therapy neurology Neuroplasticity neurorehabilitation Nutrition omega Paleo Physical Therapy Recovery Science second brain superfood synaptogenesis TBI Therapy tube feed uridine
Rabiner et al. (2009) 2007 One public and one private university undergraduates (N = 3,390) 8.9% (while in college), 5.4% (past 6 months) Most common reasons endorsed: to concentrate better while studying, to be able to study longer, to feel less restless while studying 48%: from a friend with a prescription; 19%: purchased it from a friend with a prescription; 6%: purchased it from a friend without a prescription
Historically used to help people with epilepsy, piracetam is used in some cases of myoclonus, or muscle twitching. Its actual mechanism of action is unclear: It doesn't act exactly as a sedative or stimulant, but still influences cognitive function, and is believed to act on receptors for acetylcholine in the brain. Piracetam is used off-label as a 'smart drug' to help focus and concentration or sometimes as a way to allegedly boost your mood. Again, piracetam is a prescription-only drug - any supply to people without a prescription is illegal, and supplying it may result in a fine or prison sentence.
Either way, if more and more people use these types of stimulants, there may be a risk that we will find ourselves in an ever-expanding neurological arm's race, argues philosophy professor Nicole Vincent. But is this necessarily a bad thing? No, says Farahany, who sees the improvement in cognitive functioning as a social good that we should pursue. Better brain functioning would result in societal benefits, she argues, "like economic gains or even reducing dangerous errors."
Factor analysis. The strategy: read in the data, drop unnecessary data, impute missing variables (data is too heterogeneous and collected starting at varying intervals to be clean), estimate how many factors would fit best, factor analyze, pick the ones which look like they match best my ideas of what productive is, extract per-day estimates, and finally regress LLLT usage on the selected factors to look for increases.
The U.S. Centers for Disease Control and Prevention estimates that gastrointestinal diseases affect between 60 and 70 million Americans every year. This translates into tens of millions of endoscopy procedures. Millions of colonoscopy procedures are also performed to diagnose or screen for colorectal cancers. Conventional, rigid scopes used for these procedures are uncomfortable for patients and may cause internal bruising or lead to infection because of reuse on different patients. Smart pills eliminate the need for invasive procedures: wireless communication allows the transmission of real-time information; advances in batteries and on-board memory make them useful for long-term sensing from within the body. The key application areas of smart pills are discussed below.
Coconut oil was recommended by Pontus Granström on the Dual N-Back mailing list for boosting energy & mental clarity. It is fairly cheap (~$13 for 30 ounces) and tastes surprisingly good; it has a very bad reputation in some parts, but seems to be in the middle of a rehabilitation. Seth Robert's Buttermind experiment found no mental benefits to coconut oil (and benefits to eating butter), but I wonder.
The fish oil can be considered a free sunk cost: I would take it in the absence of an experiment. The empty pill capsules could be used for something else, so we'll put the 500 at $5. Filling 500 capsules with fish and olive oil will be messy and take an hour. Taking them regularly can be added to my habitual morning routine for vitamin D and the lithium experiment, so that is close to free but we'll call it an hour over the 250 days. Recording mood/productivity is also free a sunk cost as it's necessary for the other experiments; but recording dual n-back scores is more expensive: each round is ~2 minutes and one wants >=5, so each block will cost >10 minutes, so 18 tests will be >180 minutes or >3 hours. So >5 hours. Total: 5 + (>5 \times 7.25) = >41.
If you're considering taking pharmaceutical nootropics, then it's important that you learn as much as you can about how they work and that you seek professional advice before taking them. Be sure to read the side effects and contraindications of the nootropic that you are considering taking, and do not use it if you have any pre-existing medical conditions or allergies. If you're taking other medications, then discuss your plans with a doctor or pharmacist to make sure that your nootropic is safe for you to use.
The real-life Limitless Pill? One of the newer offerings in the nootropic industry, Avanse Laboratories' new ingenious formula has been generating quite much popularity on the internet, and has been buzzing around on dedicated nootropic forums. Why do we pick this awesome formula to be the #1 nootropic supplement of 2017 and 2018? Simple, name another supplement that contains "potent 1160mg capsule" including 15 mg of world's most powerful nootropic agent (to find out, please click on Learn More). It is cheap, in our opinion, compared to what it contains. And we don't think their price will stay this low for long. Avanse Laboratories is currently playing… Learn More...
Using prescription ADHD medications, racetams, and other synthetic nootropics can boost brain power. Yes, they can work. Even so, we advise against using them long-term since the research on their safety is still new. Use them at your own risk. For the majority of users, stick with all natural brain supplements for best results. What is your favorite smart pill for increasing focus and mental energy? Tell us about your favorite cognitive enhancer in the comments below.
Nature magazine conducted a poll asking its readers about their cognitive-enhancement practices and their attitudes toward cognitive enhancement. Hundreds of college faculty and other professionals responded, and approximately one fifth reported using drugs for cognitive enhancement, with Ritalin being the most frequently named (Maher, 2008). However, the nature of the sample—readers choosing to answer a poll on cognitive enhancement—is not representative of the academic or general population, making the results of the poll difficult to interpret. By analogy, a poll on Vermont vacations, asking whether people vacation in Vermont, what they think about Vermont, and what they do if and when they visit, would undoubtedly not yield an accurate estimate of the fraction of the population that takes its vacations in Vermont.
Attention-deficit/hyperactivity disorder (ADHD), a behavioral syndrome characterized by inattention and distractibility, restlessness, inability to sit still, and difficulty concentrating on one thing for any period of time. ADHD most commonly occurs in children, though an increasing number of adults are being diagnosed with the disorder. ADHD is three times more…
Amphetamines have a long track record as smart drugs, from the workaholic mathematician Paul Erdös, who relied on them to get through 19-hour maths binges, to the writer Graham Greene, who used them to write two books at once. More recently, there are plenty of anecdotal accounts in magazines about their widespread use in certain industries, such as journalism, the arts and finance.
Now, what is the expected value (EV) of simply taking iodine, without the additional work of the experiment? 4 cans of 0.15mg x 200 is $20 for 2.1 years' worth or ~$10 a year or a NPV cost of $205 (\frac{10}{\ln 1.05}) versus a 20% chance of $2000 or $400. So the expected value is greater than the NPV cost of taking it, so I should start taking iodine.
For illustration, consider amphetamines, Ritalin, and modafinil, all of which have been proposed as cognitive enhancers of attention. These drugs exhibit some positive effects on cognition, especially among individuals with lower baseline abilities. However, individuals of normal or above-average cognitive ability often show negligible improvements or even decrements in performance following drug treatment (for details, see de Jongh, Bolt, Schermer, & Olivier, 2008). For instance, Randall, Shneerson, and File (2005) found that modafinil improved performance only among individuals with lower IQ, not among those with higher IQ. [See also Finke et al 2010 on visual attention.] Farah, Haimm, Sankoorikal, & Chatterjee 2009 found a similar nonlinear relationship of dose to response for amphetamines in a remote-associates task, with low-performing individuals showing enhanced performance but high-performing individuals showing reduced performance. Such ∩-shaped dose-response curves are quite common (see Cools & Robbins, 2004)
"As a neuro-optometrist who cares for many brain-injured patients experiencing visual challenges that negatively impact the progress of many of their other therapies, Cavin's book is a god-send! The very basic concept of good nutrition among all the conflicting advertisements and various "new" food plans and diets can be enough to put anyone into a brain fog much less a brain injured survivor! Cavin's book is straightforward and written from not only personal experience but the validation of so many well-respected contemporary health care researchers and practitioners! I will certainly be recommending this book as a "Survival/Recovery 101" resource for all my patients including those without brain injuries because we all need optimum health and well-being and it starts with proper nourishment! Kudos to Cavin Balaster!"
This formula presents a relatively high price and one bottle of 60 tables, at the recommended dosage of two tablets per day with a meal, a bottle provides a month's supply. The secure online purchase is available on the manufacturer's site as well as at several online retailers. Although no free trials or money back guarantees are available at this time, the manufacturer provides free shipping if the desired order exceeds a certain amount. With time different online retailers could offer some advantages depending on the amount purchased, so an online research is advised before purchase, as to assess the market and find the best solution.
If you could take a drug to boost your brainpower, would you? This question, faced by Bradley Cooper's character in the big-budget movie Limitless, is now facing students who are frantically revising for exams. Although they are nowhere near the strength of the drug shown in the film, mind-enhancing drugs are already on the pharmacy shelves, and many people are finding the promise of sharper thinking through chemistry highly seductive.
Remembering what Wedrifid told me, I decided to start with a quarter of a piece (~1mg). The gum was pretty tasteless, which ought to make blinding easier. The effects were noticeable around 10 minutes - greater energy verging on jitteriness, much faster typing, and apparent general quickening of thought. Like a more pleasant caffeine. While testing my typing speed in Amphetype, my speed seemed to go up >=5 WPM, even after the time penalties for correcting the increased mistakes; I also did twice the usual number without feeling especially tired. A second dose was similar, and the third dose was at 10 PM before playing Ninja Gaiden II seemed to stop the usual exhaustion I feel after playing through a level or so. (It's a tough game, which I have yet to master like Ninja Gaiden Black.) Returning to the previous concern about sleep problems, though I went to bed at 11:45 PM, it still took 28 minutes to fall sleep (compared to my more usual 10-20 minute range); the next day I use 2mg from 7-8PM while driving, going to bed at midnight, where my sleep latency is a more reasonable 14 minutes. I then skipped for 3 days to see whether any cravings would pop up (they didn't). I subsequently used 1mg every few days for driving or Ninja Gaiden II, and while there were no cravings or other side-effects, the stimulation definitely seemed to get weaker - benefits seemed to still exist, but I could no longer describe any considerable energy or jitteriness.
So it's no surprise that as soon as medical science develops a treatment for a disease, we often ask if it couldn't perhaps make a healthy person even healthier. Take Viagra, for example: developed to help men who couldn't get erections, it's now used by many who function perfectly well without a pill but who hope it will make them exceptionally virile.
Organizations, and even entire countries, are struggling with "always working" cultures. Germany and France have adopted rules to stop employees from reading and responding to email after work hours. Several companies have explored banning after-hours email; when one Italian company banned all email for one week, stress levels dropped among employees. This is not a great surprise: A Gallup study found that among those who frequently check email after working hours, about half report having a lot of stress.
First half at 6 AM; second half at noon. Wrote a short essay I'd been putting off and napped for 1:40 from 9 AM to 10:40. This approach seems to work a little better as far as the aboulia goes. (I also bother to smell my urine this time around - there's a definite off smell to it.) Nights: 10:02; 8:50; 10:40; 7:38 (2 bad nights of nasal infections); 8:28; 8:20; 8:43 (▆▃█▁▂▂▃).
Either prescription or illegal, daily use of testosterone would not be cheap. On the other hand, if I am one of the people for whom testosterone works very well, it would be even more valuable than modafinil, in which case it is well worth even arduous experimenting. Since I am on the fence on whether it would help, this suggests the value of information is high.
No. There are mission essential jobs that require you to live on base sometimes. Or a first term person that is required to live on base. Or if you have proven to not be as responsible with rent off base as you should be so your commander requires you to live on base. Or you're at an installation that requires you to live on base during your stay. Or the only affordable housing off base puts you an hour away from where you work. It isn't simple. The fact that you think it is tells me you are one of the "dumb@$$es" you are referring to above.
How should the mixed results just summarized be interpreted vis-á-vis the cognitive-enhancing potential of prescription stimulants? One possibility is that d-AMP and MPH enhance cognition, including the retention of just-acquired information and some or all forms of executive function, but that the enhancement effect is small. If this were the case, then many of the published studies were underpowered for detecting enhancement, with most samples sizes under 50. It follows that the observed effects would be inconsistent, a mix of positive and null findings.
My worry about the MP variable is that, plausible or not, it does seem relatively weak against manipulation; other variables I could look at, like arbtt window-tracking of how I spend my computer time, # or size of edits to my files, or spaced repetition performance, would be harder to manipulate. If it's all due to MP, then if I remove the MP and LLLT variables, and summarize all the other variables with factor analysis into 2 or 3 variables, then I should see no increases in them when I put LLLT back in and look for a correlation between the factors & LLLT with a multivariate regression.
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One of the most common strategies to beat this is cycling. Users who cycle their nootropics take them for a predetermined period, (usually around five days) before taking a two-day break from using them. Once the two days are up, they resume the cycle. By taking a break, nootropic users reduce the tolerance for nootropics and lessen the risk of regression and tolerance symptoms. | CommonCrawl |
Mathematical modeling of auxetic systems: bridging the gap between analytical models and observation
James N. Grima-Cornish1,
Joseph N. Grima1,2 &
Daphne Attard ORCID: orcid.org/0000-0002-6288-58071
The Poisson's ratio, a property which quantifies the changes in thickness when a material is stretched and compressed, can be determined as the negative of the transverse strain over the applied strain. In the scientific literature, there are various ways how strain may be defined and the actual definition used could result in a different Poisson's ratio being computed. This paper will look in more detail at this by comparing the more commonly used forms of strain and the Poisson's ratio that is computable from them. More specifically, an attempt is made to assess through examples on the usefulness of the various formulations to properly describe what can actually be observed, thus providing a clearer picture of which form of Poisson's ratio should be used in analytical modelling.
The link between mathematics and science is an ancient one which continues to predominate in research today. This can be attributed to the fact that mathematics gives research an angle which cannot exist with qualitative research alone, that of the removal of uncertainty in the statements and theories given. Furthermore, with the inclusion of mathematics, statements can be communicated in a factual manner which could not be done otherwise. Moreover, mathematics manages to communicate findings in a more elegant manner, which is not always the case when it is given in other ways. The elegance of mathematics is what has, in many cases, pushed forward the understanding of many subjects in science, as well as other disciplines (Russell, 1919). Famous examples, as well as everyday examples, of this can be seen practically everywhere. To name a few: high-grade commercial coffee machines are manufactured with a high level of attention and calibrated in order to produce the exact same dose of coffee at the same temperature and pressure output every time in order to completely eliminate variation from one cup of coffee to the next (Fischer & Eugster, 1994; Pandolfi, 1988); restoration of famous monuments and structures also depends greatly on mathematical models in order to get every detail correct; acoustics, such as the manufacturing of great musical halls such as the Elbphilharmonie in Hamburg (Mack, 2018), as well as manufacturing of everyday speakers for a desktop computer both involve calculations which must be made to maximise the desired effect (Juszkiewicz & Ewen, 2002), albeit one requires more perfection than the other; and the deploying of rockets and shuttles to space in order to take satellites, people and other technologies into space which require intensive mathematical calculation to a great degree of accuracy for any mission to be a success (Wilson, 1964).
This is no less important in the field of thermo-mechanical metamaterials, the field of study relating to: 'engineered materials having previously unachievable [anomalous] thermal and/or mechanical properties that are defined by their microstructural architecture rather than their composition' (Zheng et al., 2014). Here, analytical modelling is used in order to break down mechanisms relating to structure down to mathematical relationships. This way, the effect of each structural variable can be looked at individually, and this information can be used to fine tune them to the required values for applications or to apply these models to other materials in order to better understand the mechanism underlying their behaviour. The above being said, a better understanding of these materials through this type of modelling allows for improvement in the designing of future applications and materials, and therefore, the overall bettering of contemporary technology in the endless search to do so.
Thermo-mechanical anomalous properties include a variety of such properties, one of which merits a particular mention is that of auxetic behaviour (Alderson et al., 2005; Alderson, Alderson, Ravirala, Simkins, & Davies, 2012; Alderson & Evans, 1992; Allen et al., 2016; Allen et al., 2017; Attard & Grima, 2012; Azzopardi, Brincat, Grima, & Gatt, 2015; Babaee et al., 2013; Baughman & Galvão, 1993; Bertoldi, Reis, Willshaw, & Mullin, 2010; Brańka, Heyes, Makowiak, Pieprzyk, & Wojciechowski, 2012; Brańka, Heyes, & Wojciechowski, 2009; Brańka, Heyes, & Wojciechowski, 2011; Dudek et al., 2017; Dudek et al., 2018; Evans, 1991; Greaves, Greer, Lakes, & Rouxel, 2011; Grima, Farrugia, Gatt, & Attard, 2008; Grima, Gatt, Alderson, & Evans, 2006; Grima, Grech, Grima-Cornish, Gatt, & Attard, 2018; Grima, Jackson, Alderson, & Evans, 2000; Ha, Plesha, & Lakes, 2016; Harkati, Daoudi, Bezazi, Haddad, & Scarpa, 2017; Hewage, Alderson, Alderson, & Scarpa, 2016; Hoover & Hoover, 2005; Ishibashi & Iwata, 2000; Kadic, Tiemo, Schittny, & Wegener, 2013; Kolken & Zadpoor, 2017; Lakes, 1987; Lim, 2013; Lim, 2015; Liu & Hu, 2010; Mizzi et al., 2018; Pasternak & Dyskin, 2019; Poźniak, Wojciechowski, Grima, & Mizzi, 2016; Qin, Sun, Liu, Li, & Liu, 2017; Qu, Kadic, Naber, & Wegener, 2017; Ryder & Tan, 2016; Sigmund, 1995; Strek, Maruszewski, Narojczyk, & Wojciechowski, 2008; Strek, Michalski, & Jopek, 2019; Taylor et al., 2014; Tretiakov & Wojciechowski, 2014; Verma, Shofner, & Griffin, 2014; Verma, Shofner, Lin, Wagner, & Griffin, 2015; Wojciechowski, 1987; Wojciechowski, 1989; Wojciechowski, 2003a; Wojciechowski, Tretiakov, & Kowalik, 2003; Zhang, Hu, Liu, & Xu, 2013). Auxetic behaviour, or auxeticity, is the property which describes a material becoming fatter when stretched and thinner when compressed (Evans, Nkansah, Hutcherson, & Rogers, 1991). This anomalous property manifests when the material or structure has what is known as a negative Poisson's ratio as described in more detail below. Since the coining of the term by Evans et al. (1991), numerous materials have been found or been designed to manifest this property, including foams, nanolayers, crystals and constructible macrostructures.
It has long been recognised that one of the best methods to study auxetics and related systems is via the formulation of mathematical models. A typical approach involves the representation of the salient geometric features of a material through a structural model and to then analyse the deformation mechanism afforded by this model structure. The success of this approach lies partly in the fact that auxeticity requires the right synergism between 'geometry' and 'deformation mechanism' (Alderson & Evans, 1995; Grima, Alderson, & Evans, 2005), but also due to the fact that, as stated above, a mathematical model can depict a system in an unequivocal manner which removes shadows of uncertainty.
This being said, there is an issue which is so far grossly unresolved: how to compute the Poisson's ratio in a meaningful and practical manner which can adequately be used for the derivation of mathematical models as well as to report experimental data. Such Poisson's ratio should be able to clearly describe the behaviour of the system, without overlooking or over-emphasising some critical aspects of the system. This in turn poses a question how strain should be reported, a problem that is particularly pertinent in the field of auxetics when, during uniaxial loading, there is a 'non-standard' response to loads. Circumstances which offer such a challenge include, for example, situations when during stretching, the sample would initially be visibly getting thinner and then, past a certain amount of stretching, the sample starts to get 'fatter'. This problem is amply discussed by Smith, Wootton, and Evans (1999) where recommendations were made on how experimental data should be analysed and reported so as to bring to light the auxetic properties of the test sample. The present work will look at the other side of this issue, that is, how it is best to perform and report mathematical models in a meaningful and elegant manner which is easy to correlate to experimental work and at the same time does not output Poisson's ratios which look highly fascinating (e.g. gigantic auxeticity) but are mere artefacts of the reporting protocol used. This will be performed in a manner which explains, step-by-step, how analytical modelling of simple systems can be performed by looking at two of the more well-known models for mechanical metamaterials, namely the hexagonal honeycomb (Abd El-Sayed, Jones, & Burgess, 1979; Evans, Alderson, & Christian, 1995; Gibson, Ashby, Schajer, & Robertson, 1982; Masters & Evans, 1996) deforming through changes in angle (i.e. idealised hinging model as considered by (Evans et al., 1995; Masters & Evans, 1996) and the rotating rectangles model (Type I, (Grima, Alderson, & Evans, 2004; Grima, Alderson, & Evans, 2005)).
Methods used for calculation of discrete strains and Poisson's ratios
The Poisson's ratio of a sample is a property which measures the extent of change in lateral dimensions in some particular cross-section, for a given direction of uniaxial stretching in the same cross-section. The accepted convention for the reporting of the Poisson's ratio requires the identification of these directions. In fact, for a 3D sample of measurements X, Y and Z in the x-, y- and z-directions, respectively, the Poisson's ratio in the Ox1-Ox2 plane for uniaxial stretching in the Ox2 direction is defined as, ν21, where ν21 can be determined as follows:
$$ {\nu}_{21}=-\frac{\mathrm{Resultant}\ \mathrm{strain}\ \mathrm{in}\ \mathrm{lateral}\ {Ox}_1\ \mathrm{direction}}{\mathrm{Applied}\ \mathrm{axial}\ \mathrm{strain}\ \mathrm{in}\ {Ox}_2\ \mathrm{direction}}=-\frac{\varepsilon_1}{\varepsilon_2} $$
where ε2 represents the applied uniaxial strain being applied in the Ox2 direction whilst ε1 represents the strain in the perpendicular Ox1 direction, i.e. a direction which is orthogonal to that where the uniaxial stain is being applied and which must lie in the plane where the Poisson's ratio is measured. The order of suffixes for the Poisson's ratio is important, where the first suffix by convention refers to the direction of stretching (i.e. corresponds to the strain in the denominator). The negative sign ensures that the vast majority of materials which tend to get thinner (−ve ε1) when stretched (+ve ε2) would have a positive Poisson's ratio. The Poisson's ratio is one of the fundamental mechanical properties and can range from − 1 ≤ ν ≤ + 0.5 for three-dimensional isotropic materials (Lempriere, 1968), − 1 ≤ ν ≤ + 1 for two dimensional materials (Wojciechowski, 2003b) and can take any value for anisotropic materials (Wojciechowski, 2003b).
The problem that arises is that, whilst there is uniformity in how the Poisson's ratio is determined from strains, there are various accepted conventions on how strains can be reported, which unfortunately would result in very distinct forms of the Poisson's ratio. To illustrate this, one may refer to an arbitrary linear system of original length Linit = L[0] which is being stretched and re-measured n successive times such that the new lengths are L[1], L[2], L[3], …, L[n] till reaching a final length Lfin = L[n]. These length measurements are given by (see Fig. 1 for definition of terms, noting that δL[i] refers to the change in L with respect to the previous length and ΔL[i] refers to the total change in L with respect to Linit):
$$ {\displaystyle \begin{array}{l}L\left[0\right]={L}_{init}\\ {}L\left[1\right]=L\left[0\right]+\delta L\left[1\right]={L}_{init}+\delta L\left[1\right]={L}_{init}+\Delta L\left[1\right]\\ {}L\left[2\right]=L\left[1\right]+\delta L\left[2\right]={L}_{init}+\delta L\left[1\right]+\delta L\left[2\right]={L}_{init}+\Delta L\left[2\right]\\ {}L\left[3\right]=L\left[2\right]+\delta L\left[3\right]={L}_{init}+\delta L\left[1\right]+\delta L\left[2\right]+\delta L\left[3\right]={L}_{init}+\Delta L\left[3\right]\\ {}\dots \\ {}L\left[k\right]=L\left[k-1\right]+\delta L\left[k\right]={L}_{init}+\sum \limits_{i=1}^k\delta L\left[k\right]={L}_{init}+\Delta L\left[k\right]\\ {}\dots \\ {}L\left[n\right]=L\left[n-1\right]+\delta L\left[n\right]={L}_{init}+\sum \limits_{i=1}^n\delta L\left[n\right]={L}_{init}+\Delta L\left[n\right]={L}_{fin}\end{array}} $$
A hypothetical linear system being stretched from a length of 100 to 150 mm where the strains are being computed at four irregular intervals along the deformation
Obviously, this type of experiment could also have been carried out in compression. From these successive measurements, which could have easily been recorded in an experimental procedure, as noted by Smith et al. (1999), one may compute strain in a number of ways including the following:
The 'engineering strain', also commonly referred to as the Cauchy strain or the nominal strain, the strain which is most widely used by the engineering and experimental community which is reported as the ratio of the 'extension over the original' length, defined at point k as follows:
$$ {e}^{eng}\left[k\right]=\frac{\Delta L\left[k\right]}{L_{init}} $$
This strain is also sometimes expressed as a percentage. The Poisson's ratio computed using such strain is normally referred to as the 'engineering Poisson's ratio'.
The 'instantaneous strain', also sometimes referred to as the 'incremental strain' where for any two successive length measurements, the strain at point k is defined as follows:
$$ \delta \varepsilon \left[k\right]=\frac{L\left[k\right]-L\left[k-1\right]}{L\left[k-1\right]}=\frac{\delta L\left[k\right]}{L\left[k-1\right]} $$
which strain, in the limit of infinitesimal δL, will become equivalent to the strain normally used in the derivation of analytical models, commonly referred to as the 'infinitesimally small strain' defined as follows:
$$ d\varepsilon =\frac{dL}{L}=\underset{\delta L\left[k\right]\to 0}{\lim}\left(\frac{\delta L\left[k\right]}{L\left[k-1\right]}\right) $$
The Poisson's ratio computed using such strain is sometimes referred to as the 'Instantaneous Poisson's ratio' or the 'Poisson's Function'.
The 'true strain', also commonly referred to as the Hencky strain, or, logarithmic strain, defined at point k as follows:
$$ {\displaystyle \begin{array}{l}{e}^{true}\left[k\right]=\underset{L\left[0\right]}{\overset{L\left[k\right]}{\int }} d\varepsilon =\underset{L\left[0\right]}{\overset{L\left[k\right]}{\int }}\frac{dL}{L}\\ {}\kern0.75em ={\left[\ln (L)\right]}_{L\left[0\right]}^{L\left[k\right]}=\ln \left(L\left[k\right]\right)-\ln \left(L\left[0\right]\right)=\ln \left(\frac{L\left[k\right]}{L\left[0\right]}\right)=\ln \left(\frac{L\left[0\right]+\Delta L\left[k\right]}{L\left[0\right]}\right)=\ln \left(1+\frac{\Delta L\left[k\right]}{L\left[0\right]}\right)\\ {}\kern0.75em =\ln \left(1+{e}^{eng}\right)\end{array}} $$
The Poisson's ratio computed using such strain is sometimes referred to as the 'true Poisson's ratio'.
Methods used for calculation of the Poisson's ratio properties of periodic systems
This section presents a step-by-step guide for evaluating the Poisson's ratios of periodic 2D systems, exemplified through the hinging honeycomb system using the different methods stated above in a manner which can be easily reproduced and extended using the other models. This will be followed by the reporting of the equivalent expressions for strains and Poisson's ratios derived with the same procedures. These expressions for the Poisson's ratio based on the different methods to compute strain may hence be compared so as to assess their relative ability to describe the behaviour upon uniaxial loading well, in particular the Poisson's ratio. This process will then be repeated for the type I rotating rectangles system, so as to further illustrate the methodology and the concepts presented.
Hexagonal honeycombs
Over the years, there have been various studies on the re-entrant and non-re-entrant honeycomb system so as to study their mechanical behaviour, where deformation is typically assumed to be trough flexure of the ligaments (Abd El-Sayed et al., 1979; Evans et al., 1995; Gibson et al., 1982; Masters & Evans, 1996) or changes in angles between the ligaments, i.e. as an idealised hinging model (Evans et al., 1995; Masters & Evans, 1996). The honeycomb model structure, like many others studied for similar purposes, can be described as a finite system containing a finite number of honeycomb cells, or, as an infinity of cells where a 'representative repeat unit' is tessellated to form a space-filling model. The latter representation is normally being the preferred version, for various reasons ranging from mathematical elegance in the model to their ability to represent nano- or mirco-scale honeycombs where the number of cells present in a real sample is so large that it can be treated like an infinitely large system. The methodology applied in the formulation and derivations of such models typically involves the following:
Definition of the 'research problem' and identification of what needs to be studied;
Definition of the geometry of the system in a manner where every independent length and angle is uniquely identified, including the identification of a suitable periodically repeating unit (the 'unit cell');
Formulation of the assumption related to the deformation mechanism and identification of the geometric parameters which will be treated as the variables and those that will be assumed to be constant (depending on the deformation mechanism);
Expression of the unit cell parameters in terms of geometric parameters and variables;
Calculation of strains from the unit cell parameters and formulation of expressions for the Poisson's ratio (and other relevant properties).
Definition of the research problem
Any model for a given system, irrespective of how simple the system may look, is never likely to be fully comprehensive given the various aspects that one could study. In this particular case, it needs to be stated a priori that the aim of this study is to represent the deformations and properties of the system depicted in Fig. 2 in a mathematical manner, i.e. describe what happens as the system is being pulled or compressed either in the direction along the length of the vertical ligaments, or in the orthogonal direction, as depicted, and where deformations only occur through in-plane changes in angle. Whilst this statement may look trivial, it excludes the need to over define the system and can be used as a guide to appropriately select between different possible methods of definition or simplification, as indicated in more detail below. For example, by stating that only the mechanical aspects related to uniaxial loading are being studied, one may automatically preclude, for example, the need to consider any possible thermal, electrical and magnetic effects on the system.
The system being modelled which in general could have non-equal lengths and non-equal angles, but in this case, it shall be assumed that the system will have l1 = l2 = l and l3 = h (kept constant), and θ1 = θ2 = θ (a variable). For the purpose of this study, the system will only be uniaxially loaded in the vertical Ox2 direction and the ligaments will not be allowed to stretch, flex or move out of plane, i.e. θ is the only variable. Note that the parameters are defined in (a) whilst (b) shows various representations with the possible unit cells identified where (b-i) shows a re-entrant system, (b-ii) shows a non-re-entrant conventional honeycomb and (b-iii) shows a sheared system (or an irregular non-symmetric honeycomb). All of these conformations can be represented by the unit cells. Note that the convention being used to label θ is not the same as that used in previous work by (Abd El-Sayed et al., 1979; Evans et al., 1995; Gibson et al., 1982; Masters & Evans, 1996)
Definition of the geometry
The system under study is described graphically in Fig. 2 and may be described as a two-dimensionally periodic structure, where in simple terms, a periodic structure is defined as a system made from sub-structural units which are tessellated in one-dimension, two-dimensions or three-dimensions in a space filling manner. Examples of one-dimensional periodic structures include railway tracks (or, in mathematics, the trigonometric functions) whilst crystals are probably the best well-known examples of systems exhibiting three-dimensional periodicity. Examples of two-dimensional periodic systems, apart from the honeycombs studied here, include some of the well-known patterned/Islamic tiling designs.
To adequately describe the geometry of this periodic system, one should first attempt to identify the appropriate 'unit cell' which may be used to generate the full system through tessellation (i.e. via translation only). As illustrated in Fig. 2b, the smallest unit cell which may be used to describe these hexagonal honeycombs is the one highlighted in red which contains just three ligaments. This smallest unit cell is in the shape of a parallelogram for sheared systems, which becomes rhombic when the angles between the ligaments are made equal (a result of the added symmetry). Whilst, in theory, this smallest unit cell can be used to derive the mathematical models of the system, in practice, it is much more convenient to choose a larger rectangular-shaped unit cell which has the advantage that for non-sheared systems, such as the ones modelled here, the unit cell angles are 90o and one of its unit cell vectors is parallel to the ligaments of length h. As a result, when modelled with this rectangular unit cell, the structure may easily and elegantly be aligned with the Cartesian axis, as shown in Fig. 2.
Having identified these unit cells, the next step is to identify the essential geometric parameters that are needed to describe the whole system. As noted above, availability of the parallelogram-shaped unit cells dictates that the whole system, in general, can be generated through tessellation of just three ligaments of length l1, l2 and l3 joined together at their end and two angles θ1 and θ2 (the third angle is 360o − (θ1 + θ2), due to the construing of planar Euclidean geometry). Here, it should be noted that, for this particular case, the system studied (the original undeformed system) is further assumed to have lengths l1 = l2 = l and l3 = h, and angles θ1 = θ2 = θ. However, at this stage, without looking at the permitted deformation mechanism/s, one cannot assume that the condition of equality will remain valid throughout the deformation.
Definition of the deformation mechanism
Having described the general case and the geometric parameters which are needed to describe the system, one should attempt to reduce the number of variables by looking at the 'deformation mechanism' and formulating the appropriate assumptions to be used. In this particular case, as noted above, it is being assumed that deformations occur only through changes in the angles between the ligaments, i.e. an idealised hinging model. This precludes any changes in length and it may thus be assumed that the lengths l1 = l2 = l and l3 = h remain constant throughout the deformation process. This means that the lengths l and h are not to be treated as variables but as simple geometric parameters which are used to define the shape and size of the structure, but which remain constant in the deformation (idealised hinging model).
It is also assumed that this particular derivation will only look at uniaxial loading in the vertical Ox2 direction, or, the horizontal Ox1 direction. In general, the two angles θ1 and θ2 are independent of each other, and if the system had to be subjected to shear loading or off-axis loading, which is not the case in this work, θ1 ≠ θ2. However, if what is being considered is a particular scenario where two ligaments are (and remain) of equal length l, the honeycombs angles are initially at an equal magnitude θ1 = θ2. Additionally, if the system is only being subjected to uniaxial on-axis loading for the rectangular unit cell (i.e. loading in the Ox1 or Ox2 direction), the angles θ1 and θ2 remain equal to each other due to symmetry. Hence, it may be further assumed that θ1 = θ2 = θ, which is the only variable of this system. In other words, given that as noted above, the scope of this modelling exercise is to study uniaxial loading in the Ox1 or Ox2 directions where the ligament lengths do not change, for the purpose of this work, it is sufficient to define the geometry in terms of three parameters: the lengths l and h, and the angle θ, of which only θ is a variable.
Here, it must be emphasised that this condition, which provides the simplification that for uniaxial on-axis loading θ1 = θ2 = θ, only holds for on-axis loading and there is no actual physical constrain to force these angles to remain equal. In fact, had the system sheared, these angles will no longer remain of equal value. This is in sharp contrast with other 'rotating squares' (Grima & Evans, 2000), 'rotating rectangles' (Grima et al., 2004; Grima, Gatt, Alderson, & Evans, 2005; Grima, Manicaro, & Attard, 2011) or other unimode systems typically studied by Milton (2013), since this condition only applies if loading is on-axis. However, this honeycomb system can be transformed to a more constrained system by letting h = 0, which would transform the 'hinging honeycomb' to a 'wine-rack' model.
Expression of cell parameters and alignment in space
Having identified the unit cell/s that could be used for defining the system, and the geometric parameters/variables needed to fully describe the system to be modelled, the alignment of the system in the global space needs to be defined, and the unit cell parameters measured in terms of these geometric parameters/variables.
As amply described in standard crystallography textbooks, in general, any periodic 3D systems (such as crystals) can be described in terms of a unit cell in the shape of a parallelepiped which, in turn, can be described in terms of three vectors which correspond to the sides of the unit cell. By convention, these three vectors are denoted by a, b and c which have a length a, b and c, respectively. The angles between these cell vectors (the unit cell angles) are denoted as α (the angle between b and c), β (the angle between a and c) and γ (the angle between a and b). Whilst in general, this unit cell (and hence the crystal) can be aligned in the 3D global space in an arbitrary manner, as per convention proposed by the Institute of Radio Engineers (IRE) [Mason, 1950], the crystal is typically aligned in space with the c cell vector always parallel to the Ox3-direction (i.e. its components in the Ox1 and Ox2 direction are 0); the b cell vector always lying in the Ox2-Ox3 plane (i.e. its component in the Ox1 direction is 0) and the remaining a vector left free in space. With this alignment, the cell vectors are of the following forms:
$$ {\displaystyle \begin{array}{l}\mathbf{a}=\left({X}_{11},{X}_{12},{X}_{13}\right)\\ {}\mathbf{b}=\left(0,{X}_{22},{X}_{23}\right)\\ {}\mathbf{c}=\left(0,0,{X}_{33}\right)\end{array}} $$
i.e. the system will always have its unit cell projections in the Ox1, Ox2 and Ox3 directions as X11, X22 and X33. This offers significant simplification in calculating strains.
For the simpler 2D system, a similar type of alignment is normally used, where the unit cell is aligned in the plane with one of its sides aligned parallel with the Ox2 axis and the other side left free. Thus, for a parallelogrammical unit cell, referring to Fig. 3, the cell vectors would be of the form a = (X11, X12) and b = (0, X22) with the special case of a rectangular unit cell having a = (X11, 0) and b = (0, X22), where X11 and X22 are the projections of the unit cell in the Ox1 and Ox2 on-axis directions, respectively.
Alignment of a 2D unit cell in the Cartesian space based on the convention adopted by the Institute of Radio Engineers (IRE) as specified in Mason (1950)
In the particular case of the honeycomb, as discussed above, it is possible and convenient to formulate the model in terms of a rectangular unit cell, where, referring to Fig. 2, the shape and size of the representative unit cell of this hexagonal honeycomb may be described quantitatively by the unit cell projections in the Ox1 and Ox2 directions as follows:
$$ {X}_{11}=2l\sin \theta $$
$$ {X}_{22}=2\left(h-l\cos \theta \right) $$
Note that with θ as defined here, angles of 0o < θ < 90o would correspond to re-entrant honeycombs, angles 90o < θ < 180o would correspond to conventional non re-entrant honeycombs and θ = 90o would correspond to the special case of a honeycomb having T-shaped junctions.
Strains and Poisson's ratios
Given the expressions for Xii as a continuous function of a single variable, the angle θ, one may derive expressions for strains which can be considered as being the continuous equivalent of the discrete strains presented above. Throughout this derivation, recognising that the formulation of the engineering strain and true strain require and are dependent upon the identification of a starting 'original' conformation of the system, it shall be assumed that unless otherwise stated, the deformation will be performed on a honeycomb which is characterised by angle θ0 between the ligaments such that the original undeformed conformation is represented by a unit cell of dimensions X11(θ0) × X22(θ0).
With this in mind, for Xii = Xii(θ), the engineering, infinitesimal and true strains in the Oxi directions can be defined, respectively as follows:
$$ {e}_i^{eng}\left(\theta \right)=\frac{X_{ii}\left(\theta \right)-{X}_{ii}\left({\theta}_0\right)}{X_{ii}\left({\theta}_0\right)}=\frac{X_{ii}\left(\theta \right)}{X_{ii}\left({\theta}_0\right)}-1 $$
$$ d\varepsilon {}_i\left(\theta \right)=\frac{{d X}_{ii}}{X_{ii}}=\frac{1}{X_{ii}}\frac{{d X}_{ii}}{d\theta} d\theta $$
$$ {e}_i^{true}\left(\theta \right)=\underset{X_{ii}\left({\theta}_o\right)}{\overset{X_{ii}\left(\theta \right)}{\int }}\frac{1}{X_{ii}}{dX}_{ii}={\left[\ln \left({X}_{ii}\right)\right]}_{X_{ii}\left({\theta}_o\right)}^{X_{ii}\left(\theta \right)}=\ln \left({X}_{ii}\left(\theta \right)\right)-\ln \left({X}_{ii}\left({\theta}_0\right)\right)=\ln \left(\frac{X_{ii}\left(\theta \right)}{X_{ii}\left({\theta}_o\right)}\right) $$
where the respective Poisson's ratios, as a function of θ, are hence given as follows:
$$ {\nu}_{ij}^{eng}\left(\theta \right)=-\frac{e_j^{eng}\left(\theta \right)}{e_i^{eng}\left(\theta \right)}=-\left(\frac{X_{jj}\left(\theta \right)-{X}_{jj}\left({\theta}_0\right)}{X_{ii}\left(\theta \right)-{X}_{ii}\left({\theta}_0\right)}\right)\frac{X_{ii}\left({\theta}_0\right)}{X_{jj}\left({\theta}_0\right)} $$
$$ {\nu}_{ij}^{f\hbox{'}n}\left(\theta \right)=-\frac{d{\varepsilon}_j\left(\theta \right)}{d{\varepsilon}_i\left(\theta \right)}=-\frac{X_{ii}}{X_{jj}}\left(\frac{{d X}_{jj}}{d\theta}\right){\left(\frac{{d X}_{ii}}{d\theta}\right)}^{-1} $$
$$ {\nu}_{ij}^{true}\left(\theta \right)=-\frac{e_j^{true}\left(\theta \right)}{e_i^{true}\left(\theta \right)}=-\ln \left(\frac{X_{jj}\left(\theta \right)}{X_{jj}\left({\theta}_o\right)}\right)/\ln \left(\frac{X_{ii}\left(\theta \right)}{X_{ii}\left({\theta}_o\right)}\right)=-\frac{\ln \left({X}_{jj}\left(\theta \right)\right)-\ln \left({X}_{jj}\left({\theta}_0\right)\right)}{\ln \left({X}_{ii}\left(\theta \right)\right)-\ln \left({X}_{ii}\left({\theta}_0\right)\right)} $$
where the expressions can be plotted against the geometric variable, or, probably more practically for real applications, against the applied engineering strain or the applied true strain so as to obtain the strain-dependent Poisson's ratio. The latter plots would need to be obtained in a parametric manner.
The type I rotating rectangles model
The same approach was applied to the type I rotating rectangles model. In this case, it may be shown that, although the smallest unit cell is one which contains just two rectangles, it is much more convenient to model the structure through the rectangular unit cell shown and defined in Fig. 4. The unit cell projections in the Ox1 and Ox2 directions are given as follows:
$$ {X}_{11}=2\left[a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)\right] $$
$$ {X}_{22}=2\left[a\sin \left(\frac{\theta }{2}\right)+b\cos \left(\frac{\theta }{2}\right)\right] $$
The type I rotating rectangles system represented through a rectangular unit cell and its alignment in space
where in this case, if the rectangles are assumed to be perfectly rigid and simply rotate relative to each other in the plane of the structure, the system fulfils all the requirement to be a unimode mechanism where, for a given a and b, the structural properties are fully defined through the angle θ (the only variable during deformation).
Comparison of the different methods for computing strain and Poisson's ratios
Without loss of generality, the different methods for computing strains are illustrated for the system shown in Fig. 1 for n = 5 where a wire-like system is stretched from an initial length of Linit = L[0] = 100 mm to a final length Lfin = L[5] = 150 mm, in successive increments of δl[k] = 10 mm, 12 mm, 8 mm, 10 mm and 10 mm. The calculated strains using the different methods are given in Table 1. These very simple calculations highlight three important characteristics, which might sound trivial, but are worth stating:
The engineering strain, particularly when expressed as a percentage, is useful to provide a very visual representation of the deformation in a cumulative manner: indeed, it is easy to visualise that the final length is 50% longer than the starting length, a feature which no other strain measurement can provide;
The 'instantaneous strain' and the 'true strain' can highlight the fact that the 10 mm increases in length are not all equivalent: as the system is stretched longer, successive 10 mm increases will contribute to a smaller relative extension when compared to earlier ones when the length was shorter;
Only the engineering or true strains provide a cumulative measure, one which could be used, for example, to plot stress-strain curves, this being an obvious consequence of the manner how the 'instantaneous strain' is calculated.
Table 1 The linear dimensions of the system in Fig. 1 and the calculated engineering strain, the engineering strain as a percentage, the instantaneous strain and the true strain
Nevertheless, the distinction between the three different methods of computing the strain becomes more evident when used to compute the Poisson's ratio. Once again, without loss of generality, these different methods for computing strains and Poisson's ratios are illustrated in Fig. 5a for a system based on nine measurements (see Table 2), these being labelled A-I. Here, the system is being stretched in the vertical Ox2 direction so its height changes from X22 = 30 mm to X22 = 62 mm. As this is happening, the width X11 of the system in the Ox1 direction is initially decreasing as the system gets thinner (from A to C), then its width X11 remains constant (from C to D), and then X11 increases (from D to H) in such a manner that system F has the same width as the original system A, with systems G and H having a width which is larger than system A. The width X11 of the system then decreases again (from H to I) in such a manner that system I has a smaller width X11 than system H, which width is still larger when compared to that of the original system A.
a A hypothetical 2D system being stretched from a length of 30 to 62 mm in the vertical direction; b plot of the width vs the length, c, d plots of the Poisson's ratios vs. length or engineering strain, with the initial system being A for c and D for d
Table 2 The dimensions of the system (in mm) in Fig. 5 together with the strains and Poisson's ratio computed. System A is considered as the initial structure
The strains and Poisson's ratios computed through the different methods are reported in Table 2 and plotted in Fig. 5c against the vertical height Y and the engineering strain. From these results, taking system A as the initial system, it is evident that the reported Poisson's ratio is very much dependent on the manner how the strains are computed. Of the three methods, it is only the instantaneous Poisson's ratio that could capture the observation that there was no change in width upon stretching from X22 = 38 to X22 = 42 mm (C to D), something which would correspond to zero Poisson's ratio. Similarly, the very evident auxeticity which is demonstrated as the system is stretched from X22 = 42 mm is not captured by the engineering Poisson's ratio and the true Poisson's ratio, since between X22 = 42 mm and X22 = 50 mm (D to F), the width of the system would be smaller or equal to that of the original sample. All three methods of calculating Poisson's ratio report auxeticity when stretched beyond X22 = 50 mm till X22 = 58 mm, as expected, but only the instantaneous Poisson's ratio could capture the non-auxetic characteristics on stretching from X22 = 58 mm till X22 = 62 mm (H to I). From this perspective, one could easily conclude that the Poisson's ratio computed from the instantaneous strains is superior to the other methods. However, whilst from a purely scientific perspective, one cannot contradict the fact that the system is auxetic as it is stretched from X22 = 42 mm onwards, it should be equally recognised that from a practical perspective, the evident auxeticity in the region between X22 = 42 to X22 = 50 mm could well be useless. In fact, from a practical perspective it is only the widening compared to the initial system which is of important result since in a number of practical situations, it is only 'before' and 'after' states which matter, and not the path taken during the deformation. An example of this would be in devices such as rivets where, for proper affixation, the width would need to be larger than the original. Such information can be more easily obtained from the engineering Poisson's ratio or the true Poisson's ratio, since the system where the width would have once again reached that of the original sample (system F, X22 = 50 mm) would have an engineering or true Poisson's ratio of zero, beyond which auxeticity is reported. Unfortunately, this important information, at least from a practical perspective, is not easy to extrapolate from the data relating to the instantaneous Poisson's ratio as there is nothing remarkable about the instantaneous Poisson's ratio of system F. It should also be noted that although the system was not behaving in an auxetic manner as it was stretched from X22 = 58 to X22 = 62 mm (H to I), its width is still larger than that of the original sample, something which is not identifiable from the instantaneous Poisson's ratio.
However, it is equally true that knowledge of Poisson's ratio during the deformation path could be important in other practical applications. For example, had the application been a cable passing through a hole, where the sample is pulled from an initial length to a final length, and in the process change the thickness to an appropriate final thickness, knowledge of the dimensional changes as the sample is stretched would have been of paramount importance, for instance, so as to ensure that the thickness achievable through the deformation process would always be smaller than the hole the cable is passing through. Similarly, in applications where the sample needs to be "seen" getting fatter as it is stretched, then the only usable portion of the deformation would be that where the instantaneous Poisson's ratios give a negative Poisson's ratio (or positive if the sample needs to be 'seen' getting thinner).
At this point, it should be noted that the engineering and true Poisson's ratios are particularly sensitive to the choice of the initial system. In fact, very different results would have been obtained if, for example, system D is chosen as the initial structure. In this case, since system C corresponds to a conformation which represents the main 'turning point' in the behaviour, there is much less disagreement between the different formulations of the Poisson's ratio. Nevertheless, it must be emphasised that for real samples, the initial conformation cannot be arbitrarily chosen.
Obviously, it must be emphasised that the qualitative aspect of this information could also have been extracted from looking at the original data related to the dimensions, i.e. 'X11' vs 'X22', plotted in Fig. 5b. In such plots, one would need to look at the gradients to assess the sign of the instantaneous Poisson's ratio and the relative positions between two points to assess the sign of the engineering/true Poisson's ratios. More specifically, for the 'instantaneous Poisson's ratio', a positive gradient in the 'dimensions' plot would indicate auxeticity, a zero gradient would indicate a zero Poisson's ratio whilst a negative gradient would indicate positive Poisson's ratio. Similarly, the sign of the engineering/true Poisson's ratio can be worked out from the gradient of the line joining two datapoints, one of which should be the original system.
The Poisson's ratio properties of hexagonal honeycombs
Applying the method presented above, for the honeycomb system for loading in the vertical Ox2 or horizontal Ox1 direction, one obtains the following:
Engineering strains and engineering Poisson's ratio:
$$ {e}_1^{eng}\left(\theta \right)=\frac{\sin \left(\theta \right)-\sin \left({\theta}_0\right)}{\sin \left({\theta}_0\right)} $$
$$ {e}_2^{eng}\left(\theta \right)=-\frac{l\cos \left(\theta \right)-l\cos \left({\theta}_0\right)}{h-l\cos \left({\theta}_0\right)}=-\frac{\cos \left(\theta \right)-\cos \left({\theta}_0\right)}{h/l-\cos \left({\theta}_0\right)} $$
$$ {\nu}_{21}^{eng}={\left({\nu}_{12}^{eng}\right)}^{-1}=-\frac{e_1^{eng}}{e_2^{eng}}=\frac{\sin \left(\theta \right)-\sin \left({\theta}_0\right)}{\cos \left(\theta \right)-\cos \left({\theta}_0\right)}\frac{h/l-\cos \left({\theta}_0\right)}{\sin \left({\theta}_0\right)} $$
Infinitesimally small strains and Poisson's function:
$$ d{\varepsilon}_1\left(\theta \right)=\frac{{d X}_{11}}{X_{11}}=\frac{1}{X_{11}}\frac{{d X}_{11}}{d\theta} d\theta =\frac{2l\cos \left(\theta \right)}{2l\sin \left(\theta \right)} d\theta =\frac{\cos \left(\theta \right)}{\sin \left(\theta \right)} d\theta =\cot \left(\theta \right) d\theta $$
$$ d{\varepsilon}_2\left(\theta \right)=\frac{{d X}_{11}}{X_{11}}=\frac{1}{X_{11}}\frac{{d X}_{11}}{d\theta} d\theta =\frac{2l\sin \left(\theta \right)}{2h-2l\cos \left(\theta \right)} d\theta =\frac{\sin \left(\theta \right)}{h/l-\cos \left(\theta \right)} d\theta $$
$$ {\nu}_{21}={\left({\nu}_{12}\right)}^{-1}=-\frac{d{\varepsilon}_1}{d{\varepsilon}_2}=-\frac{\left(h/l-\cos \left(\theta \right)\right)\cos \left(\theta \right)}{\sin^2\left(\theta \right)} $$
True strains and true Poisson's ratio:
$$ {e}_1^{true}\left(\theta \right)=\ln \left(\frac{X_{11}\left(\theta \right)}{X_{11}\left({\theta}_o\right)}\right)=\ln \left(\frac{2l\sin \left(\theta \right)}{2l\sin \left({\theta}_o\right)}\right)=\ln \left(\frac{\sin \left(\theta \right)}{\sin \left({\theta}_o\right)}\right)=\ln \left(\sin \left(\theta \right)\right)-\ln \left(\sin \left({\theta}_o\right)\right) $$
$$ {e}_2^{true}\left(\theta \right)=\ln \left(\frac{X_{22}\left(\theta \right)}{X_{22}\left({\theta}_o\right)}\right)=\dots =\ln \left(\frac{h/l-\cos \left(\theta \right)}{h/l-\cos \left({\theta}_o\right)}\right)=\ln \left(h/l-\cos \left(\theta \right)\right)-\ln \left(h/l-\cos \left({\theta}_o\right)\right) $$
$$ {\nu}_{21}^{true}={\left({\nu}_{12}^{true}\right)}^{-1}=-\frac{e_1^{true}}{e_2^{true}}=-\frac{\ln \left(\sin \left(\theta \right)\right)-\ln \left(\sin \left({\theta}_o\right)\right)}{\ln \left(h/l-\cos \left(\theta \right)\right)-\ln \left(h/l-\cos \left({\theta}_o\right)\right)} $$
It should be noted that, due to physical constraints, if the structure had to be loaded in tension in the horizonal Ox1 direction, the system would not have been able to go past the 90o conformation (T-shaped joints) through the hinging mechanism as this would represent a locking position. There are no such locking positions for loading in the vertical Ox2 direction (apart from the extreme conformations were θ = 0o or θ = 180o).
Typical results for the hexagonal honeycombs where l = 1 and h = 2 (i.e. as in Fig. 6 below) are reported through plots of the unit cell lengths, plotted against each other and against θ (Fig. 7), as well as the so derived Poisson's ratio plotted against the applied engineering strain and the angle θ for the systems (Fig. 8). These plots were produced for various initial conformations of the honeycomb which include the ones where honeycombs are almost fully closed. In the case of the true Poisson's ratio, this is also plotted against the applied true strain.
To-scale representation of the hinging honeycombs with h = 2, l = 1 as these are stretched/compressed in the vertical or horizontal direction. Note that systems where θ < 90o cannot be pulled in the horizontal direction past the system where θ = 90o to achieve θ > 90o whilst the systems where θ > 90o cannot be pulled in the horizontal direction past the system where θ = 90o to achieve θ < 90o, i.e. θ = 90o corresponds to a 'locking conformation' for pulling in the horizontal direction
a The cell parameters for the honeycombs plotted against each other. b The cell parameters plotted against the parameter θ. Note that the plot in (a-i) would be most useful for loading in Ox2 whilst that in (a-ii) for loading in Ox1, where the turning point corresponds to the locked conformation (θ = 900) and the lower portion refers to angles less than 900 whilst the upper portion refers to angles more than 900
Plots of the Poisson's ratios for honeycombs having h = 2, l = 1 for an initial configuration having an angle between the ligaments of θo = 30o plotted against strain (left) and against the angle θ (right). The shaded region (grey) highlights a non-accessible range of angles. Note that in the case of axial straining in the Ox2 direction, a second set of plots is presented which highlights a smaller range of Poisson's ratios
The Poisson's ratio properties of the type I rotating rectangles
Using the procedure described above, the strains and Poisson's ratios for the 'rotating rectangles' system may be expressed as follows:
$$ {e}_1^{eng}\left(\theta \right)=\frac{a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)-a\cos \left(\frac{\theta_0}{2}\right)-b\sin \left(\frac{\theta_0}{2}\right)}{a\cos \left(\frac{\theta_0}{2}\right)+b\sin \left(\frac{\theta_0}{2}\right)} $$
$$ {e}_2^{eng}\left(\theta \right)=\frac{b\cos \left(\frac{\theta }{2}\right)+a\sin \left(\frac{\theta }{2}\right)-b\cos \left(\frac{\theta_0}{2}\right)-a\sin \left(\frac{\theta_0}{2}\right)}{b\cos \left(\frac{\theta_0}{2}\right)+a\sin \left(\frac{\theta_0}{2}\right)} $$
$$ {\nu}_{21}^{eng}={\left({\nu}_{12}^{eng}\right)}^{-1}=-\frac{\left[a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)-a\cos \left(\frac{\theta_0}{2}\right)-b\sin \left(\frac{\theta_0}{2}\right)\right]\left[b\cos \left(\frac{\theta_0}{2}\right)+a\sin \left(\frac{\theta_0}{2}\right)\right]}{\left[b\cos \left(\frac{\theta }{2}\right)+a\sin \left(\frac{\theta }{2}\right)-b\cos \left(\frac{\theta_0}{2}\right)-a\sin \left(\frac{\theta_0}{2}\right)\right]\left[a\cos \left(\frac{\theta_0}{2}\right)+b\sin \left(\frac{\theta_0}{2}\right)\right]} $$
$$ d{\varepsilon}_1\left(\theta \right)=\frac{-a\sin \left(\frac{\theta }{2}\right)+b\cos \left(\frac{\theta }{2}\right)}{2a\cos \left(\frac{\theta }{2}\right)+2b\sin \left(\frac{\theta }{2}\right)} $$
$$ d{\varepsilon}_2\left(\theta \right)=\frac{a\cos \left(\frac{\theta }{2}\right)-b\sin \left(\frac{\theta }{2}\right)}{2a\sin \left(\frac{\theta }{2}\right)+2b\cos \left(\frac{\theta }{2}\right)} $$
$$ {\displaystyle \begin{array}{l}{\nu}_{21}^f={\left({\nu}_{12}^f\right)}^{-1}=-\frac{\left[a\sin \left(\frac{\theta }{2}\right)+b\cos \left(\frac{\theta }{2}\right)\right]\left[-a\sin \left(\frac{\theta }{2}\right)+b\cos \left(\frac{\theta }{2}\right)\right]}{\left[a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)\right]\left[a\cos \left(\frac{\theta }{2}\right)-b\sin \left(\frac{\theta }{2}\right)\right]}\\ {}\kern4.75em =\frac{a^2{\sin}^2\left(\frac{\theta }{2}\right)-{b}^2{\cos}^2\left(\frac{\theta }{2}\right)}{a^2{\cos}^2\left(\frac{\theta }{2}\right)-{b}^2{\sin}^2\left(\frac{\theta }{2}\right)}\end{array}} $$
$$ {\varepsilon}_1^{true}\left(\theta \right)=\ln \left[\frac{a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)}{a\cos \left(\frac{\theta_0}{2}\right)+b\sin \left(\frac{\theta_0}{2}\right)}\right] $$
$$ {\varepsilon}_2^{true}\left(\theta \right)=\ln \left[\frac{b\cos \left(\frac{\theta }{2}\right)+a\sin \left(\frac{\theta }{2}\right)}{b\cos \left(\frac{\theta_0}{2}\right)+a\sin \left(\frac{\theta_0}{2}\right)}\right] $$
$$ {\nu}_{21}^{true}=\left({\nu}_{12}^{true}\right)=-\frac{\ln \left[\frac{a\cos \left(\frac{\theta }{2}\right)+b\sin \left(\frac{\theta }{2}\right)}{a\cos \left(\frac{\theta_0}{2}\right)+b\sin \left(\frac{\theta_0}{2}\right)}\right]}{\ln \left[\frac{b\cos \left(\frac{\theta }{2}\right)+a\sin \left(\frac{\theta }{2}\right)}{b\cos \left(\frac{\theta_0}{2}\right)+a\sin \left(\frac{\theta_0}{2}\right)}\right]} $$
Note that in this case, as amply discussed elsewhere (Grima et al., 2004; Grima, Alderson, & Evans, 2005), there are locking conformations which correspond to the conformations when the diagonals of rectangular units become aligned to the direction of loading. As shown in Fig. 9, for loading in the Ox2 direction, these angles correspond to 2ϕ2 + θm2 = 180o where \( {\phi}_2={\tan}^{-1}\left(\frac{b}{a}\right) \) whilst for loading in Ox1 direction, they correspond to 2ϕ1 + θm1 = 180o where \( {\phi}_1={\tan}^{-1}\left(\frac{a}{b}\right) \). Thus, for this specific case where loading is in the Ox2 direction, if the original structure has an angle between the rectangles of θ0, where 0o < θ0 < θm2, then the system cannot be stretched past θ = θm2, at which angle, X22 is at a maximum. Similarly, for θm2 < θ0 < 180o. The practical relevance of these conformations may be appreciated better in Fig. 10 below which shows a to-scale representation of the rotating rectangles model were a = 2 and b = 1. This image also visually highlights the typical Poisson's ratio properties of these systems.
The 'locking conformations' of the type I rotating rectangles system as this is stretched in a the Ox2 direction and b the Ox1 direction. Note that locking occurs when diagonals of rectangular units become parallel to the direction of loading as this would correspond to the most stretched conformation in that direction
To-scale representation of the rotating rectangles model were a = 2 and b = 1. Note that the different conformations in a correspond to 0o < θ < 126.87o whilst those in b correspond to 126.87o < θ < 180o
The intention of this discussion will not be to focus on the ability of these systems to exhibit negative Poisson's ratios as such properties are amply documented (Abd El-Sayed et al., 1979; Evans et al., 1995; Gibson et al., 1982; Grima et al., 2004; Grima, Alderson, & Evans, 2005; Masters & Evans, 1996). Instead, the focus will be to discuss the appropriateness of the various Poisson's ratios to represent what is really happening as the systems are being stretched or compressed. To do this, it is important to interpret the plots of the Poisson's ratios in conjunction with the images depicting the systems and the plots of the unit cell parameters.
Looking at the honeycomb systems shown in Fig. 6, without loss of generality, the discussion will first focus on the properties when the starting conformation is a re-entrant honeycomb having a geometry where the initial angle between the ligaments is θ0 = 30o. This is a representative dimension in the Ox2 direction of X22 ≈ 2.27 mm. Recognising that the representative dimensions in the Ox2 direction can span from X22 = 2.00 mm (fully closed re-entrant system, θ = 0o) to X22 = 6.00 (fully closed non re-entrant system, θ = 180o), this corresponds to a structure which is very near to the fully closed re-entrant conformation. From the perspective of dimensions in the horizontal direction, θ0 = 30o corresponds to X11 = 1.00 mm, which corresponds to the median size, since in this case, the permissible range of values is between 0.00 mm (corresponding to the hypothetical fully closed system) and 2.00 mm (corresponding to the fully stretched structure with T-shaped joints). This suggests that this system can be compressed by c. 0.27 mm in the vertical direction, which would result in a massive horizontal contraction of almost ×4 this amount (c. 0.97) to become, at least in theory, negligibly thin. This situation for compressing in the vertical direction presents one of the classical dilemmas on the Poisson's ratio and how this should be reported. On the one hand, one could argue that the system is simply contracting by a finite amount when it is strained by another finite amount, resulting in a Poisson's ratio which obviously should be large in magnitude, but finite. Such a Poisson's ratio is reported through the engineering or true Poisson's ratio, which in this case is negative (as it should be) since the system is contracting laterally as it is compressed, and large in magnitude, reflecting the fact that the system shrinks laterally by a very substantial amount (see structures in Fig. 6). Nonetheless, one could equally argue that since this system becomes, theoretically, infinitely thin as it is axially compressed, reporting of a Poisson's ratio ν21 which tends to −∞ would be merited. Such 'giant auxeticity' is only obtainable by the Poisson's function. Here, it should however be mentioned that the three formulations of the Poisson's ratio can be considered as relatively concordant at least when compared to other situations where the difference is much more pronounced. For example, if one looks at the same system when this is subjected to a tensile load in the vertical direction, one would notice that the three versions of the Poisson's ratios differ substantially from each other, even in sign. The 'true' and 'engineering' Poisson's ratios seem to suggest that auxetic behaviour is retained till the system is stretched from 0% strain (θ0 = 30o) till reaching an engineering strain of c. 1.50, representing an axial extension of c. 150% (θ = 150o, which is almost the fully stretched conformation in Ox2 direction). This tallies with the observation that if one had to compare a 'before' and 'after' scenario, there is always an evident widening upon stretching compared to the initial system. As noted above, from a practical perspective, this extensive widening compared to the initial system is an important result since in a number of practical situations, it is these 'before' and 'after' states which are of utmost importance, and not the path undertaken during the deformation. On the other hand, if knowledge of Poisson's ratio during the deformation path is required, for example, in applications where the sample needs to be "seen" getting fatter as it is stretched, then the only usable 'auxetic' portion of the deformation would be till stretching by c. 76% (\( {e}_2^{eng}\approx 0.76 \), θ = 90o) as it is only till that conformation that the sample is visibly getting wider, following which it would start to contract again (see Fig. 6), as expected, since such honeycombs are not re-entrant. This information is only extractable from the Poisson's function.
It is sometimes also equally important to have knowledge of the Poisson's ratio at some particular extent of deformation. For example, if the application is such that the system needs to be initially deformed, and after this deformation, the Poisson's ratio of the 'stretched (or compressed) sample' would still need to be negative, then the Poisson's function would need to be considered. A practical example of this is in the use of auxetics for the manufacture of insoles where, to be able to benefit from the advantages associated with auxeticity, the sample needs to retain the auxeticity after contracting due to the effect of the bodyweight of the individual.
All this may suggest that the Poisson's function offers the most advantages compared to other methods of reporting the Poisson's ratio. However, the results reported here also suggest that the formulation of Poisson's function is rather inadequate to describe the behaviour of systems which approach a 'locking conformation'. This can be very clearly seen by looking at the results for loading in the horizontal direction where the Poisson's function seems to suggest that the system should be exhibiting giant auxeticity as it is approaching the θ = 90o conformation. Nevertheless, when looking at the conformations in Fig. 6, this supposedly giant auxeticity is not easy to detect. In fact, an analysis of the deformations clearly suggests that there is nothing remarkable as the system is stretched on approaching θ = 90o (\( {e}_1^{eng}\to 1.00 \)) with the engineering and true Poisson's ratio reporting a more modest value of − 0.76 and − 0.82, respectively, which is more in line with what is being observed. The artificial report of 'giant auxeticity' as \( {e}_1^{eng}\to 1.00 \) may be explained by the fact that the expression for the Poisson's function involves a division by 0, since in the following:
$$ {\nu}_{12}^{f\hbox{'}n}\left(\theta \right)=-\frac{d{\varepsilon}_2\left(\theta \right)}{d{\varepsilon}_1\left(\theta \right)}=-\frac{X_{22}}{X_{11}}\left(\frac{{d X}_{22}}{d\theta}\right){\left(\frac{{d X}_{11}}{d\theta}\right)}^{-1} $$
\( \frac{{d X}_{11}}{d\theta}=\cos \left(\theta \right)=0 \) when θ = 90o, a maximum turning point in the plot of X11 vs. θ.
Unfortunately, from a physical perspective this corresponds to a 'locking conformation' (i.e. a point where deformation becomes blocked) and not one where the system gets very significantly thicker as it is stretched, and hence one may conclude that this report of supposedly 'giant auxeticity' is a mere artefact of the method used rather than a real effect. All these emphasise that no formulation of the Poisson's ratio that have been formulated so far can be considered as being optimal.
Focusing the attention on the results obtained when the starting conformation is a re-entrant honeycomb where the initial angle between the ligaments is θ0 = 1.00o, i.e. a re-entrant conformation which is practically fully closed and almost infinitly thin, differences between the output of the different formulations, and some other inadequacies, come to light. Looking first at stretching in the vertical direction, with this starting conformation, the system can be stretched from a practically infinitely thin conformation (re-entrant) to another practically infinitely thin conformation (the non-re-entrant, where θ → 180o, \( {e}_1^{eng}\to 2.00 \)) with the widest conformation being mid-way through the deformation when θ = 90o (\( {e}_1^{eng}=1.00 \)). Here, the three formulations of the Poisson's ratio differ significantly from each other with the engineering form predicting giant auxeticity over most of the range of stretching (even when the thickness is actually decreasing) to describe the fact that, compared with the initial system, the width of the stretched system would always have increased by orders of magnitude thus becoming very noticeably wider when compared to the practically infinitly thin initial conformation. From this perspective, it is the engineering Poisson's ratio which highlights best the giant auxeticity, and, as noted before, the Poisson's function reports the Poisson's ratio as it varies along the path, changing sign from negative to positive mid-way when θ = 90o. However, in this case, the most remarkable differences can be spotted when stretching in the Ox1 horizontal direction where the engineering Poisson's ratio reports near zero negative values almost throughout the deformation whilst the Poisson's function reports a much larger negative value which tends to −∞ as the system approaches the 'locking conformation' when θ = 90o. In analogy to what was discussed before, such giant auxeticity is just an artefact of the method rather than a massive expansion upon stretching. What needs to be considered well is the report of the near zero engineering Poisson's ratios, which brings its own niche of importance and practical applications (Attard & Grima, 2011; Gaal, Rodrigues, Dantas, Galvão, & Fonseca, 2020; Grima et al., 2010). One could argue that the observed increase in size in the Ox2 direction upon stretching in the Ox1 direction is not insignificant (i.e. the magnitude of the Poisson's ratio should not be too low). However, one could equally argue that over most of the deformation, this increase in width is practically insignificant when compared to the manifold increase in size that is experienced in the direction of stretching (Ox1), i.e. a Poisson's ratio close to zero is justifiable. In fact, on closer observation of Fig. 6, one may note, for example, that as θ changes from 1 to 30o, although there is a gigantic increase in the length in the direction of stretching, the width only changes by a comparatively negligible amount from c. 2 to c. 2.27 (c. 10%.).
An obvious remark that should also be made at this point is that, in addition to what is stated above, an advantage of using the Poisson's function is that it is independent of the starting conformation, i.e. in some aspects, it is more amenable for discussion and reporting. In contrast, the engineering and true Poisson's ratios are highly dependent on the initial conformation, as evident when one compares Figs. 8, 11, and 12, which consider systems with a different initial conformation, including one where the initial conformation is not re-entrant, from which plots similar conclusions to those discussed above can be drawn.
Plots of the Poisson's ratios for honeycombs having h = 2, l = 1 for an initial configuration having an angle between the ligaments of θo = 1o plotted against strain (left) and against the angle θ (right). The shaded region (grey) highlights a non-accessible range of angles. Note that a second set of plots is presented which highlights a smaller range of Poisson's ratios
Plots of the Poisson's ratios for honeycombs having h = 2, l = 1 for an initial configuration having an angle between the ligaments of θo = 150o plotted against strain (left) and against the angle θ (right). This initial conformation is a non-re-entrant system and thus not auxetic at small strains. The shaded region (grey) highlights a non-accessible range of angles
At this point, it should be remarked that the findings on the differences between the forms of the Poisson's ratios as reported for the hexagonal honeycomb are also transferable to other systems. To show this, similar plots are also reported for the type I rotating rectangles with a = 2 and b = 1 (as drawn to scale in Fig. 10) where the initial conformation has θ0 = 20o and θ0 = 160o (see Fig. 13). Note that when the rectangles have these dimensions, the locking angle for uniaxial loading in the Ox2 direction is θm2 = 126.87o, i.e. the system with θ0 = 20o can be operated through stretching/compression in the Ox2 direction within the range 0o < θ < 126.87o whilst the system with θ0 = 160o can be similarly operated within the range 126.87o < θ < 180o, see Fig. 11. Note that at θ = 126.87o, the system would be at its maximum dimensions in the Ox2 direction. As was the case for the honeycombs, an important conclusion that can be drawn is that the Poisson's function reported artificially large magnitudes of the Poisson's ratio as the system approached a locking conformation whilst the engineering and true Poisson's ratio formulations reported more realistic values. On the other hand, the Poisson's function once again offered the advantage that it permitted to easily identify when a change in behaviour (from 'getting thinner' to 'getting fatter', or vice versa) was observed whilst the deformation was taking place.
Results for a typical type I rotating rectangles system with a = 2 and b = 1: a Plot of the cell parameters plotted against each other with X22 on the x-axis since this corresponds to the direction of loading. The 'loop' in this figure indicates the presence of a locked conformation; b the cell parameters plotted against the angle θ; c the Poisson's ratios plotted against applied strain in the Ox2 direction for two different systems; d the Poisson's ratio plotted against the angular parameter θ where the shaded region indicates an inaccessible region. Plots (d-iii) and (d-iv) are equivalent to (d-i) and (d-ii) but with a larger range plotted
Given these various observations about the strengths and weaknesses of the various forms of expressing the Poisson's ratio, it is rather difficult to conclude which form is the best descriptor of reality, but one can definitely conclude that caution should be taken when interpreting any results. In view of this, it would be ideal if the report of the Poisson's ratio as the 'Poisson's function' (i.e. what is typically reported in analytical models (Alderson & Evans, 1995; Evans et al., 1995; Gatt, Attard, Manicaro, Chetcuti, & Grima, 2011; Grima & Evans, 2000; Grima, Gatt, et al., 2005) is complemented with either the report of the engineering or true Poisson's ratio, as was done in isolated cases (Alderson & Evans, 1995). An important remark that should be made is that the true Poisson's ratio formulation seems to avoid most of the 'extreme' reports of the Poisson's ratio. However, one must acknowledge that the true strains may be difficult to relate to the actual extent of deformation due to the use of the logarithmic function. For example, a 50% extension would correspond to a true strain of ln(1 + e) = ln(1 + 0.5) ≈ 0.405, a value which is not of much 'meaning'. Thus, had one to opt to report the Poisson's ratio as a 'true Poisson's ratio', it would still be preferable to plot it against the more meaningful engineering strain, which could in itself lead to confusion in interpretation. Hence, from this aspect, the engineering Poisson's ratio is preferred over the true strain.
Before concluding, it must be noted that although this work has emphasised on the best way of reporting the Poisson's ratio, it must be stressed that the plots of the cell parameters themselves, which are generally not given much importance, contain a wealth of information in a very succinct manner. For example, when a load is applied in the Oxi direction, plots of Xjj vs. Xii can yield the sign of the Poisson's function through the gradient, since from the following:
$$ {\nu}_{ij}^{f\hbox{'}n}\left(\theta \right)=-\frac{d{\varepsilon}_j\left(\theta \right)}{d{\varepsilon}_i\left(\theta \right)}=-\frac{dX_{jj}/{X}_{jj}}{dX_{ii}/{X}_{ii}}=-\frac{X_{ii}}{X_{jj}}\frac{dX_{jj}}{dX_{ii}} $$
since Xjj and Xii are always positive, a negative Poisson's ratio requires a positive gradient \( \frac{dX_{jj}}{dX_{ii}} \). Furthermore, it should be noted that such plots can be used to easily identify locking conformations, as these are the conformations where the graph 'loops'. Here, it should be noted that whenever a graph of Xjj vs. Xii 'loops' (as is the case in X22 vs. X11 for the hexagonal honeycombs), from a mathematical perspective, Xjj cannot be considered as a function of Xii. This can, however, be considered as a proper function by only considering part of the data on a particular side of the locking conformation. The magnitude of the gradient of these plots also gives a very realistic measure of what one normally expects from a Poisson's ratio. Such data can usually be obtained from both modelling and experiment (it is normally the 'raw data'), and one may argue that it should become more standard practice to report it, possibly alongside the analysis of such data.
This work has compared the various ways how strain may be defined and how the actual definition affects the Poisson's ratio being computed. It was shown that different strain formulations result in rather different values and trends in the Poisson's ratio and that both the Poisson's ratio computed using the engineering strains and the ones obtained from the 'instantaneous strains' have their advantages and disadvantages.
In particular from this work, it was concluded that the various forms of determining the Poisson's ratio complement each other in describing the behaviour of systems which do not merely undergo infinitesimal deformations, particularly those which could exhibit both positive or negative Poisson's ratio (depending on the actual conformation, or extent of applied strain). It was thus concluded that the report of the Poisson's ratio should ideally be made both as the 'Poisson's function' and as the 'engineering/true Poisson's ratio'. A case was also made for reporting the actual 'sample' dimensions (or unit cell parameters) as these contain very useful and unbiased information which describe the Poisson's ratio of the system. It was argued that given that such data is normally easy to obtain from both modelling and experiment (the 'raw data'), it should become more standard practice to report it. Such more complete reporting of the Poisson's ratio behaviour would provide a more visually descriptive and unbiased picture of the true behaviour of auxetic systems and thus be of benefit to permit further research and development of systems studied computationally.
Ox i (i = 1,2,3):
The mutually orthogonal Cartesian axis
ε i (i = 1,2,3):
Strain in the Oxi direction (general definition)
ν ij (i,j = 1,2,3):
Poisson's ratio in the Oxi-Oxj plane for loading in the Oxi direction (general definition)
\( {e}_i^{eng} \) :
Engineering strain in the Oxi direction
\( {e}_i^{true} \)(i = 1,2,3):
True strain in the Oxi direction
δε i(i = 1,2,3):
Incremental strain in the Oxi direction
dε i :
Infinitesimally small incremental strain in the Oxi direction
\( {\nu}_{ij}^{eng} \) (i,j = 1,2,3):
Engineering Poisson's ratio in the Oxi-Oxj plane for loading in the Oxi direction (computed from engineering strains)
\( {\nu}_{ij}^{true} \) (i,j = 1,2,3):
True Poisson's ratio in the Oxi-Oxj plane for loading in the Oxi direction (computed from engineering strains)
\( {\nu}_{ij}^{f\hbox{'}n} \) (i,j = 1,2,3):
Poisson's function in the Oxi-Oxj plane for loading in the Oxi direction (computed from incremental strains)
a, b, c :
Unit cell vectors
X ij (i,j = 1,2,3):
Component of the unit cell vectors, with X11 and X22 being the projections of the unit cell in Ox1 and Ox2 directions, respectively
L init, L fin :
The initial and final length of a wire-like sample (defined in Fig. 1)
L[k], k = 0, 1, 2, 3, …:
Successive length measures of a wire-like sample (defined in Fig. 1)
δL[k], k = 0, 1, 2, 3, …:
Change in length between successive measurements in a wire-like sample between measurement k-1 and measurement k (defined in Fig. 1)
Total change in length from the initial length to kth measurement in a wire-like sample (defined in Fig. 1)
l 1, l 2, h, θ 1, θ 2 :
Geometric parameters which define the shape of a generic hexagonal honeycomb (defined in Fig. 2)
l, h, θ :
Geometric parameters which define the shape of the more symmetric hexagonal honeycomb (defined in Fig. 2) where l1 = l2 = l and θ1 = θ2 = θ throughout the deformation. For the hinging model discussed here, θ is a variable whilst the other parameters are constants
a, b, θ :
Geometric parameters which define the shape of the rotating rectangles (defined in Fig. 4) For the hinging model discussed here, θ is a variable whilst the other parameters are constants.
θ o :
The value of θ for the initial structure
θ m :
A value of θ which corresponds to a 'locking position' (defined in Fig. 9 for rotating rectangles)
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This work is funded by the University of Malta and the Malta Council for Science & Technology (A-ROW, a FUSION: The R&I Technology Development Programme 2018 project).
Metamaterials Unit, Faculty of Science, University of Malta, Msida, MSD 2080, Malta
James N. Grima-Cornish, Joseph N. Grima & Daphne Attard
Department of Chemistry, Faculty of Science, University of Malta, Msida, MSD 2080, Malta
Joseph N. Grima
James N. Grima-Cornish
Daphne Attard
JNGC conceived the work and produced the first draft of this manuscript. JNG edited the manuscript. DA supervised the project. All authors where involved in the discussion. The author(s) read and approved the final manuscript.
Correspondence to Daphne Attard.
The authors declare that there are no competing interests.
Grima-Cornish, J.N., Grima, J.N. & Attard, D. Mathematical modeling of auxetic systems: bridging the gap between analytical models and observation. Int J Mech Mater Eng 16, 4 (2021). https://doi.org/10.1186/s40712-020-00125-z
DOI: https://doi.org/10.1186/s40712-020-00125-z
Mechanical metamaterials
Auxetic
Negative Poisson's ratio
Poisson's function | CommonCrawl |
Timely initiation of breastfeeding and associated factors among mothers of infants under 12 months in South Gondar zone, Amhara regional state, Ethiopia; 2013
Liyew Mekonen1Email author,
Wubareg Seifu2 and
Zemenu Shiferaw1
International Breastfeeding Journal201813:17
Received: 9 July 2017
Published: 2 May 2018
Timely initiation of breastfeeding is defined as putting the newborn to the breast within one hour of birth. Significant benefits in reducing neonatal mortality and morbidity can be attained with effective promotion of timely initiation of breastfeeding and exclusive breastfeeding during the first months of life. Therefore, this study was conducted to assess timely initiation of breastfeeding and associated factors among mothers in South Gondar, Amhara regional state, Northern Ethiopia.
A community based cross-sectional study was employed. A multistage stratified sampling technique was used to select the sample of 845 mothers with 97.4% response rate. Moreover, data were collected by face to face interview using a semi structured questionnaire.
The prevalence of timely initiation of breastfeeding was 48.7% (54.7% in urban and 25.1% in rural areas). The odds of initiation of breastfeeding within one hour was higher for urban mothers (Adjusted Odds Ratio [AOR] 2.1; 95% Confidence Interval [CI] 1.4, 3.3), multiparous mothers (AOR 2.8; 95% CI 2.0, 3.8), mothers who had antenatal care (AOR 3.2; 95% CI 2.0, 5.2), mothers delivered in health institution (AOR 3.1; 95% CI 2.2, 4.6) and mothers delivered vaginally (AOR 4.1; 95% CI 1.7, 9.8) than their respective counterparts.
This study depicts the rate of timely initiation of breastfeeding was low in south Gondar zone. Factors which were positively associated with timely initiation of breastfeeding include urban residence, multiparity, having antenatal care, mother deliver in health institution and vaginal mode of delivery. Therefore, South Gondar health office and healthcare providers have to provide breastfeeding information during antenatal care by giving special emphasis to rural and primiparous mothers in which timely initiation of breastfeeding is poorly practiced. Further study is needed to assess the implementation of policies on timely initiation of breastfeeding.
Timely initiation
Associated factors
South Gondar
Timely initiation of breastfeeding is defined as putting the newborn to the breast within one hour of birth [1]. It was one the ten steps to successful breastfeeding on which the Baby Friendly Hospital Initiative was based and launched in 1992 [2]. Substantial benefits in reducing neonatal mortality and morbidity can be achieved with effective promotion of timely initiation of breastfeeding and exclusive breastfeeding during the first months of life [3].
The World Health Organization (WHO) and United Nation Children's Fund (UNICEF) recommend initiation of breastfeeding within the first hour after birth and exclusive breastfeeding for the first six months followed by continued breastfeeding to age two years or beyond along with appropriate complementary feeding [4]. Despite these recommendations, only 39% of newborns in the developing world are put to the breast within one hour of birth, and only 37% of infants less than six months of age are exclusively breastfed [5].
The Ethiopian government developed national infant and young child feeding guidelines in 2004 and has tried behaviour change communications on breastfeeding. There are considerable variations by region on timely initiation of breastfeeding, 60% in Amhara region where South Gondar Zone is found and the highest in Dire Dawa regions and Harari (90.5% and 89.4%, respectively) [6, 7].
Therefore, this study addressed the issue of timely initiation of breastfeeding practice and associated factors in South Gondar Zone. The finding of this study can provide relevant information for future planning and interventions of appropriate strategies to promote the timely initiation of breastfeeding practices.
Study setting
The study was conducted in South Gondar Zone which is found in Amhara regional state located at 664 km from the capital city Addis Ababa. Based on the 2007 Census conducted by the Central Statistical Agency of Ethiopia, this zone had a total population of 2,051,738, of whom 1,041,061 were men and 1,010,677 were women. A total of 468,238 households were counted in this zone, which results in an average of 4.38 persons to a household. There were 3953 mothers who had children aged less than 12 months at the time of the survey.
Study design and sample
A community based cross-sectional study was conducted among randomly selected 845 mothers of infants under 12 months in South Gondar Zone, Amhara Regional State, Ethiopia from March to May 2013. Mothers of infants under 12 months who were permanent residents were included. Mothers who were not available during data collection who were unconscious, critically ill and unable to respond were excluded from this study.
Sample size determination
The required sample size was determined using single population proportion formula with the following assumptions:
Level of confidence = 95%
Type I error (α) = 0.05
5% margin of error
Design effect = 2(Multi stage sampling)
Based on the assumption the prevalence of timely initiation of breastfeeding is 50%.
$$ n=\frac{\left(Z\ a/2\right)2\times P\left(1-P\right)}{d2\ } $$
$$ n=\frac{(1.96)2\times 0.5\left(1-o.5\right)}{(0.05)2} $$
$$ n=(384) $$
Given 2 design effect and 10% non-response rate, the final sample size was 845.
A multistage random sampling technique was used for selecting mothers. There were 12 districts in South Gondar zone and from these four districts (Farta district, Estie district, Dera district, Fogera district) were selected by using a simple random sampling technique (lottery method). The sample size was proportionally allocated to the urban and rural kebeles based on the number of mothers who had infants less than 12 months. The sampling frame was prepared for both the urban and rural kebeles by doing census prior to the actual data collection period. Based on the census result there were 718 and 3235 mothers with infants less than 12 month in the rural and urban respectively. Finally 179 and 666 mothers who had infants less than 12 months were selected from rural and urban by using a computer-based generated random number respectively.
Data collection and instrument
Data were collected using structured and pretested interview questionnaires prepared from Ethiopian Demographic Health Survey and Linkage Project. Data collectors were given two days training for the questionnaires and interviewing techniques. The questionnaires were initially prepared for English and then have been translated into the local language, Amharic and again it was translated back into English to check its accuracy. The questionnaires were pretested and modified before the actual data collection. Four supervisors checked each completed questionnaire and principal investigator monitored the overall quality of the data collection.
Operational definitions
Early (timely) breastfeeding is defined as putting the newborn to the breast within one hour of birth.
Prelacteal feeding is feeding to an infant with something other than breast milk after birth to three days whereas ever breastfeeding is defined as mothers breastfeed their index baby.
Employed mothers defined as mothers that were employed in governmental, non governmental and private organization.
Formal education defined as a person who attended primary or more education.
No formal education defined as a person who might able to write or read but did not attend primary or more education.
Marital status not in union comprised single, divorced, widow, cohabited and separated.
Information access if the mother read books, listen radio or watches a television program.
Data processing and analysis
Data were cleaned, edited, and entered onto Epidata version 3.2.1 and exported to the statistical packages for social sciences (SPSS) version 20 statistical software for further analysis. Frequency distribution and cross tabulation were done against the variables of interest. Bivariate analyses were done to assess the association between explanatory variables and outcome variable of the study. All variables with a p - value of < 0.25 at the bivariate analysis were included into multivariable logistic regression model in which odds ratio with 95% confidence intervals were estimated to identify independent predictors of timely initiation of breastfeeding. A p - value less or equal to 0.05 were employed to declare the statistical significance.
A total of 823 women were participating in this study and making the response rate of 97.4%. The age of the respondents ranged from 15 to 49 with a mean (± SD) age of 27.0 (± 5.7) years. Of the total 823 respondents, 656 (79.7%) were urban dwellers. Majority 765 (93%) of mothers was married, Christian by religion were 690 (83.8%) and 819 (99.5%) were in the Amhara ethnic group. Regarding educational status, 334 (40.6%) of mothers had no formal education, 35% (288) attended primary education and 24.4% (201) respondents attended secondary and higher. Five hundred twelve (62.2%) respondents were housewives. Four hundred and forty-three (53.8%) index infants of mothers were male. The majority of the respondents 572 (69.5) had access to information (Table 1).
Sociodemographic characteristics of mothers with children less than 12 months in South Gondar Zone, Amhara regional state, Ethiopia, May 2013 (n = 823)
Age of mothers (year)
< = 20 years
> = 35 years
Not in unionc
Maternal education
No formal education
Secondary education or higher
Occupation of the mother
Employeda
Unemployedb
Child's sex
aGovernment organization employees, private organization employees
bHousewives, daily labourers, farmers, merchants, business owner, students
cSingle, divorced, widow, cohabited, separated
Health service related and obstetrics characteristics
The study revealed that 702 (85.3%) mothers had antenatal care during their last pregnancy. Correspondingly, from all mothers were attending antenatal care, 451 (64.2%) mothers were receiving any information related to breastfeeding by healthcare providers. Regarding place of the delivery, the mothers indicated that from the total sampled, the mothers 564 (68.5%) gave birth in a health institution, 362 (64.2%) occurred in the health centre and the rest 202 (35.8%) were in a hospital. Four hundred nineteen (74.3%) had postnatal counselling on breastfeeding among mothers gave birth in health institution.
Among 564 mothers who gave birth in the health institution, 315 (55.9%) bathed their baby after 24 h and 189 (33.5%) within two to 24 h. This study showed that from the 259 (32.0%) mothers who gave birth at home, 157 (60.6%) of them bathed their child within one hour after birth. Three hundred ninety-six (48.1%) of the respondent were para I (primiparous mothers) and 189 (23.0%) of them were para II. Regarding to the time of birth, 755 (91.7%) of births were a term and 53 (6.4%) were a post term. Concerning the mode of delivery from all respondents, 795 (96.6%) had a vaginal delivery and 27 (3%) had a caesarean section.
Breastfeeding related characteristics
Mothers were asked about the time she decided to breastfeed their index child. The response revealed that 505 (61.4%) decided to breastfeed their infants before they became pregnant, 183 (22.2%) decided after delivery and the rest 135 (16.4%) decided during the pregnancy. All mother's breastfed their index child. Out of all, 808 (98.2%) mothers were breastfeeding their infants at the time of interview. Two hundred and ninety (35.2%) mothers expressed and discarded the first milk (colostrum) before they gave their breast milk for their index child. The most common reason for colostrum expulsion was 'it is dirty' 122 (42.1%) followed by, it creates abdominal pain 80 (27.6%) and open the closed nipple 45 (15.5%). One hundred seventy-four (21.1%) mothers introduced prelacteal foods or fluids to their infants. The most common prelacteal food was butter which is reported by 129 (74.1%) of breastfeeding mothers followed by sugar solution 18 (10.3%) and cow milk 15 (8.6%).
Initiation of breastfeeding
The proportion of mothers who initiated breastfeeding within one hour differ by residence which was 359 (54.7%) within urban and 42 (25.1%) within rural (Fig. 1).
Distribution of breastfeeding initiation by place of residence among mothers in South Gondar zone, Amhara regional state, Ethiopia, May 2013
Factor influencing timely initiation of breastfeeding
Using multivariate analysis; residence, parity, antenatal care, place of delivery and mode of delivery were identified as an independent predictor for timely initiation of breastfeeding among mothers in South Gondar Zone. Many of independent variables which showed statistical significance with the outcome variables in the bivariate analysis remained significantly associated with outcome variable in multivariate analysis (Tables 2 and 3). In this regard, the odds initiation of breastfeeding within one hour increased for urban mothers (Adjusted Odds Ratio [AOR] 2.1; 95% Confidence Interval [CI] 1.4, 3.3), for multiparous mothers (AOR 2.8; 95% CI 2.0, 3.8), for mothers who had antenatal care (AOR 3.2; 95% CI 2.0, 5.2), among mothers delivered in a health institution (AOR 3.1; 95% CI 2.2, 4.6) and for mothers who delivered vaginally (AOR 4.1; 95% CI 1.7, 9.8) with their respective counterparts (Table 4).
Bivariate analysis shows the association between sociodemographic characteristics with initiation of breastfeeding among mothers in South Gondar zone, Amhara regional state, Ethiopia, May 2013
Sociodemographic variable
≤ 1 h
> 1 h
p - value
COR (95% CI)
Age of the mother (years)
2.1 (1.4, 3.2)
149 (44.65)
Othersc
0.95 (0.1, 6.8)
aGovernment organization employee, private organization employee
bStudent, daily labourer, housewife, farmer, business owner merchant
cMuslim, protestant
Bivariate analysis shows the association between health service related and obstetrics characteristics with initiation of breastfeeding among mothers in South Gondar Zone, Amhara regional state, Ethiopia, May 2013
Crude Odds Ratio (95% CI)
Multiparous
Primiparous
Othersa
Health Institution
Mode of delivery
Child bathing
Within one hour
More than one hour
COR Crude odds ratio, CI Confidence interval, preterm, post terma
Multivariate logistic regression showing factors independently associated with initiation of breastfeeding among mothers in the south Gondar zone, Amhara regional state, Ethiopia, May 2013
AOR (95% CI)
3.1 (2.2 4.6)
COR Crude odds ratio, AOR Adjusted odds ratio, CI Confidence Interval
The rate of timely initiation of breastfeeding in this study was 48.7%. It was consistent to findings in Brazil (47.1%) and in Ethiopia Goba Woreda (52.4%) [8, 9]. However this was much lower than findings in Nepal (72.7%), Ethiopia Arjo Woreda 62.6% and south Gondar zone Farta district (76.7%) [10–12]. The prevalence of timely initiation of breastfeeding in this study was (48.7%) higher than in Amhara region (38%) [7].
In this study, 290 (35.2%) respondents expressed and discarded the first milk (colostrum) before starting to breastfeed for their index child. It was consistent with the study conducted in northern part of Ethiopia where colostrum was said to cause abdominal problems, but discarding a portion was sufficient to mitigate this effect [13]. However, according to a study conducted in western Nepal, a significant portion of mothers gave colostrum or breast milk as the first meal to 332 (86.2%) babies, while the remaining 54 (14%) babies were given a fluid other than breast milk as their first feed [10].
This study revealed that 174 (21.1%) mothers were introducing prelacteal foods or fluids to their child. The prevalence of the prelacteal feeding practice in the current study is higher than national findings where the proportion of women who gave prelacteal feeding within the first three days of life was 13% [14]. Another study conducted in the rural northern part of Ethiopia showed that the majority of mothers practiced ritual prelacteal feeding [13]. In addition, in this study, the common prelacteal food introduced to the newborn baby was butter in 129 (74.1%) followed by sugar solution 18 (10.3%) and cow milk in 15 (8.6%). It was consistent with research findings in Mekele town which revealed the common prelacteal food introduced for the newborn babies was butter followed by sugar solution and cow milk [15].
Timely initiation of breastfeeding is influenced by varied and complex interrelated factors and multivariate logistic analysis showed that the odds of timely initiation of breastfeeding among mothers who had antenatal care was increased 3.2 times compared to mothers who had no antenatal care. Correspondingly, mothers that received antenatal care have relative reduced risks of about 8% of delaying breastfeeding initiation than mothers without antenatal care [16]. The possible reason could be that pregnant women who had antenatal care might be informed about timely initiation of breastfeeding by healthcare providers.
The odds of timely initiation of breastfeeding among mothers increased 3.1 times among mothers who had an institutional delivery compared with mothers who delivered at home. A similar study indicated that mothers who delivered their babies at home have an increased relative risk of about 12% of delaying early initiation of breastfeeding than mothers who delivered in the hospitals (or clinics) [16]. This can be explained as mothers who gave birth in health institution had healthcare provider support which helped to initiate breastfeeding timely.
In the current study, the odds of timely initiation increased 4.1 times when mothers delivered vaginally than mothers delivered through caesarean section. It is similar to a study conducted in Nigeria which says mothers that were delivered of their babies through caesarean section have about 58% increased the risk of delaying the early introduction of the first breast milk to their babies as compared to mothers who had a vaginal (normal) delivery [17]. In addition, a systematic review of 53 studies revealed that rates of early breastfeeding (any initiation or at hospital discharge) were lower after caesarean delivery compared with after vaginal delivery [18].
The odds of timely initiation of breastfeeding increased 2.1 times for mothers who resided in an urban than in rural area. Similarly, a study conducted in Ethiopia Goba district showed that urban dwellers were three times more likely to practice timely initiation of breastfeeding when compared to their rural counterparts [9]. In contrast, a study conducted in Al-Hassa province, Saudi Arabia showed that rural mothers were 4.2 times more likely to initiate breastfeeding within one hour [17]. In the current study, the lower rate of timely initiation of breastfeeding in rural areas was probably the traditional practice in the areas; such as early child bathing, colostrum expulsion and prelacteal feeding.
The odds of timely initiation of breastfeeding increased 2.8 times among multiparous mothers than primiparous mothers. In the same way, a study conducted in Turkey showed that breastfeeding initiation was later in primiparous mothers than in mothers who are multiparous [19].
This study shows that the rate of timely initiation of breastfeeding was low in south Gondar zone. Factors which were positively associated with timely initiation of breastfeeding include urban residence, multiparty, having antenatal care, mother deliver in health institution and vaginal mode of delivery. Therefore, South Gondar health office and healthcare providers have to provide breastfeeding information during antenatal care by giving special emphasis to rural and primiparous mothers where timely initiation of breastfeeding is poorly practiced. Further study is needed to assess the implementation of policies on timely initiation of breastfeeding.
AOR:
Adjusted Odds Ratio
CI:
Confidence Interval
Crude Odds Ratio
SPSS:
Statistical Package for Social Science
UNICEF:
United Nation Children's Fund
We would like to express our deepest gratitude to Jimma University and South Gondar health bureau for their facilitation during this study. We are also extremely grateful to the data collectors, supervisors and study participants.
The data of this study available from the corresponding author and will be avail on reasonable request.
LM conceived and designed the study, performed analysis and interpretation of data and drafted the first manuscript. WS and ZS participated in the critical review of the subsequent draft of the manuscript. All authors read and approved the final manuscript for publication.
The ethical approval and clearance for the study before data collection were obtained from Jimma University College of public health and medicine research ethical clearance board. Official letters were obtained from South Gondar zone health office and each study districts health bureau. At the time of data collection, Informed consent was obtained from the respondents after explaining the purpose of the study.
College of Medicine and Health Sciences, Public Health Department, Reproductive Health and Nutrition Unit, Jigjiga University, Jigjiga, Ethiopia
College of Medicine and Health Sciences, Public Health Department, Epidemiology and Biostatistics Unit, Jigjiga University, Jigjiga, Ethiopia
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Comparison of Ferguson's δ and the Gini coefficient used for measuring the inequality of data related to health quality of life outcomes
Hsien-Yi Wang1,2,
Willy Chou3,4,
Yang Shao5 &
Tsair-Wei Chien6
Health and Quality of Life Outcomes volume 18, Article number: 111 (2020) Cite this article
Ferguson's δ and Gini coefficient (GC) are defined as contrasting statistical measures of inequality among members within populations. However, the association and cutting points for these two statistics are still unclear; a visual display is required to inspect their similarities and differences.
A simulation study was conducted to illustrate the pertinent properties of these statistics, along with Cronbach's α and dimension coefficient (DC) to assess inequality. We manipulated datasets containing four item lengths with two number combinations (0 and 33%) in item length if two domains exist. Each item difficulty with five-point polytomous responses was uniformly distributed across a ± 2 logit range. A simulated response questionnaire was designed along with known different structures of true person scores under Rasch model conditions. This was done for 20 normally distributed sample sizes. A total of 320 scenarios were administered. Four coefficients (Ferguson's δ, GC, test reliability Cronbach's α, and DC) were simultaneously calculated for each simulation dataset. Box plots were drawn to examine which of these presented the correct property of inequality on data. Two examples were illustrated to present the index on Google Maps for securing the discriminatory power of individuals.
We found that 1-Ferguson's δ coefficient has a high correlation (0.95) with GC. The cutting points of Ferguson's δ, GC, test reliability Cronbach's α, and the DC are 0.15, 0.50, 0.70, and 0.67, respectively. Two applications are shown on Google Maps with GCs of 0.14 and 0.42, respectively. Histogram legends and Lorenz curves are used to display the results.
The GC is recommended to readers as an index for measuring the extent of inequality (or lower discrimination power) in a given dataset. It can also show the study results of person measures to determine the inequality in the health-related quality of life outcomes.
The required measurement properties of health-related quality of life (QoL) questionnaires are reliability, discriminatory power, and validity. Traditionally, assessment measurements are formally evaluated using the indices of reliability (the degree of measurement error) and validity (the extent to which the questionnaire measures what it is supposed to measure) [1].
Ferguson's δ [2] was applied in studies published before 2007 to measure the discriminatory power of a test [3, 4]. Hankins provided a generalized formula for calculating Ferguson's δ for questionnaires with dichotomous and polytomous items [1] and then (re-)introduced the coefficient δ as an index of discrimination to be distinguished from the well-known validity and reliability measurement properties [5, 6].
However, Hankins' paper [1] resulted in critical comments from Wyrwich [7] and Norman [8] regarding reliability issues. Hankins [1] (1) applied Ferguson's δ to identify the discrimination of GHQ-12 data using the dichotomous scoring method and four-point Likert-type scoring method, and (2) showed that the Likert-type scoring method could better discriminate between individuals compared with the dichotomous method. Moreover, as expected, the four-point Likert scale showed higher reliability than the dichotomous method. Hankins responded to the comments [9] by stating that, aside from reliability and validity, Ferguson's δ is an additional index of an instrument's measurement properties, i.e., Ferguson's δ can only be computed on the assumption that the measurement is valid and reliable [9]. Terluin and his colleagues [6] stated that the magnitude of Ferguson's δ is only determined by the distribution of the scores in a given sample. They also argued that Ferguson's δ is an unnecessary property measure because it ignores reliability, making it impossible to interpret when questionnaire reliability is considered [6].
Despite these discussions [1, 5,6,7,8,9] in 2007 and 2008, several papers [10,11,12] were published, in which Ferguson's δ was used to measure the scale discriminatory power between individuals. In Eq. (1), we list the traditional Ferguson's δ used for the binary scale and polytomous items developed by Hankins [1]. The original Ferguson's formula is simplified to the Guilford's equation (i.e., the 2nd part in Eq. (1)) [13] and developed in line with Hankins' formula (i.e., the 3rd part used for the polytomous scale) given by [1]
$$ {\displaystyle \begin{array}{l}\delta =\frac{n^2-\sum \limits_{i=0}^k{f}_i^2}{n^2-\frac{n^2}{\left(k+1\right)}}=\frac{\left(k+1\right)\left({n}^2-\sum \limits_{i=0}^k{f}_i^2\right)}{k\left({n}^2\right)}=\frac{\left(k\left(m-1\right)+1\right)\left({n}^2-\sum \limits_{i=0}^{k\left(m-1\right)}{f}_i^2\right)}{k\left(m-1\right)\left({n}^2\right)},\\ {}\end{array}} $$
where n is the number of elements (or summation of all frequencies), f is the frequency of score i, k is the number of questionnaire items, and m is the length of the scale (i.e., number of the threshold for a rating scale). Terluin et al. [6] proposed a standard computation of Ferguson's δ (see Eq. (2)) expressed as
$$ \delta =\frac{q}{q-1}\times \frac{\left({n}^2-\sum \limits_{i=1}^q{f}_i^2\right)}{\left({n}^2\right)}=\frac{q}{q-1}\times \left(1-\frac{\sum \limits_{i=1}^q{f}_i^2}{\left({n}^2\right)}\right)=\frac{q}{q-1}\left(1-\sum \limits_{i=1}^q\frac{f_i^2}{n^2}\right)=\frac{q}{q-1}\left(1-\sum \limits_{i=1}^q{p}_i^2\right), $$
where q represents the possible score categories (i.e., number of bins for all elements) and p is the proportion for each frequency to the total number of persons. Referring to the reliability, commonly represented by Cronbach's α [14] and shown in Eq. (3), where K is the item length, we can see that Eq. (3) is very similar to Eq. (2), particularly in the 2nd part.
$$ \alpha =\frac{k}{k-1}\left(1-\frac{\sum \limits_{i=1}^I{\sigma}_i^2}{\sigma_x^2}\right) $$
The difference lies in the numerator, which is the sum of identical elements across all bins in Eq. (2) and the summed variances across items in Eq. (3). As can be seen, the property of Ferguson's δ is almost involved in Cronbach's α if we reverse δ as (1 − δ).
Furthermore, the Gini coefficient (GC) [15] is a measure of statistical dispersion to represent the income or wealth distribution of a nation's residents (see Eq. (4)) and is expressed as.
$$ \mathrm{Gini}=\frac{q}{q-1}\times \frac{\sum \limits_i\sum \limits_j\mid {X}_i-{X}_j\mid }{2\sum \limits_i\sum \limits_j\overline {X_{ij}}}=\frac{q}{q-1}\times \frac{\sum \limits_i\sum \limits_j\mid {X}_i-{X}_j\mid }{2\times {q}^2\times {\overline{X}}_{ij}}, $$
where X is the frequency for each element, X-bar refers to all elements in frequencies or bins, and q is the number of bins. The numerator denotes the total absolute deviation between frequencies in bins, and the denominator represents the maximal portion of the total difference. We can also clearly see in Eq. (4) That the property of Ferguson's δ is similar to GC if we reverse δ as (1 − δ). Given that Eqs. (2)–(4) are very similar, whether they have high correlations is worthy of investigating. Thus, we aim to inspect the associations among the three indices, examine their validity and dimension coefficient (DC) [16], and verify whether GC can replace Ferguson's δ as an index of discriminatory power between individuals.
The objectives of this study are as follows: to (i) compare the relationship between GC and Ferguson's δ; (ii) to examine the associations among these four indices using simulation data; and (iii) to illustrate applications of the usefulness on GC in practice.
Simulative datasets
The simulated data contain four item lengths (i.e., 5, 10, 15, and 20) with two number combinations (i.e., 0 and 33%) in item length if two domains exist (e.g., 7 items on a domain and 3 on another when item length = 10).
Each item difficulty, with five-point polytomous responses, was uniformly distributed across a ± 2 logit range, and the questionnaire response was interacted by person ability and item difficulty under Rasch model conditions [17]. This was carried out for 20 normally distributed sample sizes (n from 50 to 1000 with an interval of 50). The detailed steps are stated below.
(A).
The questionnaire responses were determined by (1) person ability and (2) item difficulties [17].
(B).
A total of 320 simulation datasets were manipulated as follows: (1) four types of item lengths (i.e., 5, 10, 15, and 20), (2) four kinds of item loadings (i.e., 0.3, 0.5, 0.7, and 1.0) to the test domain, and (3) 20 normally distributed sample sizes (n from 50 to 1000 with an interval of 50).
(C).
Based on the terms stated in (B) above, item difficulties on each type of item length were uniformly distributed across a ± 2 logit range, and the summation was equal to zero (i.e., the mean of all item difficulties = 0). For instance, five items had difficulties of {− 2, − 1, 0, 1, 2}. Other types of item length (e.g., 10, 15, 20) were similarly assigned with diffident difficulties from − 2 to 2. The total difficulties for each dataset were equal to zero.
(D).
Item loading to the test domain refers to the correlation of responses between the specific item and the domain (≒summation across all items). If all items have a high correlation to the domain, the scale can be unidimensional and considered to have a high construct validity (e.g., the DC > 0.9 [16]).
(E).
The way to generate responses in a one-dimensional (1D) scale (i.e., all items measuring a common character or attribute, such as leadership) is to set all responses on items with high correlation. The processes are shown herein.
(F).
First, we determined the person latent trait called variable T. Assuming the standardized summation score across all items followed a normal distribution; we applied the function of random number generation in MS Excel to produce variable T.
Next, the responses by the person on the item were determined by the new variable T1 according to the formula (= T*cos (angle) + W*sin (angle)), where variable W is generated by normally distributed random numbers, and the angle is defined by RADIANS (angle degree) using the MS Excel function.
(G).
The 1D scale was formed by performing step (F) on all items when T1 = T (i.e., corr(T, T1) ≒ 1.0) and responses are determined by the item difficulty and the person's ability on T1.
The two-domain scale is yielded by T and T1, where T1 ≠ T, corr(T, T1) ≒ correlation (=0.3, 0.5, or 0.7), and the ratio of item length on two domains is 7:3. The responses are generated by the item difficulties and Ti, respectively.
(H).
Finally, Rasch data were simulated following the process described in reference [17].
The four coefficients, Ferguson's δ, GC, test reliability (Cronbach's α), and DC, were simultaneously calculated for each simulation dataset (see Fig. 1). The simulation process for this study is presented in Additional file 1.
Study flowchart with data from 320 scenarios under the Rasch rating scale model
Comparing the relationship between GC and Ferguson's δ
Five sequential scores were designed as {1,2,3,4,5}, {1,2,3,4,5(10)}, {1,2,3,4,5(50)}, {1,2,3,4,5(300)}, and {1,2,3,4,5(900)}, where the numbers in parentheses denote the occurrence of the previous number; for instance, 5(10) means 5 occurs 10 times in the sequence. The results of (1 − Delta) and GC are expected to monotonously increase as the kurtoses are raised.
