text:
Glioblastoma are malignant tumors associated with a very poor
prognosis, due to the capacity of individual glioma cells to invade
surrounding normal brain tissue, far from the tumor focal area.
This infiltration results in the inability to completely resect
this tumor and is responsible for the almost inevitable recurrence
after treatment.
Cell-cell (homotype) as well as cell-substrate (heterotype) interactions
are key events in the migration process.
We have developed a cellular automaton where the strength of each
type of interaction is ajustable, to describe the migration of
glioma cells [1,2].
From this automaton, we were able to derive a macroscopic equation
of diffusion, where the diffusion coefficient is original compared
to other classical models[3]. First, it is nonlinear as it depends
on the cell density. Second, it depends on the two parameters
measuring the strength of homotype and heterotype interactions.
Here, we use this nonlinear diffusion coefficient in a diffusion-proliferation
equation to model the growth of glioblastoma. We define two cell
populations, characterized by different homotype and heterotype
interaction parameters and a proliferation rate that depends on
the strength of heterotype interactions. First, we study the interplay
between cell-cell and cell-matrix interactions during cell migration
in vitro on different substrates, and we reproduce some experimental
results [4].
We also compare the effects of classical treatments (surgery,
radiotherapy) for different values of homotype and heterotype
interaction parameters and we show that inhibing heterotype (or
increasing homotype) interactions (by inhibiting gap-junctions
or intergrins for example) in the margin of an operated tumor
could have a clinical interest, by reducing the chances of recurrence.
[1] Aubert M, Badoual M, Fereol S, Christov C and Grammaticos
B, 2006, A cellular automaton model for the migration of glioma
cells, Phys. Biol. 3 93.
[2] Aubert M, Badoual M, Christov C and Grammaticos B, 2008,
A model for glioma cell migration on collagen and astrocytes,
J. R. Soc. Interface, 5,75-83.
[3] Tracqui P, Cruywagen GC, Woodward DE, Bartoo GT, Murray JD
and Alvord EC Jr, A mathematical model of glioma growth: the effect
of chemotherapy on spatio-temporal growth, Cell Prolif, 1995,
28, 17-31.
[4] Giese A, Loo MA, Tran N, Haskett D, Coons SW, Berens ME,
1996, Dichotomy of astrocytoma migration and proliferation, Int.
J. Cancer 67, 275-282.
Weak solutions for a two-sidedly degenerate chemotaxis model
with volume-filling effect related to the p-laplacian operator
Ricardo Ruiz Baier
Departamento de Ingenieria Matematica, Universidad de Concepcion,
Coauthors: Mostafa Bendahmane, Departamento de Ingenieria Matematica,
Universidad de Concepcion, firstname.lastname@example.com Raimund Bürger,
Departamento de Ingenieria Matematica, Universidad de Concepcion,
email@example.com José Miguel Urbano, Departamento
de Matemática, Universidade de Coimbra, Portugal, firstname.lastname@example.com
We address the question of existence and Hölder regularity
of weak solutions for a fully parabolic model for chemotaxis with
volume-filling effect, that degenerates in a two-sided fashion,
including a p-Laplacian diffusion term. The relevant system is
suplemented with nonlinear Neumann boundary conditions. For the
proof of existence of weak solutions we use a Schauder fixed-point
argument on a regularized problem and the compactness method,
and for the regularity, we use the rescaling method.
**Simulated Two-Dimensional Red Blood Cell Motion, Deformation,
and Partitioning in Microvessel Bifurcations
Jared Barber
Program in Applied Mathematics, University of Arizona
Coauthors: Jonathan P. Alberding, Juan M. Restrepo, Timothy W.
Movement, deformation, and partitioning of mammalian red blood
cells (RBCs) in diverging microvessel bifurcations are simulated
using a two-dimensional, flexible-membrane model. A set of viscoelastic
elements represents the RBC membrane and the cytoplasm. These
elements are coupled to finite elements that represent the surrounding
fluid and the coupled system is numerically solved. Simulated
isolated RBC trajectories deviate from background flow streamlines
primarily because of cell migration towards vessel centerlines
and cell obstruction of downstream vessels. Estimates of RBC distributions
at a bifurcation are determined as a function of total blood fluxes
into the two branches and upstream RBC spatial and velocity distributions.
RBCs preferentially enter the higher-flow branch, leading to unequal
RBC fluxes in the downstream branches. Cell migration gives cells
a stronger tendency to enter the high flow branch. Cell obstruction,
on the other hand, counteracts this tendency. In unequally-sized
daughter vessels, partitioning is asymmetric, with RBCs tending
to enter the smaller vessel. Partitioning is not significantly
affected by the daughter vessel orientations. Significant differences
are found between rigid particle and flexible cell distributions.
Predicted distributions with flexible cells are consistent with
experimental observations, showing that membrane flexibility is
an important factor determining realistic RBC distributions in
Modelling the effects of disease on the speed of ecological
Sally S Bell
Heriot-Watt University, Edinburgh
Coauthors: Dr. Andy White (Heriot-Watt University, Edinburgh)
and Prof. Mike Boots (University of Sheffield, Sheffield)
We use mathematical models to analyse how a shared disease affects
the spread of an invasive species. We examine the temporal dynamics
and compare the speed of invasion with and without disease. This
work highlights the importance of the relative effects of different
disease parameters when considering the speed of invasion and
the speed of replacement of the native species. A spatial extension
of the model shows a wave of disease, instigated by the initial
introduction of the invading species, spreads through the native
population in advance of the wave of invasion. The presence of
the disease allows the invading species to extend the range over
which it replaces the native species. These results therefore
have important conservation implications.
The effects of behavioural patterns on the potential success
of the HPV vaccine.
Victoria Brown
University of Bath, UK
Coauthors: Dr. K. A. J. White (University of Bath)
In recent years, links between the human papillomavirus (HPV)
and cervical cancer in women has led to the development of a vaccine
to protect against HPV as a preventative measure for cervical
cancer (Arbyn and Dillner 2007). A national vaccination pilot
in the UK targets vaccination at pre-teenage girls, the majority
of whom are not yet sexually active (JCVI HPV Sub-group minutes,
September 2006); discussion of other target groups, such as the
equivalent male cohort, is also taking place.