If we remove the adjustment of q/(q − 1) in Eq. (4), GC becomes 1, but only for a large population in which one person has all the income. For the 5-element set, where 4 has no income, and the fifth has all the income, the GC is 0.8. Thus, we adjusted Eq. (4) with the argument of q/(q − 1) ahead at the equation.
Next, we mimicked the World Bank's method of calculating global wealth inequality for each county/area and divided resident incomes into five strata (i.e., lowest fifth, second fifth, third fifth, fourth fifth, and highest fifth). As a result, we were able to compare GCs with one another on a common base of five strata with equal size.
Examining the associations among the four indices using simulation data
Ferguson's δ, GC, Cronbach's α, and DC were all examined using simulation data. Box plots were drawn to examine the property of inequality for the former two compared with the reliability and validity of the latter two. The cutting points of these four indices were also determined in this study.
Illustrating the practical applications of GC
Two examples were illustrated to present GC on Google Maps for securing the discriminatory power of individuals.
Liking for Science Questionnaire - This measures the attitudes of children to science-related activities [18]. It is an attitude survey with Likert scale ratings, where 0 = Dislike, 1 = Don't know, and 2 = Like [18, 19]. The frequency values of examinees across bins on histogram were used to compute the GCs.
International author collaborations found in published papers on health-related QoL outcomes. After searching abstracts from MEDLINE with the keywords "Health Qual Life Outcomes" [Journal], a total of 2183 research articles were downloaded. These were then plotted on Google Maps using choropleth maps and Lorenz curves [20] to display the distribution of publication outputs across countries/areas for first authors. The frequency publication outputs of members across the world were used to compute GC using the quantile classification method (i.e., equal sizes in each class).
The attributes of GC and (1-Delta) are shown in Table 1, which indicates close relations between the two coefficients. The results of (1 − Delta coefficient) and GC monotonously increase as the kurtoses are raised across scenarios (Table 1). The higher the (1 − Delta) or GC, the lesser the discriminatory power of individuals, thus indicating that unequal inequality exists.
Table 1 Comparison of (1-Delta) and Gini coefficient
Whether the higher GC (or 1-Delta) for person measures is negatively related to lower Cronbach's α will be examined in the next section.
Examining the association among the four indices using simulation data
The correlation coefficient relation between the (1 − Delta) and GC indices is 0.95, and R-square = 0.90 (Fig. 2). The box plots in Fig. 3 show that (1 − Delta) and GC are closely related in contrast to Cronbach's α and DC, which have a negative correlation, that is, the higher the Cronbach's α (or DC), the greater the discriminatory power between individuals. In contrast, the higher the GC and (1-Delta), the lesser the discriminatory power between individuals [1, 5].
The relation between these two coefficients
Comparisons of data distribution for the three study indicators
The DC is more sensitive to misfit items(i.e., with lower loadings to the test domain) than Cronbach's α, thus implying that reliability is a necessary, but not sufficient, a component of validity [21, 22]. The DC is, therefore, necessarily incorporated with Cronbach's α to completely and fully describe a scale's characteristics [23]. This is because not all reliable scales are valid [24].
The cutting points are determined at 0.15 for (1 − Delta) or 0.85 for Delta, 0.50 for GC, 0.67 for DC, and 0.70 for Cronbach's α. Particularly, we refer the common cutting point for Cronbach's α at the lower limit (0.7) to the upper limit (0.5) for GC in Fig. 3. The result of the GC cutting point setting at 0.5 is similar to the previous study [25, 26].
Illustrating the practical applications of using Gini coefficients
In Figs. 4 and 5, the GCs for the two applicable examples are shown on Google Maps [27, 28], where GCs = 0.14 and 0.42, respectively, indicating that the publication outputs based on countries/areas for Health Qual Life Outcomes and the individual performances for the Liking for Science Questionnaire present acceptable GCs (< 0.5).
Gini coefficient sfor histogram shown on Google Maps
Distribution of countries/areas for author publications on Google Maps (GC = 0.14 on the top 5 clusters)
However, the Cronbach's α and DC values for the questionnaire in Fig. 4 are 0.88 and 0.64, respectively, indicating that the GC is negatively related to Cronbach's α. Two items (Nos. 5 and 23) are a misfit to the Rasch model and lead to a lower DC (0.64 < 0.67) [16].
The top three most productive countries for Health Qual Life Outcome are the United States (224, 5.59%), the United Kingdom (205, 5.11%), and Sweden (200, 4.99%), respectively. Interested readers are invited to scan the Quick Response codes in Figs. 4 and 5 to see more details about the two practical examples.
This study finds that (i) GC can replace Ferguson's δ as an additional index of an instruments measurement properties aside from reliability and validity, and (2) the method used by the World Bank to calculate the GC for each country/area, in which sample scores are divided into five strata with equal size, is practically feasible.
What this adds to existing knowledge and what is known
Although Terluin et al. [6] argued that Ferguson's δ becomes unnecessary when reliability is considered, Ferguson's δ (or GC) can be an additional index aside from reliability and validity [9, 29]. In particular, GC at a cutting point of 0.5 is easier to use than Delta and can better determine a scale with effective discriminatory power.
For a long time, the alarming level of GC has been globally taken as 0.4, especially in World Bank calculations for global wealth inequality [25]. Although the 0.4 standards is widely accepted [26], the derivation of the value lacks rigid theoretical foundations. Our computations using simulation data show that the alarming level should be specified to be equal or larger than 0.5 based on the 95% confidence intervals rather than 0.4 at the median to the distribution. The result is similar to that presented by a previous study [25].
A normal distribution is expected to have a discrimination of Delta > 0.90 [1, 2], which is near the findings (1 − Delta = 0.09 or Delta ≥0.91) in Fig. 3. However, the lower boundary of Delta at 0.85 can provide readers an objective way to examine the discriminatory power, particularly for data following a uniform distribution. Many teachers are concerned about whether the abilities of students are equal. This is because the more equal the abilities of students, the more willing many teachers are to teach the class [30]. The GC can be used to compare the degree of equality between the academic abilities of students.
What this implies and what should be changed
Our findings in Task 2 (i.e., Comparing the relationship between GC and Ferguson's δ) corresponds with previous studies [16, 31], which suggest incorporating Cronbach's α with DC or exploratory factor analysis to jointly assess scale quality. We see in Fig. 3 that the DC can discriminate scale dimension tendency more sensitively than Cronbach's α. We also confirm that the cutting points are 0.7 for Cronbach's α [32] and 0.67 for DC [16, 33].
In Task 3 (i.e., Illustrating practical applications of the usefulness of GC), we show visual displays on Google Maps, which enable users to gain an overall geospatial visualization [34, 35]. The GC across bins on the histogram (Figs. 4 and 5) can be an additional index shown to readers. Interested readers are recommended to use the methods shown in Additional files from 2 to 4 and Ref. [36] to easily compute the GCs on their own.
The strengths of this study
Due to authors in previous five articles [5,6,7,8,9] discussing the value of Ferguson's δ in 2007 and 2008, we complemented these article in details about the feature of Ferguson's δ related to other coefficients(e.g., Gini coefficient and Cronbach's Alpha), including Equations from (1) to (4) and verifications of Ferguson's δ that can be replaced with Gini coefficient for measuring the extent of inequality (or lower discrimination power) in a given dataset based on a quality-of-care scale.
Furthermore, we applied legends along with choropleth maps [37, 38] to report GCs (< 0.50) and to ensure a higher discriminatory power (or say equal sizes in classes) on the Questionnaire (in Task 3, Fig. 4) and the international coauthor collaborations in papers extracted using the search term "Health Qual Life Outcomes" (Fig. 5) [28]. To the best of our knowledge, this is a distinct application that has never been used in previous papers.
We also developed a visual display to present the survey results in Ref. [26], that is, we presented the histograms and the GCs on Google Maps based on cloud computation. The way we incorporated choropleth maps and legends with Google Maps is a unique approach compared with other research methods [39,40,41]. This is because we used the dashboard to present the study results, which are better displayed than in traditional image formats. Interested readers can even manipulate the links according to their methods to understand the features of interest, such as the examinee distribution [26] and the international coauthor collaboration [27]. As the saying goes, "A picture is worth a thousand words," and about this, we hope that future studies can report on other types of information using Google's application programming interface.
Limitations and directions for future study
Our study has some limitations. First, only 320 simulation scenarios were conducted(i.e., equal to 4 item length × 4 numbers of misfit items × 20 sample sizes). Hence, caution should be exercised when using the inference of this study, as many other scenarios in the real world have not been included.
Second, the GC calculation followed the World Bank's GC calculation for a country/area, in which incomes are divided into five strata. For this reason, the feasibility and applicability should be further proved in the future even though any kind of data (i.e., count or continuous variables) can be easily applied to compute GC and objectively compared to others because all values are based on an equal number of observations in bins.
Third, the data in Task 3 were extracted from the MEDLINE Library and were carefully addressed. Every linkage was examined as correctly as possible. The originally downloaded contexts included some errors in symbols that might affect the resulting reports in this study, such as those shown in Fig. 5.
Fourth, the simulation data were processed under Rasch model conditions. The results in this study, such as the determination of cutting points for each index (Fig. 3) might be different from those of the other situations. For instance, a single variable is generated by a normal or uniform distribution or based on other types of item response theory. Hence, future studies regarding the determination of cutting points for indices are encouraged, and other conditions should be used to simulate data in the future.
The GC is recommended to readers as an index to measure the extent of inequality (or lower discrimination power) in a given dataset. The GCs can also show the study results of person measures to determine the inequality in health-related QoL outcomes.
All items and data can be obtained from those in additional supporting files of this study.
one dimension
dimension coefficient
EFA:
exploratory factor analysis
GC:
HQLO:
Health Quality of Life Outcomes
IRT:
item response theory
SNA:
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We thank The Enago Company (https://www.enago.tw/) for proofreading the manuscript service.
There are no sources of funding to be declared.
Department of Nephrology, Chi-Mei Medical Center, Tainan, Taiwan
Hsien-Yi Wang
Department of Sport Management, College of Leisure and Recreation Management, Chia Nan University of Pharmacy and Science, Tainan, Taiwan
Department of Physical Medicine and Rehabilitation, Chung Shan Medical University, Taichung, Taiwan
Willy Chou
Department of Physical Medicine and Rehabilitation, Chiali Chi Mei Hospital, Tainan, Taiwan
School of Fashion and Design, Jiaxing Vocational and Technical College, Jiaxing, China
Yang Shao
Departments of Medical Research, Chi-Mei Medical Center, 901 Chung Hwa Road, Yung Kung Dist., Tainan, 710, Taiwan
Tsair-Wei Chien
All authors have read and approved the final manuscript. TWC developed the study concept and design. YS and WC collected data, TWC and YS analyzed and interpreted the data. TWC drafted the manuscript, and all authors provided critical revisions for important intellectual content. The study was supervised by HYW.
HYW is a medical doctor and a specialist in biostatistics. SY is expert in computer science. TWC is an associate professor at Chi Mei Medical Center, Taiwan. He is an expert in computer science and Rasch modeling, mainly in the field of data analysis using the statistical technique.
Correspondence to Tsair-Wei Chien.
All data were simulated from computer algorithms.
The process of simulation data in this study at https://youtu.be/5BLJtiif2M4.
MP4 showing the simulation process http://www.healthup.org.tw/marketing/course/marketing/raschsimulateddata.mp4.
MS Word for showing the codes of simulation.
Excel file to calculate Delta and GC as well as contents for this study.
Wang, HY., Chou, W., Shao, Y. et al. Comparison of Ferguson's δ and the Gini coefficient used for measuring the inequality of data related to health quality of life outcomes. Health Qual Life Outcomes 18, 111 (2020). https://doi.org/10.1186/s12955-020-01356-6
Ferguson's δ
Cronbach's α | CommonCrawl |
Paclitaxel and curcumin coadministration in novel cationic PEGylated niosomal formulations exhibit enhanced synergistic antitumor efficacy
Ashraf Alemi1,
Javad Zavar Reza1,2,
Fateme Haghiralsadat3,
Hossein Zarei Jaliani4,
Mojtaba Haghi Karamallah2,
Seyed Ahmad Hosseini5 &
Somayeh Haghi Karamallah6
The systemic administration of cytotoxic chemotherapeutic agents for cancer treatment often has toxic side effects, limiting the usage dose. To increase chemotherapeutic efficacy while reducing toxic effects, a rational design for synergy-based drug regimens is essential. This study investigated the augmentation of therapeutic effectiveness with the co-administration of paclitaxel (PTX; an effective chemotherapeutic drug for breast cancer) and curcumin (CUR; a chemosensitizer) in an MCF-7 cell line.
We optimized niosome formulations in terms of surfactant and cholesterol content. Afterward, the novel cationic PEGylated niosomal formulations containing Tween-60: cholesterol:DOTAP:DSPE-mPEG (at 59.5:25.5:10:5) were designed and developed to serve as a model for better transfection efficiency and improved stability. The optimum formulations represented potential advantages, including extremely high entrapment efficiency (~ 100% for both therapeutic drug), spherical shape, smooth-surface morphology, suitable positive charge (zeta potential ~ + 15 mV for both CUR and PTX), sustained release, small diameter (~ 90 nm for both agents), desired stability, and augmented cellular uptake. Furthermore, the CUR and PTX kinetic release could be adequately fitted to the Higuchi model. A threefold and 3.6-fold reduction in CUR and PTX concentration was measured, respectively, when the CUR and PTX was administered in nano-niosome compared to free CUR and free PTX solutions in MCF-7 cells. When administered in nano-niosome formulations, the combination treatment of CUR and PTX was particularly effective in enhancing the cytotoxicity activity against MCF-7 cells.
Most importantly, CUR and PTX, in both free form and niosomal forms, were determined to be less toxic on MCF-10A human normal cells in comparison to MCF-7 cells. The findings indicate that the combination therapy of PTX with CUR using the novel cationic PEGylated niosome delivery is a promising strategy for more effective breast cancer treatment.
Chemotherapy is the standard treatment for various types of cancers. However, chemotherapy is associated with high systemic toxicity and low therapeutic effectiveness [1]. Nanotechnology has revolutionized the diagnosis and treatment of cancer [2]. A nano-sized drug delivery system (DDS), or nanocarrier, is designed to deliver therapeutic and/or diagnostic agents to their target sites [3]. Over recent decades, drug delivery systems using vesicular carriers have attracted great interest because these carriers provide high encapsulation efficiency, control drug release, enhance drug solubility, carry both hydrophilic and hydrophobic drugs, reduce side effects, prolong circulation in blood, and possess the ability to target a specific area [4, 5]. Vesicles made of natural or synthetic phospholipids are called liposomes, while transferosomes are modified liposomal systems that, in addition to phospholipids, contain a single chain surfactant as an edge activator; ethosomes contain ethanol as an edge activator instead of a single chain surfactant. Despite having some advantages over conventional dosage forms, vesicular carriers present many problems in practical applications, such as high cost, the use of organic solvents for preparation, and a limited shelf life due to lipid rancidification [6]. Therefore, a continuous endeavor has been made to find an alternative vesicular carrier. Niosomes meet this requirement [7]. Niosomes, or non-ionic surfactant vesicles, are unilateral or multilamellar spheroidal structures. Niosomes are preferred as an effective alternative to conventional liposomes, as they offer several advantages, including greater stability, lower cost, biodegradability, biocompatibility, non-immunogenic, and low toxicity, and they can be stored more easily for industrial production in pharmaceutical applications [5, 8,9,10,11,12]. To improve stability and circulation half-life, niosomes may be coated with appropriate polymer coatings, such as polyethylene glycol (PEG), creating PEGylated niosomes. PEG coating also helps reduce systemic phagocytosis, which results in prolonged systemic circulation, as well as reduced toxicity profiles [13, 14]. Paclitaxel (PTX) is an important antineoplastic drug, and it is isolated from the bark of Taxus brevifolia. PTX demonstrates an effective chemotherapeutic and cytotoxic activity against breast, ovarian, colon, lung, prostate, and brain cancers. However, the wide therapeutic effects of PTX are limited due to the low therapeutic index and poor water-solubility [15, 16]. Curcumin (CUR) is a hydrophobic polyphenol compound obtained from the rhizome of the plant Curcuma longa. CUR exhibits various pharmacological activities, such as anti-inflammatory, anti-oxidant, and anti-tumor effects. Particularly, CUR has been demonstrated to be highly effective against a variety of different malignancies, including leukemia and lymphoma, as well as colorectal, breast, lung, prostate, and pancreatic carcinoma. However, the pharmacological application of CUR has been impeded due to its extremely low aqueous solubility, instability, extremely poor bioavailability, and high metabolic rate [17,18,19]. As a result, nanotechnology is considered one of the most significant methods to design and develop various nano-carrier formulations for curcumin and paclitaxel, such as polymeric micelles, liposomes, self-assemblies, nanogels, niosome biodegradable microspheres, and cyclodextrin [18, 20, 21]. In this study, we loaded both curcumin and paclitaxel into cationic PEGylated niosomal formulations for enhanced efficacy in MCF-7 human breast adenocarcinoma cells. In addition to formulation design and optimization, we have examined release profile, intracellular delivery, and enhancement of cytotoxicity appears.
The effect of surfactant:cholesterol ratio on CUR/PTX niosome formulations
To specify the optimal formulation for attaining high entrapment efficiency, controlled release (at 37 °C and pH 7.4), and small vesicle size, various niosomal CUR/PTX formulations were evaluated (Table 1). As shown in Table 1, cholesterol had a profound effect on CUR/PTX entrapment efficiency in niosomes: by increasing the amount of cholesterol content from 10% in formulation 1 (F1) to 30% in formulation 4 (F4), PTX/CUR entrapments into nano-niosomes were constantly increased. However, adding cholesterol from F1 to F4 decreased the percentage of CUR/PTX released over 12 h. Furthermore, as can be seen from the presented results, the mean diameter of the niosomes increased with increasing the cholesterol content (F1 → F5, Table 1). However, the addition of up to 50% cholesterol to niosomes in F5 decreased niosomal efficiency in trapping curcumin/paclitaxel compared to the 30% cholesterol content in F4. Based on high entrapment efficiency and sustained drug release, the F4 formula has chosen as the formulation for further studies.
Table 1 Effect of the non-ionic surfactant Tween 60: cholesterol with various molar ratios on entrapment efficiency (EE %), size and % release (R) in CUR/PTX loaded Niosomes
The effect of DSPE-mPEG (2000) and DOTAP in niosomal formulation
For attaining less aggregation, smaller niosomes, higher entrapment efficiency, and improved stability, 5% PEG was added to F4. According to Table 2, the F6 niosomal formula containing 5% PEG showed higher drug entrapment, smaller diameter, smaller Poly-Dispersity Index (PDI), and lower drug release than the F4 formula. Table 2 shows the number of positive charge particles and the entrapment efficiency were increased by adding 10% DOTAP to F6. However, vesicle size and PDI declined with a 10% increase in the molar amount of DOTAP. The obtained results showed the CUR/PTX niosomal formulations containing Tween-60: cholesterol:DOTAP:PEG with a 59.5:25.5:10:5 molar ratio (F7) had the desired feature based on high entrapment efficiency, sustained drug release, small diameter, and improved transfection efficiency (Table 2).
Table 2 Effect cationic phospholipid DOTAP and DSPE-mPEG (2000) on entrapment efficiency (EE %), size and % release (R) in CUR/PTX loaded Niosomes
Physical characterization of niosomal vesicles
The internal structure of CUR/PTX niosomes was evaluated by cryogenic transmission electron microscopy (Cryo-TEM). As illustrated in Fig. 1a, b, the optimum formula of CUR/PTX niosomes was spherically shaped. Furthermore, the niosomes structures' rigid boundaries were indicated. According to SEM photographs, the niosomal vesicles were found to be round with smooth surfaces (Fig. 1c, d).
Morphological assessment: a niosomal paclitaxel; b niosomal curcumin by cryogenic transmission electron microscopy (Cryo-TEM). Scanning electron microscopy (SEM) of c curcumin niosome; and d paclitaxel niosome
In-vitro drug release study
Evaluation of in vitro drug release was performed using the dialysis method. The results of a 72-h release profile of CUR and PTX from the optimum formulation (F7) in PBS pH 7.4 at 37 °C are displayed in Fig. 2. After 72 h, 29.93 and 28.16% of the loaded drugs were released for CUR and PTX, respectively. The cumulative release profile of CUR and PTX was apparently biphasic, with an initial rapid release period followed by a slower release phase.
The in vitro release profile of curcumin and paclitaxel from niosomal optimum formula
Release kinetics modeling
Figure 3 shows the CUR/PTX release data were analyzed mathematically according to: zero-order, first-order, Hixson–Crowell, and Higuchi's equations. Table 3 summarizes the correlation coefficients (R2) calculated for niosomal formulations. The results revealed that the release of CUR and PTX from niosomal films is most fitted to the Higuchi model, according to the higher correlation coefficient.
Curcumin and paclitaxel comparative plots. a Zero order release kinetics; b first order release kinetics; c Higuchi (SQRT) release kinetics and d Hixson–Crowell model
Table 3 Release kinetics data of CUR and PTX from the niosomal optimum formulae
Fourier transforms infrared (FTIR) spectral evaluation
To confirm the drug presence in CUR/PTX nano-niosome formulations, FTIR analysis was performed. Figure 4a shows the FTIR spectrum of free paclitaxel. There were characteristic peaks in this spectrum: O–H stretching and N–H stretching in 2° amine at 3445 cm−1, –CH3 asymmetric and symmetric stretching at 2923 cm−1, conjugation of C=O with phenyl group at 1733 cm−1, C–O stretch at 1122 cm−1, and the C–H out-of-plane bending vibrations for monosubstituted rings in the paclitaxel molecule in the region of 900–500 cm−1.
FTIR spectra. a Free paclitaxel; b free curcumin; c blank noisome; d niosomal paclitaxel; e niosomal; curcumin; f comparison blank noisome and niosomal paclitaxel; g comparison blank noisome and niosomal curcumin
Figure 4b demonstrates the FTIR spectrum of free curcumin. The bands exhibited in this spectrum can be assigned to: C–H stretching and O–H stretching at 3507 cm−1, aromatic ring C=C stretching at 1506 cm−1, C=O stretch at 1152 cm−1, and the C–H out-of-plane bending vibrations for ortho-disubstituted rings in curcumin molecule in the region of 800–600 cm−1.
The FTIR pattern for blank niosome (Fig. 4c) demonstrates various characteristic peaks of DOTAP, Tween-60, cholesterol, and DSPE-mPEG in the range of 3500–1115 cm−1. The band observed at 3435 cm−1 was assigned to cholesterol and Tween-60 (O–H stretching in phenols and N–H stretching in 2°-amines). C–N stretch and C–O stretch occur at 1148 cm−1 and belonged to DOTAP and Tween-60, respectively. The carbonyl group exhibits a strong absorption band at 1642.15 cm−1 due to C–O stretching vibration in DSPE-mPEG, Tween-60, and DOTAP. All peaks were repeated in the FTIR spectrum of PTX/CUR niosome formulations. The niosomal paclitaxel FTIR spectrum (Fig. 1d) shows the out-of-plane bending peaks in the range of 900–500 cm−1, and it can be used to assign mono-substitution on the paclitaxel ring that confirms paclitaxel loading in the niosome formulation. Furthermore, according to the niosomal curcumin FTIR spectrum (Fig. 1e), the out-of-plane bending peaks in the 800–600 cm−1 range can be utilized to allocate ortho substitutions on the curcumin ring that corroborates curcumin loading in the niosome formulation. When compared to the blank noisome, the sharper band in the 1600 cm−1 region and the broader bands in the 3500 cm−1 and 900–500 cm−1 regions in the CUR/PTX niosomal formulations (Fig. 1f, g) affirm curcumin and paclitaxel entrapment in the nano-niosomes.
Physical stability examination
To determine physical stability, the optimum formulation of curcumin/paclitaxel-loaded niosomes, in terms of encapsulation efficiency, vesicle size, PDI, and zeta potential, were tested by storing them at 4 °C. After storage for 60 days, the encapsulation efficiency, vesicle size, PDI, and zeta potential of the optimized formulation (F7) were not significantly changed from the freshly prepared samples (p value < 0.05). These results confirmed the stability of the F7 formula.
IC50s for individual curcumin and paclitaxel on MCF-7 and MCF-10A cells
To determine the inhibitory effect of individual curcumin and paclitaxel as a free form and as a niosomal form on MCF-7 and MCF-10A cells, we first performed dose–response experiments for curcumin and paclitaxel. As indicated in Fig. 5, individual treatments with the free form and the niosomal form resulted in growth inhibition of MCF-7 and MCF-10A cells in a dose-dependent pattern. Table 4 evaluates the IC50 values of these agents. The IC50 values of free PTX solution and free CUR solution was 13.54 and 44.60 μg mL−1, respectively, against MCF-7 cells and 30.75 and 76.71 μg mL−1, respectively, against MCF-10A cells (Fig. 5a, b). This revealed that MCF-10A cells needed at least a ∼ 2.27-fold higher concentration of PTX solution and a ∼ 1.7-fold higher concentration of CUR solution to attain IC50 compared to their counterpart MCF-7 cancer cells. As depicted in Fig. 5c, d, nano-niosomes were highly efficient in delivering the PTX and CUR drugs to both MCF-7 and MCF-10A cells. A threefold and 3.6-fold reduction in CUR and PTX concentration were measured, respectively, when the CUR and PTX were administered in nano-niosomes compared to free CUR and free PTX solutions in MCF-7 cells. Similarly, the CUR and PTX delivered in nano-niosomes to MCF-10A cells demonstrated a 1.2- and 1.5-fold lowered concentration, respectively. These results indicated that PTX and CUR in free and niosomal forms had less cytotoxicity on MCF-10A cells as a model for normal human mammary epithelial cells. The IC50 concentrations were then utilized to generate fixed ratios for subsequent combination experiments and for the calculation of combination index (CI).
Inhibition of cell growth by curcumin (CUR) and paclitaxel (PTX) individual as a drug free form and drug niosomal form in MCF-7 and MCF-10A cell. a Free CUR; b Free PTX; c Nio CUR; d Nio PTX for MCF-7 (filled square) and MCF-10A (filled triangle) cells
Table 4 The IC50 values of paclitaxel, curcumin alone and in combination on MCF-7 and MCF-10A cells, administered in the forms of free drug and drug niosomal form
Growth inhibitory effects of paclitaxel in combination with curcumin
To determine the synergistic antitumor effects of curcumin and paclitaxel, we performed a combination study, and the results are presented in Table 5. Figure 6a, b showed the dose–response curves for MCF-7 and MCF-10A cell lines exposed to paclitaxel and curcumin combination therapy. According to the results, curcumin could significantly increase the cell growth inhibition of paclitaxel; in the presence of free CUR solution, the IC50 of free PTX solution was diminished to ∼ 1.6-fold in MCF-7 cells and ∼ 1.4-fold in MCF-10A cells. This combination therapy regimen was significantly efficacious (p value < 0.05) when the PTX and CUR was delivered in nano-niosome formulations compared to a free solution (Table 4). Thus, the use of PTX and CUR together resulted in enhanced therapeutic potential. Figure 6 also illustrates the combination index analysis of the PTX and CUR interaction in MCF-7 and MCF-10A cells. Values of CI < 1 were obtained from the paclitaxel and curcumin combination in both free forms and niosomal forms for MCF-7 and MCF-10A cells, demonstrating that the two drugs interact synergistically to inhibit cell growth (Fig. 6c–f).
Table 5 Paclitaxel and curcumin combination index (CI) against MCF-7 and MCF-10A cells
Analysis of synergy between curcumin and paclitaxel for MCF-7 (filled triangle) and MCF-10A (filled square) cells. a Dose–response curve of free CUR + Free PTX; b dose–response curve of Nio CUR + Nio PTX. CI values at different levels of growth inhibition effect (fraction affected, FA; c Free CUR + Free PTX in MCF-7 cells; d Nio CUR + Nio PTX in MCF-7 cells; e Free CUR + Free PTX in MCF-10A cells, f Nio CUR + Nio PTX in MCF-10A cell
Nano-niosomal CUR/PTX cellular uptake experiments
Cellular uptake experiments were performed to evaluate the cellular uptake behavior of different CUR/PTX niosomal formulations in the following cells: MCF-7 cells as a cancer cell model and MCF10A cells as a model for normal human mammary epithelial cells. Figures 7, 8 and 9 illustrates the cellular uptake images of F6 and F7 CUR/PTX-loaded niosome formulations on MCF-7 and MCF10A cell lines monitored by fluorescence microscope. As depicted in Fig. 7b, d, the MCF-7 cells treated with the CUR/PTX F7 formula containing 10% DOTAP showed greater green and cyan (blue–green) color intensity compared to cells treated with CUR/PTX F6 formula (without DOTAP, Fig. 7a, c). By adding 10% DOTAP to the F6 formula, the drug release, vesicle size, and polydispersity index decreased, while the transfection efficiency was enhanced. Similarly, these results are observed in MCF-10A cells (Fig. 9a–d); however, the intensity of the green and cyan color in these cells was much less than in the MCF-7 cells. These findings indicate that CUR/PTX-loaded niosome formulations entered healthy cells much less than cancerous cells. These results are consistent with cytotoxicity experiments.
Cellular uptake of F6 and F7 CUR/PTX loaded niosomes formulations on MCF-7cell line. MCF-7cell line [a1 F6 Nio CUR Nucleus, a2 F6 Nio CUR, a3 F6 Nio CUR merged; b1 F7 Nio CUR Nucleus, b2 F7 Nio CUR, b3 F& Nio CUR merged; c1 F6 PTX CUR Nucleus, c2 F6 Nio PTX, c3 F6 Nio CUR PTX; d1 F7 Nio PTX Nucleus, d2 F7 Nio PTX, d3 F& Nio PTX merged]
Cellular uptake of F6 and F7 CUR/PTX loaded niosomes formulations on MCF-7cell line. MCF-10A cell line [a1 F6 Nio CUR Nucleus, a2 F6 Nio CUR, a3 F6 Nio CUR merged; b1 F7 Nio CUR Nucleus, b2 F7 Nio CUR, b3 F& Nio CUR merged; c1 F6 PTX CUR Nucleus, c2 F6 Nio PTX, c3 F6 Nio CUR PTX; d1 F7 Nio PTX Nucleus, d2 F7 Nio PTX, d3 F& Nio PTX merged]
Apoptosis analysis
Apoptosis was measured by annexin V-fluorescein isothiocyanate (FITC)/propidium iodide (PI) double staining (Sigma-Aldrich, USA). MCF-7 cells were seeded in six-well plates at a density of 1 × 105 cells per well. Apoptosis was induced by treating the cells with PTX and CUR, either as single agents or as a PTX + CUR combination, administered in aqueous solution or in nano-niosome formulations at an IC50 concentration for each drug. After 24 h of incubation, the cells were detached using 0.25% trypsin/EDTA (Sigma-Aldrich, USA) and centrifuged at 1500 rpm for 3 min, after which the pellet was resuspended in ice-cold, phosphate-buffered saline (PBS, pH 7.4). Annexin V-FITC solution (3 µL) was added to each cell suspension. In addition, 3 µL of propidium iodide stock solution was added to the cells to identify necrotic cells. After 30 min of incubation on ice, the stained cells were analyzed by flow cytometry using the BD FACSCalibur instrument. Cells that did not receive any drug treatment served as the control.
Plants have been employed as medicines for centuries, and the usage of plant-derived chemicals has been extended into anticancer drugs. Lately, chemotherapeutic strategies have advanced to the utilization of combined active compounds because they are believed to be more active than a single agent. Hence, treatment effectiveness could increase, and the toxic side effects may be reduced, due to the extremely low use of drugs. Curcumin (diferuloylmethane), a yellow pigment isolated from the rhizome of turmeric, has been reported to have an extensive spectrum of pharmacological activities. Furthermore, curcumin is currently involved in the early phase of a clinical trial as a potential chemo-preventive agent [22, 23]. Therefore, it is logical to evaluate whether curcumin, as a new antiproliferative agent, can sensitize tumors to the chemotherapeutic drug paclitaxel for breast cancer cells. Paclitaxel (PTX) has been used as an effective chemotherapeutic drug for a wide range of tumors, such as breast, lung, prostate, ovarian, and pancreatic cancers [24, 25]. The CUR and PTX combination is a remarkable anticancer drug therapy. PTX is a powerful microtubule-stabilizing agent that commences cell cycle arrest, while CUR attacks biologically by regulating several signal transduction pathways [26,27,28]. Despite these good therapeutic effects, the wide therapeutic range of PTX and CUR is limited due to poor aqueous solubility and low therapeutic index. A promising approach for circumventing these issues is the use of a vesicular nanocarrier, such as niosomes, which are an alternative to phospholipid vesicles for the encapsulation of hydrophobic drugs due to providing high encapsulation efficiency, biocompatibility, biodegradation, low preparation cost, and sufficient stability, as well as being free from organic solvents and offering easy storage [7]. In this study, we have developed a novel cationic PEGylated niosomal formulation for encapsulating paclitaxel and curcumin. The vesicular systems were prepared from the nonionic surfactant Tween-60, as a commercial surfactant, and all formulations were compared in terms of entrapment efficiency, drug release, vesicle size, and polydispersity index. Furthermore, niosomes formulated without cholesterol formed a gel, and only the addition of cholesterol was a homogenous niosome obtained [29]. The hydrophilic–lipophilic balance (HLB) of the nonionic surfactant, the chemical structure of the components, and the critical packing parameter (CPP) are important in forming bilayer vesicles instead of micelles. The HLB value of a surfactant plays a key role in controlling the drug entrapment efficiency of the vesicle it forms. A surfactant, such as Tween-60, with an HLB value in the 14–17 range is inappropriate for creating niosomes. For HLB > 6, cholesterol must be added to the surfactant until forming a bilayer vesicle. Also, the presence of cholesterol in the formulation of niosomes is necessary for the physical stability of these nano-sized vesicles (i.e., suppressing the surfactant's tendency to form aggregates, decreasing drug leakage, vesicle size, and dispersion). This was primarily ascribed to the increase in hydrophobicity (particularly with higher HLB surfactant molecules, such as Tween-60) that augmented the structural affinity of the bilayer membrane for CUR/PTX molecules [6, 11, 12, 30,31,32]. Therefore, cholesterol is added to the formulations as a membrane-stabilizing factor. As a result, by increasing the amount of cholesterol content from 10 to 30%, PTX/CUR entrapments in nano-niosomes were increased, while the percentage of CUR/PTX release was decreased. Furthermore, the mean diameter of niosomes increased with increasing the cholesterol content. However, the addition of cholesterol content to niosomes up to 50% decreased niosomal efficiency in trapping curcumin/paclitaxel compared to 30% cholesterol content. This finding can be explained by the possible competition between curcumin and paclitaxel as lipophilic drugs and the cholesterol incorporation into the niosomes. A further increase in cholesterol tends to deposit between the bilayers, excluding the drug from the niosomal bilayers. Above a certain level of cholesterol, entrapment efficiency decreased possibly due to a decrease in CPP [6, 11, 12, 30,31,32]. Improving stability, increasing the drug encapsulation, decreasing mean size diameter, and reducing drug release is due to the presence of PEGylation in the niosomal formulations [33, 34]. Therefore, 5% PEG was added to the F4 formula. According to the findings, the F6 niosomal formula demonstrated higher drug entrapment, smaller diameter, smaller PDI, and lower drug release than the F4 formula. Additionally, cationic lipids added to the niosomal formulations enhanced the niosomes' physicochemical properties and the transfection efficiency. The addition of DOTAP decreased the drug's release and vesicle size due to a decline in the cholesterol content. This effect also decreased the polydispersity index, which is relevant to the further reciprocal repel force between the particles with the same sign charge in the suspension system [35,36,37,38]. To hamper the aggregation of vesicular systems, it is essential to introduce a charge on the surface of the vesicle. A good indicator for the size of this barrier is zeta potential. If all the particles possess large enough zeta potential, they presumably repel each other strongly enough that they will not have the tendency to aggregate [39]. After storage for 60 days, the presence of DOTAP and PEG in niosomal formulations of CUR and PTX demonstrated no significant changes when compared to freshly prepared samples in terms of encapsulation efficiency, vesicle size, PDI, and zeta potential of the optimized formulation (F7). This implies that the new F7 niosome formulation could minimize problems associated with niosome instability, including aggregation, fusion, and drug leakage. The rate of drug release from a delivery system is a crucial factor and must be appraised to attain an optimal system with the desired drug-release profile. The in vitro release study was conducted to predict how a delivery system may function under the ideal status, which might display some indication of its in vivo efficiency. In-vitro drug release demonstrated that the cumulative release profile of CUR and PTX were apparently biphasic, with an initial rapid release period followed by a slower release phase. Because CUR and PTX are small molecules and the permeability cut-off of the dialysis bag was 12 kDa, the released CUR and PTX poured easily from the bag. As a result, neither the dialysis bag nor the drug size restricted the drug's release. The initial fast rate of release was regulated by the diffuse mechanism (concentration gradient of CUR/PTX between noisome and buffer), while the later slow release resulted from the drug's sustained release from the inner layer [40,41,42].
The in vitro release of CUR and PTX from the niosomal formulation was assessed by fitting the cumulative drug release into mathematical release models, which are commonly applied to elucidate release kinetics and to compare release profiles. The CUR/PTX niosomal formulations followed the Higuchi model. These findings indicated that CUR and PTX molecules were diffused in the niosome matrix and that there were no possible interactions between the niosome components and the drugs [5, 43,44,45]. In this study, we have investigated the effect of PTX and CUR combination therapy, in both free forms and niosomal forms, on MCF-7 cells as a cancer cell model and MCF10A cells as a model for normal human mammary epithelial cells (Tables 4 and 5). The ratiometric combination of PTX and CUR significantly suppressed the growth of MCF-7 cells. When the free drugs were administered in nano-niosome formulations, the cytotoxicity effects manifested even more. The enhanced therapeutic activity achieved with the combination therapy was ascribed to the P-glycoprotein (P-gp) downregulation and to the inhibition of the NFκB pathway by CUR. Most importantly, CUR downregulates the NF-ĸB signaling pathways, thus inhibiting cancer cell growth and inducing apoptosis. Therefore, CUR sensitizes cancer cells to increase the cancer cells' response to anticancer drugs. Increasing the accumulation of PTX within the cancer cell due to P-gp downregulation can overcome the MDR phenomenon [1, 27, 46]. We observed a similar trend for MCF-10a cells. Nevertheless, as expected, CUR and PTX had fewer side effects in both free form and niosomal form on MCF10A human mammary epithelial cells. The cellular uptake experiments were demonstrated by the addition of DOTAP, which enhanced the transfection efficiency of the CUR/PTX F7 formula; it is well known that cationic lipids enhance the transfection efficiency of niosomal formulations [35,36,37,38]. Quantitative apoptotic activity measurements were made by flow cytometry analysis in PTX and CUR treated cells. Statistically significant when apoptotic activity of paclitaxel NanoNiosome formulation is compared with free paclitaxel and curcumin NanoNiosome formulation is compared with free curcumin solution in MCF-7 cells (p < 0.05). In addition to these findings, flow cytometry analysis also revealed that the apoptosis was significantly greater with the combination therapy and with drugs administered NanoNiosome formulations at p < 0.05. These results collaborate with the cell viability experiment to affirm that NanoNiosomes were effective in delivering the PTX and CUR to the cells, and combination therapy with PTX and CUR delivered in NanoNiosome formulations indeed demonstrated higher therapeutic efficacy in MCF-7 cells.
Our successful findings suggest novel cationic PEGylated niosomal formulations for paclitaxel and curcumin co-administration. The encapsulation efficiency of both drugs was extremely successful. The drugs' release profile demonstrated burst release followed by a sustained drug release for both agents. The combination of PTX (a powerful anticancer drug) with CUR (an effective chemosensitizer), particularly in nano-niosome formulations, can improve the therapeutic effectiveness of cancer treatments. Our experimental evidence indicated that a nanocarrier-based approach adopted for the delivery of CUR/PTX combinations was efficient in battling cancer cells in vitro.
CUR/PTX niosomes preparation
We used the thin-film hydration method to prepare the curcumin and paclitaxel-loaded niosomes [47]. Tween-60 (DaeJung Chemicals & Metals, South Korea) and cholesterol (Sigma-Aldrich, USA) were dissolved in chloroform to obtain the different molar ratio molarities (as illustrated in Table 1). PTX (Stragen, Switzerland) and CUR (Sigma-Aldrich, USA) were dissolved in chloroform and added to the mixture of surfactant and lipids. Fluorescent label Dil (Sigma, USA) was added to the lipid phase at 0.1% mol for lipid staining to evaluate cellular uptake. Niosomal formulations were screened for particle size, controlled release, and high entrapment efficiency parameters. After attaining optimized synthetic conditions, the cationic lipid DOTAP (1,2-dioleoyl-3-trimethylammonium-propane, Sigma-Aldrich, USA) and polyethylene glycol (Lipoid PE 18:0/18:0–PEG2000, DSPE-mPEG 2000, Lipoid GmbH, Germany) were added for improving stability and transfection efficiency of the niosomal formulations. Organic solvent was removed by rotary evaporator (Heidolph, Germany) at 50 °C until a thin-layered film formed. The dry lipid films were hydrated by adding phosphate-buffered saline (PBS, pH = 7.4) at 60 °C for 60 min to obtain the niosomal suspensions. After hydration, the prepared vesicles were sonicated for 30 min using a microtip probe sonicator (model UP200St, Hielscher Ultrasonics GmbH, Germany) to reduce the vesicles' mean size. Thereafter, free drugs (unloaded) were separated from niosomal vesicles using a dialysis bag diffusion technique against PBS for 1 h at 4 °C (MW = 12 kDa, Sigma-Aldrich, USA) [48]. Drug-free niosomes were produced in a similar manner without adding curcumin and paclitaxel. The dose of both drugs was 0.5 mg mL−1 for all formulations.
Analysis of encapsulation efficiency
To evaluate entrapment efficiency, spectroscopic measurements were performed. The amounts of niosomal encapsulated CUR and PTX were analyzed with a UV spectro-photometer (model T80+, PG Instruments, United Kingdom) at 429 and 236 nm (ʎmax), respectively [7]. The encapsulation efficiency was determined as follows:
$${\text{Encapsulation efficiency }}\left( \% \right) \, = \frac{{{\text{The amount of CUR}}/{\text{PTX encapsulated within niosomes}}}}{{{\text{Total amount of CUR}}/{\text{PTX added}}}} \times 100$$
The particle size distribution, zeta potential and Poly-Dispersity Index (PDI) of the obtained niosomes were measured by dynamic light scattering technique using a ZetaPALS zeta potential and particle size analyzer (Brookhaven Instruments, Holtsville, NY, USA). Scattered light was detected at room temperature at an angle of 90°, and the diluted samples in 1700 µL of deionized water (0.1 mg mL−1) were prepared and immediately measured after preparation. All measurements were carried out three times, and their mean values were calculated. The internal structure of NanoNiosome formulations was determined by cryogenic transmission electron microscopy (FEI Tecnai 20, type Sphera, Oregon, USA) equipped with a LaB6 filament at 200 kV. A drop of NanoNiosome solution was placed over a 200-mesh Copper-coated TEM grids, and TEM measurement was accomplished. Characterization of surface morphology of Niosomes was evaluated using scanning electron microscope (SEM. To prepare the sample used in SEM, a little amount of the NanoNiosome solution dispersed in water was placed on the mesh copper grid 400. Then, the copper grid was placed in an evacuated desiccator to evaporate the solvent. Finally the samples were coated with gold coater to make them conductive, followed by evaluation of the surface morphology using SEM with 100 W power instrument (model KYKY-EM3200-30 kV, China).
The in vitro release of CUR/PTX from niosomes was monitored using a dialysis bag (MW = 12 kDa) against PBS (containing 2% Tween-20 to imitate a physiological environment) for 72 h at 37 °C and 7.4 pH [42]. First, the CUR/PTX niosome samples were suspended in a dialysis tube, and the release of both drugs was evaluated in 10 mL of PBS with continuous stirring. Then, 2 mL of the sample was collected from the incubation medium at specific time intervals and immediately substituted with an equal volume of fresh PBS. The amount of CUR/PTX released was determined using a UV–Vis spectrometer at 429 and 236 nm, respectively.
Mathematical modeling of drug release kinetic
Cumulative percentages of the drug released from the niosomes were calculated by the following Eq. (1):
$${\text{Release}} = \frac{{{\text{M}}_{\text{t}} }}{{{\text{M}}_{\text{f}} }}$$
where Mt and Mf are the cumulative amounts of drug released at any time (t) and the final amounts of drug released, respectively.
To determine the release kinetic, the release data were fitted to the mathematical models by the linear regression analysis of Graph pad prism 6.0, as follows:
Zero-order rate equation:
$$Q_{t} = Q_{0} + K_{0} t$$
where Qt is the amount of the remaining drug in the formulation at time t; Q0 is the initial amount of drug in the formulation; and K0 is the zero-order release constant.
First-order rate equation:
$$\log {\text{C}} = \log {\text{C}}_{0 } - \frac{{{\text{K}}_{\text{t}} }}{2.303}$$
where C0 is the initial drug concentration; K is the first-order release constant; and t is time.
Higuchi's model:
$${\text{Q}} = {\text{ K}}_{\text{H}} {\text{t}}^{ 1/ 2}$$
where Q is the amount of drug released in time t per unit area, and KH is the Higuchi dissolution constant.
Hixson–Crowell model:
$${\text{Q}}_{0}^{ 1/ 3} - {\text{ Q}}_{\text{t}}^{ 1/ 3} = {\text{ K}}_{\text{s}} {\text{t}}$$
Q0 is the initial amount of the drug in the niosomes; Qt is the cumulative amount of the drug released at time t; and Ks is the Hixson–Crowell release constant.
Finally, the correlation coefficients' values were compared to determine the release model that best fits the data [40, 42].
The samples' functional group characterizations were investigated using FTIR spectrometer (Model 8300, Shimadzu Corporation, Tokyo, Japan) for pure CUR, pure PTX, blank noisome, niosomal-CUR, and niosomal-PTX. For preparation, the samples were lyophilized as a dry powder and mixed with potassium bromide (KBr). Then, the samples were placed in a hydraulic press to form the pellets. The FTIR spectrum was scanned in the wavelength range of 400–4000 cm−1.
To determine the physical stability of niosomal curcumin/paclitaxel during storage, the change in particle size, zeta potential, PDI, and the remaining amount of the drug in vesicle was assessed over 14-, 28-, and 60-day intervals [9, 39].
Cell lines and culture conditions
Human breast cancer MCF-7 cells (the Iranian Biological Resource Center, Tehran, Iran) were cultured in DMEM/F12 Ham's mixture (InoClon, Iran) supplemented with 2 mM GlutaMAX™-I (100X, Gibco, USA), 10% FBS (Fetal Bovine Serum, Gibco, USA), and 1 mg mL−1 penicillin/streptomycin (Gibco, USA). Non-tumorigenic human breast epithelial cell line MCF-10A (the Iranian Biological Resource Center, Tehran, Iran) was grown in DMEM/F12 Ham's mixture supplemented with 2 mM GlutaMAX™-I, 5% horse serum (Gibco, USA), EGF (Epithelial growth factor, Sigma, USA) 20 ng mL−1, insulin 10 μg mL−1 (Sigma, USA), hydrocortisone 0.5 μg mL−1 (Sigma, USA), 100 ng mL−1 cholera toxin (Sigma, USA), and 1 mg mL−1 penicillin/streptomycin. An MCF-10A cell line was used for comparison in all experiments.
The cytotoxicity of various formulations was determined by MTT (Sigma, USA) assay [49,50,51]. Briefly, MCF-7 and MCF-10A cells were seeded in 96-well plates at 10,000 cells per well. Following attachment for 24 h, the cells were treated with 200 μL fresh medium containing serial dilutions of the different drug/niosome formulations: free-PTX solution, free-CUR solution, free PTX + free CUR physical mixture, niosomal CUR, niosomal PCT, and the co-administration of niosomal CUR-niosomal PTX. After incubation for 48 h, 20 μL MTT (5 mg mL−1 in PBS) was added into each 96-well plate and incubated for 3 h at 37 °C. Finally, the medium was carefully removed, and 180 μL of DMSO was added to each well to dissolve the formazan crystals formed. Absorbance of each well was recorded by EPOCH Microplate Spectrophotometer (synergy HTX, BioTek, USA) at 570 nm. The cytotoxicity of the different formulations was expressed as the Inhibitory Concentration (IC50) value, defined as the drug concentration required for inhibiting cell growth by 50% relative to the control. The IC50 values of PTX and CUR as single drugs or in combination were calculated using GraphPad Prism 6. The curcumin and paclitaxel combination was appraised by calculating the CI value using the CompuSyn software, with the method utilized by Chou and Talalay:
$${\text{CI }} = \frac{a}{A} + \frac{b}{B}$$
where a is the PTX IC50 in combination with CUR at concentration b; A is the PTX IC50 without CUR; and B is the CUR IC50 in the absence of PTX. According to the Chou and Talalay equation, when CI < 1, the interaction between the two drugs is synergistic; when CI = 1, the interaction between the two drugs is additive; and when CI > 1, the two drugs are antagonistic [52,53,54].
MCF-7 and MCF-10A cells were seeded at a density of 2 × 105 cells per well in a 6-well plate and incubated for 24 h to allow them to attach. The cells were then treated with the different NioCUR and NioPTX formulations. After 3 h of incubation, the cells were washed three times with cold PBS and fixed with a 4% paraformaldehyde solution (Sigma, USA). Then, the cells were stained with DAPI (0.125 µg mL−1, Thermo Fisher Scientific, USA) and imaged with a fluorescence microscope (BX61, Olympus, Japan) [48, 49, 51].
An annexin V-FITC/PI double staining assay was carried out to confirm whether apoptosis was induced by curcumin or paclitaxel alone or in combination when administered in an aqueous solution and nano-niosome formulation. The results in Fig. 9 show quantitative apoptotic activity in MCF-7 cells via apoptosis assay using flow cytometry following the treatment of cells for 24 h. In apoptotic cells, the membrane phospholipid phosphatidylserine (PS) is translocated from the inner to the outer surface of the plasma membrane, thereby exposing PS to the external cellular environment. Annexin V is a 35–36 kDa Ca2+-dependent phospholipid-binding protein with high affinity for PS, and it binds to exposed apoptotic cell-surface PS. Annexin V can be conjugated to fluorochromes, such as FITC, while retaining its high affinity for PS, thus serving as a sensitive probe for the flow cytometric analysis of cells undergoing apoptosis. Furthermore, propidium iodide (PI) is a fluorescent intercalating agent that can be used as a DNA stain in flow cytometry. PI cannot pass the membrane of live cells and apoptotic cells; however, it stains dead cells, making it useful to differentiate necrotic, apoptotic, healthy, and dead cells. In the scatter plot of double variable flow cytometry, the Q4 quadrant (FITC−/PI−) shows living cells; the Q2 quadrant (FITC+/PI+) stands for late apoptotic cells; the Q3 quadrant (FITC+/PI−) represents early apoptotic cells; and the Q1 quadrant (FITC−/PI+) shows necrotic cells. The flow cytometry plots demonstrate there was enhancement in cellular apoptosis in MCF-7 cells when PTX and CUR were administered in nano-niosome formulations as compared to free drugs (p < 0.05). Furthermore, when PTX and CUR were co-administered in nano-niosome formulations, there was a significant increase in apoptosis (i.e., 15.27% early apoptosis in niosomal curcumin and 31.03% early apoptosis in niosomal paclitaxel versus 49.79% early apoptosis in niosomal curcumin + niosomal paclitaxel, p < 0.05). These results are consistent with the growth inhibitory effects of paclitaxel in combination with curcumin.
Apoptosis assay using flow cytometry following the treatment of cells for 24 h. a Control; b free curcumin + free paclitaxel; c free curcumin; d free paclitaxel; e niosomal curcumin; f niosomal paclitaxel; g niosomal curcumin + niosomal paclitaxel
Statistical data analyses were performed via GraphPad Prism 6 software and expressed as mean ± SD. A Student t test was used when comparing two independent groups, and an ANOVA test was used when comparing multiple samples. A p value < 0.05 was considered significant.
PTX:
CUR:
DDS:
drug delivery system
PEG:
EE:
entrapment efficiency
Cryo-TEM:
cryogenic transmission electron microscopy
FTIR:
Fourier transforms infrared
CI:
combination index
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All authors had equal role in design, work, statistical analysis and manuscript writing. All authors read and approved the final manuscript.
All data generated or analyzed during this study are included in this article.
The manuscript was approved by the Shahid Sadoughi University of Medical Sciences Internal Review Board. There are no human subjects or animals involved in the study.
This study was financially supported by grant from the Shahid Sadoughi University of Medical Sciences, Yazd, Iran.
Department of Clinical Biochemistry, Faculty of Medicine, Shahid Sadoughi University of Medical Sciences, Yazd, Iran
Ashraf Alemi
& Javad Zavar Reza
Biotechnology Research Center, International Campus, Shahid Sadoughi University of Medical Science, Yazd, Iran
Javad Zavar Reza
& Mojtaba Haghi Karamallah
Department of Life Science Engineering, Faculty of New Sciences & Technologies, University of Tehran, Tehran, Iran
Fateme Haghiralsadat
Protein Engineering Laboratory, Department of Medical Genetics, School of Medicine, Shahid Sadoughi University of Medical Sciences, Yazd, Iran
Hossein Zarei Jaliani
Nutrition and Metabolic Diseases Research Center, Ahvaz Jundishapur University of Medical Sciences, Ahvaz, Iran
Seyed Ahmad Hosseini
Student Research Committee, Shiraz University of Medical Sciences, Shiraz, Iran
Somayeh Haghi Karamallah
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Correspondence to Javad Zavar Reza.
Alemi, A., Zavar Reza, J., Haghiralsadat, F. et al. Paclitaxel and curcumin coadministration in novel cationic PEGylated niosomal formulations exhibit enhanced synergistic antitumor efficacy. J Nanobiotechnol 16, 28 (2018) doi:10.1186/s12951-018-0351-4
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Jon Michael Dunn
Indiana University, Bloomington
Intuitive Semantics for First-Degree Entailments and 'Coupled Trees'.J. Michael Dunn - 1976 - Philosophical Studies 29 (3):149-168.details
Logical Consequence and Entailment in Logic and Philosophy of Logic
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Entailment: The Logic of Relevance and Necessity, Vol. II.Alan Ross Anderson, Nuel D. Belnap & J. Michael Dunn - 1992 - Princeton University Press.details
Relevance Logic in Logic and Philosophy of Logic
Truth, Misc in Philosophy of Language
On the Ternary Relation and Conditionality.Jc Beall, Ross T. Brady, J. Michael Dunn, A. P. Hazen, Edwin D. Mares, Robert K. Meyer, Graham Priest, Greg Restall, David Ripley, John Slaney & Richard Sylvan - 2012 - Journal of Philosophical Logic 41 (3):595 - 612.details
One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)
Algebraic Methods in Philosophical Logic.J. Michael Dunn - 2001 - Oxford University Press.details
This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.
Nonclassical Logics in Logic and Philosophy of Logic
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Partiality and its Dual.J. Michael Dunn - 2000 - Studia Logica 66 (1):5-40.details
This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate (...) twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)
Dialetheism in Logic and Philosophy of Logic
Many-Valued Logic in Philosophy of Language
Curry's Paradox.Robert K. Meyer, Richard Routley & J. Michael Dunn - 1979 - Analysis 39 (3):124 - 128.details
Liar Paradox in Logic and Philosophy of Logic
Relevant Predication 2: Intrinsic Properties and Internal Relations.J. Michael Dunn - 1990 - Philosophical Studies 60 (3):177-206.details
Intrinsic and Extrinsic Properties in Metaphysics
Relations in Metaphysics
Star and Perp: Two Treatments of Negation.J. Michael Dunn - 1993 - Philosophical Perspectives 7:331-357.details
Logical Connectives in Logic and Philosophy of Logic
The Substitution Interpretation of the Quantifiers.J. Michael Dunn & Nuel D. Belnap - 1968 - Noûs 2 (2):177-185.details
Substitutional Quantification in Philosophy of Language
Algebraic Completeness Results for R-Mingle and its Extensions.J. Michael Dunn - 1970 - Journal of Symbolic Logic 35 (1):1-13.details
Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi.Katalin Bimbo & J. Michael Dunn - 2008 - Center for the Study of Language and Inf.details
Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.
Substructural Logic in Logic and Philosophy of Logic
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Positive Modal Logic.J. Michael Dunn - 1995 - Studia Logica 55 (2):301 - 317.details
We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.
Modal Logic in Logic and Philosophy of Logic
The Trilaticce of Constructive Truth Values.Yaroslav Shramko, J. Michael Dunn & Tatsutoshi Takenaka - 2001 - Journal of Logic and Computation 11 (1):761--788.details
Canonical Extensions and Relational Completeness of Some Substructural Logics.J. Michael Dunn, Mai Gehrke & Alessandra Palmigiano - 2005 - Journal of Symbolic Logic 70 (3):713 - 740.details
In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.
Relevant Predication 1: The Formal Theory. [REVIEW]J. Michael Dunn - 1987 - Journal of Philosophical Logic 16 (4):347-381.details
Kripke Models for Linear Logic.Gerard Allwein & J. Michael Dunn - 1993 - Journal of Symbolic Logic 58 (2):514-545.details
We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, (...) similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic. (shrink)
Model Theory in Logic and Philosophy of Logic
Negation in the Context of Gaggle Theory.J. Michael Dunn & Chunlai Zhou - 2005 - Studia Logica 80 (2-3):235-264.details
We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as "unnecessity" and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a (...) characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn 's kite and also a dual version in [12]. He also suggested combining them into a "united" kite. We give a united semantics for this united kite of negations. (shrink)
Completeness of Relevant Quantification Theories.Robert K. Meyer, J. Michael Dunn & Hugues Leblanc - 1974 - Notre Dame Journal of Formal Logic 15 (1):97-121.details
A Modification of Parry's Analytic Implication.J. Michael Dunn - 1972 - Notre Dame Journal of Formal Logic 13 (2):195-205.details
A Kripke-Style Semantics for R-Mingle Using a Binary Accessibility Relation.J. Michael Dunn - 1976 - Studia Logica 35 (2):163 - 172.details
On the Decidability of Implicational Ticket Entailment.Katalin Bimbó & J. Michael Dunn - 2013 - Journal of Symbolic Logic 78 (1):214-236.details
The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$. (shrink)
Proof Theory in Logic and Philosophy of Logic
E, R, and Γ.Robert K. Meyer & J. Michael Dunn - 1969 - Journal of Symbolic Logic 34 (3):460-474.details
Algebraic Completeness Results for Dummett's LC and Its Extensions.J. Michael Dunn & Robert K. Meyer - 1971 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 17 (1):225-230.details
New Consecution Calculi for $R^{T}_{\To}$.Katalin Bimbó & J. Michael Dunn - 2012 - Notre Dame Journal of Formal Logic 53 (4):491-509.details
The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)
Contradictory Information: Too Much of a Good Thing. [REVIEW]J. Michael Dunn - 2010 - Journal of Philosophical Logic 39 (4):425 - 452.details
Both I and Belnap, motivated the "Belnap-Dunn 4-valued Logic" by talk of the reasoner being simply "told true" (T) and simply "told false" (F), which leaves the options of being neither "told true" nor "told false" (N), and being both "told true" and "told false" (B). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this on a daily basis with the Web. But (...) the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Jøsang's Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty— that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Jøsang's "Opinion Triangle" becomes an "Opinion Tetrahedron" and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical "probability" to non-classical logic. (shrink)
E, R, and $Gamma$.Robert K. Meyer & J. Michael Dunn - 1969 - Journal of Symbolic Logic 34 (3):460-474.details
Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic
Truth-Value Semantics.J. Michael Dunn - 1978 - Journal of Symbolic Logic 43 (2):376-377.details
A Truth Value Semantics for Modal Logic.J. Michael Dunn - 1973 - In Hugues Leblanc (ed.), Journal of Symbolic Logic. Amsterdam: North-Holland. pp. 87--100.details
Modal and Intensional Logic in Logic and Philosophy of Logic
Contradictory Information: Better Than Nothing? The Paradox of the Two Firefighters.J. Michael Dunn & Nicholas M. Kiefer - 2019 - In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency. Springer Verlag. pp. 231-247.details
Prominent philosophers have argued that contradictions contain either too much or too little information to be useful. We dispute this with what we call the "Paradox of the Two Firefighters." Suppose you are awakened in your hotel room by a fire alarm. You open the door. You see three possible ways out: left, right, straight ahead. You see two firefighters. One says there is exactly one safe route and it is to your left. The other says there is exactly one (...) safe route and it is to your right. While the two firemen are giving you contradictory information, they are also both giving you the perhaps useful information that there is a safe way out and it is not straight ahead. We give two analyses. The first uses the "Opinion Tetrahedron," introduced by Dunn as a generalization of Audun Jøsang's "Opinion Triangle." The Opinion Tetrahedron in effect embeds the values of the "Belnap-Dunn 4-valued Logic" into a context of subjective probability generalized to allow for degrees of belief, disbelief, and two kinds of uncertainty—that in which the reasoner has too little information and that in which the reasoner has too much information. Jøsang had only a single value for uncertainty. We also present an alternative solution, again based on subjective probability but of a more standard type. This solution builds upon "linear opinion pooling." Kiefer had already developed apparatus for assessing risk using expert opinion, and this influences the second solution. Finally, we discuss how these solutions might apply to "Big Data" and the World Wide Web. (shrink)
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Quantum Logic as Motivated by Quantum Computing.J. Michael Dunn, Tobias J. Hagge, Lawrence S. Moss & Zhenghan Wang - 2005 - Journal of Symbolic Logic 70 (2):353 - 359.details
Generalized Onrno Negation.J. Michael Dunn - 1996 - In H. Wansing (ed.), Negation: A Notion in Focus. W. De Gruyter. pp. 7--3.details
Symmetric Generalized Galois Logics.Katalin Bimbó & J. Michael Dunn - 2009 - Logica Universalis 3 (1):125-152.details
Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the weak (...) distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them. (shrink)
A Theorem in 3-Valued Model Theory with Connections to Number Theory, Type Theory, and Relevant Logic.J. Michael Dunn - 1979 - Studia Logica 38 (2):149 - 169.details
Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...) number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)
Number Theory in Philosophy of Mathematics
Dualling: A Critique of an Argument of Popper and Miller.J. Michael Dunn & Geoffrey Hellman - 1986 - British Journal for the Philosophy of Science 37 (2):220-223.details
Popper: Induction in 20th Century Philosophy
Relevant Predication 3: Essential Properties.J. Michael Dunn - 1990 - In J. Dunn & A. Gupta (eds.), Truth or Consequences. Kluwer Academic Publishers. pp. 77--95.details
Essentialism in Metaphysics
Relevance Logics and Relation Algebras.Katalin Bimbó, J. Michael Dunn & Roger D. Maddux - 2009 - Review of Symbolic Logic 2 (1):102-131.details
Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)
Conditional Assertion and Restricted Quantification: Abstracts of Comments.J. Michael Dunn - 1970 - Noûs 4 (1):13.details
Conditional Assertion in Philosophy of Language
Quantifiers in Philosophy of Language
Truth or Consequences Essays in Honor of Nuel Belnap.L. R. S., J. M. Dunn & A. Gupta - 1990 - Wiley-Blackwell.details
Nuel Belnap: Doctoral Students.Carlos Giannoni, Robert Meyer, J. Michael Dunn, Peter Woodruff, James Garson, Kent Wilson, Dorothy Grover, Ruth Manor, Alasdair Urquhart & Garrel Pottinger - 1990 - In J. Dunn & A. Gupta (eds.), Truth or Consequences. Kluwer Academic Publishers.details
A Relational Representation of Quasi-Boolean Algebras.J. Michael Dunn - 1982 - Notre Dame Journal of Formal Logic 23 (4):353-357.details
Quantification and RM.J. Michael Dunn - 1976 - Studia Logica 35 (3):315 - 322.details
Four-Valued Logic.Katalin Bimbó & J. Michael Dunn - 2001 - Notre Dame Journal of Formal Logic 42 (3):171-192.details
Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...) and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)
Relevant Robinson's Arithmetic.J. Michael Dunn - 1979 - Studia Logica 38 (4):407 - 418.details
In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in (...) relevant arithmetic. (shrink)
A Guide to the Floridi Keys: Luciano Floridi: The Philosophy of Information. Oxford: Oxford University Press, 2011, Xx+405pp, £37.50 HB.J. Michael Dunn - 2013 - Metascience 22 (1):93-98.details
Philosophy of Information in Philosophy of Computing and Information
Philosophy of Information, Misc in Philosophy of Computing and Information
A Consecutive Calculus for Positive Relevant Implication with Necessity.Nuel D. Belnap, Anil Gupta & J. Michael Dunn - 1980 - Journal of Philosophical Logic 9 (4):343-362.details
E, R and Γ.Robert K. Meyer & J. Michael Dunn - 1971 - Journal of Symbolic Logic 36 (3):521-522.details
A Sieve for Entailments.J. Michael Dunn - 1980 - Journal of Philosophical Logic 9 (1):41 - 57.details
The validity of an entailment has nothing to do with whether or not the components are true, false, necessary, or impossible; it has to do solely with whether or not there is a necessary connection between antecedent and consequent. Hence it is a mistake (we feel) to try to build a sieve which will "strain out" entailments from the set of material or strict "implications" present in some system of truth-functions, or of truth-functions with modality. Anderson and Belnap (1962, p. (...) 47). (shrink)
Axiomatizing Belnap's Conditional Assertion.J. Michael Dunn - 1975 - Journal of Philosophical Logic 4 (4):383 - 397.details
The Impossibility of Certain Higher-Order Non-Classical Logics with Extensionality.J. Michael Dunn - 1988 - In D. F. Austin (ed.), Philosophical Analysis. Kluwer Academic Publishers. pp. 261--279.details
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Editors' Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing. [REVIEW]J. Michael Dunn, Lawrence S. Moss & Zhenghan Wang - 2013 - Journal of Philosophical Logic 42 (3):443-459.details
Quantum Computation in Philosophy of Computing and Information | CommonCrawl |
Journal of Modern Dynamics
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2017, Volume 11: 219-248. Doi: 10.3934/jmd.2017010
This volume Previous Article Effective equidistribution of circles in the limit sets of Kleinian groups Next Article Most interval exchanges have no roots
Minimality of interval exchange transformations with restrictions
Ivan Dynnikov and
Alexandra Skripchenko
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Str., Moscow 119991, Russia
Author Bio: Ivan Dynnikov <[email protected]>; Alexandra Skripchenko <[email protected]>
Revised: December 11, 2016
The work is supported by the Russian Science Foundation under grant 14-50-00005 and performed at Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.
Abstract Full Text(HTML) Figure(4) Related Papers Cited by
It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the `only if' direction and provide a series of examples to support the converse one. We show that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace (this result is due to Barak Weiss).
Interval exchange transformation,
minimality,
unique ergodicity,
measurable foliation.
Mathematics Subject Classification: Primary: 37E05; Secondary: 37E35.
Figure 1. The subset $M(\pi, \mathscr U)\subset\Delta^n\cap\mathscr U$ in Examples 1.7 (left) and 1.8 (right). Only points with $\dim_{\mathbb Q}\langle a_1, a_2, 1\rangle=3$ are considered
Figure 2. Singularities of $\mathscr F_{\pi, \mathbf a}$
Figure 3. A transversal representing a separating cycle (bold line) and the restriction cycle (dashed line) in Example 2.18
Figure 4. Double suspension surface. Each vertical straight line segment in $\partial D$ is collapsed to a point, the bottom side of the square is identified with the top one
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Modeling of vacuum grippers for the design of energy efficient vacuum-based handling processes
Felix Gabriel ORCID: orcid.org/0000-0001-5222-301X1,
Markus Fahning1,
Julia Meiners1,
Franz Dietrich2 &
Klaus Dröder1
Production Engineering volume 14, pages545–554(2020)Cite this article
Vacuum-based handling is widely used in industrial production systems, particularly for hand-ling of sheet metal parts. The process design for such handling tasks is mostly based on approximate calculations and best-practice experience. Due to the lack of detailed knowledge about the parameters that significantly influence the seal and force transmission behavior of vacuum grippers, these uncertainties are encountered by oversizing the gripping system by a defined safety margin. A model-based approach offers the potential to overcome this limitation and to dimension the gripping system based on a more exact prediction of the expected maximum loads and the resulting gripper deformation. In this work, we introduce an experiment-based modeling method that considers the dynamic deformation behavior of vacuum grippers in interaction with the specific gripper-object combination. In addition, we demonstrate that for these specific gripper-object combinations the gripper deformation is reversible up to a certain limit. This motivates to deliberately allow for a gripper deformation within this stability range. Finally, we demonstrate the validity of the proposed modeling method and give an outlook on how this method can be implemented for robot trajectory optimization and, based on that, enable an increase of the energy efficiency of vacuum-based handling of up to 85%.
Automated handling of parts, which adds up to about 50 % of all robot-guided processes in production environments [1] and usually even exceeds the time used for actual machining [2], is often realized by means of vacuum-based handling techniques [3], in particular in the automotive field and for packaging tasks [2, 4]. For industrial high-volu-me handling applications, vacuum is typically generated pneumatically through ejectors, due to their fast and wear-free operation and the direct integrability into gripper systems. Hence, ejector-based hand-ling processes are in focus of this work. With regard to typical efficiency ratios of air compression and pneumatic vacuum generation, a maximum of 2% of the initially invested electrical energy is eventually usable for the vacuum-based handling process (Fig. 1, top). In air compression, less than 10 % of the inserted energy can be used for vacuum generation [5], as the vast energy share is transformed to thermal energy by heat dissipation, which is reusable through heat recovery [6]. For vacuum ejectors, exergetic conversion efficiencies of up to 20% are calculated in [7].
A small process-related decrease in energy consumption has a huge impact on the overall energy consumption of the entire handling process
However, as major research on energy efficient compressed air generation and distribution [8,9,10,11,12,13,14,15] as well as on the optimisation of ejector performance and efficiency was shown [7, 16,17,18], there is currently a lack of generally applicable methods for the design of vacuum-based handling processes. The process design for vacuum-based handling tasks is usually based on prior experience and best practice knowledge. Uncertainties such as leakage or the unknown force transmission behavior of vacuum grippers make it necessary to roughly estimate the process-specific loads and therefore oversize the system by a defined safety margin. Subsequently, by use of the existent handling system, experimental trial-and-error tests are carried out for identification of advantageous operating parameters. Therefore, in order to eliminate the necessity of such extensive efforts to achieve a highly energy efficient vacuum-based handling process, the objective of this work is to provide a vacuum gripper model that can be applied for design optimization of a vacuum-based handling system and process.
With such a model, the gripper deformation that occurs during the process due to the applied mass and its acceleration through the implemented robot motion can be predicted. The robot trajectory can then be optimized in such a way that the loads at the gripper-object-interface (GOI) do not lead to a critical gripper deformation which would eventually result in a permanently reduced suction force. Conservative methods are based on avoiding any vacuum gripper deformation to ensure a robust handling process. However, this paper raises the research hypothesis that it is possible to reduce the overall energy consumption by deliberately allowing for a certain limited gripper deformation. By downsizing of grippers and ejector as well as load-adapted trajectory planning, the energy consumption for vacuum generation can be significantly reduced. Therefore, a model is required that allows for the prediction of the gripper behavior due to the occurring loads at the GOI and thus enables the task-specific optimization of the robot trajectory.
Section 2 gives an overview on existing modeling approaches for vacuum grippers. These approaches are evaluated with regard to their applicability for model-based trajectory planning and gripper deformation prediction. In Sect. 3, we present an experiment-based modeling method for vacuum grippers and the experimental setup for obtaining the required data from selected industrial vacuum grippers. On the basis of this method, we introduce an extension of the standard model for estimation of the maximum bearable loads for specific suction grippers. For a more detailed prediction of the gripper deformation due to process-induced loads, we introduce a dynamic surrogate mo-del. In Sect. 4, the energy savings achievable through the presented modeling approach are elaborated. Finally, Sect. 5 ends with a conclusion of the presented work and gives an outlook for future work.
Various researchers present work on determining the achievable suction force of active vacuum grippers. The suction force is determined either by the active area resulting from deformation of the sealing lip or by the volume within the suction cup [19,20,21,22,23,24]. For simplification, in these publications the force applied on the object and the suction cup is regarded as point load. In general, the seal between gripper and object is not determined explicitly, but an ideal seal is assumed. However, based on such simplified model approaches that do not include object-specific seal properties, the specific force transmission and seal at the GOI cannot be determined to a sufficient level of detail.
In [25], the mass distribution of the object and the main form of the vacuum gripper are considered in more detail. Furthermore, Braun introduces an explicit sealing force that must be present in order to provide the required suction effect. Based on a static system analysis, the required suction force for a specific hand-ling task is determined. Braun shows that pneumatic large-area suction grippers can be dimensioned adequately by the use of static characteristic maps that describe maximum holding forces depending on gripper design and material. Bahr et al. extend this static system analysis by considering moments that occur at the center of the suction cup in order to determine the required pressure difference [26]. On this basis, Mantriota focuses on the minimum value of the necessary static friction coefficient [27,28,29]. Such generalizing models enable a quick estimation of the reachable suction force but do not provide deeper insights into the object-specific seal and the dynamic force transmission.
Initial work on dynamic models for vacuum grippers can be found in [30]. Radtke predicts the deformation behaviour of the suction gripper by means of a dynamic surrogate model that includes tilting, torsion, traction and thrust. The viscoelastic behavior of the flexible part of the gripper is modeled by means of a spring-damper element with the objective to determine the natural oscillation behavior of vacuum grippers and, on this basis, to develop grippers with improved damping behavior. Radtke gives concrete re-commendations on adapted gripper designs based on experimental studies. Karako et al. propose a similar spring-damper model approach with the aim of improved acceleration parameters for the robot motion pattern. They showed that the process cycle time can be tuned successfully based on the proposed model, but encountered significant accuracy deviations regarding the prediction of grasp stability [31]. Both Radtke and Karako model the spring-damper element with linear characteristics which enables a more detailed consideration of the dynamics applied to the grasped object. However, the feasibility of these approaches is demonstrated, but it is not transferred to industrial implementation in order to achieve technical improvements or energy savings.
Based on Braun and Ratdke, Becker proposes an extended force model to identify the maximum force absorption of suction grippers for a variable surface roughness and object shapes [32]. Becker's model covers all motion directions of the manipulator and takes the relevant static and dynamic influences into account. Based on experiments, he provides general qualitative information about the examined interdependencies between certain design properties of gripper and object such as gripper design and material as well as object surface roughness and curvature. However, since only the maximum reachable force is determined for different gripper-object combinations, the detailed dynamic deformation behavior is not included. As the authors demonstrate in [30, 31], it is of great interest to gain knowledge about the exact deformation behavior of vacuum grippers due to a certain load state. Hence, in order to determine the maximum applicable suction force [33, 34], conduct finite element (FE) simulations. The setup of such an FE analysis requires many iterations because of the present non-linearities of the rubber-like gripper material. In addition, it is crucial to carefully set up the boundary conditions to achieve a stable computation of the FE simulations. In [33, 34] the authors demonstrate good results with regard to an accurate prediction of the gripper deformation compared to real experiments. However, such effort-intensive simulation methods are not feasible for a quick model-based process and system planning.
The aforementioned research regards models for single vacuum grippers, but also several publications can be found that aim at the transfer of such knowledge to the system level. In [25, 27,28,29, 35], the authors show strategies for the task-specific design of gripper systems that consist of multiple vacuum grippers. In detail, the maximum bearable forces and moments are calculated. Further, in [36, 37] methods are proposed for determining the optimal positions of multiple vacuum grippers on the object to be handled.
In summary, a substantial body of research proposes different modeling approaches to mathematically describe the behavior of vacuum grippers due to induced loads. The majority of the proposed methods are based on a static calculation method; few publications demonstrate feasible dynamic prediction models that can be implemented for model-based robot trajectory and also vacuum gripper design. The analyzed research suggests that the implementation of such a dynamic prediction model in combination with the consideration of different object geometries and roughness offer great potential for model-based system and process design with the objective of improved energy efficiency. Therefore, the gripper deformation behavior could be predicted not only depending on the gripper design, but also under consideration of the influence of the gripper-object combination on the sealing and force transmission.
Modeling method for vacuum grippers
Numerous design- and process-related parameters influence the resulting holding force that can be generated by a specific vacuum gripper. At first, the most significant influence factors are extracted in order to specify the model structure and the strategy for experiment-based data acquisition. On that basis, both the extended standard model and the analytical surrogate model are specified with concrete measurement data from experiments.
Influencing parameters and model structure
The effective holding force results from multiple interdependencies of design parameters (for gripper and object) and process parameters (Fig. 2). It defines the force with that the vacuum gripper can encounter the weight and inertial force of an attached object. For example, a good form fit at the GOI reduces the leakage flow which would reduce the pressure difference available for generation of the holding force. The form fit is mainly influenced by the specific gripper-object combination, but will potentially be affected by process-induced loads at the GOI. A straightforward approach for determining the quantitative relations between these influencing parameters is to explore the resulting parameter space experimentally.
The effective holding force is composed of both fixed design parameters and variable process control parameters
The required hardware and automation setup is time- and cost-intensive but makes up for high modeling and computational effort and—considering non-linear material behavior of the applied materials such as silicone or rubber—potential inaccuracies in simulations based on numerical models. The most simple possible approach is to measure the pull-off force of specific grippers and therefore enable a relative reduction of the oversizing margin. In this work, we firstly propose a model approach that extends the standard model by the influence of different gripper-object combinations on the pull-off force that is applied axially to the gripper. The standard model [38] calculates the maximum force F acting on the object by one vacuum gripper as given in Eq. (1):
$$\begin{aligned} F=\Delta p \times A \times z \times \frac{1}{S} \times n \times \eta \end{aligned}$$
with the pressure difference \(\Delta p\), the effective suction area A and the number of grippers z. By division by the safety margin S, the theoretical holding force is reduced. The sealing and deformation behaviors of the gripper are expressed by the deformation coefficient n and the leakage factor \(\eta\). Then, our extended standard model approach replaces the product of \(n\cdot \eta\) by considering the influence of the object geometry, the gripper material and design in a single factor \(n_{\mathrm {GOI}}\). At a relatively low modeling resp. experimental effort, this approach offers the potential to reduce the need for oversizing since more detailed knowledge about the expected maximum force is available. Equation (2) shows the extension of Equation (1) by the proposed factor \(n_{\mathrm {GOI}}\):
$$\begin{aligned} F=\Delta p \times A \cdot z \times \frac{1}{S} \times n_{\mathrm {GOI}}. \end{aligned}$$
However, an analytic surrogate model is an adequate compromise between modeling effort and the remaining need for oversizing and allows to be built-up and refined through both experimentally acquired measured data and simulation results. Thus, for a more detailed and accurate prediction of the gripper behavior, we propose an analytical spring-damper surrogate model based on a generalized Kelvin–Voigt model (parallel connection of spring and damper) as introduced in [30, 31]. In extension of the already pre-sent work, this model approach describes the gripper deformation due to axial and radial stress (in the scope of this work, we focus on the axial gripper behavior) as well as it considers the influence of the specific gripper-object combination and its influence on sealing and force transmission. The knowledge about process-induced gripper deformations and the corresponding grasp stability potentially enables to design robot trajectories and gripper systems in such a way that it can be assured that gripper-specific critical load values are never exceeded, while eliminating the need for oversizing.
Experimental setup for robot-based vacuum gripper characterization
A robot-based experimental setup has been built-up for acquisition of the required measurement data and is described in the following. Several grippers that are commonly used for metal sheet or package handling were selected for experimental characterization. For the objects to be grasped, there is no standardized definition of adequate geometries for such a fundamental characterization problem. Therefore, primitive test objects were designed parametrically to generate multiple geometry variants (Fig. 3).
Top: Parametric test objects (\(r_{\mathrm {c}}\): curvature radius), bottom: exemplary gripper-object combination
In a robot-based setup (Fig. 4), the selected grippers (ø 60 mm) can be tested in combination with the different objects and—for further experiments in the future—at arbitrarily complex load cases. The experimental setup comprises of an industrial robot (KUKA KR60-3) with compressed air supply and a vacuum ejector with a pre-connected proportional valve to vary the ejector input pressure. For measurement of the occurring loads, a 6-D force-torque-sensor is used to which single vacuum grippers of different types can be attached. The force data are acquired at a sampling rate of 500 Hz. For vacuum generation, a compact ejector with integrated pressure measurement is used. The input pressure is varied by a proportional valve.
Experimental setup for robot-based tests
In this setup, the balance of forces between robot and the fixed test object provides direct information about the reaction forces inside the vacuum gripper, since the gripping area is fixed to the test object and the pulling force that is generated by the robot can be measured. Therefore, the measured force data can be directly used for model parametrization.
Extended standard model
For the extended standard model, the maximum achievable forces just before the gripper pull-off are regarded. In this work, grippers (provided by J. Schmalz GmbH) with 1.5 foldings of different gripper materials (nitrile butadiene rubber NBR, high temperature material HT, Vulkollan VU) are considered in combination with the object geometries. As these grippers were designed for different application fields, they differ in the detailed design of support structures and inner diameters. Table 1 shows the parameters that were tested in full-factorial experiments. 30 repetitions were conducted per parameter setting to compensate for statistical variance and outliers. The test runs were packed into batches of five runs per setting and then randomized in such a way that gripper and object were changed every five test runs.
Table 1 Test plan "extended standard model"
The objective is to aggregate such influences into the above introduced factor \(n_{\mathrm {GOI}}\) for a more precise prediction of the gripper-object-specific pull-off force. Therefore, pull-off tests were conducted with all selected test objects (and a plane object as reference) at a target velocity of 10 mm/s (in orientation to the test velocity for tensile testing of rubber [39]). The test objects consist of milled aluminium and have an average surface roughness \(R_\mathrm{Z}=0.43\,\pm\,0.08\) \({\upmu }\)m (six measurements in milling direction with Hommel Wave W10 skidded roughness gage). The input pressure for the ejector has been set to 4.0 bar for all tests since this is the most efficient operating point. On the basis of the force data and the gripper specifications provided by the manufacturer, the introduced factor \(n_{\mathrm {GOI}}\) was computed for all gripper-object combinations as shown in Fig. 5. The third convex dome object could not be considered since the large curvature resulted in too strong leakage.
The object geometry influences the achievable holding force significantly
The gripper made of NBR does not reach the theoretically calculated holding force, since \(n_{\mathrm {GOI}}\) is below 1 in all cases. The gripper made of HT2 material shows significantly lower forces than theoretically calculated, except for two concave dome objects. For the gripper made of HT1 material, \(n_{\mathrm {GOI}}\) reaches the theoretical force in most cases. The gripper made of VU, however, provides exceptionally high suction forces, especially in combination with the concave dome objects. For a concrete mathematical description of the observed phenomena, a function \(n_{\mathrm {GOI}}(r_{\mathrm {c}},f_{\mathrm {s}})\) can be built via regression, where \(r_{\mathrm {c}}\) is the curvature radius and \(f_{\mathrm {s}}\) describes the degree of object symmetry. Such object-specific knowledge about the achievable pull-off force can potentially be applied to grasp planning problems where the optimal positions of the vacuum gripper on an object are determined. However, it is further intended to elaborate a dynamic surrogate model to provide for a more detailed model-based process design.
Analytical spring-damper model approach
The objective of this spring-damper model is the prediction of the gripper elongation due to process-indu-ced forces. With one selected gripper (SAB HT2), pull-off tests were conducted at 15 target velocities (0.5, 1, 5, 10, 25, 50, 75, 100, 125, 150, 175, 200, 225, 250 mm/s) at 3.0 bar in combination with only the flat object to demonstrate the modeling workflow. Here, a lower pressure level was chosen in order to achieve a relatively large gripper deformation for the model validation. 30 repetitions per parameter set were packed into six batches of five runs per target velocity setting and then randomized. Algorithm 1 shows the schematic procedure of the pull-off tests.
As a first step, the spring characteristics were extracted from the pull-off test at 0.5 mm/s, which can be regarded as quasi-static. The regression can be conducted with the following kernel function:
$$F_{{{\text{spring}}}} (x) = ab + x^{{ - b \times (x - c)}} ,$$
Figure 6 shows the regression (mean squared error, MSE, over all curves of 0.897 N\(^2\)) of the quasi-static pull-off measurement for the selected vacuum gripper.
Non-linear regression of the spring characteristics
Regression of all force measurements provides a 3D-map of the resilience inside the vacuum gripper (dashed in Fig. 7). In accordance with the equilibrium of forces that is present inside the spring-damper element during the pull-off experiments, the difference of the force curves and the quasi-static spring characteristics curve results in the damping characteristics (dotted in Fig. 7).
Multidimensional non-linear regression of the damper characteristics from the measurement data
The elongation velocity is displayed only up to 160 mm/s because higher target velocities could not be reached by the robot due to acceleration limits. Using a fourth degree polynomial as kernel function, a regression of the damping force \(F_{\mathrm {d}}(\dot{z})\) dependent on the elongation velocity \(\dot{z}\) can be calculated (MSE=7.001 N\(^2\)). By fitting of the curve-wise polynomial coefficients, a two-dimensional regression of the damping force dependent on the actual elongation z and the elongation velocity \(\dot{z}\) provides the damping force as shown in Eq. (4).
$$\begin{aligned} F_{\mathrm {d}}(z,\dot{z})=a(z)+b(z)\times \dot{z}+c(z)\times \dot{z}^2. \end{aligned}$$
Hence, the damping force can directly be calculated for a specific combination of z and \(\dot{z}\) (green in Fig. 7). For the numerical computation of the gripper elongation due to externally applied forces, the resulting damping factor can directly be extracted.
In the first section of this paper, we raised the research hypothesis that it is possible to reduce the overall energy consumption by deliberately allowing for a certain limited gripper deformation. This would require that the gripper deformation is fully reversible below this limit. In order to evaluate this assumption, further pull-off tests are conducted at a velocity of 10 mm/s (this velocity was chosen for a relatively low time required for conducting the experiments; the velocity dependency of the deformation will be examined in the future). After full attachment at \(\mathrm{POS}_{\mathrm {a}}\) the elongation is firstly generated until the target position \(\mathrm{POS}_{\mathrm {t}}\) is reached, reset to \(\mathrm{POS}_{\mathrm {a}}\) and pulled off entirely to ensure independent data. This is repeated 30 times. The z-component of \(\mathrm{POS}_{\mathrm {t}}\) is increased stepwise by 1 mm until the maximum elongation is reached.
Figure 8 shows the resilience force that is generated in the vacuum gripper (by applying an external pull force) and then decreased (due to the release motion back to the initial position \(\mathrm{POS}_{\mathrm {a}}\)). Up to a certain limit (around 13 mm), a slight hysteresis can be identified, but the resilience force in- and decreases steadily. Above that limit, the force drops significantly as soon as the release of the external force begins. The tests at all other target positions are displayed in light gray for better visibility of the highlighted curves.
The gripper deformation is determinately reversible
With regard to a handling process in industrial practice, this implies that a gripper deformation up to the identified limit can be described as reversible. Above that limit, the gripper is not able to retract immediately which indicates that a permanent deformation will remain.
Experimental validation of the spring-damper model
For experimental validation of the proposed spring-damper model, the robot end effector was equipped with an ultrasonic distance sensor (Balluff BUS0026) directed to the top face of the test object (flat surface, 2.4 kg). In order to enable a significantly visible gripper elongation, the input pressure was set to 3.0 bar for a reduced suction flow. Initially, the vacuum gripper was in full contact with the test object. Subsequently, the robot was accelerated in the z axis and stopped at a target position of \(z_{\mathrm {max}}=680\) mm. Figure 9 depicts the measured and simulated gripper elongation.
Experimental validation of the non-linear spring-damper model
The simulation of the elongation was conducted by numerical calculation of a generalized Kelvin–Voigt model where spring and dam-per are arranged in parallel. The inertial force of the attached mass was calculated by means of the acceleration that is applied due to the robot trajectory. The results of the simulation do not match the measured data. The simulated maximum elongation is however close to the measured values. The apparent prediction errors can potentially be attributed to phenomena of the gripper deformation behavior that cannot be modeled with the proposed spring-damper model. Moreover, a detailed model of the retraction behavior of the gripper may improve the simulation results. In this validation experiment, a simplified robot trajectory was used to evaluate the gripper elongation. Nevertheless, the model can be applied to determine critical deformations in the context of robot trajectory optimization, given that the prediction accuracy will be improved significantly. The proposed experiment-based modeling method is generally feasible and can also be applied to other gripper-object combinations for a prediction of the gripper deformation that is more detailed than previously available methods.
Reference scenario: potential energy savings by model-based process design
An industrial reference scenario is introduced in the following to estimate the potential energy savings that can be achieved by application of the elaborated model approaches. For a fictitious pick and place task where a 1\(\times\)1 m aluminium sheet is to be transferred between two defined positions within a cycle time of 4 s, J. Schmalz GmbH provided a gripping system including a vacuum ejector that were dimensioned conservatively by use of the standard calculation scheme as shown in Eq. (1). This gripping system comprises of four round bell suction cups (diameter 60 mm) that can be evacuated by an ejector with a suction flow rate of 36 l/min. At a recommended target pressure difference \(\Delta p=600\) mbar (and the programmed robot trajectory for completion of one handling cycle in 4 s), no gripper deformation occurs. Assuming that the resulting deformation will still remain within the specific stable range as examined in Fig. 8, it is possible to reduce the gripper size to a diameter of 50 mm (orange in Fig. 10). The gripper dead volume would be reduced by 41 % compared to the original grippers, which directly leads to a proportionally reduced consumption of compressed air that is required for evacuation of the dead volume (tubings neglected). Further, it is plausible that the ejector could also be downsized since it may take the same time for evacuating a reduced volume, and therefore a reduced suction flow rate could be sufficient. The result is a halved compressed air consumption. Downsizing both the gripper and the ejector would result in a total energy use reduction to 29.5 % of the originally required energy input.
Allowing a limited gripper elongation enables significant savings of the energy required for vacuum generation
Own experiments have shown that an adapted control strategy for the built-in air saving function of the used compact ejector has a significant impact on the compressed air use as well (green in Fig. 10). An increase in the pressure hysteresis (the range within that the pressure difference is allowed to drop before the ejector starts to generate vacuum in order to maintain a stable level of vacuum) by 100 mbar enables a relative reduction in energy consumption of 10%. When we reduced the target pressure difference from 600 to 500 mbar, this resulted in 55% of the original air consumption. Again, a combination of both control strategies achieves more than 50% energy saving. A larger increase of the pressure hysteresis and decrease of the pressure difference lead to significant gripper deformations. At the specified control parameters \(\Delta p=600\) mbar and a hysteresis of 150 mbar, the grippers did not show any noticeable deformation. In general, it must be critically evaluated whether a combination of multiple separate downsizing strategies keeps the gripper deformation within the stable range, since minimal leakage between gripper and object is a prerequisite for application of the described energy saving strategies. However, combining all proposed strategies enables an overall reduction to less than 15 % of the original energy consumption.
Summary and outlook
In this work, an experiment-based modeling method for vacuum grippers was presented. The initially introduced extended standard model considers the specific influence of a certain gripper-object-combination on the resulting maximum holding force. This approach potentially allows for a relative downsizing of a vacuum-based gripping system in comparison to the standard model. Based on present research on dynamic surrogate models for vacuum grippers, we demonstrated that on the basis of more extensive testing, a detailed spring-damper-model can be built from test data. Experimental data are publicly available at https://lnk.tu-bs.de/AmMnbw. It was further shown that the dynamic gripper elongation can be predicted qualitatively by means of this surrogate model. However, the occurring prediction errors show that not all phenomena can be modeled sufficiently with the spring-damper approach and must be further investigated in order to enable a precise model-based robot trajectory optimization. With respect to the research hypothesis raised in this work, it can be summarized that the proposed modeling method does not only provide the basis for highly task-specific process and system design optimization, but can also reliably secure these design decisions with knowledge about the gripper deformation reversibility and thus significantly reduce the need for oversizing. Hence, the energy that is needed to realize vacuum-based handling tasks can generally be decreased to a large extent.
In future work, alternative respectively more complex model structures and further regression methods will be evaluated in order to increase the prediction accuracy. In addition, we will evaluate if it is possible to reduce the experimental effort in such a way that a valid model can still be achieved. For example, this could be achieved through generalized pre-trained neural networks that are then post-trained with data from the gripper-object-specific experiments. Up to now, the industrial practicability of the proposed modeling method is limited by the required amount of test data. Whereas subsequent regressions were applied to simplify the position- and velocity-dependent determination of spring- and dam-per parameters in the dynamic simulation, the resulting regression error could be reduced by a more complex representation of the spring- and dam-per characteristics. With regard to the influence of the gripper-object-combination, both proposed model approaches will be extended by considering tests with different object surface roughness. This augments the solution space for the optimization of the robot trajectory or the determination of the optimal gripper positions on a specific object.
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The authors thank the German Federal Ministry of Economic Affairs and Energy for supporting the project BiVaS (03ET1559B). The project is mainly conducted at Open Hybrid LabFactory, a ForschungsCampus funded by the German Federal Ministry of Education and Research.
Open Access funding enabled and organized by Projekt DEAL.
Institute of Machine Tools and Production Technology, Technische Universität Braunschweig, Brunswick, Germany
Felix Gabriel, Markus Fahning, Julia Meiners & Klaus Dröder
Chair for Assembly Technology and Factory Management, Institute for Machine Tools and Factory Management, Technical University Berlin, Berlin, Germany
Franz Dietrich
Felix Gabriel
Markus Fahning
Julia Meiners
Klaus Dröder
Correspondence to Felix Gabriel.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Gabriel, F., Fahning, M., Meiners, J. et al. Modeling of vacuum grippers for the design of energy efficient vacuum-based handling processes. Prod. Eng. Res. Devel. 14, 545–554 (2020). https://doi.org/10.1007/s11740-020-00990-9
Vacuum-based handling | CommonCrawl |
Differentially mutated subnetworks discovery
Morteza Chalabi Hajkarim1,
Eli Upfal2 &
Fabio Vandin ORCID: orcid.org/0000-0003-2244-23203
We study the problem of identifying differentially mutated subnetworks of a large gene–gene interaction network, that is, subnetworks that display a significant difference in mutation frequency in two sets of cancer samples. We formally define the associated computational problem and show that the problem is NP-hard.
We propose a novel and efficient algorithm, called DAMOKLE, to identify differentially mutated subnetworks given genome-wide mutation data for two sets of cancer samples. We prove that DAMOKLE identifies subnetworks with statistically significant difference in mutation frequency when the data comes from a reasonable generative model, provided enough samples are available.
We test DAMOKLE on simulated and real data, showing that DAMOKLE does indeed find subnetworks with significant differences in mutation frequency and that it provides novel insights into the molecular mechanisms of the disease not revealed by standard methods.
The analysis of molecular measurements from large collections of cancer samples has revolutionized our understanding of the processes leading to a tumour through somatic mutations, changes of the DNA appearing during the lifetime of an individual [1]. One of the most important aspects of cancer revealed by recent large cancer studies is inter-tumour genetic heterogeneity: each tumour presents hundreds-thousands mutations and no two tumours harbour the same set of DNA mutations [2].
One of the fundamental problems in the analysis of somatic mutations is the identification of the handful of driver mutations (i.e., mutations related to the disease) of each tumour, detecting them among the thousands or tens of thousands that are present in each tumour genome [3]. Inter-tumour heterogeneity renders the identification of driver mutations, or of driver genes (genes containing driver mutations), extremely difficult, since only few genes are mutated in a relatively large fraction of samples while most genes are mutated in a low fraction of samples in a cancer cohort [4].
Recently, several analyses (e.g, [5, 6]) have shown that interaction networks provide useful information to discover driver genes by identifying groups of interacting genes, called pathways, in which each gene is mutated at relatively low frequency while the entire group has one or more mutations in a significantly large fraction of all samples. Several network-based methods have been developed to identify groups of interacting genes mutated in a significant fraction of tumours of a given type and have been shown to improve the detection of driver genes compared to methods that analyze genes in isolation [5, 7,8,9].
The availability of molecular measurements in a large number of samples for different cancer types have also allowed comparative analyses of mutations in cancer [5, 10, 11]. Such analyses usually analyze large cohorts of different cancer types as a whole employing methods to find genes or subnetworks mutated in a significant fraction of tumours in one cohort, and also analyze each cancer type individually, with the goal to identify:
pathways that are common to various cancer types;
pathways that are specific to a given cancer type.
For example, [5] analyzed 12 cancer types and identified subnetworks (e.g., a TP53 subnetwork) mutated in most cancer types as well as subnetworks (e.g., a MHC subnetwork) enriched for mutations in one cancer type. In addition, comparative analyses may also be used for the identification of mutations of clinical relevance [12]. For example: comparing mutations in a patients that responded to a given therapy with mutations in patients (of the same cancer type) that did not respond to the same therapy may identify genes and subnetworks associated with response to therapy; comparing mutations in patients whose tumours metastasized with mutations in patients whose tumours did not metastasize may identify mutations associated with the insurgence of metastases.
Pathways that are significantly mutated only in a specific cancer type may not be identified by analyzing one cancer type at the time or all samples together (Fig. 1), but, interestingly, to the best of our knowledge no method has been designed to directly identify sets of interacting genes that are significantly more mutated in a set of samples compared to another. The task of finding such sets is more complex than the identification of subnetworks significantly mutated in a set of samples, since subnetworks that have a significant difference in mutations in two sets may display relatively modest frequency of mutation in both set of samples, whose difference can be assessed as significant only by the joint analysis of both sets of samples.
Identification of subnetworks with significant difference in mutation frequency in two set of samples \({\mathcal {C}}, {\mathcal {D}}\). The blue subnetwork is significantly more mutated in \({\mathcal {D}}\) than in \({\mathcal {C}}\), but it is not detected by methods that look for the most significantly mutated subnetworks in \({\mathcal {C}}\) or in \({\mathcal {D}}\) or in \({\mathcal {C}}\cup {\mathcal {D}}\), since the orange subnetwork is in each case mutated at much higher frequency
Several methods have been designed to analyze different aspects of somatic mutations in a large cohort of cancer samples in the context of networks. Some methods analyze mutations in the context of known pathways to identify the ones significantly enriched in mutations (e.g., [13]). Other methods combine mutations and large interaction networks to identify cancer subnetworks [5, 14, 15]. Networks and somatic mutations have also been used to prioritarize mutated genes in cancer [7, 8, 16,17,18] and for patients stratification [6, 19]. Some of these methods have been used for the identification of common mutation patterns or subnetworks in several cancer types [5, 10], but to the best of our knowledge no method has been designed to identify mutated subnetworks with a significant difference in two cohorts of cancer samples.
Few methods studied the problem of identifying subnetworks with significant differences in two sets of cancer samples using data other than mutations. [20] studied the problem of identifying optimally discriminative subnetworks of a large interaction network using gene expression data. Mall et al. [21] developed a procedure to identify statistically significant changes in the topology of biological networks. Such methods cannot be readily applied to find subnetworks with significant difference in mutation frequency in two sets of samples. Other related work use gene expression to characterize different cancer types: [22] defined a pathway-based score that clusters samples by cancer type, while [23] defined pathway-based features used for classification in various settings, and several methods [24,25,26,27,28] have been designed for finding subnetworks with differential gene expression.
In this work we study the problem of finding subnetworks with frequency of mutation that is significantly different in two sets of samples. In particular, our contributions are fourfold. First, we propose a combinatorial formulation for the problem of finding subnetworks significantly more mutated in one set of samples than in another and prove that such problem is NP-hard. Second, we propose DifferentiAlly Mutated subnetwOrKs anaLysis in cancEr (DAMOKLE), a simple and efficient algorithm for the identification of subnetworks with a significant difference of mutation in two sets of samples, and analyze DAMOKLE proving that it identifies subnetworks significantly more mutated in one of two sets of samples under reasonable assumptions for the data. Third, we test DAMOKLE on simulated data, verifying experimental that DAMOKLE correctly identifies subnetworks significantly more mutated in a set of samples when enough samples are provided in input. Fourth, we test DAMOKLE on large cancer datasets comprising two cancer types, and show that DAMOKLE identifies subnetworks significantly associated with one of the two types which cannot be identified by state-of-the-art methods designed for the analysis of one set of samples.
Methods and algorithms
This section presents the problem we study, the algorithm we propose for its solution, and the analysis of our algorithm. In particular, "Computational problem" section formalizes the computational problem we consider; "Algorithm" section presents DifferentiAlly Mutated subnetwOrKs anaLysis in cancEr (DAMOKLE), our algorithm for the solution of the computational problem; "Analysis of DAMOKLE" section describes the analysis of our algorithm under a reasonable generative model for mutations; "Statistical significance of the results" section presents a formal analysis of the statistical significance of subnetworks obtained by DAMOKLE; and "Permutation testing" section describes two permutation tests to assess the significance of the results of DAMOKLE for limited sample sizes.
Computational problem
We are given measurements on mutations in m genes \(\mathcal {G}=\{1,\dots ,m\}\) on two sets \({\mathcal {C}}=\{c_1,\dots ,c_{n_C}\},{\mathcal {D}}=\{d_1,\dots ,d_{n_D}\}\) of samples. Such measurements are represented by two matrices C and D, of dimension \(m \times n_C\) and \(m \times n_D\), respectively, where \(n_C\) (resp., \(n_D\)) is the number of samples in \({\mathcal {C}}\) (resp., \({\mathcal {D}}\)). \(C(i,j)=1\) (resp., \(D(i,j)=1\)) if gene i is mutated in the j-th sample of \({\mathcal {C}}\) (resp., \({\mathcal {D}}\)) and \(C(i,j)=0\) (resp., \(D(i,j)=0\)) otherwise. We are also given an (undirected) graph \(G=(V,E)\), where vertices \(V = \{1,\dots ,m \}\) are genes and \((i,j) \in E\) if gene i interacts with gene j (e.g., the corresponding proteins interact).
Given a set of genes \(S \subset \mathcal {G}\), we define the indicator function \(c_{S}(c_i)\) with \(c_{S}(c_i)=1\) if at least one of the genes of S is mutated in sample \(c_i\), and \(c_{S}(c_i)=0\) otherwise. We define \(c_{S}(d_i)\) analogously. We define the coverage \(c_{S}({\mathcal {C}})\) of S in \({\mathcal {C}}\) as the fraction of samples in \({\mathcal {C}}\) for which at least one of the genes in S is mutated in the sample, that is
$$\begin{aligned} c_{S}({\mathcal {C}}) = \frac{\sum _{i=1}^{n_C} c_{S}(c_i)}{n_C} \end{aligned}$$
and, analogously, define the coverage \(c_{S}({\mathcal {D}})\) of S in \({\mathcal {D}}\) as \(c_{S}({\mathcal {D}}) = \frac{\sum _{i=1}^{n_D} c_{S}(d_i)}{n_D}.\)
We are interested in identifying sets of genes S, with \(|S|\le k\), corresponding to connected subgraphs in G and displaying a significant difference in coverage between \({\mathcal {C}}\) and \({\mathcal {D}}\), i.e., with a high value of \(|c_{S}({\mathcal {C}})-c_{S}({\mathcal {D}})|\). We define the differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) as \(dc_{S}({\mathcal {C}},{\mathcal {D}}) = c_{S}({\mathcal {C}})-c_{S}({\mathcal {D}}).\)
In particular, we study the following computational problem.
The differentially mutated subnetworks discovery problem: given a value \(\theta\) with \(\theta \in [0,1]\), find all connected subgraphs S of G of size \(\le k\) such that \(dc_{S}({\mathcal {C}},{\mathcal {D}}) \ge \theta\).
Note that by finding sets that maximize \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) we identify sets with significantly more mutations in \({\mathcal {C}}\) than in \({\mathcal {D}}\), while to identify sets with significantly more mutations in \({\mathcal {D}}\) than in \({\mathcal {C}}\) we need to find sets maximizing \(dc_{S}({\mathcal {D}},{\mathcal {C}})\). In addition, note that a subgraph S in the solution may contain genes that are not mutated in \({\mathcal {C}}\cup {\mathcal {D}}\) but that are needed for the connectivity of S.
We have the following.
The differentially mutated subnetworks discovery problem is NP-hard.
The proof is by reduction from the connected maximum coverage problem [14]. In the connected maximum coverage problem we are given a graph G defined on a set \(V=\{v_1,\dots ,v_n\}\) of n vertices, a family \(\mathcal {P} = \{P_1,\dots ,P_n\}\) of subsets of a universe I (i.e., \(P_i \in 2^{I}\)), with \(P_i\) being the subset of I covered by \(v_i \in V\) and value k, and we want to find the subgraph \(C^* = \{v_{i_1},\dots , v_{i_k}\}\) with k nodes of G that maximizes \(|\cup _{j=1}^k P_{i_j}|\).
Given an instance of the connected maximum coverage problem, we define an instance of the differentially mutated subnetworks discovery problem as follows: the set \(\mathcal {G}\) of genes corresponds to the set V of vertices of G in the connected maximum coverage problem, and the graph G is the same as in the instance of the maximum coverage instance; the set \({\mathcal {C}}\) is given by the set I and the matrix C is defined as \(C_{i,j}=1\) if \(i \in P_j\), while \({\mathcal {D}}=\emptyset\).
Note that for any subgraph S of G, the differential coverage \(dc_D({\mathcal {C}},{\mathcal {D}})= c_{S}({\mathcal {C}}) - c_{S}({\mathcal {D}}) = c_{S}({\mathcal {C}})\) and \(c_{S}({\mathcal {C}}) = |\cup _{g \in S} P_{g}|/|I|\). Since |I| is the same for all solutions, the optimal solution of the differentially mutated subnetworks discovery instance corresponds to the optimal solution to the connected maximum coverage instance, and viceversa. \(\square\)
We now describe DifferentiAlly Mutated subnetwOrKs anaLysis in cancEr (DAMOKLE), an algorithm to solve the differentially mutated subnetworks discovery problem. DAMOKLE takes in input mutation matrices C and D for two sets \({\mathcal {C}}\), \({\mathcal {D}}\) of samples, a (gene–gene) interaction graph G, an integer \(k>0\), and a real value \(\theta \in [0,1]\), and returns subnetworks S of G with \(\le k\) vertices and differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}}) \ge \theta\). Subnetworks reported by DAMOKLE are also maximal (no vertex can be added to S while maintaining the connectivity of the subnetwork, \(|S| \le k\) and \(dc_{S}({\mathcal {C}},{\mathcal {D}}) \ge \theta\)). DAMOKLE is described in Algorithm 1. DAMOKLE starts by considering each edge \(e=\{u,v\} \in E\) of G with differential coverage \(dc_{\{u,v\}}({\mathcal {C}},{\mathcal {D}})\ge \theta /(k-1)\), and for each such e identifies subnetworks including e to be reported in output using Algorithm 2.
GetSolutions, described in Algorithm 2, is a recursive algorithm that, give a current subgraph S, identifies all maximal connected subgraphs \(S', |S'| \le k\), containing S and with \(dc_{S'}({\mathcal {C}},{\mathcal {D}}) \ge \theta\). This is obtained by expanding S one edge at the time and stopping when the number of vertices in the current solution is k or when the addition of no vertex leads to an increase in differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) for the current solution S. In Algorithm 2, N(S) refers to the set of edges with exactly one vertex in the set S.
The motivation for design choices of DAMOKLE are provided by the results in the next section.
Analysis of DAMOKLE
The design and analysis of DAMOKLE are based on the following generative model for the underlying biological process.
For each gene \(i \in \mathcal {G}=\{1,2,...,m\}\) there is an a-priori probability \(p_i\) of observing a mutation in gene i. Let \(H\subset \mathcal {G}\) be the connected subnetwork of up to k genes that is differentially mutated in samples of \({\mathcal {C}}\) w.r.t. samples of \({\mathcal {D}}\). Mutations in our samples are taken from two related distributions. In the "control" distribution F a mutation in gene i is observed with probability \(p_i\) independent of other genes' mutations. The second distribution \(F_H\) is analogous to the distribution F but we condition on the event \(E(H)=\)"at least one gene in H is mutated in the sample".
For genes not in H, all mutations come from distribution F. For genes in H, in a perfect experiment with no noise we would assume that samples in \({\mathcal {C}}\) are taken from \(F_H\) and samples from \({\mathcal {D}}\) are taken from F. However, to model realistic, noisy data we assume that with some probability q the "true" signal for a sample is lost, that is the sample from \({\mathcal {C}}\) is taken from F. In particular, samples in \({\mathcal {C}}\) are taken with probability \(1-q\) from \(F_H\) and with probability q from F.
Let p be the probability that H has at least one mutation in samples from the control model F, \(p= 1-\prod _{j\in H} (1-p_j)\approx \sum _{j\in H} p_j.\) Clearly, we are only interested in sets \(H\subset \mathcal {G}\) with \(p\ll 1\).
If we focus on individual genes, the probability gene i is mutated in a sample from \({\mathcal {D}}\) is \(p_i\), while the probability that it is mutated in a sample from \({\mathcal {C}}\) is \(\frac{(1-q)p_i}{1-\prod _{j\in H} (1-p_j)}+qp_i.\) Such a gap may be hard to detect with a small number of samples. On the other hand, the probability of E(H) (i.e., of at least one mutation in the set H) in a sample from \({\mathcal {C}}\) is \((1-q) +q(1-\prod _{j\in H} (1-p_j)) = 1-q + qp\), while the probability of E(H) in a sample from \({\mathcal {D}}\) is \(1-\prod _{j\in H} (1-p_j) = p\) which is a more significant gap, when \(p \ll 1.\)
The efficiency of DAMOKLE is based on two fundamental results. First we show that it is sufficient to start the search only in edges with relatively high differential coverage.
If \(dc_{S}({\mathcal {C}},{\mathcal {D}}) \ge \theta,\) then, in the above generating model, with high probability (asymptotic in \(n_C\) and \(n_D\) )there exist an edge \(e \in S\) such that \(dc_{\{e\}}({\mathcal {C}},{\mathcal {D}}) \ge (\theta -\epsilon )/(k-1),\) for any \(\epsilon >0.\)
For a set of genes \(S'\subset \mathcal {G}\) and a sample \(z\in {\mathcal {C}} \cup {\mathcal {D}}\), let \(Count(S',z)\) be the number of genes in \(S'\) mutated in sample z. Clearly, if for all \(z\in {\mathcal {C}} \cup {\mathcal {D}}\), we have \(Count(S,z)=1\), i.e. each sample has no more than one mutation in S, then
$$\begin{aligned} dc_{S}({\mathcal {C}},{\mathcal {D}})=\, & {} c_{S}({\mathcal {C}})-c_{S}({\mathcal {D}}) =\,\frac{\sum _{i=1}^{n_C} c_{S}(c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} c_{S}(d_i)}{n_D} \\=\, & {} \frac{\sum _{i=1}^{n_C} \sum _{j\in S} Count (\{j\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} \sum _{j\in S} Count(\{j\},d_i)}{n_D} \\= \,& {} \sum _{j\in S} \left( \frac{\sum _{i=1}^{n_C} Count (\{j\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} Count(\{j\},d_i)}{n_D} \right) \\\ge & {} \theta . \end{aligned}$$
Thus, there is a vertex \(j^*=\arg \max _{j\in S} \left( \frac{\sum _{i=1}^{n_C} Count (\{j\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} Count(\{j\},d_i)}{n_D} \right)\) such that \(dc_{\{j^*\}}({\mathcal {C}},{\mathcal {D}}) =\frac{\sum _{i=1}^{n_C} Count (\{j^*\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} Count(\{j^*\},d_i)}{n_D} \ge \theta /k.\)
Since the set of genes S is connected, there is an edge \(e=(j^*, \ell )\) for some \(\ell \in S\). For that edge,
$$\begin{aligned} dc_{\{e \}}({\mathcal {C}},{\mathcal {D}}) \ge \frac{\theta -dc_{\{\ell \}}({\mathcal {C}},{\mathcal {D}})}{k-1} +dc_{\{\ell \}}({\mathcal {C}},{\mathcal {D}}) \ge \frac{\theta }{k-1}. \end{aligned}$$
For the case when the assumption \(Count(S,z)=1\) for all \(z \in {\mathcal {C}}\cup {\mathcal {D}}\) does not hold, let
$$\begin{aligned} Mul(S, {\mathcal {C}},{\mathcal {D}})= & {} \frac{\sum _{i=1}^{n_C} \sum _{j\in S} Count (\{j\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_C} c_{S}(c_i)}{n_C} \\&+ \frac{\sum _{i=1}^{n_D} Count(\{j\},d_i)}{n_D} -\frac{\sum _{i=1}^{n_D} c_{S}(d_i)}{n_D}. \end{aligned}$$
$$\begin{aligned} \sum _{j\in S} \left( \frac{\sum _{i=1}^{n_C} Count (\{j\}, c_i)}{n_C} - \frac{\sum _{i=1}^{n_D} Count(\{j\},d_i)}{n_D} \right) - Mul(S, {\mathcal {C}},{\mathcal {D}}) \ge \theta \end{aligned}$$
$$\begin{aligned} dc_{\{e \}}({\mathcal {C}},{\mathcal {D}})\ge \frac{\theta +Mul(S, {\mathcal {C}},{\mathcal {D}}) }{k-1}. \end{aligned}$$
Since the probability of having more than one mutation in S in a sample from \({\mathcal {C}}\) is at least as high as from a sample from \({\mathcal {D}}\), we can normalize (similar to the proof of Theorem 2 below) and apply Hoeffding bound (Theorem 4.14 in [29]) to prove that
$$\begin{aligned} Prob(Mul(S, {\mathcal {C}},{\mathcal {D}}) < -\epsilon )\le 2e^{-2\epsilon ^2 n_C n_D/(n_C+n_D)}. \end{aligned}$$
\(\square\)
The second result motivates the choice, in Algorithm 2, of adding only edges that increase the score of the current solution (and to stop if there is no such edge).
If subgraph S can be partitioned as \(S= S' \cup \{j\} \cup S'',\) and \(dc_{\mathcal {S'}\cup \{j\}}({\mathcal {C}},{\mathcal {D}})< dc_{\mathcal {S'}}({\mathcal {C}},{\mathcal {D}})- p p_j,\) then with high probability (asymptotic in \(n_{{\mathcal {D}}}\) ) \(dc_{S \setminus \{j\}}({\mathcal {C}},{\mathcal {D}}) > dc_{S}({\mathcal {C}},{\mathcal {D}}).\)
We first observe that if each sample in \({\mathcal {D}}\) has no more than 1 mutation in S then \(dc_{\mathcal {S'}\cup \{j\}}({\mathcal {C}},{\mathcal {D}})< dc_{\mathcal {S'}}({\mathcal {C}},{\mathcal {D}})\) implies that \(dc_{\{j\}}({\mathcal {C}},{\mathcal {D}})<0\), and therefore, under this assumption, \(dc_{S \setminus \{j\}}({\mathcal {C}},{\mathcal {D}}) > dc_{S}({\mathcal {C}},{\mathcal {D}})\).
To remove the assumption that a sample has no more than one mutation in S, we need to correct for the fraction of samples in \({\mathcal {D}}\) with mutations both in j and \(S''\). With high probability (asymptotic in \(n_D\)) this fraction is bounded by \(pp_j +\epsilon\) for any \(\epsilon >0\). \(\square\)
Statistical significance of the results
To compute a threshold that guarantees statistical confidence of our finding, we first compute a bound on the gap in a non significant set.
Assume that S is not a significant set, i.e., \({\mathcal {C}}\) and \({\mathcal {D}}\) have the same distribution on S, then
$$\begin{aligned} Prob( dc_{S}({\mathcal {C}},{\mathcal {D}}) > \epsilon )\le 2e^{-2 \epsilon ^2 n_{{\mathcal {C}}}n_{{\mathcal {D}}}/(n_{{\mathcal {C}}}+n_{{\mathcal {D}}})}. \end{aligned}$$
Let \(X_1,\dots , X_{n_C}\) be independent random variables such that \(X_i=1/n_C\) if sample \(c_i\) in \({\mathcal {C}}\) has a mutation in S, otherwise \(X_i=0\). Similarly, let \(Y_1,\dots , Y_{n_D}\) be independent random variables such that \(Y_i= -1/n_D\) if sample \(d_i\) in \({\mathcal {D}}\) has a mutation in S, otherwise \(Y_i=0\).
Clearly \(dc_{S}({\mathcal {C}},{\mathcal {D}}) = \sum _{i=1}^{n_C} X_i + \sum _{i=1}^{n_D} Y_i\), and since S is not significant \(E\left[\sum _{i=1}^{n_C} X_i +\sum _{i=1}^{n_D} Y_i\right]=0\).
To apply Hoeffding bound (Theorem 4.14 in [29]), we note that the sum \(\sum _{i=1}^{n_C} X_i + \sum _{i=1}^{n_D} Y_i\) has \(n_C\) variables in the range \([0,1/n_C]\), and \(n_D\) variables in the range \([-1/n_D, 0]\). Thus,
$$\begin{aligned} Prob( dc_{S}({\mathcal {C}},{\mathcal {D}}) > \epsilon )\le 2e^{(-2 \epsilon ^2 )/(n_c/n_c^2 + n_d/n_D^2)} = 2e^{-2 \epsilon ^2 n_{{\mathcal {C}}}n_{{\mathcal {D}}}/(n_{{\mathcal {C}}}+n_{{\mathcal {D}}})}. \end{aligned}$$
Let \(N_{k}\) be the set of subnetworks under consideration, or the set of all connected components of size \(\le k\). We use Theorem 2 to obtain guarantees on the statistical significance of the results of DAMOKLE in terms of the Family-Wise Error Rate (FWER) or of the False Discovery Rate (FDR) as follows:
FWER: if we want to find just the subnetwork with significant maximum differential coverage, to bound the FWER of our method by \(\alpha\) we use the maximum \(\epsilon\) such that \(N_{k} 2e^{-2 \epsilon ^2 n_{{\mathcal {C}}}n_{{\mathcal {D}}}/(n_{{\mathcal {C}}}+n_{{\mathcal {D}}})}\le \alpha .\)
FDR: if we want to find several significant subnetworks with high differential coverage, to bound the FDR by \(\alpha\) we use the maximum \(\epsilon\) such that \({N_{k} 2e^{-2 \epsilon ^2 n_{{\mathcal {C}}}n_{{\mathcal {D}}}/(n_{{\mathcal {C}}}+n_{{\mathcal {D}}})}}/n(\alpha ) \le \alpha\), where \(n(\alpha )\) is the number of sets with differential coverage \(\ge \epsilon\).
Permutation testing
While Theorem 2 shows how to obtain guarantees on the statistical significance of the results of DAMOKLE by appropriately setting \(\theta\), in practice, due to relatively small sample sizes and to inevitable looseness in the theoretical guarantees, a permutation testing approach may be more effective in estimating the statistical significance of the results of DAMOKLE and provide more power for the identification of differentially mutated subnetworks.
We consider two permutation tests to assess the association of mutations in the subnetwork with the highest differential coverage found by DAMOKLE. The first test assesses whether the observed differential coverage can be obtained under the independence of mutations in genes by considering the null distribution in which each gene is mutated in a random subset (of the same cardinality as observed in the data) of all samples, independently of all other events. The second test assesses whether, under the observed marginal distributions for mutations in sets of genes, the observed differential coverage of a subnetwork can be obtained under the independence between mutations and samples' memberships (i.e., being a sample of \({\mathcal {C}}\) or a sample of \({\mathcal {D}}\)), by randomly permuting the samples memberships.
Let \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) be the differential coverage observed on real data for the solution S with highest differential coverage found by DAMOKLE (for some input parameters). For both tests we estimate the p-value as follow:
generate N (permuted) datasets from the null distribution;
run DAMOKLE (with the same input parameters used on real data) on each of the N permuted datasets;
let x be the number of permuted datasets in which DAMOKLE reports a solution with differential coverage \(\ge dc_{S}({\mathcal {C}},{\mathcal {D}})\): then the p-value of S is \((x+1)/(N+1)\).
We implemented DAMOKLE in PythonFootnote 1 and tested it on simulated and on cancer data. Our experiments have been conducted on a Linux machine with 16 cores and 256 GB of RAM. For all our experiments we used as interaction graph G the HINT+HI2012 networkFootnote 2, a combination of the HINT network [30] and the HI-2012 [31] set of protein–protein interactions, previously used in [5]. In all cases we considered only the subnetwork with the highest differential coverage among the ones returned by DAMOKLE. We first present the results on simulated data ("Simulated data" section) and then present the results on cancer data ("Cancer data" section).
Simulated data
We tested DAMOKLE on simulated data generated as follows. We assume there is a subnetwork S of k genes with differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})= c\). In our simulations we set \(|{\mathcal {C}}|=|{\mathcal {D}}|=n\). For each sample in \({\mathcal {D}}\), each gene g in G (including genes in S) is mutated with probability \(p_g\), independently of all other events. For samples in \({\mathcal {C}}\), we first mutated each gene g with probability \(p_g\) independently of all other events. We then considered the samples of \({\mathcal {C}}\) without mutations in S, and for each such sample we mutated, with probability c, one gene of S, chosen uniformly at random. In this way c is the expectation of the differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})\). For genes in \(G \setminus S\) we used mutation probabilities \(p_g\) estimated from oesophageal cancer data [32]. We considered only value of \(n \ge 100\), consistent with sample sizes in most recent cancer sequencing studies. (The latest ICGC data releaseFootnote 3 from April 30\(^{th}\), 2018 has data for \(\ge 500\) samples for \(81\%\) of the primary sites).
The goal of our investigation using simulated data is to evaluate the impact of various parameters on ability of DAMOKLE to recover S or part of it. In particular, we studied the impact of three parameters: the differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) of the planted subnetwork S; the number k of genes in S; and the number n of samples in each class. To evaluate the impact of such parameters, for each combination of parameters in our experiments we generated 10 simulated datasets and run DAMOKLE on each dataset with \(\theta = 0.01\), recording
the fraction of times that DAMOKLE reported S as the solution with the highest differential coverage, and
the fraction of genes of S that are in the solution with highest differential coverage found by DAMOKLE.
We first investigated the impact of the differential coverage \(c = dc_{S}({\mathcal {C}},{\mathcal {D}})\). We analyzed simulated datasets with \(n=100\) samples in each class, where \(k=5\) genes are part of the subnetwork S, for values of \(c = 0.1, 0.22, 0.33, 0.46, 0.6, 0.8\),. We run DAMOKLE on each dataset with \(k=5\). The results are shown in Fig. 2a. For low values of the differential coverage c, with \(n=100\) samples DAMOKLE never reports S as the best solution found and only a small fraction of the genes in S are part of the solution reported by DAMOKLE. However, as soon as the differential coverage is \(\ge 0.45\), even with \(n=100\) samples in each class DAMOKLE identifies the entire planted solution S most of the times, and even when the best solution does not entirely corresponds to S, more than \(80\%\) of the genes of S are reported in the best solution. For values of \(c \ge 0.6\), DAMOKLE always reports the whole subnetwork S as the best solution. Given that many recent large cancer sequencing studies consider at least 200 samples, DAMOKLE will be useful to identify differentially mutated subnetworks in such studies.
a Performance of DAMOKLE as a function of the differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})\) of subnetwork S. The figure shows (red) the fraction of times, out of 10 experiments, that the best solution corresponds to S and (blue) the fraction of genes in S that are reported in the best solution by DAMOKLE. For the latter, error bars show the standard deviation on the 10 experiments. \(n=100\) and \(k=5\) for all experiments. b Performance of DAMOKLE as a function of the number k of genes in subnetwork S. \(n=100\) and \(dc_{S}({\mathcal {C}},{\mathcal {D}})=0.46\) for all experiments. c Performance of DAMOKLE as a function of the number n of samples in \({\mathcal {C}},{\mathcal {D}}\). \(k=10\) and \(dc_{S}({\mathcal {C}},{\mathcal {D}})=0.46\) for all experiments
We then tested the performance of DAMOKLE as a function of the number of genes k in S. We tested the ability of DAMOKLE to identify a subnetwork S with differential coverage \(dc_{S}({\mathcal {C}},{\mathcal {D}})=0.46\) in a dataset with \(n=100\) samples in both \({\mathcal {C}}\) and \({\mathcal {D}}\), when the number k of genes in S varies as \(k=5,7,9\). The results are shown in Fig. 2b. As expected, when the number of genes in S increases, the fraction of times S is the best solution as well as the fraction of genes reported in the best solution by S decreases, and for \(k=9\) the best solution found by DAMOKLE corresponds to S only \(10\%\) of the times. However, even for \(k=9\), on average most of the genes of S are reported in the best solution by DAMOKLE. Therefore DAMOKLE can be used to identify relatively large subnetworks mutated in a significantly different number of samples even when the number of samples is relatively low.
Finally, we tested the performance of DAMOKLE as the number of samples n in each set \({\mathcal {C}},{\mathcal {D}}\) increases. In particular, we tested the ability of DAMOKLE to identify a relatively large subnetwork S of \(k=10\) genes with differential coverage \(dc_S({\mathcal {C}},{\mathcal {D}}) = 0.46\) as the number of samples n increases. We analyzed simulated datasets for \(n=100, 250, 500\). The results are shown in Fig. 2. For \(n=100\), when \(k=10\), DAMOKLE never reports S as the best solution and only a small fraction of all genes in S are reported in the solution. However, for \(n=250\), while DAMOKLE still reports S as the best solution only \(10\%\) of the times, on average \(70\%\) of the genes of S are reported in the best solution. More interestingly, already for \(n=500\), DAMOKLE always reports S as the best solution. These results show that DAMOKLE can reliably identify relatively large differentially mutated subnetworks from currently available datasets of large cancer sequencing studies.
Cancer data
We use DAMOKLE to analyze somatic mutations from The Cancer Genome Atlas. We first compared two similar cancer types and two very different cancer types to test whether DAMOKLE behaves as expected on these types. We then analyzed two pairs of cancer types where differences in alterations are unclear. In all cases we run DAMOKLE with \(\theta =0.1\) and obtained p-values with the permutation tests described in "Permutation testing" section.
We used DAMOKLE to analyze 188 samples of lung squamous cell carcinoma (LUSC) and 183 samples of lung adenocarcinoma (LUAD). We only considered single nucleotide variants (SNVs)Footnote 4 and use \(k=5\). DAMOKLE did not report any significant subnetwork, in agreement with previous work showing that these two cancer types have known differences in gene expression [33] but are much more similar with respect to SNVs [34].
Colorectal vs ovarian cancer
We used DAMOKLE to analyze 456 samples of colorectal adenocarcinoma (COADREAD) and 496 samples of ovarian serous cystadenocarcinoma (OV) using only SNVs.Footnote 5 For \(k=5\), DAMOKLE identifies the significant (\(p<0.01\) according to both tests in "Permutation testing" section) subnetwork APC, CTNNB1, FBXO30, SMAD4, SYNE1 with differential coverage 0.81 in COADREAD w.r.t. OV. APC, CTNNB1, and SMAD4 are members of the WNT signaling and TFG-\(\beta\) signaling pathways. The WNT signaling pathway is one of the cascades that regulates stemness and development, with a role in carcinogenesis that has been described mostly for colorectal cancer [35], but altered Wnt signaling is observed in many other cancer types [36]. The TFG-\(\beta\) signaling pathway is involved in several processes including cell growth and apoptosis, that is deregulated in many diseases, including COADREAD [35]. The high differential coverage of the subnetwork is in accordance with COADREAD being altered mostly by SNVs and OV being altered mostly by copy number aberrations (CNAs) [37].
Esophagus-stomach cancer
We analyzed SNVs and CNAs in 171 samples of esophagus cancer and in 347 samples of stomach cancer [32].Footnote 6 The number of mutations in the two sets is not significantly different (t-test p = 0.16). We first considered single genes, identifying TP53 with high (\(>0.5\)) differential coverage between the two cancer types. Alterations in TP53 have then be removed for the subsequent DAMOKLE analysis. We run DAMOKLE with \(k=4\) with \({\mathcal {C}}\) being the set of stomach tumours and \({\mathcal {D}}\) being the set of esophagus tumours. DAMOKLE identifies the significant (\(p<0.01\) for both tests in "Permutation testing" section) subnetwork \(S=\) {ACTL6A, ARID1A, BRD8, SMARCB1} with differential coverage 0.26 (Fig. 3a, b). Interestingly, all four genes in the subnetwork identified by DAMOKLE are members of the chromatin organization machinery recently associated with cancer [38, 39]. Such subnetwork is not reported as differentially mutated in the TCGA publication comparing the two cancer types [32]. BRD8 is only the top-16 gene by differential coverage, while ACTL6 and SMARCB1 are not among the top-2000 genes by differential coverage. We compared the results obtained by DAMOKLE with the results obtained by HotNet2 [5], a method to identify significantly mutated subnetworks, using the same mutation data and the same interaction network as input: none of the genes in S appeared in significant subnetworks reported by HotNet2.
Results of DAMOKLE analysis of esophagus tumours and stomach tumours and of diffuse gliomas. a Subnetwork S with significant (\(p<0.01\)) differential coverage in esophagus tumours vs stomach tumours (interactions from HINT+HI2012 network). b Fractions of samples with mutations in genes of S in esophagus tumours and in stomach tumours. c Subnetwork S with significant (\(p<0.01\)) differential coverage in LGG samples vs GBM samples (interactions from HINT+HI2012 network). d Fractions of samples with mutations in genes of S in LGG samples and GBM samples
Diffuse gliomas
We analyzed single nucleotide variants (SNVs) and copy number aberrations (CNAs) in 509 samples of lower grade glioma (LGG) and in 303 samples of glioblastoma multiforme (GBM).Footnote 7 We considered nonsilent SNVs, short indels, and CNAs. We removed from the analysis genes with \(<6\) mutations in both classes. By single gene analysis we identified IDH1 with high (\(>0.5\)) differential coverage, and removed alterations in such gene for the DAMOKLE analysis. We run DAMOKLE with \(k=5\) with \({\mathcal {C}}\) being the set of GBM samples and \({\mathcal {D}}\) being the set of LGG samples. The number of mutations in \({\mathcal {C}}\) and in D is not significantly different (t-test p = 0.1). DAMOKLE identifies the significant (\(p<0.01\) for both tests in "Permutation testing" section) subnetwork \(S=\) {CDKN2A, CDK4, MDM2, MDM4, RB1} (Fig. 3c, d). All genes in S are members of the p53 pathway or of the RB pathway. The p53 pathway has a key role in cell death as well as in cell division, and the RB pathway plays a crucial role in cell cycle control. Both pathways are well known glioma cancer pathways [40]. Interestingly, [41] did not report any subnetwork with significant difference in mutations between LGG and GBM samples. CDK4, MDM2, MDM4, and RB1 do not appear among the top-45 genes by differential coverage. We compared the results obtained by DAMOKLE with the results obtained by HotNet2. Of the genes in our subnetwork, only CDK4 and CDKN2A are reported in a significantly mutated subnetwork (\(p <0.05\)) obtained by HotNet2 analyzing \({\mathcal {D}}\) but not analyzing \({\mathcal {C}}\), while MDM2, MDM4, and RB1 are not reported in any significant subnetwork obtained by HotNet2.
In this work we study the problem of finding subnetworks of a large interaction network with significant difference in mutation frequency in two sets of cancer samples. This problem is extremely important to identify mutated mechanisms that are specific to a cancer (sub)type as well as for the identification of mechanisms related to clinical features (e.g., response to therapy). We provide a formal definition of the problem and show that the associated computational problem is NP-hard. We design, analyze, implement, and test a simple and efficient algorithm, DAMOKLE, which we prove identifies significant subnetworks when enough data from a reasonable generative model for cancer mutations is provided. Our results also show that the subnetworks identified by DAMOKLE cannot be identified by methods not designed for the comparative analysis of mutations in two sets of samples. We tested DAMOKLE on simulated and real data. The results on simulated data show that DAMOKLE identifies significant subnetworks with currently available sample sizes. The results on two large cancer datasets, each comprising genome-wide measurements of DNA mutations in two cancer subtypes, shows that DAMOKLE identifies subnetworks that are not found by methods not designed for the comparative analysis of mutations in two sets of samples.
While we provide a first method for the differential analysis of cohorts of cancer samples, several research directions remain. First, differences in the frequency of mutation of a subnetwork in two sets of cancer cohorts may be due to external (or hidden) variables, as for example the mutation rate of each cohort. While at the moment we ensure before running the analysis that no significant difference in mutation rate is present between the two sets, performing the analysis while correcting for possible differences in such confounding variable or in others would greatly expand the applicability of our method. Second, for some interaction networks (e.g., functional ones) that are relatively more dense than the protein–protein interaction network we consider, requiring a minimum connectivity (e.g., in the form of fraction of all possible edges) in the subnetwork may be beneficial, and the design of efficient algorithms considering such requirement is an interesting direction of research. Third, different types of mutation patterns (e.g., mutual exclusivity) among two set of samples could be explored (e.g., extending the method proposed in [42]). Fourth, the inclusion of additional types of measurements, as for example gene expression, may improve the power of our method. Fifth, the inclusion of noncoding variants in the analysis may provide additional information to be leveraged to assess the significance of subnetworks.
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FV designed the study. FV and EU designed the algorithms. FV and MCH implemented the software and performed the computational analysis. All authors interpreted the results. All authors wrote the manuscript. All authors read and approved the final manuscript.
This work is supported, in part, by University of Padova projects SID2017 and STARS: Algorithms for Inferential Data Mining and by NSF grant IIS-1247581. The results presented in this manuscript are in whole or part based upon data generated by the TCGA Research Network: http://cancergenome.nih.gov/.
Biotech Research and Innovation Centre, University of Copenhagen, Copenhagen, Denmark
Morteza Chalabi Hajkarim
Department of Computer Science, Brown University, Providence, RI, USA
Eli Upfal
Department of Information Engineering, University of Padova, Padova, Italy
Fabio Vandin
Correspondence to Fabio Vandin.
Hajkarim, M.C., Upfal, E. & Vandin, F. Differentially mutated subnetworks discovery. Algorithms Mol Biol 14, 10 (2019). https://doi.org/10.1186/s13015-019-0146-7
Somatic mutations
Differential analysis | CommonCrawl |
DFT-based low-complexity optimal cell ID estimation in NB-IoT
Vincent Savaux1
EURASIP Journal on Advances in Signal Processing volume 2020, Article number: 14 (2020) Cite this article
This paper deals with cell identifier (ID) estimation for narrowband-Internet of things (NB-IoT) system. It is suggested to transform the usual maximum likelihood (ML) estimator expression in order to highlight a discrete Fourier transform (DFT), which can be computed with fast algorithms. Therefore, the proposed method is a DFT-based low-complexity cell ID estimator that can be qualified as optimal in the ML sense. The principle is extended to the practical case where the channel is unknown and must be estimated. In this scenario, the concentrated likelihood function needs to be maximized, in which the ML channel estimate is a function of the unknown cell ID parameter. This operation only involves a few additional multiplications. Simulation results reveal that the performance of the proposed method actually matches the optimal one of the ML cell ID estimator. Furthermore, the technique is robust to residual frequency offset up to several hundreds of Hertz. We also show that the mean square error of channel estimation reaches its Cramér-Rao bound (CRB).
The Internet of things (IoT) market is growing rapidly as the number of applications increases in various domains such as industry, smart home, smart cities, and agriculture. Among the low-power wide area (LPWA) technologies allowing for long-range applications [1–3], narrowband-IoT (NB-IoT) is a promising solution as it is inherited from long-term evolution (LTE) [4–7]. Thus, similar to LTE, NB-IoT devices work in licensed frequency bands, occupying one resource block of the LTE system. Moreover, every device must be synchronized with an evolved node B (eNB) to connect the network. The synchronization process is carried out according to two main steps: the devices physically synchronize with the eNBs using the narrowband primary synchronization signal (NPSS) [8–11] and then seek the cell identifier (ID) of the neighboring eNBs through the narrowband secondary synchronization signal (NSSS). In this paper, we focus on the cell ID estimation process.
The cell ID can take 504 different values and is transmitted through the NSSS as a combination (multiplication) of a Zadoff-Chu (ZC) sequence and a Hadamard sequence. Few papers dealing with the cell ID estimation in NB-IoT have been proposed in the literature yet. In [12], the exhaustive maximum likelihood (ML) cell ID search is suggested, and computational simplifications are established in [13, 14]. The ML leads to a cross-correlations between the observations (i.e., the received NSSS) and all the possible combinations of ZC and Hadamard sequences. The presented reductions of complexity in [13, 14] are based on the fact that the Hadamard sequences composing the NSSS consist in ± 1 elements. As a consequence, all the multiplications of the NSSS observation by the Hadamard sequences do not need to be computed as it only changes the sign of the observation samples. Despite their advantageous reduction of complexity, the methods in [13, 14] are based on cross-correlation and do not take further advantage of the NSSS features to simplify the cell ID estimation.
In this paper, we suggest to rewrite the cross-correlation in the cell ID ML estimator as a discrete Fourier transform (DFT) of the observations. To do so, we first notice that the ZC sequence consists in complex exponential samples that can then be transformed and reformulated as the coefficients \(\omega ^{\text {kn}} = e^{-2 j \pi \frac {\text {kn}}{N}}\) of the DFT. It results that algorithms for fast computations of DFTs [15, 16] can be used, therefore reducing the complexity of the ML estimator. Furthermore, the suggested method is also compatible with other simplifications based on the property of Hadamard sequences [13, 14]. In addition, we deal with the practical case where the channel is unknown and must be jointly estimated with the cell ID. It is proved that the complex argument of the channel only needs to be estimated, as the modulus of the channel is not involved in the maximization of the likelihood function. Besides, we show that the cell ID can then be estimated by maximizing the so-called concentrated-likelihood function in which the channel argument is substituted by the estimated one and whose value only depends on the cell ID value. This operation only slightly increases the complexity of the proposed ML estimator. The simulation results show that the performance of the suggested method actually matches that of the exhaustive ML and that it remains accurate for residual frequency offset of several hundreds of Hertz. Moreover, it is verified through simulations that the mean square error (MSE) of the estimator of the channel phase reaches the Cramér-Rao bound (CRB).
The rest of the paper is organized as follows: Section 3 presents the NSSS reception model, including the exhaustive ML cell ID search. Section 4 introduces the suggested low-complexity ML estimator, and it is extended to the practical case where the channel needs to be estimated. The performance of the estimator is shown and discussed in Section 5, and Section 6 concludes this paper.
Notation: the vectors and matrices are written in boldface x and uppercase boldface X, respectively, and the scalars are written in normal font x. ∥.∥ represents the Euclidian norm, |.| the modulus (of complex scalar), and Re{.} the real part. Moreover, x∗, arg(.), and (.)H stand for the conjugate, the argument, and the Hermitian (complex conjugate) operators, respectively. The modulo is written mod, the mathematical expectation is denoted by \(\mathbb {E}\{.\}\), and ⊗ is the Kronecker product.
The aim of this study is to prove that the ML estimation of the cell ID in NB-IoT system can be carried out through a simple DFT without loss of performance compared with the exhaustive search. The paper includes a theoretical development that describes the DFT-based ML cell ID estimator and simulation results that show the relevance of the suggested method. The theoretical study does not require any specific material, and the simulations have been carried out with MatLab R2016a.
NSSS reception
This section describes the received NSSS, which is also detailed in [17]. The NSSS is composed of 11 OFDM symbols of 12 subcarriers, namely 11×12=132 resource elements. It is assumed that the time-frequency synchronization has been accurately performed thanks to the NPSS [8–11], in such way that the residual synchronization errors are negligible. Otherwise, the effect of the potential residual errors will be discussed in Section 5. Furthermore, since the signal is narrowband, it is reasonable to suppose that the frequency response of the channel is constant over the NSSS resource block of 12 subcarriers. Thus, after the cyclic prefix removal and the DFT over the 11 OFDM symbols, the received NSSS column vector y of size N=132 can be expressed as
$$ \mathbf{y} = \alpha\mathbf{d} + \mathbf{w}, $$
where α is the complex channel coefficient, w is the additive white Gaussian noise (AWGN) vector of size N×1 containing the samples \(w_{n} \sim \mathcal {CN}(0,\sigma ^{2})\), and d is the transmitted NSSS sequence, whose elements dn are defined in [17] as
$$ d_{n} = b_{q}(m)e^{-2j\pi\theta_{f} n}e^{-j\frac{\pi u n'(n'+1)}{131}}, $$
where {bq(m)∈{−1,1}} is one of the four Hadamard sequences defined in Table 10.2.7.2.1-1 in [17], which has been reproduced in Table 1. Moreover, we have
$$ \theta_{f} = \frac{1}{4}((\frac{n_{f}}{2})\mod 4), $$
Table 1 Definition of bq(m)
where nf is the frame number, which is always even. Furthermore, we have:
$$\begin{array}{*{20}l} &n' = n\mod 131 \\ &m = n \mod 128 \\ &u = N_{\text{ID}}^{\text{Ncell}}\mod 126 + 3 \\ &q = \lfloor \frac{N_{\text{ID}}^{\text{Ncell}}}{126} \rfloor, \end{array} $$
with \(N_{\text {ID}}^{\text {Ncell}}\) the cell ID of the eNB.
For a sake of clarity, we can rewrite (1) in a detailed form as follows:
$$ \mathbf{y} = \alpha\underbrace{\mathbf{B} \boldsymbol{\theta}}_{\mathbf{X}(q,\theta_{f})} \mathbf{e}(u) + \mathbf{w}, $$
where B,θ, and X(q,θf) are the N×N diagonal matrix composed of the elements \(\phantom {\dot {i}\!}b_{q}(m), e^{-2j\pi \theta _{f} n}\), and \(\phantom {\dot {i}\!}b_{q}(m)e^{-2j\pi \theta _{f} n}\), respectively. Note that the diagonal elements of X(q,θf) are taken from a finite set {±1,±j}. The N×1 vector e(u) contains the ZC sequence [18, 19] \(\phantom {\dot {i}\!}e_{n}(u)=e^{-j\frac {\pi u n'(n'+1)}{131}}\), where the parameter u is called the root of the sequence. In the following, the estimation of the cell ID results in finding the parameters (u,q) from the observation y, as the cell ID can be expressed from (4) as
$$ N_{\text{ID}}^{\text{Ncell}} = 126q + u -3. $$
It can be noticed that the estimation of the cell ID does not require that of the parameter θ, but it must be nevertheless properly estimated in order to unscramble the received NSSS sequence, i.e., remove the sequence θ.
Maximum likelihood cell ID estimation
The ML estimation of u, q, and θf leads to an exhaustive search of the optimal parameters through the likelihood function of complex observation y, denoted by L, which can be expressed as
$$ (\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max}} \underbrace{C e^{-\frac{1}{\sigma^{2}}\|\mathbf{y} - \alpha\mathbf{X}(q,\theta_{f}) \mathbf{e}(u)\|^{2}}}_{L}, $$
where C is a constant that can be omitted in the following developments since it does not depend on (u,q,θf). Since |Xn|2=1 and |en(u)|2=1 for any Xn∈{±1,±j}, we can develop (7) as
$$\begin{array}{*{20}l} (\hat{u},\hat{q},\hat{\theta}_{f}) =& \underset{u,q,\theta_{f}}{\text{arg\ max}} e^{-\frac{1}{\sigma^{2}}\|\mathbf{y}\|^{2}}e^{-\frac{|\alpha|^{2}N}{\sigma^{2}}} \\ &\times e^{\frac{2}{\sigma^{2}}\text{Re}\{\mathbf{y}^{H} \alpha\mathbf{X}(q,\theta_{f}) \mathbf{e}(u)\}}\\ =& \underset{u,q,\theta_{f}}{\text{arg\ max}} \text{Re}\{\mathbf{y}^{H} \alpha\mathbf{X}(q,\theta_{f}) \mathbf{e}(u)\}, \end{array} $$
which can be rewritten in a scalar form as
$$ (\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max }} \text{Re}\left\{ \sum_{n = 0}^{131} y_{n}^{*} \alpha X_{n}(q,\theta_{f}) e_{n}(u) \right\}. $$
The cross-correlation in (9) requires Nm=16×126×132=266112 complex multiplications, where 16 corresponds to q×θf possible values, and 126 corresponds to u possible values. Furthermore, it requires the a priori knowledge of the channel coefficient α or it must be estimated. In the following, we take advantage of the ZC sequence \(e_{n}(u)=e^{-j\frac {\pi u n'(n'+1)}{131}}\) to rewrite (9) as a DFT, therefore simplifying the process. Furthermore, the estimation of the channel is tackled as well.
Suggested low-complex ML cell ID estimation
This section presents the way to reformulate the ML estimator (9) into a simplest form using DFT. The principle of the method is first detailed in an ideal case where the channel is supposed to be known (or at least accurately estimated), and we then extend to a more general case where both the cell ID and the channel are estimated through ML using DFT formulation.
Ideal case: known channel
In order to reduce the complexity of the cell ID estimation, we can rewrite (9) by means of a DFT. For clarity purpose, we note \(z_{n} = y_{n}^{*} \alpha X_{n}(q,\theta _{f})\), then (9) can be expressed as
$$\begin{array}{*{20}l} (\hat{u},\hat{q},\hat{\theta}_{f}) &= \underset{u,q,\theta_{f}}{\text{arg max}} \text{Re}\left\{\sum_{n = 0}^{131} z_{n} e^{-\frac{j\pi u n'(n'+1)}{131} }\right\} \\ &= \underset{u,q,\theta_{f}}{\text{arg max}} \text{Re}\left\{\sum_{n = 0}^{131} |z_{n}| e^{j\arg(z_{n})} e^{-\frac{j\pi u n'(n'+1)}{131} }\right\}. \end{array} $$
To further develop the sum in (10), it should be noticed that n′(n′+1) is always even, for every even or odd n′ and can be expressed as
$$ n'(n'+1) = 2 \binom{n'+1}{2}. $$
To recognize a DFT, we define \(\tilde {n} = \binom {n'+1}{2} \mod 131\) such that the sum in (10) can be rewritten as
$$ \sum_{n = 0}^{131}|z_{n}| e^{j\arg(z_{n})}e^{-\frac{j\pi u n'(n'+1)}{131}} = \sum_{n = 0}^{131}|z_{n}| e^{j\arg(z_{n})}e^{-\frac{2j\pi u \tilde{n}}{131}}. $$
We show in Fig. 1\(\tilde {n} = n'(n'+1) \mod 131\) versus n, with n′=n mod 131. Thus, we can observe that \(\tilde {n}\) is symmetrical according to n=65, i.e., for any n1<65, there exists a 65<n2≤130 such as n1+n2=130 and \(\tilde {n}_{1} = \tilde {n}_{2}\). Moreover, the set \(\Omega _{n} = \{\tilde {n} = \binom {n'+1}{2} \mod 131|n=0,1,..,N-1\}\) is a subset of [ [0,131] ]. As a consequence, we can rearrange the elements \(|z_{n}| e^{j\arg (z_{n})}\) in order to highlight the DFT. To this end, we define pk as
$$ p_{k} = \left\{\begin{array}{ll} &|z_{0}| e^{j\arg(z_{0})} + |z_{130}| e^{j\arg(z_{130})} \\ &+ |z_{131}| e^{j\arg(z_{131})}, \text{if} k=\tilde{n}=0 \\ &|z_{n_{1}}| e^{j\arg(z_{n_{1}})} + |z_{n_{2}}| e^{j\arg(z_{n_{2}})}, \\ &\text{if} k = \tilde{n}_{1} \in \{\Omega_{n}\setminus\{0\}\}, n_{1}+n_{2}=130 \\ &0, \text{if\ k} \notin \Omega_{n} \end{array}\right.. $$
\(\tilde {n} = n'(n'+1) \mod 131\) versus n, with n′=n mod 131
Finally, from (12) and the rearrangement in (13), we obtain:
$$ \sum_{n = 0}^{131}|y_{n}| e^{j\arg(z_{n})}e^{-\frac{2j\pi u \tilde{n}}{131}} = \sum_{k = 0}^{130} p_{k} e^{-\frac{2j\pi u k}{131}}, $$
where we recognize the DFT of pk. By defining the vector p=[p0,p1,…,pN−1], the ML estimator of u, q, and θf can be simply expressed as
$$ (\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max }} Re\{DFT(\mathbf{p}) \}. $$
Figure 2 illustrates Re{DFT(p)} versus u, for a ZC sequence featuring a root u=7. We can see a peak at u=7, which proves the relevance of the suggested transformation. The direct computation of (15) using DFT has a complexity of 16×66×126=133056 complex multiplications, which is very similar to the exhaustive search in (9). This is because 131 is a prime number. However, this can be reduced by using an algorithm for fast calculation of prime DFT, such as FFTWFootnote 1, chirp-z algorithm [15], or generalized Goertzel algorithm [16]. Thus, considering the complexity analysis of the chirp-z algorithm [15] for instance, the computation of (15) requires 2 multiplications of size 132, one of size 256 (the first power of 2 larger than 132), and 2 FFTs of size 256. Therefore, the complexity of (15) is of order 16×4016=73856 where 16 correspond to the estimation of (u,θf) and 4016 to the chirp-z algorithm. This is of order four times less complex than the exhaustive search (9) and could be even simplified. In fact, it must be noted that the FFT is sparse, as only 66 samples over 131 are non-zero in the input vector p; therefore, numerous multiplications can be avoided. Furthermore, we can take advantage of the fact that Xn(q,θf)∈{±1,±j} to further simplify (15) since the 16 possible combinations (q,θf) can be tested with straightforward changes of signs in the different sums that compose the FFTs. The complexity could then be reduced by eight, namely 9232 multiplications, leading to a computation cost similar to that in [13, 14]. However, note that the different FFT implementations are out of the scope of this paper and then not further dealt with in this work. In the following, we extend the suggested solution to a more realistic case where the channel is unknown and must be estimated.
Re{DFT(p)} versus u, ZC sequence has root u=7
ML cell ID and channel estimation
From (9), we can notice that the substitution \(\phantom {\dot {i}\!}\alpha = |\alpha |e^{j\phi _{\alpha }}\) with ϕα the argument of α, yields:
$$ (\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max}} \text{Re}\left\{ \sum_{n = 0}^{131} y_{n}^{*} e^{j\phi_{\alpha}} X_{n}(q,\theta_{f}) e_{n}(u) \right\}, $$
since |α| is independent of (u,q,θf) and can then be removed from (9). We deduce that it is sufficient to focus on the estimation of ϕα to, in turn, estimate \((\hat {u},\hat {q},\hat {\theta }_{f})\). Thus, the ML estimation of ϕα can be expressed, after straightforward developments, as
$$\begin{array}{*{20}l} \hat{\phi}_{\alpha} &= \max_{\phi_{\alpha}} L \\ &= \max_{\phi_{\alpha}} \underbrace{\text{Re}\left\{ \sum_{n = 0}^{131} y_{n}^{*} e^{j\phi_{\alpha}} X_{n}(q,\theta_{f}) e_{n}(u) \right\}}_{f(\phi_{\alpha})}. \end{array} $$
Hence, solving \(\frac {\partial }{\partial \phi _{\alpha }} f(\phi _{\alpha }) = 0\) leads to
$$\begin{array}{*{20}l} &e^{2j\phi_{\alpha}} = \frac{\sum_{n = 0}^{131} y_{n} X_{n}^{*}(q,\theta_{f}) e_{n}^{*}(u)}{\sum_{n = 0}^{131} y_{n}^{*} X_{n}(q,\theta_{f}) e_{n}(u)} \\ \Rightarrow& \hat{\phi}_{\alpha} = \frac{1}{2}\arg \left(\frac{\sum_{n = 0}^{131} y_{n} X_{n}^{*}(q,\theta_{f}) e_{n}^{*}(u)}{\sum_{n = 0}^{131} y_{n}^{*} X_{n}(q,\theta_{f}) e_{n}(u)} \right) \\ \Leftrightarrow& \hat{\phi}_{\alpha} = \arg \left(\sum_{n = 0}^{131} y_{n} X_{n}^{*}(q,\theta_{f}) e_{n}^{*}(u) \right), \end{array} $$
since the denominator is the conjugate of the numerator. The CRB [20, 21] of this estimator is given by \({CRB} = \frac {\sigma ^{2}}{2|\alpha |^{2}N}\), such as proved in Appendix 6. In the following, we note the estimate \(\hat {\phi }_{\alpha }(u,q,\theta _{f})\) in order to highlight its dependency to the unknown discrete parameters. Furthermore, we can rewrite the ML estimator (16) by substituting the likelihood function L by the concentrated-likelihood one (including the estimate \(\hat {\phi }_{\alpha }(u,q,\theta _{f})\)), leading to
$$\begin{array}{*{20}l} {}(\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max}} \text{Re}\left\{ e^{j\hat{\phi}_{\alpha}(u,q,\theta_{f})} \sum_{n = 0}^{131} y_{n}^{*} X_{n}(q,\theta_{f}) e_{n}(u) \right\}. \end{array} $$
Interestingly, it must be noticed that the sum \(\sum _{n = 0}^{131} y_{n}^{*} X_{n}(q,\theta _{f}) e_{n}(u)\) is found in both (18) and (19). It can then be computed only once by using the DFT-based method leading to (15) with \(z_{n} = y_{n}^{*} X_{n}(q,\theta _{f})\) and (19) then becomes
$$ (\hat{u},\hat{q},\hat{\theta}_{f}) = \underset{u,q,\theta_{f}}{\text{arg\ max}} \text{Re}\left\{ e^{j\hat{\phi}_{\alpha}(u,q,\theta_{f})}DFT(\mathbf{p})\right\}. $$
The joint ML cell ID and channel estimation algorithm can be summarized as follows:
Compute DFT(p) with \(z_{n} = y_{n}^{*} X_{n}(q,\theta _{f})\) by using an algorithm such as the chirp-z transform. This step has the same complexity as that discussed after (15).
Estimate \(\hat {\phi }_{\alpha }(u,q,\theta _{f})\) in (18). The computation of \(\hat {\phi }_{\alpha }(u,q,\theta _{f})\) requires the ratio of the real and the imaginary parts of \(\sum _{n = 0}^{131} y_{n} X_{n}^{*}(q,\theta _{f}) e_{n}^{*}(u)\) to feed the arc-tangent function. The computational cost is then 16×126=2016 multiplications, corresponding to all possible (u,q,θf) values.
Estimate (u,q,θf) in (20). This also requires 2016 multiplications.
The general algorithm for joint cell ID and channel estimation then requires 4032 more complex multiplications than the ideal case where the channel is supposed to be known. This is a reasonable additional complexity compared to the ML, even in simplified implementations such as the suggested one or those presented in [13, 14]. Table 2 summarizes the achievable complexity of the suggested DFT-based cell ID estimator compared with [12–14], given in a number of complex multiplications. The indicated value 73856 is given considering the chirp-z algorithm for the computation of the DFT. In that case, the computation cost is of the same order as [13]. Moreover, if we consider further simplifications (sparse DFT and Xn(q,θf)∈{±1,±j}, not dealt with in this paper), the complexity could be reduced to 9232, i.e., of the same order as [14]. In addition, the channel estimation has been added as well, which is not considered in [12–14].
Table 2 Complexity comparison of the suggested DFT-based cell ID estimator with [12–14], in a number of complex multiplications
Unfortunately, we cannot theoretically predict the estimation performance in case of discrete unknown parameters (except in binary detection), such as done through CRB for continuous parameters. In fact, it is only tractable to asymptotically predict the probability of estimation errors, such as hereby presented. As a consequence, we will show the accuracy of the estimators (15) and (20) through simulations in Section 5.
Asymptotic error probability analysis
In this section, we carry out an asymptotic analysis of the probability of error of cell ID estimation for the ML estimators (9) and its DFT-based version (15). An error of cell ID estimation occurs if q, u, or both parameters are badly estimated. We denote by \(\mathcal {P}\) this probability, then it is expressed as
$$\begin{array}{*{20}l} \mathcal{P} =& \mathbb{P}(\hat{N}_{\text{ID}}^{\text{Ncell}} \neq N_{\text{ID}}^{\text{Ncell}}) \\ =& \mathbb{P}(\hat{u} = u \cap \hat{q} \neq q) + \mathbb{P}(\hat{u} \neq u \cap \hat{q} = q) \\ & + \mathbb{P}(\hat{u} \neq u \cap \hat{q} \neq q). \end{array} $$
We can reasonably assume that the events are independent (this assumption will be verified in Section 5), therefore \(\mathcal {P}\) simplifies to
$$\begin{array}{*{20}l} \mathcal{P} =& \mathbb{P}(\hat{u} = u)\mathbb{P}(\hat{q} \neq q) + \mathbb{P}(\hat{u} \neq u)\mathbb{P}(\hat{q} = q) \\ & + \mathbb{P}(\hat{u} \neq u)\mathbb{P}(\hat{q} \neq q) \\ =& \mathbb{P}(\hat{q} \neq q) + \mathbb{P}(\hat{u} \neq u)\mathbb{P}(\hat{q} = q). \end{array} $$
Unfortunately, the probabilities \(\mathbb {P}(\hat {q} \neq q)\) and \(\mathbb {P}(\hat {u} \neq u)\) (respectively \(\mathbb {P}(\hat {q} = q)\) and \(\mathbb {P}(\hat {u} = u)\)) do not have tractable expressions, as they involve multiple integrals of multivariate Gaussian distributions. However, we can straightforwardly obtain the upper bound of the probabilities in (22) when σ2 tends to +∞. In that case, the events \(\hat {q} = q\) and \(\hat {u} = u\) are equiprobable for any q and u, respectively. As a consequence, we have \({\lim }_{\sigma ^{2} \to +\infty } \mathbb {P}(\hat {q} \neq q) = \frac {3}{4}\) and \({\lim }_{\sigma ^{2} \to +\infty } \mathbb {P}(\hat {u} \neq u) = \frac {125}{126}\), yielding \({\lim }_{\sigma ^{2} \to +\infty } \mathcal {P} = \frac {503}{504}\). In fact, this corresponds to the probability of missing the cell ID in a random choice of the cell ID among the 504 possible values.
Simulations and discussion
Simulations results
The simulation results have obtained using MatLab, and 105 independent runs per point have been performed. The channel coefficient obeys a zero-mean complex Gaussian distribution with unitary variance. The signal to noise ratio (SNR) is defined as \(\text {SNR} = \frac {\mathbb {E}\{\|\alpha \mathbf {d}\|^{2}\}}{\mathbb {E}\{\|\mathbf {w}\|^{2}\}}\). In the simulations, the chirp-z algorithm [15] has been used, but we remind that other fast DFT techniques could be applied. Note that for simplicity matter, we use a descriptive shortcut in all the following comments. Thus, we mention the "ML cell ID estimation" whereas we refer to the ML estimations of (u,q,θf) (e.g., in (9) or (15)). The actual cell ID estimation is obtained from the estimate \((\hat {u},\hat {q})\) using (6), i.e., \(\hat {N}_{\text {ID}}^{\text {Ncell}} = 126\hat {q} + \hat {u} -3\).
Known channel
In Fig. 3, we compare the ML exhaustive cell ID search (9) and the suggested DFT-based implementation (15) through the probability of error of estimation versus SNR in the range [−15,−2] dB. We assume the ideal case where the channel is known. It can be verified that both trajectories match, as a very slight difference (<0.1 dB) can be observed. The latter can be due to the computation in transformed domains that are involved in the suggested algorithm. In fact, it can be seen for instance in Fig. 2 that Re{DFT(p)} is not null around the peak at u=7, which may lead to few errors in low SNR range. However, this result shows that the suggested implementation of the ML estimator does not change its performance as the difference is negligible. We can observe that the probability of estimation error dives below 10−3 for SNR values larger than – 7.5 dB, showing the robustness of the ML estimator.
Error probability versus SNR (dB) of cell ID ML estimator using (9) and the suggested DFT-based implementation (15). A known channel is assumed
Figure 4 shows \(\mathcal {P}\) versus SNR from – 20 to – 8 dB, the different probabilities of error developed in (21) and (22) and the corresponding asymptotes in very low SNR range. It can be observed that it is more likely to badly estimate both u and q than one of the parameter for any SNR value. Moreover, the asymptotes 0.744, 0.248, and 0.006 well fit \(\mathbb {P}(\hat {u} \neq u \cap \hat {q} \neq q), \mathbb {P}(\hat {u} \neq u \cap \hat {q} = q)\), and \(\mathbb {P}(\hat {u} = u \cap \hat {q} \neq q)\), respectively, in a very low SNR range. This shows that the assumption of independence of the events holds, as for instance
$$\begin{array}{*{20}l} {\lim}_{\sigma^{2} \to +\infty} \mathbb{P}(\hat{u} \neq u \cap \hat{q} \neq q) &= {\lim}_{\sigma^{2} \to +\infty} \mathbb{P}(\hat{u} \neq u) \mathbb{P}(\hat{q} \neq q) \\ &= \frac{125}{126}\times\frac{3}{4} = 0.744. \end{array} $$
Probabilities of error versus SNR (dB), \(\mathcal {P}\) and developments from (21), (22), and asymptotes
The assumption of independence of the events is also verified in Fig. 4 as we can see that (22) holds. In fact, the behaviors of \(\mathcal {P}\) and \(\mathbb {P}(\hat {q} \neq q) + \mathbb {P}(\hat {u} \neq u)\mathbb {P}(\hat {q} = q)\) exactly match for any SNR value.
Estimated channel
Figure 5 compares the probability of error of the suggested cell ID estimation versus SNR in the ideal cases where the channel is supposed to be known with the more practical case where the channel phase is estimated. It can be observed that the performance of the cell ID estimation where the channel is estimated is only 0.8 dB weaker than the ideal case. This shows that the substitution of the likelihood function by the concentrated-likelihood function only slightly degrades the performance and that the channel phase is accurately estimated in (18).
Error probability versus SNR (dB) of cell ID ML estimator using the suggested DFT-based implementation with channel estimation (20)
In order to confirm the previous statement, we show in Fig. 6 the MSE of the channel phase estimator versus SNR, where the MSE is defined as
$$ \text{MSE} = \mathbb{E}\{|\hat{\phi}_{\alpha} - \phi_{\alpha}|^{2}\}. $$
MSE of channel phase estimation using (18). Comparison with the CRB
Moreover, it is compared with the CRB previously defined. It can be seen in Fig. 6 that the MSE matches the CRB, the phase estimator is then optimal in the ML sense.
Performance of the method under residual frequency offset hypothesis
In this section, we consider the non-idealistic case where a residual frequency offset still remains after the physical synchronization stage thanks to NPSS. Thus, Fig. 7 shows the performance of the suggested ML cell ID estimator (20) (i.e., the channel phase is estimated) in the presence of a phase offset Δf taken in the set {100,200,400} Hz. We can observe in Fig. 7 that the estimation error behaviors corresponding to Δf=100,200, and 400 Hz experience performance losses of 0.1, 0.4, and 1.3 dB, respectively. These results show the robustness of the DFT-based ML cell ID estimator to the residual errors, in particular, if it is below 200 Hz.
Error probability versus SNR (dB) of the suggested DFT-based implementation (15) assuming residual frequency offsets Δf∈{100,200,400} Hz
Sub-optimal ML-based estimator
In order to avoid the channel phase estimation step, we suggest to rewrite (20) by substituting the real part by the modulus operator as follows:
$$\begin{array}{*{20}l} (\hat{u},\hat{q},\hat{\theta}_{f}) &= \underset{u,q,\theta_{f}}{\text{ arg max }} | e^{j\hat{\phi}_{\alpha}(u,q,\theta_{f})}DFT(\mathbf{p})\ | \\ &= \underset{u,q,\theta_{f}}{\text{ arg max }} | DFT(\mathbf{p})\ |. \end{array} $$
This new expression is referred as the "alternative form" of the ML estimator. It is supposed to be sub-optimal compared with the DFT-based ML method in (20), as it is not obtained from the maximization of the (concentrated) likelihood function. However, it is shown in Fig. 8 that the performance of the alternative form actually matches that of the DFT-based ML estimator. This could be due to the channel model we assumed: we considered an invariant channel over 12 subcarriers. Thus, taking the modulus removes the channel phase in the expression of the estimator. We draw from Fig. 8 that in very low frequency selectivity conditions, if the channel does not need to be estimated, then the alternative ML form (24) can be used without loss of performance compared with (20).
Error probability versus SNR (dB) of the suggested DFT-based implementation (15) compared with the alternative form
Performance in frequency selective channel
The channel was considered constant over the 180 kHz bandwidth in previous simulations. This is justified by the fact that it can be reasonably assumed that the channel is shorter than the cyclic prefix duration of 4.7 μs. It results that the coherence bandwidth of the channel is larger than \(\frac {1}{4.7\times 10^{-6}} \approx 212.7\) kHz, i.e., larger than the 180 kHz bandwidth of the NSSS signal. However, more realistic channel models should be considered, taking into account the slight frequency selectivity within the 180 kHz. To this end, a four taps channel h=[h0,h1,h2,h3] has been simulated, where the coefficients hi are zero-mean complex Gaussian variables (i.e., h is a Rayleigh channel) with the same variance \(\mathbb {E}\{|h_{i}|^{2}\} = \frac {1}{4}\). The delay between two taps has been set to 1 μs, such that the maximum delay is shorter than the cyclic prefix. Note that in that case, (1) should be rewritten to highlight the frequency selectivity. Thus, if the channel is supposed to be temporarily static over the 11 OFDM symbols, then (1) becomes
$$ \mathbf{y} = \bar{\mathbf{H}}\mathbf{d} + \mathbf{w}, $$
where \(\bar {\mathbf {H}}\) is a 132×132 diagonal channel matrix which can be expressed as \(\bar {\mathbf {H}} = \mathbf {I} \otimes \mathbf {H}\), with I the 11×11 identity matrix, and H the 12×12 diagonal matrix containing the channel frequency response [H0,H1,..,H11]. Therefore, each subcarrier is weighted by a complex coefficient \(\phantom {\dot {i}\!}H_{m} = r_{m} e^{j \phi _{m}}, m=0,1,..,11\).
Figure 9 shows the corresponding error probability versus SNR (dB) for both the suggested DFT-based implementation (15) compared with the alternative form. It can be observed that the probability of error is larger in Fig. 9 than in Fig. 8, due to the effect of the frequency selectivity. The error probability is even lower bounded for higher SNR values. Once again, both trajectories match, but this behavior is not further investigated in this paper. However, the advantage of the DFT-based implementation using the channel estimation compared with the modulus form is that it can be adapted to cope with frequency selective channels. Thus, the phase estimation (18) can be redefined for any subcarrier index m∈{0,1,..,11} as
$$ \hat{\phi}_{m} = \arg \left(\sum_{n = 0}^{10} y_{m+12n} X_{m+12n}^{*}(q,\theta_{f}) e_{m+12n}^{*}(u) \right), $$
Error probability versus SNR (dB) of the suggested DFT-based implementation (15) compared with the alternative form in a frequency selective channel
where the sum corresponds to the 11 OFDM symbols. The modulus rm could be also estimated similarly (knowing that ZC sequences have unitary modulus) and integrated in the concentrated-likelihood function to jointly estimate the channel coefficients Hm and the cell ID. However, this is not further detailed in this paper.
Possible use of the method in other applications
We hereby discuss the possible application of such a suggested method in other applications. As a general comment, it must be noted that the DFT-based algorithm can be used to solve any problem where the root u of a ZC sequence must be estimated from noisy observations y. For instance in LTE, ZC sequences are used to generate signals in both downlink and uplink. Thus, in downlink, the cell ID is split into two values: one is transmitted through the PSS generated with ZC sequences and the other one is transmitted through the SSS generated with binary sequences [17]. The PSS in LTE is then used for both physical synchronization and part of the cell ID estimation, by estimating the ZC root value u among {0,1,2}. The suggested technique could then be used to estimate u from PSS, but with some limitations in the presented form. First the reduction of complexity is effective when the set of possible ZC roots is large, which is not the case as u∈{0,1,2}. Second, the proposed method requires a prior physical synchronization, and it is not adapted to perform both synchronization and estimation yet.
In uplink-LTE [17] and 5G-new radio (5G-nR) [22], ZC sequences are also used by UEs to generate preambles that are transmitted to start the random access procedure, i.e., to inform the eNBs that UEs intend to access the network. To this end, each UE randomly chooses a preamble among a predefined list, and the corresponding ZC root u defines a random access preamble identifier (RAPID). This RAPID is then re-transmitted by the eNB to initiate contention resolution among UEs. The eNBs then require to properly estimate the root u, and the suggested method could therefore be used to accurately perform this step.
In this paper, we have presented a DFT-based low-complexity estimator of the cell ID in NB-IoT, which is optimal in the ML sense. In fact, the ML estimator leads to a cross-correlation between the observation and a ZC sequence defined as a complex exponential. Thus, by transforming and interchanging the samples of the observation, the cross-correlation can be rewritten as a DFT. It follows that an algorithm for fast computation of DFTs can be used, reducing the complexity of the ML estimator. This principle has been extended to the case where the channel must be estimated. The channel estimate is then introduced in the likelihood function, leading to the maximization of the concentrated likelihood function. This operation can be carried out with a slight additional complexity. The simulation results have shown that the proposed method indeed reaches the performance of the exhaustive ML cell ID search. Furthermore, other series of simulations have revealed that the DFT-based technique is robust to the residual frequency offset, as it experiences ≤0.5 dB loss up to 200 Hz offset. Further investigations will be undertaken in future works in order to adapt the estimator to frequency selective channels.
Appendix: cramér-Rao bound of ϕα estimator
The CRB [20, 21] is defined as
$$ \text{CRB} = -\mathbb{E} \left\{ \frac{\partial^{2} \ln(L)}{\partial \phi_{\alpha}^{2}} \right\}^{-1}, $$
where L is defined in (7), and then
$$\begin{array}{*{20}l} \frac{\partial^{2} \ln(L)}{\partial \phi_{\alpha}^{2}} &= - \frac{2}{\sigma^{2}}Re \left\{ \sum_{n = 0}^{131} y_{n}^{*} |\alpha| e^{j\phi_{\alpha}} X_{n}(q,\theta_{f}) e_{n}(u) \right\}. \end{array} $$
Since for any n=0,1,..,131, we have
$$\mathbb{E}\{Re\{w_{n}^{*} X_{n}(q,\theta_{f}) e_{n}(u)\}\}=0,$$
then we obtain \(CRB = \frac {\sigma ^{2}}{2|\alpha |^{2}N}\) where N=132.
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
see FFTW site http://www.fftw.org/
AWGN:
Additive white Gaussian noise
CRB:
Cramér-Rao bound
DFT:
FFT:
Fast fourier transform
IDFT:
Inverse discrete fourier transform
eDRX:
Extended discontinuous reception
eNB:
Evolved node B
IoT:
LMMSE:
Linear minimum mean square error
LPWA:
Low -power wide area
LTE:
ML:
Maximum likelihood
MSE:
Mean square error
NB-IoT:
Narrowband Internet of things
NPSS:
Narrowband primary synchronization signal
NSSS:
Narrowband secondary synchronization signal
OFDM:
Orthogonal frequency division multiplexing
ZC:
Zadoff-Chu
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IRT b <>com, Rennes, France
Vincent Savaux
The author carried out both the theoretical developments and the simulations. The author read and approved the final manuscript.
Correspondence to Vincent Savaux.
The author declares that he has no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Savaux, V. DFT-based low-complexity optimal cell ID estimation in NB-IoT. EURASIP J. Adv. Signal Process. 2020, 14 (2020). https://doi.org/10.1186/s13634-020-00677-4 | CommonCrawl |
Tips for accurately measuring reflection in an FDTD simulation
This page describes three different methods for measuring reflectance from a structure. Using a monitor placed behind the source is the common and simplest method. Placing the monitor in front of the source can improve the accuracy in situations where there are source injection errors. We also discuss an alternative technique using a monitor placed along the source injection axis.
Monitor behind the source
The most straightforward setup technique used for measuring reflections from a structure is to place a frequency domain monitor behind the source injection plane. The incident and reflected fields exist in front of the source, but only the reflected fields will be measured if the monitor is placed behind the source.
Note: Direction of power flow
If the net power flows in the negative direction, the transmission function will return a negative number. You may need to multiply by -1 if you want the these quantities to have positive values.
Source injection errors
An ideal source would create a beam that moves in the propagation direction, without any scattering to other directions. Of course, FDTD is a numerical technique and some level of numerical error must be expected. In most simulations, these numerical source injection errors are negligible, meaning the method of placing a monitor behind the source to measure reflections generally works well. However, in some simulations, the source injection errors can be significant. These issues are more likely to occur when using a source that injects at an angle, a mode source, or a broadband source.
To understand the effect this back scatter can have on the accuracy of reflection measurements, imagine that 1% of the injected field is backscattered at the injection plane and 10% is actually reflected from the structure. In this case, the power measured by a monitor behind the source will be $$P=(0.01+0.1)^2=0.0121$$ for a source amplitude of 1. In other words, for this case, the error in the measured reflectivity is 2.1% even though the backscattered field is only 1%. Interference effects have been the two fields tends to amplify the effect of any source injection errors
Monitor in front of the source
If source injection errors are a problem in your simulation, you can measure reflected power using a monitor in front of the source.
In the usr_relection_angled.fsp example file, a broadband plane wave source is injected with a center angle of 30 degrees. One power monitor is placed in front of the source and one is placed behind the source. By taking the reflection to be 1 minus the transmission from the monitor in front of the source, the reflection obtained from the monitor in front of the source is closer to the theoretical value than the reflection obtained using the transmission from the monitor behind the source. The usr_reflection_angled.lsf script plots the reflection using the two methods and the theoretical reflection.
Note: Monitor positions
In a lossless simulation, the net power transmission through a monitor in front of the source is independent of its placement. The transmission will be the same whether it is placed between the source and the reflecting surface or past the reflecting surface.
This technique for measuring reflection assumes that the injected power is normalized to 1. Source injection errors can also lead to errors in the power normalization. In such cases, it may be necessary to measure the actual power injected by running a reference simulation and then using this to re-normalize the results of your main simulation. To do this, use the monitor in front of the source to measure the forward propagating power in reference simulation that is setup exactly the same as the main simulation, but without the structure (which could lead to back reflections).
Line monitor interference technique
An alternative technique for measuring reflectivity is to use a line monitor placed along the source injection axis. This technique is not generally recommended because the analysis is more complex than the other methods, it will only work for cases where only a single mode exists, and the accuracy of the result is dependent on the spatial sampling rate.
In reflection_interference.fsp, a plane wave in a vacuum is normally incident on a dielectric structure with index n=3. Between the source and the material interface, the amplitude of E^2 oscillates due to interference between the incident and reflected fields.
We can use the amplitude of E^2 in the region between the source and the interface to determine the power reflected:
$$ I(x) = I_1(x) + I_2(x) + 2 \sqrt{I_1(x)I_2(x)} \ cos(\varphi_1 (x) - \varphi_2(x) ) $$
I(x) will oscillate with amplitude:
$$ A(x) = 2 \sqrt{I_{1\ peak} I_{2\ peak}} = 2r$$
for incident intensity 1 and reflectivity \(R=r^2\)
$$ R = \left( \frac{A}{2} \right) ^2 $$
The following plots shows the calculated R using this technique from usr_reflection_interference.fsp and the measured transmission through a monitor behind the source. Low spatial sampling rate is a problem here because we cannot accurately measure the amplitude of the oscillation. Using a higher mesh accuracy setting, or adding a mesh override region would improve the accuracy of the result.
usr_reflection_angled.fsp
usr_reflection_angled.lsf
usr_reflection_interference.fsp
Broadband Fixed Angle Source Technique (BFAST)
Defect on a metal surface
getsourceangle - Script command
Using the data visualizer and figure windows | CommonCrawl |
Extended similarity indices: the benefits of comparing more than two objects simultaneously. Part 2: speed, consistency, diversity selection
Ramón Alain Miranda-Quintana ORCID: orcid.org/0000-0003-2121-44491,
Anita Rácz ORCID: orcid.org/0000-0001-8271-98412,
Dávid Bajusz ORCID: orcid.org/0000-0003-4277-94813 &
Károly Héberger ORCID: orcid.org/0000-0003-0965-939X2
Despite being a central concept in cheminformatics, molecular similarity has so far been limited to the simultaneous comparison of only two molecules at a time and using one index, generally the Tanimoto coefficent. In a recent contribution we have not only introduced a complete mathematical framework for extended similarity calculations, (i.e. comparisons of more than two molecules at a time) but defined a series of novel idices. Part 1 is a detailed analysis of the effects of various parameters on the similarity values calculated by the extended formulas. Their features were revealed by sum of ranking differences and ANOVA. Here, in addition to characterizing several important aspects of the newly introduced similarity metrics, we will highlight their applicability and utility in real-life scenarios using datasets with popular molecular fingerprints. Remarkably, for large datasets, the use of extended similarity measures provides an unprecedented speed-up over "traditional" pairwise similarity matrix calculations. We also provide illustrative examples of a more direct algorithm based on the extended Tanimoto similarity to select diverse compound sets, resulting in much higher levels of diversity than traditional approaches. We discuss the inner and outer consistency of our indices, which are key in practical applications, showing whether the n-ary and binary indices rank the data in the same way. We demonstrate the use of the new n-ary similarity metrics on t-distributed stochastic neighbor embedding (t-SNE) plots of datasets of varying diversity, or corresponding to ligands of different pharmaceutical targets, which show that our indices provide a better measure of set compactness than standard binary measures. We also present a conceptual example of the applicability of our indices in agglomerative hierarchical algorithms. The Python code for calculating the extended similarity metrics is freely available at: https://github.com/ramirandaq/MultipleComparisons
Molecular similarity is a key concept in cheminformatics, drug design and related subfields [1, 2]. However, the quantification of molecular similarity is not a trivial task. Generally, binary fingerprints serve to define binary similarity (and distance) coefficients [3], which are routinely used in virtual screening [4], fragment-based de novo ligand design [5,6,7,8], hit-to-lead optimization [9], etc.
It is well- known that "the results of similarity assessment vary depending on the compound representation and metric" [10,11,12]. Willett carried out a detailed comparison of a large number of similarity coefficients and established that the "well-known Tanimoto coefficient remains the method of choice for the computation of fingerprint-based similarity" [13]. He also calculated multiple database rankings using a fixed reference structure and the rank positions were concatenated, in a process called "similarity fusion" [14]. On the other hand, Martin et al. have also called for attention that the "widely and almost exclusively applied Tanimoto similarity coefficient has deficiencies together with the Daylight fingerprints" [15]. If the compounds are selected using an optimal spread design, "the Tanimoto coefficient is intrinsically biased toward smaller compounds, when molecules are described by binary vectors with bits corresponding to the presence or absence of structural features" [16].
In our earlier investigations we could prove the equivalency of several coefficients [17], as well as identify a few alternatives to the popular Tanimoto similarity [18]. We have also dedicated a paper to develop an efficient mathematical framework to study the consistency of arbitrary similarity metrics [19]. It is also worth noting that Tanimoto and other metrics can also be applied to quantify field-based representations, like shape similarity [20].
Classically, we can estimate the diversity of a compound set with binary comparisons by calculating its full similarity matrix. Likewise, popular diversity selection algorithms require pre-calculating the full similarity matrix of the compound pool. While this is fine up until a certain size, the similarity matrix calculation scales quadratically with the number of molecules, O(N2), resulting in very long computation times for larger sets. Methods to speed up these routine calculations are therefore sought after.
To note, one major train of thought for cutting down on computation times began with the introduction of the modal fingerprint [21]. Modal fingerprints are consensus fingerprints that collect the common features of a compound set, which can later be used for comparing sets, or as queries for similarity screening. The concept was further developed by the Medina-Franco group, introducing database fingerprints [22] (DFP) and statistical-based database fingerprints [23] (SB-DFP), with more sophisticated mathematical backgrounds.
By contrast, we have set out to extend the notion of similarity comparisons from two molecules (objects) to many (n). In our companion paper, we introduced the full mathematical framework for a series of new similarity indices, which are applicable for multiple (or n-ary, as opposed to pairwise) comparisons with and without weighting alike [24]. This is also briefly summarized in the "Extended similarity indices—theory" section of this article.
Our work has some common roots with modal fingerprints and its successors, chiefly in looking for the bit positions that are common to a certain percentage of a compound database (which we term similarity counters here). However, instead of identifying a consensus fingerprint to provide a simplified representation of a large compound set, we use our approach to quantify its overall similarity, extending the concept of similarity from two to many (n) molecules. With this, we avoid any information loss that is inherent to modal fingerprints and their successors, while providing a way to quantify compound set similarity with an algorithm that scales as O(N).
Here we demonstrate the (i) speed superiority of the extended similarity coefficients i.e. how the new indices outperform their binary analogues; (ii) how the new indices are superior in diversity selection; (iii) the robustness of extended coefficients, when changing the coincidence threshold (γ, a continuous meta parameter), and their consistency with the standard binary similarity indices; (iv) the behavior of extended similarity indices as compactness measures on selected datasets; and (v) their utility in hierarchical clustering by providing novel linkage criteria.
Computational methods
Extended similarity indices—theory
The companion paper contains the theoretical description and detailed statistical characterization of the extended similarity indices [24]. Nonetheless, to the convenience of the reader, a brief summary is included here.
The extended (or n-ary) similarity indices calculate the similarity of a set of an arbitrary number (n) of objects (bitstrings, molecular fingerprints), instead of the usual pairwise comparisons. To achieve that, we have extended the existing mathematical framework of similarity metrics. Whereas in binary comparisons, we can count the number of positions with 1–1, 1–0, 0–1, or 0–0 coincidences (usually termed a, b, c and d, respectively), in extended comparisons, we have more counters with the general notation \({C}_{n(k)}\), meaning k occurrences of "on" (1) bits out of a total of n objects. Let us note that a and d encode features of similarity and b and c encode features of dissimilarity in pairwise comparisons (although considering 0–0 coincidences or d as similarity features is optional, as reflected in the definition of some of the most popular similarity metrics, including the Tanimoto index [17]). By analogy, the key concept of our methodology is to classify the larger number of counters \({C}_{n(k)}\) into similarity and dissimilarity counters with a carefully designed indicator that reflects the a priori expectation for the number of co-occurring 1 bits (coincidence threshold or γ). To construct the extended similarity metrics, we simply replace the terms a, b, c and d in the definition of binary metrics with the respective sums of 1-similarity (a), dissimilarity (b + c) and, if needed, 0-similarity (d) counters. As a result, we will have a single similarity value for our set of n objects. Optionally, we can apply a weighting scheme to express the greater contributions to similarity for those counters with a larger number of co-occurrences k. To note, all of our metrics are consistent with the "traditional" binary definitions, in that they reproduce the original formulas when n = 2. The Python code for calculating the extended similarity metrics is freely available at: https://github.com/ramirandaq/MultipleComparisons
Figure 1 is an illustrative visualization of the difference between the binary comparisons and n-ary comparisons with the example of five compounds.
Illustration for the extended similarity metrics versus binary comparisons. A large number of pairwise comparisons is not necessarily able to reveal essential similarities between multiple molecules, despite of the significantly more calculations
Datasets and fingerprint generation
In order to evaluate our extended similarity metrics in real-life scenarios, we have chosen to generate popular molecular fingerprints for compound sets of various sizes, selected based on different principles—and therefore representing different levels of average similarity. Specifically, molecules were selected from the Mcule database [25] of purchasable compounds (> 33 M compounds in total) either: (i) randomly, (ii) by maximizing their similarity, or (iii) by maximizing their diversity (the latter two were achieved with the LazyPicker algorithm implemented in the RDKit, maximizing the similarity or dissimilarity of the respective sets). A fourth principle for compound set selection was assembling molecule sets, where every molecule shares a common core scaffold. For reasons of practicality, this was achieved by selecting molecules randomly from the ZinClick database: a database of over 16 M 1,2,3-triazoles. [26, 27] To ensure that the small core scaffold (5 heavy atoms) attributes to a significant portion of the molecules, we imposed a constraint that only molecules with at most 15 heavy atoms in total were picked (thus, at least 33% of the basic structures of any two molecules were identical). The resulting sets were termed "random" (R), "similar" (S), "diverse" (D), and "triazole" (T), respectively. Duplicates were removed and from each SMILES entry, only the largest molecule was kept, thereby removing any salts. For each selection principle, compound sets of 10, 100, 1000, 10,000 and 100,000 molecules were generated. The sets were stored as SMILES codes, which were, in turn, used to generate MACCS [2] and Morgan [28] fingerprints, the latter with a radius of 4 and an addressable space (fingerprint length) of either 1024, 2048 or 4096 bits. For the compound set selection and fingerprint generation tasks detailed above, the RDKit cheminformatics toolkit was utilized [29]. In the following sections, we apply our newly introduced extended similarity metrics, and also traditional pairwise similarity calculations to quantify the similarities of the resulting sets and to characterize the behavior of the extended similarity metrics on molecule sets with varying size and overall level of similarity. For the clustering case study, two compound sets were collected from recent works, corresponding to two JAK inhibitor scaffolds (25 indazoles [30] and 7 pyrrolo-pyrimidines [31]). Preparation and fingerprints generation of these sets was carried out as detailed above.
Visualization of target-specific compound sets
To highlight the applicability of the new extended similarity indices in drug design and computational medicinal chemistry, we have compiled several datasets with ligands of specific, pharmaceutically relevant protein targets. Specifically, 500 randomly selected ligands were picked for two closely related oncotargets, Bruton's tyrosine kinase (BTK) and Janus kinase 2 (JAK2) and a structurally dissimilar therapeutic target, the β2 adrenergic receptor (ligands with an experimental IC50/EC50/Kd/Ki value of 10 µM or better were picked from the ChEMBL database after duplicate removal and desalting) [32, 33]. Additionally, a larger dataset of cytochrome P450 (CYP) 2C9 ligands (2965 inhibitors with a potency of 10 µM or better and 6046 inactive species) was downloaded from Pubchem Bioassay (AID 1851) [34]. Cytochrome P450 (CYP) enzymes are of key importance for drug metabolism and are therefore heavily studied in medicinal chemistry and drug design [35].
In order to visualize the mentioned datasets, we have generated their Morgan fingerprints (radius: 4, length: 1024) and projected the datasets to two dimensions with t-distributed stochastic neighbor embedding (t-SNE), [36] as implemented in the machine learning package Scikit-learn, [37] with the following settings: perplexity = 30, metric = 'jaccard', init = 'pca' (initial embedding), n_components = 2.
Time analysis
One of the biggest practical advantages of the extended similarity indices is that now we can calculate the overall similarity of a group of molecules much more efficiently than by using the traditional binary comparisons. At a heuristic level, when we have a set with N molecules and calculate its chemical diversity using binary comparisons, we first need to select all possible pairs of molecules; then, calculate the similarity of each pair, and finally average the result [38, 39]. There will be \(N(N-1)\left/ 2 \right.\) pairs i.e. O(N2) operations are to be performed. In other words, the time required to calculate the similarity of a set of molecules is expected to grow quadratically with the size of the set. On the other hand, if we use n-ary indices, we can compare all of the molecules at the same time, which we expect to scale linearly with the size of the system, that is, in O(N).
This can be easily seen in Fig. 2, where we show the different times required to compare datasets using binary or n-ary indices when we use MACCS fingerprints (the same trends are observed for the other fingerprint types, as shown in the Additional file 1: Sect. 1). Remarkably, following these trends, estimating the similarity of one million molecules takes 400 s with n-ary comparisons, and close to 190 years with binary comparisons.
Average time required to calculate the set similarity of the different datasets using MACCS fingerprints with binary (a) and n-ary (b) similarity indices
The speed gain provided by our indices means that we can quantify the similarity of sets with our new indices that are completely inaccessible by current methods, thus allowing us to apply the tools of comparative analysis to the study of more complex databases. This can prove key in the study of chemical diversity [40,41,42]. The remarkable efficiency of our indices can be exploited in many different scenarios. For instance, the standard way to compare two sets of molecules requires us first to determine the medoid of each set. Traditional algorithms can do this in O(N2) (if we want to exactly calculate the medoid), or in O(\(\frac{NlogN}{{\varepsilon }^{2}}\)) (if we want to estimate the medoid up to a given error ε). However, with our indices we can just directly compare both sets requiring only O(N) operations. We can directly apply our indices in diversity picking, or use them with novel linkage criteria in agglomerative clustering algorithms. We demonstrate the former in the next section, and the latter application in the "Clustering based on extended similarity indices" section.
Diversity selection
The key advantage of our method in diversity selection is that we can quantify the similarity of a set in O(N) while working with the complete representation of the data. One could think of doing this using self-organizing maps [43] (SOMs), or multidimensional scaling [44] based on different molecular descriptors or fingerprint types. However, these alternatives cannot quantify the diversity in an exact way, rather they are realizing a kind of clustering or mapping of the databases and visualize the differences in a heatmap or scatterplot (thus inevitably reducing the complexity of the initial data by representing it in an approximated way). Binary similarity metrics have also been extensively used in the past decades to quantify the overall similarity/diversity of a database, but they are not a viable option for larger databases due to their time-demanding calculation process. In this sense, our method produces a fast, accurate and superior measure of the diversity of a set.
Probably the most popular way to select a diverse set of molecules from a dataset makes use of the MaxMin algorithm: [45, 46].
If no compounds have been picked so far, choose the 1st picked compound at random.
Repeatedly, calculate the (binary) similarities between the already picked compounds and the remaining compounds in the dataset (compound pool). Select the molecule from the compound pool that has the smallest value for the biggest similarity between itself and the already selected compounds.
Continue until the desired number of picked compounds has been selected (or the compound pool has been exhausted).
The MaxSum diversity algorithm [47] is closely related to MaxMin, being also based on traditional binary similarity measures, but differing in the selection step:
Repeatedly, calculate the (binary) similarities between the already picked compounds and the compound pool. Select the molecule from the pool that has the minimum value for the sum of all the similarities between itself and the already selected compounds.
Inspired by these methods, here we propose a modified algorithm that directly attempts to maximize the dissimilarity between the selected compounds (we can call this the "Max_nDis" algorithm):
Repeatedly, given the set of compounds already picked Pn = \(\left\{{M}_{1},{M}_{2},\dots ,{M}_{n}\right\}\) select the compound M' such that the set \(\left\{{M}_{1},{M}_{2},\dots ,{M}_{n},{M}^{^{\prime}}\right\}\) has the minimum similarity (as calculated using one of our n-ary indices).
The key difference between these algorithms is a conceptual one: while in MaxMin and MaxSum a new compound is added by maximizing some local (in most cases binary) criterion; in our method, the new compounds are explicitly added by directly maximizing the diversity of the new set. Our method provides a more direct route to obtaining chemically diverse sets, because this is the direct criterion in our optimization. We can compare this conceptual difference to optimization algorithms that locate either a local minimum or the global minimum of the abstract space being investigated (with the latter usually being substantially slower). In this analogy, the Max_nDis algorithm would be similar to an optimization algorithm that locates the global minimum, but with the same speed as a local optimization algorithm (which would correspond to the MaxMin and MaxSum pickers).
To illustrate this, we have compared the MaxMin, MaxSum and Max_nDis algorithms for four types of fingerprints, four datasets with varying levels of similarity, and an additional, larger dataset of cytochrome P450 2C9 inhibitors. In all cases, we ran the algorithms several times (7), so we were able to sample several random initial starting points. We report the average of the similarities obtained these different runs, and also the corresponding standard deviations, which allow us to more clearly distinguish between the different algorithms. In our first test, 10, 20, 30, …, 90 diverse molecules were selected from the "random" (R) compound set of 100 molecules. Figure 3 shows the corresponding results in the case of different fingerprint types (MACCS, Morgan-1024, Morgan-2048 and Morgan-4096). In all cases, and even with a relatively small pool for picking (80–90 selected out of 100), the Max_nDis algorithm selected more diverse sets than MaxMin and MaxSum.
n-ary Jaccard-Tanimoto (JT) similarities of diverse sets, selected with the MaxMin (orange), MaxSum (blue), and Max_nDis (green) algorithms. Error bars correspond to standard deviations derived by seven random initialization
In the next step, we have selected 100 molecules from the larger (10,000 and 100,000 molecules) "random" (R), "similar" (S), "diverse" (D), and "triazole" (T) datasets with MaxMin, MaxSum, and our algorithm, as well. Figure 4 shows that Max_nDis was consistently superior to MaxMin and MaxSum. This was particularly outstanding for the datasets that were more diverse to start with ("random" and "diverse").
Finally, we have compared the selection algorithms for a larger dataset of cytochrome P450 2C9 inhibitors (2965). The results clearly show (Fig. 5), that diversity selection based on the extended similarity metrics was able to produce drastically more diverse sets of 10, 20, 30, …, 100 molecules.
The Max_nDis algorithm has the same time scaling as MaxMin and MaxSum, but routinely resulted in compound sets that are 2–3 times more diverse. The differences were, logically, smaller, when we have selected the molecules from a smaller pool (Fig. 3), but were especially striking for the CYP 2C9 dataset, where the smallest sets (10 and 20 molecules) could be selected with n-ary similarities below 0.03, and even for 100 selected compounds, this did not increase to 0.1 (vs. close to 0.4 for MaxMin and MaxSum). We can also observe that the overall similarity increases monotonically with the size of the selected set in case of the Max_nDis algorithm (unless the compound pool is nearly exhausted, e.g. > 80 compounds selected from 100, see Fig. 3), which is consistent with the fact that it is used as the direct objective of the picking itself.
n-ary indices: robustness and consistency
A key factor in the applicability of our new indices is their robustness, which we define as their ability to provide consistent results even when we modify some of the parameters used to calculate them, for instance, when we change the coincidence threshold (γ). Let us say that we have two molecular sets, A and B (both having the same number of elements), and an n-ary similarity index \({s}_{n}\). We can measure their set similarity using a given coincidence threshold, γ1, which we will denote by: \({s}_{n}^{\left({\gamma }_{1}\right)}\left(A\right)\), \({s}_{n}^{\left({\gamma }_{1}\right)}\left(B\right)\). Without losing any generality we can say that A is more similar than B, that is: \({s}_{n}^{\left({\gamma }_{1}\right)}\left(A\right)>{s}_{n}^{\left({\gamma }_{1}\right)}\left(B\right)\). Then, the results obtained using index sn will be robust, inasmuch this relative ranking does not change, if we pick another coincidence threshold, i.e. if for \({\gamma }_{2}\ne {\gamma }_{1}\) we also have \({s}_{n}^{\left({\gamma }_{2}\right)}\left(A\right)>{s}_{n}^{\left({\gamma }_{2}\right)}\left(B\right)\). Notice that we can write this property as:
$$\left[{s}_{n}^{\left({\gamma }_{1}\right)}\left(A\right)-{s}_{n}^{\left({\gamma }_{1}\right)}\left(B\right)\right]\left[{s}_{n}^{\left({\gamma }_{2}\right)}\left(A\right)-{s}_{n}^{\left({\gamma }_{2}\right)}\left(B\right)\right]>0$$
This is highly reminiscent of the consistency relationship for comparative indices [48, 49], and for this reason, from now on we will refer to this property as internal consistency.
In order to study the internal consistency of the extended indices, we focused on the similar (S) and triazole (T) datasets with 10, 100, 1000, and 10,000 molecules. In Fig. 6 we show an example of the non-weighted extended Faith (eFai) index (eFainw) using the MACSS fingerprints for different set sizes. We see that the T (blue) and S (green) lines never cross each other, which means that the relative rankings of these sets is preserved (in other words, this index is internally consistent under the present conditions for the sets T and S).
Set similarity calculated with the eFainw index for the different datasets and sizes considered using MACSS fingerprints. The abbreviations are resolved in the Appendix 1 and also in ref. [24]
A more quantitative measure of this indicator can be obtained by calculating the fraction of times that the relative rankings of the S and T sets were preserved. This simple measure (which we call the internal consistency fraction, ICF) allows us to quickly quantify the internal consistency of an index since we can readily identify a greater value with a greater degree of internal consistency (a value of 1 corresponds to a perfectly internally consistent index, as it was the case for the eFainw index shown in Fig. 6). The detailed results are presented in the Additional file 1: Section 2. It is reassuring to notice that many of the indices identified as best in the accompanying paper (like the eBUBnw and eFainw indices) provide the highest ICF values.
Another important measure of robustness is the consistency of the extended similarity metrics with the corresponding standard binary similarity indices. Given an n-ary index calculated with a coincidence threshold γ, \({s}_{n}^{\left(\gamma \right)}\), and a binary index \({s}_{2}\), they will be consistent if for any two sets A, B we have:
$$\left[{s}_{n}^{\left(\gamma \right)}\left(A\right)-{s}_{n}^{\left(\gamma \right)}\left(B\right)\right]\left[{s}_{2}\left(A\right)-{s}_{2}\left(B\right)\right]>0$$
To avoid confusion with the previously introduced internal consistency, we will refer to Eq. (2) as the external consistency. It is obvious that the external consistency indicates whether the n-ary and binary indices rank the data in the same way. It is thus natural to use sum of ranking differences (SRD) to analyze this property. Briefly, SRD is a statistically robust comparative method based on quantifying the Manhattan distances of the compared data vectors from an ideal reference, after rank transformation (a more detailed description of the method is included in the accompanying paper). If the reference in the SRD analysis is selected to be the binary results, then the indices will be externally consistent if and only if SRD = 0.
In Fig. 7 we show how the SRD changes for several indices when we vary the coincidence threshold. We selected sets with 300 molecules to allow us to explore a large number of coincidence thresholds. As it was the case for the internal consistency (Additional file 1: Table S1), here we see once again that the choice of fingerprint greatly impacts the consistency. Remarkably, the eJTnw index is particularly well-behaved if we use Morgan4 fingerprints, being externally consistent for the vast majority (142 out of 150) of the coincidence thresholds analyzed. This is reassuring, given the widespread use of the Jaccard-Tanimoto index [13, 16, 17].
SRD variation with the coincidence threshold for the eBUBnw, eFainw, and eJTnw indices over sets with 300 molecules for the MACSS, Morgan4_1024, Morgan4_2048, and Morgan4_4096 fingerprints
Analogously to the ICF, we can define an external consistency fraction, ECF for measuring the fraction of times that the SRD is zero for all the coincidence thresholds that could be analyzed for a given set of molecules. In other words, the ECF is an indication of how often the n-ary index ranks the data in exactly the same order as the binary indices (ECF values are summarized in Table S2). Once again it is comforting to see that many of the best indices with respect to our previous SRD and ICF analyses are also the best with respect to the ECF. The detailed results on external consistency are presented in the Additional file 1: Section 3, along with SRD-based comparisons of the consistency measures according to several factors, such as the applied fingerprints and the effect of weighting (Additional file 1: Section 4).
Extended similarity indices on selected datasets
Our indices can also be used to analyze several datasets, for instance: the 100-compound selections from the commercial libraries (random, diverse, similar, triazole, see "Datasets and fingerprint generation" section), as well as 500 randomly selected ligands for three therapeutical targets, and a larger dataset (9011 compounds) from the PubChem Bioassay dataset AID 1851, containing cytochrome P450 2C9 enzyme inhibitors and inactive compounds. We have applied t-distributed stochastic neighbor embedding (t-SNE) to visualize the sets in 2D (Fig. 7) and compiled the runtimes and average similarity values calculated with the binary and the non-weighted extended similarity metrics (where n was the total number of compounds, i.e. all compounds were compared simultaneously). The t-SNE plots were generated from Morgan fingerprints (1024-bit) and are provided solely to illustrate the conclusions detailed here. The three case studies correspond to distinct scenarios. For the commercial compounds, the sets selected by maximizing similarity, or fixing the core scaffold (triazole) clearly form more compact groups than the randomly picked compounds or the diverse set (Fig. 8a). The BTK and JAK2 inhibitors, and the β2 adrenergic receptor ligands form groups of similar compactness, with moderate overlap (Fig. 8b). The CYP 2C9 enzyme inhibitors and inactive compounds form loose and completely overlapping groups (Fig. 8c).
t-distributed stochastic neighbor embedding (t-SNE) of: (a) the sets of 100 compounds selected with the different selection methods, (b) sets of 500 ligands of different pharmaceutical targets, and (c) sets of cytochrome P450 2C9 ligands and inactive compounds from PubChem Bioassay 1851. The table summarizes the number of compounds in the sets, as well as computation times and average similarities (averaged over the 19 non-weighted similarity metrics, and, for the binary comparisons, also over all possible compound pairs)
The key results are summarized in the table in Fig. 8. This lists the n-ary similarities (averaged over 19 non-weighted n-ary similarity metrics) and the corresponding binary similarities (averaged over 19 non-weighted binary similarity metrics and over all pairs of compounds). We also present the computation times for all of the clusters in the t-SNE plots, so that the reader can match the quantitative information against the visual representation of the clusters. We wanted to highlight here the utility of the new n-ary metrics to quantify the overall similarity (or conversely, diversity) of compound sets. First, it is clear that the extended similarity metrics offer a tremendous performance gain, with total computation times as low as 2–3 s even for the largest dataset (9011 compounds). By contrast, computation times for the full binary distance matrices range from 1.2 min (100 compounds), to 34–36 min (500 compounds), and to 46 h (6046 compounds). Additionally, it is worth noting that the extended metrics offer a greater level of distinction in terms of the compactness of the sets, ranging from 0.521 (diverse set) to 0.831 (similar set) in the most illustrative case, compared to a range from 0.503 (diverse) to 0.614 (similar) for binary comparisons. While there is almost no distinction in the binary case between the BTK, JAK2 and β2 sets, a minimal distinction is still retained by the extended metrics (returning a noticeably higher similarity score for the slightly more compact group of β2 ligands). The same observation goes for the CYP 2C9 dataset, where the slightly greater coherence of the group of 2C9 inhibitors is reflected at the level of the second decimal place in the n-ary comparisons, but only third decimal place for the "traditional" binary comparisons. Moreover, for the binary calculations of the 2C9 inactive set (6046 compounds), a computer with 64 GB RAM was required to avoid running out of memory and even then, the calculation took almost 2 days to complete (this is contrasted to 3 s of runtime on a more modest machine for the n-ary comparisons). In summary, our indices are much better equipped to uncover the relations between the elements of large sets because they take into account all the features of all the molecules at the same time (while scaling much better than traditional binary comparisons).
Clustering based on extended similarity indices
The success of our indices in quantifying the degree of compactness of a set suggests that they can be also applied in clustering. Traditionally, the similarity or dissimilarity between clusters is given as a function based on binary distance metrics (i.e. reversed similarity), which are then used in a linkage criterion to decide which clusters (or singletons) should be merged in each iteration. The n-ary indices, on the other hand, provide an alternative route towards hierarchical agglomerative clustering: we measure the distance (or similarity) between two sets A and B by forming the set \(C=A\cup B\), and then calculating the similarity of all the elements of C using an n-ary index. The rest of the algorithm proceeds as usual, that is, combining at each step those clusters that are more similar to (or less distant from) each other. In this approach, the n-ary similarities effectively act as novel linkage criteria. To showcase the applicability of the new extended similarity metrics in clustering, we have implemented this new agglomerative clustering algorithm based on the extended Jaccard-Tanimoto index (eJT).
For illustrative purposes, we have collected two compound sets from recent works, corresponding to two distinct JAK inhibitor scaffolds (25 indazoles [30] and 7 pyrrolo-pyrimidines [31]). Figure 9 summarizes the results obtained by two "classical" clustering approaches (based on pairwise Tanimoto distances and the single and complete linkage rules), as well as the n-ary agglomerative clustering algorithm. It is clear that all three algorithms can distinguish between the two core scaffolds. Additionally, the comparison nicely highlights the difference in the train of thought for the n-ary similarity metrics: while classical agglomerative clustering approaches operate with pairwise linkages of smaller subclusters, the n-ary algorithm "builds up" the larger, coherent clusters step by step, thereby providing a more compact visual representation for the larger groups. In other words, the n-ary indices allow us to analyze the data from a different perspective, thus facilitating to uncover other relations between the objects being studied. It is important to remark that this is merely a proof-of-principle example of the application of our indices to the clustering problem. Uncovering the general characteristics of n-ary clustering and further ideas for algorithms need to be further explored in more detail (we are currently working on this direction and the corresponding results will be presented elsewhere).
a The two core scaffolds of the JAK inhibitor dataset: pyrrolo-pyrimidine (orange) and indazole (green). b–d Results of agglomerative clustering with the n-ary Jaccard-Tanimoto metric (b), and the binary JT metric with single linkage (c) and binary JT metric with complete linkage (d)
Conclusions and summary
In the companion paper, we have introduced a full mathematical framework for extended similarity metrics, i.e. for quantifying the similarities of an arbitrary number (n) of molecular fingerprints (or other bitvector-like data structures). Here, after briefly reiterating the core ideas, we show the practical advantages and some prospective applications for the new similarity indices.
First, the calculation of extended similarity indices is drastically faster (more efficient) than the traditional binary indices used so far, scaling linearly with the number of compared molecules, as opposed to the quadratic scaling of calculating full similarity matrices with binary comparisons. To note, calculating the n-ary similarity of a set of ~ 6000 compounds took three seconds on a standard laptop, while calculating the binary similarity matrix for the same set took almost two days on a high-end computer.
An important prospective application for the new similarity indices is diversity picking. Here, our Max_nDis algorithm based on the extended Tanimoto index consistently selected much more diverse sets of molecules than currently used algorithms. The reason for this is that the Max_nDis algorithm directly maximizes the diversity (minimizes the n-ary similarity) of the selected dataset at each step, while traditional approaches like the MaxMin and MaxSum algorithms individually evaluate the similarities of the next picked compound to the members of the already picked set. It is noteworthy that this result is achieved without increasing the computational demand of the process.
Clustering, as another prospective field of application, showcases the different train of thought behind the agglomerative clustering algorithm we implemented based on the extended Tanimoto similarity, "building up" the larger, more coherent clusters step by step, rather than linking/merging smaller subclusters. Here, implications for further variations of clustering algorithms are wide, and we plan to extend upon this work in the close future.
Further on, we have demonstrated several important features of the new metrics: they are robust or "internally consistent" for different coincidence threshold settings. On the other hand, not all of them are consistent with their binary counterparts in terms of how they rank different datasets (external consistency); this is also influenced by the fingerprint used. Based on these results, a subset of the metrics can be preferred (this includes the extended Jaccard-Tanimoto index), this is detailed in the Supplementary Information. We have also provided visual examples that showcase the capacity of the new indices to distinguish between compact and more diffuse clusters of molecules.
The extended similarity indices provide a new dimension to the comparative analysis, giving us great flexibility at the time of comparing groups of molecules. Now, in this contribution we have shown that these indices are not only attractive from a theoretical point of view, but extremely convenient in practice. This combination of flexibility and unprecedented computational performance is extremely appealing and will allow us to explore the chemical space in novel, more efficient ways.
Python code for calculating the extended similarity metrics is freely available at: https://github.com/ramirandaq/MultipleComparisons
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The authors are indebted to the editor (Rajarshi Guha), for his suggestions for improving the manuscript, particularly to complete an analysis illustrating the superior performance of the extended indices in diversity picking.
National Research, Development and Innovation Office of Hungary (OTKA, contract K134260 and PD134416): AR, DB, KH. University of Florida: startup grant: RAMQ. Hungarian Academy of Sciences: János Bolyai Research Scholarship: DB.
Department of Chemistry, University of Florida, Gainesville, FL, 32603, USA
Ramón Alain Miranda-Quintana
Plasma Chemistry Research Group, Research Centre for Natural Sciences, Magyar tudósok krt. 2, 1117, Budapest, Hungary
Anita Rácz & Károly Héberger
Medicinal Chemistry Research Group, Research Centre for Natural Sciences, Magyar tudósok krt. 2, 1117, Budapest, Hungary
Dávid Bajusz
Anita Rácz
Károly Héberger
RAM-Q: theory, conceptualization, derivation, mathematical proofs, software, writing. DB: conceptualization, software, reading, writing. AR: conceptualization, calculations, methodology, writing. KH: conceptualization, calculations, validation, statistical analysis, funding acquisition, writing. All authors read and approved the final manuscript.
Correspondence to Ramón Alain Miranda-Quintana or Károly Héberger.
The authors declare no financial interest.
Part 1 is available at: https://doi.org/10.1186/s13321-021-00505-3
: Figure S1: Average time required to calculate the set similarity of the different datasets using Morgan4 fingerprints with binary similarity indices. Figure S2: Average time required to calculate the set similarity of the different datasets using Morgan4 fingerprints with n-ary similarity indices. Table S1: Average internal consistency fractions over sets with 10, 20, …, 300 molecules of all the extended similarity indices for all fingerprint types. Table S2: Average external consistency fractions over sets with 10, 20,…, 300 molecules of all the extended similarity indices for all fingerprint types. Figure S3: SRD analysis for the internal (i) and external (e) consistencies over the different fingerprint types. Figure S4: Effect of internal (i) and external consistency (e) on the extended multiple similarity indices. Notation can be found in Appendix 1, and also in the accompanying paper.4. Figure S5: Effect of weighting on the extended multiple similarity indices. Figure S6: Joint effect of internal and external consistency as well as weighting on the extended multiple similarity indices.
Extended n-ary similarity indices.