The vaccine is administered on three separate occasions and information
from a clinician suggests that if one of these is missed, then
the vaccine will not be an effective barrier to HPV. The cost
of the triple dose is £240.
We are interested in exploring the impact of the vaccine in relation
to behavioural traits demonstrated at the population level. In
particular, given that the vaccine may only be effective for 5-10
years, how does the onset of sexual activity in the population
affect the efficacy of the vaccine in eradicating the virus?
To address such questions, we develop a model system which describes
the spread of the virus in a heterosexual population (males and
females are explicitly modelled), where individuals may either
be sexually active or not. We focus the analysis of the model
on the interaction between three parameters: the proportion of
the female population that is initially vaccinated, the rate at
which females become sexually active and the proportion of vaccinated
individuals that are still sexually inactive when the vaccine
loses its efficacy. Model parameters are estimated using published
We discuss the results of our analysis; in particular, we describe
the conditions under which HPV might be eradicated from a population
with different behavioural patterns. We will discuss extensions
to this work; in particular, the extension to consider an explicit
age dependent model.
Arbyn and Dillner 2007: Marc Arbyn and Joakim Dillner, Review
of current knowledge on HPV vaccination: An Appendix to the European
Guidelines for Quality Assurance in Cervical Cancer Screening,
Journal of Clinical Virology, 38 (2007).
JCVI HPV Sub-group minutes: Department of Health, Joint Committee
on Vaccination and Immunisation, Minutes of the HPV Sub-group
meeting, Wednesday 22 September.
Quantifying the advantages of early intervention on Lupus
Nephritis flares
Paula Budu-Grajdeanu
Mathematical Biosciences Institute
Coauthors: Richard C. Schugart, Avner Friedman, Daniel J. Birmingham
and Brad H. Rovin
Although the prognosis for lupus nephritis (LN) has dramatically
improved with aggressive immunosuppressive therapies, the current
approach to treatment remains inadequate. It is likely that combinations
of biomarkers will be needed to accurately describe the evolution
of LN flare and to forecast impending LN flares. To demonstrate
the utility and feasibility of lupus biomarker development, we
develop a mathematical model of lupus renal flare that quantifies
reduction in immunosuppressive medication as a function of how
early flare is detected. The results suggest that early initiation
of treatment is advantageous. Further refinement of the model,
incorporating clinical and biomarker data, will benefit patients
by allowing physicians to individualize immunosuppressive therapies.
Phase Models with Time Delay
Sue Ann Campbell
University of Waterloo
Coauthors: Ilya Kobelevskiy and Andrew Smith
We consider a network of inherently oscillatory neurons with
time delayed connections. We reduce the system of delay differential
equations to a phase model representation and show how the time
delay enters into the reduced model. For the case of two neurons,
we show how the time delay may affect the stability of the periodic
solution leading to stability switching between synchronous and
antiphase solutions as the delay is increased. The results of
the phase model analysis are compared with numerical bifurcation
analysis of the full system of delay differential equations. Both
type I and type II oscillators are considered.
Modeling of cancer virotherapy with recombinant
measles viruses
Thomas W. Carr
Southern Methodist University
Coauthors: Zeljko Bajzer (Mayo Clinic College of Medicine) Kresimir
Josic (University of Houston) Stephen J. Russell (Mayo Clinic
College of Medicine) David Dingli (Mayo Clinic College of Medicine)
The Edmonston vaccine strain of measles virus has potent and
selective activity against a wide range of tumors. Tumor cells
infected by this virus or genetically modified strains express
viral proteins that allow them to fuse with neighboring cells
to form syncytia that ultimately die. Moreover, infected cells
may produce new virus particles that proceed to infect additional
tumor cells. We present a model of tumor and virus interactions
based on established biology and with proper accounting of the
free virus population. The range of model parameters is estimated
by fitting to available experimental data. The stability of equilibrium
states corresponding to complete tumor eradication, therapy failure
and partial tumor reduction is discussed. We use numerical simulations
to explore conditions for which the model predicts successful
therapy and tumor eradication. The model exhibits damped, as well
as stable oscillations in a range of parameter values. These oscillatory
states are organized by a Hopf bifurcation.
An epidemic model with host cross-immunity to a continuum
of strains.
Farida Chamchod
Department of Mathematical Sciences, University of Bath, Bath
Coauthors: Nicholas F Britton
Several models in multi-strain diseases have included cross-immunity.
Lots of them are complicated to deal with when the number n of
strains increases. A history-based model for example contains
2^n+n*2^(n-1) variables. However, in a status-based model the
number of variables is 2^n+n-1 and by the reduced-transmission
and polarized immunity assumptions the numbers of variables in
total is drastically reduced to 2n. In this work, the model is
based on the status-based formulation. We use a real line with
each point representing a strain with a particular antigenic make-up
to represent a phylogeny of the diseases. The number of equations
is now 2. We then study a travelling wave of the system and how
it depends on other parameters such as the mutation rate, the
basic reproductive ratio, and the cross-immunity coefficient.
From the result, we conclude that the travelling wave represents
an antigenic drift process with strains present in the population
dying out and being replaced by new ones at new points in the
antigenic space.
A Go or Rest model for
cell migration. A step forward toward the Go or Grow
Dr. Arnaud Chauvière
Technische Universitaet Dresden - Germany
Coauthors: Haralambos Hatzikirou and Andreas Deutsch
Cell migration is an essential feature of, either physiologic
or pathologic, phenomena in biology, such as embryonic development,
wound healing or tumor invasion. According to the local microenvironment
and the cell function, the characteristics of the migration may
vary considerably.
Here we look closer at the influence of the cell density on the
migration dynamics, and we assume two different regimes: when
cells are isolated, the corresponding motion is essentially characterized
by a sequence of runs separated by random reorientations
of the velocity; in denser areas, migrating cells interact with
other cells and collision effects become relevant.
Additionally a resting regime is included in the migration
modelling. This can either result from environmental conditions
or relates to a strategy of cells to fulfil efficiently their
function. As an example, cells undergo mitosis only under favourable
environmental conditions, and an immotile state is then required.
A second illustration is the Go or Grow hypothesis
currently accepted in the biology of brain tumor invasion.