Additive indices
eAC eAC_1 eACw Extended
Austin-Colwell \({s}_{eAC\left(1s\_wd\right)}=\frac{2}{\pi }\mathrm{arcsin}\sqrt{\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}}\)
eACnw \({s}_{eAC\left(1s\_d\right)}=\frac{2}{\pi }\mathrm{arcsin}\sqrt{\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}}\)
eBUB eBUB_1 eBUBw Extended
Baroni-Urbani-Buser \({s}_{eBUB\left(1s\_wd\right)}=\frac{\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]}+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\left\{\begin{array}{c}\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\right]}+\\ \sum_{1-s}{f}_{s}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}+\sum_{d}{f}_{d}\left({\Delta }_{n\left(k\right)}\right){C}_{n\left(k\right)}\end{array}\right\}}\)
eBUBnw \({s}_{eBUB\left(1s\_d\right)}=\frac{\sqrt{\left[\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]\left[\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right]}+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\left\{\sqrt{\left[\sum_{1-s}{C}_{n\left(k\right)}\right]\left[\sum_{0-s}{C}_{n\left(k\right)}\right]}+\sum_{1-s}{C}_{n\left(k\right)}+\sum_{d}{C}_{n\left(k\right)}\right\}}\)
eCT1 eCT1_1 eCT1w Extended
Consonni-Todeschini (1) \({s}_{eCT1\left(1s\_wd\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\)
eCT1nw \({s}_{eCT1\left(1s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
Consonni-Todeschini (2) \({s}_{eCT2\left(1s\_wd\right)}=\frac{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\mathrm{ln}\left(1+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\)
eCT2nw \({s}_{eCT2\left(1s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\mathrm{ln}\left(1+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
eFai eFai_1 eFaiw extended
Faith \({s}_{eFai\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+0.5\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eFainw \({s}_{eFai\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+0.5\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eGK eGK_1 eGKw Extended
Goodman–Kruskal \({s}_{eGK\left(1s\_wd\right)}=\frac{2\mathrm{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\mathrm{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eGKnw \({s}_{eGK\left(1s\_d\right)}=\frac{2\mathrm{min}\left(\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)},\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)-\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\mathrm{min}\left(\sum_{1-s}{C}_{n(k)},\sum_{0-s}{C}_{n(k)}\right)+\sum_{d}{C}_{n(k)}}\)
eHD eHD_1 eHDw Extended
Hawkins-Dotson \({s}_{eHD\left(1s\_wd\right)}=\frac{1}{2}\left(\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\end{array}\right)\)
eHDnw \({s}_{eHD\left(1s\_d\right)}=\frac{1}{2}\left(\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{0-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\end{array}\right)\)
eRT eRT_1 eRTw Extended
Rogers-Tanimoto \({s}_{eRT\left(1s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eRTnw \({s}_{eRT\left(1s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\)
eRG eRG_1 eRGw Extended
Rogot-Goldberg \({s}_{eRT\left(1s\_wd\right)}=\begin{array}{c}\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}+\\ \frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\end{array}\)
eRGnw \({s}_{eRT\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}+\frac{\sum_{0-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{0-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eSM eSM_1 eSMw Extended
Simple matching, Sokal-Michener \({s}_{eSM\left(1s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eSMnw \({s}_{eSM\left(1s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eSS2 eSS2_1 eSS2w Extended
Sokal-Sneath (2) \({s}_{eSS2\left(1s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eSS2nw \({s}_{eSS2\left(1s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
Asymmetric indices
Label Type Name Equation
Consonni-Todeschini (3) \({s}_{eCT3\left(1s\_wd\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\)
eCT3nw \({s}_{eCT3\left(1s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
eCT3_0 eCT30w \({s}_{eCT3\left(s\_wd\right)}=\frac{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\)
eCT30nw \({s}_{eCT3\left(s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
Consonni-Todeschini (4) \({s}_{eCT4\left(1s\_wd\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}\)
eCT4nw \({s}_{eCT4\left(1s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
eCT4nw \({s}_{eCT4\left(s\_d\right)}=\frac{\mathrm{ln}\left(1+\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}\right)}{\mathrm{ln}\left(1+\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}\right)}\)
eGle eGle_1 eGlew Extended
Gleason \({s}_{eGle\left(1s\_wd\right)}=\frac{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eGlenw \({s}_{eGle\left(1s\_d\right)}=\frac{2\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eGle_0 eGle0w \({s}_{eGle\left(s\_wd\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eGle0nw \({s}_{eGle\left(s\_d\right)}=\frac{2\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{2\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eJa eJa_1 eJaw Extended
Jaccard \({s}_{eJa\left(1s\_wd\right)}=\frac{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eJanw \({s}_{eJa\left(1s\_d\right)}=\frac{3\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eJa_0 eJa0w \({s}_{eJa\left(s\_wd\right)}=\frac{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eJa0nw \({s}_{eJa\left(s\_d\right)}=\frac{3\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{3\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eRR eRR_1 eRRw Extended
Russel-Rao \({s}_{eRR\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eRRnw \({s}_{eRR\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eRR_0 eRR0w \({s}_{eRR\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eRR0nw \({s}_{eRR\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eSS1 eSS1_0 eSSw Extended
Sokal-Sneath (1) \({s}_{eSS1\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eSSnw \({s}_{eSS1\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\)
eSS1_1 eSS0w \({s}_{eSS1\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+2\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eSS0nw \({s}_{eSS1\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+2\sum_{d}{C}_{n(k)}}\)
eJT eJT_1 eJTw Extended
Jaccard-Tanimoto \({s}_{eJT\left(1s\_wd\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eJTnw \({s}_{eJT\left(1s\_d\right)}=\frac{\sum_{1-s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{1-s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
eJT_0 eJT0w \({s}_{eJT\left(s\_wd\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}+\sum_{d}{f}_{d}\left({\Delta }_{n(k)}\right){C}_{n(k)}}\)
eJT0nw \({s}_{eJT\left(s\_d\right)}=\frac{\sum_{s}{f}_{s}\left({\Delta }_{n(k)}\right){C}_{n(k)}}{\sum_{s}{C}_{n(k)}+\sum_{d}{C}_{n(k)}}\)
Miranda-Quintana, R.A., Rácz, A., Bajusz, D. et al. Extended similarity indices: the benefits of comparing more than two objects simultaneously. Part 2: speed, consistency, diversity selection. J Cheminform 13, 33 (2021). https://doi.org/10.1186/s13321-021-00504-4
Computational complexity
Extended similarity indices
Molecular fingerprints
Sum of ranking differences | CommonCrawl |
thermodynamics chemistry questions and answers
$\Delta_{a} H^{\circ}(C)=715.0 \mathrm{kJ} \mathrm{mol}^{-1}$ $-92380=\Delta U-4955$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. An exothermic reaction $X \rightarrow Y$ is spontaneous in the back direction. SHOW SOLUTION (ii) $\quad \Delta S=+v e$ because aqueous solution has more disorder than solid. to do mechanical work as burning of fuel in an engine, provide electrical energy as in dry cell, etc. $-2050 k J=3312 k_{0} J+694 k J+5 B_{O=0}-4446 k J-3712 k_{\circlearrowright} J$ Q. Reaction of combustion of octane: $q=125 g \times 4.18 J / g \times(286.4-296.5)$ Given that power $=\frac{\text { Energy }}{\text { time }} \Rightarrow$ time $=\frac{\text { energy }}{\text { power }}$ Given that $S_{m}^{\circ} C(\text { graphite })=5.74 J K^{-1} m o l^{-1}$ (i) $\quad \Delta S\Delta H)$, Q. Here is a list of Thermodynamics MCQs with Answers (Multiple Choice Questions) is given below. Q. Silane $\left(S i H_{4}\right)$ burns in air as: $S(C a(s))^{\circ}=41.42 \quad J K^{-1} m o l^{-1}$ $=\left(8.93 \mathrm{kJ} K^{-1}\right) \times(6.73 \mathrm{K})=60.0989 \mathrm{kJ}$ The students those are preparing for the Govt. $N_{2}(g)+3 H_{2}(g) \rightarrow 2 N H_{3}(g)$ SHOW SOLUTION $=53.28 \mathrm{JK}^{-1} \times 20 \mathrm{K}=1065.6 \mathrm{V}$ or $1.065 \mathrm{kJ}$, Here, $C_{m}=24.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1} ; n=\frac{60}{27}=2.22 \mathrm{mol}$, \[ c=2.22 \mathrm{mol} \times 24.0 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}=53.28 \mathrm{JK}^{-1} \], Now, $q=53.28 \mathrm{JK}^{-1} \times \Delta T$, $=53.28 \mathrm{JK}^{-1} \times 20 \mathrm{K}=1065.6 \mathrm{V}$ or $1.065 \mathrm{kJ}$, Q. ( } i v)$, (i) $\quad \mathrm{CH}_{3} \mathrm{OH}(l)+\frac{3}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}(l)$, $\Delta_{r} H^{\circ}=-726 \mathrm{kJmol}^{-1}$, $C_{(G r a p h i t e)}+2 H_{2}(g)+\frac{1}{2} O_{2}(g) \longrightarrow C H_{3} O H(l)$, $\Delta_{r} H^{\circ}=-239 k J m o l^{-1}$, Q. This enthalpy change corresponds to breaking four $C-C l$ bonds $E=\frac{3}{2} R T$ Mono-atomic gas. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. For an ideal gas, from kinetic theory of gases, the average kinetic energy per mole $\left(E_{k}\right)$ of the gas at any temperature $T K,$ is given by $E_{k}=3 / 2 R T$ Also calculate enthalpy of solution of ammonium nitrate. Q. SHOW SOLUTION $\Delta_{a} H^{\circ}\left(C l_{2}\right)=242 k J m o l^{-1}$ For the water gas reaction : (ii) $\quad H C l$ is added to $A g N O_{3}$ solution and precipitate of $A g C l$ is obtained. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. (iv) $\quad \Delta S>O$. $-$ (i) $\quad S=+v e$ because liquid changes to more disordered gaseous state. Red phosphorus reacts with liquid bromine as: Standard enthalpy of vapourisation of benzene at its boiling point is $30.8 \mathrm{kJ} \mathrm{mol}^{-1} .$ For how long would a $100 \mathrm{Welectric}$ heater have to operate in order to vapourise $100 \mathrm{g}$ of benzene at its boiling point. Formula sheet. The temperature falls from $296.5 \mathrm{Kto} 286.4 \mathrm{K.Find}$ out the value of $q$ for calorimeter and its contents. In what way internal energy is different from enthalpy ? ©2020 24houranswers.com. SHOW SOLUTION $\Delta G=\Delta H-T \Delta S=(+)-T(+)$ (iii) A partition is removed to allow two gases to mix. $2 \Delta_{f} G^{\circ}\left(O_{2}\right)$ $C(\text { graphite })+2 H_{2}(g)+\frac{1}{2} O_{2}(g) \longrightarrow C H_{3} O H(l)$ The equilibrium constant for a reactions is $10 .$ What will be the value of $\Delta G^{\circ} ?$ $C(g)+O_{2}(g) \rightarrow C O_{2}(g) ; \Delta_{r} H^{\circ}=-393 k J m o l^{-1}$ (i) At what temperature the reaction will occur spontaneously from left to right? We'll send you an email right away. SHOW SOLUTION Therefore, $C_{v}=3 / 2 R$, At $(T+1) K,$ the kinetic energy per mole $\left(E_{k}\right)=3 / 2 R(T+1)$ Therefore, increase in the average kinetic energy of the gas for $1^{\circ} \mathrm{C}(\text { or } 1 \mathrm{K})$ rise in temperature $\Delta E_{k}=3 / 2 R(T+1)-3 / 2 R T=3 / 2 R$, When the gas is heated to raise its temperature by $1^{\circ} \mathrm{C},$ the increase in its internal energy is equal to the increase in kinetic energy, i.e., $\Delta U=\Delta E_{K}$, Now $C_{v}=\frac{\Delta U}{\Delta T}$ and $\Delta T=1^{\circ} \mathrm{C}$, Q. Normal response time: Our most experienced, most successful tutors are provided for maximum expertise and reliability. $\Delta H^{\circ}=-168.0 \mathrm{kJ} \mathrm{mol}^{-1}$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Why is $\Delta E=0,$ for the isothermal expansion of ideal gas? One should spend 1 hour daily for 2-3 months to learn and assimilate Thermodynamics … (ii), (ii) $C_{( \text {graphite) } }+O_{2}(g) \rightarrow C O_{2}(g) ; \Delta_{r} H^{\circ}=-393 \mathrm{kJ} \mathrm{mol}^{-1}$, (iii) $\times 2: 2 H_{2}(g)+O_{2}(g) \rightarrow 2 H_{2} O(l) ; \Delta_{r} H^{o}=-572 k J m o r^{1}$, (iv) $\quad C(g)+2 H_{2}(g)+2 O_{2}(g) \longrightarrow C O_{2}(g)+2 H_{2}^{\circ} O$, $\Delta_{r} H^{\circ}=-965 \mathrm{kJmol}^{-1}$, Subtract eq. $\therefore \quad \Delta H=+22.2 k_{0} J$ Specific heat of $L i(\mathrm{s}), N a(\mathrm{s}), K(s), R b(s)$ and $C s(s)$ at $398 K$ are $3.57,1.23,0.756,0.363$ and $0.242 \mathrm{Jg}^{-1} \mathrm{K}^{-1}$ respectively. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden. \[ \therefore \Delta U=q+w=0+w_{a d}=w_{a d} \]. Standard enthalpy of vapourisation of benzene at its boiling point is $30.8 \mathrm{kJ} \mathrm{mol}^{-1} .$ For how long would a $100 \mathrm{Welectric}$ heater have to operate in order to vapourise $100 \mathrm{g}$ of benzene at its boiling point. Therefore, the reaction will be spontaneous above $2484.8 \mathrm{K}$ (ii) Let us now calculate the $\Delta G$ value for reduction of $P b O$ In what way is it different from bond enthalpy of diatomic molecule ? $\Delta n_{g}=2-(1+3)=-2 m o l, T=298 K$ Here you can get Class 11 Important Questions Chemistry based on NCERT Text book for Class XI.Chemistry Class 11 Important Questions are very helpful to score high marks in board exams. We respect your privacy. – oxygen bond in $\mathrm{O}_{2}$ molecules. (ii) If work is done by the system, internal energy will decrease. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Let us calculate $T$ at which $\Delta_{r} G^{\circ}$ becomes zero, $\Delta_{r} G^{\circ}=\Delta_{r} H^{\circ}-T \Delta_{r} S^{\circ}=0$, $\therefore \quad T=\frac{\Delta_{r} H}{\Delta_{r} S}$, $=\frac{491.18 \mathrm{kJ} \mathrm{mol}^{-1}}{197.67 \times 10^{-3} \mathrm{kJ} \mathrm{mol}^{-1} \mathrm{K}^{-1}}=2484.8 \mathrm{K}$, Therefore, the reaction will be spontaneous above $2484.8 \mathrm{K}$, $\left(\text { or } 2211.8^{\circ} \mathrm{C}\right)$, Q. $q=125 g \times 4.18 \times 10^{-3} \mathrm{kJ} / \mathrm{g} \times-10.1=-5.28 \mathrm{kJ}$, $q=125 g \times 4.18 J / g \times(286.4-296.5)$, $q=125 g \times 4.18 \times 10^{-3} \mathrm{kJ} / \mathrm{g} \times-10.1=-5.28 \mathrm{kJ}$, Q. $\Delta S_{v a p . $=2.303\left(18 \mathrm{cal} \mathrm{K}^{-1} \mathrm{mol}^{-1}\right) \log \frac{273}{373}$ Internal energy : The energy of a thermodynamic system under given conditions is called internal energy. Q. $\Delta G=\Delta H-T \Delta S-v e=\Delta H-(+v e)(-v e)$ Our 1000+ Thermodynamics questions and answers focuses on all areas of Thermodynamics covering 100+ topics. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. $=(174.8)-(109.12+615.42)$ What type of wall does the system have? Q. $\Delta H=\Delta U+P \Delta V$. Time $=\frac{39487 \mathrm{J}}{100 \mathrm{Js}^{-1}}=394.87 \mathrm{s}$, $\left(C_{6} H_{6}\right)=(6 \times 12)+(6 \times 1)=78$, $\therefore$ Energy required to vapourise $100 g$ benzene, $=\frac{30.8}{78} \times 100=39.487 k_{U}=39487 J$, Given that power $=\frac{\text { Energy }}{\text { time }} \Rightarrow$ time $=\frac{\text { energy }}{\text { power }}$, Time $=\frac{39487 \mathrm{J}}{100 \mathrm{Js}^{-1}}=394.87 \mathrm{s}$, Q. If there is trend, use it to predict the molar heat capacity of Fr. Which formula forms a link between the Thermodynamics and Electro chemistry? (Hint. (ii) Vapours to liquid at $35^{\circ} \mathrm{C}$ What is the sign of $\Delta S$ for the forward direction? But $\mathrm{NO}_{2}(g)$ is formed. $\Delta S_{2} \int_{T_{1}}^{T_{2}} C_{p(\text {liquid})} \frac{d T}{T}=C_{p(\text {liquid})} \ln \frac{T_{2}}{T_{1}}$ Calculate standard molar entropy change of the formation of THERMODYNAMICS Mechanical Interview Questions And Answers pdf free download for gate,objective questions,mcqs,online test quiz bits,lab viva manual. $\Rightarrow$ Enthalpy change, $\Delta H=\Delta E+\Delta n_{g} R T$ $0=\Delta H-T \Delta S$ or $\Delta H=T \Delta S$ or $T=\frac{\Delta H}{\Delta S}$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Calculate the number of $k J$ necessary to raise the temperature of $60.0 \mathrm{g}$ of aluminium from $35-55^{\circ} \mathrm{C} .$ Molarheat capacity of $A l$ is $24 J m o l^{-1} K^{-1}$ $C(s)+2 C l_{2}(g) \rightarrow C C l_{4}(g) \Delta H^{\circ}=-135.5 k J m o l^{-1}$ Calculate the entropy change when $36 g$ of liquid water evaporates at $373 K\left(\Delta_{v o p} H=40.63 \mathrm{kJ} \mathrm{mol}^{-1}\right)$ Get Class 11 Chemistry Thermodynamics questions and answers to practice and learn the concepts. The temperature of calorimeter rises from $294.05 K$ to $300.78 K .$ If the heat capacity of calorimeter is $8.93 \mathrm{kJK}^{-1}$, calculate the heat transferred to the calorimeter. (Files = Faster Response). $-\Delta H_{v a p}=26.0 \mathrm{kJ} \mathrm{mol}^{-1}=26000 \mathrm{J} \mathrm{mol}^{-1}$ $2 P(s)+3 B r_{2}(l) \rightarrow 2 P B r_{3}(g) \Delta_{r} H^{o}=-243 k J m o l^{-1}$ Entropy change for evaporation of $36 g$ of wate Welcome to 5.1 THERMODYNAMICS. $S^{\circ} \mathrm{Ca}(\mathrm{OH})_{2}(a q)=-74.50 \mathrm{JK}^{-1} \mathrm{mol}^{-1}$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Please use the purchase button to see the entire solution. $=-805.0+2(-228.6)-[+52.3+2(0)]$ SHOW SOLUTION $-92380=\Delta U-2 \times 8.314 \times 298$ Click Here for Detailed Notes of any chapter. You will get here all the important questions with answers for class 11 Chemistry Therodynamics and chapters. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. $\Delta_{v a p} H^{\ominus}$ of $C O=6.04 \mathrm{kJ} \mathrm{mol}^{-1}$ So, molar heat capacity of these elements can be obtained by multiplying specific heat capacity by atomic mass. Calculate $\Delta S$ for the conversion of: Professionals, Teachers, Students and Kids Trivia Quizzes to test your knowledge on the subject. }=\frac{\Delta H_{v a p . Molar mass of phosphorus $=30 \mathrm{gmol}^{-1}$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. (iii) $\quad H_{2} O(l)$ at $0^{\circ} C \rightarrow H_{2} O(s)$ at $0^{\circ} C$ Molar heat capacity of $N a(s)=1.23 \times 23=28.3 \mathrm{J} \mathrm{mol}^{-1} \mathrm{K}^{-1}$ $\Delta G_{r}^{o}=\left[2 \Delta G_{f}^{o}\left(N O_{2}(g)\right)\right]-\left[2 \Delta G_{f}^{O}(N O(g))+\Delta G_{f}^{O}\left(O_{2}\right)\right]$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. We know (ii) Work is done by the system? SHOW SOLUTION Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Check the below NCERT MCQ Questions for Class 11 Chemistry Chapter 6 Thermodynamics with Answers Pdf free download. SHOW SOLUTION Also get to know about the strategies to Crack Exam in limited time period. $H C l(g)+H_{2} O \rightarrow H_{3} O^{+}(a q)+C l^{-}(a q)$ $=-800.78 \mathrm{kJ} \mathrm{mol}^{-1}$ A piston exerting a pressure of 1 atm rests on the surface of water at $100^{\circ} \mathrm{C} .$ The pressure is reduced to smaller extent when $10 g$ of water evaporates and $22.2 \mathrm{kJ}$ of heat is absorbed. Here, we are given, $\Delta H_{\text {Reaction}}=\Sigma \Delta H^{\circ}$ $f$ (Products) $-\Sigma \Delta H^{\circ}$ f $(\text {Reactants})$, Q. Is there any enthalpy change in a cyclic process ? $\Delta H_{\text {Reaction}}=\Sigma \Delta H^{\circ}$ $f$ (Products) $-\Sigma \Delta H^{\circ}$ f $(\text {Reactants})$ $C_{p}=\left(1.0 \mathrm{cal} K^{-1} g^{-1}\right)\left(18.0 \mathrm{g} \mathrm{mol}^{-1}\right)=18.0 \mathrm{cal} \mathrm{K}^{-1} \mathrm{mol}^{-1}$ $\Delta_{r} H^{\circ}=-239 k J m o l^{-1}$, $\mathrm{CH}_{3} \mathrm{OH}(l)+\frac{3}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)$, $\Delta_{r} H^{\circ}=-726 k J m o l^{-1}$, $C(g)+O_{2}(g) \rightarrow C O_{2}(g) ; \Delta_{r} H^{\circ}=-393 k J m o l^{-1}$, $H_{2}(g)+\frac{1}{2} O_{2}(g) \rightarrow H_{2} O(l) ; \Delta_{r} H^{\circ}=-286.0 k J m o l^{-1}$, $C(\text { graphite })+2 H_{2}(g)+\frac{1}{2} O_{2}(g) \longrightarrow C H_{3} O H(l)$, (i) $\left.\quad \mathrm{CH}_{3} \mathrm{OH}(l)+\frac{3}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)\right]$, (ii) $\quad C(g)+O_{2}(g) \longrightarrow C O_{2}(g) ; \Delta_{r} H^{\circ}=-393 k J m o r^{1}$, (iii) $H_{2}(g)+\frac{1}{2} O_{2}(g) \longrightarrow H_{2} O(\eta) ; \Delta_{r} H^{\circ}=-2860 \mathrm{kJ} \mathrm{mol}^{-1}$, Multiply eqn. Calculate the value of standard Gibb's energy change at 298 K and predict whether the reaction is spontaneous or not. Enthalpy of combustion of octane, Heat transferred $=$ Heat capacity $\times \Delta T$, $=\left(8.93 \mathrm{kJ} K^{-1}\right) \times(6.73 \mathrm{K})=60.0989 \mathrm{kJ}$, Molar mass of octane $\left(C_{8} H_{18}\right)=(8 \times 12)+(18 \times 1)=114$, $\therefore$ Internal energy change $(\Delta E)$ during combustion of one mole of, octane $=\frac{60.0989}{1.250} \times 114=5481.02 k J \mathrm{mol}^{-1}$, $C_{8} H_{18}(l)+\frac{25}{2} O_{2}(g) \rightarrow 8 C O_{2}(g)+9 H_{2} O(l)$, $\Delta n_{g}=8-\frac{25}{2}=-\frac{9}{2}$, $\Rightarrow$ Enthalpy change, $\Delta H=\Delta E+\Delta n_{g} R T$, $=\left(5481.02 \times 10^{3}\right)+(-4.5 \times 8.314 \times 300.78)$, $=\left(5481.02 \times 10^{3}\right)-11253.08 \mathrm{J} \mathrm{mol}^{-1}$, $=5492273.082 \mathrm{Jmol}^{-1}=5492.27 \mathrm{kJ} \mathrm{mol}^{-1}$, Q. Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. Molar mass of $C O=28 g \mathrm{mol}^{-1}$ Automobile radiator system is analyzed as closed system. Heat released for the formation of $35.2 g$ of $C O_{2}$ Q. You may read our privacy policy for more info. $\Delta H=\Delta E$ during a process which is carried out in a closed vessel $(\Delta v=0)$ or number of moles of gaseous products $=$ number of moles of gaseous reactants or the reaction does not involve any gaseous reactant or product. $\mathrm{SiH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{SiO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$ Calculate enthalpy change when $2.38 g$ of carbon monoxide (CO) vapourises at its normal boiling point. -condensation into a liquid. $=\frac{30.8}{78} \times 100=39.487 k_{U}=39487 J$ $\Delta G=120-380=-260 k J$ $\Delta_{r} G^{\circ}=\Delta_{r} H^{o}-T \Delta_{r} S^{\circ}$, $\Delta_{r} H^{\circ}=+491.18 k J \mathrm{mol}^{-1}$, $\Delta_{r} S^{o}=197.67 \times 10^{-3} \mathrm{kJ} \mathrm{mol}^{-1} \mathrm{K}^{-1}, T=298 \mathrm{K}$, $\Delta_{r} G^{\circ}=491.18 k J \mathrm{mol}^{-1}-298 \mathrm{Kx}$, $\left(197.67 \times 10^{-3} \mathrm{kJ} \mathrm{mol}^{-1} \mathrm{K}^{-1}\right)$, $=491.18 \mathrm{kJ} \mathrm{mol}^{-1}-58.9 \mathrm{kJ} \mathrm{mol}^{-1}=432.28 \mathrm{kJ} \mathrm{mol}^{-1}$. $-(1) \quad \Delta S=-v e$ $\Delta S_{\text {Reaction }}=\Sigma S_{m(\text { products })}^{\circ}-\Sigma S_{m(\text { reactants })}^{\circ}$ Enthalpy of solution of $N H_{4} N O_{3}=\frac{5.282}{20} \times 80$ Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...Sol. These important questions will play significant role in clearing concepts of Chemistry. $\frac{-393.5 \times 35.2}{44}=-314.8 k J$. $\Delta G_{f}^{\circ} C H_{4}(g)=-50.72 \mathrm{kJ} \mathrm{mol}^{-1}$ and $\Delta \mathrm{G}_{f}^{\circ} \mathrm{O}_{2}(g)=0$ [CBSE Sample Paper for 2006] [1] ... THERMODYNAMICS Interview Questions And Answers
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A discrete hierarchy of double bracket equations and a class of negative power series
March 2017, 7(1): 53-72. doi: 10.3934/mcrf.2017004
Control and stabilization of 2 × 2 hyperbolic systems on graphs
Serge Nicaise ,
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, F-59313 -Valenciennes Cedex 9, France
Received June 2015 Revised February 2016 Published December 2016
We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.
Keywords: Hyperbolic systems, wave equation, stabilization.
Mathematics Subject Classification: Primary: 35L50; Secondary: 93D15.
Citation: Serge Nicaise. Control and stabilization of 2 × 2 hyperbolic systems on graphs. Mathematical Control & Related Fields, 2017, 7 (1) : 53-72. doi: 10.3934/mcrf.2017004
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Figure 1. A tree shaped network: generations of edges
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Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry
September 2017, 9(3): 335-390. doi: 10.3934/jgm.2017014
Geometry of matrix decompositions seen through optimal transport and information geometry
Klas Modin ,
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Received January 2016 Revised August 2016 Published June 2017
Fund Project: This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 661482, and from the Swedish Foundation for Strategic Research under grant agreement ICA12-0052
Figure(10)
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher-Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry--the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices.
Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the $QR$, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples.
The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher-Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.
Keywords: Matrix decompositions, polar decomposition, optimal transport, Wasserstein geometry, Otto calculus, entropy gradient flow, Lyapunov equation, information geometry, Fisher-Rao metric, $QR$ decomposition, Iwasawa decomposition, Cholesky decomposition, spectral decomposition, singular value decomposition, isospectral flow, Toda flow, Brockett flow, double bracket flow, orthogonal group, Hessian metric, multivariate Gaussian distribution.
Mathematics Subject Classification: 15A23, 53C21, 58B20, 15A18, 49M99, 65F15, 65F40.
Citation: Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014
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Figure 1. Illustration of the geometry of the polar decomposition of diffeomorphisms. The element $\nabla\phi$ in the factorization $\varphi=\nabla\phi\circ\psi$ is obtained at the intersection of the polar cone and the fiber of $\mu_1=\pi(\varphi)$. To compute $\nabla\phi$, one may start at $\varphi$ and follow a gradient flow constrained to the fiber of $\mu_1$ (vertical gradient flow, see $\S 2.2.1$), or one may take a gradient flow of a functional on the space of densities that approaches $\mu_1$ (entropy gradient flow, see $\S 2.2.2$) and lift it to a corresponding gradient flow on the polar cone (lifted gradient flow, see $\S 2.2.3$).
Figure 2. Evolution of the matrix elements of $B(t)$ for the vertical gradient flow in Example 1. Notice that $B(0)=A$ and that $B(t)$ converges towards $P_{\infty}$ in (43) as $t\to\infty$.
Figure 3. Convergence towards the limit $P_{\infty}$ of the vertical gradient flow in Example 1.
Figure 4. Evolution of the lifted gradient flow in Example 2. Notice that $P(0)$ is the identity and that $P(t)$ converges towards $P_{\infty}$ in (54) as $t\to\infty$.
Figure 5. Convergence towards the limit $P_{\infty}$ of the lifted gradient flow in Example 2. Notice that the convergence of both $-F(P(t))$ and $d^{2}(P(t),P_{\infty})$ as $t\to\infty$ is exponential, as fully explained by Theorem 2.14.
Figure 6. Evolution of the lifted gradient flow in Example 3. Notice that $R(0)$ is the identity and that $R(t)$ converges towards $R_{\infty}$ in (80) as $t\to\infty$.
Figure 7. Convergence towards the limit $R_{\infty}$ of the lifted gradient flow in Example 3. The convergence of both $-F(R(t))$ and $d^{2}(R(t),R_{\infty})$ is exponential with rate $\exp(-2t)$, as ensured by Theorem 3.10.
Figure 8. Phase diagram of equation (86) for geodesics on ${\text{D}}(n)$. For every $l>0$ there is a unique integral curve $\lambda(t)$ such that $\lambda(0)=1$ and $\lambda(1)=l$. In consequence, every $\Lambda\in{\text{D}}(n)$ is connected to the identity $I$ by a unique horizontal geodesic.
Figure 9. Evolution of the horizontal gradient flow in Example 4. $\Lambda(t)$ appears to converge to the ordered sequence of eigenvalues $(1/2,1,5)$.
Figure 10. Convergence of $F(\Lambda(t))$ towards the minimum for the horizontal gradient flow in Example 4. The convergence appears to be exponential.
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Klas Modin | CommonCrawl |
A split-optimization approach for obtaining multiple solutions in single-objective process parameter optimization
Manik Rajora1,
Pan Zou2,
Yao Guang Yang3,
Zhi Wen Fan3,
Hung Yi Chen3,
Wen Chieh Wu3,
Beizhi Li2 &
Steven Y. Liang1,2
It can be observed from the experimental data of different processes that different process parameter combinations can lead to the same performance indicators, but during the optimization of process parameters, using current techniques, only one of these combinations can be found when a given objective function is specified. The combination of process parameters obtained after optimization may not always be applicable in actual production or may lead to undesired experimental conditions. In this paper, a split-optimization approach is proposed for obtaining multiple solutions in a single-objective process parameter optimization problem. This is accomplished by splitting the original search space into smaller sub-search spaces and using GA in each sub-search space to optimize the process parameters. Two different methods, i.e., cluster centers and hill and valley splitting strategy, were used to split the original search space, and their efficiency was measured against a method in which the original search space is split into equal smaller sub-search spaces. The proposed approach was used to obtain multiple optimal process parameter combinations for electrochemical micro-machining. The result obtained from the case study showed that the cluster centers and hill and valley splitting strategies were more efficient in splitting the original search space than the method in which the original search space is divided into smaller equal sub-search spaces.
In today's rapidly changing scenario in the manufacturing industries, optimization of process parameters is essential for a manufacturing unit to respond effectively to the severe competitiveness and increasing demand for quality products in the market (Cook et al. 2000). Previously, to obtain optimal combinations of input process parameters, engineers used a trial-and-error-based approach, which relied on engineers' experience and intuition. However, the trial-and-error-based approach is expensive and time consuming; thus, it is not suitable for complex manufacturing processes (Chen et al. 2009). Thus, researchers have focused their attention on developing alternate methods to the trial-and-error-based approach that can help engineers obtain the combination of process parameters that will minimize or maximize the desired objective value for a given process. The methods for obtaining these combinations of process parameters can be split into 2 main categories: 1. forward mapping of process inputs to a performance indicator with backwards optimization and 2. reverse mapping between the performance indicators and the process inputs. In forward mappings, first, a model is created between the process inputs and the performance indicators using either physics-based models, regressions models, or intelligent techniques. Once a satisfactory model has been created, it is then utilized to obtain the combination of process parameters that will lead to a desired value of the output using optimization techniques such as the Genetic algorithm (GA), Simulated Annealing (SA), Particle Swarm Optimization (PSO), etc. The desired output can either be to a. minimize a given performance indicator or b. reach a desired level of a performance indicator.
Chen et al. (2009) utilized the back propagation neural network (BPNN) and GA to create a forward prediction model and optimize the process parameters of plastic injection molding. Ylidiz (2013) utilized a hybrid artificial bee colony-based approach for selecting the optimal process parameters for multi-pass turning that would minimize the machining cost. Senthilkumaar et al. (2012) used mathematical models and ANN to map the relationship between the process inputs and performance indicators for finish turning and facing of Inconel 718. GA was then used to find the optimal combination of process parameters, with the aim of minimizing surface roughness and flank wear. Pawar and Rao (2013) applied the teaching–learning-based optimization (TLBO) algorithm to optimize the process parameters of abrasive water jet machining, grinding, and milling. They created physics-based models between the input and output parameters of each process and then utilized TLBO to minimize the material removal rate in abrasive water jets, minimize production cost and maximize production rate with respect to grinding, and minimize the production time in milling. Fard et al. (2013) employed adaptive network-based fuzzy inference systems (ANFIS) to model the process of dry wire electrical discharge machining (WEDM). This model was then used to optimize, using artificial bee colony (ABC), the process inputs that would minimize surface roughness and maximize material removal rate. Teixidor et al. (2013) used particle swarm optimization (PSO) to obtain optimal process parameters that would minimize the depth error, width error, and surface roughness in the pulsed laser milling of micro-channels on AISI H13 tool steel. Katherasan et al. (2014) used ANN to model the process of flux cored arc welding (FCAW) and then utilized PSO to minimize bead width and reinforcement and maximize depth of penetration. Yusup et al. (2014) created a regression model for the process parameters and process indicators of an abrasive waterjet (AWJ) and then used ABC to minimize the surface roughness. Panda and Yadava (2012) used ANN to model the process of die sinking electrochemical spark machining (DS-ESM) and then used GA for multi-objective optimization of the material removal rate and average surface roughness. Maji and Pratihar (2010) combined ANFIS with GA to create forward and backward input–output relationships for the electrical discharge machining process (EDM). In their proposed methodology, GA was used to optimize the membership functions of the ANFIS, with the aim of minimizing the error between the predicted and actual outputs. Cus et al. (2006) developed an intelligent system for online monitoring and optimization of process parameters in the ball-end milling process. Their objective was to find the optimal set of process parameters, using GA to achieve the forces selected by the user. Raja et al. (2015) optimized the process parameters of electric discharge machining (EDM) using the firefly algorithm to obtain the desired surface roughness in the minimum possible machining time. Raja and Baskar (2012) used PSO to optimize the process parameters to achieve the desired surface roughness while minimizing machining time for face milling. Rao and Pawar (2009) developed mathematical models using response surface modeling (RSM) to correlate the process inputs and performance indicators of WEDM. They then used ABC to achieve the maximum machining speed that would give the desired value of the surface finish. Lee et al. (2007) modeled the process of high-speed finish milling using a 2 stage ANN and then used GA to maximize the surface finish while achieving the desired material removal rate. Teimouri and Baseri (2015) used a combination of fuzzy logic and the artificial bee colony algorithm to create a forward prediction model between input and output parameters for friction stir welding (FSW). This trained model was then utilized to find the optimal input parameters that would give the desired output value by minimizing the absolute error between the predicted and specified output using the imperialist competitive algorithm (ICA).
An ample amount of work has also been done to create a reverse mapping model between the process parameters and the performance indicators. Parappagoudar et al. (2008) utilized the back-propagation neural network (BPNN) and a genetic-neural network (GA-NN) for forward and reverse mapping of the process parameters and performance indicators in a green sand mold system. Parappagoudar et al. (2008) also extended their application of BPNN and GA-NN to create forward and backward mappings for the process of the Sodium Silicate-Bonded, Carbon Dioxide Gas Hardened Molding Sand System. Amarnath and Pratihar (2009) used radial basis function neural networks (RBFNNs) for forward and reverse input–output mapping of the tungsten inert gas (TIG) welding process. In their proposed methodology, the structure and the parameters of the RBFNN were modified using a combination of GA and the fuzzy C-means (FCM) algorithm for both the forward and reverse mapping. Chandrashekarappa et al. (2014) used BPNN and GA-NN for forward and reverse mappings of the squeeze casting process. Kittur and Parappagoudar (2012) utilized BPNN and GA-NN for forward and reverse mapping in the die casting process. Because batch training requires a tremendous amount of data, they used previously generated equations to supplement the experimental data. Malakooti and Raman (2000) used ANN to create forward- and backward-direction mappings between the process outputs and inputs for the cutting operation on a lathe.
Even though extensive research has been done regarding optimization of the process parameter for different processes, the current algorithms used for the optimization procedure are limited to finding only one set of optimal process parameter combinations for a single-objective optimization problem each time the algorithms are executed. Though this process parameter combination may achieve the desired output, it may not always be suitable for actual production or may lead to undesirable experimental conditions. It can also be observed from the experimental data of different processes that different process parameter combinations may lead to the same or similar performance indicators. For example, in turning, multiple combinations of process parameters may lead to the same or similar value of surface roughness. In EMM, multiple combinations of process parameters may lead to the same or similar value of taper and overcut. Therefore, there is a possibility to develop a method that can provide multiple optimal process parameter combinations for a single-objective optimization problem.
In this paper, the presented method is to obtain multiple optimal process parameter combinations for a single-objective optimization problem by splitting the original search space into smaller sub-search spaces and finding the optimal process parameter combinations in each sub-search space. Two different methods are used to split the original search space, and GA is utilized to optimize the process parameters in each sub-search space. The optimization results obtained after using the two search space splitting methods are compared to the optimization results obtained when the original search space was divided equally into smaller sub-search spaces; GA was used to optimize the process parameters in each sub-search space. EMM of SUS 304 is used as a case study because its experimental data shows that multiple process parameter combinations can lead to the same performance indicators. Due to the lack of physics-based models, a general regression neural network (GRNN) is used to create a forward prediction model between the input process parameters and the performance indicators for the process of EMM. The rest of the paper is organized as follows: section "Modeling" describes the modeling stage of the method. Section "Case study" presents and discusses the results obtained. Section "Conclusion" presents conclusions from the presented work and mentions future directions for the proposed approach.
Split-optimization approach
Traditional GA, when used in a single-objective optimization, only converges to a single local optima or near-optimum solution, while the search space might consist of multiple local optima that can satisfy the given criteria. Multi-objective GA, on the other hand, does provide multiple solutions, but each solution satisfies each objective to a different degree. A possible method to obtain multiple solutions for a single-objective optimization problem is to split the original search space into several smaller sub-search spaces, with each sub-search space containing a possible solution to the given objective. Once these sub-search spaces have been identified, GA can then be used in each sub-search space to find the possible solution. The procedure of the proposed split-optimization approach consists of two parts: splitting of the original search spaces into sub-search spaces and the application of GA to find the solution in each sub-search space. Because any optimization function needs a fitness function as an input, in this paper, a generalized neural network (GRNN) was used as the fitness function due to a lack of physics-based models for the given process. The flow chart of this proposed split-optimization approach is shown in Fig. 1.
The basic structure of the split-optimization approach
Because the results obtained after using GA depend on the training accuracy of the GRNN, it is important to train the GRNN sufficiently so that it can predict the performance indicators with a high degree of accuracy. As there will always be some degree of error associated with the outputs of the GRNN, a possible method to cope with these errors is to take into consideration the significance level of the optimization problem. The significance level here is defined as a customized parameter that allows solutions with a fitness value better than or equal to it to be counted as final optimal solutions. The significance level by default is regarded as zero, which indicates that only solutions with the same minimum fitness value can be regarded as the final optimum solutions.
Splitting strategies
As mentioned earlier, two strategies are used to split the original search space into sub-search spaces. The details of the two strategies are highlighted below.
Hill and valley splitting strategy
The steps of the splitting strategy are as follows:
Identify two data points, A and B, from the experimental data set whose input values are furthest away from each other. Here, \(A = \left( {a_{1} ,a_{2} , \cdots ,a_{n} ,y_{a} } \right)\) and \(B = \left( {b_{1} ,b_{2} , \cdots ,b_{n} ,y_{b} } \right)\), indicating that all the data points have n inputs and 1 output.
Select a random data point C 1 from the remaining data points and determine whether it is a hill, valley, or neither compared to the initial points, i.e., A and B, based on the value of its output. For example, if \(y_{a}\) < \(y_{b}\) < \(y_{{c_{1} }}\), then C 1 is a hill; if \(y_{{c_{1} }}\) < \(y_{a}\) < \(y_{b}\), then C 1 is a valley; if \(y_{a}\) < \(y_{{c_{1} }}\) < \(y_{b}\), then C 1 is neither.
Select a random data point C 2 from the remaining data points; find a pair of previously selected data points whose input values encompass the input values of C 2 .
Compare the output value of C 2 with the data points selected in step c and determine whether it is a hill, valley, or neither.
Repeat step c and d until all the data points have been identified as a hill, valley, or neither.
After the classification of all the experimental data points is completed, the input values of the original points (A and B) and all the points classified as either a hill or valley are used to split the original search space into smaller sub-search spaces. This is done by dividing the original range of the input parameters of the experimental data into sub-ranges by using the input values of the points classified as hill or valley and then finding all the combinations of the sub-ranges for all the inputs. Once the search space has been split into sub-search spaces, GA is used to optimize each search space individually. Figure 2 shows the flow chart of the hill and valley splitting strategy.
The flow chart of the hill and valley splitting strategy
Cluster centers splitting strategy
In this strategy, the k-means clustering algorithm is used to divide the experimental data set into k clusters. Once the k cluster centers are identified, they are used to split the original search space into smaller sub-search spaces. This is accomplished by dividing the original range of input process parameters of the experimental data into smaller sub-ranges using the values of the k cluster centers. Next, the original search space is divided by using all the combinations of the sub-ranges for all the inputs. Figure 3 shows the flow chart of the cluster centers splitting strategy.
The flow chart of the cluster centers splitting strategy
GRNN-GA optimization
As mentioned earlier, a forward prediction model was created using GRNN (Specht 1991). The inputs of the GRNN were voltage, pulse on time, and feed rate; the outputs were D in and D out . During the training of the GRNN, the original data was split into training, validation, and testing data sets, and tenfold cross validation was used during the training phase of the GRNN to avoid overfitting and to find the optimal value of the spread parameter that would minimize the mean squared error (MSE). Once the GRNN was trained sufficiently, it was then utilized as the fitness function for GA during the optimization procedure.
An input parameter optimization problem in EMM was utilized as a case study because it can be seen from Table 1 that multiple combinations of input process parameters lead to the same or similar values of taper and overcut.
Table 1 Original range of the controllable process parameters
Description of the case
Figure 4 schematically depicts the EMM experimental setup. The system consists of a three-dimensional movement device, a small-scale power supply of 100 A, and an electrolyte pump and filter. The feeding system is controlled by a PC-Based CNC Controller, RTX real-time windows kernel program, and a motion card that drives the linear motor precisely. A pulse generator supplies a periodic current to the experimental model. A digital oscilloscope ensures that the pulse generator produces a rectangular waveform with accurate amplitude. If the tool feed rate is excessive, the tool will contact the workpiece and cause a short circuit; thus, an oscilloscope is employed to detect any short circuits. Whenever the oscilloscope detects a short circuit, a signal is sent rapidly to the PC and the tool is extracted automatically until the measured voltage returns to the applied voltage. The micro array holes electrode module includes the multiple nozzle tool electrodes, PVC mask and tool fixture. The electrolyte is pumped to a multiple electrode cell and exits through the small nozzle in the form of a free standing jet directed towards the anode workpiece.
Schematic diagram of electrochemical micromachining system (left) and micro array hole electrode module (right)
Other basic information and settings are as follows: the electrolyte velocity was 10 m/s, electrolyte temperature was 27 °C, the initial gap between the tool and the workpiece was 100 µm, tool moving distance was 800 µm, the workpiece material was SUS 304, the electrolyte used was 10 %wt. NaNO3, the nominal diameter of the hole was 900 µm, and the depth of the hole was 500 µm.
Voltage, pulse on time, and feed rate were used as the controllable process parameters, while the inner diameter of the micro-hole D in and the outer diameter D out were the measurable performances. The range of each process parameter is shown in Table 1. The range of the variables was fixed by taking into consideration two factors: 1. limitation of the devices used for EMM and 2. making sure that the experimental conditions would be stable within the chosen range. The resolution of the process parameters were was 0.1 V for the voltage, 0.1 µs for pulse on time, and 0.1 µm/s for the feed rate. This indicates that there are close to 3 million possible combinations of all the process parameters. Therefore, the proposed method was applied for this particular case study.
The process of EMM has two responses, i.e., taper and overcut. When drilling micro-size holes in thin metallic foils, a major requirement is for the holes to have straight walls. The straightness of a wall can be represented by the taper and is given by:
$$Taper = \left| {(D_{in} - D_{out} )/depth_{{}} } \right|$$
In critical applications, particularly in micro instruments, the straightness of a drilled hole is also very important. Overcut, as given by Eq. (2), is the difference between the aim holes' diameters and actual hole diameter and is a good representation of the straightness of a drilled hole. A small overcut value represents a more precise EMM process.
$$Overcut = \left| {(D_{in} - D)/2_{{}} } \right|$$
In the process of EMM, the aim is to find the set of process parameter combinations that will minimize both taper and overcut. Though EMM has two responses, for the purpose of this case study, the two responses were combined into a single-objective by the use of weight values. Before combining them into a single objective, the values of taper and overcut were normalized between 0 and 1. Equation (3) shows how the taper and overcut were normalized, while Eq. (4) shows the objective function.
$$T_{{normalized}} = \frac{{T_{{predicted}} - T_{{\min ,experimental}} }}{{T_{{\max ,experimental}} - T_{{\min ,experimental}} }},O_{{normalized}} = \frac{{O_{{predicted}} - O_{{\min ,experimental}} }}{{O_{{\max ,experimental}} - O_{{\min ,experimental}} }}$$
where \(T_{predicted}\) is the taper value predicted by the GRNN, \(T_{{\min} ,experimental}\) is the minimum taper value in the experimental data, \(T_{{\max} ,experimental}\) is the maximum taper value in the experimental data, and \(T_{normalized}\) is the normalized predicted taper value. Similarly, \(O_{predicted}\) is the overcut value predicted by the GRNN, \(O_{{\rm min} ,exp erimental}\) is the minimum overcut value in the experimental data, \(O_{{\rm max} ,exp erimental}\) is the maximum overcut value in the experimental data, and \(O_{normalized}\) is the normalized predicted overcut value.
$$Objective = 0.5 \times T_{normalized} + 0.5 \times O_{normalized}$$
To create a forward prediction model for the process of EMM, three different sets of experiments were created. In the first experimental set, voltage and feed rate had 3 levels each, while pulse on time was constant, which resulted in a total of 9 combinations of input parameters. These combinations of input experiments were used to perform the process of EMM, and for each combination, D in and D out were recorded. In the second and third experimental sets, voltage, pulse on time, and feed rate had 3 levels each, which resulted in 27 combinations of input process parameters for both experimental sets 2 and 3. The process of EMM was performed using the combination of inputs; D in and D out were again recorded. The levels of voltage, pulse on time, and feed rate are given in Table 2.
Table 2 Levels of voltage, pulse on time, and feed rate values used for the three experimental sets
In the experiments, the Charge Coupled Device (CCD) camera was utilized to measure all the workpieces after the process of EMM. Figure 5 shows the pictures taken using the CCD camera. The CCD images were then processed through a software, which provided the average value of the diameters of the holes on the front and back of the workpiece. The experimental data obtained is shown in Table 3.
Pictures taken using the CCD camera. a The front of the workpiece, while b. The back of the workpiece
Table 3 The 63 groups of experimental data
As stated earlier, to compensate for the errors associated with the trained GRNN, a significance level needs to be specified. In this case study, if the value of the objective function, given by Eq. (4), after optimization was less than 0.5, then the solution of that particular sub-search space was said to be a final optimal solution. The only changeable parameter for the GRNN was the spread value, which was obtained after the training process. The changeable parameters for GA are listed in Table 4.
Table 4 Parameter values used for GA
The methods mentioned above were used to split and optimize the search space 10 times independently, and the average value of the objective function for the best solutions of each run was calculated. The run that had the lowest average value of the objective function was used as the best run; its results are presented here.
Hill and valley spitting strategy
As mentioned previously, the first step in this method is to find two data points that are furthest away from each other. To accomplish this task, the distance from the origin to every data point was obtained after each input was normalized using Eq. (6). The equations used to normalize the inputs are given in Eq. (5). The two data points with distances d min and d max were the inputs furthest away from each other. Then, the steps outlined in the previous section were followed to split the original search space into several sub-search spaces.
$$V_{i,normalized} = \frac{{V_{i} - V_{{\rm min} ,exp erimental} }}{{V_{{\rm max} ,\exp erimental} - V_{{\rm min} ,exp erimental} }},\;P_{i,normalized} = \frac{{P_{i} - P_{{\rm min} ,exp erimental} }}{{P_{{\rm max} ,exp erimental} - P_{{\rm min} ,exp erimental} }},\;F_{i,normalized} = \frac{{F_{i} - F_{{\rm min} ,exp erimental} }}{{F_{{\rm max} ,exp erimental} - F_{{\rm min} ,exp erimental} }}$$
where \(V_{i,normalized}\) is the normalized values of voltage in the ith experimental data, \(V_{{\rm min} ,exp erimental}\) is the minimum voltage value in the experimental data, \(V_{{\rm max} ,exp erimental}\) is the maximum voltage value in the experimental data, and \(V_{i}\) is the voltage in the ith experimental data. Similarly, \(P_{i,normalized}\) is the normalized values of pulse on time in the ith experimental data, \(P_{{\rm min} ,exp erimental}\) is the minimum pulse on time value in the experimental data, \(P_{{\rm max} ,exp erimental}\) is the maximum pulse on time value in the experimental data, and \(P_{i}\) is the pulse on time in the ith experimental data. \(F_{i,normalized}\) is the normalized values of feed rate time in the ith experimental data, \(F_{{\rm min} ,exp erimental}\) is the minimum feed rate value in the experimental data, \(F_{{\rm max} ,\exp erimental}\) is the maximum feed rate value in the experimental data, and \(F_{i}\) is the feed rate in the ith experimental data.
$$D_{i} = \sqrt {\left( {V_{i,normalized} } \right)^{2} + (P_{i,normalized} )^{2} + (F_{i,normalized} )^{2} }$$
Table 5 provides the ranges for each of the input values. For each of the sub-search spaces, GA was utilized to find the optimal process parameter combination. The optimization results are shown in Table 6.
Table 5 Splitting result of the hill and valley splitting strategy
Table 6 Optimization results obtained after using the hill and valley splitting strategy
The k value in the k-means clustering algorithm is a user dependent parameter; an inappropriate choice of k may yield poor results. However, so far there is no clear guideline for choosing the value of k. In this case study, the value of k was varied from 2 to 6; the corresponding splitting and optimization results are shown in Table 7. The maximum number of optimal solutions was obtained when the value of k was 6. These results are shown in Table 8.
Table 7 Splitting result obtained using cluster centers strategy
Table 8 The optimization results obtained using the cluster centers splitting strategy with k = 6
Equally splitting strategy
The results obtained using the two previous splitting strategies were compared to the results obtained when the original search space was split equally into smaller sub-search spaces. In the equally splitting strategy, each process parameter was equally split into 4 sub-ranges, as shown in Table 9. The optimization results obtained using the equally splitting strategy are shown in Table 10.
Table 9 Splitting result of equally splitting strategy
Table 10 The optimization result obtained using the equally splitting strategy
Comparison and analysis
These three splitting strategies provide different ways to split the search space into smaller sub-search spaces. To evaluate the efficiency of a strategy, the percentage of useful sub-search spaces was calculated using Eq. (7).
$$percentage\,of\, useful\, sub\,- search\, spaces = \frac{{No.\,of\, optimal\, solutions}}{{No.\, of\, sub - search\, spaces}} \times 100\,\%$$
Table 11 shows the comparison between the 3 strategies based on Eq. (7).
Table 11 Comparison of three splitting strategies
It can be observed that the equally splitting strategy is the least efficient way because its percentage of useful sub-search spaces is the lowest (7.8 %). The efficiency of hill and valley splitting is fixed because it lacks any controllable parameters and because the sequence in which points are selected can affect their classification. It can be seen from Table 10 that there is a correlation between the efficiency of cluster centers splitting and the value of k. However, there is no clear understanding between the value of k and the efficiency of the method and there is also no guideline for selecting the optimal value of k.
This case study utilized a trained NN prediction model in the evaluation of input parameter combinations. Therefore, to validate the optimization result, one additional experiment with the randomly chosen optimized input parameter combination was done. The data of the validation experiment is shown in Table 12.
Table 12 Result of an additional validation experiment
Based on the validation experimental result, it can be seen that the prediction error of the NN prediction model used in this case study is quite low and the results obtained using the proposed approach are better than the results shown in the initial experimental data. Noteworthy, the optimized input process parameter combination was not in the initial training dataset and the optimization algorithm was able to find a better-than-ever objective value. Therefore, the optimization result is verified.
In this paper, a split-optimization approach was proposed for obtaining multiple solutions for a single-objective process parameter optimization problem. The proposed approach consisted of two stages: splitting of the original search space into smaller sub-search spaces and optimization of process parameters in each of the smaller sub-search spaces. Two splitting strategies, i.e., hill and valley splitting strategy and cluster centers splitting strategy, were used to split the original search space into smaller sub-search spaces efficiently. Next, GA was used in each sub-search space to find multiple combinations of process parameters that minimized the single-objective value, one from each sub-search space. The efficiency of these two strategies was verified by comparing them with a method in which the original search space is divided into smaller and equal sub-search spaces. The comparison of the results from the different splitting methods showed that the hill and valley splitting strategy and cluster centers splitting strategy were more efficient than the equal splitting strategy. Among all the methods, the cluster centers splitting strategy, for a k value of 6, was able to achieve the most optimal solutions. The results obtained from the hill and valley splitting strategy showed that though it is an efficient method, its efficiency depends on the order in which the points are classified as a hill or valley.
Possible future work includes a study of the relationship between the efficiency of the cluster centers splitting strategy and the k value; a guideline should be to choose an optimal value of k. Future works also include experimentally validating the multiple solutions obtained using the proposed approach, applying the proposed approach to more case studies, and refining the proposed approach based on the results of the experimental validation and other case studies.
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MR and ZP—analysis of data, development of the required code, and writing of the manuscript. YGY, ZWF, HYC, and WCW—study, collection and analysis of data. BL—comments for the paper. SYL—guideline for the proposed approach and comments for the paper. All authors read and approved the final manuscript.
We would like to acknowledge the Metal Industries Research & Development Center for collecting the data and providing us background knowledge regarding EMM.
George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA
Manik Rajora
& Steven Y. Liang
Mechanical Engineering College, Donghua University, Songjiang District, Shanghai, 201620, China
Pan Zou
, Beizhi Li
Regional R&D Services Department, Metal Industries Research and Development Center, Taichung, 407, Taiwan, ROC
Yao Guang Yang
, Zhi Wen Fan
, Hung Yi Chen
& Wen Chieh Wu
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Correspondence to Steven Y. Liang.
Rajora, M., Zou, P., Yang, Y.G. et al. A split-optimization approach for obtaining multiple solutions in single-objective process parameter optimization. SpringerPlus 5, 1424 (2016) doi:10.1186/s40064-016-3092-6
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Symmetric interval identification systems of order three
On strange attractors in a class of pinched skew products
February 2012, 32(2): 619-641. doi: 10.3934/dcds.2012.32.619
The periodic-parabolic logistic equation on $\mathbb{R}^N$
Rui Peng 1, and Dong Wei 2,
Department of Mathematics, Xuzhou Normal University, Xuzhou, 221116, China
Hebei University of Engineering, Handan City, Hebei Province, 056038, China
Received August 2010 Revised June 2011 Published September 2011
In this article, we investigate the periodic-parabolic logistic equation on the entire space $\mathbb{R}^N\ (N\geq1)$: $$ \begin{equation} \left\{\begin{array}{ll} \partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \ & {\rm in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N, \end{array} \right. \end{equation} $$ where the constants $T>0$ and $p>1$, and the functions $a,\ b$ with $b>0$ are smooth in $\mathbb{R}^N\times\mathbb{R}$ and $T$-periodic in time. Under the assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$ are bounded away from $0$ and infinity for all large $|x|$, where the constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existence and uniqueness of positive $T$-periodic solutions. In particular, we determine the asymptotic behavior of the unique positive $T$-periodic solution as $|x|\to\infty$, which turns out to depend on the sign of $\gamma$. Our investigation considerably generalizes the existing results.
Keywords: uniqueness, positive periodic solution, entire space, Periodic-parabolic logistic equation, asymptotic behavior..
Mathematics Subject Classification: Primary: 35K20, 35B10; Secondary: 35K60, 35B0.
Citation: Rui Peng, Dong Wei. The periodic-parabolic logistic equation on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 619-641. doi: 10.3934/dcds.2012.32.619
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Simons inequality
From Encyclopedia of Mathematics
2010 Mathematics Subject Classification: Primary: 53A10 [MSN][ZBL]
An inequality proved by Simons in his fundamental work [Si] on minimal varieties, which played a pivotal role in the solution of the Bernstein problem. The inequality bounds from below the Laplacian of the square norm of the second fundamental form of a minimal hypersurface $\Sigma$ in a general Riemannian manifold $N$ of dimension $n+1$. More precisely, if $A$ denotes the second fundamental form of $\Sigma$ and $|A|$ its Hilbert-Schmidt norm, the inequality states that, at every point $p\in \Sigma$, \[ \Delta_\Sigma |A|^2 (p) \geq - C (1 + |A|^2 (p))^2 \] where $\Delta_\Sigma$ is the Laplace operator on $\Sigma$ and the constant $C$ depends upon $n$ and the Riemannian curvature of the ambient manifold $N$ at the point $p$. When $N$ is the Euclidean space, a more precise form of the inequality is \[ \Delta_\Sigma |A|^2 \geq - 2 |A|^4 + 2 \left(1+\frac{2}{n}\right) |\nabla_\Sigma |A||^2 \] (see Lemma 2.1 of [CM] for a proof and [SSY] for the case of general ambient manifolds). Moreover, the inequality is an identity in the special case of $2$-dimensional minimal surfaces of $\mathbb R^3$ (cf. [CM]).
The inequality was used by Simon in [Si] to show, among other things, that stable minimal hypercones of $\mathbb R^{n+1}$ must be planar for $n\leq 6$ and it was subsequently used to infer curvature estimates for stable minimal hypersurfaces, generalizing the classical work of Heinz [He], cf. [SSY], [CS] and [SS]. Simons also pointed out that there is a nonplanar stable minimal hypercone in $\mathbb R^8$, cf. Simons cone.
[CS] H. I. Choi, R. Schoen, "The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature", Invent. Math., 81, (1985) pp. 387-394.
[CM] T. H. Colding, W. P. Minicozzi III, "A course in minimal surfaces", Graduate Studies in Mathematics, AMS, (2011).
[He] E. Heinz, "Ueber die Loesungen der Minimalflaechengleichung" Nachr. Akad. Wiss. Goettingen Math. Phys. K1 ii, (1952) pp. 51-56
[SS] R. Schoen, L. Simon, "Regularity of stable minimal hypersurfaces" Comm. Pure App. Math., 34, (1981), pp. 741-797.
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[Si] J. Simons, "Minimal varieties in riemannian manifolds" Ann. of Math., 88 (1968) pp. 62-105 MR233295 Zbl 0181.49702
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CDK4/6 inhibitor palbociclib suppresses IgE-mediated mast cell activation
Yi-Bo Hou1 na1,
Kunmei Ji1 na1,
Yue-Tong Sun1,
Li-Na Zhang1 &
Jia-Jie Chen ORCID: orcid.org/0000-0002-1372-84501
Journal of Translational Medicine volume 17, Article number: 276 (2019) Cite this article
Mast cell activation causes degranulation and release of cytokines, thereby promoting inflammation. The aim of this study was to investigate the inhibitory effect of CDK4/6 inhibition on mast cell activation in vitro and in vivo.
RBL-2H3 rat basophilic leukemia cells (BLCs) and mouse bone marrow-derived mast cells (BMMCs) were sensitized with anti-dinitrophenol (DNP) immunoglobulin (Ig)E antibodies, stimulated with DNP-human serum albumin (HSA) antigens, and treated with the CDK4/6 inhibitor palbociclib. Histological stains were applied to reveal cytomorphological changes. Murine IgE-mediated passive cutaneous anaphylaxis (PCA) and ovalbumin (OVA)-induced active systemic anaphylaxis (ASA) models were used to examine palbociclib effects on allergic reactions in vivo. Western blots were performed to detect the expression of cell signaling molecules associated with mast cell activation.
Activated BLCs and BMMCs released copious granule-related mediators (histamine and β-hexosaminidase), which was reduced by palbociclib in a concentration-dependent manner. Palbociclib inhibited expression of the mast cell activation marker CD63 in activated BLCs and inhibited granule release (visualized with toluidine blue staining) while preventing morphological changes, (elongated shape maintained) and filamentous actin (F-actin) reorganization. Palbociclib suppressed molecular Lyn and/or mitogen-activated protein kinase (MAPK) signaling associated with mast cell activation in stimulated BLCs and attenuated allergic reactions in PCA mice dose dependently. Palbociclib attenuated body temperature reduction and diminished serum histamine levels in ovalbumin OVA-challenged ASA mice.
Palbociclib suppresses IgE-mediated mast cell activation in vitro and in vivo, suggesting that it may be developed into a therapy for mast cell-mediated allergic diseases via inhibition of mast cell degranulation.
Common allergic diseases, including asthma, allergic rhinitis, and specific dermatitis, are consequent to hypersensitive immune reactions [1]. In a given year, approximately one in five people in the world are affected by allergic diseases [2]. Socioeconomic development has been associated with an increasing incidence of allergic diseases year over year [3, 4]. Importantly, mast cells, which are major innate immunity effector cells, play a principal role in inducing allergic inflammation by releasing various mediators, including lipid mediators, chemokines, and cytokines [5]. Thus, mast cells are an attractive target for the treatment of allergic inflammation.
Mast cell activation, which plays a key role in inducing IgE-mediated allergic inflammation, depends on cross-linking of antigen immunoglobulin (Ig)E complexes with the high affinity IgE receptor, commonly referred to as FcεRI, on the surface of mast cells [1, 6]. The subsequent mast cell degranulation that ensues can trigger acute inflammatory reactions and promote chronic allergy progression by secreting histamine, proteases, and chemotactic factors, as well as by engaging in de novo synthesis of inflammatory cytokines [5, 7]. During an acute allergic response, histamine, which is a well-established vasodilator, also acts to increase vascular permeability, leading to a low body temperature and leukocyte extraversion from the circulation into local tissues [8]. Therefore, suppression of mast cell activation has the potential to attenuate allergic inflammation [9].
Antihistamine and steroid drugs are common clinical therapies used to treat allergic diseases [10, 11]. Additionally, small molecule inhibitors targeting leukotrienes or histamine receptors have been developed to treat allergic diseases [12]. Mast cell stabilizers that inhibit activated mast cell release (e.g. sodium cromoglycate, nedocromil, and lodisa) have emerged as another potential allergy treatment approach [13, 14]. Whereas these treatments target allergy symptom control, blockade of mast cell activation represents an opportunity to alleviate the immune dysfunction underlying allergic diseases more directly [15].
Palbociclib (IBRANCE; PD0332991; Pfizer; C24H29N7O2) is an orally available drug approved by the US FDA for the treatment of cancers [16]. Notably, it was approved as a first-line treatment of estrogen receptor-positive (ER+)/human epidermal growth factor receptor 2-negative (HER-) advanced breast cancer based on PALOMA-1 study findings [16, 17]. Palbociclib, is a selective cyclin-dependent kinase (CDK)4/6 inhibitor, with low enzymatic half-maximal inhibitory concentrations for CDK4 (11 nM) and CDK6 (15 nM), that inhibits retinoblastoma protein phosphorylation in early G1 phase, leading to cell cycle arrest and thus suppression of cell proliferation [17].
The effects of CDK4/6 inhibitors, such as palbociclib, on mast cell activation and allergic reactions remain to be clarified.
The aim of this study was to investigate potential anti-allergic effects of palbociclib on IgE-mediated mast cell activation. We sensitized mast cells with anti-dinitrophenol (DNP) IgE antibodies and then used DNP-human serum albumin (HSA) antigen stimulation to activate the sensitized mast cells in vitro. We used a murine IgE-mediated passive cutaneous anaphylaxis (PCA) model and ovalbumin (OVA)-induced active systemic anaphylaxis (ASA) model to examine the effects of palbociclib on allergic reactions in vivo. Finally, we explored the molecular mechanisms underlying palbociclib effects on IgE-mediated mast cell activation.
Reagents and antibodies
Palbociclib was purchased from Med Chem Express (Monmouth Junction, NJ). Monoclonal DNP-specific IgE, DNP-HSA, and 4-nitrophenyl N-acetyl-β-D-glucosaminide were obtained from Sigma-Aldrich (St. Louis, MO). Evans blue, formamide, toluidine blue and mast cell stabilizer ketotifen were obtained from Dalian Meilun Biotechnology Co. Ltd. (Dalian, China). Antibodies targeting the tyrosine-protein kinase Lyn, Tyr397 phosphosphorylated (p)-Lyn, mitogen-activated protein kinase (MAPK) p38, c-Jun N-terminal kinase (JNK) (Abcam, Cambridge, MA), extracellular signal-regulated kinase (ERK)1/2, p-p38 (Thr180/Tyr182), p-JNK (Thr183/Tyr185), glyceraldehyde 3-phosphate dehydrogenase (GAPDH; Santa Cruz Biotechnology, Santa Cruz, CA), and p-ERK1/2 (Thr202/Tyr204) (p-ERK1/2) (Cell Signaling Technology, Beverly, MA) were used. Fluorescein isothiocyanate (FITC)-phalloidin was purchased from Yeasen Biotech Co. Ltd. (Shanghai, China). Lyn inhibitor Bafetinib, ERK inhibitor U0126, JNK inhibitor SP600125 and p38 inhibitor SB203580 were purchased from MedChem Express (Monmouth Junction, NJ, USA). Silencing RNAs (SiRNAs) for knocking down the expression of Lyn were designed and obtained from Gene Pharma (Shanghai, China).
Female BALB/c mice (4–5 weeks old) were purchased from Guangdong Medical Laboratory Animal Center (Foshan, China), and housed in a specific pathogen-free environment with a relatively stable temperature (24 ± 1 °C) and humidity (55 ± 10%) for 1 week before experimentation. The mice were used to isolate bone marrow-derived mast cells (BMMCs) as well as for our PCA and ASA models. All studies involving mice were performed according to protocols approved by the Animal Care and Use Committee of the School of Medicine of Shenzhen University.
RBL-2H3 rat basophilic leukemia cells (BLCs; Cellcook Biotechnology Co., Guangzhou, China) were cultured in complete Dulbecco's modified eagle medium with 4.0 mM l-glutamine with sodium pyruvate penicillin (100 U/ml), 100 μg/ml streptomycin, non-essential amino acids, and 10% fetal bovine serum in a humidified incubator at 37 °C, 5% CO2. Mouse bone marrow-derive mast cells (BMMCs) were isolated from BALB/c mouse femurs and cultured in complete RPMI-1640 with 100 U/ml penicillin, 100 μg/ml streptomycin, 2 mM l-glutamine, 1 mM sodium pyruvate, 10 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid, 10% fetal bovine serum, and 10 ng/ml interleukin (IL)-3 and 10 ng/ml stem cell factor. After 4–6 weeks in culture, mast cell purity reached ≥ 95%, as indicated by flow cytometry detection of cell-surface CD117 and FcεRI expression [18, 19].
Measurement of β-hexosaminidase
BLCs and BMMCs were dispensed into 24 well-plates, sensitized with 50 ng/ml anti-DNP IgE for 12 h, and washed with Tyrode's buffer. Palbociclib or other small molecule inhibitors (Lyn inhibitor bafetinib, ERK inhibitor U0126, JNK inhibitor SP600125 and p38 inhibitor SB203580) was then applied for 1 h. Subsequently, after stimulating with 100 ng/ml DNP-HSA for 30 min, the supernatant and cell lysate were reacted for 1.5 h with 1 mM -nitrophenyl-N-acetyl-β-d-glucosaminide at 37 °C, followed by quenching with 150 μl carbonate buffer. Optical density (OD) at 405 nm was measured with microplate reader (Bio-Rad, USA) [19,20,21]. We used the following equation to calculate percent β-hexosaminidase release (% degranulation):
$$ \frac{{\left( {\text{Experimiental release} - \text{Tyrode's release}} \right)}}{{\left( {\text{Triton-X-100\,release} - \text{Tyrode's release}} \right)}}{ \times 100}. $$
Cell viability assay
BLCs (2 × 103/well) and BMMCs (1 × 104/well) were cultured in 96-well plates for 24 h and then treated with palbociclib for 24 h (as above). Cell viability was measured with Cell Counting Kit 8 (Med Chem Express, Monmouth Junction, NJ) according to the manufacturer's protocol.
Histamine release measurement
BLCs were placed in 24 well-plates (2 × 105 cells/well), sensitized with 50 ng/ml anti-DNP IgE for 12 h, washed twice with Tyrode's buffer, and then treated with experimentally indicated concentrations of palbociclib for 1 h. After stimulating with 100 ng/ml DNP-HSA for 20 min, histamine levels in culture media were determined with enzyme-linked immunosorbent assay (ELISA) kits (IBL, Germany), according to the manufacturer's protocol [21].
CD63 expression can be used as an index of mast cell degranulation [22]. After stimulation with DNP-HSA, BLCs pretreated with or without palbociclib were washed with phosphate buffered saline (PBS), labeled with PE-CD63 antibody (Millenia Biotec, Auburn, CA), and detected by flow cytometric analysis (Cytoflex flow analyzer, Beckman Coulter, CA).
Toluidine blue staining
Toluidine blue staining was used to reveal mast cell activation as evidenced by deposition of metachromatic granules against a pale blue background [23, 24]. BLCs were incubated with 250 μl of 4% paraformaldehyde/PBS for 30 min at room temperature. The fixed cells were stained with 300 μl of toluidine blue dye (1% w/v in 0.9% saline solution, pH 2.5) for 30 min. The stained cells were observed with an inverted microscope (Carl Zeiss, Goettingen, Germany) [23].
Microfilament staining
Filamentous actin (F-actin) is involved in mast cell degranulation [24, 25]. Because phalloidin binds F-actin specifically, we used FITC-phalloidin staining to observe F-actin changes. BLCs were fixed in 4% paraformaldehyde in PBS for 30 min, then washed twice with PBS and permeated with 0.1% T-X100 PBS for 3 min. After washing with PBS, cellular F-actin was stained with 100 nM FITC-phalloidin for 30 min. Cells were imaged with a fluorescence microscope (Carl Zeiss, Goettingen, Germany) via a FITC channel (excitation/emittance = 496/516 nm) [25].
SiRNA transfection
Rat Lyn-specific siRNA sequences with the following sequences were designed and synthesized by GenePharma Co. (Shanghai, China): 5′-GGACAUAACAAGGAAA GAUTTAUCUUUCCUUGUUAUGUCCTT-3′(siRNA-1) and 5′-CCAUGGGAUAA AGAUGCUUTTAAGCAUCUUUAUCCCAUGGTT-3′ (siRNA-2). The negative control siRNA sequence was 5′-UUCUCCGAACGUGUCACGUTT-3. These siRNAs were transfected into BLCs with the aid of lipofectamine® RNAiMAX reagent (Invitrogen, Carlsbad, CA) according to the manufacturer's protocol. Following confirmation of the siRNA effect by western-blotting assay, activated siRNA-transfected BLCs were subjected to β-hexosaminidase assays following experimentally indicated pretreatments.
We used western blots to investigate the effects of palbociclib on the activation of the FcεRI signaling regulatory proteins Lyn and the MAPK signaling proteins p38, JNK, and ERK1/2, which have bene reported to be involved in mast cell activation [26]. BLCs were sensitized with 50 ng/ml DNP-IgE overnight and washed twice with PBS and placed in fresh media. After incubating with or without palbociclib at 37 °C for 1 h, BLCs were stimulated with 100 ng/ml DNP-HSA and then washed twice with PBS. Cells were lysed in RIPA buffer (Beyotime, Beijing, China) with a protease-inhibitor cocktail (Med Chem Express, Monmouth Junction, NJ). Cell lysates were centrifuged at 12,000 rpm for 15 min. Supernatants were mixed with a loading sample buffer (Thermo Fisher Scientific) and denatured by heating for 10 min at 100 °C. Proteins were separated by sodium dodecyl sulfate–polyacrylamide gel electrophoresis and transferred to polyvinylidene difluoride membranes (Merck Millipore, Billerica MA). After incubating with a primary antibody in tris-buffered saline with 0.1% tween 20 buffer that contained 5% bovine serum albumin or skim milk. The membranes were incubated with horse-radish peroxidase-conjugated secondary antibodies for 1 h at room temperature in the same buffer. Chemiluminescence reagents (Meilun, Dalian, China) were applied according to the manufacturer's protocol.
PCA model
PCA mice were used to examine the effects of palbociclib on IgE-mediated allergic reaction in vivo. The PCA model is an acute allergic animal model wherein allergic reactions are induced by antigen stimulation of mast cells in ear skin [19, 20]. Briefly, BALB/c mice (female, 4–5 weeks-old, 5 per group) were injected intradermally with 0.5 μg of DNP-IgE in the left ear. After a 24-h infiltration period, palbociclib (25 mg/kg or 50 mg/kg) or ketotifen (100 mg/kg, positive control) dissolved in physiological saline was injected intraperitoneally [19,20,21]. One hour later, 200 μl of 5 mg/ml Evans blue solution containing 0.1 mg/ml DNP-HSA was administered into the tail vein. The mice were euthanized by cervical dislocation an hour after Evans blue injection. The dye was extracted from dissected ears in 700 μl of formamide for 12 h at 62 °C and quantitated by a spectrophotometer at 620 nm. Ear thickness was measured with a dial thickness gauge.
ASA model
The ASA model has been used to examine immediate-type hypersensitivity, which has been shown to be strongly associated with mast cells [21]. BALB/c mice (female, 4–5 weeks-old, 5 per group) were sensitized with OVA (100 μg OVA and 2 mg alum adjuvant in 200 μl PBS), by intraperitoneal injection on day 0 and day 7 as described previously [21]. Subsequently, palbociclib (50 mg/kg) and ketotifen (50 mg/kg) injections were given intraperitoneally on days 9, 11, and 13. The animals were challenged with OVA injections following sensitization. On day 14, 200 μg of OVA was injected intraperitoneally, and rectal temperature was measured every 10 min for 90 min. After 90 min, a blood sample was obtained from the tail of each mouse. Blood IL-4 and IL-10 levels were measured by ELISA (Shanghai Huzhen Biotechnology Co., Ltd. Shanghai, China) according to the manufacturer's protocol.
The data are presented as the means with standard deviations (SDs) of at least three independent experiments for cell experiments and of 5 mice per group for animal experiments. Statistical analyses were performed in Prism 7.0 (GraphPad Software, Inc.). One-way analyses of variance (ANOVAs) and Dunnett's post‑hoc tests for multiple comparisons were applied to detect inter-group differences with a significance criterion of p < 0.05.
Palbociclib inhibited mast cell degranulation without cytotoxicity
Cell viability assays revealed no evidence of palbociclib cytotoxicity in BLCs or BMMCs (Fig. 1a) following 24 h of exposure at concentrations < 100 μM. Palbociclib treatment reduced levels of released granule-related mediators (histamine and β-hexosaminidase) dose dependently in DNP-HSA activated BLCs (Fig. 1b, c). Similar degranulation inhibition was seen in BMMCs treated with palbociclib (Fig. 1e). Cytometric analysis of palbociclib-treated BLCs showed dose-dependent inhibition of the upregulation of the expression of the mast cell activation marker CD63 (Fig. 1d).
Palbociclib attenuation of mast cell degranulation. a Cell viability of B LCs following DNP-HSA challenge with or without 1-h palbociclib pretreatment (BLCs had been anti-DNP IgE-sensitized and incubated with palbociclib for 24 h prior to the challenge). b β-Hexosaminidase release from BLCs in a. c Histamine release from BLCs in A. d Flow cytometric analysis of CD63 expression in BLCs. e β-Hexosaminidase release from BMMCs stimulated with 100 ng/ml DNP-HSA for 30 min (prior 50 ng/ml DNP-specific IgE priming with or without 1-h palbociclib pretreatment). Means ± SDs of 3 independent experiments are shown; *p < 0.05, **p < 0.01 vs. the HSA-DNP group
Palbociclib inhibited activation-associated morphological changes in mast cells
Non-activated BLCs had an elongated shape with purple intra-cellular particles, whereas activated BLCs had irregular shapes and released purple particles extracellularly. Palbociclib inhibited activation-associated morphological changes and degranulation (particle release) in activated BLCs significantly (Fig. 2a). Non-activated BLCs had a spindle shape and uniformly distributed F-actin. Activated BLCs became elliptical in accordance with their F-actin cytoskeletal changes. Pretreatment with palbociclib inhibited activation-associated shape changes and F-actin cytoskeleton decomposition in activated BLCs (Fig. 2b).
Palbociclib inhibition of morphological changes associated with activation in mast cells. Anti-DNP IgE-sensitized BLCs were pretreated (or not) with palbociclib for 1 h and then challenged with DNP-HSA (100 ng/ml) for 30 min. a Toluidine blue-stained RBCs. Blue arrows indicate irregular cell morphology and the release purple particles. b The statistical data were from 3 independent experiments. c FITC-phalloidin stained BLCs. Red arrow indicates cell morphology irregularity due to decomposition of F-actin cytoskeleton. d The data are summarized from 3 independent experiments, *p < 0.05, **p < 0.01, compared to the HSA-DNP group
Palbociclib inhibited signaling pathways involved in mast cell activation
Palbociclib pretreatment, prior to a DNP-HSA challenge, inhibited activation of Lyn as well as activation of MAPKs (p38, JNK, and ERK) in BLCs, as evidenced by decreased levels of p-Lyn, p-p38, p-JNK, and p-ERK1/2 (Fig. 3a, b). The effects of palbociclib on Lyn and MAPK signaling molecules exhibited dose dependence. Experiments with small molecule inhibitors or siRNAs intended to investigate whether palbociclib suppression of mast cell degradation involves down-regulation of Lyn and MAPK signaling showed no statistically significant differences in β-hexosaminidase release levels in DNP-HSA activated BLCs between cells treated with palbociclib alone and cells treated with palbociclib plus the Lyn signaling inhibitor bafetinib (Fig. 3c). Lyn expression was reduced in siRNA-treated BLCs (Fig. 3d). Similarly, down-regulation of Lyn expression in BLCs did not affect palbociclib inhibition of mast cell activation, as evidenced by detection of β-hexosaminidase release (Fig. 3e). Similar results were obtained with the ERK inhibitor U0126, the JNK inhibitor SP600125, and the p38 inhibitor SB203580 (Fig. 3f). These results suggest that palbociclib inhibition of mast cell activation may involve suppression of Lyn and MAPK signaling.
Palbociclib-mediated reduction in levels of phosphorylated proteins that are involved in mast cell activation. Western blots of total lysate samples from activated mast cells (see "Materials and methods") demonstrating palbociclib treatment (0 μM, 10 μM, 20 μM, and 40 μM) inhibition of Lyn activation (a) and MAPK (p38, JNK, and ERK1/2) activation (b). c β-Hexosaminidase release from activated BLCs subjected to 1-h pretreatments (as experimentally indicated) prior to the challenge. d Western blot analysis of Lyn expression in RBL-2H3 cells transfected with siRNA-Lyn. e β-Hexosaminidase release from activated BLCs pretreated with siRNA-Lyn and palbociclib. f β-Hexosaminidase release from activated BLCs pretreated with signaling inhibitors (ERK inhibitor U0126, JNK inhibitor SP600125, and p38 inhibitor SB203580) for 1 h prior to the challenge
Palbociclib attenuated PCA in vivo
Inhibition of mast cell activation was confirmed in a positive control group treated with the mast cell stabilizer ketotifen. When ears were injected with 4% Evans blue dye mixed with antigen in PCA tests, they showed marked thickening in response and Evans blue solution poured out from PCA reaction sites, demonstrating vascular hyperpermeability. Dye solution extravasation and ear thickening were inhibited by 25 mg/kg or 50 mg/kg palbociclib (Fig. 4).
Palbociclib suppression of PCA in vivo. Mouse ear skin (N = 5/group) was sensitized with anti-DNP IgE, treated (or not) with palbociclib, and injected with 20 μg DNP-HSA containing 1% Evans blue as described in detail in "Materials and methods". Ketotifen treatment was as positive control. a Representative images of PCA mouse ears. b Representative photomicrographs of PCA ear tissue sections. Palbociclib reverses PCA-induced ear thickening (c) and PCA-induced increases in Evans Blue OD620nm. Means ± SDs of 3 independent experiments are shown; *p < 0.05, **p < 0.01 vs. control
Palbociclib attenuated ASA in vivo
Mice were sensitized by repeated administration of OVA with alum adjuvant, and anaphylaxis was induced with an intraperitoneal OVA challenge, as shown in Fig. 5a. Ketotifen treatment was as positive control. OVA mice exhibited decreasing rectal temperatures 30–50 min after the OVA challenge injection, and these temperature reductions were attenuated by palbociclib (Fig. 5b). Concomitantly, total serum IL-4 and IL-10 levels reflective of inflammation were increased after the OVA challenge and those increases were suppressed by palbociclib (Fig. 5c, d).
Palbociclib suppression of ASA in vivo. a ASA model protocol (N = 5/group). Ketotifen treatment was as positive control. Palbociclib prevents ASA-induced reductions in body temperature b as well as ASA-induced increases in IL-10 (c) and IL-4 (d) serum levels in ASA mice (determined by ELISA). Means ± SDs of 3 independent experiments are shown; *p < 0.05, **p < 0.01 vs. control
In the present study, palbociclib exhibited inhibitory effects on mast cell degranulation, as evidenced by reduced release of histamine and β-hexosaminidase by DNP-IgE/HAS-stimulated BLCs and BMMCs. In our in vivo experiments, palbociclib attenuated the DNP-IgE/HSA-induced PCA reaction (Evans blue extravasation) dose-dependently and suppressed ASA responses (i.e. OVA challenge-induced body temperature reduction and serum IL-4 or IL-10 level increases). Together, these results show that the CDK4/6 inhibitor palbociclib can suppress IgE-mediated mast cell activation in vitro and in vivo.
During mast cell degranulation, morphological changes occur due to the actions of contractile microfilaments [29]. Mast cell activation via aggregation of IgE-FcεRI complexes causes degranulation and release of proinflammatory mediators, and these processes involve F-actin reorganization [25, 29]. Our findings showing that pretreatment with palbociclib inhibited cell shape changes and cytoskeletal decomposition in BLCs resemble previously reported inhibitory effects of coptisine on mast cell degranulation [24].
Activation of protein tyrosine kinases is the earliest detectable signaling response to FcεRI cross-linking on mast cells [26]. Upon tyrosine kinase activation, molecular mechanisms involving the Src family kinase Lyn [30] and/or intracellular signaling via MAPKs [26, 31] promotes mast cell degranulation. Mast cell activation inhibitors, including the corticosteroid dexamethasone [21], have been shown to suppress mast cell activation by down-regulating signaling pathways. Here, similarly, we observed that palbociclib suppressed Lyn activation and MAPK pathway signaling.
Drug repurposing, wherein previously developed pharmacotherapies are used in new applications, can accelerate clinical drug discovery and development [32]. The major advantages of drug repurposing are previously established low cytotoxicity and pharmacokinetic activity [33]. For example, metformin, a widely used anti-diabetic drug [34] that also has anti-cancer effects [35,36,37], has been shown to inhibit IgE and aryl hydrocarbon receptor-mediated mast cell activation in vitro and in vivo [38]. Additionally, the antidiarrheal medicine berberine, which is widely used for gastrointestinal ailments such as bacterial gastroenteritis and dysentery [39], has been found to suppress mast cell-mediated allergic responses via down-regulation of FcɛRI activation and MAPK signaling [40]. Hence, searching among existing drugs for pharmacotherapies that can be repurposed into novel allergic disease treatments that target mast cell activation represents a promising strategy.
Currently, palbociclib is being used as an anti-cancer drug. It inhibits the growth of cancer cells by inhibiting CDK4/6. In 2017, it was approved by the US FDA as an adjuvant therapy, with endocrine blockers, for ER+/HER2− advanced breast cancer in postmenopausal women. Pooled safety analysis (PALOMA trial) revealed a peak incidence of adverse events in the first 6 months of treatment, with a subsequent decrease in incidence over time [41]. In clinical use, oral palbociclib is generally well-tolerated at its oncological dosage of 125 mg daily on a 21-day on, 7-day off schedule [42, 43]. When administered at a dose of 150 mg/kg, palbociclib does not affect body weight in mice [44]. Palbociclib's low toxicity profile and ability to inhibit mast cell activation in our murine models suggest that palbociclib could be used to alleviate IgE-mediated allergic diseases in human patients.
Palbociclib inhibition of cell growth by way of cell cycle arrest [45, 46], suggests that mast cell activation may require unimpeded cell cycle progression. Upregulation of CDK6 expression can enhance MAPK and NF-κB signaling [47, 48], both of which have been associated with mast cell activation [26]. Previous demonstrations showing that palbociclib can inhibit neoplastic mast cell proliferation [49] did not indicate whether palbociclib also inhibits mast cell activation, and specifically cell degradation, a key allergic disease treatment target. Hence, given palbociclib's CDK inhibitory actions and its ability to suppress signaling molecule expression and/or activation, we hypothesize that palbociclib administered at a low-cytotoxicity dose may exert anti-IgE-mediated allergy effects by down-regulating of MAPK and/or NF-κB signaling downstream of CDK6 inhibition via FcεRI, whose activation may be modulated by the Src family kinase Lyn [30]. Lyn, interacting with FcεRIβ, is indispensable for FcεRI-mediated human mast cell activation, and specific inhibition of Lyn signaling may represent a new therapeutic strategy for the treatment of human allergic diseases [50]. Our results showed that palbociclib inhibition of mast cell activation may involve suppression of Lyn and/or MAPK signaling.
Interestingly, palbociclib reduced the total serum IL-10 levels in the ASA model. IL-10 is secreted by a wide variety of cell types, even including mast cells [51]. IL-10 enhances IgE-mediated mast cell responses and is necessary mediator of allergy development in vivo [51]. Additionally, IL-10 has been shown to be critical for Th2 responses in a murine allergy model [52]. The present results suggest that palbociclib inhibition of mast cells may reduce IL-10 levels, or even Th2 cells. These possibilities need to be examined directly with further experimentation.
In conclusion, our findings showing that the CDK4/6 inhibitor palbociclib can suppress IgE-mediated mast cell activation in vitro and in vivo suggest that palbociclib is a potential therapeutic candidate for treating mast cell-mediated allergic diseases, including allergic rhinitis, atopic dermatitis, and anaphylaxis [27, 28]. Inhibition of mast cell activation—an important pathogenic process in IgE-induced allergic reactions [11]—represents a novel strategy for relieving allergic symptoms and treating allergic reactions.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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We thank the other members of Ji Kunmei lab for their critical comments.
The present study was supported in part by research funding from the National Natural Science Foundation of China (Grant Nos. 81571570 and 81602595), Guangdong Province (Grant Nos. 2016A030313039 and 2017A010105014), and Shenzhen City 2016 Biochemistry Discipline Construction, the Natural Science Foundation of Shenzhen City (JCYJ20170818142053544).
Yi-Bo Hou and Kunmei Ji contributed equally to this work
Department of Biochemistry and Molecular Biology, School of Medicine, Shenzhen University, Shenzhen, 518060, People's Republic of China
Yi-Bo Hou
, Kunmei Ji
, Yue-Tong Sun
, Li-Na Zhang
& Jia-Jie Chen
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Conceived and designed the study: JKM and CJJ. Analyzed the data: CJJ and CJJ. Performed the studies: CJJ, HYB and ZLN. Contributed reagents/materials/analysis tools: CJJ and JKM. Wrote the manuscript: HYB, CJJ and JKM. All authors read and approved the final manuscript.
Correspondence to Jia-Jie Chen.
All studies involving mice were performed according to protocols approved by the Animal Care and Use Committee of the School of Medicine of Shenzhen University.
Hou, Y., Ji, K., Sun, Y. et al. CDK4/6 inhibitor palbociclib suppresses IgE-mediated mast cell activation. J Transl Med 17, 276 (2019). https://doi.org/10.1186/s12967-019-2026-9
Accepted: 14 August 2019
Mast cells
CDK inhibitor
Drug repurposing
Clinical translation | CommonCrawl |
A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity
Mathematical analysis of an age-structured HIV model with intracellular delay
doi: 10.3934/dcdsb.2021149
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Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system
Gengen Zhang ,
South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China
* Corresponding author: Gengen Zhang
Received October 2020 Early access May 2021
Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No.11701110), the China Postdoctoral Science Foundation (Grant No.2020M682746)
Figure(7) / Table(3)
In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.
Keywords: Quantum Zakharov system, time splitting method, exponential wave integrator method, soliton-soliton collisions, pattern dynamics.
Mathematics Subject Classification: Primary: 35Q55, 65M06, 65M70; Secondary: 65M12.
Citation: Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021149
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Figure 1. Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.1 $
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Figure 2. Inelastic collision between two solitons in Example $ 4.2 $ under case (i)
Figure 3. Inelastic collision between two solitons in Example $ 4.2 $ under case (ii)
Figure 4. Inelastic collision between two solitons in Example $ 4.2 $ under case (iii)
Figure 5. Pattern dynamics of QZS in Example $ 4.3 $, contours of $ |E| $
Figure 7. Plot of energy density $ |E|^2 $ and ion density fluctuation $ N $ in Example $ 4.5 $, $ \mu = \nu = 20 $, $ \varepsilon = \frac{1}{2^5} $
Table 1. Spatial errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ \tau = 10^{-5} $
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
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Table 2. Temporal errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
Table 3. Temporal errors of the scheme at $ T = 1 $ for Example $ 4.4 $ with different $ \varepsilon $, $ h = 1/2^4 $
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
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Gengen Zhang | CommonCrawl |
Molecular generative model based on conditional variational autoencoder for de novo molecular design
Jaechang Lim1,
Seongok Ryu1,
Jin Woo Kim1 and
Woo Youn Kim1, 2Email authorView ORCID ID profile
Journal of Cheminformatics201810:31
Received: 14 March 2018
We propose a molecular generative model based on the conditional variational autoencoder for de novo molecular design. It is specialized to control multiple molecular properties simultaneously by imposing them on a latent space. As a proof of concept, we demonstrate that it can be used to generate drug-like molecules with five target properties. We were also able to adjust a single property without changing the others and to manipulate it beyond the range of the dataset.
Molecular design
Conditional variational autoencoder
The ultimate goal of molecular design for new materials and drugs is to directly generate molecules with the desired properties. This is apparently challenging work because a molecular space is extraordinarily vast, discrete, and disorganized with diverse types of molecules. For instance, \(10^{8}\) molecules have been synthesized [1], whereas it is estimated that there are \(10^{23}\)–\(10^{60}\) drug-like molecules [2]. Despite advances in experimental techniques, it is too demanding to find molecules suitable for specific applications only through experiments.
Computer-aided molecular design has attracted much attention as a promising solution to overcome the experimental limitation [3–6]. Fast calculation methods along with reasonable accuracy and very low cost enable high-throughput virtual screening to find molecules with target properties. A common strategy is to select computationally top molecules out of millions of molecules in a virtual library and then verify them experimentally, leading to a significant reduction in time and efforts. Molecules in the library may not meet the given criteria. In this case, traditional optimization methods such as a genetic algorithm can be used to further improve molecular properties beyond the criteria by structural modifications [7–9]. However, they have a fundamental limitation in terms of efficiency because many trials and errors are inevitable to optimize molecular properties in a huge molecular space.
Recently emerging generative models based on deep learning techniques may offer a viable solution for more efficient molecular design. Gómez-Bombarelli et al. adopted a variational autoencoder [10] to optimize the molecular properties in a latent space in which molecules are expressed as a real vector [11]. The key advantage of this method is that a gradient-based optimization becomes feasible because the latent space is continuous and differentiable. It has been successfully applied to improving the partition coefficient of drug candidates and the delayed fluorescent emission rate of organic light emitting diode candidates. Blaschke et al. employed the adversarial autoencoder [12] (AAE) and the Bayesian optimization to generate ligands specific to the dopamine type 2 receptor [13]. Kadurin et al. [14] compared the VAE and AAE as a molecular generation model in terms of the reconstruction error and variability of the output molecular fingerprints. In addition to those autoencoder-based models, a generative model developed for natural language processing has also been used for molecular design [15–18]. Molecular structures can be expressed with SMILES. Then, this model learns the probability distribution of the next character of a given piece of SMILES. Yuan et al. [16] designed potential inhibitors for a target protein and tested them in experiments. Based on the natural language processing model, Segler et al. [17] and Gupta et al. [18] applied transfer learning to molecular design for certain biological activities. This approach is especially useful when there is not enough data to train artificial neural networks in the normal way. Olivecrona et al. [19], Guimaraes et al. [20], and Jaques et al. [21] proposed a reinforcement learning method to modify a pre-trained molecular generative model so as to impose several properties in molecules generated from the generative model.
We note that various molecular properties are correlated with each other. Therefore, adjusting one target property by structural modifications may cause an undesired change in other properties. To avoid this problem in rational molecular design, one has to control several properties at the same time. Here, we propose a molecular generative model using the conditional variational autoencoder (CVAE) [22] suitable for multivariable control. In addition to the advantages of using the latent space, our method can incorporate the information of molecular properties in the encoding process and manipulate them in the decoding process.
As a proof of concept, we used the CVAE to generate drug-like molecules satisfying five target properties at the same time: molecular weight (MW), partition coefficient (LogP), number of hydrogen bond donor (HBD), number of hydrogen acceptor (HBA), and topological polar surface area (TPSA). We were able to produce a number of molecules with the specific values of the five target properties within a given range. It was also possible to adjust a single target property without changing the others. Furthermore, we were able to generate molecules with properties beyond the range of the database.
Conditional variational autoencoder (CVAE)
We selected the CVAE as a molecular generator. It is one of the most popular generative models which generates objects similar to but not identical to a given dataset. In particular, it is distinguished from the VAE in that it can impose certain conditions in the encoding and decoding processes. To elucidate the difference between VAE and CVAE, we compared their objective functions with one another. The objective function of the VAE is given by
$$\begin{aligned} E[\text {log}P(X|z)]-D_{KL}[Q(z|X)\parallel P(z)], \end{aligned}$$
where E denotes an expectation value, P and Q are probability distributions, \(D_{KL}\) is the Kullback-Leibler divergence, and X and z indicate the data and latent spaces, respectively. The first and second terms are often called the reconstruction error and the KL term, respectively. In an autoencoder, Q(z|X) and P(X|z) are approximated by an encoder and a decoder, respectively. A key difference of the CVAE from the VAE is to embed the conditional information in the objective function of the VAE, leading to the revised objective function as follow:
$$\begin{aligned} E[\text {log}P(X|z,c)]-D_{KL}[Q(z|X,c)\parallel P(z|c)], \end{aligned}$$
where c denotes a condition vector. The condition vector c is directly involved in the encoding and decoding processes. In our model, the molecular properties we want to control were represented as the condition vector. As a result, the CVAE can generate molecules with the target properties imposed by the condition vector.
Incorporating molecular properties in the VAE to generate molecules with desirable properties are also possible through a two-step model proposed by Gómez-Bombarelli et al. In this method, the VAE is trained jointly with an additional neural network for property prediction. Subsequently, a Gaussian process model creates a mapping from the resulting latent space to the associated molecular properties. Finally, property optimization in the resulting latent space is performed by a gradient descent optimization method.
The key difference of our CVAE model from the jointly trained VAE model is that the molecular properties are directly incorporated into both the encoder and decoder. The resulting latent vector is composed of two parts: the first part is for the target molecular properties, while the second part involves the molecular structures and the other properties. Therefore, the desired molecular properties can be embedded in a target molecular structure simply by setting a condition vector. In other words, one can control the structure and the properties independently except for some cases in which the properties are strongly coupled to a molecular scaffold. This is particularly useful to incorporate a certain property in a given molecule just with a marginal structure modification. After all, the CVAE is less sensitive to the continuity and smoothness of the latent space, because it does not require the derivative of the latent space with respect to the latent vector of the molecular structure. Another technical difference of the CVAE from the jointly trained VAE is that it does not need any further optimization process, which is inevitable in the jointly trained VAE for each different property value.
Molecular representation and model construction
We represented molecules with SMILES codes to take advantage of state-of-the-art deep learning techniques that are specialized in dealing with texts and sequences. Each SMILES code was canonicalized for a unique molecular representation. One 'E' was padded on the end of the SMILES code to indicate the end of the string. Subsequently, each character including 'E' is represented with a one-hot vector, resulting in an input matrix. Each one-hot vector of the input matrix is transformed to an embedding vector with the size of 300, and then the input matrix is concatenated with a predefined condition vector. The first, second, and last entries of the condition vector are filled with information consisting of the MW, LogP, and TPSA, respectively, while the remaining two entries are labeled by the HBD and HBA as shown in Fig. 1. The values of MW, logP, and TPSA are normalized from -1.0 to 1.0. HBD and HBA are expressed with a one-hot vector, because they are integer numbers.
The resulting matrix is subjected to the encoder of the CVAE to generate a latent vector. We adopted the so-called recurrent neural network (RNN) with an LSTM cell for both the encoder and decoder of the CVAE [23]. They are made of a 3-layer RNN with 500 hidden nodes on each layer. A softmax layer was used in each output of the decoder cell, and a cross entropy was used as the cost function of the reconstruction error. The latent vector concatenated with the condition vector becomes an input of the decoder at each time step of the RNN cell. Finally, the output vector of each decoder cell is transformed to a vector whose size is equal to that of the one-hot vector of the input matrix. The softmax activation function is applied to each transformed vector. The encoder and decoder are optimized to minimize the cost function of the CVAE. To generate a molecule with the target properties imposed by the condition vector, the cell of the RNN decoder are unrolled for 120 times. All characters before 'E' were taken in the stochastic write-out process, and if 'E' did not appear in the 120 characters, the result was considered as invalid. Each output vector of the decoder cell represents the probability distribution of the SMILES code characters and 'E'. Finally, the output vector is converted to a SMILES code. It should be noted that even a single wrong character in the resulting SMILES code gives rise to an invalid molecule. To increase the rate of valid SMILES codes, we used the stochastic write-out method which samples each character of SMILES according to a probability distribution. As a result, a single set of latent and condition vectors may give a number of different molecules. We performed 100 times the stochastic write-out per one latent vector and took all valid molecules except duplicated ones for later analysis.
Dataset and hyperparameters
RDKit [24], an open source cheminformatics package, was used for checking out the validity of the generated SMILES codes and calculating the five target properties of the molecules.
The total dataset is made of molecules randomly selected from the ZINC dataset [25]. Generally, with more data, the performance becomes better. Typical deep learning models need hundreds of thousands of data points. We checked out the convergence of the results with respect to the size of the data in our case. The use of 5,000,000 ZINC molecules did not increase both the validation and the success rates of generating molecules with the target properties compared to those from 500,000 ZINC molecules. Thus, we adopted the dataset of the 500,000 molecules, 80% of which were used for training, and the rest was used for the test. The distribution of the five target properties in the total dataset is shown in Fig. 2. The learning rate was set to 0.0001 and exponentially decayed at a rate of 0.97. The model was trained until converged. In the performance evaluation of the CVAE, if each target property of the generated molecules was different from the given target value with the 10% error range of the average value of the total dataset, we regarded those molecules as successful. The source code is available from GitHub (https://github.com/jaechanglim/CVAE).
Schematic representation of conditional variational autoencoder for molecular design
Distribution of molecular weight, LogP, HBD, HBA, and TPSA in the total dataset (500,000)
As the first application, we demonstrated that the CVAE method can generate molecules with specific values for the five target properties by applying it to Aspirin and Tamiflu. The values of the (MW, LogP, HBD, HBA, and TPSA) for Aspirin and Tamiflu are (180.04, 1.31, 1, 3, and 63.6) and (312.2, 1.285, 2, 5, and 90.64), respectively. The condition vector of each molecule was made by those values. Latent vectors to be concatenated with the condition vector were sampled by adding a Gaussian type noise to the latent vector of a molecule selected randomly in the training set. Figure 3a, b show nine molecules produced with the condition vector of Aspirin and Tamiflu, respectively. All of them had similar properties to those of Aspirin and Tamiflu within an error range of 10%, respectively. However, the molecular structures in Fig. 3 are considerably different from those of the original molecules because of the latent vectors chosen randomly from the training set.
The second application was to generate molecules similar in both properties and structure to the mother molecule by sampling latent vectors around that of the mother. Figure 4 shows the molecules generated in such a way from Aspirin. They look very similar to Aspirin and also have similar properties with those of Aspirin within an error range of 10%.
Molecules generated by the CVAE with the condition vector made of the five target properties of a Aspirin and b Tamiflu
Molecules generated by the CVAE with the condition vector made of the five target properties of Aspirin and the latent vector slightly modified from that of Aspirin
As the third case study, we tested whether the CVAE method can change only a single property without changing the others. The condition vector was constructed with the MW, HBD, HBA, and TPSA of Tamiflu, and we varied LogP from 0.0 to 3.0. Latent vectors were sampled around that of Tamiflu. Figure 5 shows the result. All the molecules have similar properties to the original ones except LogP as desired. The molecules from the top left to the bottom right have gradually increasing LogP values from − 0.23 to 3.55. In some cases, however, such a delicate control of individual properties was not possible. For instance, we could not generate molecules with a LogP beyond 4.0. It is probably because LogP is not completly independent from the other four properties, so a substantial change in LogP entails a change in the other properties. Moreover, it was difficult to adjust the MW and TPSA independently because the MW and TPSA are highly correlated with one another.
Molecules generated by the CVAE with the condition vector made of MW, HBD, HBA, and TPSA of Tamiflu and continuously changing LogP
Finally, we investigated the possibility to change a specific molecular property beyond the range of a training set. Latent vectors were sampled around molecules in the training set. In the condition vector, the four properties were given randomly except for a single target property. The target property was set to 10% larger than its maximum value in the training set (e.g., 5.5 for LogP and 165 for TPSA). Figure 6 shows the resulting molecules. Indeed, it was able to generate molecules with a LogP larger than 5.5 (Fig. 6a) and molecules with a TPSA larger than 165 (Fig. 6b). We compared the distribution of the LogP and TPSA for 1000 randomly selected molecules from the training set and 1000 generated molecules with property values outside of the range of the dataset (toward larger values). Figure 7 shows that the distribution of the target properties are shifted to larger values, leading to an increased ratio of molecules with property values outside of the range. The rate of valid molecules is relatively low compared to the case of generating molecules with property values in the range of the dataset.
a Molecules with LogP larger than 5.5. b Molecules with TPSA larger than 165
Distribution of a LogP and b TPSA for 1000 randomly selected molecules in training set and 1000 generated molecules with LogP and TPSA outside of the range of the dataset, respectively
Numbers of attempts and valid molecules for generating 100 molecules whose five properties are the same with those of Aspirin, Tamiflu, Lenalidomide, Rivaroxaban, and Pregabalin
Number of valid molecules
Success rate (100/attempts, %)
Rivaroxaban
We analyzed the latent space constructed by the CVAE. Two principle axes were extracted by principal component analysis. Figure 8 shows the two components of the latent vectors of 1000 randomly selected molecules from the test set with their MW, LogP and TPSA values. Molecules with similar properties are likely located around a same region of the latent space in the jointly trained VAE. In our CVAE model, the latent vector is comprised of two parts as explained in the method section. Therefore, a specific region in the latent space does not necessarily have a correlation with the target molecular properties which are controlled by the condition vector. This is good because the separation of information enables a more flexible control of the molecular structure and properties when generating new molecules.
The latent space of 1000 randomly selected molecules with MW, LogP and TPSA values
Apart from the successful applications of the CVAE method, it has a drawback that should be resolved. The success rate of generating desirable molecules is very low. We tested how many attempts were required to generate 100 molecules with the five desired properties and how many valid molecules were generated during those attempts. We also compared when the condition vector is set randomly or to target properties to show the effect of the condition vector for generating desirable molecules.
Table 1 summarizes the number of attempts for generating 100 molecules whose five properties are same as those of aspirin, Tamiflu, Lenalidomide, Rivaroxaban, and Pregabalin, respectively. Lenalidomide, Rivaroxaban, and Pregabalin are top selling small molecule drugs in 2016 [26]. In Table 1, 'condition' means that the condition vector was set as the five properties of the target molecules, whereas 'random' means that the condition vector was randomly made. The number of valid molecules in Table 1 indicates the number of valid molecules generated during the attempts to create molecules with the five desired properties. For example, 100 aspirin-like molecules and 32,567 valid moleculces were obtained from 28,840 attempts to create aspirin-like molecules. The reason why the number of valid molecules is larger than the number of attempts is that the stochastic write-out process is performed 100 times for each attempt. All successful molecules (100 per each target molecule) are reported in the Supporting Information. It should be noted that the success rate dramatically dropped when the condition vector is randomly set. It clearly manifests that the successful molecules generated by the CVAE in the example studies were not the result of many random trials.
Number of generation attempts and number of valid molecules for three different sampling methods of latent vectors
Average number of valid molecules
Average number of attempts
Around target molecules
Around known molecules
The generation process was continued until 100 molecules with the five target properties were successfully created from a single target molecule, and it was repeated for 100 target molecules selected randomly from the ZINC dataset. The table shows the average values over the 100 target molecules
We further analyzed the performance of the CVAE by investigating the change in the success rate and the number of valid molecules according to latent vector sampling methods. We employed three different sampling methods: random, around the latent vectors of known molecules, and around the latent vectors of target molecules. For all the sampling methods, the condition vector was constructed using the five properties of the target molecules. The generation process was continued until 100 molecules with the five target properties were successfully created from a single target molecule, and it was repeated for 100 different target molecules selected randomly from the ZINC dataset. Table 2 shows the average values for the success rate and the number of valid molecules over the 100 target molecules. It was unexpected that sampling latent vectors around a target molecule was the most ineffective in terms of the success rate and valid molecules because of the high rate of duplicated molecules. In this case, the structure of the generated molecules was very similar to that of the target molecule as shown in Fig. 4. Sampling latent vectors around those of known molecules performed best. Because the known molecules were randomly selected from the ZINC set, their structures and properties would be considerably different from those of a target molecule. Nonetheless, we were able to generate molecules with the desired properties from those latent vectors with a relatively high success rate. It manifests that the condition vector appropriately modified the molecular structures to have the target properties. Finally, it was also possible to generate desirable molecules from completely random latent vectors but with a low success rate.
We suspect that at some part the overall low success rates regardless of the latent vector sampling methods are due to the strong correlation between the five target properties. In addition, it is known that the discrete nature of SMILES causes a high rate of invalid molecules in the decoding process from latent vectors to molecules [27]. The stochastic write-out method circumvents this problem, but more fundamental solutions should be devised. More severely, SMILES does not have the 3D conformational information of molecular structures. Therefore, it must have limitations in applications in which conformational effects are critical. Molecular graph representation incorporating conformational information can be a promising alternative. Encoding molecular graphs seems to be straightforward, but decoding from a latent space to molecular graphs is still an open problem. Recently, significant progress along this line has been made [28–30]. Such a better molecular representation may also improve the success rate of molecular generation. We expect that the success rate may be further improved by using the grammar variational autoencoder [27] and the reinforcement learning [19, 20].
We proposed a new molecular design strategy based on the conditional variational autoencoder. Instead of high-throughput virtual screening, our method as one of the deep learning-based generative models directly produces molecules with desirable target properties. In particular, its strength is controlling multiple target properties simultaneously by imposing them on a condition vector. We demonstrated that it was possible to generate drug-like molecules with specific values for the five target properties (MW, LogP, HBD, HBA, and TPSA) within an error range of 10%. In addition, we were able to selectively control LogP without changing the other properties and to increase a specific property beyond the range of the training set. Thus, this new method has attractive applicability for efficient molecular design.
Author's contributions
Jaechang Lim, Seongok Ryu, Jin Woo Kim, and Woo Youn Kim organized this work. Jaechang Lim and Woo Youn Kim wrote the paper. All authors read and approved the final manuscript.
This work was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1E1A1A01078109).
Below is the link to the electronic supplementary material.
13321_2018_286_MOESM1_ESM.docx Supplementary material 1 (docx 791 KB)
Department of Chemistry, KAIST, 291 Daehak-ro, Daejeon, 34141, Republic of Korea
KI for Artificial Intelligence, KAIST, 291 Daehak-ro, Daejeon, 34141, Republic of Korea
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Gravity duals of conformal field theories without relevant operators.
Could the full theory of quantum gravity just be a nonrenormalizable quantum field theory?
The Reeh-Schlieder theorem and quantum geometry
How closely can string theory mimic asymptotic safety for gravity?
Background dependence of local field operators inside a black hole and non renormalizability of gravity?
Euclidean quantum gravity and gravitational instantons
ADM Hamiltonian formalism and Quantum gravity
Is there any quantum-gravity theory that has flat space-time and gravitons?
Open problems in Loop Quantum Gravity and Superstring theories.
Is there a simple way to understand why SUGRA is two-loop renormalisable?
Quantum gravity and relevant/irrelevant operators
I am familiar with the casual dichotomy in QFT between coupling with positive dimensions in energy implies relevant operator on one side and negative dimension implies irrelevant operator on the other side (forgetting about marginal operators) once you have set $\hbar$ and c equal to 1, and then I thought than in a theory of quantum gravity you would have also set G equals to 1 (or $8\pi G = 1$), so it's no more possible to count in energy dimensions, as there are no more dimensions (scales). Does it only mean that our little trick is now over, or is there any conceptual novelty regarding the RG flow?
This post imported from StackExchange Physics at 2014-05-01 12:21 (UCT), posted by SE-user toot
quantum-gravity
asked Jun 6, 2012 in Theoretical Physics by toot (445 points) [ no revision ]
retagged May 1, 2014
One doesn't have to set $G=1$ or $8\pi G=1$ in a quantum theory of gravity; it's just one possible choice that may be convenient.
If one wants to study relevant and irrelevant operators in general relativity and its extensions, it's useful not to set $G=1$ or $8\pi G=1$ because by doing so, we would make all quantities dimensionless.
Instead, it is a good idea to reformulate general relativity in the same way as other quantum field theories. Quantum field theories with a weakly coupled classical limit are usually described by Lagrangians $$ {\mathcal L} = {\mathcal L}_\text{free} + {\mathcal L}_\text{interactions} $$ The fields are normalized and redefined so that the kinetic terms (those with 2 derivatives in the case of bosons, 1 derivative in the case of fermions) have the usual normalization, schematically $(\partial_\mu \phi)^2/2$ for bosonic fields and $\bar\Psi \partial_\mu \gamma^\mu \Psi$ for fermionic fields.
This is possible for general relativity, too. Note that its Lagrangian is the integrand of the Einstein-Hilbert action $$ {\mathcal L}_{\rm GR} = \frac{1}{16\pi G} R $$ and it is proportionally to Ricci scalar that schematically contains terms $g \partial^2 g$, among others. We may expand the metric around a background, in the simplest case the flat background $$ g_{\mu\nu} = \eta_{\mu\nu} + \sqrt{8\pi G} \cdot h_{\mu\nu} $$ The $\eta$ tensor is the flat Minkowski metric; $h$ is the perturbation away from it which carries the "operator character". I have conveniently added the coefficient in front of $h$ because when we insert it to the Einstein-Hilbert action, the leading term will generate $${\mathcal L} \sim (\partial h)^2 $$ and the coefficients involving $G$ will cancel; let me be sloppy about the numerical coefficients of order one. However, the nonlinear Einstein-Hilbert action will also produce terms that are of higher order in $h_{\mu\nu}$ but each $h$ will appear together with a factor of $\sqrt{G}$, too.
So the cubic and higher interaction vertices in general relativity are weighted by $\sqrt{G}$ and its higher powers. Because $G$ has a positive dimension of length for $d\gt 2$, the interactions in general relativity are irrelevant (non-renormalizable). That's true even for $d=3$. However, gravity in $d=3$ has no local excitations, it is kind of vacuous, so the non-renormalizability problem may be, to some extent, circumvented.
At any rate, for $d=4$ and higher, even the leading interaction is irrelevant and leads to non-renormalizable divergences already at 2-loop level (and even in ${\mathcal N}=8$ supergravity, which offers as many supersymmetric cancellations as possible, there are new divergences requiring counterterms at the 7-loop level). So the theory breaks down at a cutoff scale that isn't too far from the Planck scale, $m_{\rm Pl} = G^{1/(d-2)}$.
That's where a consistent theory of quantum gravity i.e. string/M-theory has to cure the problems by providing the theory with new states (strings, branes, and – more universally – black hole microstates) and new constraints. Only if we try to study distances shorter than the Planck distance (a sort of a meaningless exercise for many reasons) or energy scales higher than the Planck scale (where the typical "particles" really look like ever larger black holes), we find out that the RG flows break down and become meaningless as a methodology. However, at distances much longer than the Planck length, the RG flows for GR are just fine and behave as they do in any non-renormalizable theory. The naively quantized Einstein's theory isn't predictive at the Planck scale but at much lower energies, one may systematically add new corrections, with a gradually increasing number of derivatives, to make an effective field theory ever more accurate.
Some people, most famously Steven Weinberg, have speculated that there could be a "zero-distance" ultraviolet limit that could run to general relativity at long distances. I think that the research attempting to find evidence for this conjecture remains inconclusive, to say the least. Most folks in quantum gravity are actually convinced that this can't work not only because of the apparent absence of the required scale-invariant field theory describe the UV; but also because such a picture of gravity would contradict holography and black hole thermodynamics.
This post imported from StackExchange Physics at 2014-05-01 12:21 (UCT), posted by SE-user Luboš Motl
answered Jun 6, 2012 by Luboš Motl (10,278 points) [ no revision ]
p$\hbar$ysicsOverfl$\varnothing$w | CommonCrawl |
The Surprising Flavor of Infinite Series
1+2+4+8+16+...=-1, as proven by Henry Reich on Minute Physics! Now, as a mathematician, I must say that his proof is far from being rigorous. In fact, anyone familiar with the surprising flavor of infinite series should not find it convincing. Surprisingly though, his proof can be rigorously and naturally justified! Find out how!
July 1, 2013 ArticleAnalysis, Infinity, Mathematics, Numbers, SequencesLê Nguyên Hoang 11134 views
Addition is the simplest thing in mathematics, right? Wrong! On Minute Physics, Henry Reich unveils some of the most weirdest aspects of addition:
After a few basic operations, Henry ends up proving $1+2+4$+$8+16$+$…=-1$!
That's clearly wrong! Isn't it?
Hehe… It's a bit like eating the Japanese poisonous fugu fish: If it's done by a qualified cook, then it's good. Otherwise, don't put it in your mouth!
So, Henry Reich, is he a qualified cook?
Hummm… He's a physicist, isn't he? More seriously though, these manipulations are extremely hard to justify, even though they do unveil something fundamental. Anyways, his not-even-one-minute crash course is definitely not enough for you to start your own fugu restaurant! Let's see what I mean in more details…
I don't aim at discrediting Henry Reich at all. He's doing an exceptional work on Minute Physics and Minute Earth and I am a huge fan! And his awesome video above, although not rigorous, has made me savor the surprising and very tricky flavor of infinite series! Frankly, I wasn't expecting writing this article to be giving me so many headaches. In fact, there are still so many interesting issues about infinite series I don't understand. And I'd love your help! I left these troubling issues in the last section of this article…
Positive Convergent Series
Obviously, additions aren't that complicated when they involve a finite number of terms. These are not what's of interest for us in this article. Rather, let's have fun with the cool infinite sums, also known as series. Now, some series aren't as tricky as Henry's. The simplest kinds of series are the positive convergent series.
What are the positive convergent series?
They are sort of the non-poisonous fishes. Unless you forget them in your stove, they'll be edible! The harder thing though is to distinguish them from other fishes.
So what are positive convergent series like?
They are series such that terms are positive and get very quickly small. The most famous example is based on Zeno's paradoxes. One of these paradoxes is illustrated by the following video by the Open University:
The key message of this video is that adding up an infinite number of numbers can make a lot of sense. In fact, when the terms are positive and get very quickly small, it's a very natural thing to do.
Yes! A variant of Zeno's paradox illustrates himself walking the 2 miles to his house. He always has to walk half of the distance to the house before he gets there. So, first, he walks 1 mile. Then half a mile. Then a fourth of a mile… As you've guessed, each term is half of the previous one. Eventually, the total walked distance is $1+^1/_2$+$^1/_4+^1/_8$+$^1/_{16}$+$^1/_{32}+…$, which thus adds up to 2 miles he had to walk.
So $1+^1/_2$+$^1/_4+^1/_8$+$^1/_{16}+^1/_{32}$+$…=2$?
Yes! Here's another illustration. Assume your restaurant (which is still not licensed to cook fugu!) is given orders of a cake. The first order is 1 cake, the second is half a cake, the third a fourth… and so on. How many cakes do you have to make? The answer is 2, as we've discussed and it's illustrated by the following figure:
The series we are discussing here is a particular case of geometric series which often come up in mathematics. There's a lot to say about them. If you can, please a write an article about that!
Positive Divergent Series
So the nice positive convergent series you're talking about are those where the terms eventually become 0?
NO!!! No. No, no, no, no… No! NO. No.
Wow… I said something bad…
I'm sorry for being so emotional here. But it's the source of sooo many mistakes! And mistakes like that can have you cooking poisonous fishes! So let me phrase it clearly. Not all series with terms getting to zero converge.
In fact, I want you to promise me! Promise me you'll never assert that series whose terms go to zero converge. Promise me!
Huh… Fine. I promise.
Good. Let me show why such a thing is false. Consider the series $1+^1/_2$+$^1/_3+^1/_4$+$^1/_5+^1/_6…$. This series is called the harmonic series, and has played a key role in the understanding of infinity.
And I guess that this harmonic series isn't a nice convergent series…
Yes. Let me prove it! Suppose the orders of cakes are those of the harmonic series. Let's bake pieces accordingly. Now, if we are shrewd about the way we arrange these pieces, we can make an interesting pattern emerge! Here's how I've done it:
So what I've done here is put the 1 aside. Then, I made a cluster of pieces containing the next piece of cake. Then a cluster with the 2 following pieces. Then with the 4 next ones. Then with the 8 next ones… And so on. Do you see the pattern I'm trying to highlight here?
Sorry… But no.
Keep in mind that I'm trying to show you why the harmonic series isn't a nice convergent series!
Oh, yes! All the clusters of pieces you've made is at least half a cake!
Bingo! But why?
Take your time to find it out by yourself!
Humm… Can you help me please?
Sure. Look at the cluster of 8 pieces. Why do the pieces add up to more than half of the cake?
I know! There are 8 of them, and all are greater than 1/16!
There you go! In fact, that's why I arranged the pieces like that! As you can imagine, the next cluster will have 16 pieces, all of which are of greater size than 1/32. And so on! This gives us an infinite number of clusters of pieces, all of which are at least half a cake. Thus…
The amount of cake is infinite! Brilliant!
Isn't it? Mathematically, we say that the harmonic series diverges. That's because its terms don't get small quickly enough. I hope you'll keep that in mind for the rest of your life!
So how do we know if the terms of the series get quickly enough?
There's no general result for such a thing. In other words, you can always find a fish in the see which will not have been classified as safe to cook or poisonous yet. But there are several tests you can make, which would work for most of the series you'll meet in the classrooms. I'm not going to explain these tests here, but if you can, you should write an article to explain them!
Convergent Series
So far, we've only discussed series with positive numbers. But additions of a finite number of terms are actually defined for many more mathematical objects. Obviously, there are also the negative numbers, but we can also add complex numbers, or even vectors! But as we try to extend the concept of infinite series for these objects, things get tricky…
Wait! You're going to fast! What's a complex number? What's a vector?
You're reminding me that no Science4All article has been written yet on these major mathematical objects… I'll try to fix that some day (you can fix it too, by writing your own article)! For our purpose here, we need to consider these objects (positive and negative numbers, complex numbers and vectors) as motions.
A positive number is a motion? That makes no sense!
Imagine yourself in an elevator. Each number can be associated with a motion. $+2$ corresponds to a motion of 2 floors up. $-5$ is a motion of 5 floors down. Similarly, complex numbers and vectors are motions in higher dimension spaces. And, as we add up vectors, we add up the motions. In the figure below, on the left, the motion given by positive and negative numbers is a left-right motion. On the right is a motion in the plane, as expressed by complex numbers or, equivalently, 2-dimensional vectors.
Now, the figure only displays sums of a finite number of motions. But, just as we did it for positive numbers, it's often useful to consider a sum of an infinite number of motions. In other words, let's try to define series of motions!
The concept of series of motions I'm using here is not conventional. But I think it's a relevant and nice geometrical visualization of the concept of series of vectors.
Now, for the series of motions to have a value, the sum must take us towards one precise location. If a series of motions does take us to one precise location, then we say that it converges.
Absolutely Convergent Series
So how do we know if a series of motions converges?
Hum… It's usually hard to know! One thing that's relevant is to look at the successive lengths of the motions.
I guess the lengths must get smaller and eventually become 0?
You're totally right! In particular, the series $1-1$+$1-1$+$1-1+…$ doesn't converge because the sum of motions never takes us anywhere precisely! It always makes us zigzag between values 1 and 0. But it never gets to a precise location. And that's because the lengths of the motions don't get to 0. Therefore, as you said, for a series to converge, the lengths of motions must get to 0. But that's not enough!
What about if these lengths get very quickly small?
Hehe… You're getting close to something fundamental! Lengths are positive numbers. How could you say that these positive numbers get very quickly small?
Hummm…
Come on! You're so close! Look at what we've talked about so far!
I know! What if the series of the lengths converge?
Bingo! The series of the lengths is a positive series. And if it converges, then the series of motions can't be moving too far away. Eventually, the motions will slow down so much that the sum of motions will be hardly moving. It will get stuck somewhere. And the series of motions will thus converge!
More accurately, for what I say to be true, lengths must satisfy the properties of a metric and, in particular, satisfy the triangle inequality. But that's a given in normed vector spaces which are studied in classrooms.
Sure! What about the series of positive and negative numbers of the figure above? This series is $8-4$+$2-1$+$^1/_2-^1/_4+…$. The lengths of the motions are 8, 4, 2, 1… thus, the series of lengths is $8+4$+$2+1$+$^1/_2$+$^1/_4+…$. If we get rid of the 3 first terms, then we obtain the geometric series we discussed at the beginning of this article! This series converges, thus, so does the series $8-4$+$2-1$+$^1/_2-^1/_4+…$.
So, to test if a series of motions converges, we merely need to study the series of lengths?
No! The two conditions are not equivalent! The convergence of the series of lengths is a sufficient condition for the series of motions to converge, but it's not necessary. In other words, if the series of lengths converges, then so does the series of motions. But the series of lengths doesn't need to converge for the series of motions to converge.
OK… It think I get it…
Great! Now, the case where the series of lengths do converge too is so interesting that mathematicians have named it. Series whose series of lengths converge are called absolutely convergent series.
For some motions, we often rather talk about uniformly convergent series. This is actually the same concept, as long as the concept of length refers to a norm of the vector space on which the series is defined.
In the fish kingdom, these absolutely convergent series are like sweet harmless tuna: They are easy to cook and relatively easy to recognize. But they are plenty of other edible non-poisonous fishes in the see! And some are delicious!
Conditionally Convergent Series
As I said, not all convergent series are absolutely convergent. These convergent-but-not-absolutely series are more simply called conditionally convergent series. These aren't as dangerous as the fugu fish, but they are already quite tricky to cook.
For a start, they may be harder to identify in general. There is the special case of alternating series which is easy to recognize, as well as a few other particular cases. But, apart from that, proving that a series is conditionally convergent is often quite difficult. But that's not why conditionally convergent series are very tricky.
What's the reason?
Let's take an example. And I can't resist to involving one particular famous conditionally convergent series… This famous series is $4-^4/_3$+$^4/_5-^4/_7$+$^4/_9-^4/_{11}+…$. This series is known as Leibniz series, and it first was studied by Madhava in India. Surprisingly, it equals $\pi$!
Waw! That's awesome! But why?
Hummm… It'd be a bit long to explain it. It has to do with the Taylor series of the function $arctan$. But that's not what I want to talk about.
Taylor series, trigonometry and Leibniz series are important objects in mathematics. If you can, you should write an article about these!
Oh yeah! You wanted to talk about this series being tricky…
Yes. What's extremely weird with conditionally convergent series is that the order in which the addition is made strongly matters!
Really? That sounds weird!
It is weird! But it's quite understandable when you think about it! Leibniz series has an infinite number of positive terms, as well as an infinite number of negative terms. And since it is not absolutely convergent, the series of positive terms must diverge.
If the series of positive terms converged, then, for Leibniz series to converge, the series of the negative terms should converge too. But the series of lengths is the sum of the series of positive terms and the series of the absolute values of the negative terms, and would thus be finite. That's in contradiction with the conditional convergence!
You're right! So the series of positive terms diverges…
Yes! And so does the series of negative terms. This had a dramatic consequence: By reordering the terms of the series, we can make the series equal anything we want! This is known as the Riemann series theorem.
Really? Can you make it equal 1 billion?
Easy! Depending on whether the current position is below or above 1 billion, we add a positive or a negative motion to get it close to 1 billion. It's as simple as that, and is done in the figure below:
Eventually, we'll be zigzagging around 1 billion. But, because the initial series of motions is convergent, the lengths of motions eventually go to 0. Since we cannot exceed 1 billion by more than the lengths of the motions, we won't be exceeding it by much eventually. Likewise, we'll eventually never be much below. This argument indicates that the series equals 1 billion.
What I've said here is not a rigorous proof, but is the essence of the proof. If you're familiar with the concept of limit, you should try to write the rigorous proof. It's a great exercise!
Wow! Indeed! These conditionally convergent series are tricky!
I know! That's why you must be cautious with infinite series, or you'll end up proving $\pi=4-^4/_3$+$^4/_5-^4/_7$+$^4/_9-^4/_{11}+…$=$1,000,000,000$.
Could divergent series even be any trickier than that?
They are! Not only can you not rearrange the terms, you can't combine them to simplify the sum either! Otherwise, you could even prove $1=0$, as James Grime does it on Numberphile:
Be careful with what you do with series, and think twice before buying a result, especially if its yours. Just like you should think twice before putting fugu in your mouth!
So how can we cook divergent series properly?
We're getting there…
Super-Summation
The trick is to consider the summation of a series as an operator which transforms a sequence of motions into an overall motion… when it's possible. Below are some of the mappings made by this operator we have discussed so far.
Wait… You've added a last one.
I have. This last one can be shown to be conditionally convergent. It thus has a value, but I don't know it. Still, it's important to note that it has some value. On the contrary, the first, third and fourth series have no value, because they are divergent.
Can't we say that the first and third ones equal infinity?
We could. But this would conceal something amazing about infinite series! That's why it's better to give them no value… so far.
I see! You want to give the first series the same value as Henry did, don't you?
Hehe… You see right through me! And the key idea is to define a better concept of summation. Let's call it a super-summation. This terminology is absolutely not conventional, but I find it very appropriate.
But how do you define this super-summation?
Let's rather focus on what our super-summation needs to be capable of. First, just like Clark Kent, the super-summation must be able to pass exactly for a classical summation for the classical situations the classical summation can handle. This means that the super-summation must operate identically to the classical summation on convergent series. This condition is known as regularity.
But that's not enough to give a value to Henry's series, is it?
You're totally right! There are in fact plenty of super-summations which satisfy the condition I've given so far. And they sometimes give different values to a same series!
It sounds like all these extensions are arbitrary…
Some super-summations are more natural than others though, and they have applications to physics! Some of them are obtained by averaging the successive overall motions. James Grime explains such a super-summation, known as the Cesaro summation in the sequel of the Numberphile video. And there are more powerful such averaging super-summations, like the Abel summation or the Borel summation. However, these technics can't sum Henry's series…
So how can Henry's summation be justified?
Henry mentions it briefly and indirectly… But it's impossible for you to guess if you've never heard of it! The more powerful super-summation Henry hints at is based on analytic continuation. Analytic continuation is a very useful technic in mathematics. For instance, it's the core idea of the construction of the famous Riemann zeta function. But it's also a bit long to explain, and I won't do it here.
But if you can, you should write about analytic continuation! Or on Riemann zeta function!
So, using analytic continuation, we can compute Henry's series?
Yes! And after a few pages of rigorous proof, we can prove that indeed, it's natural to assume $1+2+4$+$8+16$+$…=-1$!
Linearity and Stability
But that doesn't say if Henry's trick is right… right?
No, indeed. To better understand what he did, we need to introduce each rule of manipulations he used.
So what are these manipulations?
There are three of them, and it's natural to extend them, as they already work on convergent series. First, if we multiply all the terms of a convergent series by a constant number, then the sum must be multiplied by this constant number. As done below:
Second, if we take two convergent series and we add their terms one by one, the series we obtain must have a sum which is the sum of the two initial convergent series.
The two manipulations here are known as linearity. In other words, the summation is linear. Because of that, we want the super-summation to be linear too.
Learn more about linearity with my article on linear algebra.
The third and last manipulation is stability. It's the most natural one. It says that if we insert a 0 in front of a series, then the summation must keep the same value.
I'm not sure I see how these manipulations relate to Henry's calculation…
The manipulations enable us to obtain an equation which Henry's series must satisfy. Come on! Give it a try!
OK, I'll do it. But that's only because we're getting to the end of this article! Insert a zero before Henry's series. You obtain $0+1+2+4$+$8+16+…$. Multiply that by 2. You obtain the series $0+2+4+8$+$16+32+…$. Now, subtract Henry's series term by term, and you end up with $(-1)+0+0$+$0+0+…$, which is a convergent series adding up to -1. Thus, twice Henry's series minus itself equals -1. This indicates that Henry's series equals -1. Sweet, isn't it?
But are these manipulations allowed?
Hummm… It took me a while to be sure of it, but the answer is yes, because there exists a unique natural regular linear stable super-summation extended to all divergent geometric series but $1+1+1$+$1+…$. And the justification is very tricky. You'd better be a damn' good cook before you try such manipulations! Indeed, you can't do these manipulations on just any divergent series!
The reason why Henry's manipulations can be allowed is pretty technical. It also raises plenty of interesting questions about the algebraic structure of series, most of which I have no answer to. I've left these concerns in the last section, and I could use your help! Beware it's high level math!
Take the series $1+1+1$+$1+1+…$. Add a zero and subtract itself. You obtain $(0+1+1+1+1+…)$-$(1+1+1+1+1+…)$=$(-1)+0+0$+$0+0+…$, which is a convergent series which equals -1. But we've subtracted the series to itself, and we should have therefore obtained 0. That's contradictory and shows that the series $1+1$+$1+1+1+…$ is not summable by any regular linear stable super-summation, even though it can be reduced to a convergent series using our manipulations. That's how tricky divergent series are! Also, beware that you can't add an infinite number of 0s without affecting the sum. Indeed, using Cesaro's super-summation, you can prove that $1+0-1$+$1+0-1$+$1+0-1+…$=$2/3$! Thus, $1+0-1$+$1+0-1$+$…\neq 1$-$1+1-1$+$1-1+…$. Disturbing, right? The key is to understand the signs $+$ no longer stand for the addition we are familiar with!
So how do we know what we can or can't do?
An easy answer is to tell you to apply proven theorems only! If you can't manage them, don't cook divergent series! However, as long as you don't eat them, you can always play with them (even though that goes against my mom saying I should never play with food!). This is what the greatest mathematicians like Euler have been doing for years, even though they had no justifications for that. This led him to amazing results, such as $1+^1/_4+^1/_9$+$^1/_{16}+^1/_{25}$+$…$=$\pi^2/6$. And, hopefully, after some playtime, you'll end up with something fundamental… The more modern mathematician method though would be to define a super-summation as a mapping of sequences with a number, and to prove some convincing properties of this super-summation, as done by Hardy and Ramanujan.
I bet you didn't expect addition to be that complicated! I know I didn't… To sum up, know that if a series is not absolutely convergent, then it's a tricky series. This is even more true if it's not convergent at all! In this case, be careful with what you can and what you can't do! More often than not, the manipulations you undertake may lead you to some contradictory $1=0$… That's why, at school, you won't be dealing with divergent series, even though they are so cool!
Why bother with super-summation? Is it just for the fun of the math behind that?
You've got to admit that it's amusing that simple manipulations like Henry's retrieve a value for divergent series! But that's not all! Surprisingly, these super-summations also have incredible applications in physics, and quantum field theory in particular, with astonishingly accurate predictions! There's this great article by Christiane Rousseau also on using divergent series to solve and understand differential equation. Unfortunately, my knowledge is more than limited in this area… at least so far.
But if you know about the applications of super-summations or about quantum field theory, you should write an article!
For Math Guys…
The internet is quite poor regarding divergent series, so I had to do some thinking on my own. The following contains mostly my personal reasonings, although I have been greatly helped by quality discussions with David Louapre and Rémi Peyre below this great (but flawed) article in French.
In the general case, manipulations like Henry's can't be allowed. They may lead to paradoxical conclusions such as $1=0$, like when we're applying them to series like $1+1+1$+$1+…$.
So how come can they be allowed in Henry's case?
OK. I'll have to be technical here… Let $\mathbb R^{\mathbb N}$ the set of infinite real sequences. It's a vector space. The set $C$ of sequences whose series converge forms a vector subspace, on which the classical definition $S$ is well-defined and is a linear form. Plus, let $e$ be the insertion of a zero ahead of a sequence. It's a linear mapping for which $C$ stable. Now, what's important to note is that the vector space spanned by the sequence $u=(1,2$,$4,8,16,…)$ is in direct sum with $C$. Thus, any vector of $v \in C\oplus u\mathbb R$ is uniquely decomposable into $v=c$+$\alpha u$, where $c \in C$ and $\alpha \in \mathbb R$. We can thus define the super-summation on $T$ on $C\oplus u\mathbb R$ by $T(v)$=$S(c)-\alpha$. This super-summation is linear by construction.
But we still need to prove that it satisfies the stability property, right?
Exactly! Now, note that $T(e(v)) $=$ T(e(c)+\alpha e(u)) $=$ T(e(c)) $+$ \alpha T(e(u))$. Now, since $c \in C$ and $C$ is stable by $e$, we have $e(c) \in C$, thus $T(e(c)) $=$ S(e(c)) $=$ S(c) $=$ T(c)$. What's more, note that $e(u) $=$ u/2-{\bf 1}/2$, where ${\bf 1} $=$ (1,0,0$,$0,0,…)$. Thus, by linearity, $T(e(u)) $=$ T(u)/2 $-$ T({\bf 1})/2$. But ${\bf 1}$ is convergent, thus $T({\bf 1}) $=$ S({\bf 1})=1$. Therefore, using $T(u)=-1$, we have $T(e(u)) $=$ T(u)/2 $-$ T({\bf 1})/2 $=$ -^1/_2 – ^1/_2 $=$ -1$… which is $T(u)$! Phew… It works! We have $T(e(v))=T(v)$ for all $v \in C\oplus u\mathbb R$.
Can this be generalized to all divergent geometric series?
All but $1+1+1$+$1+…$! This is because the set of divergent geometric series $u_x$=$(1,x,x^2$,$x^3,x^4,…)$ with $x \leq -1$ or $x > 1$ forms a free family (hence enabling a well-defined super-summation), and that they are related to convergent series by $e$ with $e(u_x) $=$ u_x/x$-${\bf 1}/x$ (hence guaranteeing $T(e(u_x)) $=$ T(u_x)$, provided that $T(u_x) $=$ 1/(1-x)$). Hence, a unique natural linear regular stable super-summation exists for $C \oplus (\oplus_{x \notin ]-1,1]} u_x\mathbb R$).
Can we extend this set?
Hehe… For one thing, we can include bounded series which can be summed using Cesaro or other regular linear stable super-summation, as the terms of these series are in direct sum with divergent geometric series. We can do even better! In fact, it seems that all sequences $u$ for which the next element $u_{n+1}$ is obtained by a linear combination of a finite number of the previous ones (for $n$ high enough) can be added, if the sum of the coefficients of the linear combination isn't 1. In other words, if $u_{n+1} $=$ \sum_{0\leq j \leq k} a_j u_{n-j}$ with $\sum a_j \neq 1$, then I think $u$ can be added (but I'm not sure as I haven't tried to prove that!).
And apart from that?
Apart from that, I have no idea! What I do know is that there is no linear regular stable super-summation defined on a space that would contain series like $1+1+1+1+…$ or $1+2+3+4+5+…$. Surely, enough, you can follow the manipulations explained by Tony Padilla on Numberphile.
Still, this calculation is fundamentally wrong (as well as the other first proof by Ep Copeland). As Rémi Peyre commented it on ScienceÉtonnante, $S(1,2,3,4,…) $-$ 2S(e(1,2,3,4,…)) $+$ S(e^2(1,2,3,4,…)) $=$ S(1,2,3,4,…) $-$ 2S(0,1,2,3,4,…) $+$ S(0,0,1,2,3,4,…) $=$ S(1,0,0,0,…) $=$1$ using linearity, but also equals $S(1,2,3,4,…)$-$2S(1,2,3,4,…)$+$S(1,2,3,4,…) = 0$ using stability, hence proving $1=0$. Thus, once again, there is no linear regular stable super-summation defined on a space that would contain series like $1+1+1+1+…$ or $1+2+3+4+5+…$.
Yet, what's particularly troubling is that several methods lead to the result $1+2+3+4$+$5+…$=$-^1/_{12}$. Worse, this very formula finds applications in string theory and in computing the Casimir force in theoretical physics! In other words, it really seems true… But I don't know why!
Isn't it because of analytic continuation?
Analytic continuation applied to the Riemann zeta function does sort of indicate this result. However, I don't know if there doesn't exist any other analytic continuation which would produce another result for the series $1+2+3+4$+$5+…$. I have no idea! As a mathematician, what I do claim is that I haven't been convinced so far mathematically by the relevancy of defining $1+2+3+4$+$5+… = -1/12$. Rather, and that's strongly based on my ignorance of the topic, I'd say that there is an underlying algebra we haven't uncovered yet… But this is the voice of my young audacious naive arrogant mind speaking!
It's worth noting that the equality $1+2+4+8$+$16+…=-1$ is obvious for 2-adic numbers… Which makes me wonder if real numbers really are the appropriate numbers to describe some physical phenomenons. After all, quantum mechanics is already weirdly described by complex numbers…
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One comment to "The Surprising Flavor of Infinite Series"
neurofuzzy says:
Excellent. I started researching quite a bit after the numberphile video and came across this .pdf: http://math.berkeley.edu/~theojf/DivergentSeries.pdf which takes a completely different route to defining these series!
So, what you do is define allowed algebraic operators on stable sequences, and THEN you sum them, presumably using the standard definition that "The sum of P where P is some infinite sequence is the limit of partial sums of its terms".
The approach in the paper I linked goes a different route. It says that if p is a sequence and A is a linear operator (lower triangular linear operator, in the article) with the property that for every convergent sequence p, Ap also converges and to the same limit (where "converges" = "limit of partial sums exists"), then "A" is regular, and for ANY sequence p it is satisfactory to ignore p and consider only Ap.
I'm not sure if this method is equivalent to what this post presents. I had assumed this allowed the summation of "1+2+3+4+…" but I only just realized that wasn't explicitly stated in the article! In fact, perhaps it can't; the article states that one particular linear operator can sum (z+z^2+z^3+…) in the entire half plane Re(z)<1. Which is powerful but just weak enough to leave us hanging! Hmp.
You should also read http://math.stackexchange.com/questions/39802/why-does-123-dots-1-over-12 and Luboš Motl's answer/sums. Of course he's very nonrigorous – he doesn't say anything like "According to thm. abc we can't apply the same operations and reach a different result or a contradiction" (and I think what's in your article shows that such a statement isn't true). | CommonCrawl |
maclaurin series for sinx3 inch metal rings for crafts
An example where the Maclaurin series is useful is the sine function. Vote. Search: Taylor Series Ode Calculator. The Taylors series is given by the formula. f(x) = f (x) + f (x) * x + f (x) * x 2 / 2! % 'n' is the number of expansion terms. The pink curve is a polynomial of degree seven: The pink curve is a polynomial of degree seven: x 2 n + 1. This Maclaurin series solver expands the given function by differentiating it up to the nth order. 13 sin ( x 67.4) + 1 = 14. sin ( x 67.4) = 1. x = 157.4, 337.4. We know the MacLaurin series for cos(x) is sin (x) (1)nx2n+1 1) Find a MacLaurin series for these functions. % calculating factorial for the expression. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Since someone asked in a comment, I thought it was worth mentioning where this comes from. taylor-maclaurin-series-calculator.
Maclaurin series expansion calculator is an easy-to-use tool. . . In this tutorial we shall derive the series expansion of the hyperbolic sine function by using Maclaurins series expansion function. Maclaurin series sin(3x) Natural Language; Math Input; Extended Keyboard Examples Upload Random. This question hasn't been solved yet Ask an expert Ask an expert Ask an expert done loading. For the given function, find its power series (in powers of x) and the interval of convergence. x. 2 months ago. Once we have their values we simply plug them in this general formula to find the series expansion. Solution for . \displaystyle {x}= {0} x = 0.
Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . To find the Maclaurin Series simply set your Point to zero (0) Since the differential equation has non-constant coefficients, we cannot assume that a solution is in the form \(y = e^{rt}\) which arise by separating variables in Laplace's equation in parabolic cylindrical coordinates, are also known as the Weber differential equations Explore f(x) = f (x) + f (x) * x + f (x) * x 2 / 2! + f (x) * x 3 / 3! Expert Answer. Consider the function of the form. On the other hand, it is easy to calculate the values of sin (x) \sin(x) sin (x) and all of its derivatives when x = 0 x=0 x = 0. f ( 0) = sin. However , the answer is only157.4 why is that the case ? The first term is simply sin x with x = 0. Maclaurin Series. The first term is simply sin x with x = 0. Maclaurin Series for sin x. f ( x) = sin. The Maclaurin series for sin ( x) is: sin ( x) = x x 3 3! From these, we show how to divide one series into another to obtain the first few terms for the series of tan (x). In this tutorial we shall derive the series expansion of the trigonometric function sine by using Maclaurins series expansion function. ( 0) = 0. + now putting f (x) It would typically be covered in a second-semester calculus class, but its possible to understand the idea with only a very basic knowledge of derivatives. ( 0) = 0. Transcribed image text: The Maclaurin series expansion for sin x is sin x = x - 3! The formula for the Maclaurin series. Search: Taylor Series Ode Calculator.
Find the maximum value of 5 sin x 12 cos x + 1 and the corresponding value of x from 0 to 360. 2. Now as: sinhx = ex ex 2. Consider the function of the form. From these, we show how to divide one series into another to obtain the first few terms for the series of tan (x). Title: challenge-10-7-1 Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies Inc,SMART Board Interactive Whiteboard Since sin 0 = 0, it is the cosine derivatives, which will yield a result. Homework help starts here! Embed. Maclaurin sin 2x. Maclaurin Series function in matlab. The Maclaurin series of e^x is: = 6. Maclaurin series is a special case of Taylors series that gives us the value of a function at a point (0). Maximum value = 13+1=14. An example where the Maclaurin series is useful is the sine function. That allows us to specify a bit cleaner what we want: macResult n x = sum (zipWith (*) (take n (map (x^^) [0..])) sinMacFactors) notice the second argument to zipWith. The Taylors series is given by the formula. = sum_(n=0)^oo (-1)^n x^(2n)/((2n+1)!)
(Type an expression in terms of ag that includes all terms up to order 8.) Related Symbolab blog posts. To expand any function, follow the below steps. Search: Taylor Series Ode Calculator. Res=0; % loop to calculate factorial and add the element to fact. : Then plug in the expansion for sin x and collect the terms. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. f ( x) = sinh. the maclaurin series for sinx Run Reset Share Import Link. The Maclaurin series can be expressed in the following way: f (x) = f (0) + f '(0) 1! If the principal part of the Laurents series vanishes, then the Laurents series reduces to sinx.
Find the Taylor series for f (x) = x^5 5x^3 + x around x = 2.
It should be true for 1 < sin(x) < 1 To get the Maclaurin series for xsin x, all you have to do is to multiply the series with x throughout, as indicated by the formula above. The first thing we need to do is to find out the values of the derivatives. However , the answer is only157.4 why is that the case ?
Show transcribed image text Expert Answer. maclaurin \sin(x) en. Maximum value = 13+1=14.
Corresponding value of x. Then do the i Then do the i Q: Evaluate the indefinite integral using power series. The formula for the Maclaurin series. Find the maximum value of 5 sin x 12 cos x + 1 and the corresponding value of x from 0 to 360. I found the value of x and theres 2 values.
7. m (a) Find Maclaurin expansions for sin 2.x , cos 2.x and e * up to and including the term in x5 . The series for cos (x) is obtained by differentiation. Taylor/Maclaurin Series Calculator Find the Taylor/Maclaurin series representation of functions step-by-step Suppose we wish to find the Taylor series of sin(x) at x = c, where c is any real number that is not zero. The first term is simply sin x with x = 0. Step-by-step math courses covering Pre-Algebra through Calculus 3. 2. 6.3.3 Estimate the remainder for a Taylor series approximation of a given x. Math Advanced Math Q&A Library 2. Search: Taylor Series Ode Calculator. f ( 0) = sin. Analytic functions The Pictured on the right is an accurate approximation of sin x around the point x = 0. The Maclaurin series is just a Taylor series centered at Sries de Taylor/Maclaurin But all of that was focused on approximating the function around x is equal to 0 Taylor Calculator Real 27 v Donaldina Cameron was an illustration of this kind of angel Donaldina Cameron was an illustration of this kind of angel. . Title: challenge-10-7-1 Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies Inc,SMART Board Interactive Whiteboard
5 x 2 sin 2 x d x Write the integrand for the given indefinite integral as a Maclaurin series. Search: Taylor Series Ode Calculator. Answer (1 of 4): The series for arctan(x) is So to find arctan(sinx) you could just substitute sin(x) in the above. The series expansion of \(\frac{{\sin x}}{x}\) near origin is Q5. The Maclaurin series of sin(x) is only the Taylor series of sin(x) at x = 0.
x2 + f '''(0) 3! for i = 0:n. Res = Res + a^i/factorial (i); A Taylor series provides us a polynomial approximation of a function centered around point a Taylor and Maclaurin Series interactive applet Enter your calculator's 14-digit ID# (F1:Tools About) 3 Worksheet - Calculus Maximus, Kevin W Example 6: The differential equation Futuristic Logo Maker Example 6: The differential equation. Step 2: Now click the button Calculate to get the result. A: We need to find the area between two curves.The area between two For the given function, find its power series (in powers of x) and the interval of convergence. A Taylor series provides us a polynomial approximation of a function centered on the point a, whereas a Maclaurin series is always centered on a = 0. This is a very nice and easy one for beginner students. 2. The series for cos (x) is obtained by differentiation. This calculator for to calculating the sum of a series is taken from Wolfram Alpha LLC Free Substitution differential equations calculator - solve differential equations using the substitution method step-by-step This website uses cookies to ensure you get the best experience Show Instructions In general, you can skip the Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more edu Department of Mathematics University of California, Santa Barbara These lecture notes arose from the course \Partial Di erential Equations" { Math 124A taught by the author in the Department of Mathematics at UCSB in the fall quarters of
How do I obtain the Maclaurin series for #f(x)= 2xln(1+x3)#? In the last section, we learned about Taylor Series, where we found an approximating polynomial for a particular function in the region near some value x = a. Find more Mathematics widgets in Wolfram|Alpha. The first thing we need to do is to find out the values of the derivatives.
}-+\ \cdots\ . However, the pattern is very simple as you can see. Such a polynomial is called the Maclaurin Series. f ( x) = sin.
0. Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . :) https://www.patreon.com/patrickjmt !! 1. 1. Functions. Using x = 0, the given equation function becomes. f(x) = ln(2x + 3) 2 months ago Find the Maclaurin series of the following function: e^3x^2 The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid 1700s. Expert Answer. When finding the Maclaurin series representation for sin (x)/x, I decided to multiply the Maclaurin series for each individual function first. Using this general formula, derive the Maclaurin expansion of sin 2x. . The Taylor Series for f (x) = ln (x) at x = 1. The Maclaurin series of sine is: = 4. Find the maximum value of 5 sin x 12 cos x + 1 and the corresponding value of x from 0 to 360. Corresponding value of x. Find the truncation error of the Maclaurin series sinx cosx given that is truncated into three (3) terms. Explore how the steps of the Taylor Series are used to find and evaluate derivatives through an example using the Maclaurin series, a specific type of Taylor series, to solve for sin(x). Python Fiddle Python Cloud IDE. The MacLaurin series for sin (x) In my previous post I said recall the MacLaurin series for :. The formula for the Maclaurin series. Homework help starts here! = = x -1/(3!)x^3+1/(5!) +-+ 5! However, the pattern is very simple as you can see.
E.g. I found the value of x and theres 2 values. f ( 0) = sinh. We can fix that by swapping those two around like: sinMacFactors = zipWith (/) sinZeroDerivations factorials. 13 sin ( x 67.4) + 1 = 14. sin ( x 67.4) = 1. x = 157.4, 337.4. Maclaurin sin 2x. The xsin x series is the most easiest to derive.
The Maclaurin Series for f (x) = 1/ (1-x)^2. }-+\ \cdots\ . Calculus: We compute the Maclaurin series for f (x) = sin (x) using the Taylor coefficient formula. It turns out that this series is exactly the same as the function itself! Maclaurin series expansion of sinx up to a number of significant figures. Free Taylor/Maclaurin Series calculator - Find the Taylor/Maclaurin series representation of functions step-by-step. the below code gives the answer for the sine of an angle using Maclaurin series. Related Symbolab blog Since sin 0 = 0, it is the cosine derivatives, which will yield a result. That wants to be written as:
The sequence of steps is very similar to the sin x derivation that was shown earlier. Language English. Since someone asked in a comment, I thought it was worth mentioning where this comes from. Maclaurin Series for sin x. LIM8.F (LO) , LIM8.F.2 (EK) Transcript. Don't try to find it by determining the derivatives. Consider the MacLaurin series for sinx: sinx = sum_(n=0)^oo (-1)^n x^(2n+1)/((2n+1)!) Corresponding value of x. For the given function, find its power series (in powers of x) and the interval of convergence. If we wish to calculate the Taylor series at any other value of x, we can consider a variety of approaches. . % 'a' is the value whose exponential is to be found. We can fix that by swapping those two around like: sinMacFactors = zipWith (/) sinZeroDerivations factorials. + f (x) * x 3 / 3! Starting with the on-term approximation, sin x = x, add terms one at a time to estimate sin (7/3).
The series expansion for sin x is given for sin x = sum_{k=0}^oo(-1)^k (x^{2k+1})/((2k+1)!) The process to find the Taylor series expansion for {eq}sin (x) {/eq} will follow the same procedure used to find the Maclaurin series representation. You already know the expansions of the function sin x and e^y. There are five types of problems in this exercise: Determine the first three non-zero terms of the Maclaurin polynomial: The user is asked to find the first three non-zero terms of the Maclaurin polynomial for the Created by Sal Khan. Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . function result = MacLaurin1 (a,n) % Program to calculate MacLaurin expression. Thanks to all of you who support me on Patreon. Q: 0 1 . The Taylor and Maclaurin series gives a polynomial approximation of a centered function at any point a, while the Maclaurin is always centered on a = 0.
image/svg+xml. k=0. $1 per month helps!! The definition of the sine function does not allow for an easy method of computing output values for the function at arbitrary input values. Math Advanced Math Q&A Library 2. Once we have their values we simply plug them in this general formula to find the series expansion. mohamed on 17 May 2013. For the given function, find its power series (in powers of x) and the interval of convergence. and divide by x term by term: (sinx)/x = sum_(n=0)^oo (-1)^n 1/x x^(2n+1)/((2n+1)!) (d) Let Px4( ) be the fourth-degree Taylor polynomial for f about 0 Included are derivations for the Taylor series of \({\bf e}^{x}\) and \(\cos(x)\) about \(x = 0\) as well as showing how to write down the Taylor series for a polynomial The following example should help to make this idea clear, using the sixth-degree Taylor polynomial for cos x Math Advanced Math Q&A Library 2. 6.3.2 Explain the meaning and significance of Taylors theorem with remainder. Step 3: Finally, the expansion series for the given function will be displayed in the new window. We know the MacLaurin series for cos(x) is sin (x) (1)nx2n+1 1) Find a MacLaurin series for these functions. f(x) = f (x) + f (x) * x + f (x) * x 2 / 2! The Maclaurin Series for sin (x), cos (x), and tan (x) The Maclaurin Series of f (x) = (1+x)^ {1/2} 1a. A Maclaurin series is a special subset of the Taylor series. f(x) = ln(2x + 3) 2 months ago Find the Maclaurin series of the following function: e^3x^2 Included are derivations for the Taylor series of \({\bf e}^{x}\) and \(\cos(x)\) about \(x = 0\) as well as showing how to write down the Taylor series for a polynomial Taylor series is a way to representat a function as a sum of terms calculated based on the function's derivative values at a given point as shown on the image below In this post 66 This page shows how to derive the Maclaurin expansion for sin x. Math Calculus Q&A Library (1) Use the Maclaurin series of sin x to evaluate the limit ,3 sin x x + lim. Using x = 0, the given equation function becomes. Maximum value = 13+1=14. Euler's identity: 8. Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . Search: Taylor Series Ode Calculator. The Maclaurin series of cosine is: = 5. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0. I found the value of x and theres 2 values. + now putting f (x) We know the MacLaurin series for cos(x) is sin (x) (1)nx2n+1 1) Find a MacLaurin series for these functions. You start with the series expansion of sin x as shown in the Maclaurin series for sin x article. Vote. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . Once we have their values we simply plug them in this general formula to find the series expansion. xn. We've got the study and writing resources you need for your assignments.Start exploring!
Find the fourth degree Maclaurin polynomial for the function f(x) = ln(x+ 1) n maths an infinite sum giving the value of a function f in the neighbourhood of a point a in terms of the derivatives of the function evaluated at a You can specify the order of the Taylor polynomial The hyperbolic tangent satisfies the second-order x. 2. 3 marks (b) Hence obtain an expansion for e"* cos -+ 2x| up to and including the term in x3 . Which according to wikipedia is: n = 0 ( 1) n ( 2 n + 1)! It would typically be covered in a second-semester calculus class, but its possible to understand the idea with only a very basic knowledge of derivatives.
We have: sinhx = 1 2[ n=0 xn n! Apr 17, 2009. How do you use a Maclaurin series to find the derivative of a function? Use your calculator to determine the true value. We can derive the McLaurin series for sinh(x) from the one othe exponential function: as for every n: [ dn dxn ex]x=0 = e0 = 1. the Mc Laurin series for ex is: ex = n=0 xn n! + now putting f (x) The sequence of steps is very similar to the sin x derivation that was shown earlier. That allows us to specify a bit cleaner what we want: macResult n x = sum (zipWith (*) (take n (map (x^^) [0..])) sinMacFactors) notice the second argument to zipWith. The Maclaurin series for sin (x) is: n = 0 ( 1) n x 2 n + 1 ( 2 n + 1)! Get the free "Maclaurin Series" widget for your website, blog, Wordpress, Blogger, or iGoogle. Taylor Series, Laurent Series, Maclaurin Series TIDES integrates by using the Taylor Series method with an optimized variable-stepsize and variable-order formulation, and extended formulas for variational equations nth Degree Taylor Polynomial If there is a jump discontinuity, the partial sum of the Fourier series has oscillations near the jump, which might increase the Calculus: We compute the Maclaurin series for f (x) = sin (x) using the Taylor coefficient formula. The procedure to use the Maclaurin series calculator is as follows: Step 1: Enter two functions in the respective input field. 13 sin ( x 67.4) + 1 = 14. sin ( x 67.4) = 1. x = 157.4, 337.4. A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function. .
The region in the first quadrant bounded above by the line y = 2, below by the curve y=2 sinx, A: Click to see the answer Q: Solve for the value of constant(C) by obtaining the orthogonal trajectory of the given equation Present the following function as power series and determine its intervals of convergence: x/x^2 + 9 (in powers of x) 2 months ago. . See all questions in Because the behavior of polynomials can be easier to understand than functions such as sin(x), we can use a Maclaurin series to help in solving differential Answered: (1) Use the Maclaurin series of sin x | bartleby. Title: challenge-10-7-1 Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies Inc,SMART Board Interactive Whiteboard Homework help starts here!
and it is easy to see that for n even the terms are the same and just cancel each other, so that just the odd The Maclaurin Series for f (x) = (1+x)^ {1/2} 1b. Just plug them in, and expand up till 4th order (or better to just look what terms contribute to the x^4 coefficient). Question: Use the Maclaurin series sin x = (a) x sin(x) (b) x cos x (1)kxk+1 to find the Maclaurin series for the following. . Homework help starts here! 3 marks (Leave all answers in exact form.) We also note from the graph that f is even, so we expect all odd powers of x in the series to vanish. x = 0. Transcribed image text: Substitute y (x) = ax and the Maclaurin series for 3 sinx info y-3xy-3 sinx and equate the coeficients of like powers of x on both sides of the equation to find the first four nonzero term A=0 solution to the differential equation. + f (x) * x 3 / 3! The Maclaurin Series for f (x) = e^x. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid 1700s. Maclaurin Expansion of sin(x) | The Infinite Series Module 6.3.1 Describe the procedure for finding a Taylor polynomial of a given order for a function. Modified 9 years ago. x3 + (f (4)) 0 4!x4 + = n=0 f (n)(0) n! n=0 ( x)n n!] I however don't understand why this notation is correct at all. Answer (1 of 4): #4623 If f(x)=\dfrac{\sin{x}}{x}\text{ for }x\ne0,\ f(0)=1, then the Maclaurin series for f is 1-\dfrac{x^2}{3!}+\dfrac{x^4}{5! In this tutorial we shall derive the series expansion of the trigonometric function sine by using Maclaurins series expansion function. sin(x) 2 sin(x) 3 arctan(x) A: Consider f(x)=sinxx Maclaurin series for the function is given by Q: Find the first three non-zero terms of the Maclaurin series for the function f(x) = cos(), and w The power series of {eq}sin(x) {/eq} is simply the Maclaurin series expansion for {eq}sin(x) {/eq}. x + f ''(0) 2! The point a = 0 is the fixed point in the Maclaurin series. f (x) = ln (2x + 3) ln a b" B 914 Find the Taylor series for without using the Taylo0 Ba b r series formula Taylor series for a function f(x) is given as follows Question 1: I was trying to find the derivative of log(x) You can specify the order of the Taylor polynomial Limit Calculator How To Clock Piston Rings Limit Calculator. Maclaurin series is a special case of Taylors series that gives us the value of a function at a point (0). How does the Maclaurin series calculator work? the below code gives the answer for the sine of an angle using Maclaurin series. The MacLaurin series for sin (x) In my previous post I said recall the MacLaurin series for :. The Maclaurin series for sin x, cos x, and e^x exercise appears under the Integral calculus Math Mission. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, Consider the function of the form. Euler's formula: = 7. Answer (1 of 4): #4623 If f(x)=\dfrac{\sin{x}}{x}\text{ for }x\ne0,\ f(0)=1, then the Maclaurin series for f is 1-\dfrac{x^2}{3!}+\dfrac{x^4}{5! url: Go The Maclaurin series is a special case of the Taylor series. You da real mvps! On the other hand, it is easy to calculate the values of sin (x) \sin(x) sin (x) and all of its derivatives when x = 0 x=0 x = 0. This is the first derivative. Maclaurin series: = Ratio = 2. Q: Find the Maclaurin Series of the following functions. The Taylors series is given by the formula. First, find x^7 + cdots then (sinx-x)/x^3 = sum_{k=1}^oo(-1)^k (x^{2k-2})/((2k+1)!) After each new term is added, compute the true and approximate percent relative errors. That wants to be written as: Answer (1 of 2): We can prove the expansion of circular functions by using indeterminate coefficients and repeated differentiation. In order to find these things, well first have to find a power series representation for the Maclaurin series, which we can do by hand, or using a table of common Maclaurin series. However , the answer is only157.4 why is that the case ? Viewed 4k times. (2k + 1)! This website uses cookies to ensure you get the best experience. + x 5 5! Follow 299 views (last 30 days) Show older comments. Calculus We now take a particular case of Taylor Series, in the region near. 7! This is the first derivative. The first thing we need to do is to find out the values of the derivatives. Solutions for Chapter 11.R Problem 49E: Find the Maclaurin series for f and its radius of convergence. Maclaurin series is a special case of Taylors series that gives us the value of a function at a point (0). So, Let us find the derivatives, and compute the values at x = 0. About Pricing Login GET STARTED About Pricing Login. This exercise shows user how to turn a function into a power series. This page shows how to derive the Maclaurin expansion for sin x. Using x = 0, the given equation function becomes. Approximating cos (x) with a Maclaurin series (which is like a Taylor polynomial centered at x=0 with infinitely many terms). Using this general formula, derive the Maclaurin expansion of sin 2x. By M. Bourne. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Determine the Maclaurin Series (5th approximation) of the function defined by f (x) = sin x . x^5 -1/(7!) en. Analytic functions The Pictured on the right is an accurate approximation of sin x around the point x = 0. This page shows how to derive the Maclaurin expansion for sin x. It only seems to work if you consider that x is the zeroth term. The Maclaurin series of sin(x) is only the Taylor series of sin(x) at x = 0. Maclaurin Series for sin x. Q: Find the three areas of the region bounded by y= 2x+10 , y=4x+1 and the lines x=-2 and x=5. f(x) = ln(2x + 3) 2 months ago Find the Maclaurin series of the following function: e^3x^2 Solution for Find the Maclaurin series of sin(x) 28 TL=0. Complex functions can be converted to power series by using substitution. ( 0) = 0. 3. Follow @python_fiddle.
- GitHub - zeyveli/Maclaurin-series-expansion-of-sinx: Maclaurin series expansion of sinx up to a number of significant figures. The definition of the sine function does not allow for an easy method of computing output values for the function at arbitrary input values.
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maclaurin series for sinx 2022 | CommonCrawl |
GTC/CanariCam Deep Mid-infrared Imaging Survey of Northern Stars within 5 pc
2021ApJ...923..119G
10.3847/1538-4357/ac2c0a
Gauza, Bartosz; Béjar, Víctor J. S.; Rebolo, Rafael; Álvarez, Carlos; Zapatero Osorio, María Rosa; Bihain, Gabriel; Caballero, José A.; Pinfield, David J.; Telesco, Charles M.; Packham, Christopher
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In this work we present the results of a direct imaging survey for brown dwarf companions around the nearest stars at the mid-infrared 10 micron range (λ c = 8.7 μm, Δλ = 1.1 μm) using the CanariCam instrument on the 10.4 m Gran Telescopio Canarias (GTC). We imaged the 25 nearest stellar systems within 5 pc of the Sun at declinations δ > -25° (at least half have planets from radial-velocity studies), reaching a mean detection limit of 11.3 ± 0.2 mag (1.5 mJy) in the Si-2 8.7 μm band over a range of angular separations from 1″ to 10″. This would have allowed us to uncover substellar companions at projected orbital separations between ~2 and 50 au, with effective temperatures down to 600 K and masses greater than 30 M Jup assuming an average age of 5 Gyr and masses down to the deuterium-burning mass limit for objects with ages <1 Gyr. From the nondetection of such companions, we determined upper limits on their occurrence rate at depths and orbital separations yet unexplored by deep imaging programs. For the M dwarfs, the main component of our sample, we found with a 90% confidence level that fewer than 20% of these low-mass stars have L- and T-type brown dwarf companions with m ≳ 30 M Jup and T eff ≳ 600 K at ~3.5-35 au projected orbital separations.
Very Low Mass Stars, Brown Dwarfs and Planets
Our goal is to study the processes that lead to the formation of low mass stars, brown dwarfs and planets and to characterize the physical properties of these objects in various evolutionary stages. Low mass stars and brown dwarfs are likely the most numerous type of objects in our Galaxy but due to their low intrinsic luminosity they are not so
Rebolo López
Exoplanets and Astrobiology
The search for life in the universe has been driven by recent discoveries of planets around other stars (known as exoplanets), becoming one of the most active fields in modern astrophysics. The growing number of new exoplanets discovered in recent years and the recent advance on the study of their atmospheres are not only providing new valuable
Pallé Bago
Refereed
Stellar & Interstellar Physics (FEEI)
Exoplanetary Systems & Solar System (SEYSS)
The impact of two massive early accretion events in a Milky Way-like galaxy: repercussions for the buildup of the stellar disc and halo
We identify and characterize a Milky Way-like realization from the Auriga simulations with two consecutive massive mergers $\sim 2$ Gyr apart at high redshift, comparable to the reported Kraken and Gaia-Sausage-Enceladus. The Kraken-like merger (z = 1.6, $M_{\rm Tot}=8\times 10^{10}\, \rm {M_{\odot }}$) is gas-rich, deposits most of its mass in the
Orkney, Matthew D. A. et al.
2022MNRAS.517L.138O
The Circular Polarization of the Mn 1 Resonance Lines around 280 nm for Exploring Chromospheric Magnetism
We study the circular polarization of the Mn I resonance lines at 279.56, 279.91, and 280.19 nm (hereafter, UV multiplet) by means of radiative transfer modeling. In 2019, the CLASP2 mission obtained unprecedented spectropolarimetric data in a region of the solar ultraviolet including the Mg II h and k resonance lines and two lines of a subordinate
del Pino Alemán, Tanausú et al.
2022ApJ...940...78D
Euclid preparation. XXI. Intermediate-redshift contaminants in the search for z > 6 galaxies within the Euclid Deep Survey
Context. The Euclid mission is expected to discover thousands of z > 6 galaxies in three deep fields, which together will cover a ∼50 deg2 area. However, the limited number of Euclid bands (four) and the low availability of ancillary data could make the identification of z > 6 galaxies challenging. Aims: In this work we assess the degree of
van Mierlo, S. E. et al.
2022A&A...666A.200V | CommonCrawl |
Fibonacci subset fun
Michael Lugo Uncategorized July 21, 2012 July 21, 2012 4 Minutes
A week ago I wrote a post on bitwise set trickery in which I asked a question from James Tanton: how many subsets S of {1, 2, …, 10} have the property that S and S+2 cover all the numbers from 1 to 12? To solve this is a one-liner in R. Slightly generalizing, we can replace 10 by n:
library(bitops)
g = function(n){sum(bitOr(0:(2^n-1), 4*(0:(2^n-1))) == (2^(n+2)-1))}
Then it's easy to compute g(n) for n = 1, 2, …, 20:
n 1 2 3 4 5 6 7 8 9 10
g(n) 0 1 1 1 2 4 6 9 15 25
n 11 12 13 14 15 16 17 18 19 20
g(n) 40 64 104 169 273 441 714 1156 1870 3025
and if positive integers are at least your casual acquaintances you'll recognize a lot of squares here, and in particular a lot of squares of Fibonacci numbers:
The numbers in between the squares are a little trickier, but once we're primed to think Fibonacci it's not hard to see
So this leads to the conjecture that
for positive integers $n$. If you're allergic to cases you can write this as
So how to prove these formulas? We can explicitly list, say, the g(8) = 9 sets that we need.
sets = function(n){
indices = which(bitOr(0:(2^n-1), 4*(0:(2^n-1))) == (2^(n+2)-1))-1; #like the bit trickery above
for (i in 1:length(indices)) {
print(which(intToBits(indices[i]) == 01)) #convert from integer to vector of its 1s
(The "-1" is because lists in R are 1-indexed.) Then, for example, sets(9) outputs the fifteen sets
and now we examine the entrails. Each row consists of a subset S of {1, … 9} such that, when we take its union with S + 2, we get all the integers from 1 up to 11. Now when we add 2 to every element of S, we don't change parity, so it makes sense to look at even and odd numbers separately. If we extract just the even numbers from each of the sets, we get {2, 4, 8}, {2, 6, 8}, or {2, 4, 6, 8}, each of which occurs five times; if we extract just the odd numbers we get {1, 5, 9}, {1, 3, 5, 9}, {1, 3, 7, 9}, {1, 5, 7, 9} or {1, 3, 5, 7, 9}, each of which occurs three times. Every possible combination of one of the even subsets with one of the odd subsets occurs exactly once!
What to make of this? Well, in order for to satisfy , we must have that S contains 2; at least one of 2 or 4; at least one of 4 or 6; at least one of 6 or 8; and 8. Similarly, looking at just the odd numbers, S must contain 1; at least one of 1 or 3; at least one of 3 or 5; at least one of 5 or 7; at least one of 7 or 9; and 9. And will satisfy the overall condition if and only if its even part and its odd part do what they need to do; there's no interaction between them.
So now what? Consider the set in bold above; call it S. Its even part is {2, 4, 8} and its odd part is {1, 5, 7, 9}. We can take the even part and divide everything by 2 to obtain a set T = {1, 2, 4}; we can take the odd part, divide through by 2, and round up to get U = {1, 3, 4, 5}. The conditions transform similarly; in order for to satisfy , we have to have that:
the set T, obtained from S by taking its even subset and dividing through by 2, contains 1; at least one of 1 or 2; at least one of 2 or 3; at least one of 3 or 4; and 4;
the set U, obtained from S by taking its odd subset and dividing through by 2, contains 1; at least one of 1 or 2; at least one of 2 or 3; at least one of 3 or 4; at least one of 4 or 5; and 5.
Of course it's easier to say that T is a subset of {1, 2, 3, 4} containing 1, 4, and at least one of every two consecutive elements, and U is a similar subset of {1, 2, 3, 4, 5}. So how many subsets of {1, 2, 3, …, n} contain 1, n, and at least one of every two consecutive elements?
This is finally where the Fibonacci numbers come in. It's easier to count possible complements of T. If T satisfies the condition given above, then its complement with respect to {1, 2, …, n} is a subset of {2, 3, …, n-1} containing no two consecutive elements. Call the number of such sets f(n). For n = 3 there are two such subsets of {2}, namely the empty set and {2} itself; for n = 4 there are three such subsets of {2, 3}, namely the empty set, {2}, and {3}. So f(3) = 2, f(4) = 3. Now to find f(n) if n > 4. This is the number of subsets of {2, 3, …, n-1} which contain no two consecutive elements. These can be divided into two types: those which contain n-1 and those which don't. Those which contain n-1 can't contain n-2, and so are just subsets of {2, 3, …, n-3} which contain no two consecutive elements, with n-1 added. There are f(n-2) of these. Those which don't contain n-1 are just subsets of {2, 3, …, n-2} with no two consecutive elements; there are f(n-1) of these. So f(n) = f(n-1) + f(n-2); combined with the initial conditions we see that f(n) is just the nth Fibonacci number.
So we can finally compute g(n). For a set S to satisfy the condition defining g(n) — that is, to have and $S \cup (S+2) = {1, 2, \ldots, n+2}$ — we have to have that the corresponding is a subset of containing no two consecutive elements, and the corresponding is a subset of containing no two consecutive elements. The number of ways to do this is exactly , which is what we wanted.
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2 edition of Alpha particle scattering from 24șMg nuclei found in the catalog.
Alpha particle scattering from 24șMg nuclei
Anne Gerben Drentje
by Anne Gerben Drentje
Published 1972 by Veenstra-Visser in Groningen .
Alpha rays -- Scattering
Pagination 105 p.
Question: In A Scattering Experiment, An Alpha Particle A Is Projected With The Velocity Uo ( M/s)i ( M/s)j ( M/s)k Into A Stream Of Oxygen Nuclei Moving With A Common Velocity Vo ( M/s)j. After Colliding Successively With Nuclei B And C, Particle A Is Observed To Move Along The Path Defined By The Points A1(, , ) And A2(, , ), While. Alpha Particle Scattering and Rutherford's Nuclear Model of an Atom Class 12 Video | EduRev video for Class 12 is made by best teachers who have written some of the best books of Class It has gotten views and also has rating.
A focused and energy-analyzed Mev alpha-particle beam is used to bombard the targets. The elastically and inelastically scattered alpha particles from the gaseous N/sup 14/ target are detected by means of nuclear photographic plates spaced every deg in a multiplate scattering chamber covering the laboratory angular range of 10 to deg. The extracted potentials have been introduced in the analysis of α-particle elastic scattering from 12 C, 16 O, 20 Ne, 24 Mg, and 28 Si targets. Forty-seven sets of data through a wide range of energy (31– MeV) have been investigated using the derived potentials.
A thin gold foil was placed in the beam, and the scattering of the alpha particles was observed by the glow they caused when they struck a phosphor screen. Figure \(\PageIndex{7}\): Rutherford's experiment gave direct evidence for the size and mass of the nucleus by scattering alpha particles from a thin gold foil. Ignore the alpha particle's size and assume that all gold nuclei are exposed to it; that is, no gold nuclei are hidden behind other nuclei. Problem 23 In Rutherford's famous scattering experiments that led to the planetary model of the atom, alpha particles (having charges of $+2 e$ and masses of $ \times 10^{} \mathrm{kg}$) were fired.
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Alpha particle scattering from 24șMg nuclei by Anne Gerben Drentje Download PDF EPUB FB2
Try the new Google Books. Check out the new look and enjoy easier access to your favorite features. Try it now. No thanks. Try the new Google Books Get print book. No eBook available Alpha Particle Scattering from 24Mg Nuclei.
Anne Gerben Drentje. Veenstra-Visser, - Alpha rays. Additional Physical Format: Online version: Drentje, Anne Gerben. Alpha particle scattering from 24 Mg.
nuclei. Groningen, Veenstra, (OCoLC) A COMPLEX NUCLEUS ASPECTS OF ALPHA-PARTICLE SCATTERING was published in Proceedings of the Third Conference on Reactions Between Complex Nuclei on page Author: R. Davis. The Alpha Particle Scattering Experiment. They took a thin gold foil having a thickness of × m and placed it in the centre of a rotatable detector made of zinc sulfide and a microscope.
Then, they directed a beam of MeV alpha particles emitted from a radioactive source at the foil. In Rutherford's now-famous paper of May on the scattering of alpha particles by gold foil, he included this sketch of the hyperbolic path of a particle.
Credit: E. Rutherford, "The Scattering of α and β Particles by Matter and the Structure of Matter," Philosophical Magazine,– An ab initio calculation of alpha–alpha scattering is described for which the number of computational operations scales approximately quadratically with particle number and which uses lattice.
Rutherford's alpha particle scattering experiment changed the way we think of atoms. Before the experiment the best model of the atom was known Alpha particle scattering from 24șMg nuclei book the Thomson or "plum pudding" model.
The atom was believed to consist of a positive material "pudding" with negative "plums" distributed throughout. /**/ Rutherford directed beams of alpha particles (which are the nuclei of helium atoms and hence. the unattenuated intensity of the alpha particle beam.
Using classical mechanics to calculate the effect on al-pha particles as they approached atomic nuclei, Ruther-ford derived an equation to describe the scattering of particles at large angles, where scattering is mostly due to a single scattering event rather than multiple small an.
Electron Scattering from Nuclei. The scattering of electrons from nuclei has given us the most precise information about nuclear size and charge distribution. The electron is a better nuclear probe than the alpha particles of Rutherford scattering because it is a point particle and can penetrate the nucleus.
Rutherford scattering was the first method used to measure the size of nuclei. More precise measurements are made with electron scattering, and it was discovered that the density of nuclei is approximately has made possible the modeling of.
The radius obtained in the present work from the 4He scattering by silver is shown in fig. 7 and is seen to be close to be alpha-particle line (dashed) given by Kerlee et al. Typical errors obtained in the latter results are shown on the point for Zn.
The interaction radii for e are found to be very similar to those of the alpha particle. The alpha particle is tightly bound, but there are no stable A= 5 nuclei. 5He (2p+ 3n) has a half-life of only ×10−22 seconds, while 5Li (3p+ 2n) has a half-life of only ≈ 3×10−22 seconds.
Those lifetimes are so short, that the unbalanced nucleon can only make a. This is an experiment which studies scattering alpha particles on atomic nuclei.
You will shoot alpha particles, emitted by Am, at thin metal foils and measure the scattering cross section of the target atoms as a function of the scattering angle, the alpha particle. The deflected alpha-particle beam of the University of Washington inch cyclotron has been used to study the elastic scattering of to Mev alpha particles by Ag, Ta, Pb, and Th at 60°.
The alphaparticle energy was decreased in steps of approximately 1 Mev by placing remotely controlled absorbers in the cyclotron beam. Elastically scattered alpha particles were selected and counted by. Nuclear Instruments and Methods in Physics Research B12 () North-Holland, Amsterdam ON THE INVESTIGATION OF ALPHA-PARTICLE RESONANCE ELASTIC SCATTERING FROM THE " NUCLEUS R.A.
JARJIS * Department of Physics, Schuster Laboratory, The University o/Manchester, Manchester M] 3 9PL, England Received 26 April Solid target. Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modified (by interaction with other particles and/or potentials).Furthermore, while the particle's kinetic energy in the center-of-mass frame is constant.
A 4MeV alpha particle is scattered by, when it approaches a gold nucleus. Calculate the impact parameter if Z for gold is Calculate the impact parameter if Z for gold is Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 are generally produced in the process of alpha decay, but may also be produced in other particles are named after the first letter in the Greek alphabet, symbol for the alpha particle is α or α 2+.
The Rutherford Scattering Experiment Tony Tyson Ap 1 Introduction The foundations of modern ideas about atomic structure are considered to have been laid by Sir Ernest Rutherford inwith his postulates concerning the scattering of alpha particles by atoms.
alpha-particle resonances as evidence of clustering at high excitation in 40ca, The European Physical Journal A - Hadrons and Nuclei47 () 1–7. [5] E. Rutherford, The scattering of alpha and beta particles by matter and the structure of the atom. Experiments in which beams of particles such as electrons, nucleons, alpha particles and other atomic nuclei, and mesons are deflected by elastic collisions with atomic nuclei.
Much is learned from such experiments about the nature of the scattered particle, the scattering .The information which can be obtained from studies of low energy alpha-particle scattering from heavy nuclei and from alpha-decay is discussed.
The sensitivity of calculated widths and lifetimes for alpha-decay to the real nuclear potential is examined in detail using a formalism based on the unified theory of nuclear reactions.Elastic scattering of alpha particles — helium-4 nuclei — is pivotal in countless nuclear processes, from stellar nucleosynthesis to supernovae.
Despite their importance, it has so far been.
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Is there a more rigorous way to show these two sums are exactly equal?
I would like to have a more rigorous proof of the hypothesis: The Crandall eta derivative series is equal to the following more elementary one.
The following two series give the same sum.
$$-\sum _{n=1}^{\infty } \frac{(-1)^n \eta ^{(n)}(n)}{n!}$$
$$\sum _{n=1}^{\infty } (-1)^n \left(n^{\frac{1}{n}}-1\right).$$
@Gottfried Helms
made the following proof that is correct as far as the arithmetic and numeric evaluation of the eta derivatives from the logs. However, even as a novice, I think it lacks any formal proving power to show, logarithms do add up to exactly the eta derivatives! I think he was just pointing out to me that there is some trivialness in equating the two sums.
By use a series for
$n^{1/n} = \exp(\log n/n)$ and laying the two series: $$\sum _{n=1}^{\infty } (-1)^n \left(n^{\frac{1}{n}}-1\right)$$ and $$-\sum _{n=1}^{\infty } \frac{(-1)^n \eta ^{(n)}(n)}{n!}$$ out in two dimensions:
$$ \begin{array} {rclll} -\exp( {\log(1) \over 1})+1 & = & -{\log(1) \over 1} &-{\log(1)^2 \over 1^2 2!} &-{\log(1)^3 \over 1^3 3!} & - \cdots \\ +\exp( {\log(2) \over 2})-1 & = &+{\log(2) \over 2} &+{\log(2)^2 \over 2^2 2!} &+{\log(2)^3 \over 2^3 3!} & + \cdots \\ -\exp( {\log(3) \over 3})+1 & = &-{\log(3) \over 3} &-{\log(3)^2 \over 3^2 2!} &-{\log(3)^3 \over 3^3 3!} & - \cdots \\ \vdots \qquad & \vdots & \quad\vdots & \quad\vdots& \quad\vdots & \ddots \\ \hline \\ B \qquad & = & {\eta^{(1)}(1) \over 1!} &- {\eta^{(2)}(2) \over 2!} &+ {\eta^{(3)}(3) \over 3!} & - \cdots \end{array}$$
Can you improve his partial proof?
proof-verification alternative-proof
Marvin Ray Burns
Marvin Ray BurnsMarvin Ray Burns
Even though one has cause to be a little bit wary around formal rearrangements of conditionally convergent sums (see the Riemann series theorem), it's not very difficult to validate the formal manipulation of Helms. The idea is to cordon off a big chunk of the infinite double summation (all the terms from the second column on) that we know is absolutely convergent, which we are then free to rearrange with impunity. (Most relevantly for our purposes here, see pages 80-85 of this document, culminating with the Fubini theorem which is essentially the manipulation Helms is using.)
So, by definition the MRB constant $B$ is the conditionally convergent sum $\sum_{n \geq 1} (-1)^n (n^{1/n} - 1)$. Put $a_n = (-1)^n (n^{1/n} - 1)$, so $B = \sum_{n \geq 1} a_n$. Looking at the first column, put $b_n = (-1)^n \frac{\log(n)}{n}$, so $\eta^{(1)}(1) = \sum_{n \geq 1} b_n$ as a conditionally convergent series.
$$B - \eta^{(1)} = \sum_{n \geq 1} a_n - b_n = \sum_{n \geq 1} \sum_{m \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!}$$
(The first equation is an elementary limit statement that says if $\sum_{n \geq 1} a_n$ converges and $\sum_{n \geq 1} b_n$ converges, then also $\sum_{n \geq 1} a_n - b_n$ converges and $\sum_{n \geq 1} a_n - \sum_{n \geq 1} b_n = \sum_{n \geq 1} a_n - b_n$. It doesn't at all matter whether the convergence of either series is conditional or absolute.)
So now we check the absolute convergence of the right-hand side, i.e., that $\sum_{n \geq 1} \sum_{m \geq 2} \frac{(\log n)^m}{n^m m!}$ converges. (Remember what this means in the case of infinite sums of positive terms: it means that there is a number $K$ such that every finite partial sum $S$ is bounded above by $K$; the least such upper bound will be the number that the infinite sum converges to.) So take any such finite partial sum $S$, and rearrange its terms so that the terms in the $m = 2$ column come first, then the terms in the $m = 3$ column, and so on. An upper bound for the terms of $S$ in the $m = 2$ column is $\frac{\zeta^{(2)}(2)}{2!}$. Put that one aside.
For the $m = 3$ column, an upper bound is $\sum_{n \geq 2} \frac{(\log n)^3}{n^3 3!}$ (we drop the $n=1$ term which is $0$). By calculus we have $\log n \leq n^{1/2}$ for all $n \geq 2$, so this has upper bound $\frac1{3!} \sum_{n \geq 2} \frac1{n^{3/2}} \leq \frac1{3!} \int_1^\infty \frac{dx}{x^{3/2}}$ by an integral test, which yields $\frac{2}{3!}$ as an upper bound. Applying the same reasoning for the $m$ column from $m = 4$ on, an upper bound for that column would be $\frac1{m!} \int_1^\infty \frac{dx}{x^{m/2}} = \frac{2}{m!(m-2)}$. Adding all those upper bounds together, an upper bound for the entire doubly infinite sum would be
$$\frac{\zeta^{(2)}(2)}{2!} + \sum_{m \geq 3} \frac{2}{m!(m-2)}$$
which certainly converges. So we have absolute convergence of the doubly infinite sum.
Thus we are in a position to apply the Fubini theorem, which justifies the rearrangement expressed in the first of the following equations
$$\sum_{n \geq 1} \sum_{m \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!} = \sum_{m \geq 2} \sum_{n \geq 1} (-1)^n \frac{(\log n)^m}{n^m m!} = \sum_{m \geq 2} (-1)^{m+1} \frac{\eta^{(m)}(m)}{m!}$$
giving us what we wanted.
$\begingroup$ I understood most of that. Thanks! I will take apart until I fully understand. $\endgroup$
– Marvin Ray Burns
$\begingroup$ Take your time; glad to help. $\endgroup$
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High rate of linezolid intermediate susceptibility and resistance among enteric vancomycin-resistant Enterococcus (VRE) recovered from hospitalized patients actively screened for VRE colonization
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Journal: Infection Control & Hospital Epidemiology / Volume 40 / Issue 7 / July 2019
Published online by Cambridge University Press: 15 May 2019, pp. 821-822
Establishment of Salsola tragus on aeolian sands: A Southern Colorado Plateau case study
Kathryn A. Thomas, Margaret Hiza Redsteer
Journal: Invasive Plant Science and Management , FirstView
Published online by Cambridge University Press: 02 May 2019, pp. 1-9
Russian-thistle (Salsola tragus L.), is a nonnative, C4 photosynthesizing, annual plant that infests disturbed and natural areas in the arid U.S. Southwest. Land managers of natural areas may need to decide whether a S. tragus infestation is potentially harmful and whether it should be actively managed. One factor informing that decision is an understanding of the conditions under which this weed emerges and establishes and how those processes affect where and when infestations occur. We studied S. tragus establishment on aeolian (windblown) sandy soils at Petrified Forest National Park, AZ. Our sites were a previously disturbed sand sheet and a semistabilized sand dune. Measurements in plots on these sites over two growing seasons revealed a similar number of S. tragus seedlings emerging on both sites early in the 2015 growing season. As the season progressed, S. tragus cover (seedling survival and growth) was lower on the sand dune, except for a plot placed entirely on a coppice mound. In 2016, S. tragus seedling emergence and development of cover, measured on plots at both sites, was exceptionally low, as was summer rainfall. A growth chamber assay of seedling emergence from soil and litter samples collected at each site showed emergence was greatest from samples collected where S. tragus litter remained on the soil surface, and otherwise was infrequent. Our study suggests that S. tragus emergence and early establishment are sensitive to low precipitation and that soil-surface microtopography and grass and shrub cover may be determinants of the spatial pattern of infestation on sandy soils. As aeolian sands occur throughout drylands of the U.S. Southwest, deeper understanding of the conditions under which S. tragus seedlings emerge and establish can inform management of this invasive annual in those habitats.
RAMSEY GROWTH IN SOME NIP STRUCTURES
MSC 2010: Model theory
MSC 2010: Extremal combinatorics
MSC 2010: Graph theory
Artem Chernikov, Sergei Starchenko, Margaret E. M. Thomas
Journal: Journal of the Institute of Mathematics of Jussieu , First View
Published online by Cambridge University Press: 19 February 2019, pp. 1-29
We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek (Duke Mathematical Journal163(12) (2014), 2243–2270) from the semialgebraic case to arbitrary polynomially bounded $o$ -minimal expansions of $\mathbb{R}$ , and show that it does not hold in $\mathbb{R}_{\exp }$ . This provides a new combinatorial characterization of polynomial boundedness for $o$ -minimal structures. We also prove an analog for relations definable in $P$ -minimal structures, in particular for the field of the $p$ -adics. Generalizing Conlon et al. (Transactions of the American Mathematical Society366(9) (2014), 5043–5065), we show that in distal structures the upper bound for $k$ -ary definable relations is given by the exponential tower of height $k-1$ .
Throw the Blasphemer off a Cliff: Luke 4.16–30 in Light of the Life of Aesop
Margaret Froelich, Thomas E. Phillips
Journal: New Testament Studies / Volume 65 / Issue 1 / January 2019
Published online by Cambridge University Press: 29 November 2018, pp. 21-32
In Jesus' sermon at Nazareth in Luke (4.16–30), his reminder that Elijah had aided non-Jews (vv. 26–7) is met with an unusual death sentence – to throw Jesus from a cliff. This has been conceptually and geographically vexing for scholars. This paper reads the passage beside the Life of Aesop, in which the Delphians condemn the fabulist to the same fate for blasphemy (130–42). Aesop's offence, like Jesus', is to malign the special status of the Delphians before their god. The Lukan Evangelist's use of the same manner of death for the same type of speech act indicates that the crowd at Nazareth has condemned Jesus for blasphemy.
Education in Twins and Their Parents Across Birth Cohorts Over 100 years: An Individual-Level Pooled Analysis of 42-Twin Cohorts
Karri Silventoinen, Aline Jelenkovic, Antti Latvala, Reijo Sund, Yoshie Yokoyama, Vilhelmina Ullemar, Catarina Almqvist, Catherine A. Derom, Robert F. Vlietinck, Ruth J. F. Loos, Christian Kandler, Chika Honda, Fujio Inui, Yoshinori Iwatani, Mikio Watanabe, Esther Rebato, Maria A. Stazi, Corrado Fagnani, Sonia Brescianini, Yoon-Mi Hur, Hoe-Uk Jeong, Tessa L. Cutler, John L. Hopper, Andreas Busjahn, Kimberly J. Saudino, Fuling Ji, Feng Ning, Zengchang Pang, Richard J. Rose, Markku Koskenvuo, Kauko Heikkilä, Wendy Cozen, Amie E. Hwang, Thomas M. Mack, Sisira H. Siribaddana, Matthew Hotopf, Athula Sumathipala, Fruhling Rijsdijk, Joohon Sung, Jina Kim, Jooyeon Lee, Sooji Lee, Tracy L. Nelson, Keith E. Whitfield, Qihua Tan, Dongfeng Zhang, Clare H. Llewellyn, Abigail Fisher, S. Alexandra Burt, Kelly L. Klump, Ariel Knafo-Noam, David Mankuta, Lior Abramson, Sarah E. Medland, Nicholas G. Martin, Grant W. Montgomery, Patrik K. E. Magnusson, Nancy L. Pedersen, Anna K. Dahl Aslan, Robin P. Corley, Brooke M. Huibregtse, Sevgi Y. Öncel, Fazil Aliev, Robert F. Krueger, Matt McGue, Shandell Pahlen, Gonneke Willemsen, Meike Bartels, Catharina E. M. van Beijsterveldt, Judy L. Silberg, Lindon J. Eaves, Hermine H. Maes, Jennifer R. Harris, Ingunn Brandt, Thomas S. Nilsen, Finn Rasmussen, Per Tynelius, Laura A. Baker, Catherine Tuvblad, Juan R. Ordoñana, Juan F. Sánchez-Romera, Lucia Colodro-Conde, Margaret Gatz, David A. Butler, Paul Lichtenstein, Jack H. Goldberg, K. Paige Harden, Elliot M. Tucker-Drob, Glen E. Duncan, Dedra Buchwald, Adam D. Tarnoki, David L. Tarnoki, Carol E. Franz, William S. Kremen, Michael J. Lyons, José A. Maia, Duarte L. Freitas, Eric Turkheimer, Thorkild I. A. Sørensen, Dorret I. Boomsma, Jaakko Kaprio
Journal: Twin Research and Human Genetics / Volume 20 / Issue 5 / October 2017
Published online by Cambridge University Press: 04 October 2017, pp. 395-405
Print publication: October 2017
Whether monozygotic (MZ) and dizygotic (DZ) twins differ from each other in a variety of phenotypes is important for genetic twin modeling and for inferences made from twin studies in general. We analyzed whether there were differences in individual, maternal and paternal education between MZ and DZ twins in a large pooled dataset. Information was gathered on individual education for 218,362 adult twins from 27 twin cohorts (53% females; 39% MZ twins), and on maternal and paternal education for 147,315 and 143,056 twins respectively, from 28 twin cohorts (52% females; 38% MZ twins). Together, we had information on individual or parental education from 42 twin cohorts representing 19 countries. The original education classifications were transformed to education years and analyzed using linear regression models. Overall, MZ males had 0.26 (95% CI [0.21, 0.31]) years and MZ females 0.17 (95% CI [0.12, 0.21]) years longer education than DZ twins. The zygosity difference became smaller in more recent birth cohorts for both males and females. Parental education was somewhat longer for fathers of DZ twins in cohorts born in 1990–1999 (0.16 years, 95% CI [0.08, 0.25]) and 2000 or later (0.11 years, 95% CI [0.00, 0.22]), compared with fathers of MZ twins. The results show that the years of both individual and parental education are largely similar in MZ and DZ twins. We suggest that the socio-economic differences between MZ and DZ twins are so small that inferences based upon genetic modeling of twin data are not affected.
2511: Use of an online provider learning community to assess clinical HIV/HCV/STDs-related training needs
Cabiria Monica Barbosu, Jose G. Perez-Ramos, Margaret Demment, Thomas Fogg, Jack Chang, Beatrice Aladin, Cheryl Smith, Timothy De Ver Dye, Terry Doll
Journal: Journal of Clinical and Translational Science / Volume 1 / Issue S1 / September 2017
Published online by Cambridge University Press: 10 May 2018, p. 51
Print publication: September 2017
OBJECTIVES/SPECIFIC AIMS: The prevention, management, and treatment of HIV, STDs, and HCV requires continuous training that reflects contemporary best-practice and innovative care models. In order to improve the NYS AIDS Institute's comprehensive web-enabled training program, which enhances the capacity of a diverse healthcare workforce, a needs assessment (NA) of our community of practice (CoP) is needed to better understand their training needs, circumstances, and instructional modalities preferences. The goal of the assessment was to better understand our CoP's preferences of online trainings, and as a result to develop a "responsive design" system that will enhance user's learning experience thus improving patient care. METHODS/STUDY POPULATION: We developed and deployed an NA survey using REDCap. The instrument consisted in 27 questions related to providers' preferences on receiving continuing educational training and their use of technologies, including mobile platforms, online modules, webinars, and telehealth. As part of the recruitment strategy, several resources were deployed over a 1-month recruitment period including sequential email blasts, website promotion, and assessment links included in newsletters and social media. Weekly reminders were also used to promote the participation from our CoP. RESULTS/ANTICIPATED RESULTS: A total of 310 respondents participated in the NA, with 85.8% from NYS. 177 were clinicians (20.5% MD, 2.9% PA, 17.3% NP, and 16.3% RN) and 133 nonclinical providers (case/care managers, social workers, public health professionals, coordinators/administrators, and other). The participants worked in hospitals, community health centers, substance use centers, private practices, and state/local health departments. More than 90% of respondents indicated that they preferred both live/in-person and online training, and participants most strongly indicated that they stayed up-to-date on current developments through CDC, the AIDS Institute, and conferences. More than 60% of respondents considered that receiving CE credit for the training was very important and 28% indicated they would use training materials in Spanish if offered. In terms of technology, over 80% of the respondents preferred computers, but more 50% also used mobile devices (computer at home 61.8%, computer at work 85%, tablet 29.9%, iPhone 20.9%, Android 16.6%, other device 2.3%). DISCUSSION/SIGNIFICANCE OF IMPACT: Accessing an online CoP provided a useful opportunity to assess training needs and preferences of clinical and nonclinical providers. Most providers indicated that they were primarily likely to use a work computer to complete online training or secondarily a home computer. With a significant portion of respondents indicating use of tablets, smartphones, and other devices, online training opportunities should be developed with responsive design to assure flexibility and access. In addition to online training, participants indicated that they also strongly valued live, in-person training. Offering training with CDC and the NYS AIDS Institute branding, in Spanish, together with offering continuing education credit, were all seen as desirable training elements. Accessing this online CoP helped streamline and target training priorities and logistics.
2514: Governance for a decentralized informatics academic environment
Thomas Fogg, Margaret Demment, Jack Chang, Kathleen Holt, Dongmei Li, Helene McMurray, David Pinto, Timothy De Ver Dye
OBJECTIVES/SPECIFIC AIMS: Due to scope and breadth of research activity and infrastructure capacities at academic medical centers, the discipline of Biomedical Informatics is often deployed in a decentralized manner through geographically dispersed and unrelated organizational units. As a result, without a conscious strategy, an academic medical center risks redundant effort and gaps in resources, and perhaps poor coordination. A mechanism to bring together disparate organizational entities to identify, discuss, and negotiate Informatics-related concerns may produce a better institutional research environment. The University of Rochester (UR) has implemented such a strategy of Informatics governance, adapting tactics from team science, diplomacy, and deliberative engagement. METHODS/STUDY POPULATION: Based on current needs and institutional Informatics priorities, the UR's Clinical and Translational Science Institute (CTSI) established 6 Informatics "clusters" in distinct but deliberately overlapping focal areas: (1) Data—capture, management, and analysis of all types of data for research. (2) Analytics—quantitative research across the spectrum of translational research. (3) Infrastructure—technical and computing infrastructure to support informatics. (4) Electronic health records (EHR)—(i) features within the EHR explicitly designed to address the needs of research; (ii) accessing and procuring EHR data for research. (5) Population health—Informatics design and systems expertise relevant to population health research (a key CTSI focus area). (6) Education—development, deployment, and assessment of Informatics learning opportunities for learners at all levels. Each cluster facilitates access to expertise and resources around the institution, promotes collaboration, identifies redundancy, and serves as a forum to strategize regarding institutional needs related to Biomedical Informatics. A CTSI faculty or staff member leads each cluster. To maximize effectiveness of the cluster, other members are decision-makers in the organizations they represent, or serve in a critical staff function. Clusters meet in person on a quarterly basis with more frequent electronic interaction. The clusters share documents via Box, a secure online file sharing app. The cluster coordinators meet as a group on a biweekly basis to monitor progress and make plans. RESULTS/ANTICIPATED RESULTS: There were 45 different people representing 46 distinct centers, departments or offices, and 2 outside agencies agreed to participate in the clusters. In total, 20 people represented a single organizational unit; 15 represented 2 units; 8 represented 3 units, and 2 represented 4 units. The richness and complexity of these organizational linkages illustrates the decentralized nature of Informatics at the institution and the promise of the cluster approach. DISCUSSION/SIGNIFICANCE OF IMPACT: Adapting to a decentralized Informatics environment, the CTSI established clusters that recognize and respect autonomy and capacity of a wide range of units throughout the university, creating a collaborative atmosphere for steering and implementing an overall Informatics vision. As Informatics capacity rapidly expands throughout growing biomedical research institutions without a centralized Informatics hub, this distributed, deliberative approach could offer an effective governance solution that promotes cooperation. In this model, the CTSI provides the leadership and staffing necessary to ensure progress at the institutional level around Informatics and creates a venue for communication and coordination on Informatics-related topics.
2492: Leveraging CTSA informatics capacity to expand global health engagement and research capacity in Latin America and the Pacific
Timothy De Ver Dye, Thomas Fogg, Margaret Demment, José Pérez-Ramos, Scott McIntosh, Deborah Ossip, Angela Sy, Carmen Velez Vega, Karen Peters, Haq Nawaz
OBJECTIVES/SPECIFIC AIMS: The objective of this partnership was to create a global network of clinical and public health researchers and communities conducting technology-assisted research in noncommunicable disease. METHODS/STUDY POPULATION: The University of Rochester's Clinical and Translational Science Institute (CTSI) has successfully leveraged the informatics core's capacity into an emerging network of organizations that focus on technology and health in settings outside of the mainland United States. The CTSI coordinated with another NIH-funded infrastructure program [the RCMI Translational Research Network (RTRN)] to identify partner institutions interested in technology and health. RTRN identified the University of Puerto Rico and the University of Hawaii, both of which serve as hubs for common research interests in technology and health throughout the Caribbean and the Pacific. This network was formalized as the CDC's Coordinating Center for its Global and Territorial Health Research Network (the "Global Network"), with additional US partners (Yale, University of Illinois at Chicago, University of North Caroline Chapel Hill, and the University of South Florida) within a wider scope of the CDC's Prevention Research Centers (PRC) program. RESULTS/ANTICIPATED RESULTS: Through combining 2 main NIH-funded research infrastructure networks (CTSA and RTRN), with a large CDC-funded PRC, the University of Rochester's Informatics Core was successful in establishing a new productive global health network throughout Latin America and the Caribbean, and in the Pacific, garnering additional research support from NIH Fogarty and other programs. The resulting network not only supports locally-important research in technology and health on compelling health issues (eg, diabetes, ZIka, participation in research), but also facilitates community engagement through local partnerships and the cores of the involved networks. In addition, much of the information and communications technology (ICT)-related research and learnings from the Global Network activity is immediately applicable to populations in the United States, served by the various collaborative networks. In total, while new, the Global Network supports a wide range of projects and engagements throughout the world that expand local informatics capacity and use of technology in the research process and to address global health problems, further enhancing the CTSI's informatics core to serve the needs of its own constituency and promote research engagement with technology within this population. Local research collaborative projects reinforce the utility of the network and its resources, evidenced by tools, publications, partnerships, and conference presentations that have arisen. Lessons to date from this Global Network collaboration include: specific global research projects provide opportunities for partnership building and meaningful collaboration, team science is of central importance in distributing the work of the network, synergy is multidirectional with expertise and need flowing in all directions, and project team members in all locales learned and contributed substantially in ways that carried into their other responsibilities. DISCUSSION/SIGNIFICANCE OF IMPACT: The overall partnership has created opportunity for South-South collaboration, for adaptation of projects among locales, and has helped boost reputational value for all partners involved. Implications for other CTSA awardees include: global collaboration can serve core research and technical needs for the CTSA itself and its local partners, CTSA status can be leveraged to access resources to support local research, and collaboration in other federally-funded research networks helps expand the insight, scope, and potential for new research.
Psycho-historical linguistics: its context and potential
MARGARET E. WINTERS
Journal: English Language & Linguistics / Volume 21 / Issue 2 / July 2017
Published online by Cambridge University Press: 07 July 2017, pp. 413-421
Historical linguistics is a field that, perhaps more than other branches of linguistics, can be said to exhibit a certain conservatism. To be clear, this term is not meant in any traditional political sense. Rather it is meant to capture the notion that, as a discipline, diachronic studies seem to accept and build on previous theories and empirical findings to a greater extent than do most synchronic subdisciplines. This may be because data are comparatively rare and hard to come by. One result of this scarcity is that, once analyzed, there are fewer opportunities for reanalysis predicated on new data. There are, of course, occasions when more or less radical proposals are brought forward subsequently, which result in debates of the kind which are much more common in synchronic syntax, say, or phonology. The reconstruction of the Indo-European consonant system (Beekes 1995: 132–4 provides a summary), for example, continues to be debated almost two hundred years after it was first proposed.
Epidemiological Effectiveness and Cost of a Fungal Meningitis Outbreak Response in New River Valley, Virginia: Local Health Department and Clinical Perspectives
Nargesalsadat Dorratoltaj, Margaret L. O'Dell, Paige Bordwine, Thomas M. Kerkering, Kerry J. Redican, Kaja M. Abbas
Journal: Disaster Medicine and Public Health Preparedness / Volume 12 / Issue 1 / February 2018
Published online by Cambridge University Press: 05 June 2017, pp. 38-46
Print publication: February 2018
We evaluated the effectiveness and cost of a fungal meningitis outbreak response in the New River Valley of Virginia during 2012-2013 from the perspective of the local public health department and clinical facilities. The fungal meningitis outbreak affected 23 states in the United States with 751 cases and 64 deaths in 20 states; there were 56 cases and 5 deaths in Virginia.
We conducted a partial economic evaluation of the fungal meningitis outbreak response in New River Valley. We collected costs associated with the local health department and clinical facilities in the outbreak response and estimated the epidemiological effectiveness by using disability-adjusted life years (DALYs) averted.
We estimated the epidemiological effectiveness of this outbreak response to be 153 DALYs averted among the patients, and the costs incurred by the local health department and clinical facilities to be $30,413 and $39,580, respectively.
We estimated the incremental cost-effectiveness ratio of $198 per DALY averted and $258 per DALY averted from the local health department and clinical perspectives, respectively, thereby assisting in impact evaluation of the outbreak response by the local health department and clinical facilities. (Disaster Med Public Health Preparedness. 2018;12:38–46)
Education and training of clinical and translational study investigators and research coordinators: A competency-based approach
Nancy A. Calvin-Naylor, Carolynn Thomas Jones, Michelle M. Wartak, Karen Blackwell, Jonathan M. Davis, Ruthvick Divecha, Edward F. Ellerbeck, Karl Kieburtz, Margaret J. Koziel, Katherine Luzuriaga, Jennifer Maddox, Nancy A. Needler, Susan Murphy, Kieran Pemberton, Catherine Radovich, Eric P. Rubinstein, Harry P. Selker, Pamela Tenaerts, Kelly Unsworth, Kay Wilson, Jonelle E. Wright, Richard Barohn, Thomas P. Shanley
Journal: Journal of Clinical and Translational Science / Volume 1 / Issue 1 / February 2017
Published online by Cambridge University Press: 13 January 2017, pp. 16-25
Training for the clinical research workforce does not sufficiently prepare workers for today's scientific complexity; deficiencies may be ameliorated with training. The Enhancing Clinical Research Professionals' Training and Qualifications developed competency standards for principal investigators and clinical research coordinators.
Clinical and Translational Science Awards representatives refined competency statements. Working groups developed assessments, identified training, and highlighted gaps.
Forty-eight competency statements in 8 domains were developed.
Training is primarily investigator focused with few programs for clinical research coordinators. Lack of training is felt in new technologies and data management. There are no standardized assessments of competence.
Enhancing Clinical Research Professionals' Training and Qualifications (ECRPTQ): Recommendations for Good Clinical Practice (GCP) training for investigators and study coordinators
Thomas P. Shanley, Nancy A. Calvin-Naylor, Ruthvick Divecha, Michelle M. Wartak, Karen Blackwell, Jonathan M. Davis, Edward F. Ellerbeck, Karl Kieburtz, Margaret J. Koziel, Katherine Luzuriaga, Jennifer Maddox, Nancy A. Needler, Susan Murphy, Kieran Pemberton, Catherine Radovich, Eric P. Rubinstein, Harry P. Selker, Pamela Tenaerts, Kelly Unsworth, Kay Wilson, Jonelle E. Wright, Richard Barohn
Published online by Cambridge University Press: 13 January 2017, pp. 8-15
The translation of discoveries to drugs, devices, and behavioral interventions requires well-prepared study teams. Execution of clinical trials remains suboptimal due to varied quality in design, execution, analysis, and reporting. A critical impediment is inconsistent, or even absent, competency-based training for clinical trial personnel.
In 2014, the National Center for Advancing Translational Science (NCATS) funded the project, Enhancing Clinical Research Professionals' Training and Qualifications (ECRPTQ), aimed at addressing this deficit. The goal was to ensure all personnel are competent to execute clinical trials. A phased structure was utilized.
This paper focuses on training recommendations in Good Clinical Practice (GCP). Leveraging input from all Clinical and Translational Science Award hubs, the following was recommended to NCATS: all investigators and study coordinators executing a clinical trial should understand GCP principles and undergo training every 3 years, with the training method meeting the minimum criteria identified by the International Conference on Harmonisation GCP.
We anticipate that industry sponsors will acknowledge such training, eliminating redundant training requests. We proposed metrics to be tracked that required further study. A separate task force was composed to define recommendations for metrics to be reported to NCATS.
Paleogroundwater in the Moutere Gravel Aquifers Near Nelson, New Zealand
Michael K Stewart, Joseph T Thomas, Margaret Norris, Vanessa Trompetter
Journal: Radiocarbon / Volume 46 / Issue 2 / 2004
Radiocarbon, 18O, and chemical concentrations have been used to identify groundwater recharged during the last ice age near Nelson, New Zealand. Moutere Gravel underlies most of the Moutere Depression, a 30-km-wide system of valleys filled with Plio-Pleistocene gravel. The depression extends northwards into Tasman Bay, which was above sea level when the North and South Islands of New Zealand were connected during the last glaciation. The aquifers are tapped by bores up to 500 m deep. Shallow bores (50–100 m) tap "pre-industrial" Holocene water (termed the "modern" component) with 14C concentrations of 90 ± 10 percent modern carbon (pMC) and δ18O values of −6.8 ± 0.4, as expected for present-day precipitation. Deeper bores discharge water with lower 14C concentrations and more negative 18O values resulting from input of much older water from depth. The deep end-member of the mixing trend is identified as paleowater (termed the "glacial" component) with 14C concentration close to 0 pMC and more negative 18O values (-7.6). Mixing of the modern and glacial components gives rise to the variations observed in the 14C, 18O, and chemical concentrations of the waters. Identification of the deep groundwater as glacial water suggests that there may be a large body of such water onshore and offshore at deep levels. More generally, the influence of changing sea levels in the recent past (geologically speaking) on the disposition of groundwaters in coastal areas of New Zealand may have been far greater than we have previously realized.
Abnormalities in serum biomarkers correlate with lower cardiac index in the Fontan population
Bradley S. Marino, David J. Goldberg, Adam L. Dorfman, Eileen King, Heidi Kalkwarf, Babette S. Zemel, Margaret Smith, Jesse Pratt, Mark A. Fogel, Amanda J. Shillingford, Barbara J. Deal, Anitha S. John, Caren S. Goldberg, Timothy M. Hoffman, Marshall L. Jacobs, Asher Lisec, Susan Finan, Lazaros K. Kochilas, Thomas W. Pawlowski, Kathleen Campbell, Clinton Joiner, Stuart L. Goldstein, Paul Stephens, Alvin J. Chin
Journal: Cardiology in the Young / Volume 27 / Issue 1 / January 2017
Published online by Cambridge University Press: 05 July 2016, pp. 59-68
Fontan survivors have depressed cardiac index that worsens over time. Serum biomarker measurement is minimally invasive, rapid, widely available, and may be useful for serial monitoring. The purpose of this study was to identify biomarkers that correlate with lower cardiac index in Fontan patients.
Methods and results
This study was a multi-centre case series assessing the correlations between biomarkers and cardiac magnetic resonance-derived cardiac index in Fontan patients ⩾6 years of age with biochemical and haematopoietic biomarkers obtained ±12 months from cardiac magnetic resonance. Medical history and biomarker values were obtained by chart review. Spearman's Rank correlation assessed associations between biomarker z-scores and cardiac index. Biomarkers with significant correlations had receiver operating characteristic curves and area under the curve estimated. In total, 97 cardiac magnetic resonances in 87 patients met inclusion criteria: median age at cardiac magnetic resonance was 15 (6–33) years. Significant correlations were found between cardiac index and total alkaline phosphatase (−0.26, p=0.04), estimated creatinine clearance (0.26, p=0.02), and mean corpuscular volume (−0.32, p<0.01). Area under the curve for the three individual biomarkers was 0.63–0.69. Area under the curve for the three-biomarker panel was 0.75. Comparison of cardiac index above and below the receiver operating characteristic curve-identified cut-off points revealed significant differences for each biomarker (p<0.01) and for the composite panel [median cardiac index for higher-risk group=2.17 L/minute/m2 versus lower-risk group=2.96 L/minute/m2, (p<0.01)].
Higher total alkaline phosphatase and mean corpuscular volume as well as lower estimated creatinine clearance identify Fontan patients with lower cardiac index. Using biomarkers to monitor haemodynamics and organ-specific effects warrants prospective investigation.
Clinical Response, Outbreak Investigation, and Epidemiology of the Fungal Meningitis Epidemic in the United States: Systematic Review
Kaja M. Abbas, Nargesalsadat Dorratoltaj, Margaret L. O'Dell, Paige Bordwine, Thomas M. Kerkering, Kerry J. Redican
Published online by Cambridge University Press: 18 December 2015, pp. 145-151
We conducted a systematic review of the 2012–2013 multistate fungal meningitis epidemic in the United States from the perspectives of clinical response, outbreak investigation, and epidemiology. Articles focused on clinical response, outbreak investigation, and epidemiology were included, whereas articles focused on compounding pharmacies, legislation and litigation, diagnostics, microbiology, and pathogenesis were excluded. We reviewed 19 articles by use of the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) framework. The source of the fungal meningitis outbreak was traced to the New England Compounding Center in Massachusetts, where injectable methylprednisolone acetate products were contaminated with the predominant pathogen, Exserohilum rostratum. As of October 23, 2013, the final case count stood at 751 patients and 64 deaths, and no additional cases are anticipated. The multisectoral public health response to the fungal meningitis epidemic from the hospitals, clinics, pharmacies, and the public health system at the local, state, and federal levels led to an efficient epidemiological investigation to trace the outbreak source and rapid implementation of multiple response plans. This systematic review reaffirms the effective execution of a multisectoral public health response and efficient delivery of the core functions of public health assessment, policy development, and service assurances to improve population health.(Disaster Med Public Health Preparedness. 2016;10:145–151)
Zygosity Differences in Height and Body Mass Index of Twins From Infancy to Old Age: A Study of the CODATwins Project
Aline Jelenkovic, Yoshie Yokoyama, Reijo Sund, Chika Honda, Leonie H Bogl, Sari Aaltonen, Fuling Ji, Feng Ning, Zengchang Pang, Juan R. Ordoñana, Juan F. Sánchez-Romera, Lucia Colodro-Conde, S. Alexandra Burt, Kelly L. Klump, Sarah E. Medland, Grant W. Montgomery, Christian Kandler, Tom A. McAdams, Thalia C. Eley, Alice M. Gregory, Kimberly J. Saudino, Lise Dubois, Michel Boivin, Adam D. Tarnoki, David L. Tarnoki, Claire M. A. Haworth, Robert Plomin, Sevgi Y. Öncel, Fazil Aliev, Maria A. Stazi, Corrado Fagnani, Cristina D'Ippolito, Jeffrey M. Craig, Richard Saffery, Sisira H. Siribaddana, Matthew Hotopf, Athula Sumathipala, Fruhling Rijsdijk, Timothy Spector, Massimo Mangino, Genevieve Lachance, Margaret Gatz, David A. Butler, Gombojav Bayasgalan, Danshiitsoodol Narandalai, Duarte L Freitas, José Antonio Maia, K. Paige Harden, Elliot M. Tucker-Drob, Bia Kim, Youngsook Chong, Changhee Hong, Hyun Jung Shin, Kaare Christensen, Axel Skytthe, Kirsten O. Kyvik, Catherine A. Derom, Robert F. Vlietinck, Ruth J. F. Loos, Wendy Cozen, Amie E. Hwang, Thomas M. Mack, Mingguang He, Xiaohu Ding, Billy Chang, Judy L. Silberg, Lindon J. Eaves, Hermine H. Maes, Tessa L. Cutler, John L. Hopper, Kelly Aujard, Patrik K. E. Magnusson, Nancy L. Pedersen, Anna K. Dahl Aslan, Yun-Mi Song, Sarah Yang, Kayoung Lee, Laura A. Baker, Catherine Tuvblad, Morten Bjerregaard-Andersen, Henning Beck-Nielsen, Morten Sodemann, Kauko Heikkilä, Qihua Tan, Dongfeng Zhang, Gary E. Swan, Ruth Krasnow, Kerry L. Jang, Ariel Knafo-Noam, David Mankuta, Lior Abramson, Paul Lichtenstein, Robert F. Krueger, Matt McGue, Shandell Pahlen, Per Tynelius, Glen E. Duncan, Dedra Buchwald, Robin P. Corley, Brooke M. Huibregtse, Tracy L. Nelson, Keith E. Whitfield, Carol E. Franz, William S. Kremen, Michael J. Lyons, Syuichi Ooki, Ingunn Brandt, Thomas Sevenius Nilsen, Fujio Inui, Mikio Watanabe, Meike Bartels, Toos C. E. M. van Beijsterveldt, Jane Wardle, Clare H. Llewellyn, Abigail Fisher, Esther Rebato, Nicholas G. Martin, Yoshinori Iwatani, Kazuo Hayakawa, Joohon Sung, Jennifer R. Harris, Gonneke Willemsen, Andreas Busjahn, Jack H. Goldberg, Finn Rasmussen, Yoon-Mi Hur, Dorret I. Boomsma, Thorkild I. A. Sørensen, Jaakko Kaprio, Karri Silventoinen
Published online by Cambridge University Press: 04 September 2015, pp. 557-570
A trend toward greater body size in dizygotic (DZ) than in monozygotic (MZ) twins has been suggested by some but not all studies, and this difference may also vary by age. We analyzed zygosity differences in mean values and variances of height and body mass index (BMI) among male and female twins from infancy to old age. Data were derived from an international database of 54 twin cohorts participating in the COllaborative project of Development of Anthropometrical measures in Twins (CODATwins), and included 842,951 height and BMI measurements from twins aged 1 to 102 years. The results showed that DZ twins were consistently taller than MZ twins, with differences of up to 2.0 cm in childhood and adolescence and up to 0.9 cm in adulthood. Similarly, a greater mean BMI of up to 0.3 kg/m2 in childhood and adolescence and up to 0.2 kg/m2 in adulthood was observed in DZ twins, although the pattern was less consistent. DZ twins presented up to 1.7% greater height and 1.9% greater BMI than MZ twins; these percentage differences were largest in middle and late childhood and decreased with age in both sexes. The variance of height was similar in MZ and DZ twins at most ages. In contrast, the variance of BMI was significantly higher in DZ than in MZ twins, particularly in childhood. In conclusion, DZ twins were generally taller and had greater BMI than MZ twins, but the differences decreased with age in both sexes.