A kinetic (mesoscopic) model is first derived and a continuous
(macroscopic) model is deduced as its diffusive limit. This so-called
Go or Rest model provides anomalous diffusion which
is furthermore analyzed.
The model is then extended to include proliferating phenomena.
The study of the invasive front will aim to understand heterogeneous
patterns observed in tumor invasion.
Turing Pattern Formation
in Stochastic Reaction-Subdiffusion Systems
Jiawei Chiu
A*STAR Institute of High Performance Computing
Coauthors: K.-H. Chiam
We investigate the formation of spatial
Turing patterns in stochastic reaction-diffusion systems
that arise commonly in biology, such as the motion of proteins
in a crowded cytoplasm, or the migration of epithelial cells driven
by active biological processes. Here, we consider the case of
two species of "particles, " be they proteins or cells,
performing continuous random time walks specified by the probability
distribution function yi(x, t) = m(x)Wi(t), where the Laplace
transform of the Wi's is ~ 1 - (hi s)a and a < 1 denotes subdiffusion.
We seek to understand under what conditions there is Turing instability
by performing linear stability analysis. We find that, if we fix
the ratios between the diffusive constants, there is a critical
value of a below which there is no pattern. We discuss the relevance
of this critical value to several biological processes. In addition,
we carry out simulations based on Gillespie's algorithm to study
the effect of noise, induced by the low copy number of particles,
on the conditions for Turing pattern formation. We find that Turing
patterns can survive in a very noisy system.
Low red cell production may protect against
severe anemia during a malaria infection Insights from
Deborah Cromer
Imperial College, London
Coauthors: Jaroslav Stark, Miles P Davenport
The malaria parasite causes lysis of red blood cells, resulting
in anemia, a major cause of mortality and morbidity. Intuitively,
one would expect the production of red blood cells to increase
in order to compensate for this loss. However, it has been observed
that this response is weaker than would be expected. Furthermore
iron supplementation for iron deficient children in malaria endemic
regions can paradoxically adversely affect the clinical outcome
of malaria infection. A possible explanation may lie in the preference
that some malaria parasites show for infecting immature red blood
cells (reticulocytes). In the presence of a parasite preference
for immature red cells, a rise in red cell production can fuel
the fire of infection by increasing the availability of
the parasites preferred target cell.
We present a mathematical model of red blood cell production and
infection in order to explore this hypothesis. We assess the effect
of varying the reticulocyte replacement rate and preference of
the parasite for reticulocytes on four key outcome measures assessing
anemia and parasitemia.
For a given level of parasite preference for reticulocytes we
uncover an optimal erythropoietic response which minimizes disease
severity. Increasing red blood cell production much above this
optimum confers no benefit to the patient, and in fact can increase
the degree of anemia and parasitemia. These conclusions are consistent
with epidemiological studies demonstrating that both iron deficiency
and anemia are protective against severe malaria, whilst iron
supplementation in malaria endemic regions is with an increased
number of malaria related adverse effects. Thus, suppression of
red blood cell production, rather than being an unfortunate side
effect of inflammation, may be a host protective effect against
severe malarial anemia.
Ecosystem engineering in predator-prey
Kim Cuddington
Ohio University (University of Waterloo as of August 2008)
Coauthors: Alan Hastings (UCDavis) Theresa Talley (UCDavis)
Predator-prey interactions are described as having a negative
impact on the prey species and a positive impact on the predator
species. We develop a general model to describe how ecosystem
engineering may alter the net effect of trophic species interactions.
We modify a standard predator (P) - prey (N) population model
to include the effects of an environmental state (E). We describe
the modification of the environmental state as a linear function
of predator or prey density. The environment has a normal state,
E0, to which it will return in the absence of environmental modification
by either of the two species, and either or both of these species
can move the environment away from this state. Throughout we make
the simple assumption that the modified environmental state could
linearly modify all population parameters as:
dN/dt=[(a0 + a1 E) - (b0 + b1 E)N - (c0+c1E)P]N
dP/dt = [(f0+f1E)-(g0+g1E)P+(h0+h1E)N]P
dE/dt = -k(E-E0)+mP+nN.
We apply this model to a conservation problem in a crayfish-dragonfly
system. We derive the conditions where the crayfish predator may
benefit the endangered dragonfly species through the building
of burrows. The net impact of the crayfish on the dragonfly will
be positive where the product of the environmental modification
rate and the benefit of the burrow to the dragonfly species is
greater than the product of the rate at which the environment
returns to its unmodified state and the predation rate (a1 m >
c0 k). As a result, we predict that the relationship between the
predator and prey species could swing back and forth between a
net positive and a net negative interaction with environmental
Using Phylogenetics and Mutual Information
to Identify Coevolving Sites in Protein Families
Christopher DeHaan
The University of Western Ontario
Coauthors: Lindi Wahl, Andrew Fernandes
Proteins with similar functions found in different organisms
can be aligned in a multiple sequence alignment (MSA), and standard
techniques are now available to infer the phylogenetic tree which
relates these sequences in evolutionary history. A number of recent
papers have elucidated the use of Mutual Information (MI) in identifying
positions within such a protein family which co-evolve or are
in contact in the folded structure. However, to date, none of
these MI methods have made use of the phylogenetic history of
the proteins in the MSA. I have used the inferred phylogeny of
an MSA to simulate realistic random protein sequences, but with
no interdependence on any of the individual positions. Initial
amino acid distributions and their mutation probabilities are
based on observed data (a contingency table). This allows the
computation of a distribution of MI in the absence of interdependence,
which serves as a null distribution which is specific to a particular
MSA. Positions in the real MSA which share significantly more
MI than the simulated null distribution can then be identified
with some certainty. This method leads to more accurate identification
of protein active sites and sites which are in contact, both of
which are critical in determining the protein mechanism and structure.
Modeling the migration of cancer cells:
from microscopic to macroscopic models
Christophe Deroulers
Laboratory IMNC, Campus d'Orsay, 91406 Orsay Cedex, France
Coauthors: Mathilde BADOUAL, Marine AUBERT, Basile GRAMMATICOS
It is well known that the migration of cancer cells plays a key
role in the development of some brain tumors, such as gliomas.
Because cancer cells invade tissues far from the tumor center,
the tumors have no sharp boundary. A surgeon cannot remove all
cancer cells, and cancer reccurs invariably.
Therefore, it is crucial to take cell migration into account
in the modelling of glioma. This is easy in a cellular automaton-like
model, where some stochastic rules tell how the individual cells
move, duplicate and die, and some of us used such a model to reproduce
in vitro experiments of cancer cells migration and to show that
cells interact while migrating [1]. However, a cellular automaton
is not so convenient as partial differential equations (PDEs)
that are often used to model the density of cancer cells in the
brain. Especially, it is not suited for the study of real-size
tumors with several millions of cells. Therefore, it would be
nice to derive some PDE that takes the migration and interaction
of cancer cells into account.
We give one analytic technique to go from the definition of a
microscopic model (a cellular automaton) to a macroscopic model
(a PDE). We apply this technique to the situation of [1] and we
show that our PDE reproduces both simulation and in vitro experimental
results [2]. We notice that the PDE we obtain is, because of the
interaction of cancer cells, a nonlinear diffusion equation (belonging
to the family of the porous media equations), whereas it is often
postulated that diffusion of cancer cells is linear. Interestingly,
we notice that our model is closely related to some kinetically
constrained models introduced for the study of the physics of
glasses, supercooled liquids and jamming systems.
We also give some results on the effect of taking into account
the shape of cancer cells, which are not point-like as assumed
in [2] but elongated, and on dealing with statistical correlations
of the position of cancer cells.
[1] M. Aubert, M. Badoual, S. Fereol, C. Christov and B. Grammaticos,
A cellular automaton model for the migration of glioma cells,
Phys. Biol. vol. 3 p. 93 (2006).
[2] C. Deroulers, M. Badoual, M. Aubert and B. Grammaticos, Modelling
tumour cell migration: from microscopic to macroscopic, submitted
Stoichiometric network analysis and
graph theoretic methods for studying spatial models of chemical
reaction networks
Mirela Domijan
Unversity of Warwick
Chemical reaction networks play an important role in understanding
the biological processes that take part on a cellular level. However,
these networks still continue to be a challenge to model and analyse.
If we take a deterministic approach and we assume that the chemicals
involved cannot diffuse, the chemical reaction networks are modelled
by systems of ODEs. Modelling of chemical interactions can involve
many chemicals and complex interactions, resulting in large ODE
systems with nonlinear terms. If there is lack of quantitative
information about the interactions, there will also be parameter
uncertainty in the models. These issues can make it impossible
to numerically simulate the system dynamics or to perform bifurcation
analysis. In such cases, analysis can be successfully performed
via graph theory and stoichiometric network analysis. Analysis
via these methods does not depend on the unknown parameters. Instead,
they relate system's dynamic properties to network structure and
more specifically, to reaction stoichiometry.
Yet, in certain cellular processes reactions may not proceed
in "well-mixed" environments and chemicals may diffuse.
Then, spatial effects on the chemical dynamics need to be considered
and analysis of appropriately-constructed systems of reaction-diffusion
PDEs needs to be performed. Diffusion can lead to new behaviors
in the PDE systems, such as diffusion-driven (Turing) instability.
This instability constitutes the basis of Turing's mechanism for
pattern formation. Turing instability has been well described
for systems of two chemicals, however, theory is still lacking
for large systems. We will present concepts from stoichiometric
network analysis and related graph theory that can be applied
to analysis of spatial reaction networks. The conditions will
address the occurrence of Turing instability. A Turing unstable
two-chemical system relies on the activation and inhibition between
the two chemicals. The conditions that we will give generalise
the conditions of positive/negative feedback cycles for Turing
Random Spatial Networks: A Biological Solution
to the Structure/Transport/Connection Problem
Donald A. Drew
Rensselaer Polytechnic Institute
Coauthors: Yanthe Pearson
Spatial networks are collections of fibers occupying a spatial
region. The network is called random if we are interested in an
ensemble of equivalent such networks, where the positions of the
individual fibers are not identical but have some statistical
sameness. Examples of random statistical networks
include microtubule structures, capillaries, neurons, and trees.
Each of these networks has a function that it can fulfill by producing
a realization out of an ensemble of such networks having certain
properties. For example, a microtubule network that is responsible
for cell integrity must support the forces that maintain the cell
shape; capillaries must deliver and/or absorb chemical species
to the surrounding matrix. We discuss microscale (individual fiber)
and structural (probability density function) models to describe
random spatial networks. so as to relate network statistical structure
to assembly dynamics of individual network fibers, and to determine
the biological functionality of the network from network statistical
properties. In addition, network assembly, disassembly, and interactions
with the surroundings during network formation and structure can
add to the understanding of the biochemistry and biophysics of
fiber formation and guidance.
We shall focus on axonogenesis. Axons are the propagation elements
in neurostructures in all higher species. During formation of
the brain and nervous system, axons are generated by extension
of processes from neural cells with dynamics determined at the
growth cone. The progress of the growth cone is determined by
the response of surface receptors to gradients of signaling molecules.
Surface structures bind the signaling molecules, leading to changes
in the assembly of the actin/microtubule structure that drives
axonal growth cone motion. In this paper we introduce a two-dimensional
stochastic model which captures the random behavior of axon growth
to simulate axonal trajectories for cells in a homogeneous medium.
We use data to evaluate the standard deviation of the angle changes
on the axonal trajectories and to verify the validity of the structure
of the stochastic differential equations for the axonal trajectories.
For the model of growth of axons, we analyze trajectory data consisting
of measurements of the position of the axon tip at different frames
in the time sequence of micrographs. This data shows that the
axon tip changes direction randomly, but the data is noisy due
to the data collection procedures. We develop algorithms to filter
out noise while maintaining the underlying dynamics of the axon
growth process. We perform statistical analyses on relevant variables
generated from our filtered data. We present Monte Carlo simulations
of stochastic differential equation systems.
Analysis of serial engagement and peptide-MHC
transport in T cell receptor microclusters
Omer Dushek
University of British Columbia
Coauthors: Daniel Coombs
During stimulation of a T cell by an antigen-presenting-cell
(APC) bearing cognate peptide-major-histocompatibility complexes
(pMHC), T cell receptors (TCR) have been shown to form stable
micrometer-scale clusters in the contact region. pMHC molecules
diffusing in the APC membrane may bind and unbind from multiple
TCR in a cluster. Such serial engagements of multiple TCR by a
single pMHC have been hypothesized to be important in T cell signal
amplification. We use mathematical modeling to characterize the
number of clustered TCR bound by a single pMHC. We show that the
TCR-pMHC bond kinetics alone do not allow substantial serial engagement
of TCR and suggest molecules that could enhance TCR engagements.