By Margaret Bent, Anna Maria Busse Berger, Lawrence F. Bernstein, Bonnie J. Blackburn, M. Jennifer Bloxam, Philippe Canguilhem, Julie E. Cumming, Anthony M. Cummings, David Fallows, David Fiala, Alison K. Frazier, James Hankins, Leofranc Holford-Strevens, Deborah Howard, Andrew Kirkman, Michael Long, Laurenz Lütteken, Evan A. MacCarthy, Patrick Macey, Honey Meconi, John Milsom, Klaus Pietschmann, Alejandro Enrique Planchart, Yolanda Plumley, Keith Polk, Anne Walters Robertson, Jesse Rodin, David J. Rothenberg, Thomas Schmidt-Beste, Peter Schubert, Nicole Schwindt, Richard Sherr, Pamela F. Starr, Anne Stone, Reinhard Strohm, Richard Taruskin, Blake Wilson, Emily Zazulia
Edited by Anna Maria Busse Berger, University of California, Davis, Jesse Rodin, Stanford University, California
Book: The Cambridge History of Fifteenth-Century Music
Published online: 05 July 2015
Print publication: 16 July 2015, pp xix-xxvi
Establishment and preliminary outcomes of a palliative care research network
Peter Hudson, Annette Street, Suzanne Graham, Sanchia Aranda, Margaret O'Connor, Kristina Thomas, Kate Jackson, Odette Spruyt, Anna Ugalde, Jennifer Philip
Journal: Palliative & Supportive Care / Volume 14 / Issue 1 / February 2016
The difficulties in conducting palliative care research have been widely acknowledged. In order to generate the evidence needed to underpin palliative care provision, collaborative research is considered essential. Prior to formalizing the development of a research network for the state of Victoria, Australia, a preliminary study was undertaken to ascertain interest and recommendations for the design of such a collaboration.
Three data-collection strategies were used: a cross-sectional questionnaire, interviews, and workshops. The questionnaire was completed by multidisciplinary palliative care specialists from across the state (n = 61); interviews were conducted with senior clinicians and academics (n = 21) followed by two stakeholder workshops (n = 29). The questionnaire was constructed specifically for this study, measuring involvement of and perceptions of palliative care research.
Both the interview and the questionnaire data demonstrated strong support for a palliative care research network and aided in establishing a research agenda. The stakeholder workshops assisted with strategies for the formation of the Palliative Care Research Network Victoria (PCRNV) and guided the development of the mission and strategic plan.
Significance of results:
The research and efforts to date to establish the PCRNV are encouraging and provide optimism for the evolution of palliative care research in Australia. The international implications are highlighted.
The CODATwins Project: The Cohort Description of Collaborative Project of Development of Anthropometrical Measures in Twins to Study Macro-Environmental Variation in Genetic and Environmental Effects on Anthropometric Traits
Karri Silventoinen, Aline Jelenkovic, Reijo Sund, Chika Honda, Sari Aaltonen, Yoshie Yokoyama, Adam D. Tarnoki, David L. Tarnoki, Feng Ning, Fuling Ji, Zengchang Pang, Juan R. Ordoñana, Juan F. Sánchez-Romera, Lucia Colodro-Conde, S. Alexandra Burt, Kelly L. Klump, Sarah E. Medland, Grant W. Montgomery, Christian Kandler, Tom A. McAdams, Thalia C. Eley, Alice M. Gregory, Kimberly J. Saudino, Lise Dubois, Michel Boivin, Claire M. A. Haworth, Robert Plomin, Sevgi Y. Öncel, Fazil Aliev, Maria A. Stazi, Corrado Fagnani, Cristina D'Ippolito, Jeffrey M. Craig, Richard Saffery, Sisira H. Siribaddana, Matthew Hotopf, Athula Sumathipala, Timothy Spector, Massimo Mangino, Genevieve Lachance, Margaret Gatz, David A. Butler, Gombojav Bayasgalan, Danshiitsoodol Narandalai, Duarte L. Freitas, José Antonio Maia, K. Paige Harden, Elliot M. Tucker-Drob, Kaare Christensen, Axel Skytthe, Kirsten O. Kyvik, Changhee Hong, Youngsook Chong, Catherine A. Derom, Robert F. Vlietinck, Ruth J. F. Loos, Wendy Cozen, Amie E. Hwang, Thomas M. Mack, Mingguang He, Xiaohu Ding, Billy Chang, Judy L. Silberg, Lindon J. Eaves, Hermine H. Maes, Tessa L. Cutler, John L. Hopper, Kelly Aujard, Patrik K. E. Magnusson, Nancy L. Pedersen, Anna K. Dahl Aslan, Yun-Mi Song, Sarah Yang, Kayoung Lee, Laura A. Baker, Catherine Tuvblad, Morten Bjerregaard-Andersen, Henning Beck-Nielsen, Morten Sodemann, Kauko Heikkilä, Qihua Tan, Dongfeng Zhang, Gary E. Swan, Ruth Krasnow, Kerry L. Jang, Ariel Knafo-Noam, David Mankuta, Lior Abramson, Paul Lichtenstein, Robert F. Krueger, Matt McGue, Shandell Pahlen, Per Tynelius, Glen E. Duncan, Dedra Buchwald, Robin P. Corley, Brooke M. Huibregtse, Tracy L. Nelson, Keith E. Whitfield, Carol E. Franz, William S. Kremen, Michael J. Lyons, Syuichi Ooki, Ingunn Brandt, Thomas Sevenius Nilsen, Fujio Inui, Mikio Watanabe, Meike Bartels, Toos C. E. M. van Beijsterveldt, Jane Wardle, Clare H. Llewellyn, Abigail Fisher, Esther Rebato, Nicholas G. Martin, Yoshinori Iwatani, Kazuo Hayakawa, Finn Rasmussen, Joohon Sung, Jennifer R. Harris, Gonneke Willemsen, Andreas Busjahn, Jack H. Goldberg, Dorret I. Boomsma, Yoon-Mi Hur, Thorkild I. A. Sørensen, Jaakko Kaprio
Journal: Twin Research and Human Genetics / Volume 18 / Issue 4 / August 2015
Print publication: August 2015
For over 100 years, the genetics of human anthropometric traits has attracted scientific interest. In particular, height and body mass index (BMI, calculated as kg/m2) have been under intensive genetic research. However, it is still largely unknown whether and how heritability estimates vary between human populations. Opportunities to address this question have increased recently because of the establishment of many new twin cohorts and the increasing accumulation of data in established twin cohorts. We started a new research project to analyze systematically (1) the variation of heritability estimates of height, BMI and their trajectories over the life course between birth cohorts, ethnicities and countries, and (2) to study the effects of birth-related factors, education and smoking on these anthropometric traits and whether these effects vary between twin cohorts. We identified 67 twin projects, including both monozygotic (MZ) and dizygotic (DZ) twins, using various sources. We asked for individual level data on height and weight including repeated measurements, birth related traits, background variables, education and smoking. By the end of 2014, 48 projects participated. Together, we have 893,458 height and weight measures (52% females) from 434,723 twin individuals, including 201,192 complete twin pairs (40% monozygotic, 40% same-sex dizygotic and 20% opposite-sex dizygotic) representing 22 countries. This project demonstrates that large-scale international twin studies are feasible and can promote the use of existing data for novel research purposes.
By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O'Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
Edited by Robert Audi, University of Notre Dame, Indiana
Book: The Cambridge Dictionary of Philosophy
Print publication: 27 April 2015, pp ix-xxx | CommonCrawl |
Genetic parameters and associated genomic regions for global immunocompetence and other health-related traits in pigs
Genome-wide association studies for production, respiratory disease, and immune-related traits in Landrace pigs
Yoshinobu Uemoto, Kasumi Ichinoseki, … Keiichi Suzuki
Genome wide association study of passive immunity and disease traits in beef-suckler and dairy calves on Irish farms
Dayle Johnston, Robert Mukiibi, … Bernadette Earley
Genome wide association study of body weight and feed efficiency traits in a commercial broiler chicken population, a re-visitation
Wossenie Mebratie, Henry Reyer, … Just Jensen
Detection of loci exhibiting pleiotropic effects on body weight and egg number in female broilers
Eirini Tarsani, Andreas Kranis, … Antonios Kominakis
Genome-wide association study reveals the genetic determinism of growth traits in a Gushi-Anka F2 chicken population
Yanhua Zhang, Yuzhe Wang, … Xiangtao Kang
Estimates of genomic heritability and genome-wide association studies for blood parameters in Akkaraman sheep
Yunus Arzik, Mehmet Kizilaslan, … Mehmet Ulas Cinar
Genetic variations for egg quality of chickens at late laying period revealed by genome-wide association study
Zhuang Liu, Congjiao Sun, … Ning Yang
Whole-genome re-sequencing association study for direct genetic effects and social genetic effects of six growth traits in Large White pigs
Pingxian Wu, Kai Wang, … Guoqing Tang
Genic and non-genic SNP contributions to additive and dominance genetic effects in purebred and crossbred pig traits
Mahshid Mohammadpanah, Ahmad Ayatollahi Mehrgardi, … Llibertat Tusell
Maria Ballester1,
Yuliaxis Ramayo-Caldas1,
Olga González-Rodríguez1,
Mariam Pascual1,
Josep Reixach2,
Marta Díaz2,
Fany Blanc3,
Sergi López-Serrano4,
Joan Tibau5 &
Raquel Quintanilla1
Scientific Reports volume 10, Article number: 18462 (2020) Cite this article
Genetic association study
Quantitative trait
The inclusion of health-related traits, or functionally associated genetic markers, in pig breeding programs could contribute to produce more robust and disease resistant animals. The aim of the present work was to study the genetic determinism and genomic regions associated to global immunocompetence and health in a Duroc pig population. For this purpose, a set of 30 health-related traits covering immune (mainly innate), haematological, and stress parameters were measured in 432 healthy Duroc piglets aged 8 weeks. Moderate to high heritabilities were obtained for most traits and significant genetic correlations among them were observed. A genome wide association study pointed out 31 significantly associated SNPs at whole-genome level, located in six chromosomal regions on pig chromosomes SSC4, SSC6, SSC17 and SSCX, for IgG, γδ T-cells, C-reactive protein, lymphocytes phagocytic capacity, total number of lymphocytes, mean corpuscular volume and mean corpuscular haemoglobin. A total of 16 promising functionally-related candidate genes, including CRP, NFATC2, PRDX1, SLA, ST3GAL1, and VPS4A, have been proposed to explain the variation of immune and haematological traits. Our results enhance the knowledge of the genetic control of traits related with immunity and support the possibility of applying effective selection programs to improve immunocompetence in pigs.
Over the last decades, the genetic selection in commercial pig breeds has greatly improved traits directly related with production performance1, while health-related traits have traditionally played a minor role in breeding programs. Nowadays, the emergence of antibiotic resistance and society demands for healthier livestock products and for more sustainable production systems2 represent new challenges for the pig production industry. Animal health is one of the most important contributors to productivity, profitability, and welfare, with multiple factors being involved in the maintenance of high health herd status such as co-infections of viral or bacterial pathogens, environmental stressors, and management practices. In the midst of strong investment for designing alternatives to antimicrobials in veterinary medicine3, the incorporation of health-related traits in pig breeding programs has become an emerging and challenging trend to produce more resilient, wellbeing and disease resistant pig populations.
Breeding approaches to improve animal robustness and disease resistance have been mainly focused on direct and indirect methods4. Direct methods usually rely on targeting the genetic resistance/susceptibility to specific diseases, requiring exposition to the infectious agents. This approach is expensive, time-consuming and information demanding. An indirect approach focused on the determination of the global immunocompetence of animals with no sign of infection has become a good alternative, but requires detailed knowledge of the different components of the immune system4,5. In this approach, immunity traits (ITs) may be considered as biologically relevant parameters to measure immunocompetence4. These traits may be classified into the two major components of the immune system, innate (or natural) immunity or acquired (adaptive) immunity, although there are also traits which are considered a bridge between both components6.
The innate immune system is the host's first line of defence against infectious agents. In addition, haematological traits and stress parameters are also important indicators of the physiological and health status of farm animals7,8,9. During last years, several studies have reported medium to high heritabilities for several immune and haematological traits in pigs, suggesting an important genetic contribution to the phenotypic variability of these traits4,10,11,12,13,14. Since the pioneering work on quantitative trait loci (QTLs) mapping for general immune-capacity performed by15 in a wild boar × Swedish Yorkshire crossbred population, other groups have reported QTLs for traits related to immune-capacity in pigs16,17,18,19,20,21,22,23. More recently, with the development of high-density genotyping SNP chips, analyses applying genome wide association study (GWAS) have been performed to identify genetic markers associated with health-related traits. These studies have been however mainly addressed on haematological traits24,25,26,27,28,29,30,31,32,33,34. To date, genetic information in pigs on stress parameters focused primarily on acute adrenal activity35,36,37,38 with little or no genetic study on chronic stress parameters, such as cortisol measured in hair.
The present work aimed to study the genetic architecture of 30 health-related traits covering immune (mainly innate), haematological, and stress parameters associated to immunocompetence in a Duroc commercial line by estimating their genetic parameters and identifying associated genomic regions and candidate genes.
Descriptive statistics and phenotypic correlations
In the present study we measured a set of 30 traits related with immune, haematological and stress parameters on a commercial Duroc pig line comprising 432 individuals. The data on descriptive statistics, as well as the abbreviated name of the analysed measured traits are shown in Table 1. Among the haematological traits, the erythrocyte-related traits presented the lowest phenotypic dispersion, with a coefficient of variation (CV) below 0.1, while the leukocyte and platelet-related traits presented CV ranging from 0.34 (leukocytes count) to 0.63 (eosinophils count). Regarding the ITs, most phagocytosis traits presented limited dispersion (CV ≤ 0.2), whereas the highest CVs were obtained for the acute-phase proteins CRP and HP (CV = 0.73 and 0.67, respectively). Finally, the stress parameters presented a moderate phenotypic variation with a CV = 0.44.
Table 1 Descriptive statistics of the analysed immunity, haematological and well-being traits.
The network based on phenotypic correlation coefficients (Fig. 1) identified five interconnected clusters of correlated traits. The central cluster grouped five hemogram leukocyte-related traits (LEU, NEU, EO, LYM and MON), with rp ranging from 0.41 to 0.90. Another cluster connected with the previous one included traits related to phagocytic capacity (PHAGO_FITC, LYM_PHAGO_FITC, MON_PHAGO_FITC and GRANU_PHAGO_FITC) with rp > 0.5 among them. The phagocytosis traits related to percentage of phagocytic cells (PHAGO_%, GRANU_PHAGO_%, LYM_PHAGO_%, MON_PHAGO_%) clustered also together. Finally, two more clusters were found, the plasma concentration of Ig (IgA, IgM and IgG), with a rp = 0.73 between IgG and IgM, and the haematological erythrocyte-related traits (HB, ERY, MCV, MCH, HCT, MCHC), with negative and positive correlation coefficients. It is worth to highlight that HP generally correlated negatively with this last cluster. More details about the phenotypic correlation coefficients are provided in Supplementary Table S1.
Network based on phenotypic correlation coefficients (|r| ≥ 0.3) among the immunity-related traits. Red lines indicate positive correlated traits while blue dashed lines indicate negative correlations. The numbers along the lines and the width of the lines indicate correlation coefficients. The shape and color of nodes indicate the different classes of the analysed traits.
Genetic parameters of immunity-related traits: heritability and genetic correlations
The genetic determinism of immunocompetence was first explored by the heritability of these immunity and health-related phenotypes. Heritability (Table 2) ranged between 0.092 and 0.786. Most traits exhibited moderate to high heritability values, 22 out of 30 traits showing h2 values above 0.4. These heritabilities had relatively wide confidence intervals, that in some cases encompasses more than half heritability parameter space. Despite this, the h2 confidence intervals did not overlap zero but for the heritability of MON, MON_PHAGO_FITC and GRANU_PHAGO_%.
Table 2 Heritability values (\(\widehat{{h}^{2}}\)) for the immunity, haematological and well-being analysed traits, plus standard errors (SE) and confidence intervals at 95% (CI95) of the estimates.
Among analysed ITs, plasma concentrations of Ig showed the highest heritabilities (from 0.652 to 0.786), followed by the percentage of γδ T cells (h2 = 0.613). Focusing traits related to acute phase proteins, HP exhibited a relatively high heritability (h2 > 0.4), whereas more limited genetic contribution was estimated for CRP (h2 = 0.245) and also for NO (h2 = 0.256). For phenotypes related to phagocytosis, low to moderate heritabilities were obtained (from 0.118 to 0.495). Several haematological traits also exhibited high heritabilities, being the MCHC the most heritable among them (h2 = 0.767), followed by the quantity of ERY and PLA in blood and by the MCV (h2 ≥ 0.65 in both cases). Other erythrocyte-related phenotypes, such as total and mean corpuscular HB as well as HCT, also showed significant heritabilities above 0.4. Regarding white blood cells counts, the quantity of NEU and EOS in blood showed heritabilities above 0.55, whereas lowly values (h2 below 0.3) were obtained for the number of LYM and total LEU, and no significant additive genetic contribution to the number of MON could be assessed. Concerning stress parameters, a medium heritability (h2 = 0.456) was obtained for the CORT levels in hair (chronic stress indicator), whereas NLR showed a particularly high heritability (h2 = 0.731).
Genetic relationship among the immunity-related phenotypes was analysed through estimating the genetic correlations between each pairwise combination of traits; a heatmap showing the magnitude of the estimated correlations between the 30 traits is presented in Fig. 2. More details about these genetic correlation coefficients and their estimation standard errors are provided in Supplementary Table S2. It should be mentioned that the limited population size resulted in a limited precision of genetic correlations estimates, which generally showed high SE. The reliability of parameter estimation in the bivariate models was particularly compromised when they involved traits with heritabilities close to zero (i.e. MON, MON_PHAGO_FITC and GRANU_PHAGO_%), so these genetic correlations should be taken with caution.
Heatmap of genetic correlations estimated by pairwise combination among immunity, haematological and stress related traits in pigs.
The three plasma Ig (IgA, IgG and IgM) clustered together, showing positive genetic correlations among them but lower than phenotypic correlations (Fig. 1); only the genetic correlation between plasma IgM and IgG was relevant and significant (rg = 0.662, SE = 0.118). In general, plasma Ig concentrations did not show important genetic correlations with other ITs but positive correlations between IgG and the percentage of phagocytic cells. Scarcely related with plasmatic Ig levels, the acute phase proteins concentrations in serum clustered together with NO and γδ T cells, with moderate to high positive genetic correlations among them but between γδ T cells and NO. CRP exhibited a very strong negative genetic associations with white blood cells counts, particularly LEU and LYM. Weaker but also negative genetic associations with red cells parameters (ERY, HB and HCT) were estimated for both CRP and HP.
A large cluster grouped together white cells counts and most phagocytosis capacity phenotypes quantified as mean fluorescence in FITC among the total phagocytic cells. The different leucocyte types (LYM, EOS, NEU, MON) showed a positive and high genetic correlation with total LEU count (rg from 0.72 to 0.92), whereas genetic correlations between the different LEU types varied largely. Phagocytosis-related traits showed a complex and not always consistent (i.e. with large estimation error) picture of genetic associations. The FITC measurements of phagocytosis correlated positively between them and with leucocytes counts, with some exceptions involving lymphocytes that in turn showed a pattern of genetic associations relatively different from the rest of leukocyte types. Conversely, negative genetic associations between most leukocyte subsets and the proportion of phagocytic cells (PHAGO_%, GRANU_PHAGO_%, LYM_PHAGO_%, MON_PHAGO_%) were obtained. These percentages of phagocytic cells correlated positively between them (rg between 0.35 and 0.75) but also with LYM_PHAGO_FITC and IgG.
As far as erythrocytes-related parameters is concerned, the HTC, HB and ERY constituted a cluster, with rg between them ranging between 0.69 and 0.86. No evidences of relevant genetic associations between these traits and white blood cells were obtained but for HTC, that showed moderate positive genetic correlations with LYM, LEU and MON. Separately and opposed to the previous cluster, PLA, MCH and MCV also grouped together (rg from 0.81 to 0.94); all three traits correlated negatively with ERY count.
Finally and regarding stress-related parameters, the CORT concentration in hair showed negative genetic correlations with CRP and γδ T cells (rg = − 0.59 and − 0.50, respectively) and positive correlation with PLA (rg = 0.50); associations with the rest of haematological and ITs were generally weak and/or no significant. Regarding NLR, expectedly it was strongly correlated to NEU (rg = 0.90), but also in a lesser extent to MON and LEU (rg = 0.55 and 0.41, respectively), and negatively to LYM (rg = − 0.52).
Genomic regions and candidate genes associated with immunity traits, stress indicators and haematological parameters
To identify genomic regions associated with health-related traits, a GWAS was performed using the 30 phenotypic traits and the genotypes of 42,641 SNPs of the Porcine GGPSNP70 BeadChip (Illumina) in the 432 Duroc pigs. Significant associations at whole-genome level (FDR ≤ 0.1) were detected for IgG, γδ T cells, LYM_PHAGO_FITC, LYM, CRP, MCV and MCH. A total of 31 significant associated SNPs located in six chromosomal regions on pig chromosomes SSC4, SSC6, SSC17 and SSCX were identified (Table 3). In addition, a genomic region in SSC12 and SSC14 for the total number of LEU and in SSC13 for the total number of NEU passed the FDR threshold of < 0.2. In those regions, we identified nine associated SNPs (Table 3). The full list of associated SNPs, with their predicted consequences, is shown in Supplementary Table S3. In addition, graphical representation of the GWAS results for the traits are displayed in Manhattan plots in Supplementary Figure S1 (FDR ≤ 0.2) and Fig. 3 (FDR ≤ 0.1). It is worth to highlight that almost half of the QTLs identified (4 out of 9) were associated with lymphocytes-related traits.
Table 3 Description of the nine chromosomal regions associated with health-related traits and the annotated candidate genes.
Manhattan plots representing the association analysis between the health-related traits and SNPs distributed along the pig genome. Red line indicates those SNPs that are below the genome-wide significance threshold (FDR ≤ 0.1).
In SSC4, two regions at 8.3 Mb and 90.5–91.2 Mb associated with IgG plasma levels and CRP (Fig. 3A,B), respectively, were identified. In the proximal region of SSC4, an SNP (rs319560097) was associated with the IgG plasma levels. In this region, the candidate genes Src like adaptor (SLA) and ST3 beta-galactoside alpha-2,3-sialyltransferase 1 (ST3GAL1) related to B cell development, differentiation and function were identified. Ten SNPs in quite linkage disequilibrium (Dʹ = 0.203–0.999) were identified associated with CRP levels in a more distal region of SSC4. It is worth highlighting that two of these associated SNPs (rs81285109 and rs80958253) were located inside the CRP locus (Ensembl gene id: ENSSSCG00000006403), the main candidate gene identified in this region.
In SSC6, two regions at 17.11–17.18 Mb and 164.85–165.78 Mb were identified associated with three traits. In the proximal region of SSC6, two SNPs (rs338661853 and rs81285171) were associated with LYM_PHAGO_FITC (Fig. 3C). In this region, three candidate genes were annotated (CDH1, COG8 and VPS4A), with the vacuolar protein sorting 4 homolog A (VPS4A) gene involved in the endosomal multivesicular bodies (MVB) pathway. In the second region, we identified ten SNPs in strong linkage disequilibrium (Dʹ > 0.9), except the rs81394075 that showed Dʹ > 0.48 with the rest of SNPs, associated with MCH and/or MCV traits (Fig. 3D,E). Remarkably, a strong candidate gene, the peroxiredoxin 1 (PRDX1), associated with haematocrit levels and haemoglobin concentration functions, was mapped in this pleitropic region. Another candidate gene also mapped in this region was the phosphoinositide-3-kinase regulatory subunit 3 (PIK3R3) which was identified as component of multiple canonical pathways of which erythropoietin signalling was among them.
Three SNPs (rs80924885, rs80899023, rs80803525) at 52.46–52.51 Mb of SSC17 were associated with LYM trait (Fig. 3F). In this region, we also identified a promising candidate gene, the nuclear factor of activated T cells 2 (NFATC2), directly related with the quantity of lymphocytes, quantity of T and B lymphocytes and size of thymus cortex functions. For the percentage of γδ T cells, five SNPs located at 33.51–33.64 Mb of SSCX were found to be significantly associated to this trait (Fig. 3G). In this region, we did not identify any candidate gene among the annotated protein coding genes. However, four lncRNA and ssc-mir-9786-1 were annotated in this region. Both types of RNAs have been directly implicated in the innate immune response39,40, although there is still a lack of information about the mechanism of action of lncRNAs. In contrast, miRNAs are better characterized and there are tools and databases that allow to perform an in-silico target prediction. In this way, ssc-mir-9786-1 was predicted to target a total of 528 genes by RNAhybrid. Noteworthy, among the biological functions represented in the list of targeted genes there was T cell differentiation (Supplementary Table S4).
Regarding LEU trait, two SNPs (rs323856019 and rs343667976) at 3.25 Mb and 123.89 Mb on SSC12 and SSC14, respectively, were associated with this trait. Two candidate genes (SOCS3 and BIRC5) located on SSC12 and associated with quantity of leukocytes, lymphocytes, B and T lymphocytes, and peripheral blood leukocytes among other related diseases and functions were identified. On SSC14, no candidate genes were found among the annotated genes according to their functional information. Finally, seven SNPs with a Dʹ > 0.68 and located at 69.03–71.96 Mb on SSC13 were associated with NEU trait. In this region, several candidate genes (GATA2, PPARG, RAF1 and SEC61A1) associated with functions such as quantity of leukocytes or neutropenia were identified.
Comparison with other GWAS studies
A comparison was performed between the QTLs identified in our study and those previously published for health-related traits to identify overlapping chromosomal regions. We only identified two overlapping regions for LYM and MCH traits. Regarding LYM trait, the overlapping region (SSC17: 48.7–66.9 Mb) was identified in a F2 population from a Meishan/Pietrain family after 42 days post-infection with the protozoan pathogen Sarcocystis miescheriana41. The region for MCH at 166.3 Mb on SSC6 was less than 1 Mb away from the region identified in our analysis (164.8–165.8 Mb). This QTL was identified at chromosome-level in a GWAS analysis performed on animals of three breeds, Large-White, Landrace and Songliao Black27.
Robustness and resilience, together with well-being, are becoming a priority in livestock breeding. In our study, 30 traits including immunity, haematological and stress parameters have been measured in 432 healthy Duroc pigs to analyse individual's immunocompetence. Most of these health-related traits presented medium to high heritability values, confirming a relevant genetic determinism in the phenotypic variation of the global immunocompetence in pigs, as had been suggested by several authors10,11,12,13,14. Overall, the heritabilities in our study are in close agreement with results previously reported in other pig populations10,11,12,13,14, very particularly those estimates showing high heritabilities for phagocytosis, γδ T cells, haematological erythrocyte-related traits, and IgA antibody levels, and the lower heritability for CRP. The LEU and LYM heritabilities in our study (low-to-medium) were also similar to those previously reported by10,11,12,14, but lower to the high heritabilities reported by13. In contrast, other haematological leukocyte-related traits (i.e. EO and NEU counts) together with HP and IgG antibody levels showed in our study higher heritabilities than those reported by10,11,12, but similar to the values of13. Finally, our heritability for total IgM antibody levels was higher than previously published13. Discrepancies with other studies were however not unexpected; they could be partially attributable to differences regarding age of animals, environmental factors or the lack of protocols standardization between laboratories, but it could also denote differences in the genetic determinism of immune capacity between different pig populations.
Besides former immunity and haematological-related traits, we also analysed the genetic determinism of two stress parameters, CORT and NLR, obtaining for them medium to high heritabilities. To the best of our knowledge, this is the first study reporting a heritability for the cortisol measured in hair, suggesting the existence of genetic determinism in the susceptibility of animals to chronic stress. In line with our results, some studies in humans determined low-to-medium heritabilities for acute plasma and saliva cortisol levels42, and post-ACTH cortisol levels in blood were reported to be highly heritable in pigs43.
Heritabilities previously stated would support the possibility of genetically improving the analysed immune and health-related traits in the studied Duroc pig population. Besides the possibility of applying a multi-trait selection for immune response (ability to respond immunologically) already reported in Yorkshire pigs by44,45, the alternative of applying selection on global immunocompetence of healthy animals is worthy to be explored4,5. The complex map of genetic interactions depicted by the estimated genetic correlations should be considered to design a global strategy to improve global immunocompetence for more robust and resilient pig populations. Estimates of genetic correlations between immunity and haematological traits reported here and in previous studies should be interpreted with caution, given the low estimabitity for such multi-trait models with limited sample sizes. We found however evidences of some relevant (and indisputably different from zero) genetic correlations. They allowed inferring a map of genetic associations among these traits that differed substantially from the phenotypic association map, and was in partial agreement with previous studies in other pig population (e.g.13), that generally found weak and mostly positive genetic correlations among ITs. In our study, we confirmed positive genetic correlations between several ITs, e.g. between IgG and IgM plasmatic concentrations or between leukocyte cells subsets, and of them with their phagocytosis capacity. But we also reported strong negative genetic correlations, for instance between leukocytes counts and CRP levels or between lymphocytes and percentage of phagocytic cells. Interestingly, the chronic stress indicator (CORT levels in hair) showed also genetic antagonism with relevant immunity parameters such as the percentage of γδ T-lymphocytes or CRP basal levels.
As signalled by5, major queries about the possibility and consequences of using genetic variation in immunocompetence in breeding programs should be addressed. According to our results, applying a selection program to increase the immunocompetence of the analysed Duroc population focusing for instance in lymphocyte related traits and/or immunoglobulins is feasible, but could be accompanied by correlated responses in other immunity parameters related to inflammation and stress that are worthy to be further explored. The question about putative correlated responses in (re)production performance should also be raised. Yorkshire pigs selected for high humoral and cellular immune responses had increased weight gains but were also prone to develop more severe arthritis after infection with Mycoplasma hyorhinis44,45. Further studies and functional validations are needed to determine the best combination of ITs and to assess the effects of selecting these ITs on global animal health and well-being, as well as on production performance. In this context and considering the time and cost-demanding phenotyping of ITs, the possibility of identifying genetic variants functionally related with immunity that could be implemented in the breeding schemes assumes paramount relevance.
GWAS analyses followed by gene annotation in the significantly associated genomic regions led to identify 16 promising candidate genes that may be implicated in the phenotypic variation of nine health-related traits. Remarkably, four out of nine of the traits with significant associated signals in the pig genome were related to lymphocytes, performing functions in the innate (percentage of γδ T cells in peripheral blood and lymphocytes phagocytic capacity) or the adaptive (total concentration of IgG in plasma and total number of lymphocytes) immune systems. In the opposite, our study did not allow identifying any SNP associated to CORT stress parameter. Genetic variants associated with plasma cortisol levels have been identified in pigs35,36,37,38, but there is a lack of GWAS studies with cortisol level measured in hair samples.
Among genes identified in the lymphocyte-signalled genomic regions, the NFATC2 was mapped in the region associated with the total number of lymphocytes. This gene encodes a transcription factor that is expressed in peripheral blood lymphocytes, among others, and was firstly identified in T cells. NFATC2 plays a critical role in regulating the expression of cytokine genes in T cells during the immune response46,47 and is required for B cell development and function46,48. It is worth mentioning that knockout NFATC2 mouse displayed enhanced immune response49 and hyperproliferation of primary B cells48, which suggest a negative regulatory function in the immune system.
Other two candidate genes, SLA and ST3GAL1, were located in the genomic region associated with the total concentration of IgG, the predominant serum isotype produced by B-lymphocytes. Remarkably, both genes have been implicated in the B cell differentiation process50,51. Specifically, expression of SLA is required to optimally regulate BCR levels and signal strength during B-cell development50, while ST3GAL1 modulates the plant lectin peanut agglutinin (PNA) binding phenotype of activated B-cells, through O-glycan remodelling on CD4551.
As far as lymphocytes phagocytic capacity, three candidate genes were identified: cadherin 1 (CDH1), component of oligomeric golgi complex 8 (COG8) and VPS4A. Several studies have determined the phagocytic capacity of B-cells, mainly B1-cells but also follicular B-cells, playing an important role in innate immunity and the development of a strong humoral response52,53,54. VPS4A and COG8 have been involved in the generation of multivesicular bodies (MVBs) during phagosome maturation55, and retrograde intracellular membrane trafficking at the Golgi56, respectively. Furthermore, CDH1, a cellular receptor found on epithelial cells that can mediate entry of bacteria, is also expressed in other cells such as macrophages57.
Among the lymphocyte lineage there are cells such as the γδ T cells considered to be a bridge between innate and adaptive immunity58. Unlike in humans and mice, γδ T cells represent a prominent population in pigs' peripheral blood59. In the genomic region associated with γδ T cells, we have identified a promising miRNA (ssc-mir-9786-1) which was predicted to target genes implicated in the T cell differentiation process. This miRNA was previously identified in porcine milk exosomes60 but there is still a lack of functional validation of the direct ssc-mir-9786-1-target mRNA interaction involving genes related with the immune system.
Also related to white cells-mediated immunity, we identified promising candidate genes annotated in the regions associated with the total number of leukocytes (SSC12 and SSC14) and neutrophils (SSC13). Remarkably, one of the candidate genes selected for the total number of leukocytes, baculoviral IAP repeat containing 5 (BIRC5), also known as Survivin, is essential for T cell maturation and proliferation61. This result is in accordance with the phenotypic and genetic correlations of rp = 0.9 and rg = 0.92 observed between the total number of leukocytes and lymphocytes. In fact, when we look in detail at the regions associated with the number of lymphocytes, the same signals previously observed for leukocytes are identified at chromosome level; therefore, these regions may be more specifically affecting lymphocytes. Among the candidate genes annotated in the region associated with NEU, it is worth to highlight the peroxisome proliferator activated receptor gamma (PPARG) gene. This gene encodes a nuclear hormone receptor with a wide variety of biological functions, including a critical role in modulating inflammatory processes of the innate immune system through regulation of neutrophil trafficking and apoptosis, among other functions62.
A particularly remarkable result arising in this study was the identification of CRP as candidate gene, annotated in the region associated with variation in its traduced protein levels. CRP is highly expressed during the acute-phase response, playing an important role in host defence through activating the complement system and cell-mediated pathways63. CRP is considered a blood biomarker of inflammation, although clinical studies in humans have determined that small elevation in baseline concentration of CRP is a powerful and specific predictor of cardiovascular event risk in healthy adults64. Remarkably, differences in CRP blood level have been associated with polymorphisms in the CRP gene, and some large-scale studies have provided evidence between the relationship of CRP polymorphisms, CRP blood levels and disease risk in humans (reviewed in65). In our study, we identified two associated SNPs in the intron 2 of the isoform ENSSSCT00000083957.1 and the 3′ UTR region (exon 2) of the isoform ENSSSCT00000007016.4. Further studies are warranted to determine the role of CRP polymorphisms in the variation of CRP serum levels in our Duroc population. Moreover, taking into account the higher resemblance of the immune responses of pigs with humans compared to mice66, the present results may contribute to the implementation of pigs as large animal models for cardiovascular diseases.
Finally, two interesting candidate genes (PRDX1 and PIK3R3) were also identified in the region associated with both MCH and MCV. These red cell parameters are highly related, showing positive phenotypic and genetic correlations between them (rp = 0.89, rg = 0.81), which is concordant with the identification of this pleiotropic region. PRDX1 is a member of the peroxiredoxin family of antioxidant enzymes. Severe haemolytic anaemia characterized by marked decrease in haematocrit and haemoglobin in peripheral blood has been observed in mice lacking PRDX167. Remarkably, MCV is among the 15 traits with the highest number of QTLs identified so far, with 546 associations (PigQTLdatabase, release 41, April 26, 2020). Nonetheless, we only identified a previously published QTL region associated with MCH27, which was proximal to the region for MCH/MCV identified in our study. This result agrees with previous studies in which few overlapping QTL regions for health-related traits have been identified so far, reinforcing the specificity of the genomic architecture of immunological parameters depending on the pig population (reviewed in4).
This study focuses on the genetic basis of 30 phenotypes associated to health and well-being in a Duroc pig population. The medium-to-high heritability estimates confirmed the existence of genetic determinism in most traits related to global immunocompetence in pigs. Positive genetic correlations but also strong negative genetic correlations between several immunity traits were reported. We also identified nine chromosomal regions associated with the variation of nine immune traits, highlighting 16 promising candidate genes, including CRP, NFATC2, PRDX1, SLA, ST3GAL1, and VPS4A, functionally related to these traits. Overall, our results provide new insights into the genetic control of traits related with immunity and support the possibility of applying effective selection programs to improve immunocompetence in pigs.
All experimental procedures with pigs were performed according to the Spanish Policy for Animal Protection RD1201/05, which meets the European Union Directive 86/609 about the protection of animals used in experimentation. The experimental protocol was approved by the Ethical Committee of the Institut de Recerca i Tecnologia Agroalimentàries (IRTA).
Animal material
A total of 432 animals (217 males and 215 females) from a commercial Duroc pig line were used for this study. The pigs came from six batches (72 ± 1 animals per batch) and belonged to 134 litters (two to four piglets by litter, balancing gender when possible) obtained from 132 sows and 22 boars (all active boars in the commercial population). All animals were raised in the same farm and fed ad libitum with a commercial cereal-based diet. All animals were apparently healthy, without any sign of infection.
Samples of blood, saliva and hair were collected at 60 ± 8 d of age from all animals. Blood was collected via the external jugular vein into vacutainer tubes with or without anti-coagulants (Sangüesa S.A., Spain), according to the requirements for further immunity measurements. Saliva was collected with Salivette tubes (Sarstedt S.A.U., Germany) according to the protocols recommended by the manufacturer. Hair was collected with scissors from the dorsal area of the neck behind the ears and placed in numbered bags. All samples were transported with ice blocks to the laboratory for later processing.
Phenotypic parameters
Haematological parameters
Hemograms were measured in the Laboratory Echevarne (Spain; Barcelona) from blood sampled in 4 ml EDTA tubes. The following haematological traits were included in the genetic analyses: haematocrit (HCT), haemoglobin (HB), mean corpuscular volume (MCV), mean corpuscular haemoglobin (MCH), mean corpuscular haemoglobin concentration (MCHC), total number of leukocytes (LEU), eosinophils (EO), lymphocytes (LYM), monocytes (MON), neutrophils (NEU), erythrocytes (ERY) and platelets (PLA).
Immunity parameters
Immunity parameters were measured from plasma or serum depending on the trait. Blood samples for serum were collected in 6 ml tubes with gel serum separator and centrifuged at 1600g for 10 min at RT. Plasma was collected from blood sampled in 6 ml heparinised tubes and centrifuged at 1300g for 10 min at 4 °C. Plasma and serum samples were collected, aliquoted, and stored a − 80 °C until use.
Total concentrations of immunoglobulins IgA, IgG and IgM in plasma, and IgA in saliva, were measured by ELISA with commercial kits (Bethyl laboratories Inc., Bionova, Spain), following the manufacturer's instructions. Plasma samples were diluted 1:10,000, 1:50,000 and 1:500,000 to detect IgA, IgG and IgM, respectively, while saliva samples were diluted 1:100 to detect IgA. Samples, in duplicate, were quantified by interpolating their absorbance from the standard curves constructed with known amounts of each pig immunoglobulin class and corrected for sample dilution. Absorbance was read at 450 nm using an ELISA plate reader (Bio-Rad) and analysed using the Microplate manager 5.2.1 software (Bio-Rad).
Acute-phase proteins
C-reactive protein (CRP) levels were measured in serum samples diluted 1:3000 by ELISA kit (Abcam Plc., Spain) following manufacturer's instructions. Haptoglobin (HP) concentration was measured in undiluted serum samples by colorimetric assay (Tridelta Development Limited, Ireland) following manufacturer's instructions. All samples were quantified in duplicate using standard curves constructed by plotting absorbance against CRP or HP concentration, respectively. Absorbance was read at 450 nm for CRP and 630 nm for HP using an ELISA plate reader (Bio-Rad) and analysed using the Microplate manager 5.2.1 software.
Gamma-delta T cells (γδ T cells)
Peripheral blood mononuclear cells (PBMCs) were separated from heparinised peripheral blood by density-gradient centrifugation with Histopaque-1077 (Sigma, Spain) at 450 g for 30 min. The cells were resuspended in RPMI 1640 medium supplemented with 5% foetal bovine serum (FBS) (Sigma, Spain), 1% Penicillin–Streptomycin (10,000 U/mL–10 mg/mL) and 1% l-Glutamine (200 mM) (Cultek, Spain). For PBMCs staining, the monoclonal antibody APC Rat Anti-Pig γδ T Lymphocytes (MAC320 clone, BD Pharmigen, Spain) and the APC Rat IgG2a κ isotype control (R35-95 clone, BD Pharmigen, Spain) were used. Briefly, 106 PBMCs were stained with the primary-conjugated antibodies (1:100) in 1×PBS-1% FBS for 20 min at 4 °C. After two washes with 1×PBS-1% FBS at 4 °C, cells were resuspended in 1×PBS-1% FBS and analysed by flow cytometry using the MACSQuant Analyzer 10 Flow cytometer (Miltenyi Biotec GmbH, Bergisch Gladbach, Germany) and the MACSQuantify sofware v2.6 (Miltenyi Biotec GmbH, Bergisch Gladbach, Germany). For automated flow cytometry analysis, files were imported in R environment (v3.6.1)68 with the read.flowSet function implemented in flowCore package (v1.50.0)69. Fluorescence was transformed using arcsinhTransform function. Doublets were removed using gate_singlet function (flowStats package v3.42.070), margin events using boundaryFilter function (flowCore package v1.50.069). Gatings were then performed using gate_mindensity2 function (openCyto package v1.22.271) on FSC channel to remove FSC low events corresponding to debris, SSC high events corresponding to residual granulocytes and to gate γδ T cells as APC positive. Parameters were adjusted for each day of lab analyses on a representative sample pooling all the data into one using getGlobalFrame function.
Phagocytosis assay was carried out in heparinized whole blood samples incubated with fluorescein (FITC)-labelled opsonized E. coli bacteria by using the Phagotest kit (BD Pharmigen, Spain) and according to the protocol recommended by the manufacturer. Samples were analysed by flow cytometry using the MACSQuant Analyzer 10 Flow cytometer (Miltenyi Biotec GmbH, Bergisch Gladbach, Germany) and the MACSQuantify sofware v2.6 (Miltenyi Biotec GmbH, Bergisch Gladbach, Germany). With this assay we analyzed the percentage of cells having performed phagocytosis as well as their mean fluorescence intensity (number of ingested bacteria). Phagocytosis assay analyses were performed in R (v3.6.0)68. Doublets were removed using gate_singlet function (flowStats package v3.42.070), margin events using boundaryFilter function (flowCore package v1.50.069). Gatings were then performed using either gate_mindensity2 or gate_flowClust_2d functions (openCyto package v1.22.071) on propidium iodure (PI) channel to gate blood cells as PIhi and cells having performed phagocytosis as FITC+ . Granulocytes, monocytes, and lymphocytes were gated based on their FSC SSC properties. Parameters were adjusted for each day of lab analyses on a representative sample pooling all the data into one using getGlobalFrame function. The following phagocytosis traits were quantified: percentage of total phagocytic cells (PHAGO_%), percentage of phagocytic cells among granulocytes (GRANU_PHAGO_%), monocytes (MON_PHAGO_%), and lymphocytes (LYM_PHAGO_%), mean fluorescence in FITC among the total phagocytic cells (PHAGO_FITC), and mean fluorescence in FITC among the granulocytes (GRANU_PHAGO_FITC), monocytes (MON_PHAGO_FITC) and lymphocytes (LYM_PHAGO_FITC) that phagocyte.
Total concentrations of Nitric Oxide (NO) were measured by colorimetric assay (Thermo Fisher Scientific, Spain) following manufacturer's instructions. Serum samples were ultrafiltered through a 10,000 molecular weight cut-off (MWCO) and diluted 1:10. Samples were quantified by reference to standard curves constructed with known amounts of Nitrate Standard solution. Absorbance was read at 540 nm using a microplate reader (LUMistar Omega, BMG Labtech) and analysed using the Omega MARS software (BMG Labtech).
Stress indicators
One hundred and fifty mg of hair were weighted from each sample and placed into a 50-ml conical tube. After three washes with 3 ml of isopropanol, all samples were left to dry in an extractor hood during 12 h. Dried hair samples were cut into 2–3 mm pieces using scissors, and 50 mg were transferred into 2 ml eppendorf. One ml of methanol was added to each sample and incubated 18 h at 37 °C under moderate shaking (100 rpm). After incubation, extracted samples were centrifuged at 7000g for 2 min and 700 µl of supernatant was transferred to a new 1.5 ml tube. The supernatant was then placed into a speed vac for 2 h to evaporate the methanol. The dried extracts were stored at − 20 °C until analysis. Total concentrations of cortisol (CORT) were measured by ELISA kit (Cusabio Technology LLC., Bionova, Spain) with dried samples reconstituted with 210 µl of sample diluent. Samples were quantified by reference to standard curves constructed with known concentrations of pig cortisol dilutions of the Standard. Absorbance was read at 450 nm using an ELISA plate reader (Bio-Rad) and analysed using the Microplate manager 5.2.1 software (Bio-Rad).
Neutrophil to lymphocyte ratio
The neutrophil to lymphocyte ratio (NLR) was calculated as a ratio of NEU and LYM.
Exploratory and phenotypic analyses
Descriptive statistics of the formerly described immunity, haematological and well-being traits in our studied Duroc population are shown in Table 1. Exploratory analyses of these phenotypes were carried out for investigating both the raw data distribution and the best fitting model for subsequent analyses. Systematic non-genetic putative effects (sex, batch and day of lab analyses within batch) on each trait were tested by using a linear model (lm) in R. Normal probability plots and Shapiro–Wilk test were performed to investigate the goodness-of-fit of the residuals with the normal distribution. For most phenotypes but the percentage of phagocytic cells, data in raw form and its residuals were quite skewed to the right; in those cases, log-transformation of data corrected these departures from normality. A filtered dataset of log-transformed data (most phenotypes) and raw-data (% of phagocytosis) was employed to perform further analyses. Subsequently, pairwise phenotypic correlations (rp) among all analysed phenotypes were computed after adjusting for significant environmental factors, and a correlation network was built up with Cytoscape72, considering those Pearson's correlation coefficients with absolute value ≥ 0.3.
Estimation of genetic parameters
The heritability \(({h}^{2})\), i.e. the proportion of phenotypic variance attributable to additive genetic effects, was estimated for the 30 immune, haematological and stress traits showed in Table 1. Variance components and the corresponding h2 were estimated from an univariate animal model as follows:
$${\varvec{Y}}={\varvec{X}}{\varvec{\beta}}+{\varvec{Z}}{\varvec{u}}+{\varvec{e}}$$
where \({\varvec{Y}}\) is the vector of phenotypic observations of all individuals for the health-related trait (either log-transformed or raw data, depending upon the trait); \({\varvec{\beta}}\) is the vector of systematic (fixed) effects on the trait, including effect of sex (2 levels) plus batch effects (6 levels) for most traits but for phagocytosis-related traits, for which the data of laboratory analysis (12 levels, two by batch) was considered instead; \({\varvec{X}}\) is the corresponding incidence matrix; \({\varvec{u}}\) is the vector of animal's genetic additive (random) effects on the trait, and \({\varvec{Z}}\) the corresponding incidence matrix; and \({\varvec{e}}\) is the vector of random residual terms. The assumed distribution of additive genetic effects was u∼N(0,A \({\sigma }_{u}^{2}\)), where A is the numerator relationship matrix computed on the basis of pedigree (1388 individuals, five generations) and \({\sigma }_{u}^{2}\) is the additive genetic variance; random errors were distributed as e ∼ N(0,I \({\sigma }_{e}^{2}\)). Estimation of the model variance components and the corresponding heritability \(({h}^{2}= {\sigma }_{u}^{2}/\left({\sigma }_{u}^{2}+{\sigma }_{e}^{2}\right))\) for each trait was performed by REML using the aireml program included in the BGF90 package73; the standard errors (SE) of the heritability estimates were computed by repeated sampling from their asymptotic normal distribution following74, thus obtaining the corresponding confidence intervals at 95% (CI95).
Subsequently, pairwise genetic correlations (for each two traits combination) were estimated in a two-traits animal model described as follows in matrix notation:
$$\left \lceil\begin{array}{c}{{\varvec{Y}}}_{t1}\\ {{\varvec{Y}}}_{t2}\end{array} \right \rceil= \left \lceil\begin{array}{cc}{{\varvec{X}}}_{t1}& 0\\ 0& {{\varvec{X}}}_{t2}\end{array}\right \rceil\left[\begin{array}{c}{{\varvec{\beta}}}_{t1}\\ {{\varvec{\beta}}}_{t2}\end{array}\right]+ \left \lceil\begin{array}{cc}{{\varvec{Z}}}_{t1}& 0\\ 0& {{\varvec{Z}}}_{t2}\end{array} \right \rceil\left[\begin{array}{c}{{\varvec{u}}}_{t1}\\ {{\varvec{u}}}_{t2}\end{array}\right]+\left[\begin{array}{c}{{\varvec{e}}}_{t1}\\ {{\varvec{e}}}_{t2}\end{array}\right]$$
where \({{\varvec{Y}}}_{t1}\) and \({{\varvec{Y}}}_{t2}\) are the vectors of phenotypic observations for trait 1 and trait 2, respectively; \({{\varvec{\beta}}}_{t1}\) and \({{\varvec{\beta}}}_{t2}\) are the vectors of systematic (fixed) effects on each trait previously described, and \({{\varvec{X}}}_{t1}\) and \({{\varvec{X}}}_{t2}\) the correspondent incidence matrices; \({{\varvec{u}}}_{t1}\) and \({{\varvec{u}}}_{t2}\) are the vectors of animal genetic additive effects on trait 1 or trait 2 (random effects), and \({{\varvec{Z}}}_{t1}\) and \({{\varvec{Z}}}_{t2}\) the corresponding incidence matrices; finally \({{\varvec{e}}}_{t1}\) and \({{\varvec{e}}}_{t2}\) are the vectors of residual errors for each trait, and 0 a matrix of zeros, assumed independent. The (co)variance matrix of random genetic effects was defined as:
$$Var\left[\begin{array}{c}{{\varvec{u}}}_{t1}\\ {{\varvec{u}}}_{t2}\end{array}\right]= \left \lceil\begin{array}{cc}{\varvec{A}}{\sigma }_{u1}^{2}& {\varvec{A}}{\sigma }_{u1,u2}\\ {\varvec{A}}{\sigma }_{u1,u2}& {\varvec{A}}{\sigma }_{u1}^{2}\end{array} \right \rceil$$
where \({\sigma }_{u1}^{2}\) and \({\sigma }_{u2}^{2}\) are the additive genetic variance of traits 1 and 2, respectively, \({\sigma }_{u1,u2}\) is the genetic covariance between the traits, and A is the numerator relationship matrix as defined above. Estimation of the (co)variance components of each pairwise analysis was also performed by REML using the aireml program included in the BGF90 package73. Genetic correlation between traits were obtained as \({{\text{r}}_{\text{g}}} = \left( {{\raise0.7ex\hbox{${{\sigma _{u1,u2}}}$} \!\mathord{\left/ {\vphantom {{{\sigma _{u1,u2}}} {{\sigma _{u1}}{\sigma _{u2}}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{\sigma _{u1}}{\sigma _{u2}}}$}}} \right)\), and the SE of the genetic correlation estimates were also computed following74.
DNA extraction and SNP genotyping
Genomic DNA was extracted from blood samples using the NucleoSpin Blood (Macherey–Nagel). DNA concentration and purity were measured in a Nanodrop ND-1000 spectrophotometer.
The 432 animals were genotyped for 68,516 single nucleotide polymorphisms (SNPs) with the GGP Porcine HD Array (Illumina, San Diego, CA) using the Infinium HD Assay Ultra protocol (Illumina). Plink software75 was used to remove SNPs with a minor allele frequency (MAF) less than 5%, SNPs with more than 10% missing genotype data, and SNPs that did not map to the porcine reference genome (Sscrofa11.1 assembly). After quality control a subset of 42,641 SNPs were retained for subsequent analysis.
Genome-wide association studies (GWAS)
GWAS was carried out between the 42,641 filtered SNPs and the 30 health-related traits described in Table 1. For this purpose, the genome-wide complex trait analysis (GCTA) software76 was employed using the following model for each trait across all SNPs:
$$y_{ijk} = sex_{j} + b_{k} + u_{i} + s_{li} a_{l} + e_{ijk}$$
where yijk corresponds to the phenotypic trait (either log-transformed or raw data) of the ith individual of sex j in the kth batch; sexj corresponds to the jth sex effect (2 levels); bk corresponds to the kth batch effect (6 levels) for most traits but for phagocytosis related traits, for which the data of laboratory analysis (12 levels, two by batch) was considered instead; ui is the infinitesimal genetic effect of individual i, with u∼N(0,Gσ2u), where G is the genomic relationship matrix (GRM) calculated using the filtered autosomal SNPs based on the methodology of76 and σ2u is the additive genetic variance; sli is the genotype (coded as 0,1,2) for the lth SNP, and al is the allele substitution effect of the SNP on the trait under study; and eijk is the residual term. The false discovery rate (FDR) method of multiple testing described by Benjamini and Hochberg77 was used to measure the statistical significance for association studies at genome-wide level with the p.adjust function of R. The significant association threshold was set at FDR ≤ 0.1. Manhattan plots based on the significance of the associations across the whole genome were generated using the R package qqman78.
Comparative QTL analysis between our results and previous published data was performed by retrieving all pig QTL and association data on SSC11.1 for health traits from the pigQTL database79.
Gene annotation and SNP functional prediction
Biomart software80 was used to retrieve gene annotations from the Ensembl Genes 98 Database using the Scrofa11.1 reference assembly, considering 1 Mb downstream/upstream of around the candidate chromosomal regions. Furthermore, functional predictions of the significantly associated SNPs were performed with Variant Effect Predictor software81.
For functional categorisation of the annotated genes, data were analyzed through the use of IPA82 (QIAGEN Inc., https://www.qiagenbioinformatics.com/products/ingenuitypathway-analysis) to obtain gene ontologies (GO), biological functions, gene networks and canonical pathways in which the genes annotated in the associated regions were involved. Orthologous human gene names were retrieved from the Ensembl Genes 98 Database for functional categorisation when a pig gene name was not assigned to the gene stable id. Furthermore, information from Mouse Genome Database83 and Genecards84 was used to identify gene functions affecting the analysed phenotypes.
miRNA target prediction and functional annotation
Porcine mRNA 3′UTR sequences from the whole pig genome (assembly 11.1) were downloaded from Ensembl Gene 100 database85, and Seqkit tool86 was used to search 3′ UTR seed matches with the 7mer seed miRNA sequence. Subsequently, the obtained list of 3′ UTR was used to predict miRNA target using the RNA hybrid software87 with the following criteria: energy threshold of no more than − 25 kcal/mol and perfect match of 2–8 nt in the seed region. Enriched GO terms and pathways of the predicted miRNA target genes was performed with the ClueGO v2.5.7 plug-in of Cytoscape v3.8.0 software88.
The results from all data generated or analysed during this study are included in this published article (and its Supplementary Information files). However, the datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
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The study was funded by grant AGL2016-75432-R awarded by the Spanish Ministry of Economy and Competitivity (MINECO). Maria Ballester is financially supported by a Ramon y Cajal contract (RYC-2013-12573) from the Spanish Ministry of Economy and Competitiveness. YRC was funded by Marie Skłodowska-Curie grant (P-Sphere) agreement No 6655919 530 (EU). The authors belong to Consolidated Research Group AGAUR, ref. 2017SGR-1719. We gratefully acknowledge to technical staff from IRTA and Selección Batallé S.A for their collaboration in the experimental protocols at farm and slaughterhouse, as well as to Albert Bensaid and Virginia Aragón for valuable discussions and comments on the study.
Animal Breeding and Genetics Program, IRTA, Torre Marimon, 08140, Caldes de Montbui, Spain
Maria Ballester, Yuliaxis Ramayo-Caldas, Olga González-Rodríguez, Mariam Pascual & Raquel Quintanilla
Department of Research and Development, Selección Batallé S.A., 17421, Riudarenes, Spain
Josep Reixach & Marta Díaz
Université Paris‐Saclay, INRAE, AgroParisTech, GABI, 78350, Jouy‐en‐Josas, France
Fany Blanc
IRTA, Centre de Recerca en Sanitat Animal (CReSA, IRTA-UAB), Campus de la Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain
Sergi López-Serrano
Animal Breeding and Genetics Program, IRTA, Finca Camps i Armet, 17121, Monells, Spain
Joan Tibau
Maria Ballester
Yuliaxis Ramayo-Caldas
Olga González-Rodríguez
Mariam Pascual
Josep Reixach
Marta Díaz
Raquel Quintanilla
M.B. and R.Q. designed the study. M.B. and J.R. supervised the generation of the material animal used in this work. M.B., Y.R.C., O.G.R., M.P., J.R., M.D., J.T. and R.Q. performed the sampling. M.B., O.G.R. and S.L.S. carried out the laboratory analyses. M.B., Y.R.C., F.B. and R.Q. analyzed the data. M.B. and R.Q. interpreted the results and wrote the manuscript. All the authors read and approved the final version of the manuscript.
Correspondence to Maria Ballester or Raquel Quintanilla.
Supplementary Information.
Supplementary Table S1.
Ballester, M., Ramayo-Caldas, Y., González-Rodríguez, O. et al. Genetic parameters and associated genomic regions for global immunocompetence and other health-related traits in pigs. Sci Rep 10, 18462 (2020). https://doi.org/10.1038/s41598-020-75417-7
A genome-wide screen for resilient responses in growing pigs
Houda Laghouaouta
Lorenzo Fraile
Ramona Natacha Pena
Genetics Selection Evolution (2022)
Genome-wide associations for immune traits in two maternal pig lines
Christina M. Dauben
Maren J. Pröll-Cornelissen
Christine Große-Brinkhaus
BMC Genomics (2021) | CommonCrawl |
Ladislav Polak1,
Lukas Klozar1,
Ondrej Kaller1,
Jiri Sebesta1,
Martin Slanina1 &
Tomas Kratochvil1
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 114 (2015) Cite this article
Nowadays, the demand for high-quality multimedia services (video, audio, image, and data) is rapidly increasing. The Digital Video Broadcasting - terrestrial (DVB-T) standard, its second-generation version (DVB-T2), and the Long-Term Evolution (LTE) standard are the most promising systems to fulfill the demand for advanced multimedia services (e.g., high-definition image and video quality), especially in Europe. However, LTE mobile services can operate in a part of the UHF band allocated to DVB-T/T2 TV services previously. The main purpose of this work is to explore the possible coexistences of DVB-T2-Lite and LTE systems in the same shared frequency band (co-channel coexistence) under outdoor-to-indoor and indoor reception conditions. Furthermore, an applicable method for evaluating coexistence scenarios between both systems is shown with a particular example. These coexistence scenarios can be noncritical and critical. In the first case, both systems can coexist without significant performance degradation. In the second one, a partial or full loss of DVB-T2-Lite and/or LTE signals can occur. We consider an indoor LTE femtocell and outdoor-to-indoor DVB-T2-Lite signal reception in a frequency band from 791 up to 821 MHz. Simulations of combined indoor and outdoor signal propagation are performed in MATLAB using 3rd Generation Partnership Project (3GPP) channel models, separately for both DVB-T2-Lite and LTE systems. Correctness of path loss simulation results is verified by measurements. Afterwards, an appropriate linear model is proposed which enables to evaluate the impact of coexistence on performance of both systems in outdoor-to-indoor and indoor-to-indoor reception scenarios. The results are related to an actual location in the building and are presented in floor plans. The floor plans include different coexistence conditions (different power imbalance and different amount of overlay of the radio channels). Service availability of both systems is verified again by measurements. The resulting maps help better understand the effect of coexistence on achievable system performance under different indoor/outdoor reception situations considering real transmission conditions.
Advanced wireless communication systems can provide users with any type of multimedia. Thanks to this, the idea to 'connect, upload, download, share and transfer anything at anytime and anywhere' is not a futuristic vision [1,2]. From the viewpoint of service providers, efficient usage of limited resources in the radio frequency (RF) spectrum is one of the biggest challenges. Hence, the increasing density of wireless networks and the increasing volume of user equipment (UE) terminals in use escalate the risk of unwanted coexistence scenarios [3,4].
In the near future, the next-generation digital terrestrial television broadcasting (DVB-T2/T2-Lite) and Long-Term Evolution (LTE) systems will be deployed to provide multimedia services for mobile and portable scenarios, mainly in Europe. DVB-T2-Lite [5-8] is a new profile which was added to the DVB-T2 system specification in April 2012. This subset within DVB-T2 is very perspective for mobile and portable TV broadcasting as it is designed to support low-capacity applications for advanced handheld receivers [9]. It is based on the same core of technologies as the DVB-T2 standard but uses only a limited number of available modes. By avoiding the modes, which require the most computational power and memory [6], the necessary complexity of T2-Lite-only receivers is reduced. DVB-T2-Lite, compared to the first-generation DVB-T/H [10], can support TV content delivery with higher flexibility. Moreover, it can operate in VHF (from 174 up to 230 MHz) and UHF (from 470 up to 870 MHz) bands, allocated earlier for DVB-T/H. From the viewpoint of system flexibility, spectral efficiency, and available transmission scenarios, DVB-T2-Lite is the system of choice for the next-generation terrestrial mobile and portable digital TV broadcasting.
Third Generation Partnership Project (3GPP) LTE [11-13] technology brings a new concept, based on the Orthogonal Frequency Division Multiple Access (OFDMA), into mobile communications. LTE supports high data rates and flexible system configuration in order to adapt transmission parameters to the actual state of a radio link. LTE architecture involves a specific type of cells called femtocells. These short ranges, mainly indoor cells, improve coverage in desired areas, especially buildings. Femtocells are served by a special type of base station called Home eNodeB (HeNB). LTE can exploit the same UHF frequency bands which are already available for existing 2G/3G networks (e.g., bands: 800, 900, 1,800, and 2,600 MHz). Moreover, additional ranges (from 2.5 up to 2.7 GHz) and the 700-MHz band are also allocated for LTE usage. The European Union decided to harmonize the '800 MHz band' in favor of the LTE services, starting from January 2013 [4]. Consequently, DVB-T/T2 and LTE services can occupy either the same or adjacent frequency spectrum. As a result, unwanted coexistence between DVB-T/T2 and LTE services can occur [4,14].
This work deals with the study of possible co-channel coexistence between DVB-T2-Lite (outdoor-to-indoor reception) and LTE services (provided by the femtocell) under fixed indoor reception conditions.
The paper is organized as follows. An overview of related work in the field of different wireless standards' coexistence, especially DVB and LTE, is presented in Section 2. This section also includes a detailed list of aims and contributions of this work. A description of the explored coexistence scenario and the considered DVB-T2-Lite system parameters are presented in Section 3. Section 4 contains a description of the applied simulation method and the proposed measurement testbed together with its detailed setup. Results obtained from simulation and measurements are presented and discussed in Section 5. Finally, Section 6 concludes the paper.
Undesirable interactions between similar or different kinds of wireless communication systems, operating in adjacent or shared frequency bands, are not a new phenomenon [3,15-18]. The exploration, monitoring, measurement, and possible suppression of interferences are a hot topic. This fact is also evidenced by many published studies available. Authors of [19] studied the possible inter-band interferences between UMTS and GSM systems. In another work [20], the coexistence between advanced wireless systems and International Mobile Telecommunication-Advanced (IMT-A) services is explored. Different kinds of coexistence scenarios in LTE networks are analyzed in [21-23]. Possible methods to mitigate or suppress interferences from coexistence between two different wireless systems are outlined in [24-26].
In the last decade, researchers' attention has been devoted to the study of different coexistence scenarios between the DVB-T/T2 and LTE/LTE-A standards.
Table 1 summarizes the previously explored coexistence scenarios between such systems. From the presented works, it is clearly seen that many times the researchers use either only simulation tools or only different measuring methods to explore the coexistence scenarios. Furthermore, in most works, a scenario is considered in which macrocells are used to provide LTE service coverage, coexisting with DVB-T2-Lite services in the same or adjacent frequency band. The main aim of this research article is to explore the interaction of DVB-T2-Lite and LTE in a shared frequency band, such that femtocells (HeNB) are used to provide LTE indoor coverage. Attention is devoted to availability monitoring of DVB-T2-Lite and LTE services in different locations under fixed indoor reception conditions. For this purpose, an appropriate simulation model is proposed and verified by measurement. Based on these results, noncritical (both DVB-T2-Lite and LTE system working) and critical (partial or full loss of DVB-T2-Lite and/or LTE signals) coexistence scenarios can be identified and the general conclusions are outlined. To the best of our knowledge, no similar exploration in this form has been presented in any scientific or technical paper so far.
Table 1 Comparison of explored coexistence scenarios between DVB-T/T2 and LTE systems
Considered coexistence scenario
The investigated coexistence scenario between the DVB-T/T2 and LTE RF signals in the fixed indoor transmission scenario is shown in Figure 1. The main system parameters of DVB-T2-Lite and LTE systems, considered in this work, can be found in Table 2. The DVB-T2-Lite TV signal is broadcast in a single frequency network (SFN) at a center frequency of 794 MHz and received by UE1 in a building. In the same building, LTE femtocells are deployed and the HeNB provides mobile connectivity in a channel belonging to Band 20 (from 791 up to 821 MHz). A user of UE2 establishes connection with HeNB at downlink frequency band from 795 (797.2 MHz) to 805 MHz (817.2 MHz). We consider that the bandwidth of the LTE signal is 10 or 20 MHz, and intersystem frequency overlapping is from 0.8 up to 3 MHz. Consequently, coexistence between HeNB (supporting 3GPP LTE Release 9) and DVB-T2-Lite system can occur. As a specific type of coexistence, a partial overlapping scenario is assumed. It means that the channel of the interferer (in this case LTE) partially overlaps with the channel of the victim (in this case DVB-T2-Lite) [27]. It is assumed that both UEs are stationary.
Unwanted coexistences between DVB-T2-Lite and LTE services at fixed indoor transmission scenario. Supposed scenario where LTE femtocell is indoors and DVB-T2-Lite signal penetrates from outdoor transmitter and affects performance.
Table 2 DVB-T2-Lite and LTE main system parameters considered in this study
Simulation and measurement setup
In this section, the simulation method, used to explore the coexistence of digital TV and mobile RF signals under outdoor-to-indoor and indoor-to-indoor conditions, is presented. Furthermore, the proposed measurement testbed and its setup, used in this work, are introduced. The simulation and measurement campaign consists of the following:
Simulation (propagation loss) and measurement of LTE performance in different locations (indoor and outdoor environment);
Simulation (propagation loss) and measurement of DVB-T2-Lite performance in different locations (indoor and outdoor environment);
Simulation and measurement of simultaneous transmission (signal propagation) of both LTE and DVB-T2-Lite RF signals in order to evaluate the influence of coexistence on the performance of both systems (on physical layer (PHY) level); and
Identification of the noncritical (both systems can coexist) and critical (partial or full loss of DVB-T2-Lite and LTE signal) coexistence scenarios for both systems.
Simulation setup
The considered coexistence scenario was briefly outlined in the previous section. In this work, we assume that transmitters and receivers are located on the seventh floor (the top floor) in the building of Brno University of Technology (BUT), Faculty of Electrical Engineering and Communications (FEEC) in Brno. Laboratories of Digital TV Technology and Radio Communications, and Mobile Communications of the Department of Radio Electronics (DREL) are located on this floor. The floor plan of the seventh floor is shown in Figure 2. Approximate dimensions of the floor are 50 × 25 m. The HeNB is located in the Laboratory of Mobile Communication Systems (room 7107), and the DVB-T2-Lite transmitter is located outdoor on the terrace.
Floor plan and general block diagram. Floor plan of the seventh floor in the building of BUT, FEEC, DREL and general block diagram of the measurement testbed.
The whole simulation model is realized in MATLAB. Propagation of the LTE and DVB-T2-Lite RF signals are simulated separately. The simulation of separate propagation loss of LTE and DVB-T2-Lite RF signals will be used as the reference (no coexistence).
The simulation model consists of three main parts for both LTE and DVB-T systems. The first part represents the simulation of a link budget, according to the 3GPP recommendation for system level simulations [28,29] for both coexisting systems. Signal strength in the receiver can be expressed as follows:
$$ {P}_{\mathrm{RX}}={P}_{\mathrm{TX}}-{L}_{\mathrm{TX}\mathrm{C}}+{G}_{\mathrm{TX}\mathrm{A}}- PL+{G}_{\mathrm{RX}\mathrm{A}}-{L}_{\mathrm{RX}\mathrm{C}} $$
where P TX is transmitter power, L TXC are wiring losses, G TXA is transmitting antenna gain, G RXA is receiving antenna gain, L RXC are wiring losses, and finally, P RX is received signal level. Path losses in wireless transmission are denoted as PL (for details see Equation 3). Value of P TX is known from the transmitter setup. Values of L TXC, L RXC, G TXA, and G RXA are constants depending on the used equipment (for details see Subsection 4.2). The second part represents the validation of obtained results from the simulation according to the performed measurement and their interpretation in a map. Details are in Subsections 4.1 and 4.2. The last part compares power imbalance of tested radio channels and computed achievable performance of both systems in certain locations. Details are given in Section 5.
The propagation scenario of the LTE RF signal in femtocell involves indoor-to-indoor line-of-sight (LOS) propagation for the same room where HeNB is located (room 7107) and non-LOS (NLOS) for other indoor locations. Path losses are modeled according to the 3GPP recommendation for indoor LTE femtocell as described in [28], denoted as UE to HeNB, where UE is inside the same building as HeNB. In order to model indoor-to-outdoor propagation from the HeNB to the measurement points on the terrace, the original equation was extended with outdoor wall penetration loss. On the other hand, the recommendations in [28] are generally valid for frequencies around 2 GHz, but we exploit an 800-MHz band in this study. Therefore, it is necessary to perform a correction as described in [29]. This correction defines the correction factor for 800 MHz as follows:
$$ P{L}_{\mathrm{COR}}=20{ \log}_{10}\left({f}_c\right) $$
where f c is carrier frequency in MHz.
The resulting path loss equation is:
$$ \mathrm{PL}=38.46+20 \log (d)+0.7{d}_{\mathrm{in}}+\mathrm{L}{\mathrm{P}}_{\mathrm{floor}}+q\;{L}_{\mathrm{INwall}}+n{L}_{\mathrm{OUTwall}}+\mathrm{P}{\mathrm{L}}_{\mathrm{COR}} $$
where d is the distance between the HeNB and the UE, d in is indoor propagation distance, LP floor is penetration loss due to propagation through the floor (it is equal to zero because a single-floor propagation scenario is assumed), parameter q is the number of indoor walls separating the transmitter and receiver, L INwall is the penetration loss due to walls inside the building, n is the number of outside walls, L OUTwall is the penetration loss of the exterior wall, and PL COR is the frequency correction factor as defined in [29] and shown in Equation 2. In our case d in = d, LP floor = 0 (single floor), q > 0 (in the case of the NLOS scenario), L INwall = 5 dB, n is between 0 and 5 (depending on the concrete position on the floor) and L OUTwall = 10 dB.
The propagation scenario between the DVB-T2-Lite transmitter and TV receiver is considered as outdoor-to-indoor urban femtocell propagation, where the UE is outside as described in [29]. The DVB-T2-Lite RF signal attenuation with frequency correction can be calculated similarly to Equation 3 as:
$$ \begin{array}{l}\mathrm{PL}\kern0.5em = \max \left(15.3+37.6{ \log}_{10}(d),38.46+20{ \log}_{10}(d)\right)+\\ {}0.7{d}_{\mathrm{in}}+L{P}_{\mathrm{floor}}+q\;{L}_{\mathrm{INwall}}+n{L}_{\mathrm{OUTwall}}+\mathrm{P}{\mathrm{L}}_{\mathrm{COR}}\kern0.5em \end{array} $$
where all variables have the same meaning as in Equation 3.
No fading was included in the data displayed in Figures 3 and 4, however, both fast fading and shadowing were computed according to recommendations in [29].
Simulation of LTE RF signal propagation. The HeNB is located in room 7107 (the blue triangle), and network parameters are as described in Table 2. The path loss model was adopted from [28] and extended for the 800-MHz band according to [29] (see Equations 2 and 3). All values are in dBm.
Simulation of DVB-T2-Lite RF signal propagation. The DVB-T2-Lite transmitter is located on the terrace (the blue triangle), and its parameters are described in Table 2. The path loss model was adopted from [28] and extended for the 800-MHz band according to [29] (see Equations 2 and 4). All values are in dBm.
Figures 3 and 4 show the results of LTE and DVB-T2-Lite radio signal propagation obtained from simulation, respectively. System parameters determined according to the simulation results prove accessibility of wireless services in all tested locations.
Path loss model data provides the basis for coexistence simulations. We have provided a detailed description of LTE and DVB-T2-Lite coexistence in our previous works [30] and [31]. Based on data collected from the mentioned measurements, we made a dense description (linear model) of coexistence. There are two types of input parameters for the models: global and local. The global parameters are mainly represented by the settings of both systems' PHY, such as modulation used in DVB-T2-Lite, inverse fast Fourier transform (IFFT) size, and the Forward Error Correction (FEC) code rate of both systems. Obviously, the overlapping bandwidth is also a global parameter. Local parameters, used as the model input, are mainly local power levels of signals, background noise, and the local fading model employed. These parameters are input into the linear model, which maps them to the Quality-of-Service (QoS) parameters. More details can be found in Subsection 5.1.
Measurement setup
For evaluating the interaction of the described coexistence scenarios between DVB-T2-Lite and LTE RF signals, the same measurement testbed was used as described in our previous works ([30] and [31]). The whole measurement campaign was implemented on the seventh floor of the building of BUT, FEEC, DREL (see Figure 2). The measurement campaign and the basic principle of our measurement method are as follows.
Firstly, the parameters and performance of the 3GPP LTE network are measured in different locations on the seventh floor. At the time of LTE measurement, T2-Lite services were not broadcasted. The HeNB is located in room 7107, and its antennas are placed on top of a table (approximately 1 m above the floor). The HeNB consists of two main hardware components, namely a PC with the Fedora Linux operating system and universal software radio peripheral (USRP) N210 from Ettus, equipped with an SBX daughter card. The PC runs the commercial software package Amari LTE [32], implementing functions of LTE Mobile Management Entity (MME) and eNB (both are 3GPP LTE Release 9 compliant). A detailed configuration of the LTE network is summarized in Table 2. The receiving UE is Huawei e398-u15 (Huawei, Shenzhen, China) (LTE UE Cat. 3) [33], connected via USB port to a laptop equipped with the Rohde & Schwarz drive test software ROMES4. For receiving LTE services, the TechniSat Digiflex TT1 mobile antenna (TechniSat, Vulkaneifel, Germany) was used (G < 2 dBi). The length of its feed line is 3 m. The UE is connected to an external antenna placed on a wooden cart approximately 1.0 m above the floor. We set up the connection between UE and HeNB and performed simultaneous full buffer transmissions in uplink and downlink. The measurement was carried out in fixed points distributed on the seventh floor as shown in Figure 2. The receiving antenna was kept still for 2 min at each measurement point and in each location we have collected approximately 100 samples of each network parameter of interest (including RSS, Channel Quality Indicator (CQI), Error Vector Magnitude (EVM), etc.).
Secondly, we have measured the performance of the DVB-T2-Lite signal in different locations on the seventh floor. At the time of T2-Lite measurement, LTE services were not provided. By using the R&S single frequency unit (SFU) broadcast test system, an appropriate video transport stream for portable TV scenarios was generated. Then, the DVB-T2-Lite complete system configuration was set up, and the output signal was RF modulated (to the frequency of 794 MHz). For its amplification, a custom-built RF power amplifier (PA), based on hybrid module Mitsubishi RA20H8087M (Mitsubishi Electric, Tokyo, Japan) [34], was applied. This RF three-stage module is primarily destined for transmitters using FM modulation that operate in the range 806 up to 870 MHz, but it may also be applied in linear systems by setting the proper drain quiescent current with externally settable gate voltage. The PA was assembled according to the recommendations of the producers and thoroughly tested. The comprehensive measurement demonstrates that this PA can be used in a wider band, circa from 650 to 900 MHz, and can be used in the presented coexistence test. The gain of the PA strongly varies in the introduced frequency range from 36 to 50 dB, but in a narrow band, the gain is quite stable (max. 1.5 dB in 10 MHz bandwidth). The maximum output power of this amplifier is around 30 W. However, we practically used only 1 W (5 W was used for the scenario where power imbalances were equal to 20 dB) with quiescent drain current 4 A, gate bias voltage 4.3 V, and supply voltage 13.8 V to achieve high linearity for reliable application in the mentioned setup. Accordingly, the power efficiency in this setting is only 2%. On the other hand, reaching linearity is the fundamental parameter which needs to be set for minimizing any nonlinear distortion. For the used testing DVB-T2-Lite frequency (794 MHz), the measured 1 dB compression of this PA is 37.9 dBm (6.2 W), two-tone third-order intermodulation distortion (IMD3) (tone offset 1 MHz) is better than −38 dBc at the output power of 1 W, which corresponds to output intercept point (OIP3) 49 dBm. Between the PA and the antenna, there is an attenuator in the signal path. It serves as PA protection in the case of antenna switch-off or strong reflections in the antenna near the field. The JFW Industries 50BR-104 N attenuator (JFW Industries, Indianapolis, IN, USA) was used which was set to 0 dB during measurement. The mentioned nonlinear distortions caused by PA are not considered in our simulation model.
The used antenna is a multi-element Yagi antenna (G max = 15.4 dBi) whose horizontal radiation pattern is shown in Figure 2. The feed line for the TV transmitter chain is a coaxial cable RG58 C/U which has a power loss of approximately 0.35 dB/m on the tested bandwidth. Attenuation of the auxiliary connection between 'N' and 'BNC' connectors is approximately 0.5 dB/m.
For the LTE system (HeNB), the Sectron AO-ALTE-MG5S antenna (Sectron Inc., Ormond Beach, FL, USA) was used. In our case, it was used as an omnidirectional antenna in vertical polarization (G < 3 dBi). After setting up the testbed, we moved with the Sefram 7866HD-T2 analyzer (Sefram Instruments and Systems, Saint-Étienne, France) to measure the received TV signal through all measuring points. The same antenna setup was used as is outlined above for LTE downlink. Once again, we spent 2 min at each measurement point for correctly evaluating the performance of the received DVB-T2-Lite RF signal (to avoid fast fading by averaging).
Figures 5 and 6 show measured and extrapolated values of RSS. Figure 5 shows the results of LTE radio signal propagation while Figure 6 shows the results of T2-Lite radio signal propagation obtained from measurement. System parameters determined according to the simulation results proves accessibility of wireless service in all tested locations. As we can see, results from measurement, shown in Figures 5 and 6, correspond with simulation results shown in Figures 3 and 4. This experimental result proves our simulation technique valid for coexistence applications.
Measurement of LTE RF signal propagation. The HeNB is located in room 7107 (the blue triangle), and network parameters are described in Table 2. The measurement was carried out in highlighted points and extrapolated using MATLAB. All values are in dBm.
Measurement of DVB-T2-Lite RF signal propagation. The HeNB is located in room 7107 (the blue triangle), and network parameters are described in Table 2. The measurement was carried out in highlighted points and extrapolated using MATLAB. All values are in dBm.
Afterwards, the whole measurement campaign was repeated, but now both wireless services (DVB-T2-Lite and LTE) were provided together at the same time. The above outlined QoS parameters of both services, caused by coexistence between them, were measured separately with Rohde & Schwarz devices.
Experimental results
Parameters to evaluate the performance of DVB-T2-Lite and LTE
Before evaluating and discussing the obtained results, it is necessary to briefly define the most important measured parameters which were used to evaluate the performance of T2-Lite and LTE systems. To evaluate the quality of the received and decoded TV services, the Quasi Error-Free (QEF) reception conditions were monitored. QEF is a minimal limit defined in the DVB-T2-Lite standard for achieving video service availability without noticeable errors in the video. To fulfill such requirements, the bit error ratio (BER) after FEC decoding must be less than or equal to 1 × 10−7 [6].
To evaluate the performance of LTE, the RSS, CQI, and EVM parameters were monitored. The CQI contains information sent from the UE to the HeNB to indicate a suitable downlink transmission data rate. It is based on the observed signal-to-interference-plus-noise ratio (SINR) and used by the HeNB for downlink scheduling and link adaptation [28]. There are 15 different CQI values (numbered from 1 up to 15). The connection between them and the modulation scheme can be found in [35] (Table 7.2.3-1).
EVM, the second parameter, is a measure used to quantify the performance of an LTE communication link. It is the RMS value of the distance in the IQ constellation diagram between the ideal constellation point and the point received by the receiver. For each modulation, there is a defined EVM limit, for which the transmitted signal has an acceptable quality. This limit is equal to 17.5% for quadrature phase-shift keying (QPSK), 12.5% for quadrature amplitude modulation (16QAM), and 8.0% for 64QAM [11,28].
DVB-T2-Lite and LTE performance evaluation
In Subsection 4.1, it has been mentioned that the linear coexistence model maps input parameters from simulations and measurements to the area of QoS states. We have defined the following QoS states for the coexisting services. For DVB-T2-Lite, there are two states: correct reception and no reception. In the case of correct reception, the above defined condition for QEF reception is satisfied. For LTE, we have defined four QoS states which differ in user bitrate and potential radio access network (RAN) throughput. These parameters obviously increase with M in M-QAM modulation of subcarriers. The LTE system changes the modulation scheme adaptively according to the channel parameters (e.g., CQI, EVM). To be more precise, the highest useable M-state for the defined interfered radio channel sets the QoS state of LTE. Four states correspond to maximal M equaling 64 (64QAM), 16 (16QAM), and 4 (QPSK), and the state when providing LTE services is not possible.
The considered coexistence scenarios between DVB-T2-Lite and LTE services were described above. Furthermore, we also consider various system parameters. The complete list of assumed scenarios is clearly summarized in Table 3. There are three main parameters: bandwidth of the LTE RF channel (marked as B LTE), overlap of coexisting channel (B OVER), and the power imbalance between transmitted powers (ΔP).
Table 3 Variable parameters of DVB-T2-Lite and LTE for assumed coexistence scenarios
The last one is calculated as follows:
$$ \varDelta P\;\left[\mathrm{dB}\right]=\mathrm{E}\mathrm{I}\mathrm{R}{\mathrm{P}}_{\mathrm{LTE}}-\mathrm{E}\mathrm{I}\mathrm{R}{\mathrm{P}}_{\mathrm{TV}} $$
where equivalent isotropically radiated power (EIRP)LTE and EIRPTV denote the channel power of LTE and T2-Lite RF signals, respectively.
Figure 7 shows the simulated results of six map representations of QoS states in DVB-T2-Lite and LTE systems. Each map (from (a) to (f)) corresponds to the considered system parameters and coexistence scenarios which are presented in Table 3. In the floor plan of the university, for each point in the explored areas, the state of both coexisting systems is indicated.
Simulation - the map representation of QoS states of coexisting systems (a-f). Specific map parameters are summarized in Table 3.
Performances of T2-Lite and LTE systems can be clearly explained in the legend of Figure 7. Four colors represent the LTE maximum useable internal modulations: orange - 64QAM, yellow - 16QAM, and green - QPSK, and unavailable LTE services are indicated by a cyan color. The performance of DVB-T2-Lite services is indicated by a crosshatch in the same maps. The presence of a hatch means that the QEF limit of mobile TV reception is fulfilled. For a better explanation of the obtained results, we describe a specific example.
For example, we consider a partial overlapping coexistence scenario between T2-Lite and LTE services when B LTE = 20 MHz, B OVER = 1,600 kHz, and ΔP is equal to 0 dB (see line (c) in Table 3). Performances of coexisting systems for these parameters are plotted in Figure 7c. As can be seen from the legend, at the 1.6-MHz channel overlapping, in the LTE system, only sub-frames using QPSK and 16QAM modulations will be received and demodulated correctly (yellow color in the legend) on the left side of the corridor. It means that only at these modulations EVM errors do not exceed the permitted limit values [11]. In the remaining rooms, the highest 64QAM modulation (highlighted by orange color) is used in the LTE system. Consequently, CQI values can be 10 or higher. Furthermore, this field also indicates that the services of DVB-T2-Lite are highly noised and conditions for QEF reception are not fulfilled (there are no hatched parts). The situation result is the opposite on the terrace where DVB-T2-Lite services are broadcasted. At this place, the provided LTE services are not available (blue color). In this case, the LTE system could not decode the received signal and the CQI value is the lowest. Interestingly, in the small corridor, located between the terrace and the main floor corridor, partial coexistence between T2-Lite and LTE systems is possible. It means that at this place, both wireless systems can coexist. The QEF limit for DVB-T2-Lite is still fulfilled. However, in the LTE system, only sub-frames using QPSK modulation can be successfully processed. Hence, the CQI indicator values will be in the range from 1 up to 6 [35]. Similar graphical representations of considered coexistences are plotted in Figure 7a,b,c,d,e,f.
Now, let us focus on the first two charts (see Figure 7a,b). Their parameters differ just in the used LTE channel bandwidth (B OVER), but the disparity in state map is high. From the point of B LTE = 20 MHz LTE channel (see Figure 7b), the 800 kHz interference bandwidth is quite narrow and almost no effect can be seen on LTE inside the building. Outside, LTE works correctly with 16QAM. However, when B LTE is equal to 10 MHz (see Figure 7a), then the LTE channel, affected by the same interference bandwidth (800 kHz), is occupied by almost twice the interfering RF power. In this case, the LTE system still works correctly, but only 16QAM and QPSK (indoor/outdoor) modulations can be used. Furthermore, mobile TV reception is also more affected by LTE services because LTE interference power is concentrated into a narrower channel. In real RANs, where power limits are more likely set to 1 Hz of occupied bandwidth, the impact on the reception of mobile TV services would be the same. The influence of channels overlapping and the effect of different EIRP unbalances could be investigated from the remaining charts.
Figure 8 shows six map representations of QoS states in DVB-T2-Lite and LTE systems from measurements. In general, in most measuring points, the defined states of QoS correspond with simulation results. However, there are some minor differences caused by the accumulation of two types of uncertainties. The first ones are caused by path loss channel modeling, and these are even multiplied by the second ones, caused by the proposed linear model. Most probably, the largest influences are due to inhomogeneity in walls (doors, windows and various types of material), underestimation of noise level, and impact of multipath propagation. It is obvious that the simulation and measurement results in scenario (e) have the lowest difference. This state is caused by the highest signal level (in the above mentioned scenario) which brings reduction of noise background impact and increase the influence of intersystem jamming simultaneously for all transmission paths.
Measurement - the map representation of QoS states of coexisting systems (a-f). Specific map parameters are summarized in Table 3.
The main aim of this paper is to investigate the impact of coexisting DVB-T2-Lite and LTE systems in a shared frequency band on their system performances in the outdoor-to-indoor reception scenario. To be more precise, a scenario was considered where an indoor LTE femtocell (HeNB) and outdoor-to-indoor DVB-T2-Lite services are provided in an 800-MHz frequency band (see Figures 1 and 2). We have performed separate simulations of both LTE and DVB-T2-Lite RF signal propagation in MATLAB. Further, we have carried out measurements of both wireless systems in order to evaluate the reliability of the simulation model. Results are shown in Figures 3, 4, 5, 6 and correlate well.
According to the achieved results in our previous works ([31] and [28]), we have created a linear model to map outputs of the path loss model to defined QoS states. This model considers the relation between the value of RF channels overlapping and the power imbalance of the investigated radio channels. A detailed description is outlined in Subsection 4.1.
The presented results are expressed in a set of maps (floor plans of the building) with colored areas which determine availability or non-availability of coexisting services and achievable performance. Specific values of these parameters in the considered scenarios are presented in Table 3. The effect of coexistence on valid signal reception is quantified by the change of used modulation scheme and simultaneous availability of services. A detailed description of the color maps is described in Section 5. In the proposed linear model for both systems, we assume good channel conditions (global parameters): signal-to-noise ratio (SNR) ≥35 dB for both systems and also no multipath propagation and no Doppler frequency have been set. Once again, the proposed linear model was proved by measurements (see Figures 7 and 8). In several cases, less correspondence between the simulation and measurement results is explained.
An analysis of the obtained results from the considered coexistence scenarios leads to the following general conclusions:
The impact of DVB-T2-Lite system performance on the LTE system performance and vice versa in their co-channel coexistence scenario in a shared frequency band highly depends on the level of their channels overlapping and on the power imbalance between RF signals.
The outdoor-to-indoor penetration of the T2-Lite signal is highly critical on indoor-to-indoor reception of LTE services when the power imbalance between the RF levels is high. In these cases, the T2-system acts as a co-channel interferer to indoor LTE femtocell and vice versa.
Digital TV fixed indoor reception is more vulnerable to interferences than fixed outdoor reception.
The main aim of our future work will be to extend our proposed linear coexistence model with more global parameters (different kinds of fading channel models and Doppler shift [36-39]) for more realistic modeling of different coexistence scenarios between DVB-T2-Lite and LTE services and vice versa. Moreover, in our future work, we will consider a larger range of system parameters (code rate, IFFT length, guard interval, and higher M-QAM modulations and bandwidth) [40,41].
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This work is supported by the Cluster for Application and Technology Research in Europe on Nanoelectronics (CATRENE) under the project named CORTIF CA116 - Coexistence of Radio Frequency Transmission in the Future, the MEYS of the Czech Republic no. LF14033, no. CZ.1.07/2.3.00/20.0007 and CZ.1.07/2.3.00/30.0005, and finally by the BUT project no. FEKT-S-14-2177. The described research was performed in laboratories supported by the SIX project; no. CZ.1.05/2.1.00/03.0072, the operational program Research and Development for Innovation. Research described in this paper was financed by Czech Ministry of Education in frame of National Sustainability Program under grant LO1401. For research, infrastructure of the SIX Center was used.
Department of Radio Electronics, SIX Research Center, Brno University of Technology, Technicka 3082/12, 616 00, Brno, Czech Republic
Ladislav Polak, Lukas Klozar, Ondrej Kaller, Jiri Sebesta, Martin Slanina & Tomas Kratochvil
Ladislav Polak
Lukas Klozar
Ondrej Kaller
Jiri Sebesta
Martin Slanina
Tomas Kratochvil
Correspondence to Ladislav Polak.
Polak, L., Klozar, L., Kaller, O. et al. Study of coexistence between indoor LTE femtocell and outdoor-to-indoor DVB-T2-Lite reception in a shared frequency band. J Wireless Com Network 2015, 114 (2015). https://doi.org/10.1186/s13638-015-0338-x
Channel model
DVB-T2-Lite
Indoor and outdoor-to-indoor propagation
LTE femtocell
Path loss
QEF
EVM
CQI
RF measurement
Experimental Evaluation in Wireless Communications | CommonCrawl |
Towards in vivo estimation of reaction kinetics using high-throughput metabolomics data: a maximum likelihood approach
Methodology Article
Weiruo Zhang1,
Ritesh Kolte1 &
David L Dill2
High-throughput assays such as mass spectrometry have opened up the possibility for large-scale in vivo measurements of the metabolome. This data could potentially be used to estimate kinetic parameters for many metabolic reactions. However, high-throughput in vivo measurements have special properties that are not taken into account in existing methods for estimating kinetic parameters, including significant relative errors in measurements of metabolite concentrations and reaction rates, and reactions with multiple substrates and products, which are sometimes reversible. A new method is needed to estimate kinetic parameters taking into account these factors.
A new method, InVEst (In Vivo Estimation), is described for estimating reaction kinetic parameters, which addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on a model where all measurements have relative errors. Simulations show that InVEst produces accurate estimates for a reversible enzymatic reaction with multiple reactants and products, that estimated parameters can be used to predict the effects of genetic variants, and that InVEst is more accurate than general least squares and graphic methods on data with relative errors. InVEst uses the bootstrap method to evaluate the accuracy of its estimates.
InVEst addresses several challenges of in vivo data, which are not taken into account by existing methods. When data have relative errors, InVEst produces more accurate and robust estimates. InVEst also provides useful information about estimation accuracy using bootstrapping. It has potential applications of quantifying the effects of genetic variants, inference of the target of a mutation or drug treatment and improving flux estimation.
High-throughput assays such as mass spectrometry are improving rapidly, which creates an opportunity for large scale in vivo measurements of the metabolome. Those in vivo data could enable estimation of kinetic parameters of metabolic reactions which are hard to estimate using in vitro data.
Metabolic reactions are normally enzyme-catalyzed reactions, and quantitative estimates of their kinetic parameters could be very useful. Knowledge of kinetic parameters allows estimation of reaction rates directly from concentration measurements. Comparing the estimated kinetic parameters of a reaction in the wild type and mutant cells permits quantification of the effects of genetic variants, which may change the abundance or activity of a metabolic enzyme. Similarly, the effect of a drug that targets a particular enzyme could be estimated. If parameters can be estimated for many reactions in a pathway, it would enable inference of the target of a mutation or drug treatment – if the estimates show that one enzyme is particularly strongly affected, that enzyme is probably the target. Finally, estimated parameters also allow estimation of maximum reaction rates, which can then be used as constraints to improve flux balance analysis [1].
We explore the central problem of how to estimate the kinetic parameters of individual reactions using in vivo high-throughput measurements of metabolite concentrations and reaction rates at steady state, obtained by mass spectrometry or by nuclear magnetic resonance. The method requires metabolite concentration and reaction rate data in multiple experiments under varying conditions. For example, data could consist of several experiments obtained by perturbing the system through changes in nutrient media, drug treatment, or genetic alterations. From such data, the kinetics of many individual reactions can potentially be estimated.
Enzyme kinetic parameters have been measured for at least a century [2]. The basic method involves mixing a measured amount of substrate and enzyme, and measuring the concentration of product at various points in time, creating a progress curve [3]. In this setting, the experimenter has control over the initial concentrations of enzyme and substrate and thus can obtain relatively accurate measurements for concentrations. Although the experimental conditions are not at steady state, the mathematical formula for the kinetics can be simplified to the familiar Michaelis Menten kinetics by assuming that some elementary reactions are in near-equilibrium (this is called the quasi-steady-state assumption).
In contrast with an in vitro experiment, one major challenge with in vivo measurements of concentrations and reaction rates is the presence of significant error. Except for very low abundance metabolites, the errors are normally relative, meaning that they are proportional to the metabolite concentrations, instead of additive. (Relative error is shown in available experimental data in Additional file 1: Figure S1.) To quantify measurement precision with relative errors, experimentalists often use the coefficient of variation (CV), which is calculated by dividing the standard deviation of peak area/height by the mean peak area/height [4–6]. Methods such as least squares, which assume additive errors, are often not going to produce accurate estimates of parameters with relative errors. Because of such significant relative errors, it might not be reasonable to assume that errors are only in reaction rates as most of the in vitro enzyme kinetics methods assume. Relative errors in both concentrations and reaction rates need to be considered. Furthermore, many in vivo experiments are not time courses, so the data are assumed to be at steady-state. Another challenge with in vivo measurements is the difficulty of measuring enzyme and intermediate enzyme complex concentrations [7, 8], so these are typically unknown. Finally, control over metabolite concentrations in the cell is limited, so the range of experimental data points may be suboptimally distributed for accurate estimation of all parameters, making it difficult to estimate some parameters of a reaction.
A new estimation method, InVEst, standing for In Vivo Estimation, is described for estimating reaction parameters that addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation, based on a model where all measurements have relative errors. As described, InVEst uses a family of reversible reaction mechanisms with multiple reactants and products with a single displacement mechanism. It is not always possible to obtain data from the entire range of metabolite concentrations and reaction rates, so some parameters may not be identifiable. InVEst estimates the standard deviations of parameter estimates using bootstrapping (a method of estimating variation in statistics by random subsampling of a data set), so that the user can understand the range of errors for the estimates.
Many methods for estimating kinetic parameters have been proposed, ranging from informal graphical plotting to sophisticated statistical non-linear regression methods. However, none have addressed all of the problems of in vivo estimation discussed above. Many methods are based on the Michaelis Menten equation which are normally applied to irreversible single substrate, single product reactions. Standard graphical plotting methods, such as the double reciprocal plot [9] and direct linear plot [10], are not based on statistical estimation and yield unnecessarily inaccurate parameter estimates. Some more statistically-based methods deal with relative error or errors in all measurements – but not both. Specifically, weighted least squares [11] is a general method often used in non-linear regression that can be applied to various kinds of reactions, however, it assumes the errors are additive and that only reaction rates have errors. Total least squares [12] improves ordinary least squares by dealing with errors in all measurements, but the errors are still assumed to be additive. Raaijmakers' maximum likelihood estimation method [13] can deal with relative errors, but assumes that errors are in reaction rates only. Liebermeister et al. [14] have developed a method that integrates knowledge from many sources, along with in vivo measurements, to estimate kinetic parameters using Bayesian methods. However, this method still assumes only additive errors and requires a lot of prior information about the parameters. Only InVEst deals with relative errors in all measurements as well as reversible reactions with multiple substrates and products. A summary of existing methods appears in Table 1.
Table 1 Features of different enzyme kinetic parameter estimation methods. "WLS" stands for the weighted least squares method. "TLS" stands for the total least squares method. "Raaijmakers" is the maximum likelihood method of Raaijmakers
In this paper, our goal is to focus on the specific problem of estimating kinetic parameters as accurately as possible, given realistic assumptions about data errors. We discuss the formulation of InVEst, and evaluate the method on simulated data. We show that InVEst works well on data with relative errors in all measurements. We also demonstrate the application of InVEst and discuss the parameter identifiability issue.
Like most methods of kinetic parameter estimation, we assume that temperature and pressure are constant, so rate constants in mass action kinetic equations are constant, and the Gibbs Free Energy of Formation is constant. We also assume that the measured system is at steady state, meaning that the time derivatives of metabolite concentrations and reaction rates are zero.
Also, we assume that there are measurements of stable reactants and products of enzyme reactions, but not substrate-enzyme complexes, product-enzyme complexes and free enzyme concentrations, as they are generally difficult to measure experimentally. It is assumed that metabolite concentrations are obtained by high-throughput methods, such as chromatography, mass spectroscopy, or nuclear magnetic resonance spectroscopy [15]. For example, reasonably accurate concentration data can be obtained by mass spectroscopy with internal standards. Normally, average value of coefficient of variation for mass spectrometry below 0.2 is considered as good measurements [16–18], and thus it is not unreasonable to expect such data to have a constant coefficient of variation (i.e., normally distributed relative error) of 20 %.
We also assume that it is possible to obtain measurements of reaction rates. For steady state reaction rate measurement, one widely used method is C 13 labeling, which uses a cell culture at steady state in a medium with labeled-carbon substrates. Reaction rates can be determined by analyzing the labeling pattern of targeted metabolites from mass spectrometry [19]. In addition, we assume that the Gibb's Free Energies of Formation of metabolites are known, since these are used to compute the equilibrium constants (K eq ) for enzymatic reactions.
Single substrate and product reversible reactions
We use a standard simple but general reaction mechanism to represent most metabolic reversible reactions [20]. This subsection considers single reactant/product case. The more general case consisting of multiple reactants and multiple products will be discussed later. The reaction is a three step process, namely binding, conversion and release:
$$ \mathrm{a} + \mathrm{E}\; {\underset{k_{\text{-}1}}{\overset{k_{1}}\rightleftharpoons}} \; \text{aE} \;{\underset{k_{\text{-}2}}{\overset{k_{2}}\rightleftharpoons}}\; \text{bE} \; {\underset{k_{\text{-}3}}{\overset{k_{3}}\rightleftharpoons}} \; \mathrm{b} + \mathrm{E} $$
((1))
where a is the reactant, b is the product, E is the free enzyme, aE and bE are the intermediate complexes, and k i and k −i are reaction rate constants for i∈{1,2,3}.
Assuming the reaction is at steady state, an equation for the reaction rate can be written as:
$$ v = \frac{K_{eq}[\!a]-[\!b]}{c_{1} + c_{2}[\!a] + c_{3}[\!b]} $$
where \(K_{\textit {eq}} = \frac {k_{1}k_{2}k_{3}}{k_{-1}k_{-2}k_{-3}}\) is an equilibrium constant, obtained from the Standard Gibbs Free Energy of Formation of the reactants and products.
c 1 is
$$\left(\frac{k_{2}k_{3}}{k_{-1}k_{-2}k_{-3}} + \frac{k_{3}}{k_{-2}k_{-3}} + \frac{1}{k_{-3}}\right)/\left[E_{tot}\right], $$
$$\left(\frac{k_{1}k_{2}}{k_{-1}k_{-2}k_{-3}} + \frac{k_{1}k_{3}}{k_{-1}k_{-2}k_{-3}} + \frac{k_{1}}{k_{-1}k_{-3}}\right)/[\!E_{tot}], $$
$$\left(\frac{1}{k_{-2}} + \frac{1}{k_{-1}} + \frac{k_{2}}{k_{-1}k_{-2}}\right)/[\!E_{tot}], $$
and [ E tot ], the total enzyme, is [ E]+[ a E]+[ b E].
If K eq is very large and the reversible reactions' rate constants (k −2 and k −3) are small, c 3 can be neglected and the rate Eq. 2 can be reduced to standard irreversible Michaelis Menten equation.
This rate equation can be derived from the ordinary differential equations for mass action kinetics of a reaction (1), by setting the derivatives of the concentrations of all chemical species to zero (since the system is assumed to be at steady state) and solving for [ E tot ]. The detailed derivation and calculation for the steady state equation and equilibrium constant are presented in Additional files 2 and 3.
Parameter estimation by maximum likelihood for single substrate/product reversible reaction
The InVEst method estimates the parameters of kinetic rate Eq. (2) using maximum likelihood, assuming relative error in all measurements. Parameters are estimated from a set of n experiments, each with data values for a i (substrate), b i (product), v i (reaction rate), for experiment i.
Each data value has some known relative error. Specifically, we have a i =a i0 ε a , b i =b i0 ε b and v i =v i0 ε v , where a i0, b i0, and v i0 are latent variables representing the data values without measurement error, multiplied by a normally distributed error with mean 1 and standard deviation σ: \(\epsilon _{x} \sim N \left (1,{\sigma _{x}^{2}}\right)\) (where x is a, b, or v).
The likelihood function is:
$$\begin{aligned} L(a_{i0},b_{i0},v_{i0},c_{1},c_{2},c_{3} ; a_{i},b_{i},v_{i})= \\ f(a_{i},b_{i},v_{i} ; a_{i0},b_{i0},v_{i0},c_{1},c_{2},c_{3}) \end{aligned} $$
Since each data acquisition can be carried out independently [21], errors in a, b and v can be assumed to be independent of c 1,c 2 and c 3 and each other, the likelihood function can be written as
$$\begin{aligned} & f(a_{i},b_{i},v_{i} ; a_{i0},b_{i0},v_{i0}) = \\ & \prod f(a_{i} ; a_{i0}) \prod f(b_{i} ; b_{i0}) \prod f(v_{i} ; v_{i0}) \end{aligned} $$
The distribution of a i is
$$N\left(a_{i0},a_{i0}^{2}{\sigma_{a}^{2}}\right) = \frac{1}{\sqrt{2\pi a_{i0}^{2}{\sigma_{a}^{2}}}}\exp\left(-\frac{(a_{i}-a_{i0})^{2}}{2a_{i0}^{2}{\sigma_{a}^{2}}}\right) $$
The distributions of the other data values are similar.
The parameters that maximize the likelihood also maximize the log of the likelihood, which is
$${} {\fontsize{9.4pt}{9.6pt}\selectfont{\begin{aligned} \log(L)= &\sum_{i=1}^{n}\left(-\log\left(a_{i0}\sigma_{a}\sqrt{2\pi}\right)\right) + \sum_{i=1}^{n}\left(-\frac{(a_{i}-a_{i0})^{2}}{2a_{i0}^{2}{\sigma_{a}^{2}}}\right) \\ &+\sum_{i=1}^{n}\left(-\log\left(b_{i0}\sigma_{b}\sqrt{2\pi}\right)\right) + \sum_{i=1}^{n}\left(-\frac{(b_{i}-b_{i0})^{2}}{2b_{i0}^{2}{\sigma_{b}^{2}}}\right) \\ &+\sum_{i=1}^{n}\left(-\log\left(v_{i0}\sigma_{v}\sqrt{2\pi}\right)\right) +\sum_{i=1}^{n}\left(-\frac{(v_{i}-v_{i0})^{2}}{2v_{i0}^{2}{\sigma_{v}^{2}}}\right) \end{aligned}}} $$
Negating the log likelihood and dropping constant factors yields an objective function to minimize, subject to the constraints of Eq. 2.
$$\begin{aligned} \min &\left(\sum_{i=1}^{n}(\log(a_{i0})+\log(b_{i0})+\log(v_{i0})) \right.\\ &+ \frac{1}{2{\sigma_{a}^{2}}}\sum_{i=1}^{n}\left(\frac{a_{i}}{a_{i0}}-1\right)^{2} + \frac{1}{2{\sigma_{b}^{2}}}\sum_{i=1}^{n}\left(\frac{b_{i}}{b_{i0}}-1\right)^{2} \\ &\left. + \frac{1}{2{\sigma_{v}^{2}}}\sum_{i=1}^{n}\left(\frac{v_{i}}{v_{i0}}-1\right)^{2} \right)\\ \text{s.t.}& \quad v_{i0} = \frac{K_{eq}a_{i0} - b_{i0}}{c_{1} + c_{2}a_{i0} +c_{3}b_{i0}}, \text{\ where \(i = 1,2,\cdots,n\)} \end{aligned} $$
where all the a i , b i and v i are experimental measurements, all the relative errors σ are known and a i0, b i0, v i0 are latent variables, and c 1, c 2 and c 3 are the parameters to be estimated by solving the optimization problem.
In the implementation, this is simplified to an unconstrained optimization problem by substituting the right-hand side of Eq. 2 for v i0.