Mathematical tools: MFP calculations, asymptotic analysis, numerical
solutions of PDEs.
Epidemic control in weighted social networks
Ken Eames
University of Cambridge
Social networks provide a valuable tool for understanding the
link between population mixing behaviour and epidemic dynamics.
This understanding motivates the design and development of targeted
epidemic control through the identification of the most high-risk
individuals. Social mixing surveys demonstrate clearly that not
all interactions are of equal strength and suggest the use of
weighted networks to capture variations in contact intensity.
Here, we present the results of simulation models used to investigate
targeted interventions in weighted networks; we compare the use
of different individual-based measures of risk to prioritise individuals
for intervention and conclude that the existence of weighted social
networks offers new challenges and new opportunities for disease
control. We discuss the data requirements of proposed intervention
strategies and present some new data on the mixing patterns of
one of the most epidemiologically significant population subgroups:
school children from the ages of 5 to 11.
A Mathematical Model for the Effects
of HER2 Over expression on Cell Cycle Progression in Breast Cancer
Amina Eladdadi
Department of Mathematical Sciences, Rensselaer Polytechnic Institute,
Troy, NY 12180, USA
Coauthors: David Isaacson
We present a mathematical model to study the effects of HER2
over-expression on cell-cycle progression in breast cancer. The
model addresses the following question: How do changes in the
number of HER2 and EGFR receptors during the cell-cycle affect
the cell proliferation rate? In order to characterize the effects
of HER2 over-expression on the cell cycle progression, we use
a three-compartment cell cycle model with non-constant transition
rates. Our new hypothesis is that the transition rates depend
on the number of the cell surface HER2 receptors and their signaling
properties. The model relates the different phases of the cell
cycle transition rates to the signaling properties of the EGFR-HER2
receptors (through their binding kinetics), and the population
dynamics of cells in the corresponding cell-cycle phase.
Modeling early tumor development dynamics
- implications for treatment design
Heiko Enderling
Center of Cancer Systems Biology, Caritas St. Elizabeth's Medical
Center, Tufts University School of Medicine
Coauthors: Afshin Beheshti, Lynn Hlatky, Philip Hahnfeldt
Cancer development may be considered an evolutionary process
whereby genetically unstable cell clones compete under selective
influences of the local environment and succeed in accordance
with the relative fitnesses of their expressed phenotypes. The
competition process involves traversal through a number of bottleneck
challenges at all phases of tumor development. One factor limiting
tumor cell proliferation is space to grow, a condition which may
be alleviated by cell death within the mass. We show theoretically
how tumor populations devoid of stem cells could still persist
as long-term dormant lesions, and offer a possible explanation
for the incidence of dormant tumors observed in recent autopsy
studies. This finding questions the notion that tumors escaping
dormancy will necessarily become symptomatic. Finally, if the
tumor population is assumed to contain cancer stem cells, we show
1) that certain conditions may paradoxically limit the growth
of the lesion, even if it escapes dormancy, and 2) the number
of stem cells can be amplified through adjustments in other parameters
that reduce the local density of progeny cells. The latter observation
lends support to the theory that tumors grow in part through the
creation and merging of local metastases. From the presented model
we derive implications for treatment.
Synchronization Of Insulin Secretion
Through Intrapancreatic Ganglia
Bernard Fendler
Florida State University
ß-cells are cells located in the human pancreas and are
known to produce electrical activity. When these cells are electrically
active, they secrete a hormone necessary for maintaining glucose
homeostasis in the blood called insulin. The ß-cells are
located in the pancreas in small micro-organs called islets of
Langerhans. There are thousands of islets in the pancreas which
are known to produce oscillatory insulin secretion. Measurements
of insulin have shown that oscillatory secretion also occurs in
the blood. Since plasma insulin reflects the secretion from the
entire islet population, oscillations in plasma insulin levels
suggest that islet oscillations must be largely synchronized.
Bertram et al. Bio. Phys. Jour., 92, 1544-1555, 2007
has developed a mathematical model of the ß-cell which reproduces
many of the measured electrical and calcium properties of the
ß-cell. We use this model to investigate methods of synchronization
of the islet population. The islet is known to be innervated by
neurons, in ganglia, interspersed throughout the pancreas. We
investigate the viability of islet synchronization by coordinated
action of the intrapancreatic ganglia.
Myotome formation and the importance of
Edward Flach
Innovative Methods of Computing, Technical University Dresden
Coauthors: Lutz Brusch, Andy Oates, Andreas Deutsch
Dramatic cell shape change and relative movement are readily
observable in the differentiation of muscle cells during early
embryonic development of zebrafish. An initial population of cuboidal
adaxial cells specified in the early embryo differentiate
into rod-shaped cells that form the slow-twitch muscle fibres
used by the fish to swim through the water. There is a complex
and stereotypical shape change and spatial rearrangement of the
cells at the collective level.
We consider the underlying dynamics of the cell required to effect
this change. We produce a minimal model which can reproduce the
behaviour. Using a simulation we investigate the balance of forces
Modeling Mammalian Circadian Rhythms
Daniel B Forger
University of Michigan
Biological circadian (~24-hour) clocks time many biological processes
that must occur at specific times of the day. Circadian behavior
in mammals is co-ordinated by a group of ~20,000 neurons in the
suprachiasmatic nucleus (SCN). The molecular basis for these clock
within each SCN neuron is a complex network of genetic feedback
Several large-scale detailed mathematical models of the molecular
biology and electrophysiology of the SCN will be presented. Predictions
from these models have large implications for sleep disorders,
the effects of genetic mutations, and how timekeeping in encoded
in firing rates.
I will also present several experimental studies which have confirmed
model predictions.
Channel mechanisms for structural basis
for the Hodgkin and Huxley relation
Charles M. Fortmann
Stony Brook university
Coauthors: Yeona Kang
Neural channel transport was analyzed using a previously reported
relation for charged particle transport in two energy-type gradients.