Generalization to multiple substrates and products
For reactions with multiple substrates and products, there are two possible mechanisms, namely single-displacement and double-displacement. For single-displacement reactions, the order of substrates binding to the enzyme can be random or ordered. Those two type of reactions can be approximated by following reaction [22]:
$$\begin{aligned} & \mathrm{a}_{1} + \mathrm{a}_{2} + \cdots + \mathrm{a}_{\mathrm{m}} + \mathrm{E} \; {\underset{k_{\text{-}1}}{\overset{k_{1}}\rightleftharpoons}} \; \mathrm{a}_{1}\mathrm{a}_{2}\cdots \mathrm{a}_{\mathrm{m}}\mathrm{E} \\ & {\underset{k_{\text{-}2}}{\overset{k_{2}}\rightleftharpoons}} \; \mathrm{b}_{1}\mathrm{b}_{2}\cdots \mathrm{b}_{\mathrm{p}}\mathrm{E}\; {\underset{k_{\text{-}3}}{\overset{k_{3}}\rightleftharpoons}} \;\mathrm{b}_{1} + \mathrm{b}_{2} + \cdots + \mathrm{b}_{\mathrm{p}} + \mathrm{E} \end{aligned} $$
where m is the number of reactants and p is the number of products in this reaction.
A steady state equation can be derived as in the single reactant/product case:
$$ v = \frac{K_{eq}\prod\limits_{j=1}^{m}[\!a_{j}]-\prod\limits_{j=1}^{p}[\!b_{j}]} {c_{1} + c_{2}\prod\limits_{j=1}^{m}[\!a_{j}] + c_{3}\prod\limits_{j=1}^{p}[\!b_{j}]} $$
where c 1, c 2, c 3, K eq , and E tot are as before.
The derivation of the objective function to minimize in order to find the parameters that maximize the likelihood is a straightforward generalization of the single substrate/product case.
$${} {\fontsize{9.4pt}{9.6pt}\selectfont{\begin{aligned} \min &\left(\sum_{i=1}^{n}\sum_{j=1}^{m}\log(a_{ij0}) +\sum_{i=1}^{n}\sum_{j=1}^{p}\log(b_{ij0}) +\sum_{i=1}^{n}\log(v_{i0}) \right.\\ &+ \frac{1}{2{\sigma_{a}^{2}}}\sum_{i=1}^{n}\sum_{j=1}^{m}\left(\frac{a_{ij}}{a_{ij0}}-1\right)^{2} + \frac{1}{2{\sigma_{b}^{2}}}\sum_{i=1}^{n}\sum_{j=1}^{p}\left(\frac{b_{ij}}{b_{ij0}}-1\right)^{2} \\ &+ \left. \frac{1}{2{\sigma_{v}^{2}}}\sum_{i=1}^{n}\left(\frac{v_{i}}{v_{i0}}-1\right)^{2} \right) \end{aligned}}} $$
which is maximized subject to the constraints of Eq. 3.
In the implementation, this can also be simplified to an unconstrained optimization problem by substituting the right-hand side of Eq. 3 for v i0.
Parameter identifiability
It is sometimes not possible to obtain in vivo data whose values are well enough distributed to estimate all parameters accurately. In this section, we characterize some cases when parameters cannot be accurately estimated. From Eq. (2), it is clear that when one term in the denominator is much smaller than the others, v is relatively insensitive to the corresponding parameter. For example, if c 1,c 2 a≫c 3 b, then Eq. 2 will be approximately
$$ v = \frac{K_{eq}[\!a]-[\!b]}{c_{1} + c_{2}[\!a]}, $$
So changes in c 3 will have little effect on v. More importantly, changes in data values resulting from erroneous estimates of c 3 will be small relative to the noise in the data, so estimates of c 3 tend to have large errors. Similarly, estimates of c 1 tend to have large errors when c 2 a+c 3 b≫c 1 and estimates of c 2 have large errors when c 1+c 3 b≫c 2 a.
For illustration, consider the simpler case when K eq is very large and the rate Eq. (2) can be approximated by the standard Michaelis Menten equation. In Fig. 1(a), two data sets derived from the same actual parameters have large a i , so the v i values lie near the maximum value of the curve. We call this region as saturation region since reaction rates asymptotically approach a maximum level, and additional increases in the substrate concentration do not lead to an increase in the reaction rates. In this case, c 2, which determines the maximum value, is the only parameter that affects the curve fit, so estimates of c 1 from both data sets have large errors. In Fig. 1(b), all of the substrate concentration a i values are small, so the points lie near the region where the curve is increasing linearly. We call this region as linear region since reaction rates increase in almost a linear fashion with increasing substrate concentrations. The slope in this region is determined by c 1 almost independently of c 2 so estimates of c 2 have large errors.
Identifiability issue in two parameter case. When data points are not well-distributed, parameter identification can be difficult. This shows the curve for parameters predicted from two possible data sets, one with points gathered in the saturation region (where reaction rates asymptotically approach a maximum level) in (a) and in the other in the linear region (where substrate concentrations are small and reaction rates increase almost linearly with substrate concentrations) in (b)
Estimates of the accuracy of parameter estimates must be obtained using the available data. InVEst uses bootstrapping to estimate the variance of the parameter estimates.
Bootstrap estimation of standard error
The c parameter estimates can vary widely in accuracy, depending on the experimental data. Bootstrapping [23] is used to estimate the relative standard errors and bias of the parameter estimates, so users can tell whether the parameter estimation is good or not. Let \(\hat {c}\) be the estimate from the data, and \(\hat {c_{i}}^{*}\) be the estimate from a bootstrap sample. A typical recommendation is to use N=n 2 bootstrap samples for n experimental measurements [24]. The bootstrap estimation of standard errors is calculated from \(SE_{B}(\hat {c}) = \left [\frac {1}{N}\sum (\hat {c_{i}}^{*}-\hat {c})^{2}\right ]^{\frac {1}{2}}\) and bias estimation is calculated by \(\mathit {Bias} = \frac {1}{N}\sum \hat {c_{i}}^{*}-\hat {c}\)[25]. As the c parameters have a large range of possible values, it is more appropriate to use relative errors and relative bias to describe the estimate. The relative standard error is calculated by \(SE_{B}/\hat {c}\) and the relative bias is calculated by \(\mathit {Bias}/\hat {c}\).
Estimation of total enzyme change
Estimating kinetic parameters can be useful for identifying the effects of genetic changes or drug treatments that target metabolic enzymes. The total concentration of the enzyme in the cell may change because of changes in gene expression or loss of function in one or more copies of the gene coding for the enzyme, or the activity may change because of changes in the protein sequence or post translational modifications. Estimating these changes for specific enzymes in each sample can help identify the target of a mutation or drug (it's the enzyme whose activity changes the most), and may be useful for estimating the impact of such a change on flux through a network.
Since each of the kinetic parameters c i is of the form c i′/E tot , where c i′ is independent of the enzyme concentration, E tot can be estimated from the ratio
$$ \frac{E_{tot}^{wt}}{E_{tot}^{mt}} = \frac{c_{i}^{mt}}{c_{i}^{wt}} $$
where \(c_{i}^{wt}\) and \(c_{i}^{mt}\) are corresponding c i parameters (i = 1, 2 or 3) for wild type and mutant (or drug treated) samples. Note that it is possible to obtain a reliable estimate for E tot whenever there are reliable estimates for one of the three parameters in both samples.
Evaluate InVEst using simulated data
We evaluate the parameter estimation method on simulated data. For MATLAB code for reproducing the results of this work, please refer to [26]. The simulations were carried out in MATLAB on a laptop computer with an Intel Core i5-4200u 2.3 GHz processor and 8 GB installed memory.
Many reactions in metabolic pathways have multiple substrates and products and are reversible reactions. The simulation is based on the reaction acetylornithine aminotransferase from Saccharomyces Cerevisiae Arginine biosynthesis pathway with Arg8 [27]. Kinetic parameters and the total enzyme concentration are not available, and thus we use some heuristic numbers for them. The experimental data are chosen to be well-distributed, since poorly distributed data would guarantee inaccurate parameter estimates even for the best possible estimation method.
The reaction is:
$$\text{AcGLU-SA} + \text{GLU} \rightleftharpoons \text{AcORN} + {2-\text{oxoglutarate}} $$
Abbreviations [28]: AcGLU-SA, N-acetyl-glutamate-semialdehyde; GLU, L-glutamate; AcORN, N-acetyl- ornithine.
The standard Gibbs Free Energy of Formation for the metabolites are taken from MetaCyc database [29], and are provided in the Additional file 4. The standard Gibbs Free Energy of Formation can be used to compute K eq =1.7281, and, assuming E tot =1 M, the c parameters are c 1=2.5783, c 2=3.7327 and c 3=3.5238.
To characterize the amount of data for effective use of InVEst, we evaluated the accuracy of parameter estimates for varying numbers of simulated experiments. Data sets of 12, 24 and 30 experiments were generated by choosing values for substrate and product concentrations and computing v exactly for each choice based on Eq. 3. Relative errors were introduced by multiplying a random value from the normal distribution of N(1,σ 2). A value of 0.2 was used for σ for metabolites, and σ v of 0.2 was used for reaction rates.
For each number of experiments, 1,000 simulated data sets were generated, the c parameters were estimated using InVEst, and the mean and standard error were calculated. The results are shown in Table 2. With increasing sample size, the relative standard errors and bias in the estimates are improved. It is evident that the results for sample size of 24 and 30 are quite accurate with relative standard error near 10 % and very small relative bias. Twenty to thirty samples seems to be a reasonable sample size to choose for accurate estimations.
Table 2 Average c parameter estimates, relative standard errors and relative bias as a function of number of experiments for acetylornithine aminotransferase when σ v =0.2. Results are based on 1,000 simulated data sets. "n" is the number of experiments. "Avg Est" is the average value of the estimates. "Rel SE" is the relative standard error, and "Rel bias" is the relative bias
Second, we consider the effect of greater error in reaction rate estimates, with σ v =0.5. The results are shown in Table 3. The relative standard errors increase, but are still below 20 %. The relative bias values are also low. This shows that InVEst is robust to different measurement errors.
Table 3 c parameter estimates for acetylornithine aminotransferase when σ v =0.5. Results are based on 1000 simulated data sets of 30 experiments, each
It is also possible to evaluate the accuracy of estimates when there is only one data set (with multiple experiments) available, as would be the case in normal use of InVEst in practice. The bootstrap method is used to estimate relative standard errors in parameter estimates. To evaluate the bootstrap method, we generated a single data set of 30 experiments as the input data for parameter estimation and randomly subsampled the 30 data points 1,000 times. Each bootstrap subsample simulation took around 10 sec. The estimates for σ v =0.2 and σ v =0.5 are shown in Tables 4 and 5 respectively. As expected, the bootstrap estimates are very similar to the previous estimates from 1,000 simulated data sets.
Table 4 c parameter estimates for acetylornithine aminotransferase when σ v =0.2. Estimates are from a single simulated data set of 30 experiments. The bootstrap method was used to estimate relative standard error ("Rel SE") and relative bias ("Rel bias")
Comparison of InVEst with prior methods
Most current methods produce optimal estimates only when errors are additive and when errors occur only in reaction rate measurements. These assumptions are generally not true with in vivo data. In this subsection, we compare InVEst to some existing methods and show that InVEst produces better estimates when data have relative errors in all measurements.
As some of the existing methods only work on irreversible enzymatic reactions, we use the two parameter case of Eq. 2 for comparison. In this case, there are two parameters to be estimated, namely c 1 and c 2.
$$v = \frac{K_{eq}[\!a]}{c_{1} + c_{2}[\!a]}, $$
We first simulate the data with relative errors to both substrate a and reaction rate v, and second apply InVEst and prior methods to obtain estimates for the Michaelis Menten like curve. One thousand simulated data sets of 30 experiments each are used. The results are summarized in Table 6. InVEst has superior performance in the estimates and relative standard errors.
Table 6 Comparison of the accuracy of prior methods: total least square (TLS), ordinary least square (OLS), direct linear plot (DLP), double reciprocal plot(DRP) and InVEst. True c 1=1.5, True c 2=0.8. Data have relative errors in all variables. Results are based on 1,000 simulated data sets of 30 experiments, each. "Avg Est" is the average value of the estimates. "Rel SE" is the relative standard error
Predicting total enzyme concentration change
As noted above, the relative difference in E tot between wild type and mutant or drug-treated samples can be estimated from the estimate of any of the c i parameters from two sets of experiments.
$$\frac{E_{tot}^{wt}}{E_{tot}^{mt}} = \frac{c_{i}^{mt}}{c_{i}^{wt}}. $$
We illustrate estimation of E tot change using the Arg8 reaction. For the wild type samples, the total enzyme concentration is \(E_{\textit {tot}}^{wt} = 1 \hspace {1 mm} M\), and for the mutant/drug treated samples, the total enzyme concentration is \(E_{\textit {tot}}^{mt} = 0.1\hspace {1mm} M\). Results of the wild-type estimate appear in the previous section. Additional data for the mutant were generated as above based on the c parameter values of mutant/drug treated sample and 1,000 simulated data sets are used. The estimates for mutant/drug treated sample are shown in Table 7.
Table 7 c parameter estimates for acetylornithine aminotransferase from mutant/drug treated sample. Results are based on 1,000 simulated data sets
To obtain the prediction of total enzyme change, we take \(c_{i}^{mt}/c_{i}^{wt}\). The results are shown in Table 8.
Table 8 E tot change prediction based on 1,000 simulated data sets
Since any of the c i parameters can be used to estimate the change in E tot , the one that gives minimum standard error, c 2, was chosen. This also demonstrates that even though sometimes identifiability issues can occur and some parameters cannot be estimated, our method could still be very useful if one parameter can be estimated accurately.
This work is intended to be a first step towards estimating parameters for reactions in large metabolic networks in vivo. In vivo estimation will need to be based on data that have relatively large relative errors in all measured parameters, and will have to deal with a variety of reaction kinetics, including reactions that are reversible and have multiple substrates and/or products. Although measurement and estimation of enzyme kinetics has been studied for many decades, there is no single existing estimation method that addresses all of these issues. We have proposed a maximum likelihood approach to estimate kinetic parameters using nonlinear optimization, with estimates on the standard error and bias of the results using the bootstrap.
Simulations show that InVEst produces accurate estimates for realistic high-throughput metabolomics data. For example, with 20–30 samples with coefficients of 20 % in metabolite concentrations and 50 % in reaction rate estimates, estimates have a relative standard error of less than 20 %. Collecting data of this quality would be technically difficult, but is within the current state of the art.
An advantage of the method is that it estimates each set of reaction parameters independently. If measurements are not available for some metabolites, it can still estimate parameters for those reactions for which the data include all substrates and products.
Solving the problem of in vivo parameter estimation in its full generality will require meeting a number of additional challenges. Some reactions have more complex kinetics than those we consider, especially various kinds of inhibition. When the inhibiting metabolite and mechanism of inhibition are known, the approach described here can probably be generalized to accommodate the inhibition mechanism in our future work. Otherwise, a process of model selection may be necessary, where competing models are estimated and the quality of the results compared, with appropriate adjustments for model complexity. In addition, it will be necessary to deal with the kinetics of transport reactions, and to take account of different compartments in the cell.
Parameter identifiability is a difficult issue in in vivo estimation. We have shown that accurate estimates of all parameters require data that is well-distributed over the kinetics curve, but such data will not often be obtainable for several reasons. Experimental data must be obtained by perturbing metabolites and fluxes, for example, by adjusting nutrient media, testing mutants, and targeting reactions with drugs. First, accurate estimation may require non-physiological concentrations of metabolites – estimating c 3 for a reaction that is nearly irreversible being an example. More generally, there is usually inadequate controllability of metabolite concentrations and reaction fluxes to obtain the experimental values needed for accurate estimation, for many reasons including concentrations are toxic or inadequate to sustain life, and rate-limiting reactions that make high fluxes in other reactions impossible to obtain. Since we can't estimate everything accurately, it is important to produce estimates of the standard errors of parameter estimates, so we can tell which ones are meaningful. Also, as we note above, if some but not all parameters of a reaction can be estimated accurately, the results still may be useful. For example, it is possible to estimate the total concentration or relative activity of an enzyme in wild-type vs. mutant cells when only one of the kinetic parameters is accurately estimated.
In conclusion, a new method, InVEst, is developed for estimating reaction kinetic parameters in metabolic networks that addresses the specific challenges of in vivo data. InVEst uses maximum likelihood estimation based on models where all measurements have potentially relative errors. It can be applied to multiple substrate/product reversible enzymatic reactions with a generalized single displacement mechanism. Because it is not always possible to obtain good data covering full range of possible metabolite concentrations and reaction rates, certain parameters may be non-identifiable. InVEst uses bootstrap to estimate the standard errors of parameter estimations that can tell which estimates are reliable.
InVEst enables the estimation of reaction rates directly from concentration measurements. Also, comparing the estimated kinetic parameters of a reaction in the wild type and mutant cells can quantify enzyme abundance or activity change due to genetic variants. The same method can also be used to measure the effect of a drug that targets a particular enzyme. Moreover, estimated parameters can be used to estimate maximum reaction rates, which could be used as constraints to improve flux-balance analysis.
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D.L.D. and W.Z. were supported by a King Abdullah University of Science and Technology (KAUST) research grant under the KAUST Stanford Academic Excellence Alliance program. R. K. was supported by Stanford Graduate Fellowship.
We thank Prof. Chaitan Khosla, Chemical Engineering, Stanford University and Prof. Douglas Brutlag, Biochemistry, Stanford University, for their valuable advice and comments on our work. We also thank Prof. Chao Du, Statistics, University of Virginia, for his suggestions on the bootstrap.
Department of Electrical Engineering, Stanford University, 450 Serra Mall, Stanford, CA94305, USA
Weiruo Zhang & Ritesh Kolte
Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA94305, USA
David L Dill
Weiruo Zhang
Ritesh Kolte
Correspondence to David L Dill.
MATLAB code for reproducing the results of this work is available at Stanford Digital Repository, http://purl.stanford.edu/bg158sn4020.
WZ and DLD defined the problem. WZ did the mathematical derivations, implemented the method, and performed the simulations. RK suggested the maximum likelihood approach and worked out key ideas about the approach. DLD oversaw the work. The authors collaborated on the writing and all authors have read and approved the final manuscript.
Experimental data support on relative error model. Figure S1 Noise errors in high-throughput metabolomic data tend to be relative. The plot shows the empirical standard deviation vs. mean of metabolite concentrations in a publicly available mass spectrometry data set of 40 human urine samples [30]. Each sample has 3 technical replicates, which were used to calculate the standard deviation and mean of metabolite concentrations. The data for "peak 105" were chosen because the chromatographic peak appears in all three replicates of the sample and the measurements cover a wide range of concentrations across different samples. Low concentrations are omitted because they are highly inaccurate due to background noise. There is a linear relationship (R 2=0.71) between standard deviation and concentration mean, showing that errors are proportional to measured concentration. (PDF 1894 kb)
Derivation for steady state rate equation. This file provides a detailed derivation for steady state rate equation of a single reactant and single product reversible metabolic reaction presented in Methods section. (PDF 92.8 kb)
Equilibrium constant K eq . The equilibrium constant K eq is assumed to be a known constant. This file provides the calculation of equilibrium constant K eq based on standard Gibbs Free Energies of Formation. (PDF 98.8 kb)
Standard Gibbs Free Energy of Formation MetaCyc. This file provides standard Gibbs Free Energy of Formation taken from MetaCyc database [29] for metabolites used in the simulation example in Results section. (PDF 51.4 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Zhang, W., Kolte, R. & Dill, D.L. Towards in vivo estimation of reaction kinetics using high-throughput metabolomics data: a maximum likelihood approach. BMC Syst Biol 9, 66 (2015). https://doi.org/10.1186/s12918-015-0214-7
Relative error
Enzymatic reaction
Maximum likelihood
Error-in-all-measurements
In vivo data
Methods, software and technology | CommonCrawl |
Optimal control applied to collective behaviour
Authors: Ramon Escobedo, Aitziber Ibáñez, Enrique Zuazua - 05 November 2017
Solving the driving problem with a repulsion force
Figure1: The model describes the interaction between two agent: a driver (in red) and an evader (in blue).
The standard approach for solving a driving problem is a leadership strategy, based on the attraction that a driver agent exerts on other agents, [1],[2],[3]. Repulsion forces are mostly used for collision avoidance, defending a target or describing the need for personal space [3], [4]. We present a "guidance by repulsion" model [5] describing the behaviour of two agents, a driver and an evader. The driver follows the guided but cannot be arbitrarily close to it, while the evader tries to move away from the driver beyond a short distance. The key ingredient of the model is that the driver can display a circumvention motion around the evader, in such a way that the trajectory of the evader is modified due to the repulsion that the driver exerts on the evader. We propose different open loop strategies for driving the evader from any given point to another assuming that both switching the control and keeping the circumvention mode active have a cost. However, numerical simulations show that the system is highly sensitive to small variations in the activation of the circumvention motion, so a general open-loop control would not be of practical interest. We then propose a feedback control law that avoids an excessive use of the circumvention mode, finding numerically that the feedback law significantly reduces the cost obtained with the open-loop control.
The "Guidance by Repulsion" model
where $\vec{u}_d, \vec{v}_d \in \mathbb{R}^2$ are driver's position vector and velocity vector respectively. $\vec{u}_e, \vec{v}_e \in \mathbb{R}^2$ are evader's position vector and velocity vector respectively (Figure 1).
Figure 2: When the distance between both agents is larger than the critical distance $\delta_c$, the force acting on the driver attracts it toward the evader. If the driver is closer than $\delta_c$ from the evader, the force acting on the driver becomes a repulsion force. Note that, independently of the distance between agents, the force exerted by the driver on the evader is repulsive.
The control variable is $\kappa(t) \in {-1, 0,1 }$. $\delta_1, \delta_2, \delta_c$ are distances. $C_R, C_D^E, C_E^D$ are coefficients of the atraction-repulsion force and circumvection force. $m_e, m_d$ are the masses of the evader and the driver. $\nu_e, \nu_d$ are the frictions of the evader and the driver (see Figure 2 and Figure 3).
Figure 3: The circunvection force acting on the driver is effective only when both agents are closes that $\delta_1$, while $\delta_2$ is the distance left by the driver when it makes the circunvection manoeuvre.
The optimal control problem and the feedback law
Let denote by $B_\rho(T)$ a ball of radius $\rho$ centered in the target $T$, $N_{\rm ig}(\kappa)$ is the number of times that $\kappa(t)$ changes from 0 to $\pm 1$ and ${\cal C}(\kappa) = \int_{t_0}^{t_f} | \kappa(t) | dt$. We define the cost functional:
and we formulate the optimal control problem:
For systems that are subject to conditions of high sensitivity, closed-loop or feedback controls offer the possibility of correcting the state of the system for deviations from the desired behaviour instantaneously. The feedback control law is based on: 1- the alignment of the driver and the evader with the target point $T$ is easier to observe the orientation of the vector $\vec{v}_e(t)$ 2- when the driver is sufficiently far from the evader, $\kappa(t)$ can be set to zero. We define the alignment of the agents and the characteristic function:
The feedback control law can then be written as follows:
Figure 4: Agents' trajectories for the feedback law.
Formulate the problem for N evader and study the evolution of the cost depending on the number on evaders
The sheepherding problem is an example of the use of repulsion forces for guiding a flock. One of the close future work is building a sheepherding model, including stochastic terms in it, using experimental data.
Prove the controllability of the problem.
Study the sensibility and the efficiency of the model depending on the parameters of the model, in special on the critical distances.
Mean--field theorem for the guidance by repulsion model passing to the limit in the number of evaders, $N \rightarrow \infty$, as in [6].
Study of the problem for N evaders and M drivers.
Formulate the problem in $\mathbb{R}^3$.
[1] Alfio Borzì and Suttida Wongkaew. Modeling and control through leadership of a refined flocking system. Mathematical Models and Methods in Applied Sciences,25(02):255–282, 2015.
[2] Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, and Emmanuel Trélat. Sparse stabilization and control of alignment models. Mathematical Models and Methods in Applied Sciences, 25(03):521–564, 2015.
[3] Francesco Ginelli, Fernando Peruani, Marie-Helène Pillot, Hugues Chaté, Guy Theraulaz, and Richard Bon. Intermittent collective dynamics emerge from conflicting imperatives in sheep herds. Proceedings of the National Academy of Sciences, 112(41):12729–12734, 2015.
[4] R Escobedo, C Muro, L Spector, and RP Coppinger. Group size, individual role differentiation and effectiveness of cooperation in a homogeneous group of hunters. Journal of the Royal Society Interface, 11(95):20140204, 2014.
[5] Ramón Escobedo, Aitziber Ibañez, and Enrique Zuazua. Optimal strategies for driving a mobile agent in a guidance by repulsion model. Communications in Nonlinear Science and Numerical Simulation, 39:58–72, 2016.
[6] José Antonio Carrillo, Young-Pil Choi, and Maxime Hauray. The derivation of swarming models: mean-field limit and wasserstein distances. In Collective dynamics from bacteria to crowds, pages 1–46.Springer, 2014. | CommonCrawl |
Geometric Functional Analysis & Probability Seminar
Thursday 13:30-15:30, Ziskind 155 map, directions.
To suggest a talk (yours or somebody else's) contact Ronen Eldan, Itai Benjamini, Gady Kozma or Gideon Schechtman.
To join the mailing list, contact Amir Gonen ([email protected]). This seminar also has a google calendar entry: Weizmann - GFAP Seminar (Geometric Functional Analysis and Probability)
Shamgar Gurevich (Madison)
Harmonic Analysis on GL_n over finite fields.
here are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: $trace(\rho(g))/dim(\rho)$, for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. This talk will discuss the notion of rank for GLn over finite fields, and explain how one can apply the results to verify mixing time and rate for certain random walks. The talk will assume basic notions of linear algebra in Hilbert spaces, and the definition of a group. This is joint work with Roger Howe (Yale and Texas AM).
Jonathan Hermon (Cambridge)
Anchored expansion in supercritical percolation on nonamenable graphs.
Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that: 1. Every infinite cluster has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997). 2. Various observables, including the percolation probability and the truncated susceptibility are analytic functions of p throughout the entire supercritical phase. Joint work with Tom Hutchcroft.
List of previous talks sorted backwards
Eviatar Procaccia (Texas A&M)
Stationary Hastings-Levitov model
We construct and study a stationary version of the Hastings-Levitov(0) model. We prove that unlike the classical model, in the stationary case particle sizes are constant in expectation, yielding that this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). The stationary setting together with a geometric interpretations of the harmonic measure yields new geometric results such as stabilization, finiteness of arms and unbounded width in mean of arms. Moreover we can present an exact conjecture for the fractal dimension.
Mark Rudelson (UMich) + Serguei Popov (IMECC)
Mark Rudelson: Circular law for sparse random matrices.
Consider a sequence of $n$ by $n$ random matrices $A_n$ whose entries are independent identically distributed random variables. The circular law asserts that the distribution of the eigenvalues of properly normalized matrices $A_n$ converges to the uniform measure on the unit disc as $n$ tends to infinity. We prove this law for sparse random matrices under the optimal sparsity assumption. Joint work with Konstantin Tikhomirov.
Serguei Popov: On the range of a two-dimensional conditioned random walk
We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely, it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient and we show that it is ``almost recurrent'' in the sense that each infinite set is visited infinitely often, almost surely. We prove that, for a "typical large set", the proportion of its sites visited by the conditioned walk is approximately a Uniform$[0,1]$ random variable. Also, given a set $G\subset\R^2$ that does not "surround" the origin, we prove that a.s.\ there is an infinite number of $k$'s such that $kG\cap \Z^2$ is unvisited. These results suggest that the range of the conditioned walk has "fractal" behavior. This is a joint work with Nina Gantert and Marina Vachkovskaia, see arxiv.org/abs/1804.00291 Also, there is much more about conditioned walks in my new book (www.ime.unicamp.br/~popov/2srw.pdf, work in progress). Comments and suggestions on the latter are very welcome!
Yotam Smilansky (HUJI)
Statistics of colored Kakutani sequences of partitions
We consider statistical questions concerning colored sequences of partitions, produced by applying a partition process which was first introduced by Kakutani for the 1-dimensional case. This process can be generalized as the application of a fixed multiscale substitution rule, defined on a finite set of colored sets in R^d, on elements of maximal measure in each partition. Colored sets appearing in the sequence are modeled by certain flows on an associated directed weighted graph, and natural statistical questions can be reformulated as questions on the distribution of paths on graphs. Under some incommensurability assumptions, we show that special properties of Laplace transforms of the relevant counting functions imply explicit statistical results.
7th Student Probability Day
Gregory Berkolaiko (Texas A&M)
Nodal statistics of graph eigenfunctions
Understanding statistical properties of zeros of Laplacian eigenfunctions is a program which is attracting much attention from mathematicians and physicists. We will discuss this program in the setting of "quantum graphs", self-adjoint differential operators acting on functions living on a metric graph. Numerical studies of quantum graphs motivated a conjecture that the distribution of nodal surplus (a suitably rescaled number of zeros of the n-th eigenfunction) has a universal form: it approaches Gaussian as the number of cycles grows. The first step towards proving this conjecture is a result established for graphs which are composed of cycles separated by bridges. For such graphs we use the nodal-magnetic theorem of the speaker, Colin de Verdiere and Weyand to prove that the distribution of the nodal surplus is binomial with parameters p=1/2 and n equal to the number of cycles. Based on joint work with Lior Alon and Ram Band.
Michal Strzelecki (Warsaw)
On modified log-Sobolev inequalities
In order to prove concentration estimates for (products of) measures with heavier tails than the standard Gaussian measure one can use several variants of the classical log-Sobolev inequality, e.g., Beckner-type inequalities of Latala and Oleszkiewicz or modified log-Sobolev inequalities of Gentil, Guillin, and Miclo. The main result I plan to present asserts that a probability measure on R^d which satisfies the former inequality satisfies also the latter. Based on joint work with Franck Barthe.
Lisa Hartung (Mainz)
From 1 to 6 in branching Brownian motion
Brownian motion is a classical process in probability theory belonging to the class of �Log-correlated random fields'. It is well known do to Bramson that the order of the maximum has a different logarithmic correction as the corresponding independent setting. In this talk we look at a version of branching Brownian motion where we slightly vary the diffusion parameter in a way that, when looking at the order of the maximum, we can smoothly interpolate between the logarithmic correction for independent random variables ($\frac{1}{2\sqrt 2}\ln(t)$) and the logarithmic correction of BBM ($\frac{3}{2\sqrt 2}\ln(t)$) and the logarithmic correction of 2-speed BBM with increasing variances ($\frac{6}{2\sqrt 2}\ln(t)$). We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard BBM. We will see that the key to the above results is a precise understanding of the entropic repulsion experienced by an extremal particle. (joint work with A. Bovier)
Zemer Kosloff (HUJI)
On the local limit theorem in dynamical systems
In 1987, Burton and Denker proved the remarkable result that in every aperiodic dynamical systems (including irrational rotations for example) there is a square integrable, zero mean function such that its corresponding time series satisfies a CLT. Subsequently, Volny showed that one can find a function which satisfies the strong (almost sure) invariance principle. All these constructions resulted in a non-lattice distribution. In a joint work with Dalibor Volny we show that there exists an integer valued cocycle which satisfies the local limit theorem. The first hour will involve painting (Rokhlin towers) while the second one will be mainly concerned with the proof of the local CLT.
Ami Viselter (Haifa)
Convolution semigroups and generating functionals on quantum groups
The theory of locally compact quantum groups grew out of the need to extend Pontryagin's duality for locally compact abelian groups to a wider class of objects, as well as from a modern "quantum" point of view suggesting the replacement of some algebras of functions on a group by non-commutative objects, namely operator algebras. In this talk, which will be split into two parts, we will show how several fundamental notions from probability and geometric group theory fit in this framework. The first part will be an introduction to locally compact quantum groups. We will present the rationale and the definitions, give examples, and explain how the theory is related to other branches of math. If time permits, we will also touch upon more specific notions related to the second part. In the second part we will discuss convolution semigroups of states, as well as generating functionals, on locally compact quantum groups. One type of examples comes from probability: the family of distributions of a L\'evy process form a convolution semigroup, which in turn admits a natural generating functional. Another type of examples comes from (locally compact) group theory, involving semigroups of positive-definite functions and conditionally negative-definite functions, which provide important information about the group's geometry. We will explain how these notions are related and how all this extends to the quantum world; derive geometric characterizations of two approximation properties of locally compact quantum groups; see how generating functionals may be (re)constructed and study their domains; and indicate how our results can be used to study cocycles. Based on joint work with Adam Skalski. No background in operator algebras will be assumed.
Marta Strzelecka (Warsaw)
On k-maxima of log-concave vectors
We establish two-sided bounds for expectations of order statistics (k-th maxima) of moduli of coordinates of centred log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all k and in the isotropic case for $k\leq n- c n^{5/6}$. We also derive two-sided estimates for expectations of sums of k largest moduli of coordinates for some classes of random vectors. The talk will be based on the joint work with Rafal Latala.
Ori Gurel Gurevich (HUJI)
Random walks on planar graphs
We will discuss several results relating the behavior of a random walk on a planar graph and the geometric properties of a nice embedding of the graph in the plane (specifically, circle packing of the graph). One such a result is that for a bounded degree graph, the simple random walk is recurrent if and only if the boundary of the nice embedding is a polar set (that is, Brownian motion misses it almost surely). If the degrees are unbounded, this is no longer true, but for the case of circle packing of a triangulation, there are weights which are obtained naturally from the circle packing, such that when the boundary is polar, the weighted random walk is recurrent (we believe the converse also hold). These weights arise also in the context of discrete holomorphic and harmonic functions, a discrete analog of complex holomorphic functions. We show that as the sizes of circles, or more generally, the lengths of edges in the nice embedding of the graph tend to zero, the discrete harmonic functions converge to their continuous counterpart with the same boundary conditions. Equivalently, that the exit measure of the weighted random walk converges to the exit measure of standard Brownian motion. This improves previous results of Skopenkov 2013 and Werness 2015, who proves similar results under additional local and global assumptions on the embedding. In particular, we make no assumptions on the degrees of the graph, making the result applicable to models of random planar maps. Based of joint works with Daniel Jerison, Asaf Nachmias, Matan Seidel and Juan Souto.
Fanny Augeri (WIS) + Elliot Paquette (Ohio)
Fanny Auegei: Nonlinear large deviations bounds with applications to sparse Erdos-Renyi graphs.
In this talk, I will present the framework of the so-called nonlinear large deviations introduced by Chatterjee and Dembo. In a seminal paper, they provided a sufficient criterion in order that the large deviations of a function on the discrete hypercube to be due by only changing the mean of the background measure. This sufficient condition was formulated in terms of the complexity of the gradient of the function of interest. I will present general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. The approach relies on convex analysis arguments and is valid for a broad class of distributions. Then, I will detail an application of this nonlinear large deviations bounds to the problem of estimating the upper tail of cycles counts in sparse Erdos-Renyi graphs down to the connectivity parameter $n^{-1/2}$.
Elliot Paquette: The Gaussian analytic function is either bounded or covers the plane.
The Gaussian analytic function (GAF) is a power series with independent Gaussian coefficients. In the case that this power series has radius of convergence 1, under mild regularity assumptions on the coefficients, it is a classical theorem that the power series is a.s. bounded on open disk if and only if it extends continuously to a function on the closed unit disk a.s. Nonetheless, there exists a natural range of coefficients in which the GAF has boundary values in L-p, but is a.s. unbounded. How wild are these boundary values? Well, Kahane asked if a GAF either a.s. extends continuously to the closed disk or a.s. has range covering the whole plane. We show partial progress in establishing this in the affirmative. Joint with Alon Nishry.
Feb 21st, 2019
Houcein Abdalaoui (Rouen)
On Banach problem and simple Lebesgue spectrum.
Following Ulam, Banach asked on the existence of dynamical system on real line with simple Lebesgue spectrum. I discuss the connection of this problem to the famous Banach problem in ergodic theory due essentially to Rohklin. I will further present my recent contribution to this problem and the connection to so called Erd?s flat polynomials problem in Harmonic analysis due to J. Bourgain . My talk is based on my recent work and joint work with Mahendra Nadkarni (CBS Mumbai, India).
Rafael Butez (WIS)
On the extremal particles of a Coulomb gas and random polynomials
The purpose of this talk is to understand the behavior of the extremal zeros of random polynomials of the form $ P_N(z) = \sum_{k=0}^{N} a_k R_k(z)$ where the family $(R_k)_{k \leq N}$ is an orthonormal basis for the scalar product $\langle P,Q \rangle = \int P(z) \overline{Q(z)} e^{-2N V^{\nu(z)}} d\nu(z)$ with $\nu$ a radial probability measure on $\CC$ and $V^{\nu}(z)= \int \log |z-w|d\nu(w)$. Although the zeros of these polynomials spread like the measure $\nu$, the zeros of maximum modulus lie outsite of the support. More precisely, the point process of the roots outside of the support of the equilibrium measure converges towards the Bergman point process of the complement of the support. We also study similar results on a model of Coulomb gases in dimension $2$ where the confining potential is generated by the presence of a fixed background of positive charges. If $\nu$ is a probability measure, we study the system of particles with joint distribution on $\CC^N$, $\frac{1}{Z_N} \prod_{i \leq j} |x_i-x_j|^2 e^{-2(N+1)\sum_{k=1}^{N}V^{\nu}(x_k)} d\ell_{\CC^N}(x_1,\dots,x_N).$ This model in closely related to the study of the zeros of random polynomials. We show that the extremal particles of this Coulomb gas present a similar behavior to the random polynomial setting. All the results mentioned above are done in collaboration with David Garcia-Zelada.
Dan Mikulincer (WIS)
Quantitative high-dimensional CLTs via martingale embeddings
We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in the high dimensional regime. The method is based on the notion of a martingale embedding, a multivariate analogue of Skorokhod's embedding. Using the method we are able to obtain several new bounds for convergence in transportation distance and in entropy, and in particular: (a) We improve the best known bound, for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) We derive a non-asymptotic convergence rate for the entropic CLT in arbitrary dimension, for log-concave random vectors; (c) We give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. In this talk, we will review the method, and explain how one might use it in order to prove quantitative statements about rates of convergence.
Tsviqa Lakrec (HUJI)
Scenery Reconstruction for a Random Walk on Random Scenery with Adversarial Error Insertion
Consider a simple random walk on $\mathbb{Z}$ with a random coloring of $\mathbb{Z}$. Look at the sequence of the first $N$ steps taken and colors of the visited locations. From it, you can deduce the coloring of approximately $\sqrt{N}$ integers. Suppose an adversary may change $\delta N$ entries in that sequence. What can be deduced now? We show that for any $\theta<0.5,p>0$, there are $N_{0},\delta_{0}$ such that if $N>N_{0}$ and $\delta<\delta_{0}$ then with probability $>1-p$ we can reconstruct the coloring of $>N^{\theta}$ integers.
Eliran Subag (Courant)
Optimization of random polynomials on the sphere in the full-RSB regime
To compute the spectral norm of a p-tensor one needs to optimize a homogeneous polynomial of degree p over the sphere. When p=2 (the matrix case) it is algorithmically easy, but for p>2 it can be NP-hard. In this talk I will focus on (randomized) optimization in high-dimensions when the objective function is a linear combination of homogeneous polynomials with Gaussian coefficients. Such random functions are called spherical spin glasses in physics and have been extensively studied since the 80s. I will describe certain geometric properties of spherical spin glasses unique to the full-RSB case, and explain how they can be used to design a polynomial time algorithm that finds points within a small multiplicative error from the global maximum.
Noam Lifshitz (BIU)
Sharp thresholds for sparse functions with applications to extremal combinatorics.
The sharp threshold phenomenon is a central topic of research in the analysis of Boolean functions. Here, one aims to give sufficient conditions for a monotone Boolean function f to satisfy $\mu p(f)=o(\mu q(f))$, where $q = p + o(p)$, and $\mu p(f)$ is the probability that $f=1$ on an input with independent coordinates, each taking the value $1$ with probability $p$. The dense regime, where $\mu p(f)=\Theta(1)$, is somewhat understood due to seminal works by Bourgain, Friedgut, Hatami, and Kalai. On the other hand, the sparse regime where $\mu p(f)=o(1)$ was out of reach of the available methods. However, the potential power of the sparse regime was envisioned by Kahn and Kalai already in 2006. In this talk we show that if a monotone Boolean function $f$ with $\mu p(f)=o(1)$ satisfies some mild pseudo-randomness conditions then it exhibits a sharp threshold in the interval $[p,q]$, with $q = p+o(p)$. More specifically, our mild pseudo-randomness hypothesis is that the $p$-biased measure of $f$ does not bump up to $\Theta(1)$ whenever we restrict $f$ to a sub-cube of constant co-dimension, and our conclusion is that we can find $q=p+o(p)$, such that $\mu p(f)=o(\mu q(f))$ At its core, this theorem stems from a novel hypercontactive theorem for Boolean functions satisfying pseudorandom conditions, which we call `small generalized influences'. This result takes on the role of the usual hypercontractivity theorem, but is significantly more effective in the regime where $p = o(1)$. We demonstrate the power of our sharp threshold result by reproving the recent breakthrough result of Frankl on the celebrated Erdos matching conjecture, and by proving conjectures of Huang--Loh--Sudakov and Furedi--Jiang for a new wide range of the parameters. Based on a joint work with Keevash, Long, and Minzer.
Godofredo Iommi (PUC Chile) + Paul Dario (ENS)
Godofredo Iommi: Upper semi-continuity of the entropy map for Markov shifts
In this talk I will show that for finite entropy countable Markov shifts the entropy map is upper semi-continuous when restricted to the set of ergodic measures. This is joint work with Mike Todd and Anibal Velozo.
Paul Dario: Homogenization on supercritical percolation cluster
The standard theory of stochastic homogenization requires an assumption of uniform ellipticity on the environment. In this talk, we investigate how one can remove this assumption in a specific case: the infinite cluster of the supercritical Bernouilli percolation of Zd. We will present a renormalization argument for the infinite cluster and show how one can use it to adapt the theory developped in the uniformly elliptic setting. We will then review some results which can be obtained through this technique: homogenization theorem, large scale regularity, Liouville theorem and bounds on the corrector.
Jan 3rd, 2019
Chaim Even-Zohar (UC Davis)
Patterns in Random Permutations
Every $k$ entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size $n$? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order $1/\sqrt(n)$, while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into $k$ asymptotically uncorrelated components of different orders in $n$, that correspond to representations of $S_k$. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.
Max Fathi (Toulouse)
Stability in the Bakry-Emery theorem
The Bakry-Emery theorem asserts that uniformly log-concave probability measures satisfy certain functional inequalities, with constants that are better than those associated with the Gaussian measure. In this talk, I will explain how if the constant is almost that of the Gaussian, then the measure almost splits off a Gaussian factor, with explicit quantitative bounds. The proof is based on a combination of Stein's method and simple arguments from calculus of variations. Joint work with Thomas Courtade
Yury Makarychev (TTIC)
Performance of Johnson-Lindenstrauss Transform for k-Means and k-Medians Clustering
Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\epsilon)$ under a projection onto a random $O(log(k/\epsilon)/\epsilon^2)$-dimensional subspace whp. Further, the cost of every clustering is preserved within $(1+\epsilon)$. Crucially, the dimension does not depend on the total number of points n in the instance. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant $p$. For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan. Joint work with Konstantin Makarychev and Ilya Razenshteyn.
Manuel Stadlbauer
Exponential decay of quotients of Ruelle operators
Ruelle�s operator theorem states that the Ruelle operator $L$, which is a positive operator acting on Holder functions, is conjugated to $P+R$ where $R$ is a one-dimensional projection and the norm of $R$ is smaller than 1. This decomposition, also known as spectral gap, is of interest as it allows to characterise the underlying dynamical system through, e.g., central limit theorems or continuous response to perturbations. However, the conjugation depends on the existence of a positive eigenfunction of $L$, which might not exist in more general, fibred situations due to purely functorial reasons. A possibility to circumvent this problem is to consider quotients of operators of the form $f \mapsto \frac{L^m(f L^n (1))}{L^{m+n}(1)}.$ In fact, it is possible to provide reasonable conditions such that their dual operators contract the Wasserstein distance exponentially in $m$. The result gives rise, for example, to a law of the iterated logarithm for continued fractions with sequentially restricted entries or a topology on the set of equilibrium states for semigroups of expanding maps. This is joint work with Paulo Varandas and Xuan Zhang.
Boaz Slomka (WIS)
Improved bounds for Hadwiger�s covering problem via thin-shell estimates
A long-standing open problem, known as Hadwiger's covering problem, asks what is the smallest natural number $N(n)$ such that every convex body in {\mathbb R}^n can be covered by a union of the interiors of at most $N(n)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n} \ln n$. In this talk, I will discuss some history of this problem and present a new result in which we improve this bound by a sub-exponential factor. Our approach combines ideas from previous work, with tools from Asymptotic Geometric Analysis. As a key step, we use thin-shell estimates for isotropic log-concave measures to prove a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$. We further show that the same bound holds for the volume of $K\cap(-K)$ if the center of mass of $K$ is at the origin. If time permits we shall discuss some other methods and results concerning this problem and its relatives. Joint work with H. Huang, B. Vritsiou, and T. Tkocz
Nov 29h, 2018
Amir Dembo (Stanford)
Large deviations of subgraph counts for sparse random graphs
For fixed t>1 and L>3 we establish sharp asymptotic formula for the log-probability that the number of cycles of length L in the Erdos - Renyi random graph G(N,p) exceeds its expectation by a factor t, assuming only that p >> log N/sqrt(N). We obtain such sharp upper tail bounds also for the Schatten norms of the corresponding adjacency matrices, and in a narrower range of p=p(N), also for general subgraph counts. In this talk, based on a recent joint work with Nick Cook, I will explain our approach and in particular our quantitative refinement of Szemeredi's regularity lemma for sparse random graphs in the large deviations regime.
Nov 8h, 2018
Snir Ben Ovadia (WIS)
What are SRB and GSRB measures, and a new characterisation of their existence
SRB measures are an important object in dynamical systems and mathematical physics. Named after Sinai , Ruelle and Bowen, these measures have important properties of being equilibrium states which describe chaotic behaviour, yet may also describe the asymptotic of ``observable� events in the phase space. An open and important question, is in what generality do systems admit SRB measures? We present the notion of generalised SRB measures (GSRB in short), mention some of their important properties, and present a new condition to characterise their existence on a general setup. The first part of the talk will describe some of the motivation leading to define and to study SRB measures; and so we will define GSRB measures and compare their properties with the properties sought for SRB measures. We will also describe a case study of examples motivating to study GSRB measures. Our new result is a characterisation of systems admitting GSRB measures. In the second part of the talk, as much as time permits, we will present some key steps in the construction of GSRB measures.
Daniel Dadush (CWI Amsterdam)
Balancing vectors in any norm
In the vector balancing problem, we are given N vectors v_1,..., v_N in an n-dimensional normed space, and our goal is to assign signs to them, so that the norm of their signed sum is as small as possible. The balancing constant of the vectors is the smallest number beta, such that any subset of the vectors can be balanced so that their signed sum has norm at most beta. The vector balancing constant generalizes combinatorial discrepancy, and is related to rounding problems in combinatorial optimization, and to the approximate Caratheodory theorem. We study the question of efficiently approximating the vector balancing constant of any set of vectors, with respect to an arbitrary norm. We show that the vector balancing constant can be approximated in polynomial time to within factors logarithmic in the dimension, and is characterized by (an appropriately optimized version of) a known volumetric lower bound. Our techniques draw on results from geometric functional analysis and the theory of Gaussian processes. Our results also imply an improved approximation algorithm for hereditary discrepancy. Joint work with Aleksandar Nikolov, Nicole Tomczak-Jaegermann and Kunal Talwar.
Assaf Naor (Princeton)
Coarse (non)Universality of Alexandrov Spaces
We will show that there exists a metric space that does not admit a coarse embedding into any Alexandrov space of global nonpositive curvature, thus answering a question of Gromov (1993). In contrast, any metric space embeds coarsely into an Alexandorv space of nonnegative curvature. Based on joint works with Andoni and Neiman, and Eskenazis and Mendel.
David Ellis (Queen Mary U)
Random graphs with constant r-balls
Let $F$ be a fixed infinite, vertex-transitive graph. We say a graph $G$ is {\em $r$-locally $F$} if for every vertex $v$ of $G$, the ball of radius $r$ and centre $v$ in $G$ is isometric to the ball of radius $r$ in $F$. For each positive integer $n$, let $G_n$ be a graph chosen uniformly at random from the set of all unlabelled, $n$-vertex graphs that are $r$-locally $F$. We investigate the properties that the random graph $G_n$ has with high probability --- i.e., how these properties depend on the fixed graph $F$. We show that if $F$ is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer $r_0$ such that for every integer $r \geq r_0$, with high probability the random graph $G_n = G_n(F,r)$ defined above has largest component of size between $n^{c_1}$ and $n^{c_2}$, where $0 < c_1 < c_2 < 1$ are constants depending upon $F$ alone, and moreover that $G_n$ has a rather large automorphism group. This contrasts sharply with the random $d$-regular graph $G_n(d)$ (which corresponds to the case where $F$ is replaced by the infinite $d$-regular tree). Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics. We obtain somewhat more precise results in the case where $F$ is $\mathbb{L}^d$ (the standard Cayley graph of $\mathbb{Z}^d$): for example, we obtain quite precise estimates on the number of $n$-vertex graphs that are $r$-locally $\mathbb{L}^d$, for $r$ at least linear in $d$. Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs. This is joint work with Itai Benjamini (WIS).
Eliran Subag (NYU)
Free energy landscapes in spherical spin glasses
I will describe a new approach to the study of spherical spin glass models via free energy landscapes, defined by associating to interior points of the sphere the free energy computed only over the spherical band around them. They are closely related to several fundamental objects from spin glass theory: the TAP free energy, pure states decomposition, overlap distribution, and temperature chaos. I will explain some of of those connections.
The exclusion process (usually) mixes faster than independent particles.
The exclusion process is one of the most basic and best studied processes in the literature on interacting particle systems, with connections to card shuffling and statistical mechanics. It has been one of the major examples driving the study of mixing-times. In the exclusion process on an n-vertex graph we have k black particles and n-k white particles, one per site. Each edge rings at rate 1. When an edge rings, the particles occupying its end-points switch positions. Oliveira conjectured that the order of the mixing time of the process is at most that of the mixing-time of k independent particles. Together with Richard Pymar we verify this up to a constant factor for d-regular (or bounded degree) graphs 1 in various cases: (1) the degree d is at least logarithmic in n, or (2) the spectral-gap of a single walk is small (at most log number of vertices to the power 4) or (3) when the number of particles k is roughly $n^a$ for some constant $a \in (0,1)$. In these cases our analysis yields a probabilistic proof of Aldous' famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound which (when $k \geq n^c$) is within a $\log \log n$ factor from Oliveira's conjecture. As applications we get new mixing bounds: (a) $O(\log n \log \log n)$ for expanders, (b) order $ \log (dk)$ for the hypercube ${0,1}^d$ and (c) order $(diameter)^2 \log k$ for vertex-transitive graphs of moderate growth and for the giant component of supercritical percolation on a torus.
Omer Bobrowski (Technion)
Homological connectivity and percolation in random geometric complexes
Connectivity and percolation are two well studied phenomena in random graphs. In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes. Simplicial complexes are a natural generalization of graphs, consisting of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes. We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration. Our generalized notions of connectivity and percolation use the language of homology - an algebraic-topological structure representing cycles of different dimensions. In this talk we will review some recent progress in characterizing and analyzing these phenomena, as well as describing related phase transitions.
Ohad Feldheim (HUJI) + Eviatar Procaccia (Texas A&M)
Ohad Feldheim: Convergence of a quantile admission processes.
Consider the following stochastic model for a growing set. At time $0$ the model consists of the singleton $S = \{-\infty\}$. At every subsequent time, two i.i.d. samples, distributed according to some distribution $D$ on $\mathbb{R}$, are suggested as candidates for $S$. If the smaller among the two is closer to at least a fraction of $r$ of the current elements of $S$ (in comparison with the larger one), then it is admitted into $S$. How will the distribution of the members of $S$ evolve over time as a function of $r$ and $D$? This model was suggested by Alon, Feldman, Mansour, Oren and Tennenholtz as a model for the evolution of an exclusive social group. We�ll show that the empirical distribution of the elements of $S$ converges to a (not-necessarily deterministic) limit distribution for any $r$ and $D$. This we do by relating the process to a random walk in changing environment. The analysis of this random walk involves various classical exponential concentration inequalities as well as a new general inequality concerning mean and minimum of independent random variables. Joint work with Naomi Feldheim.
Eviatar Procaccia: Stabilization of Diffusion Limited Aggregation in a Wedge.
We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.
Emanuel Milman (Technion)
The Gaussian Double-Bubble and Multi-Bubble Conjectures
The classical Gaussian isoperimetric inequality, established in the 70�s independently by Sudakov-Tsirelson and Borell, states that the optimal way to decompose $\mathbb{R}^n$ into two sets of prescribed Gaussian measure, so that the (Gaussian) area of their interface is minimal, is by using two complementing half-planes. This is the Gaussian analogue of the classical Euclidean isoperimetric inequality, and is therefore referred to as the ``single-bubble� case. A natural generalization is to decompose $\mathbb{R}^n$ into $q \geq 3$ sets of prescribed Gaussian measure. It is conjectured that when $q \leq n+1$, the configuration whose interface has minimal (Gaussian) area is given by the Voronoi cells of $q$ equidistant points. For example, for $q=3$ (the ``double-bubble� conjecture) in the plane ($n=2$), the interface is conjectured to be a ``tripod� or ``Y� - three rays meeting at a single point in 120 degree angles. For $q=4$ (the ``triple-bubble� conjecture) in $\mathbb{R}^3$, the interface is conjectured to be a tetrahedral cone. We confirm the Gaussian double-bubble and, more generally, multi-bubble conjectures for all $3 \leq q \leq n+1$. The double-bubble case $q=3$ is simpler, and we will explain why. None of the numerous methods discovered over the years for establishing the classical $q=2$ case seem amenable to the $q \geq 3$ cases, and our method consists of establishing a Partial Differential Inequality satisfied by the isoperimetric profile. To treat $q > 3$, we first prove that locally minimimal configurations must have flat interfaces, and thus convex polyhedral cells. Uniqueness of minimizers up to null-sets is also established. This is joint work with Joe Neeman (UT Austin).
No seminar (Conference at BIU)
Bhaswar Bhattacharya (UPenn) + Elliot Paquette (Ohio)
Bhaswar Bhattacharya: Large Deviation Variational Problems in Random Combinatorial Structures
The upper tail problem in the Erdos-Renyi random graph $G\sim\mathcal{G}_{n,p}$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor of $1+\delta$. The arithmetic analog of this problem counts the number of $k$-term arithmetic progressions in a random subset of $\{1, 2, \ldots, N\}$, where every element is included independently with probability $p$. The recently developed framework of non-linear large deviations (Chatterjee and Dembo (2016) and Eldan (2017)) shows that the logarithm of these tail probabilities can be reduced to a natural variational problem on the space of weighted graphs/functions. In this talk we will discuss methods for solving these variational problems in the sparse regime ($p \rightarrow 0$), and show how the solutions are often related to extremal problems in combinatorics. (This is based on joint work with Shirshendu Ganguly, Eyal Lubetzky, Xuancheng Shao, and Yufei Zhao.)
Elliot Paquette: Random matrix point processes via stochastic processes
In 2007, Virag and Valko introduced the Brownian carousel, a dynamical system that describes the eigenvalues of a canonical class of random matrices. This dynamical system can be reduced to a diffusion, the stochastic sine equation, a beautiful probabilistic object requiring no random matrix theory to understand. Many features of the limiting eigenvalue point process, the Sine--beta process, can then be studied via this stochastic process. We will sketch how this stochastic process is connected to eigenvalues of a random matrix and sketch an approach to two questions about the stochastic sine equation: deviations for the counting Sine--beta counting function and a functional central limit theorem.
No seminar (IMU meeting in Technion)
Ron Peled (TAU)
The fluctuations of random surfaces
Random surfaces in statistical physics are commonly modeled by a real-valued function phi on a lattice, whose probability density penalizes nearest-neighbor fluctuations. Precisely, given an even function $V$, termed the potential, the energy $H(\phi)$ is computed as the sum of $V$ over the nearest-neighbor gradients of phi, and the probability density of phi is set proportional to $\exp(-H(\phi))$. The most-studied case is when $V$ is quadratic, resulting in the so-called Gaussian free field. Brascamp, Lieb and Lebowitz initiated in 1975 the study of the global fluctuations of random surfaces for other potential functions and noted that understanding is lacking even in the case of the quartic potential, $V(x)=x^4$. We will review the state of the art for this problem and present recent work with Alexander Magazinov which finally settles the question of obtaining upper bounds for the fluctuations for the quartic and many other potential functions.
David Jerison (MIT) + Ron Rosenthal (Technion)
David Jerison: Localization of eigenfunctions via an effective potential
We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda. Consider the Neumann boundary value problem for the operator $L u = - \mbox{div} (A \nabla u) + Vu$ on a Lipschitz domain $\Omega$ and, more generally, on a manifold with or without boundary. The eigenfunctions of $L$ are often localized, as a result of disorder of the potential $V$, the matrix of coefficients $A$, irregularities of the boundary, or all of the above. In earlier work, Filoche and Mayboroda introduced the function $u$ solving $Lu = 1$, and showed numerically that it strongly reflects this localization. Here, we deepen the connection between the eigenfunctions and this {\em landscape} function $u$ by proving that its reciprocal $1/u$ acts as an {\em effective potential}. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.
Ron Rosenthal: Eigenvector correlation in the complex Ginibre ensemble
The complex Ginibre ensemble is a non-Hermitian random matrix on C^N with i.i.d. complex Gaussian entries normalized to have mean zero and variance 1=N. Unlike the Gaussian unitary ensemble, for which the eigenvectors are orthogonal, the geometry of the eigenbases of the Ginibre ensemble are not particularly well understood. We will discuss a some results regarding the analytic and algebraic structure of eigenvector correlations in this matrix ensemble. In particular, we uncover an extended algebraic structure which describes the asymptotic behavior (as N goes to infinity) of these correlations. Our work extends previous results of Chalker and Mehlig [CM98], in which the correlation for pairs of eigenvectors was computed. Based on a joint work with Nick Crawford.
Anirban Basak (WIS)
Local weak limits of Ising and Potts measures on locally tree-like graphs.
Consider a sequence of growing graphs converging locally weakly to an infinite (possibly random) tree. As there are uncountably many Ising and Potts Gibbs measures on the limiting tree in the low-temperature regime it is not apriori clear whether the local weak limit of such measures exists and if so, identifying the limit remains a challenge. In this talk, I will describe these limits. The talk is based on joint works with Amir Dembo and Allan Sly.
David Ellis (Queen Mary U) + Benjamin Fehrman (Max Planck Institute)
David Ellis - The edge-isoperimetric problem for antipodally symmetric subsets of the discrete cube.
A major open problem in geometry is to solve the isoperimetric problem for n-dimensional real projective space, i.e. to determine, for each real number V, the minimum possible size of the boundary of a (well-behaved) set of volume V, in n-dimensional real projective space. We study a discrete analogue of this question: namely, among all antipodally symmetric subsets of {0,1}^n of fixed size, which sets have minimal edge-boundary? We obtain a complete answer to the second question. This is joint work with Imre Leader (Cambridge)
Benjamin Fehrman - Well-posedness of stochastic porous media equations with nonlinear, conservative noise.
In this talk, which is based on joint work with Benjamin Gess, I will describe a pathwise well-posedness theory for stochastic porous media equations driven by nonlinear, conservative noise. Such equations arise in the theory of mean field games, as an approximation to the Dean-Kawasaki equation in fluctuating hydrodynamics, to describe the fluctuating hydrodynamics of a zero range process, and as a model for the evolution of a thin film in the regime of negligible surface tension. Our methods are loosely based on the theory of stochastic viscosity solutions, where the noise is removed by considering a class of test functions transported along underlying stochastic characteristics. We apply these ideas after passing to the equation's kinetic formulation, for which the noise enters linearly and can be inverted using the theory of rough paths.
Faculty retreat - no seminar
Yinon Spinka (TAU) + Mark Rudelson (UMich)
Mark Rudelson: Invertibility of the adjacency matrices of random graphs.
Consider an adjacency matrix of a bipartite, directed, or undirected Erdos-Renyi random graph. If the average degree of a vertex is large enough, then such matrix is invertible with high probability. As the average degree decreases, the probability of the matrix being singular increases, and for a sufficiently small average degree, it becomes singular with probability close to 1. We will discuss when this transition occurs, and what the main reason for the singularity of the adjacency matrix is. This is a joint work with Anirban Basak.
Yinon Spinka: Finitary codings of Markov random fields
Let X be a stationary Z^d-process. We say that X can be coded by an i.i.d. process if there is a (deterministic and translation-invariant) way to construct a realization of X from i.i.d. variables associated to the sites of Z^d. That is, if there is an i.i.d. process Y and a measurable map F from the underlying space of Y to that of X, which commutes with translations of Z^d and satisfies that F(Y)=X in distribution. Such a coding is called finitary if, in order to determine the value of X at a given site, one only needs to look at a finite (but random) region of Y. It is known that a phase transition (existence of multiple Gibbs states) is an obstruction for the existence of such a finitary coding. On the other hand, we show that when X is a Markov random field satisfying certain spatial mixing conditions, then X can be coded by an i.i.d. process in a finitary manner. Moreover, the coding radius has exponential tails, so that typically the value of X at a given site is determined by a small region of Y. We give applications to models such as the Potts model, proper colorings and the hard-core model.
Erwin Bolthausen (UZH)
On the high temperature phase in mean-field spin glasses
We present a new way to derive the replica symmetric solution for the free energy in mean-field spin glasses. Only the Sherrington-Kirpatrick case has been worked out in details, but the method also works in other cases, for instance for the perceptron (work in progress), and probably also for the Hopfield net. The method is closely related to the TAP equations (for Thouless-Anderson-Palmer). It does not give any new results, presently, but it gives a new viewpoint, and it looks to be quite promising. As the TAP equations are widely discussed in the physics literature, also at low temperature, it is hoped that the method could be extended to this case, too. But this is open, and probably very difficult.
Elie Aidekon (Paris)
Points of infinite multiplicity of a planar Brownian motion.
Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often. Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points. Joint work with Yueyun Hu and Zhan Shi.
Oren Louidor (Technion) + Alexander Glazman (TAU)
Oren Louidor: Dynamical freezing in a spin-glass with logarithmic correlations.
We consider a continuous time random walk on the 2D torus, governed by the exponential of the discrete Gaussian free field acting as potential. This process can be viewed as Glauber dynamics for a spin-glass system with logarithmic correlations. Taking temperature to be below the freezing point, we then study this process both at pre-equilibrium and in-equilibrium time scales. In the former case, we show that the system exhibits aging and recover the arcsine law as asymptotics for a natural two point temporal correlation function. In the latter case, we show that the dynamics admits a functional scaling limit, with the limit given by a variant of Kolmogorov's K-process, driven by the limiting extremal process of the field, or alternatively, by a super-critical Liouville Brownian motion. Joint work with A. Cortines, J. Gold and A. Svejda.
Alexander Glazman: Level lines of a random Lipschitz function
We consider the uniform distribution on Lipschitz functions on the triangular lattice, i.e. all integer-valued functions which differ by 0 or 1 on any two adjacent vertices. We show that with a positive probability such a function exhibits macroscopic level lines. Instead of working directly with Lipschitz functions we map this model to the loop $O(2)$ model with parameter $x=1$. The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. A main tool in the proof is a positive association (FKG) property that was recently shown to hold when $n \ge 1$ and $0
Sebastien Bubeck (Microsoft) + Percy Deift (NYU)
Sebastien Bubeck: k-server via multiscale entropic regularization
I will start by describing how mirror descent is a natural strategy for online decision making, specifically in online learning and metrical task systems. To motivate the k-server problem I will also briefly recall what we know and what we don't know for structured state/action spaces in these models. Using the basic mirror descent calculations I will show how to easily obtain a log(k)-competitive algorithm for k-paging. I will then introduce our new parametrization of fractional k-server on a tree, and explain how to analyze the movement cost of entropy-regularized mirror descent on this parametrization. This leads to a depth*log(k)-competitive (fractional) algorithm for general trees, and log^2(k) for HSTs. I will also briefly mention dynamic embeddings to go beyond the standard log(n) loss in the reduction from general metrics to HSTs. Joint work with Michael B. Cohen, James R. Lee, Yin Tat Lee, and Aleksander Madry.
Percy Deift: Universality in numerical analysis with some examples of cryptographic algorithms.
We show that a wide variety of numerical algorithms with random data exhibit universality. Most of the results are computational, but in some important cases universality is established rigorously. We also discuss universality for some cryptographic algorithms. Joint work with C. Pfrany, G. Menon, S. Olver, T. Trogdan and S. Miller.
Amir Dembo (Stanford) + Yuval Peres (Microsoft)
Amir Dembo: Large deviations theory for chemical reaction networks.
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Andrea Agazzi and Jean-Pierre Eckman, we utilize Lyapunov stability theory to bypass this challenge and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction.
Yuval Peres: Trace reconstruction for the deletion channel
In the trace reconstruction problem, an unknown string $x$ of $n$ bits is observed through the deletion channel, which deletes each bit with some constant probability q, yielding a contracted string. How many independent outputs (traces) of the deletion channel are needed to reconstruct $x$ with high probability? The best lower bound known is linear in $n$. Until 2016, the best upper bound was exponential in the square root of $n$. We improve the square root to a cube root using statistics of individual output bits and some inequalities for Littlewood polynomials on the unit circle. This bound is sharp for reconstruction algorithms that only use this statistical information. (Similar results were obtained independently and concurrently by De, O�Donnell and Servedio). If the string $x$ is random, we can show a subpolynomial number of traces suffices by comparison to a random walk. (Joint works with Fedor Nazarov, STOC 2017, with Alex Zhai, FOCS 2017 and with Nina Holden & Robin Pemantle, preprint (2017).)
Nadav Yesha (King's College London)
CLT for small scale mass distribution of toral Laplace eigenfunctions
In this talk we discuss the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position. For the 2-dimensional torus, under certain flatness assumptions on the Fourier coefficients of the eigenfunctions and generic restrictions on energy levels, both the asymptotic shape of the variance and the limiting Gaussian law are established, in the optimal Planck-scale regime. We also discuss the 3-dimensional case, where the asymptotic behaviour of the variance is analysed in a more restrictive scenario. This is joint work with Igor Wigman.
Matan Harel (TAU)
Discontinuity of the phase transition for the planar random-cluster and Potts models with $q > 4$
The random-cluster model is a dependent percolation model where the weight of a configuration is proportional to q to the power of the number of connected components. It is highly related to the ferromagnetic q-Potts model, where every vertex is assigned one of q colors, and monochromatic neighbors are encouraged. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever $q > 4$ - i.e. there are multiple Gibbs measures at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter's formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion.
Fanny Augeri (WIS)
Large deviations principles for random matrices
In this talk, I will try to present some techniques to handle the problem of large deviations of the spectrum of random matrices. I will focus on the case of macroscopic statistics of the spectrum of Hermitian matrices - in particular Wigner matrices - as the empirical distribution of the eigenvalues, the largest eigenvalue or the traces of powers. In a first part, I will be concerned with the so-called ``objective method''. Coined by David Aldous, this method consists in introducing, given a sequence of random objects, like random finite graphs, a new infinite random object from which one can deduce asymptotic properties of the original sequence. In the context of random matrices, this method has been mainly advertised by Balint Virag, and proven effective in showing universality results for the so-called beta-ensembles. Regarding large deviations of random matrices, this ``objective method'' amounts to embed our sequence of matrices with growing size into an appropriate space on which one is able to understand the large deviations, and carry out a contraction principle. I will review several large deviations principles obtained by this method, given by interpretations of random matrices as either dense or sparse graphs, and point out the limits of this strategy. In a second part, I will present a different approach which is inspired from Ledoux's proof of the large deviations of Wiener chaoses. I will give a large deviations principles for the traces of Gaussian Wigner matrices using this strategy. Similarly as for Wiener chaoses, where the deviations are obtained by translations in the direction of the reproducing kernel, the large deviations of the traces of Gaussian Wigner matrices are due to additive perturbations of the underlying matrix. If time permits, I will explain how this approach can be generalized to large deviations governed by the same phenomenon. In particular, this approach enables us to partially recover some large deviations results for a family of Wigner matrices which exhibit a ``heavy-tail phenomenon'', meaning that the deviations of their spectrum are due to the deviations of a negligible proportion of the entries.
Nov 23rd, 2017
Naomi Feldheim (WIS)
Persistence of Gaussian Stationary Processes
Consider a real Gaussian stationary process, either on $Z$ or on $R$. What is the probability that it remains positive on $[0,N]$ for large $N$? The relation between this probability, known as the persistence probability, and the covariance kernel of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel is either non-negative or summable. In the first hour of the talk we will discuss new spectral methods which greatly simplify the analysis of persistence. We will then describe its qualitative behavior in a very general setting. In the second hour, we will describe (very) recent progress. In particular we will show the proof of the ``spectral gap conjecture'', which states: if the spectral measure vanishes on an interval containing 0 then the persistence is less then $e^{-cN^2}$, and this bound is tight if the measure is non-singular and compactly supported. Time permitting, we will also discuss ``tiny persistence'' phenomena (of the order of $e^{-e^{cN}}$). Based on joint works with Ohad Feldheim, Benjamin Jaye, Fedor Nazarov and Shahaf Nitzan.
Ilya Goldsheid (Queen Many University)
Real and complex eigenvalues of the non-self-adjoint Anderson model
The criticality of a randomly-driven front
Consider independent continuous-time random walks on the integers to the right of a front R(t). Starting at R(0)=0, whenever a particle attempts to jump into the front, the latter instantaneously advances k steps to the right, absorbing all particles along its path. Sly (2016) resolves the question of Kesten and Sidoravicius (2008), by showing that for k=1 the front R(t) advances linearly once the particle density exceeds 1, but little is known about the large t asymptotic of R(t) at critical density 1. In a joint work with L-C Tsai, for the variant model with k taken as the minimal random integer such that exactly k particles are absorbed by the move of R(t), we obtain both scaling exponent and the random scaling limit for the front at the critical density 1. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the initial local fluctuations (with the scaling limit oscillating between instantaneous super and sub-critical phases).
Pyotr Nayar (Technion)
Gaussian mixtures with applications to entropy inequalities and convex geometry
We say that a symmetric random variable X is a Gaussian mixture if X has the same distribution as YG, where G is a standard Gaussian random variable, and Y is a positive random variable independent of G. In the first part of the talk we use this simple notion to study the Shannon entropy of sums of independent random variables. In the second part we investigate, using Gaussian mixtures, certain topics related to the geometry of $B_p^n$ balls, including optimal Khinchine-type inequalities and Schur-type comparison for volumes of section and projections of these sets. In the third part we discuss extensions of Gaussian correlation inequality to the case of p-stable laws and uniform measure on the Euclidean sphere. Based on a joint work with Alexandros Eskenazis and Tomasz Tkocz.
Nishant Chandgotia (Tel Aviv University)
Irrational rotations, random affine transformations and the central limit theorem
It is a well-known result from Hermann Weyl that if alpha is an irrational number in [0,1) then the number of visits of successive multiples of alpha modulo one in an interval contained in [0,1) is proportional to the size of the interval. In this talk we will revisit this problem, now looking at finer joint asymptotics of visits to several intervals with rational end points. We observe that the visit distribution can be modelled using random affine transformations; in the case when the irrational is quadratic we obtain a central limit theorem as well. Not much background in probability will be assumed. This is in joint work with Jon Aaronson and Michael Bromberg.
Sasha Shamov
Conditional determinantal processes are determinantal
A determinantal point process governed by a locally trace class Hermitian contraction kernel on a measure space $E$ remains determinantal when conditioned on its configuration on an arbitrary measurable subset $B \subset E$. Moreover, the conditional kernel can be chosen canonically in a way that is "local" in a non-commutative sense, i.e. invariant under "restriction" to closed subspaces $L^2(B) \subset P \subset L^2(E)$. Using the properties of the canonical conditional kernel we establish a conjecture of Lyons and Peres: if $K$ is a projection then almost surely all functions in its image can be recovered by sampling at the points of the process.
Alexander Fish (Sydney)
The values of quadratic forms on difference sets, measure rigidity and equidistribution.
Given a quadratic form Q in d variables over the integers, e.g. Q(x,y,z) = xy - z^2, and a set of positive density E in Z^d, we investigate what kind of structure can be found in the set Q(E-E). We will see that if d >= 3, and Q is indefinite, then the measure rigidity, due to Bourgain-Furman-Lindenstrauss-Mozes or Benoist-Quint, of the action of the group of the symmetries of Q implies that there exists k >=1 such that k^2*Q(Z^d) is a subset of Q(E-E). We will give an alternative proof of the theorem for the case Q(x,y,z) = xy - z^2 that uses more classical equidistribution results of Vinogradov, and Weyl, as well as a more recent result by Frantzikinakis-Kra. The latter proof extends the theorem to other polynomials having a much smaller group of symmetries. Based on joint works with M. Bjorklund (Chalmers), and K. Bulinski (Sydney).
Jay Rosen
Tightness for the Cover Time of $S^{2}$
Let M be a smooth, compact, connected two-dimensional, Riemannian manifold without boundary, and let $ C_{\epsilon}$ be the amount of time needed for the Brownian motion to come within (Riemannian) distance $\epsilon$ of all points in M. The first order asymptotics of $ C_{\epsilon}$ as $\epsilon$ goes to 0 are known. We show that for the two dimensional sphere $\sqrt{C_{\epsilon}}-2\sqrt{2}\left( \log \epsilon^{-1}- \frac{1}{4}\log\log \epsilon^{-1}\right)$is tight. Joint work with David Belius and Ofer Zeitouni.
Ran Tessler (ETH) + Assaf Naor (Princeton)
Ran Tessler (11:00): A sharp threshold for Hamiltonian spheres in a random 2-complex
We define the notion of Hamiltonian sphere - a 2-complex homeomorphic to a sphere which uses all vertices. We prove an explicit sharp threshold for the appearance of Hamiltonian spheres in the Linial-Meshulam model for random 2-complexes. The proof combines combinatorial, probabilistic and geometric arguments. Based on a joint work with Zur luria.
Assaf Naor (12:00): A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms.
In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its "vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on $W^{1,1}$), metric geometry (sharp nonembeddability into $L_1$) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.
Amir Dembo
Title: Walking within growing domains: recurrence versus transience
When is simple random walk on growing in time d-dimensional domains recurrent? For domain growth which is independent of the walk, we review recent progress and related universality conjectures about a sharp recurrence versus transience criterion in terms of the growth rate. We compare this with the question of recurrence/transience for time varying conductance models, where Gaussian heat kernel estimates and evolving sets play an important role. We also briefly contrast such expected universality with examples of the rich behavior encountered when monotone interaction enforces the growth as a result of visits by the walk to the current domain's boundary. This talk is based on joint works with Ruojun Huang, Ben Morris, Yuval Peres, Vladas Sidoravicius and Tianyi Zheng.
For previous years see: 2010-2016, 2007-2010, 2005-2007 (maintained by Boaz Tsaban) 2000-2005 (maintained by Gideon Schechtman) | CommonCrawl |
An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex
Mayte Bonilla-Quintana ORCID: orcid.org/0000-0002-7195-11831,
Kyle C. A. Wedgwood ORCID: orcid.org/0000-0002-8109-27652,
Reuben D. O'Dea ORCID: orcid.org/0000-0002-1284-91031 &
Stephen Coombes ORCID: orcid.org/0000-0003-1610-76651
Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an \(I_{\text{h}}\) current in response to hyperpolarising synaptic input. A computational modelling study by Hasselmo (Philos. Trans. R. Soc. Lond. B, Biol. Sci. 369:20120523, 2013) showed that an inhibitory network of such cells can support periodic travelling waves with a period that is controlled by the dynamics of the \(I_{\text{h}}\) current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in \(I_{\text{h}}\) resonance frequency along the dorsal to ventral axis can explain the observed size and spacing between grid cell firing fields. Here we develop a biophysical spiking model within a framework that allows for analytical tractability. We combine the simplicity of integrate-and-fire neurons with a piecewise linear caricature of the gating dynamics for HCN channels to develop a spiking neural field model of MEC. Using techniques primarily drawn from the field of nonsmooth dynamical systems we show how to construct periodic travelling waves, and in particular the dispersion curve that determines how wave speed varies as a function of period. This exhibits a wide range of long wavelength solutions, reinforcing the idea that rebound spiking is a candidate mechanism for generating grid cell firing patterns. Importantly we develop a wave stability analysis to show how the maximum allowed period is controlled by the dynamical properties of the \(I_{\text{h}}\) current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions.
The ability to remember specific events occurring at specific places and times plays a major role in our everyday life. The question of how such memories are established remains an active area of research, but several key facts are now known. In particular, the 2004 discovery of grid cells in the medial enthorinal cortex (MEC) by Fyhn et al. [2], supports the notion of a cognitive map for navigation. This is a mental representation whereby individuals can acquire, code, store, recall, and decode information about relative spatial locations in their environment. The concept was introduced by Tolman in 1948 [3], with the first neural correlate being identified as the place cell system in the hippocampus [4]. Place cells are found in the hippocampus and fire selectively to spatial locations, thereby forming a place field whose properties change from one environment to another. More recently, a second class of cells was identified that fire at the nodes of a hexagonal lattice tiling the surface of the environment covered by the animal—these are termed grid cells [5]. As an animal approaches the centre of a grid cell firing field, the spiking output of grid cell will increase in frequency. The grid field size and spacing increases from dorsal to ventral positions along the MEC and is independent of the animal's speed and direction (even in the absence of visual input) and independent of the arena size. In rats, the grid field spacing can range from roughly 30 cm up to several meters [6]. Other grid cell properties include firing field patterns that manifest instantly in novel environments and maintain alignment with visual landmarks. Furthermore, neighbouring grid cells have firing fields with different spatial phases, whilst grid cells with a common spacing also have a common orientation (overturning an original suggestion that they have different orientations) [7].
May-Britt Moser and Edvard Moser shared the 2014 Nobel Prize in Physiology or Medicine with John O'Keefe for their discoveries of cells (grid and place cells, respectively) that subserve the brain's internal positioning system. From a modelling perspective grid cells have attracted a lot of attention, due in part to their relatively recent and unexpected discovery, but also due to the very geometric firing patterns that they generate. There are now three main competing mathematical models for the generation of grid-like firing patterns: oscillatory interference models, continuous attractor network models, and "self-organised" models—see Giocomo et al. [8] and Schmidt-Hieber and Häusser [9] for excellent reviews. The first class of model uses interference patterns generated by multiple oscillations to explain grid formation [10]. They have been especially fruitful in addressing the theta rhythmic firing of grid cells (5–12 Hz) and their phase precession. Here spikes occur at successively earlier phases of the theta rhythm during a grid field traversal, suggestive of a spike-timing code [11]. The second class uses activity in local networks with specific connectivity to generate the grid pattern and its spacing. In this category, the models can be further sub-divided into those that utilise spatial pattern formation across the whole tissue (possibly arising via a Turing instability), such as in the work of Fuhs and Touretzky [12] and Burak and Fiete [13], and those that rely only upon spatially localised pattern states (or bumps) in models with (twisted) toroidal connectivity as described by McNaughton et al. [14] and Guanella et al. [15]. The third class proposes that grid cells are formed by a self-organised learning process that borrows elements from both former classes [16–18]. Recent experiments revealing the in vivo intracellular signatures of grid cells, the primarily inhibitory recurrent connectivity of grid cells, and the topographic organisation of grid cells within anatomically overlapping modules of multiple spatial scales along the dorsoventral axis of MEC provide strong constraints and challenges to all three classes of models [18–20]. This has led to a variety of new models, each with a focus on one or more aspects of biophysical reality that might underlie the functionality of grid cell response. For example Couey et al. [21] have shown that a continuous attractor network with pure inhibition can support grid cell firing, with the caveat that there is sufficient excitatory input to the MEC, supposedly from hippocampus, to cause principal cells to fire. However, recent optogenetic and electrophysiological experiments have challenged this simple description [22], highlighting the importance of intrinsic nonlinear ionic currents and their distribution amongst the main cell types in MEC.
Stellate and pyramidal cells constitute the principal neurons in layer II of medial enthorinal cortex (MEC II) that exhibit grid cell firing. The former comprise approximately 70% of the total MEC II neural population and are believed to represent the majority of the grid cell population. Even before the discovery of grid cells stellate cells were thoroughly studied because of their rapid membrane time constants and resonant behaviour. Indeed, they are well known to support oscillations in the theta frequency range [23, 24]. Interestingly the frequency of these intrinsic oscillations decreases along the dorsal-ventral axis of MEC II [25], suggestive of a role in grid field spacing. The resonant properties of stellate cells have been directly linked to a high density of hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels [26], underlying the so-called \(I_{\text{h}}\) current. The time constant of both the fast and slow component of \(I_{\text{h}}\) is significantly faster for dorsal versus ventral stellate cells, providing a potential mechanism for the observed difference in the resonant frequency along the dorsal-ventral axis. However, perhaps of more importance is the fact the \(I_{\text{h}}\) current can cause a depolarising rebound spike following a hyperpolarising current injection. Given that stellate cells are mainly interconnected by inhibitory interneurons [21], this means that rebound can play an important role in shaping spatio-temporal network rhythms. The inclusion of important intrinsic biophysical properties into a network has been emphasised by several authors, such as Navratilova et al. [27] regarding the contribution of after-spike potentials of stellate cells to theta phase precession, and perhaps most notably by Hasselmo and colleagues for the inclusion of HCN channels [28–31]. This has culminated in a spiking network model of MEC that supports patterns whose periodicity is controlled by a neuronal resonance frequency arising from an \(I_{\text{h}}\) current [1]. The model includes many of the features present in the three classes described above, and is able to replicate behaviour from several experiments, including phase precession in response to a phasic medial septum input, theta cycle skipping, and the loss of the spatial periodicity of grid cell firing fields upon a reduction of input from the medial septum. Simulations of the model in one spatial dimension show that the spacing of grid firing fields can be controlled by manipulating the speed of the rebound response. We note that a change that affects the rebound response would also affect the resonant properties of the cell. In contrast to continuous attractor models that rely on the spatial scale of connectivity to control grid spacing, a change in rebound response provides a mechanism of local control via changes in the expression levels of HCN channels. This fascinating observation warrants a deeper mathematical analysis. In this paper we introduce a spiking network model that shares many of the features of the Hasselmo model [1], focussing on the formation of travelling waves that can arise in the absence of (medial septum) input. Importantly our bespoke model is built from piecewise linear and discontinuous elements that allows for an explicit analysis of the periodic waves that arise in an inhibitory network through rebound spiking. In particular our wave stability analysis demonstrates clearly that the maximum allowed period is strongly controlled by the properties of our model \(I_{\text{h}}\) current. This gives further credence to the hypothesis that HCN channels can control the properties of tissue level periodic waves that may underpin the spacing of grid cell firing in MEC.
In Sect. 2 we introduce our model of a network of stellate cells in MEC II. The single neuron model is a simple leaky integrate-and-fire (LIF) model with the inclusion of a synaptic current mediated by network firing events, and a model of \(I_{\text{h}}\) based on a single gating variable. For ease of mathematical analysis we focus on a continuum description, so that the model may be regarded as a spiking neural field. Simulations of the model in two spatial dimensions are used to highlight the genericity of spike rebound mediated spatio-temporal patterns. To uncover the systematic way in which a cellular \(I_{\text{h}}\) current can control the properties of patterns at the tissue level, we focus next on a one-dimensional version of the model without external input. By further developing a piecewise linear (pwl) caricature of the activation dynamics of a HCN channel we show in Sect. 3 how explicitly to construct the dispersion curve for periodic travelling waves. This gives the speed of a wave as a function of its period, and shows the possibility of a wide range of wave periods that could be selected. Next in Sect. 4 we exploit techniques from the analysis of nonsmooth systems to determine the Evans function for wave stability. Importantly an investigation of this function shows that the maximum allowed period can be controlled both by the overall conductance strength of the \(I_{\text{h}}\) current as well as the time-scale for activation of HCN channels. Finally in Sect. 5 we discuss natural extensions of our work, and its relevance to further models of grid cell firing.
The original work of Hasselmo [1] considered both simple resonant models as well as spiking nonlinear integrate-and-fire Izhikevich units to describe MEC stellate cells. The former incorporated a model for \(I_{\text{h}}\) using a single gating variable with a linear dynamics whilst the latter were tuned to capture the resonant and rebound spiking properties from experimental data. Synaptic interactions between cells in a discrete one-dimensional network were modelled with a simple voltage threshold process. These models were subsequently studied in more detail in [28], with a further focus on two-dimensional models and travelling waves. Here, we consider a biophysically realistic spiking model in a similar spirit to that of Hasselmo, but within a framework that will allow for a subsequent mathematical analysis. In particular, we consider a spiking model of stellate cells using a generalised LIF model that includes a nonlinear \(I_{\text{h}}\) current. Moreover, we opt for a description of synaptic interactions using an event-based scheme for modelling post-synaptic conductances.
We first consider a continuum description defined on the plane and introduce a voltage variable \(V=V(\boldsymbol {r},t)\), where \(\boldsymbol {r} \in \mathbb{R}^{2}\) and \(t \geq0\). The subthreshold LIF dynamics is given by
$$ C \frac{\partial}{\partial t} V (\boldsymbol {r},t) = - g_{\text{l}} V (\boldsymbol {r},t) + I_{\text{h}} (\boldsymbol {r},t) + I_{\text{syn}} (\boldsymbol {r},t) + I_{\text{hd}} (\boldsymbol {r},t), $$
with a set of spike times at position r generated according to
$$ T^{m}(\boldsymbol {r}) = \inf\bigl\{ t \mid V(\boldsymbol {r},t) \geq V_{\text{th}} ; t \geq T^{m-1}(\boldsymbol {r}) + \tau_{\text{R}} \bigr\} , \quad m \in \mathbb{Z}. $$
Here \(\tau_{\text{R}}\) is a refractory time-scale. Upon reaching the threshold \(V_{\text{th}}\) the membrane potential is reset to the value \(V_{\text{r}} < V_{\text{th}}\). The infimum operation ensures that a firing time is determined by the first time that the voltage variable (at a fixed position) crosses threshold (from below, remembering the IF reset process) subject to refractoriness. To model the refractory process we hold the voltage variable at the reset value \(V_{\text{r}}\) for a duration \(\tau_{\text{R}}\) after a firing event. The left-hand side of (1) describes a membrane current with capacitance C. The first term on the right-hand side of (1) represents a leak with a constant conductance \(g_{l}\) (and we have set the leakage reversal potential to zero without loss of generality). The terms \(I_{\text{h}}\), \(I_{\text{syn}}\), and \(I_{\text{hd}}\) represent currents arising from HCN channels, synaptic input, and head-direction input, respectively. \(I_{\text{h}}\) is a slow inward current with a reversal potential \(V_{\text{h}}\) that is substantially above resting levels, but which requires hyperpolarisation to become active; that is, the activation curve is monotone decreasing in V. Furthermore, the \(I_{\text{h}}\) current does not inactivate, even with prolonged (minutes) hyperpolarisation. Thus it is often modelled with a single gating variable \(n_{\text{h}}\) such that \(I_{\text{h}} (\boldsymbol {r},t)= g_{\text{h}} n _{\text{h}} (\boldsymbol {r},t) (V_{\text{h}}-V (\boldsymbol {r},t))\), where
$$ \tau_{\text{h}}(V) \frac{\partial}{\partial t} n_{\text{h}} = n_{\text{h},\infty}(V) - n_{\text{h}} . $$
Here the shape for the activation function \(n_{\text{h},\infty}\) is the sigmoid:
$$ n_{\text{h},\infty}(V)= \frac{1}{1+\exp((V-V_{1/2})/k)}, $$
and fits to experimental data give \(V_{1/2}\approx-10\mbox{ mV}\) (with respect to rest) and \(k\approx10\) [32, 33]. The time constant for activation and deactivation can vary from tens to hundreds of milliseconds [34]; however, here we fix \(\tau_{\text{h}}=\text{constant}\) for simplicity and ignore any detailed dependence on voltage. To model synaptic interactions we consider a simple effective anatomical model whereby stellate cells interact directly through inhibition. In reality inhibitory interactions are actually mediated by interneurons and there is no direct synaptic coupling between stellate cells. Introducing an overall strength of synaptic conductance \(g_{\text{syn}}\) we then write \(I_{\text{syn}}(\boldsymbol {r},t)=g_{\text{syn}} \psi(\boldsymbol {r},t)\), where
$$ \psi(\boldsymbol {r},t) = \int_{\Omega} \mathrm {d}\boldsymbol {r}' W\bigl(\boldsymbol {r},\boldsymbol {r}'\bigr) E\bigl(\boldsymbol {r}', t\bigr), $$
where the function W represents anatomical connectivity, \(\Omega \subseteq \mathbb{R}^{2}\) is the spatial domain, and the function E represents the shape of a post-synaptic response to a train of incoming spikes. We write this in the form
$$ E(\boldsymbol {r},t) = \sum_{m \in \mathbb{Z}} \eta\bigl(t-T^{m}( \boldsymbol {r})\bigr) , $$
for a given temporal filter shape η with the property that \(\eta (t)=0\) for \(t<0\) (so that interactions are causal). For convenience we will work with normalised responses such that \(\int_{0}^{\infty} \mathrm {d}t \eta(t) = 1\). As a concrete choice for the function W we shall take a smoothed bump shape \(W(\boldsymbol {r},\boldsymbol {r}')=w(|\boldsymbol {r}-\boldsymbol {r}'|)\), with
$$ w(x) = \frac{w_{0}}{2} \bigl[ \tanh\bigl(\beta(\sigma-x)\bigr) + \tanh\bigl( \beta (\sigma+x)\bigr) \bigr] , \quad\beta,\sigma>0 . $$
Here \(w_{0}<0\) in accordance with the predominantly inhibitory interactions of MEC, σ controls the spatial scale of interaction, and β the steepness of the surround inhibition function as shown in Fig. 1. The model is completed with the choice of the synaptic filter η, which we shall take to be an α-function of the form
$$ \eta(t) =\alpha^{2} t \mathrm {e}^{-\alpha t} H(t), $$
where \(\alpha^{-1}\) is the time-to-peak of the synapse, and H is the Heaviside step function. Note that we work in a regime where the model is excitable, as we do not include any background drive that would be able to make the single neuron model fire in the absence of synaptic coupling or head-direction input.
Connectivity function w in (7), with \(\sigma= 25\), \(\beta= 0.5\) and \(w_{0} = -10\)
An animal's speed \(v=v(t)\) and direction of motion \(\varPhi=\varPhi(t)\) generates input to the MEC that is modelled by the head-direction current \(I_{\text{hd}}\). For example, this could be of the form \(I_{\text{hd}}(\boldsymbol {r},t)= \boldsymbol {v} \cdot \boldsymbol {r}_{\phi}\) where \(\boldsymbol {v} = v(\cos{\varPhi}, \sin{\varPhi})\) and \(\boldsymbol {r}_{\phi}= l(\cos\phi(\boldsymbol {r}), \sin\phi(\boldsymbol {r}))\) describes a head-direction preference for the orientation \(\phi(\boldsymbol {r})\) at position r [13]. Here l is a constant that sets the magnitude of the head-direction current. The hidden assumption here is that head direction matches the direction of motion. However, this is not necessarily true behaviourally, and head direction cells may not code for motion direction [35]. Nonetheless it is a common assumption in most grid cell models, and so we adopt it here too. Fuhs and Touretzky [12] have shown that choosing an anisotropic anatomical connectivity of the particular form \(W(\boldsymbol {r},\boldsymbol {r}') = w(|\boldsymbol {r}-\boldsymbol {r}_{\phi}-\boldsymbol {r}'|)\) can then induce a spatio-temporal network pattern to flow in accordance with the pattern of head-direction information generated when traversing an environment. For continuous attractor network models that can generate, via a Turing instability, static hexagonal patterns with regions of high activity at the nodes of a triangular lattice, the induced movement of these hot-spots over a given point in the tissue gives rise to grid-like firing patterns. In this case the spacing of the firing field is hard to change, as it is mainly fixed by the spatial scale of the chosen connectivity; however, the mechanism of inducing pattern flow is robust to how the tissue pattern is generated. Thus, given the established success of the Fuhs and Touretzky mechanism we will not focus on this here, and instead turn our attention to the formation of relevant tissue patterns and, in particular, how local control of firing field spacing may be effected.
To investigate the types of solution the spiking neural field model supports, we perform simulations over a two-dimensional square domain. Since the action of the head-direction current is merely to induce a flow of emergent patterns we restrict our attention to the case without such input, i.e. \(I_{\text{hd}}=0\). See Appendix A for details of our bespoke numerical scheme implemented on a GPU architecture, and Additional file 1 for C++/CUDA code. We observe three general classes of coherent behaviour that take the form of spatially periodic non-travelling structures (though which oscillate in time), travelling periodic waves, and lurching waves. The latter are also generically found in neural systems with rebound currents, such as in models of thalamic slices [36–38]. Unlike traditional smoothly propagating waves, which exhibit a stationary profile in a co-moving frame, lurching waves consist of patterns of activity in a localised region of the domain, which after some period of time, decay and an adjacent region of the domain becomes active. These waves appear to 'lurch' across the domain. Whilst interesting in their own right, we focus in this article on the analysis of smoothly propagating periodic waves, since these have been suggested by Hasselmo [1] to play a dominant role in the formation of grid-like firing patterns in MEC. We show an example of a non-travelling periodic structure in Fig. 2, at two different time points to illustrate that the pattern is not static, but oscillates in time. An example of a smoothly propagating travelling wave is shown in Fig. 3. The model also supports more exotic spatio-temporal structures, including hexagonal patterns, and for a movie showing a dynamic state with a hexagonal sub-structure see Additional file 2.
A simulation of spatially periodic non-travelling patterns in a two-dimensional spiking neural field model with an \(I_{\mathrm{h}}\) current, solved on a spatial grid of 1000 × 1000 points. Displayed is the voltage component across the entire network at \(t=7000\mbox{ ms}\) (left) and \(t=10\mbox{,}000\mbox{ ms}\) (right). The model supports periodic patterns of localised activity. Note that these patterns are not static, but oscillate in time. Parameters: \(C = 1\mbox{ }\upmu \mbox{Fcm}^{-2}\), \(\tau_{\text{h}}=400\mbox{ ms}\), \(V_{\text{h}}=40 \mbox{ mV}\), \(g_{l} = 0.25\mbox{ mS/cm}^{-2}\), \(g_{h} = 1\mbox{ mS/cm}^{-2}\), \(\tau_{\mathrm{R}} = 200\mbox{ ms}\), \(V_{\text{th}} = 14.5\mbox{ mV}\), \(V_{\text{r}} = 0\mbox{ mV}\), \(V_{1/2} = -10\mbox{ mV}\), \(k=10\), \(g_{\text{syn}} = 15\mbox{ mS/cm}^{-2}\), \(w_{0} = -10\), \(\sigma= 25\), \(\beta ^{-1}=0\), and \(\alpha^{-1} = 20\mbox{ ms}\). The choice of a long refractory time-scale in the model is useful for eliciting a single (rather than multiple) spike rebound response. Spatial domain \(\Omega= [-L,L] \times[-L,L]\) where \(L=10 \sigma\). See also the video in Additional file 2, showing the emergence of more exotic spatio-temporal structures, including hexagons
Periodic travelling wave solutions in the spiking neural field model, solved on a spatial grid of 1000 × 1000 points. Displayed is the voltage component across the entire network at \(t=7000\mbox{ ms}\) (left) and \(t=10\mbox{,}000\mbox{ ms}\) (right). Using different initial data, the spatially periodic pattern observed in Fig. 2 is replaced by a periodic travelling wave (moving downward in the figure). Parameters as in Fig. 2, with \(w_{0}=-0.1\)
Wave Construction
To understand more fully how \(I_{\text{h}}\) controls the emergent scale of periodic waves seen in the simulations of Sect. 2 we now turn to a one-dimensional version of the model defined on the infinite domain. As in Sect. 2, we consider \(I_{\text{hd}}=0\), in which case the model is isotropic and (5) reduces to
$$ \psi(x,t) = \sum_{m \in \mathbb{Z}} \int_{-\infty}^{\infty} \mathrm {d}y w\bigl(|x-y|\bigr) \eta\bigl(t-T^{m}(y) \bigr), \quad x \in \mathbb{R}, t>0 . $$
To reduce the model to a more mathematically convenient form we make two observations about the \(I_{\text{h}}\) current. The first is that \(V_{\text{h}}\) is typically larger than V, which suggests the approximation \(V_{\text{h}} - V \simeq V_{\text{h}}\). The second is that the nonlinear activation function \(n_{\text{h},\infty}\) can be approximated by a pwl function, as illustrated in Fig. 4. Here we match the slope at \(V=V_{1/2}\), and otherwise saturate the function to one or zero, so that
$$ n_{\text{h},\infty}(V) = \textstyle\begin{cases} 1 , & V \leq V_{-}\\ \frac{1}{2} - \frac{V-V_{1/2}}{4 k} , & V_{-} < V < V_{+}\\ 0 , & V \geq V_{+}, \end{cases}\displaystyle ,\quad V_{\pm}= V_{1/2} \pm2 k. $$
Red line: Nonlinear activation function for \(n_{\text{h}}\) with \(V_{1/2}=-10\mbox{ mV}\) and \(k=10\). Green line: Piecewise linear fit of \(n_{\text{h},\infty}\) given by Eq. (10)
Simulations of the model with the reduced form of the \(I_{\text{h}}\) are in qualitative agreement with simulations of the full nonlinear model, and indeed wherever tested the same repertoire of behaviour is always found. In both instances, travelling wave behaviour with a well-defined speed and period is easily initiated; Figs. 5 and 6 compare simulation results arising in the fully nonlinear and reduced pwl model.
Simulations of a one-dimensional spiking neural field model with an \(I_{\text{h}}\) current solved on a spatial grid of 5000 points. Left: simulations of the model with the full nonlinear dynamics for \(I_{\text{h}}\). Right: simulations of the model with the reduced pwl dynamics for \(I_{\text{h}}\) given by (10). Here we show the voltage traces at an illustrative set of locations throughout the system as a function of time. Note that each single spike is mediated by rebound in response to inhibitory synaptic input. Parameters are as in Fig. 2 with \(\beta=0.5\), \(V_{\text{th}} = 14\mbox{ mV}\). The activity propagates from bottom to top after an initial hyperpolarisation current of \(-30\mbox{ mV}\) is given to a set of neurons (in yellow) from \(t=1000\mbox{ ms}\) to \(t=1250\mbox{ ms}\)
Voltage traces observed at \(x=0\), when the periodic travelling wave is fully developed. Top: simulations of the model with the reduced pwl dynamics for \(I_{\text{h}}\) given by (10). Bottom: simulations of the model with the full nonlinear dynamics for \(I_{\text{h}}\). All parameters as in Fig. 5
If we introduce the vector \(X=(V,n_{\text{h}}) \in \mathbb{R}^{2}\) then we may write the reduced model in a more abstract setting, namely in terms of the pwl evolution equation that governs the system behaviour between one spiking event and the next:
$$ \frac{\partial}{\partial t} X(x,t) = A X(x,t) + \Psi(x,t) , \quad T^{m-1}(x) \leq t < T^{m}(x). $$
In (11), A is a \(2 \times2\) matrix that is defined in a piecewise constant fashion, with dependence on the value of the voltage (in particular, which of the three domains detailed in equation (10) pertains), or whether the system is in the refractory state. In the latter case, A is defined according to
$$ A= A_{\text{R}} = \begin{bmatrix} 0 & 0\\ 0 & -1/\tau_{\text{h}} \end{bmatrix} , \quad T^{m-1}(x) \leq t < T^{m-1}(x)+\tau_{\text{R}}, $$
while for \(T^{m-1}(x) + \tau_{\text{R}} \leq t < T^{m}(x)\)
$$ A= \textstyle\begin{cases} A_{0} = \Bigl[{\scriptsize\begin{matrix}{} -1/\tau& g_{\text{h}} g_{\text{l}}^{-1}/\tau\cr -1/(4 k \tau_{\text{h}}) & -1/\tau_{\text{h}} \end{matrix}}\Bigr], & V_{-} < V < V_{+}, \\ A_{-} = \Bigl[{\scriptsize\begin{matrix}{} -1/\tau& g_{\text{h}} g_{\text{l}}^{-1}/\tau\cr 0 & -1/\tau_{\text{h}} \end{matrix}}\Bigr] , & V \leq V_{-}, \\ A_{+} = A_{-}, & V \geq V_{+}, \end{cases} $$
where we have assumed the ordering \(V_{-} < V_{\text{r}} < V_{+} < V_{\text{th}}\), introduced \(\tau=C/g_{\text{l}}\), and absorbed the factor \(V_{\text{h}}\) within \(g_{\text{h}}\). Similarly we define Ψ according to
$$ \Psi= \Psi_{\text{R}} = \begin{bmatrix} 0 \\ (1/2-(V_{\text{r}}-V_{1/2})/(4k))/\tau_{\text{h}} \end{bmatrix} , \quad T^{m-1}(x) \leq t < T^{m-1}(x)+ \tau_{\text{R}}, $$
and, for \(T^{m-1}(x) + \tau_{\text{R}} \leq t < T^{m}(x)\),
$$ \Psi= \frac{g_{\text{syn}} g_{\text{l}}^{-1}}{\tau} \psi \begin{bmatrix} 1\\ 0 \end{bmatrix} + \textstyle\begin{cases} b_{0} , & V_{-} < V < V_{+}, \\ b_{-} , & V \leq V_{-}, \\ b_{+} , & V \geq V_{+}, \end{cases} $$
$$ b_{0} = \begin{bmatrix} 0\\ (1/2+V_{1/2}/(4k))/\tau_{\text{h}} \end{bmatrix} , \qquad b_{-}= \begin{bmatrix} 0\\ 1/\tau_{\text{h}} \end{bmatrix} , \qquad b_{+}= \begin{bmatrix} 0\\ 0 \end{bmatrix} . $$
We highlight that in (12)–(16) we have introduced the subscripts \(\{R,0,-,+ \}\) to indicate the state of the system, namely whether it is refractory (labelled by R) or is not refractory and has a voltage in the range \((V_{-},V_{+})\) (labelled by 0), \((-\infty,V_{-}]\) (labelled by −), \([V_{+},\infty)\) (labelled by +).
Travelling Wave Analysis
We now seek travelling wave solutions of (11) of the form \(\widehat{X}(\xi,t)\), where \(\xi=t-x/c\) and c is the (constant) wave speed. In this case (11) transforms to
$$ \biggl(\frac{\partial}{\partial t} + \frac{\partial}{\partial\xi} \biggr) \widehat{X}(\xi,t) = A \widehat{X}(\xi,t) + \widehat{\Psi }(\xi,t). $$
A stationary travelling wave \(\widehat{X}(\xi,t) = Q(\xi) = (V(\xi ),n_{\text{h}}(\xi))\) satisfies the travelling wave equation
$$ \frac{\mathrm {d}Q}{\mathrm {d}\xi} = A Q(\xi) + \widehat{\Psi}(\xi). $$
In terms of firing events a periodic wave is described by \(T^{m}(x) = x/c + m \varDelta\), where Δ is the period of the wave such that \(Q(\xi+\varDelta) = Q(\xi)\). Substitution of this firing ansatz into (9) allows us to determine the function \(\widehat{\psi}(\xi ) = \psi(x,t) \vert _{t=T^{m}(x)}\) where \(\widehat{\Psi}(\xi )\) is obtained from (15) under the replacement of ψ by ψ̂. The function ψ̂ is easily determined as
$$ \widehat{\psi}(\xi) = c \sum_{m \in \mathbb{Z}} \int_{0}^{\infty} \mathrm {d}s \eta(s) w\bigl(\big|c(s-\xi) + c m \varDelta\big|\bigr) , $$
and is Δ-periodic, and can therefore be expressed in terms of a Fourier series as
$$ \widehat{\psi}(\xi) = \sum_{p \in \mathbb{Z}} \psi_{p} \mathrm {e}^{-2 \pi i p \xi/\varDelta}, \quad\psi_{p} = \frac{1}{\varDelta} \widetilde{w} \biggl(\frac{2 \pi p}{c \varDelta} \biggr) \widetilde{\eta} \biggl(-\frac{2 \pi p}{\varDelta} \biggr) . $$
In (20) tildes denote the Fourier integral representation, such that for a given function \(a(x)\)
$$ a(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \mathrm {d}k ~ \widetilde {a}(k) \mathrm {e}^{ikx}, \quad \widetilde{a}(k) = \int_{-\infty}^{\infty} \mathrm {d}x ~ a(x) \mathrm {e}^{-ikx} , $$
and we have made use of the result that \(c \varDelta\sum_{m} \mathrm {e}^{ikcm \varDelta} = 2 \pi\sum_{p} \delta(k-2 \pi p/(c \varDelta))\).
For the bump function (7) and the α-function (8) we have
$$ \widetilde{w}(k) = w_{0} \frac{\pi}{\beta} \frac{\sin k \sigma }{\sinh(\pi k /(2 \beta))} , \qquad \widetilde{\eta}(k) = \frac{\alpha^{2}}{(\alpha+ik)^{2}} . $$
Thus, given the decay properties of (22) as a function of k, the sum in (20) can be naturally truncated.