One energy type gradient is the electric field, expressible as
a concentration gradient along the axis of transport, the second
results from the transporting cation coupling with water and with
a neural channel deformation. Neural channels are lined with alpha
helix protein secondary structure that have near neutral charge
and are filled with water vapor and sequestered hydrophobic amino
acids arranged to present minimum water vapor and water-hydrophobic
interface. Cation point charges generate enormous electric fields
on sub-nanometer distances. Electrostatic energy reduction is
characterized by water, a strong dielectric, being pulled toward
the transporting ion, thereby deforming the neural channel structure.
An energy gradient results whenever the ion-water-structure coupling
energy is modified by changes in channel diameter and/or channel
deformation in the axial direction. The resultant two energy gradient
relation for cation transport: reduces to the Hodgkin-Huxley relation,
explains channel selectivity and environmental sensitivity, and
predicts fast non-dispersive transport under a narrow range of
conditions. The transporting cation-water-deformation model produces
current-voltage characteristics consistent with observation.
The effect of antioxidant supplementation
on the viral load of HIV
R D van Gaalen
Dept. of Applied Mathematics, University of Western Ontario
Coauthors: L M Wahl
A byproduct of cellular respiration, small highly reactive molecules
called reactive oxygen species (ROS) play a crucial role in cell
signalling and infection control. However, high levels of ROS
can cause significant damage to cell structure and function. Under
normal conditions, a healthy diet supplying adequate quantities
of antioxidants helps to maintain a safe level of ROS. Studies
have shown that infection with the human immunodeficiency virus
(HIV) results in an increase in oxidative stress. Acting like
a catalyst in nuclear factor kB activation, ROS in turn lead to
faster progression of HIV infection, and cause CD4+ T-cell apoptosis.
Clinical studies have explored the possibility of raising antioxidant
levels with mixed results. In this talk, a mathematical model
is used to explore this contested therapy.
Sexual Moran Model: Theory and Simulation
J M Grant
Applied Mathematics, University of Western Ontario
Coauthors: L Wahl (Western) and G Wild (Western)
We describe an extension of the birth-death Moran model, incorporating
sexual reproduction and a finite population size. In this model,
one male and one female parent are chosen to give birth, and neither
can be displaced by their offspring (in other words, neither parent
can be chosen for death in that time step). If the sex ratio is
fixed, we demonstrate analytically that the limiting distribution
for the number of females (or males) in the population is binomial.
We confirm this result numerically, by considering the eigenvalues
of the associated Markov transition matrix, and by individual-based
simulation. We also investigate social evolutionary questions
in the context of this model.
A Stochastic Survival Analysis for Phase
II Clinical Trials in Relapsed Ovarian Cancer - Non-Stationary
Poisson Process -
by R Gunawan
University of Waterloo
Coauthors: Tenti, G; Oza, A; Sivaloganathan, S
Ovarian cancer has long been known to be one of the leading causes
of cancer death. It commonly strikes women who are 50 years of
age or older and it has a poor prognosis. Our collaborators at
Princess Margaret Hospital in Toronto, Ontario, have conducted
various extensive clinical trials investigating molecularly targeted
agents in relapsed ovarian cancer patients. Our data were obtain
from four groups of phase II clinical trials: PHL019 (UCN-01 and
Topotecan), PHL025 (Sorafenib and Gemcitabine), PHL037 (AZD2171),
and PHL041 (PXD101). Patients participating in the clinical trials
were monitored regularly* according to the Response Evaluation
Criteria in Solid Tumor (RECIST) and the Gynecologic Cancer Intergroup
(GCIG). During each follow-up, the check-up date, the longest
diameter (LD) of the target lesion and the level of Cancer Antigen-125
serum, a surrogate marker for drug efficacy, were recorded. Survival
data were collected at the end of the study. If a patient was
still alive at the end of the study, she was assigned a 0;
whereas, if a patient died by the end of the study, she was assigned
a 1. From the data, we have demonstrated that the
survival time of the patients follows a non-stationary Poisson
process. This stochastic formulation leads us to a variety of
useful information such as average survival rates in each clinical
trial group. This could help determine the effectiveness of a
particular drug or combination of drugs. Additionally, we verified
whether CA125 was truly a marker for drug efficacy in these clinical
trial groups.
"Go or Grow": the key to
the emergence of invasion in tumor progression?
Haralambos Hatzikirou
TU Dresden
Coauthors: D. Basanta, M. Simon, C. Schaller, A. Deustch
Uncontrolled proliferation and abnormal cell migration are two
of the main characteristics of tumor growth. Of ultimate importance
is the question: what are the mechanisms that trigger the progression
from benign neoplasms (high proliferation) to malignant invasive
tumors (high migration)?. We show with a lattice-gas cellular
automaton that the transition to invasive tumor phenotypes can
be explained solely on the basis of the microscopic "Go or
Grow" mechanism (migration/proliferation dichotomy) and the
oxygen shortage, i.e. hypoxia, in the tumor environment. This
result challenges the currently prevailing view that the emergence
of invasiveness is mainly the consequence of acquired cancer cell
mutations. Moreover, we provide a theoretical explanation of our
results by means of a cut-off mean-field approach. Finally, we
suggest possible therapies that could help prevent the progression
towards malignancy and invasiveness of benign tumors.
The Inseparability of Spatial and Temporal
Clustering in a Population Model with Spatially Correlated Disturbances
David Hiebeler
Dept. of Mathematics and Statistics, University of Maine
Coauthors: Isaac Michaud (University of Maine) Nicholas Millett
(University of Maine)
Our prior work studied the effects of large-scale disturbance
events in a locally-dispersing spatial patch-occupancy population
model, where contiguous blocks of sites were simultaneously disturbed
in such a way that the per-site disturbance rate was kept fixed.
Results indicated that increasing the spatial scale of disturbance
events had a negative effect on equilibrium population density.
However, the reason for this effect was speculated to be possibly
due to different factors, such as slow recolonization of disturbed
regions via only local dispersal, or the increased temporal variability
that accompanied the increased spatial scale of disturbances.
Here, several variations of the model are explored via simulations,
to further explore why spatially correlated disturbances adversely
affect a locally-dispersing population, and to try to separate
the effects of spatial and temporal clustering in such disturbances.
Some issues related to efficient simulation of such models will
also be discussed.
Feedback, ratios and robustness in Hedgehog
David Irons
University of Sheffield, UK
Coauthors: Nick Monk (University of Nottingham, UK)
The secreted protein Hedgehog (Hh) acts a morphogen, forming
a concentration gradient and controlling cell fate decisions in
various developmental stages in many animals. Here we consider
Hh gradient formation in one such developmental context, the Drosophila
wing disc.
One evolutionarily conserved component of this pathway, of particular
interest, is a feedback loop where Hedgehog signalling up-regulates
its own receptor, Patched (Ptc). It has been suggested that this
feedback loop could enhance the robustness of the steady-state
Hh gradient against variability in Hh production levels. However,
previous models have failed to take into account two important
issues in the biological system. Firstly, interpretation of the
gradient by responsive cells (and hence Ptc production) is dependent
on the 'ratio' of unbound Ptc to bound Hh-Ptc in the cell. Here,
the Hh-Ptc complex 'dilutes' the inhibitive effect that Ptc has
on its production, altering the very nature of the feedback loop.
Secondly, total Hh levels and the expression range of downstream
target genes, such as Decapentaplegic (Dpp), are growing over
time, implying that both the formation and interpretation of the
gradient are dynamic throughout wing development.
In order to re-examine the formation and interpretation of the
Hh gradient, we present a new multi-cellular differential equation
model that represents the core logic of the feedback loop. The
model is centred on the three primary factors, Hh, Ptc and Hh-Ptc
complex; other components of the signaling pathway (such as Smo,
Ci and Cos2) are grouped together into a single 'pathway activity'
variable. We use this model to investigate the two biological
issues discussed above. In particular, we show that making Hh
gradient interpretation dependent on Hh-Ptc levels (as well as
on Ptc levels) results in a number of beneficial changes, including:
A refinement in signal interpretation, so that cells receiving
Hh levels just above / below a threshold are more likely to respond
in the appropriate way,
A boost in Ptc production in response to high levels of Hh, strengthening
the effect of the feedback loop,
Enhanced robustness of the expression ranges of downstream target
genes, in response to variable Hh levels.
We also use the model to examine how the expression ranges of
downstream target genes (e.g. Dpp) expand as the wing disc grows.
We show that making interpretation dependent on Hh-Ptc levels
(as well as on Ptc levels) has a limiting effect on this expansion,
indicating that the 'dilution' of Ptc function by Hh-Ptc also
plays a role in size regulation.
Evaluation of screening strategies for
pre-malignant lesions using a biomathematical approach
Jihyoun Jeon
Program in Biostatistics and Biomathematics, Fred Hutchinson Cancer
Research Center, Seattle, USA
Coauthors: 1. Rafael Meza, Division of Mathematical Modeling,
UBC Centre for Disease Control, Vancouver, Canada & Program
in Biostatistics and Biomathematics, Fred Hutchinson Cancer Research
Center, Seattle, USA 2. Suresh H. Moolgavkar, Program in Biostatistics
and Biomathematics, Fred Hutchinson Cancer Research Center, Seattle,
USA 3. E. Georg Luebeck, Program in Computational Biology, Fred
Hutchinson Cancer Research Center, Seattle, USA
We present mathematical expressions for the size distribution
of screen-detectable pre-malignant lesions, conditional on no
prior detection of cancer in the tissue of interest, based on
a general multistage clonal expansion model of carcinogenesis.
We apply these expressions to simulate the natural history of
colorectal cancer and to evaluate the effect of a screen for adenomatous
polyps and concomitant intervention on cancer risk. Our approach
allows the efficient simulation of multiple screens and interventions
and determination of the optimal timing of the screens. We further
demonstrate the utility of our approach by computing the benefits
of up to two colonoscopies on the lifetime risk of colorectal
cancer. If time permits, I will present some preliminary results
of the analysis of a screening trial using our methodology.
Can delays replace Phosphorylation
dephosphorylation cycles in signaling cascades?
Srividhya Jeyaraman
Indiana University School of Informatics and Biocomplexity Institute,
Bloomington, IN, USA
Coauthors: M. S. Gopinathan (Indian Institute of Information Technology
and Management, Kerala, India) Schnell, Santiago (Indiana University
School of Informatics and Biocomplexity Institute)
Many biochemical pathways operate on signaling cascades consisting
of a series of phosphorylationdephosphorylation (PD) cycles.
These cycles are coupled with several positive and negative regulations
causing inherent delays during signal propagation. Modeling these
complex signaling cascades with ordinary differential equations
(ODE) often requires a large number of variables and parameters.
Simplifying these ODE models with delays could prevent spending
time on redundant mechanisms and focus on the key regulators of
the dynamics.
Delay differential equation (DDE) models have been helpful in
the description of inherent time delays and in the reduction of
the number of variables [1]. However the consequences of model
reduction via DDEs have not been fully explored. We have systematically
examined the effect of delays in a complex network of PD cycles
[2], which commonly occur in many biochemical pathways [3]. By
introducing delays in the positive and negative regulatory interactions,
we show that a delay model can indeed reduce the number of PD
cycles and still describe the dynamics of the network effectively.
In my presentation I will show the effects of the delays and how
the results of this study can be extended to model complex biochemical
pathways [4] .
1. Smolen, P., D.A. Baxter, and J.H. Byrne, Biophys. J., 2002.
83(5): p. 2349-2359.
2. Gonze, D. and A. Goldbeter, Journal of Theoretical Biology,
2001. 210(2): p. 167-186.
3. Srividhya, J., M.S. Gopinathan, and S. Schnell, Biophysical
Chemistry, 2007. 125(2-3): p. 286-297.
4. Srividhya, J. and M.S. Gopinathan, Journal of Theoretical Biology,
2006. 241(3): p. 617-627.