The formal solution to (18) can be constructed using variation of parameters as
$$ Q(\xi) = G(\xi,\xi_{0}) Q(\xi_{0}) + \int_{\xi_{0}}^{\xi}G\bigl(\xi,\xi'\bigr) \widehat{\Psi}\bigl(\xi'\bigr) \,\mathrm {d}\xi', $$
where G is a matrix exponential given by
$$ G\bigl(\xi,\xi'\bigr) = \mathcal{T} \biggl\{ \exp \biggl( \int_{\xi'}^{\xi} \mathrm {d}s A(s) \biggr) \biggr\} . $$
Here \(\mathcal{T}\) is a time-ordering operator \(\mathcal {T} A(t) A(s) = H(t -s)A(t) A(s) + H(s-t) A(s)A(t)\). In general the issue of time-ordering makes it very difficult to evaluate G. However, in our case A is piecewise constant and so we easily may break the solution up into parts distinguished by the label \(\mu\in\{R, 0, -,+\}\). In each case trajectories are given explicitly by (23) with \(G(\xi,\xi')=G(\xi-\xi')\) and \(G=G_{\mu}\) where \(G_{\mu}(\xi) = \exp(A_{\mu}\xi)\). A global trajectory may then be obtained by patching together solutions, denoted by \(Q_{\mu}\), from each domain. It is in this fashion that we now construct the shape of a periodic travelling wave in a self-consistent manner. Of use will be matrix exponential decomposition \(\mathrm {e}^{A t} = P \mathrm {e}^{\Lambda t} P^{-1}\), where \(\Lambda= \operatorname{diag} (\lambda^{+},\lambda^{-})\) are the eigenvalues of A with associated eigenvectors \(q^{\pm}= (1, (\lambda^{\pm}- A_{11})/A_{12}, )^{T}\). Here the eigenvalues of A are given explicitly by
$$ \lambda^{\pm}= \frac{1}{2} \bigl( \operatorname{Tr}A \pm\sqrt {( \operatorname{Tr}A)^{2} - 4 \det A} \bigr) . $$
Using (20) we may then write a domain specific trajectory for \(\mu\in\{0,+,-\}\) in the form
$$\begin{aligned}[b] Q_{\mu}(\xi) ={}& G_{\mu}(\xi-\xi_{0}) Q_{\mu}(\xi_{0}) +A_{\mu}^{-1} \bigl[G_{\mu}(\xi-\xi_{0}) -I_{2} \bigr] b_{\mu}\\ & + \frac{g_{\text{syn}} g_{\text{l}}^{-1}}{\tau} \sum_{p \in \mathbb{Z}} \psi_{p} P_{\mu}\operatorname{diag} \bigl( Z_{\mu}^{+}(\xi,\xi_{0}), Z_{\mu}^{-}(\xi, \xi_{0})\bigr) P_{\mu}^{-1} \begin{bmatrix} 1\\ 0 \end{bmatrix} ,\end{aligned} $$
$$ Z_{\mu}^{\pm}(\xi,\xi_{0}) = \frac{\mathrm {e}^{\lambda_{\mu}^{\pm}(\xi-\xi_{0})} \mathrm {e}^{-2 \pi i p \xi_{0}/\varDelta} - \mathrm {e}^{-2 \pi i p \xi/\varDelta }}{\lambda_{\mu}^{\pm}+2 \pi i p/\varDelta} . $$
Here \(\Lambda_{\mu}= \operatorname{diag} (\lambda^{+}_{\mu},\lambda ^{-}_{\mu})\) with \(\lambda^{\pm}_{\mu}\) representing the eigenvalues of \(A_{\mu}\) and
$$ P_{\mu}= \begin{bmatrix} 1 & 1 \\ (\lambda^{+}_{\mu}- [A_{\mu}]_{11})/[A_{\mu}]_{12} & (\lambda^{-}_{\mu}- [A_{\mu}]_{11})/[A_{\mu}]_{12} \end{bmatrix} . $$
When \(\mu=R\) we adopt an alternative strategy (since \(A_{R}\) is singular), remembering that when the system is refractory then \(V(\xi )\) is clamped at the value \(V=V_{\text{r}}\). In this case, we only have to consider the evolution of the gating variable \(n_{\text{h}}\), which is obtained from (3) and (10) to give
$$ n_{\text{h}}(\xi) = n_{\text{h}}(\xi_{0}) \mathrm {e}^{-(\xi-\xi_{0})/\tau_{\text{h}}} + \bigl(1/2-(V_{\text{r}}-V_{1/2})/(4k)\bigr) \bigl[1-\mathrm {e}^{-(\xi-\xi_{0})/\tau _{\text{h}}}\bigr] . $$
Now let us consider the form of a periodic wave which elicits a single spike for every period, much like the ones seen in Fig. 5. An example of such a travelling wave orbit in the \((V,n_{\text{h}})\) phase plane is shown in Fig. 7. The corresponding evolution of \(V(\xi)\) and \(n_{\text{h}}(\xi)\) is shown in Fig. 8.
Orbit of a periodic travelling wave. Parameters as in Fig. 5, with \(\varDelta= 450\), \(c = 0.0669\), \(n_{h}(0) = 0.3815\) and \(\xi_{1} = 225.4223\). The travelling wave starts at \(\xi=0\) where \(V=V_{r}\) is clamped and \(n_{h}\) evolves according to (29) until \(\xi= \tau_{\mathrm{R}}\) and the system is released from the refractory period (blue line). Then it evolves clockwise according to (26) with \(\mu= 0\) until \(V(\tau_{\mathrm{R}} + \xi_{1})=V_{+}\) when it switches (red line) to \(\mu=+\) (green line). The orbit ends when \(V=V_{\text{th}}\) and V is reset. The green horizontal line for \(V > V_{\text{th}}\) is not part of the solution. It is simply a marker for the spiking event (and the model does not generate an explicit shape for an action potential). Black dotted lines represent (from left to right) \(V=V_{-}\), \(V=V_{+}\) and \(V=V_{\text{th}}\)
Profile of the two components of the periodic travelling wave \(Q(\xi)\) defined by (26) and illustrated in Fig. 7. Solid lines correspond to \(V(\xi)\) (left-hand axis) and dotted ones to \(n_{h}(\xi)\) (right-hand axis). Colour-code and parameters as in Fig. 7. Dotted black lines indicate the values where the system changes dynamics
Given the translational invariance of the system we are free to choose a travelling wave origin such that \(\xi=0\) corresponds to the system immediately after firing. For a duration \(\tau_{\text{R}}\) it will then remain clamped at \(V_{\text{r}}\) with \(n_{\text{h}}\) evolving according to (29) with \(\xi_{0}=0\) (as shown in blue in Fig. 7 and Fig. 8). From here it then evolves according to (26) with \(\mu=0\), with initial data determined by \(Q_{R}(\xi_{0})=(V_{\text{r}},n_{\text{h}}(\tau_{\text{R}}))\), until \(V(\xi)\) reaches \(V_{\pm}\), after which we set \(\mu=-\) or \(\mu =+\) in (26) and select appropriate initial data for (26), depending on the value of V achieved first. For simplicity, and since this is reliably observed in numerical simulations for a wide range of parameters (red line in Fig. 7 and Fig. 8, though the argument is easily generalised), we assume \(V_{+}\) is the relevant choice. The final piece of the orbit is then obtained from (26) with \(\mu=+\) and initial data determined by \(Q_{+}(\xi_{0})=Q_{0}(\xi_{1}+\tau _{\text{R}})\) (green line in Fig. 7 and Fig. 8) and evolving the system until \(V(\xi)=V_{\text{th}}\). Denoting the time of flight for the trajectory such that \(V_{\text{r}} \leq V < V_{+}\) by \(\xi_{1}\), and that, for \(V_{+} \leq V < V_{\text{th}}\) by \(\xi_{2}\), the period of the orbit is given by \(\varDelta=\tau_{\text{R}} + \xi_{1} + \xi_{2}\). We note that the orbit is discontinuous because of the reset of the voltage variable after one period. Thus we have four unknowns \((n_{\text{h}}(0),c,\xi_{1},\varDelta)\) related by three nonlinear algebraic equations
$$ \textstyle\begin{cases} V(\varDelta)=V_{\text{th}} &\mbox{(firing condition),}\\ n_{\text{h}}(\varDelta)=n_{\text{h}}(0) &\mbox{(periodicity condition),}\\ V(\tau_{\text{R}}+\xi_{1})=V_{+} &\mbox{(switching condition),} \end{cases} $$
whose simultaneous solution determines the dispersion relationship for the wave speed as a function of the period \(c=c(\varDelta)\). An example of a dispersion curve constructed in this way is shown in Fig. 9. Here we see that a wide range of allowed wavelengths can co-exist (with differing speeds). Note that in Fig. 9 we also include solutions that visit the region of phase-space where \(V < V_{-}\), and these solutions typically only occur for small values of Δ. Our constructive theory does not provide a wave selection principle; however, by varying initial data in direct numerical simulations we were able to find solutions in excellent agreement with the theoretical predictions up to some maximal value of Δ. The determination of this value is the subject of the next section, where we show how to analyse wave stability.
Dispersion curve \(c=c(\varDelta)\) for a periodic travelling wave. Here c is the speed of a wave with period Δ. For small periods and on the upper branch waves are constructed that visit the domain where \(V< V_{-}\). Parameters as in Fig. 5. Solid lines represent periods where the system is stable while dashed lines represent where it is unstable; red dots represent the maximum stable period of the orbit, highlighting its increase with \(\tau_{h}\). Note that waves (on the lower branch of solutions) are stable for a large range of wave periods, and that the actual value of \((c,\varDelta)\) that would be observed in a simulation are dependent upon the choice of initial data
Wave Stability
To determine the stability of a periodic travelling wave we must not only treat perturbations of the state variables, but also the associated effects on the times of firing. Moreover, one must remember that because the model is nonsmooth (due to the switch at \(V=V_{\pm}\)) and discontinuous (because of reset whenever \(V=V_{\text{th}}\)) standard approaches for analysing smooth dynamical systems cannot be immediately applied. Nonetheless, as we show below, the wave stability can in fact be explicitly determined. We do this by constructing the so-called Evans function. This has a long history of use in the analysis of wave solutions to partial differential equations, dating back to the work of Evans on the stability of action potentials in the Hodgkin–Huxley model of a nerve fibre [39], has been extended to certain classes of firing rate neural field model [40], and is developed here for spiking neural fields.
We begin our analysis by exposing the spike-train that determines the synaptic drive in (9) by writing it in the equivalent form
$$ \psi(x,t) = \sum_{m \in \mathbb{Z}} \int_{-\infty}^{\infty} \mathrm {d}y w\bigl(|x-y|\bigr) \int_{-\infty}^{t} \mathrm {d}s \eta(t-s)\delta \bigl(s-T^{m}(y)\bigr), $$
where the firing times are defined according to the threshold condition \(V(x,T^{m}(x)) = V_{\text{th}}\). We may relate spike times to voltage threshold conditions using the result that for fixed x
$$ \delta\bigl(t-T^{m}(x)\bigr) = \big|V_{t}\bigl(x,T^{m}(x) \bigr)\big| \delta\bigl(V(x,t)-V_{\text{th}}\bigr) , $$
and \(V_{t}\) denotes partial differentiation of V with respect to t. Hence
$$\begin{aligned}[b] \psi(x,t) ={}& \sum_{m \in \mathbb{Z}} \int_{-\infty}^{\infty} \mathrm {d}y w\bigl(|x-y|\bigr)\\ &\times\int_{-\infty}^{t} \mathrm {d}s \eta(t-s) \big|V_{t} \bigl(y,T^{m}(y)\bigr)\big| \delta\bigl(V(y,s)-V_{\text{th}}\bigr) . \end{aligned} $$
Consider again travelling wave solutions of the form \(V(x,t)=\widehat {V}(\xi,t)\), where \(\xi=t-x/c\). In this co-moving frame we can define a set of firing event functions \(\xi^{m}(t)\) according to the threshold condition \(\widehat{V}(\xi^{m}(t),t)=V_{\text{th}}\). These event times can be related to the co-moving voltage threshold condition using the result that, for fixed t,
$$ \delta\bigl(\xi-\xi^{m}(t)\bigr) = \big|\widehat{V}_{\xi}\bigl( \xi^{m}(t),t\bigr)\big| \delta \bigl(\widehat{V}(\xi,t)-V_{\text{th}} \bigr) . $$
Substitution into (33) and using \(\widehat{V}_{\xi}\simeq V_{t}\) close to a periodic orbit we find
$$\begin{aligned}[b] \psi(x,t) &= \sum_{m \in \mathbb{Z}} \int_{-\infty}^{\infty} \mathrm {d}y w\bigl(|x-y|\bigr) \int_{-\infty}^{t} \mathrm {d}s \eta(t-s) \delta\bigl(s-y/c- \xi^{m}(s)\bigr) \\ &= c \sum_{m \in \mathbb{Z}} \int_{0}^{\infty} \mathrm {d}s \eta(s) w\bigl(\big|c(s-\xi) + c \xi^{m} (t-s)\big|\bigr) \equiv\widehat{\psi}(\xi,t). \end{aligned} $$
Noting that, for a periodic wave \(\xi^{m}(t) = m \varDelta\), ψ̂ is independent of t and Eq. (35) recovers Eq. (19) as expected.
We now analyse the stability of such a periodic wave by perturbations such that \(\xi^{m}(t) = m \varDelta+ \delta\xi^{m}(t)\), with \(|\delta\xi ^{m}(t)| \ll1\). Writing the corresponding perturbation of \(\widehat {\psi}(\xi,t)\) as \(\widehat{\psi}(\xi,t) = \widehat{\psi}(\xi) + \delta\widehat{\psi}(\xi,t)\) we find
$$ \delta\widehat{\psi}(\xi,t) = c^{2} \sum_{m \in \mathbb{Z}} \int _{0}^{\infty} \mathrm {d}s \eta(s) w' \bigl(\big|c(s-\xi) + c m \varDelta\big|\bigr) \delta\xi^{m} (t-s) . $$
It remains to determine the relationship between \(\delta\xi^{m}(t)\) and the perturbations of the shape of the travelling wave. In Appendix B we show that we can relate \(\delta\xi^{m}(t)\) to the deviation in the voltage, denoted by \(\delta\widehat{V}(m \varDelta,t)\), via the simple relationship
$$ \delta\xi^{m}(t) = - \frac{\delta\widehat{V}(m \varDelta,t)}{V_{\xi}(m \varDelta_{-})} . $$
Thus for solutions of the form \(\delta\widehat{V}(\xi,t) = \delta \widehat{V}(\xi) \mathrm {e}^{\lambda t}\), \(\delta\widehat{V}(\xi) = \delta\widehat{V}(\xi+\varDelta)\) we find \(\delta\widehat{\psi}(\xi,t) = \delta\widehat{\psi}(\xi;\lambda ) \mathrm {e}^{\lambda t}\), with \(\delta\widehat{\psi}(\xi;\lambda) =\delta\widehat{V}(0) f(\xi ;\lambda)\), and
$$ f(\xi;\lambda) = -\frac{c^{2}}{V_{\xi}(0)} \sum_{m \in \mathbb{Z}} \int _{0}^{\infty} \mathrm {d}s \eta(s) w' \bigl(\big|c(s-\xi)+cm\varDelta\big|\bigr) \mathrm {e}^{-\lambda s}. $$
Returning to the more abstract setting given by Eq. (11) we linearise around the travelling wave by setting \(\widehat{X}(\xi,t) = Q(\xi) + \delta\widehat{X}(\xi) \mathrm {e}^{\lambda t}\), with \(\delta \widehat{X}(\xi)=\delta\widehat{X}(\xi+\varDelta)\). This yields the variational equation
$$ \frac{\mathrm {d}}{\mathrm {d}\xi} \delta\widehat{X}(\xi) = A(\xi;\lambda )\delta\widehat{X}( \xi) + \delta\widehat{\Psi}(\xi;\lambda), \quad\delta\widehat{\Psi}(\xi;\lambda) = \frac{g_{\text{syn}} g_{\text{l}}^{-1}}{\tau} \delta\widehat{\psi}(\xi;\lambda) \begin{bmatrix} 1 \\0 \end{bmatrix} , $$
where \(A(\xi;\lambda) = A(Q(\xi)) -\lambda I_{2}\) (and we use the argument of A to emphasise that it depends on position along the periodic orbit). We may write the solution to (39) in much the same way as for the periodic orbit problem given by (18), namely with the use of a variation of parameters formula, matrix exponentials and (38):
$$ \delta\widehat{X}(\xi) = \textstyle\begin{cases} G_{\text{R}}(\xi;\lambda) \delta\widehat{X}(0) & 0 \leq\xi< \tau _{\text{R,}} \\ G_{0}(\xi-\tau_{\text{R}};\lambda) \delta\widehat{X}(\tau_{\text{R}}) \\\quad{}+ \int_{\tau_{\text{R}}}^{\xi} \mathrm {d}\xi' G_{0}(\xi-\xi';\lambda) J f(\xi';\lambda) \delta\widehat{X}(0) & \tau_{\text{R}} \leq\xi< \tau_{\text{R}} +\xi_{1}, \\ G_{+}(\xi-(\tau_{\text{R}} +\xi_{1});\lambda) \delta\widehat{X}(\tau _{\text{R}} +\xi_{1}) \\ \quad{}+\int_{\tau_{\text{R}} +\xi_{1}}^{\xi} \mathrm {d}\xi' G_{+}(\xi-\xi';\lambda) J f(\xi';\lambda) \delta\widehat{X}(0) & \tau_{\text{R}} +\xi_{1} \leq\xi< \varDelta. \end{cases} $$
Here \(G_{\mu}(\xi;\lambda)=\exp([A_{\mu}- \lambda I_{2} ] \xi)\) and
$$ J= \frac{g_{\text{syn}} g_{\text{l}}^{-1}}{\tau} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} . $$
However, the evolution of perturbations through the switching manifolds \(V=V_{\pm}\), the firing threshold \(V=V_{\text{th}}\) and the release from the refractory state requires care, since in all these cases there is a jump in the Jacobian. The theory of nonsmooth dynamical systems gives a prescription for handling this using so-called saltation matrices dating back to the work of Aizerman and Gantmakher in the 1950s [41]. For a more recent perspective we recommend the paper by Leine et al. [42] and the book by di Bernardo et al. [43], particularly in engineering applications, and the paper by Coombes et al. [44] for applications in neuroscience. The \(2 \times2\) saltation matrices for handling switching, firing, and refractoriness are constructed in Appendix C and denoted \(K_{\text{switch}}\), \(K_{\text{fire}}\), and \(K_{\text{ref}}\), respectively. In essence they map perturbations through the region of nonsmooth behaviour to give \(\delta\widehat{X}(0_{+}) = K_{\text{fire}} \delta \widehat{X}(0)\), \(\delta\widehat{X}({\tau_{\text{R}}}_{+}) = \delta \widehat{X}({\tau_{\text{R}}})+ K_{\text{ref}} \delta\widehat {X}(0)\), and \(\delta\widehat{X}((\tau_{\text{R}}+\xi_{1})_{+}) = K_{\text{switch}} \delta\widehat{X}(\tau_{\text{R}}+\xi_{1})\). The saltation matrices are given explicitly by \(K_{\text{switch}} = I_{2}\) and
$$ \begin{aligned} K_{\text{fire}} &= \begin{bmatrix} 0 & 0 \\ (n_{\text{h},\xi}(0_{+})-n_{\text{h},\xi }(0_{-}))/{V_{\xi}(0_{-})} & 1 \end{bmatrix} , \\ K_{\text{ref}}& = \begin{bmatrix} {V_{\xi}({\tau_{\text{R}}}_{+})}/{V_{\xi}(0_{-})} & 0 \\ 0 & 0 \end{bmatrix} .\end{aligned} $$
If we now introduce the function \(\mathcal{F}_{\mu}(\xi,\xi_{0};\lambda)\):
$$ \mathcal{F}_{\mu}(\xi,\xi_{0};\lambda) = \int_{\xi_{0}}^{\xi} \mathrm {d}\xi' G_{\mu}\bigl(\xi-\xi';\lambda\bigr) J f\bigl(\xi';\lambda\bigr) , \quad\mu\in\{0, +,-\}, $$
then Eq. (40) may be used to generate the perturbation after one period as \(\delta\widehat{X}(\varDelta) = \varGamma(\lambda,\varDelta) \delta\widehat{X}(0)\), where
$$\begin{aligned} \varGamma(\lambda, \varDelta) ={}& \mathcal{F}_{+} (\varDelta, \tau_{\text{R}}+ \xi_{1};\lambda) \\ &+ G_{+}\bigl(\varDelta-(\tau_{\text{R}}+\xi_{1});\lambda\bigr) K_{\text{switch}} \bigl[ \mathcal{F}_{0} (\tau_{\text{R}}+ \xi_{1},\tau_{\text{R}};\lambda) \\ & + G_{0}(\xi_{1};\lambda) \bigl[G_{\text{R}} (\tau_{\text{R}};\lambda) K_{\text{fire}} + K_{\text{ref}}\bigr] \bigr] .\end{aligned} $$
Enforcing that perturbations be Δ-periodic (i.e. \(\delta\widehat{X}(\varDelta) = \delta\widehat{X}(0)\)) we obtain the spectral condition \(\mathcal{E}(\lambda, \varDelta) = 0\) where
$$ \mathcal{E}(\lambda, \varDelta) = \big|\varGamma(\lambda, \varDelta) -I_{2}\big| . $$
We identify (45) as the Evans function for the periodic wave. To determine (43) in a computationally useful form we use a Fourier representation to represent (38) (cf. (20) from (19)) as \(f(\xi ;\lambda) = \sum_{p \in \mathbb{Z}} f_{p}(\lambda) \exp(- 2 \pi i p \xi /\varDelta)\) where
$$ f_{p}(\lambda) = -\frac{1}{V_{\xi}(0)} \frac{2 \pi}{\varDelta^{2}} ip \widetilde{\eta}(-i \lambda- 2 \pi p/\varDelta) \widetilde{w}\bigl(2 \pi p /(c \varDelta)\bigr) , $$
for \(\operatorname{Re} (\lambda+\alpha) >0\). Then in a similar way to the construction of (26) we find the useful representation for (43) as
$$ \mathcal{F}_{\mu}(\xi,\xi_{0};\lambda) = \sum _{p} f_{p}(\lambda) P_{\mu}\operatorname{diag} \bigl( S_{\mu}^{+}(\xi,\xi_{0};\lambda), S_{\mu}^{-}(\xi ,\xi_{0};\lambda)\bigr) P_{\mu}^{-1} J , $$
$$ S_{\mu}^{\pm}(\xi,\xi_{0};\lambda) = \frac{\mathrm {e}^{(\lambda_{\mu}^{\pm}-\lambda)(\xi-\xi_{0})} \mathrm {e}^{-2 \pi i p \xi_{0}/\varDelta} - \mathrm {e}^{-2 \pi i p \xi/\varDelta}}{\lambda_{\mu}^{\pm}-\lambda+2 \pi i p/\varDelta} . $$
The eigenvalues of the spectral problem can be practically constructed by considering the decomposition \(\lambda= \nu+ i \omega\) and simultaneously solving the pair of equations \(\mathcal{G}(\nu, \omega )=0\) and \(\mathcal{H}(\nu, \omega)=0\), where \(\mathcal{G}(\nu, \omega)=\operatorname{Re} \mathcal{E}(\nu+ i \omega, \varDelta)\) and \(\mathcal{H}(\nu, \omega)=\operatorname{Im} \mathcal{E}(\nu+ i \omega, \varDelta)\), subject to the constraint \(\nu+\alpha>0\). Figures 10 and 11 show the zero level sets of \(\mathcal {G}\) and \(\mathcal{H}\) in the \((\nu,\omega)\) plane for two different points on the dispersion curve of Fig. 9. The intercepts when \(\nu+\alpha>0\) provide the zeros of the Evans function and here highlight clearly that, for Δ sufficiently large, the zeros of the Evans function can cross to the right-hand complex plane signalling a wave instability.
Zeros of the Evans function (45) with \(\varDelta= 460\). These occur at the intersection (green dots) of \(\mathcal{G}(\nu, \omega)=0\) (red curve) and \(\mathcal{H}(\nu, \omega)=0\) (blue curve) where \(\mathcal{G}\) is the real part of \({\mathcal{E}}\) whereas \(\mathcal {H}\) is the imaginary part. Here we can see that all the eigenvalues, except the zero eigenvalue, have negative real part, so that the periodic wave is stable. Parameters as in Fig. 5
Zeros of the Evans function (45) with \(\varDelta= 470\). These occur at the intersection (green dots) of \(\mathcal{G}(\nu, \omega)=0\) (red curve) and \(\mathcal{H}(\nu, \omega)=0\) (blue curve) where \(\mathcal{G}\) is the real part of \({\mathcal{E}}\) whereas \(\mathcal{H}\) is the imaginary part. Here we see a complex conjugate pair of eigenvalues with positive real part, so that the periodic wave is unstable. Parameters as in Fig. 5
Figure 10 also shows that there is always a zero eigenvalue. It is simple to establish the persistence of this zero under parameter variation. Differentiating (18) with respect to ξ gives
$$ \frac{\mathrm {d}}{\mathrm {d}\xi} \frac{\mathrm {d}Q}{\mathrm {d}\xi} = A \frac{\mathrm {d}Q}{\mathrm {d}\xi} + \frac{\mathrm {d}}{\mathrm {d}\xi} \widehat{\Psi}(\xi). $$
From (38) and (19) we may establish that
$$ \delta\widehat{\psi}(\xi;0) = \frac{\delta\widehat{V}(0)}{V_{\xi}(0)} \frac{\mathrm {d}}{\mathrm {d}\xi} \widehat{ \psi}(\xi) . $$
Hence for \(\lambda=0\) we see that a solution to (39) is the eigenfunction
$$ \delta\widehat{X}(\xi) = \frac{\mathrm {d}Q}{\mathrm {d}\xi} , $$
as expected from translation invariance of the system (so that a perturbation tangential to the travelling wave orbit is neutrally stable).
This paper is motivated by recent work in computational neuroscience that has highlighted rebound firing as a mechanism for wave generation in spiking neural networks that can underlie the formation of grid cell firing fields in MEC [1]. We have presented a simple spiking model with inhibitory synaptic connections and an \(I_{\text{h}}\) current that can generate smoothly propagating activity waves via post-inhibitory rebound. These are qualitatively of the type observed in previous computational studies [36, 37], yet are amenable to an explicit mathematical analysis. This is possible because we have chosen to work with piecewise linear discontinuous models, and exploited methodologies from the theory of nonsmooth systems. In particular we have exploited the linearity of our model between events (for firing, switching, and release from a refractory state) to construct periodic solutions in a travelling wave frame. To assess the stability of these orbits we have treated the propagation of perturbations through event manifolds using saltation operators. Using this we have constructed dispersion curves showing a wide range of stable periods, with a maximum period controlled by the time-scale of the rebound current. This gives further credence to the idea that the change in grid field scale along the dorsal-ventral axis of the MEC is under local control by HCN channels.
A number of natural extensions of the work in this paper suggest themselves. We briefly outline them here, and in no particular order. For simplicity we have focussed on the analysis of waves in a homogeneous model with only one spatial dimension. The analysis of the corresponding travelling waves, with hexagonal structure, in two spatial dimensions is more challenging, though is an important requirement for a complete model of grid cell firing. Moreover, the assumption of homogeneity should be relaxed. In this regard it would be of interest to understand the effects of a heterogeneity in the time constants (for voltage response, synaptic time-scale, and the time-scale of the \(I_{\text{h}}\) current) on the properties of spatio-temporal patterns. Furthermore, it would be biologically more realistic to consider two sets of inhibitory interneurons, as in [1, 28, 30]. As well as depending on the \(I_{\text{h}}\) current, grid field spacing changes as a function of behavioural context. This is believed to occur through the activation of neuromodulators [32], and simple regulation of our \(I_{\text{h}}\) current model would allow a systematic study of this, even before considering the more important issue of structured input. The work in this paper has focussed on spontaneous pattern formation that occurs in the absence of such input. Given the importance of the head-direction input for driving grid cell firing fields it would be natural to consider a mathematical analysis for the case with \(I_{\text{hd}} \neq0\). For the standard Fuhs–Touretzky mechanism of inducing firing patterns this would further require the treatment of an anisotropic interaction kernel with a dependence on a head-direction preference map. One way to address this would be via a perturbation theory around the limiting case treated in this paper, and use this to calculate a tissue response parametrised by the animal's speed and direction of motion. The same methodology would also allow an investigation of phase precession during a grid field traversal. Our simulations have also shown the possibility of 'lurching waves' and it would be interesting, at least from a mathematical perspective, to analyse their properties (speed and stability) and compare them to the co-existing smoothly propagating waves. It would further be pertinent in this case to pay closer attention to any possible nonsmooth bifurcations that could give rise to wave instabilities, such as grazing bifurcations. Finally we note that grid cells are grouped in discrete modules with common grid spacing and orientation [20]. It has recently been suggested that coupling between modules or via feedback loops to the hippocampus may help to suppress noise and underpin a robust code (with a large capacity) for the representation of position [45]. Another extension of the work in this paper would thus be to consider the dynamics of networks of interacting modules, paying closer attention to the details of MEC microcircuitry [46].
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SC was supported by the European Commission through the FP7 Marie Curie Initial Training Network 289146, NETT: Neural Engineering Transformative Technologies. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Tesla K40 GPU used for this research. KCAW was generously supported by the Wellcome Trust Institutional Strategic Support Award (WT105618MA). MBQ gratefully acknowledge the support of CONACyT and The University of Nottingham during her PhD studies. We thank the anonymous referees for their helpful comments on our manuscript.
Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD, Nottingham, UK
Mayte Bonilla-Quintana, Reuben D. O'Dea & Stephen Coombes
Centre for Biomedical Modelling and Analysis, University of Exeter, Living Systems Institute, Stocker Road, EX4 4QD, Exeter, UK
Kyle C. A. Wedgwood
Mayte Bonilla-Quintana
Reuben D. O'Dea
Stephen Coombes
Correspondence to Mayte Bonilla-Quintana.
MBQ, KCAW, RDO and SC contributed equally. All authors read and approved the final manuscript.
Code for 2D simulations. Code written in C++/CUDA programming language. Equipment requirements: Nvidia CUDA drivers and Nvidia graphics card with compute capability ≥3.5. The Armadillo C++ package must be installed. Run commands: $ cmake . $ make $./SpatialNavigationCUDA. (zip)
13408_2017_51_MOESM2_ESM.m4v
Movie for 2D patterned state evolution. A movie of the voltage component in a planar spiking neural field model with an \(I_{\mathrm{h}}\) current. Here we see the emergence of a rich patterned state. Parameters as in Fig. 2 and the system is initialised with a localised region of hyperpolarisation. Activity spreads from this initial state generating spatio-temporal patterns that interact in a complicated fashion to generate hexagonal structures. (m4v)
Appendix A: Numerical Scheme
Here, we describe the numerical scheme that we have developed to evolve the spiking neural field model. Given the large computational overheads for simulating the latter we have focussed on implementing the algorithm on a GPU system. In essence, we exploit the piecewise linear nature of the dynamics to obtain trajectories in closed form and then use a root-finding scheme to find the timings of events (switching, firing or refractory). The dynamics is updated at an event (generating synaptic currents and resetting at firing events, changing the gating dynamics at switching events, and releasing from reset at refractory events), and the process repeated. For clarity we describe our approach for two spatial dimensions as it is easily adapted to treat just one spatial dimension.
A.1 Algorithm
We solve the system over a 2D discretised grid of size \(L \times L\) with N (numerical) cells in each spatial dimension and enforce periodic boundary conditions. Cells are equi-spaced with a spatial separation of \(\varDelta x = 2L/N\) and the state of the cells in the network are evaluated over \(T+1\) time steps, discretising time as \(t_{i} = i \Delta t\), \(i=0,\dots,T\). To evaluate these states, we analytically integrate (11) to provide a closed-form expression for the trajectory of the system. Our closed-form expression allows us, for cell j, to write its state at time \(t_{i+1}\) as a function of its state at time \(t_{i}\):
$$X_{j}(t_{i+1}) = \varphi_{\Delta t} \bigl(t_{i},X_{j}(t_{i}) \bigr), $$
where \(\varphi_{h}\) is the evolution operator acting over a time h. Note that this expression assumes that no firing events have occurred between \(t_{i}\) and \(t_{i+1}\), but can account for switching events, as detailed below. We shall describe later how the algorithm handles firing events.
For convenience, we shall divide the state space of an individual cell into three regions, based on the instantaneous value of V: Region I, with \(V\leq V_{-}\); Region II, with \(V_{-}< V\leq V_{+}\); and Region III, with \(V>V_{+}\). In addition, we define Region IV to be that where the cell is in the refractory state (recall that a cell is in the refractory state at time t if \(T^{m-1}\geq t+\tau_{\mathrm{R}}\), where \(T^{m-1}< t\) is the last spiking time of that cell). In each of these regions, the equation governing the dynamics for the gating variable \(n_{\mathrm{h}}\) is different, and this needs to be reflected in our algorithm.
At a given time t, each cell in the network is in precisely one of the regions outlined in the preceding paragraph. We store a global array that tracks which region each of the cells is in and use this to select the appropriate equation to integrate, based on those presented in (13). Over an interval Δt, the region of a subset of the cells may change as the local voltage variable, V, crosses switching manifolds. Note that the continuity of trajectories restricts which transitions between regions are possible (for example, a transition between Region I and Region III is not permitted). Our evolution operator \(\varphi_{h}\) must account for this.
We identify where switching events occur by comparing the region of all of the cells before and after the operator \(\varphi_{\Delta t}\) is applied. Where switching events have occurred, we locate the switching time for cell j by searching for roots of the transcendental equation \(X_{j}^{1}(s) - V_{x} = 0\), \(t_{i} \leq s < t_{i+1}\) using Newton's method, where the superscript denotes that we only consider the V component of the state vector \(X_{j}\) and \(V_{x}\) is the switching manifold that has been crossed. We remark that the Fréchet derivatives for use in Newton's method are obtainable in closed form, and so the application of Newton's method does not rely on numerical finite difference approximations for these derivatives.
After the switching time, s, has been found, the state of the cell is updated via \(X_{j}(s) = \varphi_{\Delta s}(t_{i},X_{j}(t_{i}))\), where \(\Delta s = s-t_{i}\), according to the governing equation for that region. Following this, the regional variable is updated to reflect the fact that the cell has transitioned to a new region. Finally, the remainder of the update step is taken, according to \(X_{j}(t_{i+1}) = \varphi_{\Delta s^{*}} (s,X_{j}(s))\), where \(\Delta s^{*} = t_{i+1}-s\), by integrating the governing equation for new region. It is possible that between times s and \(t_{i+1}\), the cell passes through other switching manifolds, in which case, the procedure for identifying switching times and updating the region variable is repeated as many times as necessary.
Cells in Region IV are in the refractory state, governed by (12) and (14), in which the voltage variable is held fixed at \(V=V_{\mathrm{r}}\) for a total duration of length \(\tau_{\mathrm{R}}\). To account for this, we introduce for each cell, a counter that tracks how much time a cell has spent in the refractory period. Following a firing event, the spiking cell enters the refractory state, whereupon this counter is reset to zero (and the cell's region is changed to Region IV). The cell will remain in the refractory state until the counter reaches \(\tau_{\mathrm{R}}\). At this time, the region variable for that cell is updated to the appropriate value, based on the location of \(V_{\mathrm{r}}\) relative to \(V_{-}\) and \(V_{+}\). For computational efficiency, we can take advantage of the fact that V is unchanging in the refractory period by replacing the voltage dynamics with \(\dot{V}=1\), thus letting V implement our refractory time counter in Region IV. Thus, if the cell spikes at time \(T^{m}\), we have \(V(T^{m})=0\) upon entering Region IV. To ensure continuity of solutions upon exiting the refractory period, we must also ensure to set \(V(T^{m}+\tau_{\mathrm{R}}) = V_{\mathrm{r}}\).
Since cells are independent of one another outside of firing times, this above computation to update the state of the system can be performed in parallel across the GPU architecture in the absence of firing events. Where firing events occur, we must ensure that we correctly update the state of the network at the firing time to reflect the coupling in the network. As for the switching times, we determine firing times by searching for roots of the transcendental equation \(X_{j}^{1}(s) - V_{\text{th}} = 0\), \(t_{i} \leq s < t_{i+1}\) using Newton's method. We then amalgamate all firing events between times \(t_{i}\) and \(t_{i+1}\) and find the minimum firing time, \(t^{*}\), of this list. The states of all cells are updated to this time from \(t_{i}\):
$$X^{-}(t_{*}) = \varphi_{\Delta t^{*}} \bigl( t_{i}, X(t_{i}) \bigr), $$
where \(\Delta t^{*} = t^{*}-t_{i}\) and the superscript denotes that this corresponds to the state of the network in the limit as t approaches \(t^{*}\) from the left. Following this, the reset conditions are applied across the entire domain. Our approach allows for simultaneous firing events to occur by also applying reset conditions for all events that occurred with the time interval \([t^{*},t^{*}+\varepsilon)\), where ε is a small positive real number. After the reset conditions have been applied, the state of the network is updated again as
$$X(t_{i+1}) = \varphi_{\Delta t^{*}_{c}} \bigl( t_{*}, X^{+}(t_{*}) \bigr), $$
where \(\Delta t^{*}_{c} = t_{i+1} - t^{*}\) and the superscript denotes that the state is evaluated after the reset conditions have been applied. Note that during the completion of the timestep, we must again check to see if other cells have reached threshold, and apply reset conditions as necessary. At the completion of a timestep, the state of the network is saved and the state at the next time step is computed.
We note that a further improvement in computational efficiency can be achieved in the limit of high gain for the kernel function (7). Namely in the limit \(\beta\rightarrow\infty\) the bump function becomes a Top-hat function:
$$ w(r) = \textstyle\begin{cases} w_{0} , & r\leq\sigma, \\ 0 , & r>\sigma. \end{cases} $$
Thus, cells are only coupled with a single strength \(w_{0}\), over a finite distance, σ, and we need only consider the behaviour of other cells within this distance when applying reset conditions.
A.2 Implementation
To take full advantage of the parallel capabilities of GPUs, we must sensibly organise the processes associated with simulating the network through the construction of appropriate kernels to best to perform tasks in parallel. In addition to this, the memory associated with state variables must be managed appropriately given that memory access can present a significant bottleneck during GPU-based computations.
As discussed in the preceding section, the evolution operator \(\varphi _{h}\) has to account for the differing functional forms of the governing equations as V crosses thresholds. This is done by using Newton's method to find the times of any crossings and then by 'patching' together solution orbits on either side of the threshold crossing time. Associated with Newton's method is a tolerance for finding the crossing times, which must be specified along with other parameters. In our implementation, we take advantage of the double precision capability of our GPU. Since we are using closed-form solutions for the orbits of our system, the only source of numerical error other than machine error arises due to the tolerance selected when computing the threshold crossing times for V. In all of our simulations, we prescribe this tolerance to be 10−10 so that we have precise specification of these times.
During an update step, there may be multiple cells that reach the firing threshold at differing times. To preserve the correct dynamics in spite of this, we need to update the state of the network to the minimum of this set of firing times. To identify this, we define a variable that stores the minimum firing time across the network, which is updated by the subset of spiking cells as firing events are found. Without intervention, this procedure can lead to race conditions, in which one spiking cell will query the stored value of the minimum firing time (to compare with its own), whilst another spiking cell is replacing that same value. Race conditions mean that we cannot guarantee that the value stored in our firing time variable truly represents the minimum over all firing events. To address this problem, we use atomic events, which allow only one cell to read from and write to the stored firing time variable at once. In this way, we are guaranteed to find the true minimum across the set of firing times.
In Algorithms 1 and 2, we detail how to evolve the dynamics of the 2D network. Note that all of the for loops, and the use of \(\varphi _{h}\) to update the state of the network are performed in parallel using the GPU. The full code for our simulations, written in the C++/CUDA programming language, is available in the supplementary material.
Evolution of 2D network
EvolutionOperator
Appendix B: Perturbations at Switching Conditions
The model undergoes switching in its dynamics as the voltage variable passes through the values V+, \(V_{-}\), and \(V_{\text{th}}\). With the introduction of a set of indicator functions \(h(X(\xi,t);\mu ) = V(\xi,t)-V_{\mu}\), where \(\mu\in\{+,-,\text{th} \}\) we can define the travelling wave coordinate values at which these switching events occur according to \(h(X(\xi,t);\mu) = 0\). Now suppose that we have two trajectories: an unperturbed trajectory \(X(\xi,t)=(V(\xi ,t),n_{\text{h}}(\xi,t))\) and a perturbed trajectory \(\widetilde {X}(\xi,t)\) such that \(\delta{X}(\xi,t) = \widetilde{X}(\xi ,t)-X(\xi,t)\), with δX small. Moreover, let us consider the unperturbed trajectory to pass through the switching manifold when \(\xi = \xi^{m}(t)\), \(m \in \mathbb{Z}\). Similarly we shall consider the perturbed trajectory to switch when \(\xi= \widetilde{\xi}^{m}(t) = {\xi}^{m}(t) +\delta\xi^{m}(t)\). The indicator function for the perturbed trajectory may be Taylor expanded as (suppressing the dependence on μ for clarity):
$$\begin{aligned}[b] h\bigl(\widetilde{X}\bigl(\widetilde{\xi}^{m},t\bigr)\bigr) ={}& h\bigl( \widetilde{X}\bigl({\xi}^{m} +\delta\xi^{m}\bigr),t\bigr) \\ ={}& h \bigl(X\bigl({\xi}^{m} +\delta\xi^{m},t\bigr)+\delta X\bigl({ \xi}^{m} +\delta\xi^{m},t\bigr)\bigr) \\ \simeq{}& h\bigl(X\bigl({\xi}^{m},t\bigr) +X_{\xi}\bigl({ \xi}^{m}_{-},t\bigr)\delta\xi^{m}\bigr)\\ &+\nabla_{X} h \bigl(X\bigl({\xi}^{m} +\delta\xi^{m},t\bigr)\bigr) \cdot \delta X\bigl({\xi}^{m} +\delta\xi ^{m},t\bigr) \\ \simeq{}& h\bigl(X\bigl({\xi}^{m},t\bigr)\bigr) + \nabla_{X} h\bigl(X\bigl({\xi}^{m},t\bigr)\bigr) \cdot X_{\xi}\bigl({\xi }^{m}_{-},t\bigr)\delta\xi^{m}\\ & + \nabla_{X} h\bigl(X \bigl({\xi}^{m},t\bigr)\bigr) \cdot\delta X\bigl({\xi }^{m},t \bigr) .\end{aligned} $$
Here we have introduced the notation \(X({\xi}^{m}_{\pm},t) = \lim_{\epsilon\searrow0} X({\xi}^{m} \pm\epsilon,t)\), to make sure that the partial derivative in ξ is well defined. Using the fact that \(h(\widetilde{X}(\widetilde{\xi }^{m},t))=0=h({X}({\xi}^{m},t))\) we obtain
$$ \nabla_{X} h\bigl({X}\bigl({\xi}^{m},t\bigr)\bigr) \cdot \bigl[ \delta X\bigl({\xi}^{m},t\bigr) + X_{\xi}\bigl({ \xi}^{m}_{-},t\bigr)\delta\xi^{m} \bigr]=0 . $$
Using the result that \(\nabla_{X} h(X;\mu) = (\partial_{V},\partial _{n_{\text{h}}}) (V-V_{\mu})=(1,0)\) the above can be re-arranged to give the perturbation in the switching coordinate in terms of the difference between the perturbed and unperturbed trajectories as
$$ \delta\xi^{m}(t) = - \frac{\delta V(\xi^{m},t)}{V_{\xi}(\xi^{m}_{-},t)} . $$
Appendix C: Saltation Matrices
Saltation matrices allow us to handle any jumps in the system (or its linearisation) when it changes from one dynamical regime to another. As well as occurring at the switching manifolds this also happens when the voltage is unclamped and released from the refractory state. Using the notation of Appendix B let us first consider the deviation between the two trajectories at a switching event defined by \(\xi=\xi _{\text{s}}\), with a set of perturbed switching events at \(\xi=\xi _{\text{s}} + \delta\xi\), as
$$\begin{aligned}[b] \delta X(\xi_{\text{s}}+\delta\xi,t) &= \widetilde{X}(\xi_{\text{s}}+\delta\xi,t) - {X}(\xi_{\text{s}}+\delta\xi,t) \\ &\simeq\widetilde{X}(\xi_{\text{s}},t) + \widetilde{X}_{\xi}(\xi _{\text{s}},t) \delta\xi-\bigl[{X}(\xi_{\text{s}},t) + {X}_{\xi}(\xi_{\text{s}},t) \delta\xi\bigr] \\ &=\delta{X}(\xi_{\text{s}},t) +\bigl[\widetilde{X}_{\xi}( \xi_{\text{s}},t)-{X}_{\xi}(\xi_{\text{s}},t)\bigr]\delta \xi.\end{aligned} $$
If \(\delta\xi>0\) then the unperturbed trajectory will already have transitioned through the switch (from below), in which case the two trajectories are governed by different dynamics. A similar argument holds for \(\delta\xi<0\). Thus we may write
$$ \delta X(\xi_{\text{s}}+\delta\xi,t) \simeq\delta X(\xi_{\text{s}},t) + \bigl[{X}_{\xi}({\xi_{\text{s}}}_{-},t)-{X}_{\xi}({ \xi_{\text{s}}}_{+},t) \bigr] \delta\xi. $$
Combining (55) and (57) gives
$$ \begin{bmatrix} \delta V(\xi_{\text{s}}+\delta\xi,t) \\ \delta n_{\text{h}}(\xi_{\text{s}}+\delta\xi,t) \end{bmatrix} = \begin{bmatrix} \delta V(\xi_{\text{s}},t) \\ \delta n_{\text{h}}(\xi_{\text{s}},t) \end{bmatrix} - \frac{\delta V(\xi_{\text{s}},t)}{V_{\xi}({\xi_{\text{s}}}_{-},t)} \begin{bmatrix} V_{\xi}({\xi_{\text{s}}}_{-},t)-V_{\xi}({\xi_{\text{s}}}_{+},t)\\ n_{\text{h},\xi}({\xi_{\text{s}}}_{-},t)-n_{\text{h},\xi}({\xi_{\text{s}}}_{+},t) \end{bmatrix} . $$
We may write the above in matrix form as
$$ \begin{bmatrix} \delta V(\xi_{\text{s}}+\delta\xi,t) \\ \delta n_{\text{h}}(\xi_{\text{s}}+\delta\xi,t) \end{bmatrix} = \begin{bmatrix} 1 - \frac{V_{\xi}({\xi_{\text{s}}}_{-},t)-V_{\xi}({\xi_{\text{s}}}_{+},t)}{V_{\xi}({\xi_{\text{s}}}_{-},t)} & 0 \\ -\frac{n_{\text{h},\xi}({\xi_{\text{s}}}_{-},t)-n_{\text{h},\xi }({\xi_{\text{s}}}_{+},t)}{V_{\xi}({\xi_{\text{s}}}_{-},t)} & 1 \end{bmatrix} \begin{bmatrix} \delta V(\xi_{\text{s}},t) \\ \delta n_{\text{h}}(\xi_{\text{s}},t) \end{bmatrix} , $$
or equivalently as \(\delta X(\xi_{\text{s}}+\delta\xi,t)=K(\xi_{\text{s}}) \delta X(\xi_{\text{s}},t)\) with
$$ K(\xi_{\text{s}}) = \begin{bmatrix} {V_{\xi}({\xi_{\text{s}}}_{+},t)}/{V_{\xi}({\xi_{\text{s}}}_{-},t)} & 0 \\ (n_{\text{h},\xi}({\xi_{\text{s}}}_{+},t)-n_{\text{h},\xi}({\xi _{\text{s}}}_{-},t))/{V_{\xi}({\xi_{\text{s}}}_{-},t)} & 1 \end{bmatrix} . $$
Now since the voltage is clamped immediately after a firing event (so that \({V_{\xi}({\xi_{\text{s}}}_{+},t)}=0\)) and the dynamics for \(n_{\text{h}}\) jumps (since it depends on V which is discontinuously reset from \(V_{\text{th}}\) to \(V_{\text{r}}\)) then the saltation matrix for firing is given by
$$ K_{\text{fire}}(\xi_{\text{s}}) = \begin{bmatrix} 0 & 0 \\ (n_{\text{h},\xi}({\xi_{\text{s}}}_{+},t)-n_{\text{h},\xi}({\xi_{\text{s}}}_{-},t))/{V_{\xi}({\xi_{\text{s}}}_{-},t)} & 1 \end{bmatrix} . $$
At a switching event whenever \(V=V_{\pm}\) we note that the voltage and gating dynamics are both continuous. Thus the saltation matrix for switching is given simply by \(K_{\text{switch}}(\xi_{\text{s}})=I_{2}\), namely there is no effect.
The use of saltation matrices to propagate perturbations through the refractory state is a little more subtle than through switching and firing events, since the former occur over a finite time-scale \(\tau_{\text{R}}\) whilst the latter are instantaneous. In this case the perturbation δξ at a firing event \(\xi=\xi_{\text{f}}\) is propagated for a time \(\tau_{\text{R}}\) before a new dynamical regime is encountered. From (55) \(\delta\xi= - \delta V(\xi_{\text{f}},t)/V_{\xi}({\xi_{\text{f}}}_{-},t)\). Setting \(\xi_{\text{s}}=\xi_{\text{f}}+\tau_{\text{R}}\) and combining the above with (57) gives
$$ \delta X(\xi_{\text{s}}+\delta\xi,t) \simeq\delta X(\xi_{\text{s}},t) - \frac{\delta V(\xi_{\text{f}},t)}{V_{\xi}({\xi_{\text{f}}}_{-},t)} \begin{bmatrix} V_{\xi}({\tau_{\text{R}}}_{-},t)-V_{\xi}({\tau_{\text{R}}}_{+},t)\\ n_{\text{h},\xi}({\tau_{\text{R}}}_{-},t)-n_{\text{h},\xi}({\tau _{\text{R}}}_{+},t) \end{bmatrix} . $$
Hence we see that \(\delta X({\xi_{\text{s}}}_{+},t) = \delta X(\xi_{\text{s}},t) + K_{\text{ref}}(\xi_{\text{s}}) \delta X(\xi_{\text{f}},t)\) where
$$ K_{\text{ref}}(\xi_{\text{s}}) = \begin{bmatrix} {V_{\xi}({\xi_{\text{s}}}_{+})}/{V_{\xi}({\xi_{\text{f}}}_{-})} & 0 \\ 0 & 0 \end{bmatrix} . $$
Here we have used the fact that \({V_{\xi}({\xi_{\text{s}}}_{-},t)}=0\) (since the system is in its refractory state), and that the dynamics for \(n_{\text{h}}\) is continuous at \(\xi=\xi_{\text{s}}\).
Bonilla-Quintana, M., Wedgwood, K.C.A., O'Dea, R.D. et al. An Analysis of Waves Underlying Grid Cell Firing in the Medial Enthorinal Cortex. J. Math. Neurosc. 7, 9 (2017). https://doi.org/10.1186/s13408-017-0051-7
Grid cell
Medial enthorinal cortex
h-current
Rebound spiking
Integrate-and-fire neural field model
Nonsmooth dynamics
Travelling wave
Evans function | CommonCrawl |
The bar chart below shows the marks scored by a student in an examination. Study it carefully and answer the questions that follow.
(a) What was his average score for his best 3 subjects?
(b) What percentage of his total score was the English score?
(a) 80 marks and (b) 26 2/3%
The graph below shows the number of patients who visited a clinic during a certain week.
(a) Find the total number of patients who visited the clinic on Wednesday and Thursday.
(b) There were 40% fewer patients on Saturday than on ----------
(b) Monday
(a) 80 and
(a) 100 and
Mr Chee wanted to measure the amount of rainfall during a rainy season. He placed an empty beaker and observed the water level of the beaker and the results are shown in the graph below.
(a) What is the increase in water level from Day 1 to Day 2?
(b) Find the average water level in the beaker over 4 days.
(a) 2.1cm
(b) 4.1 cm
The bar graph shows the number of students playing in the various sports during the school`s games day. of the student play soccer. Draw the bar that shows the number of students who play soccer.
The graph below shows the favourite sports of the students in the yellow house. Study it and answer the following question.
What percentage of the students in Yellow House like swimming?
The graph below shows the number of durians sold from Monday to Friday.
What is the ratio of the number of durians sold on Wednesday to the total number of durians sold over the five days?
The bar graph below shows the number of visitors who visited a carnival for the first 6 months of the year.
In June, $\frac{1}{6}$ of the visitors were adults, and the number of boys was the same as the number of visitors in March. What fraction of the visitors in June were girls?
$\frac{7}{18}$
$\frac{7}{9}$
The bar graph below shows the height of 5 boys. Based on the information below, put a tick mark in the correct box.
(a) Leslie`s height is less than Rashid`s height.
(b) The average height of the 5 boys is more than Rashid`s height but less than Daniel`s height
(a) False
(b) True
The bar chart above shows the age of 7 children in months
(a) What is the age of the oldest child in years and months?
(b) What is the age difference in years between Danny and the youngest child?
(c) In another 3 year 10 month, what will be Calvin`s age in years?
(a) 2 years 10 months
(b) 1 1/4 months
(c) 1.5 years
(a) 2 years 9 months
The bar graph below shows the time taken by 4 boys to complete a race.
Which boy finished third in the race?
38 children were asked what they had for lunch. The bar graph below shows the children`s. Study it carefully before answering question.
What fraction of the pupils chose vegetarian food?
Give your answer in the simplest form
At a fruit stall, there are 4 types of fruits: apples, oranges, pears and durians. The bar graph below shows the number of each type of fruits at the stall. The bar that shows the number of durians has not been drawn. If 20% of the fruits at the stall are durians, how many durians are there?
100 Durians
A Class of 36 pupils sat for a test.
Their results are shown in the graph below.
To pass the test, a pupil must obtain at least 4 correct answers.
What fraction of the class failed the test?
The graph below illustrates the survey findings of the weekly pocket money of a primary 6 class.
(a) Find all number of students in the class.
(b) Calculate the percentage of students whose allowance is $30 or more
b 65%
The bar chart shows the number of toys sold by Mr Tan for each month over a 3-month period. A part of the bar chart was torn. The difference in the number of toys sold in March and May was 56. How many toys were sold in April?
40 Toys
Shi Jin has 4 bottles labelled E, F, G and H respectively. The bar graph below shows the volume of water in each bottle. The bars that show the volume of water in Bottle E and bottle F have not been drawn.
The ratio of the volume of water in Bottle E to the total volume of water in the 4 bottles is 2:9. Bottle F contains 40ml more water than bottle E. Find the total volume of water in the 4 bottles.
\begin{array}{rcl} E&\rightarrow&2u\\ F&\rightarrow&2u+40\\ G&\rightarrow&90\\ H&\rightarrow&120\\ Total&\rightarrow&9u \end{array}
The graph below shows the number of books the pupils borrowed from a school library in five days. The bar that shows the number of books borrowed on Wednesday has not been drawn.
a The average number of books borrowed each day from Monday to Friday was 64. Find the total number of books borrowed on Wednesday.
b What was the percentage decrease in the number of books borrowed from Monday to Tuesday?
b 37.5%
The graph below shows the number of story books read by the various classes in a school over a period of time.
(a) What was the total number of classes in the school?
(b) How many classes in the school read more than 40 story books?
(c) What percentage of the class in the school read fewer than 40 story books?
(a) 40 classes
(b) 18 classes
(c) 31.5%
The graph below shows the mass of newspapers collected by five classes in a recycling project. Another two classes, 6F and 6G collected a total of 36 kg of newspapers. What was the average mass of newspapers collected by all the 7 classes?
The graph below shows the family size of the pupils in a class, Study the graph carefully and answer.
There are how many pupils in the class altogether?
20 Pupils
6 Pupils
The number of pupils with a family size of 4 is (how) (many) times those with a family size of 3.
Junming had $6 with him when he went to school every morning. The bar graph below shows the amount of money he had left at the end of each school day. Study it carefully and answer question.
On which day did Junming spend the least amount of his pocket money and how much was this amount?
Tuesday, $4
The graph below shows the number of points Sam scored in the basketball matches he played in from January to May.
What is the average number of points Sam scored per month from January to May
The graph below shows the amount of money collected from 4 games stalls at a carnival.
(a)What is the average amount of money collected from each games stall?
(b) If Games stall C wants its collection to be 20% more than games stall B, how much more money must it earn?
(a) 100
(b) 465
A school conducted an immunisation exercise for its primary 5 pupils from Monday to Thursday. Each of them had their immunisation on one of the four days.
The bar graph below shoes the number of pupils that had immunisation from Monday to Thursday
(a) What percentage of the pupils had their immunisation on Wednesday?
(b) Express the number of pupils who had their immunisation on Thursday as a fraction of those who had theirs on Tuesday. Leave you answer in its simplest form
(a) 29.5%
(b) 3/5
The graph below shoes the number of pupils who shopped at the 5 stalls in a certain school on Friday. Study the graph carefully and answer the questions.
(a) How many more pupils preferred stall 5 to stall 3?
(b) What was the percentage of the pupils who shopped at stall 4?
The graph below shows the weekly expenditure of three children, James, Pei Hua and Raman. If their weekly allowance is $10 each, what is their total savings?
The bar graph below shown the number of stamps collected by 5 members of a local stamp club in the month of October.
a Find the average number of stamps the members collected in the month of October
b The total number of stamps collected by the 5 members in November is the same as in October. However, a new member joined the stamp club and the average member of stamps collected by each member became 89. How many stamps did the new member collect in November?
a 87 stamps
b 99 stamps
The bar graph shows the number of visitors to a zoo from 2013 to 2017. During which one-year period was the increase in the number of visitors the greatest?
The bar graph below shows the number of television sets owned by some families in a neighbourhood.
a What percentage of the families owned at least 2 television sets?
b What is the average number of television sets owned by the families in the neighbourhood?
round off your answer to the nearest whole number
a 75%
The bar graph shows the number of students playing in the various sports during the school`s games day. $\frac{1}{4}$ of the students play soccer. Draw the bar that shows the number of students who play soccer
How many months did sam score at least 10 points?
The bar graph shows the number of each brand of pen sold in a shop.
The prices of the pens are shown in the table below.
(a) How may brand B pens were sold?
(b) There were twice as many Brand D pens as Brand A pens sold. Draw the bar to show the number of Brand D pens sold.
(c) Each statement below is either true, false or not possible to tell from the graph. For each statement, put a tick mark in the correct column OR option in A B C D E.
(ci) The greatest amount of money is collected from the sale of Brand B pens. choose a option True OR False OR not possible to tell
(cii) The shop makes the most amount of money from the sale of Brand D pens. choose a option True OR False OR not possible to tell
(b) 160 as shown
(ci) True
(cii) not possible to tell
The average number of households in a block of flats in 58. The bar graph shows the number of households which own a vehicle. How many households do not own a vehicle?
The bar graph shows the number of cups of ice-cream sold over 6 months. A total of $1800 was collected. How much does each cup of ice-cream cost?
A class of 36 pupils sat for a test. Their results are shown in the graph below. To pass the test, a pupil must obtain at least 4 correct answers. What fraction of the class failed the test? Give your answer in the simplest form
The bar graph shows the number of stickers each child has. How many stickers must Dina give Ali so that both of them will have the same number of stickers
The bar graph below shows the number of pupils in four different classes with computers at home. Study the graph and answer the question below.
If there are 30 pupils in each class, what percentage of the total number of pupils have computer at home?
Give your answer in the simplest form.
The graph below shows the time taken by 7 girls to complete the 100 m race. The table below shows the names of the winners in the same event. Study them carefully and answer.
Result of 100m race
First-- Helen
Second-- Ivy
Third-- Joey
Forth--Kelly
Which letter on the graph shows the time taken by joey?
The bar graph below shows the numbers of cars sold by ABC company. Given that a total of 129 cars were sold from January to April, how many cars were sold in April?
The graph below shows the number of story books read by the pupils in a class.
(a) What was the total number of pupils in the class?
(b) What fraction of the pupils in the class read more than 3 story books?
(c) What is the total number of story books read?
(b) $\frac{6}{20}$
(b) 10+8=18
$\frac{18}{40}$=$\frac{9}{20}$
The fraction is $\frac{9}{20}$
(1x6)+(2x7)+(3x9)+(4x10)+(5x8)
=6+14+27+40+40
=127
The graph below shows the time taken by 6 children to complete a race. What is the total time taken by the top 4 runners to complete the race?
180 pupils were asked to choose their favourite colour. The bar graph below shows the number of pupils who chose each of the colours.
What percentage of the pupils chose red as their favourite colour?
The bar graph below shows the number of books and toys sold by a bookstore over a period of five months. Use the graph to answer.
What was the percentage increase in the number of books sold from January to February?
The table shows the number of students who travels to school using different modes of transport during schools days. Which pie chart represents the date correctly?
If you think (Pie) (chart) (no.) (1) matching with bar graph then mark Option (A)
If you think (Pie) (chart) (no.) (2) matching with bar graph then mark Option (B)
If you think (Pie) (chart) (no.) (3) matching with bar graph then mark Option (C)
If you think (Pie) (chart) (no.) (4) matching with bar graph then mark Option (D)
The graph below shows the points scored by pupils who participated in a Math competition. 2 points were awarded for each correct answer and 1 point was awarded for a partially correct answer. What is the maximum number of questions answered correctly by Suling?
5 answered correctly
Aggie had a roll of ribbon. She used some of it each day for 4 days. At the end of each day, she measured and recorded the length of ribbon left in the bar graph below. Based on the information below, put a tick in the correct option.
(a) The length of the original roll of ribbon is 80cm.
(b) The total length of ribbon used over the 4 days is 60 cm
(a) Impossible to tell
(b) Impossible to tell
The line graph shows the savings of 6 pupils in August. Each pupil had $500 at first. Study the graph carefully and answer the questions that follow.
(a) What was the total amount of money saved by the children?
(b)What fraction of the total amount of money was spent?
(Give your answer in its lowest terms)
(a) 1100 dollars
The graph below shows the number of stamps collected by 4 boys.
(a) What is the average number of stamps collected by each boy?
(b) How many stamps must Daryl give Albert so that both boys will get the same number of stamps?
(a) 135 stamps
(b) 45 stamps
Some shoppers in a supermarket were asked to sample four different brands of chocolates A, B, C and D and select their favourite brand of chocolate. The results are shown in the bar graph below.
(a) What was the total number of shoppers who sampled the different brands of chocolate?
(b) What fraction of the male shoppers selected Brand C as their favourite brand of chocolate?
(a) 130 shoppers
The bar graph below shows the number of car accidents in a country from 2008 to 2011.
In which year was there $\frac{5}{6}$ as many car accidents as in 2010?
A child from every unit in a block of flats was asked how many other children lived in the same flat. The bar graph below presents the results. Study it carefully and answer Question
What is the total number of children living in that block of flats?
A school conducted an immunisation exercise for its primary 5 pupils from Monday to Thursday. Each of them had their immunisation oin one of the four days.
The bar graph below shows the number of pupils that had immunisation from Monday to Thursday.
a What percentage of the pupils had their immunisation on Wednesday
b Express the number of pupils who had their immunisation on Thursday as a fraction of those who had theirs on Tuesday. Leave your answer in its simplest form
a The percentage is 32.5% and b $\frac{3}{5}$
a The percentage is 32.5% and b $\frac{6}{10}$
The bar graph shows the number of visitors to Sentosa from 2002 to 2005. How many visitors were there in 2004?
The graph below shows the number of participants in 4 different sports.
Study the graph carefully before you answer the question.
(a) What fraction of the participants in Badminton were men?
(b) What percentage of the total number of participants in Bowling and Basketball were women?
(c) If the average number of participants in all four sports was 78, how many women had participated in Tennis?
(a) $\frac{4}{15}$
(b) 52$\frac{1}{1}$%
(a) $\frac{12}{15}$
The bar graph shows the number of each type of cutlery sold in a shop. The table shows the prices of the cutlery.
(a) How many more spoons than knives were sold?
(b) Find the average amount of money collected from the cutlery sold. Round off your answer to the nearest dollar
(b) $4
The bar graph shows the number of pets owned by families in a neighbourhood.
(a) How many pets are there in the neighbourhood altogether?
(b) What fraction of the families who own pets, have at least 3 pets?
(a) 1150 pets
(b )$\frac{8}{31}$
(a) 950 pets
The bar graph below shoes the earning made by a furniture company from 2000 to 2004. In which year was the increase in the earnings the greatest?
A survey was conducted on a group of 40 boys to find out the number of siblings they have. The results of the survey are showing in the bar graph below.
Based on the results, how many boys have the greatest number of siblings?
In which month was the ratio of the number of books to the number of toys sold 9:8?
The graph below shoes the number of apples sold in Stall A, Stall B and Stall C. Study the graph below carefully.
(a) What is the average number of apples sold in Stall A, Stall B and Stall C?
(b) How many percent more apples did Stall A sell than Stall B?
(c) The number of apples sold in Stall D was $\frac{2}{5}$ of the total number of apples sold in Stall A, Stall B and Stall C. Draw the bar representing the number of apples sold in Stall D in the bar graph above. 1
The bar graph below shows the number of mobile phones sold over a period of 4-5 months. Between which two months was there a 50% increase in the sales?
December and January
January and February
February and March
March and April
The bar graph below shows the number of tickets sold for a concert to a group of children. How many children purchased more than 2 tickets?
The graph below shows the monthly household income of some families in a housing estate.
How many families have a monthly household income of more than or equal to $4000 per month?
The bar graph below shows the number of durians Mr Tan sold from June to September. The total number of durians sold by Mr Tan from June to Septembers was 200. How many durians were sold in July?
30 durians
The graph shows the amount of money collected by 3 class for a charity.
What was the total amount of money collected by the 3 classes?
The graph shows the amount of money Jing Hua spent over five days.
Jing Hua had $20 at first. How much money did he have at the end of Wednesday?
The graph below shows the number of cars sold from July to December.
(a) Find the percentage decrease in sales from November to December, correct your answer to 1 decimal place.
(b) In which months was the number of cars sold more than the average number of cars sold from July to December?
(b) July and December
(a) 50.48%
(b) October and December
(b) November and December
(b) October and November
There are twice as many boys as girls. There are twice as many adults as chldren.
Which one of the following bar graphs shows the above information correctly?
If you think number 1 bar graph is correct then mark Option 1
The bar graph shows the number of students who took different types of transport to school.
Which pie chart best represents the information in the bar graph?
If you think number 1 pie chart matching with bar graph then mark Option (A)
If you think number 2 pie chart matching with bar graph then mark Option (B)
If you think number 3 pie chart matching with bar graph then mark Option (C)
If you think number 4 pie chart matching with bar graph then mark Option (D)
The bar graph below shows the number of marks some pupils scored in a Math quiz. Given that the passing mark was 150, how many pupils passed the Math quiz?
The bar graph shows a total of 400 pupils involved in a Fund raising project over a period of 8 weeks. Study the graph carefully and answer the question
In which week was there 3 times as many pupils involved in the Fund Raising Project as in the 7th week?
5th week
The graph shows the number of books borrowed from the library by 5 classes in a week. The average number of books borrowed by the 5 classes was 80. How many books did class 5E borrow from the library?
Keegan drew up a bar graph based on the amount of time he spent on each of the activities in 1 week.
a) After totaling up the number of hours that he had recorded in the graph for that week, he realised that he had recorded the time spent on one of the activities wrongly. The time recorded was 10% more than the actual time for that activity. Which activity did he record wrongly?
b) What was the actual time taken for the activity that was recorded wrongly in (a)?
a Sleeping
b 60 Hours
a Studying
a Playing
a Outdoors
The bar graph below shows the results of a survey on the favourite sports of a group of students. $\frac{1}{6}$ of the students chose volleyball as their favourite sport.
Draw the bar in the graph to show the number of students who chose volleyball as their favourite sport
68 students chose volleyball as their favourite sport
The bar graph shows how pupils of Champion primary School went to school on a certain day.
Which pie chart represents the information given in the bar graph?
If you think number 1 pie chart matching with bar graph then mark Option no. 1
The graph below shoes the height of 3 boys Ali, Bala and Charles. Find the total height of Ali and Charles.
The bar graph shows the number of visitors to a zoo from 2013 to 2017. From 2013 to 2017, for how many years did the zoo receive more than 30000 visitors
The bar graph below shows the number of handphones sold over a 6 month period in a shop.
In which month did the shop sell 20% of the total number of handphones?
From the survey, what is the largest number of children living in a flat?
The graph shows the number of students at each level in Rose Primary School.
Find the total number of pupils in Rose Primary School
1110 Pupils
The bar graph below shows the timing (in) (minutes) taken by 4 girls to complete a 800 m race.
Mark the proper option the time taken by mala to complete the race
The bar graph below shows the number of food items sold at a carnival.
a Find the total number of items sold at the carnival.
b $97.50 was collected from selling the donuts, chicken wings and fishballs at the carnival. The prices of these items were in the ration 2:3:1.
How much was each chicken wings sold for
a: 125 items
b $1.30
The following bar graph shows the number of items sold in a day. Each book cost 1 dollar, each pen cost 80 cents, each file cost 2 dollars and each eraser cost 50 cents. which of the 4 items has the highest total sale price? | CommonCrawl |
IMPRS-HD alumni 2018
Rainer Weinberger (17.1.) - Roxana Chira (22.1.) - Adriana Pohl (23.1.) - Valeriy Vasilyev (7.2.) - Johannes King (9.2.) - Carolin Wittmann (9.2.) - Clio Bertelli Motta (20.2.) - Robert Reischke (18.4.) - Thales Gutcke (2.5.) - Tim Tugendhat (16.5.) - Andreas Schreiber (18.5.) - Carsten Littek (29.6.) - Jonas Frings (11.7.) - Chiara Mazzucchelli (12.7.) - Gustavo Morales (25.7.) - Fabian Klein (17.10.) - Michael Walther (18.10.) - Tobias Buck (19.10.) - Svenja Jacob (23.10.) - Jolanta Zjupa (26.10.) - Sabina Puerckhauer (7.11.) - Sara Rezaei Khoshbakht (9.11.) - Michael Rugel (21.11.) - Kseniia Sysoliatina (21.11.) - Bekdaulet Shukirgaliyev (29.11.)
Bekdaulet Shukirgaliev (KazakhsTan) 29.11.2018
The life of star clusters, from birth to dissolution: a new approach ( thesis pdf, 17 MB )
We study the evolution of star clusters, starting from their birth in molecular gas clumps until their complete dissolution in the Galactic tidal field. We have combined the "local-density-driven cluster formation" model of Parmentier and Pfalzner (2013) with direct N-body simulations of star clusters following instantaneous expulsion of their residual star-forming gas. Our model clusters are formed with a centrally peaked star-formation efficiency (SFE) profile, that is, the residual gas has a shallower density profile than stars. We build a large grid of simulations covering the parameter space of global SFEs, cluster masses, sizes and galactocentric distances. We study the survivability of our model clusters in the solar neighborhood after instantaneous gas expulsion and find that a minimum global SFE of 15 percent is sufficient to produce a bound cluster. Then studying their long-term evolution we find that our simulations are able to reproduce the cluster dissolution time observed for the solar neighborhood, provided that the cluster population is dominated by those formed with a low global SFE (about 15%). Finally, we find that the cluster survivability after instantaneous gas expulsion, as measured by cluster bound mass fraction at the end of violent relaxation, is independent of the Galactic tidal field impact.
Supervisor: Genevieve Parmentier (ARI)
Kseniia Sysoliatina (Ukraine) 21.11.2018
The life of star clusters, from birth to dissolution: a new approach ( thesis pdf, 8 MB )
In this work we study the spatial structure as well as the chemical and kinematic properties of the Milky Way disk on the basis of a semi-analytic chemo-dynamical model from Just and Jahreiß (2010) (JJ model). Assuming inside-out formation and a constant thickness of the MW disk, we generalise the local JJ model to Galactocentric distances of R = 4-12 kpc. At each radius we assume a star formation rate (SFR) with a peak shifting to younger ages for the outer disk and use the four-slope broken power-law initial mass function (IMF) from Rybizki and Just (2015). The age-velocity dispersion and age-metallicity relations (AVR and AMR) are then obtained self-consistently; the latter is constrained by metallicity distributions of the Red Clump stars from the Apache Point Observatory Galactic Evolution Experiment (APOGEE, Eisenstein et al., 2011). Within a forward modelling framework, we validate the local JJ model in the solar cylinder by testing it against the samples from the Radial Velocity Experiment (RAVE, Steinmetz et al., 2006) and the first and second Gaia data releases (DR1 and DR2, Lindegren et al., 2016; Gaia Collaboration et al., 2018). We find an overall mismatch in star counts up to ~9%, with a statistically significant discrepancy identified for the dynamically cold populations. We also develop a new treatment of the asymmetric drift and apply it in three metallicity bins to the RAVE local sample and G-dwarfs from the Sloan Extension for Galactic Understanding and Exploration (SEGUE, Yanny et al., 2009). The tangential component of the solar peculiar motion obtained from the RAVE sample is V = 4.47 ± 0.8 km/s. The rotation curve reconstructed from the SEGUE G-dwarfs in a range of distances R = 7-10 kpc has a near-zero slope of 0.033 ± 0.034. The thick disk G-dwarfs are found to be kinematically homogeneous with a scale length of 2.05 ± 0.22 kpc, which is in agreement with values from the literature.
Supervisor: Andreas Just (ARI)
Michael Rugel (Germany) 21.11.2018
On the formation and destruction of molecular clouds with the Galactic plane survey THOR ( thesis pdf, 13 MB )
This thesis investigates the properties of molecular clouds with THOR (The HI, OH and Radio Recombination Line (RRL) survey of the Milky Way). We analyze OH absorption at 18 cm within THOR and follow-up observations. We derive the abundance with respect to molecular hydrogen and the total number of hydrogen nuclei: 1) We find a decreasing OH abundance with increasing column density of molecular hydrogen. 2) Due to significant column densities of atomic hydrogen at low N_OH, the OH abundance with respect to N_H is approximately constant. 3) We detect OH components which are associated with gas that is not predominantly molecular or even CO-dark. We conclude that OH is a potential tracer for diffuse gas. Regarding the impact of star clusters on molecular clouds, we detect signatures of feedback in RRL emission in the star forming region W49A. A comparison to the WARPFIELD models (one-dimensional models of feedback-driven shells) indicates that feedback is not yet strong enough to disperse its molecular cloud and that the shell is either in process of re-collapsing to initiate a new event of star formation or has already re-collapsed. This suggests that at least parts of the star formation in W49A is regulated by feedback.
Supervisor: Henrik Beuther (MPIA)
Sara Rezaei Khoshbakht (Iran) 09.11.2018
3D map of the dust distribution in the Milky Way ( thesis pdf, 10MB )
In this thesis, I present a new non-parametric model for inferring the three-dimensional (3D) distribution of dust density in the Milky Way. Our approach uses the extinction measured towards stars at different locations in the Galaxy at known distances. Each extinction measurement is proportional to the integrated dust density along its line of sight (l.o.s). Making simple assumptions about the spatial correlation of the dust density, we infer the most probable 3D distribution of dust across the entire observed region, including along sight lines which were not observed. This is possible because our model employs a Gaussian process to connect all l.o.s. The result is a smooth, 3D map of the dust density, which is the local property of the interstellar medium (ISM) rather than an integrated quantity. Owing to our smoothness constraint and its isotropy, the method provides one of the first maps without "fingers of God" artefact. I then present the first continuous map of the dust distribution in the Galactic disk out to 7 kpc within 100 pc of the Galactic midplane, using red giant stars from SDSS APOGEE DR14. The resulting map traces some features of the local Galactic spiral arms, even though the model contains no prior suggestion of spiral arms, nor any underlying model for the Galactic structure. This is the first time that such evident arm structures have been captured by a dust density map in the Milky Way. Our resulting map also traces some of the known giant molecular clouds in the Galaxy and puts some constraints on their distances, some of which were hitherto relatively uncertain. I also demonstrate a map of the 3D distribution of dust in the Orion complex. Orion is the closest site of high-mass star formation, making it an excellent laboratory for studying the ISM and star formation. We use data from the Gaia-TGAS catalogue combined with photometry from 2MASS and WISE to get the distances and extinctions of individual stars in the vicinity of the Orion complex. We find that the distance and depth of the cloud are compatible with other recent works, which show that the method can be applied to local molecular clouds to map their 3D dust distribution. We also use data from the recent second Gaia data release (GDR2) to update the map that shows complex dust clouds in the Orion region. I finally show a 3D map of hydrogen density in the local ISM. The hydrogen equivalent column densities were obtained from the Exploring the X-ray Transient and variable Sky project (EXTRAS), which provides equivalent NH values from X-ray spectral fits of observations within the XMM-Newton Data Release. A cross-correlation between the EXTRAS catalogue and the first Gaia Data Release was performed in order to obtain accurate parallax and distance measurements. The resulting map shows small-scale density structures which can not be modelled using analytic density profiles.
Supervisor: Coryn Bailer-Jones (MPIA)
Sabina Puerckhauer (Germany) 07.11.2018
Characterising light concentrators for CTA and optimising the data selection to improve angular resolution and sensitivity ( thesis pdf)
The Cherenkov Telescope Array (CTA) is the next generation ground-based observatory for gamma-ray astronomy that will reach a performance unprecedented in the field. This thesis focuses on optimising this performance in the core energy range of CTA around 1 TeV. In a first part, the wavelength-dependent angular efficiency of light concentrators for the camera system FlashCam, proposed for the medium size telescopes of CTA, is determined with a dedicated test system. For hexagonal concentrators with three reflective coatings a signal-to-noise ratio enhancement of 2.2 compared to a camera without concentrators is observed. By varying the distance between concentrators and photo-sensors, a fine-adjustment of the angular efficiency increases this ratio by another 3%-5%. In a second part, the angular resolution and the sensitivity of CTA are studied by means of simulations. By optimising different quality selection cuts on telescope data, sensitivity enhancements of 20%-40% compared to the CTA requirements are reached and 30%-40% for the angular resolution. With the optimised cuts, spatially extended emission models of the radio galaxy Centaurus A are investigated and it is found that the optimised angular resolution of CTA allows for different theoretical emission models to be discriminated based on the predicted emission regions.
Supervisor: James Hinton (MPIK)
Jolanta Zjupa (Poland) 26.10.2018
The impact of feedback on galactic and extra-galactic scales ( thesis pdf, 7 MB)
Recent cosmological hydrodynamical simulations were for the first time able to produce galaxy populations with realistic sizes and morphologies. This success can be attributed to the inclusion of subgrid models for supernovae winds and active galactic nuclei (AGN) feedback. In this thesis, we investigate the impact of feedback driven galactic outflows. First, the expulsion of gas proves to be crucial for the rotational support of haloes hosting realistic galaxies. We employ the state-of-the-art hydrodynamical simulation suites Illustris and IllustrisTNG to characterise the amount of specific angular momentum in the baryonic component of haloes. We find the baryonic spin at z = 0 to be a factor of ∼ 2 higher than the dark matter spin, which is due to the transfer of a constant cumulative spin of Δλ = 0.0013 by z = 0 from dark matter to the gas during mergers, and to the preferential expulsion of low angular momentum gas by mostly AGN feedback. Second, galactic outflows impact the state of the diffuse gas on large scales. We employ the Lyman-α forest to examine the feedback induced changes in the inter-galactic medium (IGM) that serves as gas reservoir for accretion onto galaxies. For a clean comparison, we have run a suite of simulations with both galaxy formation physics and with the Quick Lyman-α (QLA) technique yielding an unperturbed IGM. We find the Lyman-α flux power spectrum to exhibit increasingly more power at large scales and correspondingly less power at small scales in the presence of outflows, as well as the IGM to be generally hotter. Employing IllustrisTNG we investigate the excess Lyman-α absorption as a function of impact parameter for haloes exhibiting strong and weak feedback and find significant differences that can largely be explained by the higher temperature of the perturbed gas.
Supervisor: Volker Springel (MPIK)
Svenja Jacob (Germany) 23.10.2018
Cosmic ray feedback in galaxy formation and a numerical model for turbulence ( thesis pdf, 6 MB)
Feedback processes play an important role in galaxy formation since they regulate star formation both in low mass galaxies and in massive galaxy clusters. Which mechanisms dominate and how the feedback couples to the surrounding medium, are still open questions. In this thesis, we study the feedback from cosmic rays in different environments in more detail. We develop steady state models for a sample of galaxy clusters, in which cosmic ray heating together with thermal conduction prevents large cooling flows. Observational constraints reveal that cosmic ray heating is only viable in clusters that do not show signatures of enhanced cooling. This might indicate a self-regulated feedback cycle. On galactic scales, cosmic rays can drive winds if they are allowed to diffuse or stream out of the galaxy. We demonstrate in simulations of isolated galaxies that cosmic rays are able to regulate star formation in low mass galaxies but the wind efficiency drops rapidly with increasing galaxy mass. Furthermore, almost all astrophysical flows are highly turbulent. This is a challenge for numerical simulations, which cannot resolve all scales of the turbulent cascade. Therefore, we implement a model for turbulence on subgrid scales into the hydrodynamics code AREPO. We validate our model in idealized test cases and apply it to simulations of turbulent boxes.
Tobias Buck (Germany) 19.10.2018
On the formation of the Milky Way system in cosmological context - A numerical study ( thesis pdf, 300 MB)
State-of-the-art cosmological hydrodynamical simulations have succeeded in modelling realistic Milky Way (MW) type galaxies with spatial resolution of the order of a few hundred parsec, similar to the scale-height of MW's stellar disc and the half-light radius of classical satellite galaxies. I divide the present study into two parts, the build-up of MW's stellar disc and bulge and the formation and evolution of its satellites and dwarf galaxies.
In the first part I show that observed clumpy stellar discs in the early phases of the formation of the Galaxy are dynamically unimportant for its further evolution. This confirms recent observational results where a non linear mapping between stellar mass and light causes stellar discs to appear clumpy. I turn then to explore the formation mechanism of a peanut bulge in cosmological context. I study the kinematical properties of the central stellar populations of a model galaxy using a kinematical decomposition technique and find that the observed kinematic features of the (MW) bulge can only be explained if it consists of both, a peanut bulge and a spherically symmetric bulge both formed via disc instabilities. Observing and disentangling both components will soon be possible thanks to large scale Galactic surveys like Gaia.
In the second part I study the dwarf galaxy population of (MW) mass galaxies. The simulations presented here are among the first to be able to study the formation of dwarf satellite galaxies in a realistic cosmological environment. The employed sub-grid models of the simulations reconcile simulated and observed Local Group satellite mass functions and produce dwarf galaxies whose central stellar velocity dispersion agrees with observations. Using the dwarf galaxies, I test the observational prospects of identifying tidally affected dwarfs in the Local Group using three observables: their distance, line-of-sight velocity and central velocity dispersion. Finally, I investigate the evolution of planes-of-satellites in the framework of the Cold Dark Matter model with a cosmological constant (ΛCDM). These planes quickly dissolve because they consist of a large fraction of chance aligned satellites as recently confirmed with the proper motions of the classical satellite galaxies derived from Gaia data.
Supervisor: Andrea Maccio (MPIA, NYAD)
Michael Walther (Germany) 18.10.2018
Monitoring Thermal Evolution in the Intergalactic Medium over 12 Billion Years ( thesis pdf, 25 MB)
The thermal state of the intergalactic medium (IGM) is an important probe of physical properties for the bulk of gas in the universe. Here, we perform a new measurement of the thermal state for redshift z ≤ 5.4 covering 12 billion years from the endstage of reionization to the present day. For this purpose we measure the Lyman-α forest flux power spectrum based on high resolution quasar spectra from different ground- and space-based spectrographs, combine this analysis with archival measurements of percent level precision, analyze hydrodynamical simulations, use powerful statistical techniques for interpolation, and perform Bayesian inference via Markov chain monte carlo. We observe a rise in the temperature at mean density from 6000 K at z = 5.4 towards 14 000 K at z = 3.4 followed by a cooldown phase reaching 6000 K at z = 0.03. This evolution is provides conclusive evidence for photoionization heating due to reionization of He II, as well as the subsequent cooling of the IGM due to an expanding universe in concordance with model predictions. The agreement with previous measurements is good as well, but our analysis supercedes those by accounting for additional parameters that we marginalize over, and by the vast cosmological timespan our measurement spans. At the highest redshifts z > 5 we infer lower temperatures than expected from the standard picture of IGM heating allowing leaving little room for additional smoothing due to warm dark matter free streaming. Additionally, our measurement for z < 0.5 allows additional constraints on the ultraviolet background in contradiction to previous claims of a UV underproduction crisis.
Supervisor: Joe Hennawi (MPIA, UCSB)
Fabian Klein (Germany) 17.10.2018
Simulations of an accretion disk surrounding a supermassive black hole and its interaction with a nuclear star cluster ( thesis pdf, 34 MB)
We investigate the time evolution of an AD surrounding a SMBH in an active galactic nucleus (AGN) and its dynamical interactions with a the nuclear star cluster (NSC). The AD is important in these interactions because of its dissipative force acting on the NSC stars, resulting in an increased mass flow to the SMBH and asymmetries in the phase space distribution due to its rotation. As the StarDisk project (Just et al., Kennedy et al.) only treated a static AD, viscous hydrodynamical simulations including gravity and self-gravity are used in this thesis to take dissipative feedback and lifetime checks of the AD into account. These simulations were performed using the PLUTO code along with additional modules written by Rolf Kuiper and equilibrium initial condition similar to Shakura & Sunyayev. The results were a quasi-static state as well as the confirmation of the scale-height assumptions from Kennedy et al. and the estimation of the accretion rate reproducing the expected result from Shakura & Sunyayev. Furthermore, the obtained data was used to interpolate the dissipative forces in the direct N-body code NBODY6++GPU and carry out a first test. The inclusion of more physics into the hydrodynamics as well as the advancement of the NBODY6++GPU project to real applications are both tasks for future research.
Supervisor: Rainer Spurzem (ARI)
Gustavo Morales (Chile) 25.07.2018
Stellar Tidal Streams as Cosmological Diagnostics: Comparing data and simulations at low galactic scales ( thesis pdf, 7 MB)
In hierarchical models of galaxy formation, stellar tidal streams are expected around most galaxies. Although these features may provide useful diagnostics of the LCDM model, their observational properties remain poorly constrained. Statistical analysis of the counts and properties of such features is of interest for a direct comparison against results from numerical simulations. In this work, we aim to study systematically the frequency of occurrence and other observational properties of tidal features around nearby galaxies. The approach featured here is based on a visual classification of diffuse features around a sample of nearby galaxies, using a post-processing of optical survey imaging optimized for the detection of low-surfacebrightness stellar structure. At the limiting surface brightness of this sample, 14 − 17% of the galaxies exhibit evidence of diffuse features likely to have arisen from minor merging events. For simulated images, the frequency is 16 − 19%. Our technique recovers all previously known streams in the selected sample and yields a number of new candidates. We conclude that this methodology provides a reliable foundation for the statistical analysis of diffuse circumgalactic features in wide-area imaging surveys, and for the identification of targets for follow-up studies.
Supervisor: David Martinez-Delgado, Eva Grebel (ARI)
Chiara Mazzucchelli (Italy) 12.07.2018
The Physical Properties and Cosmic Environments of Quasars in the First Gyr of the Universe ( thesis pdf, 10 MB)
Luminous quasars at redshift z > 6, i.e. < 1 Gyr after the Big Bang, are formidable probes of the early universe, at the edge of the Epoch of Reionization. These sources are predicted to be found in high–density peaks of the dark matter distribution at that time, surrounded by overdensities of galaxies. In this thesis, we present a search for and study of the most distant quasars, from the properties of their innermost regions, to those of their host galaxies and of their Mpc–scale environments. We search for the highest redshift quasars in the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS1, PS1), discovering six new objects at z >6.5. Using optical/near–infrared spectroscopic data, we perform a homogeneous analysis of the properties of 15 quasars at z> 6.5. In short : 1) The majority of z >6.5 show large blueshifts of the broad CIV 1549 Å emission line, suggesting the presence of strong winds/outflows; 2) They already host supermassive black holes (M_BH = 0.3 - 5 x 10^9 M_sun) in their centers, which are accreting at a rate comparable to a luminosity–matched sample at z =1-3) No evolution of the Fe II/MgII abundance ratio with cosmic time is observed; 4) The sizes of their surrounding ionized bubbles weakly decrease with redshift. We present new millimeter observations of the dust continuum and of the [CII] 158 mm emission line (one of the main coolant of the intergalactic medium) in the host galaxies of four quasars, providing new accurate redshifts and [CII]/infrared luminosities. We study the Mpc–scale environment of a z =5.7 quasar, via observations with broad– and narrow–band filters. We recover no overdensities of galaxies. Among the potential explanations for these findings, are that the ionizing radiation from the quasar prevents galaxy formation, the sources in the fields are dust–obscured, or quasars do not live in the most massive dark matter halos. Finally, we report sensitive optical/near–infrared follow–up observations of gas–rich companion galaxies to four quasars at z >6, firstly detected with the Atacama Large Millimeter Array (ALMA).With the exception of one source, we detect no emission from the stellar population of these galaxies. Our limits on their stellar masses (< 10^10 M_sun) and unobscured star formation rates.
Supervisor: Fabian Walter, Hans-Walter Rix (MPIA)
Jonas Frings (Germany) 11.07.2018
Structure and evolution of simulated dwarf galaxies and Milky Way satellites in Cold and Warm dark matter models ( thesis pdf, 4 MB)
The satellite galaxies and dwarf galaxies in the neighborhood of the Milky Way provide us with detailed observations that can be used to test our standard model of cosmology and structure formation, the $\Lambda$CDM model. I present a sample of 27 cosmological hydrodynamical simulations with virial masses between $5\times 10^8$ and $10^{10}\,\Msun$ that are aimed to study the properties of dwarf galaxies before accretion. The simulated galaxies are able to reproduce observed scaling relations like the dispersion - size and metallicity - stellar mass relation. The stochasticity of merger induced star formation causes a large scatter in the stellar mass - halo mass relation. In galaxies of stellar masses below $10^6 \, \Msun$ stellar feedback is unable to affect the dark matter halo and hence those galaxies retain a cuspy profile. A subsample of 7 halos is used as initial conditions for simulations of satellite - host galaxy interactions in a Milky Way mass halo. The mass removal due to tidal forces creates flat stellar velocity dispersion profiles and efficiently decreases the circular velocity at $0.5\,\mathrm{kpc}$ without stripping a large amount of stars. Additionally the stripping seems to happen in a way that effectively steepens the central dark matter density slope. To investigate the effects of warm dark matter on dwarf galaxies and Milky Way satellites I repeat the study in a $3\,\mathrm{keV}$ warm dark matter scenario. I present the simulations of 21 halos in both CDM and WDM. In WDM the critical halo mass for the onset of star formation is shifted towards higher masses, while the simulations that do produce stars, reproduce the same scaling relations as their CDM counterparts. However, WDM seems to delay the bulk star formation, making galaxies in CDM look about $2\,\mathrm{Gyr}$ older. While halo concentrations are significantly lower in WDM, the central dark matter density slope is slightly steeper for the low mass end. For four halos (in WDM and CDM) I present their evolution as Milky Way satellites. In contrast to the CDM halos, WDM halos are stripped more effective due to their lower concentrations. The survival probability for WDM satellites, on the other hand, is not necessarily lower because of their steeper central slope. Again, the WDM as well as CDM satellites end up with very cuspy profiles after being stripped. I come to the conclusion that the predictions from my simulations do not challenge the $\Lambda$CDM compared to current observational data of dwarf galaxies and Milky Way satellites. An observation of a cored density profile in one of the very low mass objects, however, would force us reconsider the dark matter model. Also WDM would not pose a solution to this problem.
Supervisor: Andrea Maccio (NYU Abu Dhabi, MPIA)
Carsten Littek (Germany) 29.06.2018
Kinetic Field Theory: Momentum-Density Correlations and Fuzzy Dark Matter ( thesis pdf, 1 MB)
Building upon the recent developments of Kinetic Field Theory (KFT) for cosmic structure formation we develop a systematic way to calculate correlation functions of the momentum-density field. We show that these correlators can be calculated from the factorised generating functional after application of partial derivatives with respect to the momentum shift. For visual aid and in order to facilitate an automatic evaluation of corrections by particle interactions we introduce a diagrammatic representation of terms. We employ this formalism to calculate the 2-point momentum-density correlation tensor including initial correlations to quadratic order and completely. A comparison of the results shows that the initial correlations are responsible for the deformation of the power-spectrum on small scales rather than the particle interactions. In the spirit of the Born approximation we use an effective force term to calculate the corrections due to gravity. Our results are in good agreement with previous analytic and simulation results. Recently, Fuzzy Dark Matter models such as Ultra-Light Axions have caught a lot of interest. Their dynamics is described by the classical equations of a condensate. This introduces a quantum potential in the Euler equation and is generally repulsive. We have developed an extension to KFT treating the effects of the quantum potential on the dynamics and on the initial density fluctuation power-spectrum. We find the effects to be largest on scales in the range of 3h/Mpc > k > 0.3h/Mpc, close to the onset of non-linear structures.
Supervisor: Matthias Bartelmann (ITA)
Andreas Schreiber (Germany) 18.05.2018
Diffusion Limited Planetesimal Formation ( thesis pdf, 60 MB)
Planets are surprisingly abundant in our own solar system, but also in extrasolar systems. It is striking to find no explanation for them, as dust in protoplanetary disks was found to not outgrow metres in size. The growth barrier of dust to km-sized planetesimals thus states a missing link onto their formation mechanisms. It is evident for planetesimals to have been present in the early solar system, as their remnants prowl the solar system today in the form of asteroids, Kuiper belt objects, and comets. Of them, many were found to be pristine, giving a hint on what once populated the early solar nebula. Studying the sizes of these pristine objects revealed for all of them a characteristic diameter of 100 km. It is stunning to find this feature independent of distance from the Sun in most pristine object families, hence this feature has to be an imprint of their formation mechanism. This thesis derives a formation criterion for planetesimals out of particle cloud collapse within protoplanetary disks. The found mechanism is capable of reproducing the characteristic sizes of these pristine objects, as it is to first order independent of radial distance from the star. By comparing collapse timescale with turbulent particle diffusion timescale, a minimum size criterion for a dust cloud to collapse is found and investigated. Naturally, dust cloud collapse happens at high dust-to-gas ratios, thus the streaming instability is a good candidate for this turbulent process. Hence, the streaming instability is studied in 2-d and 3-d simulations at dust-to-gas ratios well above unity and on typical collapse length scales. This study found a new instability, namely the azimuthal streaming instability. It operates in the radial-azimuthal plane and has characteristics similar to the streaming instability, thus its name. Subsequent collapse simulations in 2-d and 3-d proved the diffusion limited planetesimal formation to produce planetesimals right at the expected 100 km diameter. It is the conclusion of this thesis to have shown a fundamental concept to be applied in future studies on planetesimals. It has the prospect to make verifiable predictions which can proof this mechanism to have shaped the solar system as we see it today.
Supervisor: Hubert Klahr (MPIA)
Tim Tugendhat (Germany) 15.05.2018
On the Impact of Intrinsic Alignments of Galaxies on Measurements of Weak Gravitational Lensing ( thesis pdf, 8 MB)
Future weak lensing surveys will be too precise for their own good: due to their impressive statistical precision, their systematics must be treated with extreme care. In particular, intrinsic alignments (IA) of galaxies, which cause galaxy ellipticities to be correlated, are indistinguishable from gravitational lensing. I present two established models for IAs and will link them to the physical properties of two galaxy morphologies, namely spiral and elliptical galaxies. Their amplitudes relative to the lensing spectrum is determined and a realistic mix of galaxies and thus utilised to predict the impact of IAs on a Euclid-like tomographic survey and its inferences on a cosmological parameter set. It will be shown that IAs will give rise to significant systematic errors (biases) in future surveys. Furthermore, I present a method for suppressing the IA signal by using colour information and separating the survey into subsamples of blue and red galaxies. This method can also be used to suppress the lensing signal instead, enabling the treatment of just IAs. Finally, I will show a possible way of constrain deviations from General Relativity (Gravitational Slip) by using the model for elliptical galaxies and leveraging its signal against the one from a known weak lensing signal.
Supervisor: Bjoern Malte Schaefer (ARI)
Thales Gutcke (Germany / USA) 02.05.2018
The quenching of star formation in galaxies ( thesis pdf, 20 MB)
This thesis is concerned with investigating what makes star formation inefficient in galaxies. Cosmological, hydrodynamical simulations of galaxy formation show that the energy produced in stars, supernova explosions and active galactic nuclei must couple back into a galaxy's interstellar medium to prevent excess star formation. However, the physical processes at work in this feedback loop are not well understood. This thesis unravels the details of the baryon cycle to constrain the strength of feedback. The first part explores a phenomenological model of star formation quenching in massive galaxies, showing that gas starvation is a viable pathway to realistic elliptical galaxies. In the second part, a state-of-the-art implementation of stellar feedback is put to the test by comparing the chemical composition of the circum-galactic medium with the latest observations. The simulations exhibit a deficiency in highly ionized oxygen, indicating that models of thermally coupled feedback may be insufficient. The last part delves into the star formation prescription itself, since this directly affects the resulting stellar feedback cycle. An empirical model of a metallicity-dependent stellar initial mass function reveals the significant uncertainty resulting from the common assumption of its universality. Thus, this analysis links star formation processes with stellar feedback and shows how they affect the baryon cycle of entire galaxy ecosystems.
Supervisor: Andrea Maccio (MPIA/NYAD)
Robert Reischke (Germany) 18.04.2018
Cosmology as a probe of gravity with future surveys ( thesis pdf, 7 MB)
The subject of this thesis are different aspects of cosmology as a probe of the underlying gravitational theory with future surveys. In the first part of this work we discuss the parameter dependence of covariance matrices of the power spectrum estimator of the large-scale structure. Its variation across parameter space is calculated analytically by constructing a suitable basis and is then compared with numerical simulations. The method presented is applicable to any matrix-valued function which is everywhere positive-definite. The second part investigates the influence of tidal gravitational fields on the formation of dark matter halos at peaks in the density field of the large-scale structure. We extend the spherical collapse model to incorporate the influence of shear and rotation by treating them as inhomogeneities in the non-linear evolution equation. We investigate the statistics of the tidal field and how it is inherited to the statistics of the critical over-density δc. It is shown that the collapse in a tidal field will always proceed faster than the collapse in a homogeneous background. The last part investigates the combination of observations of weak gravitational lensing, galaxy clustering and the cosmic microwave background and the cross-correlations between the probes to investigate scalar-tensor theories of gravity. We carry out a Fisher analysis as well as a Monte-Carlo-Markov-chain to estimate the expected statistical errors. The analysis shows that gravitational theories can be constrained very well with future surveys.
Supervisor: Bjoern-Malte Schaefer (ARI)
Clio Bertelli Motta (Italy) 20.02.2018
The footprints of stellar evolution on the chemical composition of the Galactic old open cluster M67 ( thesis pdf, 40 MB)
In this work we investigate how stellar evolutionary processes change the surface chemical composition of stars. As a test-bench, we use the old open cluster M67, for whose stars high-resolution spectroscopic data are available in many different evolutionary stages, from the main sequence to the red clump. In particular, we use data retrieved from the archives of large spectroscopic surveys such as APOGEE and Gaia-ESO. First we investigate the effects of the so-called first dredge-up on the surface [C/N] abundance of M67 stars. We then analyse variations in the surface abundances of several elements from the main-sequence to the red-giant phase of M67 stars and discuss how these can be explained by atomic diffusion effects. We also present the results of these investigations in the broader context of Galactic archaeology studies. Furthermore, we investigate the chemical composition of three blue straggler stars and two evolved blue straggler stars in M67 in order to find hints for their formation scenario and discuss the results from the point of view of stellar evolution. Finally, we present an experiment based on TGAS data for the study of the dynamical evolution of OB associations.
Supervisor: Anna Pasquali, Eva Grebel (ARI)
Johannes King (Germany) 09.02.2018
Hochenergetische Gammastrahlung aus dem Galaktischen Zentrum ( thesis pdf, 10 MB)
Multi-wavelength observations of the centre of our Galaxy have revealed a number of energetic processes that are related to the central supermassive black hole. Gamma-ray astronomy contributes to these observations with measurements from satellite and ground based instruments such as Fermi-LAT and HESS. These measurements can make important contributions towards answering some of the key open questions. An analysis of the source HESS J1745-290, potentially linked to acceleration of cosmic rays to PeV-energies, is presented in this thesis. The analysis is performed using the HESS II array, which includes the largest existing Cherenkov telescope, in order to measure photon energies below 100 GeV. Furthermore, the Galactic centre is a promising target for indirect dark matter searches. Such a search is performed in this thesis based on HESS and Fermi-LAT data. Due to the limited angular resolution of current instruments in gamma-ray astronomy, spectral analysis is often the only tool for the investigation of gamma-ray sources. Therefore this thesis includes studies to test which spectral analysis methods can lead to biased parameter estimators.
Supervisor: Werner Hofmann (MPIK)
Carolin Wittmann (Germany) 09.02.2018
Ultra-compact and ultra-diffuse stellar systems in nearby galaxy clusters: signs of environmental influence? ( thesis pdf, 40 MB)
In this thesis we investigate ultra-compact and ultra-diffuse stellar systems in the cores of the nearby Perseus and Fornax galaxy clusters for signs of environmental influences. We search for possible disturbances of their stellar structures by examining their light distributions in deep optical wide field imaging data. In the Fornax cluster we analyse a sample of 355 spectroscopically confirmed compact stellar systems. Our data reveal that many objects show distorted outer structures, although we do not find long tidal streams around any of them. We investigate their spatial and phase-space distributions, and interpret our results in the framework of proposed formation scenarios. In the Perseus cluster we identify a population of 89 diffuse low surface brightness galaxy candidates for which we perform photometry. The majority of the diffuse candidates appear unperturbed based on their stellar structures. We find, however, that galaxies with large sizes seem to be absent in the dense cluster core region. We discuss possible implications for the dark matter content of these systems and compare our sample to faint low surface brightness galaxies in the Coma cluster. Our data reveal a few low-mass galaxies with tidal tails or disturbed morphology, and several diffuse streams and tidal debris. Nevertheless, the number of recent galaxy disruption events seems to be very low in both the Perseus and Fornax galaxy cluster cores, indicating that most of the low-mass galaxy population was probably shaped at earlier epochs.
Supervisor: Thorsten Lisker (ARI)
Valeriy Vasilyev (Russia) 07.02.2018
Dynamical model atmospheres for the abundance analysis of pulsating stars ( thesis pdf, 15 MB)
The chemical composition of Cepheid variables can provide information on the chemo-dynamical evolution of the Galaxy and beyond. The standard method for determining atmospheric parameters and abundances of Cepheids is based on one-dimensional plane-parallel hydrostatic model atmospheres, where convection is treated by Mixing Length Theory. The aim of the thesis is to investigate the impact of the atmospheric dynamics on observable spectroscopic properties. Two approaches are followed: firstly, I construct one-dimensional pulsating atmosphere models implementing a non-local, time-dependent theory of convection, and secondly check the validity of the quasi-static approach against a t wo-dimensional dynamical Cepheid model.
The spectroscopic analysis of the classical Cepheid KQ Scorpii with my one-dimensional model showed that pulsations do not produce strong enough velocity gradients in the line-formation region to explain the estimated microturbulent velocities. The spectroscopic investigation of the two-dimensional Cepheid model allowed to explain the residual line-of-sight velocity of Galactic Cepheids, long known as the "K-term", by lineshifts of convective origin. Moreover, hydrostatic 1D model atmospheres can provide unbiased estimates of stellar parameters and abundances of Cepheids for particular phases of their pulsations. Summarizing, the main result is a change of paradigm in the context of spectroscopic investigations of Cepheids toward a greater importance of convection than thought previously.
Supervisor: Hans-Guenther Luwdig, Norbert Christlieb (LSW)
Adriana Pohl (Germany) 23.01.2018
Structure of Planet-forming Disks: Multi-wavelength Polarization Diagnostics ( thesis pdf, 10 MB)
The study of dynamic processes that drive the evolution of planet-forming disks is fundamental to understand the origin and diversity of planetary systems. This requires observations at high spatial resolution and sensitivity, which nowadays typically reveal intriguing disk substructures including gaps, rings, spirals, and shadows. This thesis investigates the capability of polarization observations at multiple wavelengths to trace the earliest stages of planet formation. In-depth radiative transfer calculations are carried out in order to link numerical simulations of dust and gas evolution in disks with their observational indicators. This approach demonstrates that measuring polarization is a powerful tool to identify the shaping effects that possible embedded planets have on the density distribution of different dust grain sizes. On the observational part, this work presents several case studies of individual planet-forming disks that were observed with polarimetric imaging by the VLT/SPHERE instrument and subsequently modeled to quantify their structure. A particular focus is the characterization of spiral and ring/gap structures in the context of dust growth, planet-disk interactions, and dust dynamics near ice lines. Furthermore, a modeling study of marginally gravitationally unstable disks is presented to study the influence of the disk self-gravity on the shape and contrast of planet-induced spiral arms in scattered light images. Additionally, it is demonstrated that polarized emission of disks at millimeter wavelengths can be caused by self-scattered thermal dust emission. It is shown that the latter is a viable method to constrain grain properties and identify dust concentrations of different origin. New ALMA observations are presented that offer the first look at a dust trap in polarized scattered light in the sub-millimeter range.
Supervisor: Thomas Henning (MPIA)
Roxana Chira (Germany) 22.01.2018
On Filaments within Molecular Clouds and their Connection to Star Formation ( thesis pdf, 15 MB)
In recent years, there have been many studies on the omnipresence and structures of filaments in star-forming regions, as well as the role of their fragmentation in the process of star formation. However, only a few comprehensive studies have analysed the evolution of filaments and their distribution with the Galactic disk where the filaments form self-consistently as part of large-scale molecular cloud evolution. In this thesis, I study the effect of inclination on dust observations of filaments to evaluate whether the variations would enable the identification of further filaments in existing dust surveys. I address the early evolution of pc-scale filaments that form within individual clouds and focus on the questions how and when the filaments fragment, and how the fragmentation relates to typically used observables of the filaments. I perform dust radiative transfer calculations on models of cylinders and reconstructions of observed star-forming regions. For evaluating the equilibrium state of filaments and the nature of their fragmentation I examine three simulated molecular clouds formed in kpc-scale numerical simulations modelling a self-gravitating, magnetised, stratified, supernova-driven interstellar medium. I find that the observables of filaments in dust emission are on average on small scales influenced by inclination; yet the variations strongly depend on the structure of the object. The first fragments appear when the line masses of the simulated filaments lie well below the critical line mass of Ostriker's isolated hydrostatic equilibrium solution. This indicate that, although the turbulence of the entire clouds is mostly driven by gravitational contraction, fragmentation does not occur do to gravitational instability, but is supported by colliding flow motions. I conclude that there is no single quantity in my analysis that can uniquely trace the inclination and 3D structure of a filament based on dust observations alone. A simple model of an isolated, isothermal cylinder may not provide a good approach for fragmentation analysis, independently of the dominant driving source of the parental cloud.
Rainer Weinberger (Germany) 17.01.2018
Supermassive black holes and their feedback effects in galaxy formation ( thesis pdf, 9 MB)
Supermassive black holes play a key role in modern galaxy formation research. They are conjectured to be present in almost all massive galaxies, and through the release of enormous amounts of energy triggered by gas accretion, they are able to substantially change the properties of the host galaxy. To which extent and how the interaction mechanisms work is an open question. In this thesis, I review the current state of galaxy formation research with a focus on cosmological simulations of structure formation as well as the basic theories of supermassive black holes as far as they are important for galaxy formation. Subsequently, I discuss a new model for black hole growth and feedback in cosmological simulations, along with its application in large cosmological volume simulations. I show how supermassive black holes affect the formation and evolution of their host galaxy as well as their own growth. Furthermore, I present a model for supermassive black hole jets in a galaxy cluster environment. Applying this model, I study the coupling between the jet and the surrounding intra-cluster gas.
Supervisor: Volker Springel (HITS) | CommonCrawl |
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