A mathematical model for the formation
of feather germs
Charlotte Jupp
Centre for Mathematical Biology, Mathematical Institute, University
of Oxford
Coauthors: Dr. Ruth Baker, Centre for Mathematical Biology, Mathematical
Institute, University of Oxford. email@example.com; Prof. Philip
Maini, Centre for Mathematical Biology, Mathematical Institute,
University of Oxford. firstname.lastname@example.com
In developmental biology, a number of theories, none of which
fully explain biological results, have been suggested for the
mechanisms involved in the generation of spatial patterns. We
develop a cell-chemotaxis model to describe the pattern of forming
feather germs in the skin of chicken embryos. In our model the
cells are responding to a chemical gradient created by two morphogens
which react and diffuse according to a Turing mechanism. The model
will be used to analyse the effects of an increase in cell density
and morphogen concentration upon the system. We will show that
these results match the findings of biological experimentation
with regards to the number, size and spacing of feather germs.
We shall also demonstrate that our model is capable of replicating
both the simultaneous and sequential generation of feather germs
as occurs in vitro and in vivo, respectively.
Synchronization in inhibitory networks
Abdoul Kane
University of Toronto
Coauthors: Jonathan O. Dostrovsky, Frances K. Skinner
There is increasing evidence that large scale brain rhythms play
an important role in the execution and regulation of cognitive
and behavioral functions. Recent studies also highlight the importance
of inhibitory subnetworks in rhythm generation. This has motivated
the development of many experimental and computational models
seeking insights into the mechanisms underlying rhythms and pattern
generation. However very few analytical results are available.
We consider a biophysical model describing an inhibitory network
of neurons and investigate conditions under which certain specific
modes of activity can be observed. By applying techniques from
dynamical systems theory we derive a pared-down model that captures
the essential features of the model interneuron and also allows
an analytical treatment. We then consider a pair of such neurons
coupled through GABA-A type synapses and describe how the synaptic
time scales interact with the intrinsic dynamics to generate various
stable configurations depending on initial conditions and parameters.
Static Representation of Epidemics on Well-characterised
Dynamic Networks
Rowland R. Kao
University of Glasgow
Social network representations of populations are becoming an
increasingly useful tool when analysing epidemic structures. Most
analyses of social networks assume that contacts between nodes
(individuals) are fixed, i.e. a static network. In reality, most
links between individuals are dynamic, appearing and disappearing
over time. Here, I discuss a class of dynamic networks where epidemic
properties can be defined by an equivalent "static snapshot"
taken over the infectious periods of the individuals in the network,
using small-world networks and real datasets where contacts structure
and timing are precisely known, as examples.
Mathematical Modeling of the Blood-Atheroma
Plaque Interaction
Nader El Khatib
University of Lyon, France
Coauthors: Stephane Genieys firstname.lastname@example.com and Vitaly
Volpert email@example.com
The inflammatory reaction of atherosclerosis leads to the formation
of an atheroma plaque in the blood vessel. The interaction between
the blood and the plaque may have very dangerous consequences
such as the rupture of the plaque liberating solid parts in the
blood flow that can lead to a heart attack. The blood-plaque interaction
also produces some recirculations downstream of the plaque, and
these recirculations can give rise to the coagulation of the blood
and the formation of a clot that can block the blood flow too.
In this paper we study the interaction between the blood flow
and the atheroma plaque using a fluid-structure interaction model.
The blood is considered as a non-Newtonian fluid with a variable
viscosity defined by the Carreau's law. We investigate the influence
of this Non-newtonian variable viscosity on the plaque displacement
(and hence the risk of plaque rupture) and on the blood flow recirculations
(and hence the risk of blood clot formation). The atheroma plaque
is composed of a lipid pool and a fibrous cap and both are considered
as hyper elastic materials. The parameters of these materials
are taken from experimental data, as well as the parameters of
Carreau's law for the blood.
The simulations show that the usual Newtonian models significantly
underestimate the recirculations and overestimate the plaque displacement.
Analytical Study of Blood Flow with Periodic
Body Acceleration in the Presence of Magnetic Field
Dr Anil Kumar
Mathematics , Dornacharya College of Engineering , Greater Noida
UP India
Coauthors: Dr CL Varshney and Veer Pal Singh, Mathematics , SV
College Aligarh, UP India
In this paper we consider a human body which is quite often subjected
to acceleration under magnetic effects. The induced magnetic field
is neglected. Such acceleration can generate significant effects
on blood circulation depending on the configuration and geometry
of the blood arteries. The current study is concerned with a mathematical
model of the study of blood flow in the presence of magnetic effects
subject to externally imposed periodic body acceleration. By using
the finite Hankel Transform, an exact solution of the steady flow
of blood considered as an incompressible, couple stress fluid
has been obtained. The results have been compared with other existing
models. The effect of body acceleration on blood flow in the presence
of magnetic fields is analyzed.
Keywords: Magnetic Field; Couple Stress; Blood Flow; Periodic
Body Acceleration; Hypertension.
Tubuloglomerular Feedback Signal Transduction
In a Model of a Compliant Thick Ascending Limb
Anita T. Layton
Duke University, Department of Mathematics
We used a mathematical model to predict the impact of tubular
compliance on tubuloglomerular feedback (TGF) signal transduction
in the thick ascending limb (TAL). In several previous studies,
we used a mathematical model that represented the TAL as a rigid
tube. That model predicts that TGF signal transduction by the
TAL is a generator of nonlinearities: if a sinusoidal oscillation
is added to constant intratubular flow, the NaCl concentration
alongside the macula densa will be nonsinusoidal owing to an accumulation
of harmonics. We have hypothesized that complexity found in power
spectra based on in vivo time series of key TGF variables arises
in part from those harmonics and that nonlinearities in TGF-mediated
oscillations may result in increased NaCl delivery to the distal
nephron. To address the concern that a more realistic TAL would
damp harmonics, we have conducted simulations in a model TAL that
has compliant walls and thus a tubular radius that depends on
transmural pressure. These simulations predict that compliant
TAL walls do not damp, but instead, intensify the harmonics.
Reduction methods for multiple-time-scale
biochemical reaction networks
Chang Hyeong Lee
Worcester Polytechnic Institute
In this talk, we consider deterministic and stochastic descriptions
of reaction networks in which different reactions occur on at
least two distinct time scales. In the deterministic description,
based on perturbation analysis and complex reaction network theory,
we derive a necessary and sufficient condition under which there
is a complete separation of slow and fast variables, and we discuss
network topological properties which guarantee that the condition
is satisfied. Given this condition, we obtain an explicit expression
for the reduced equation on the slow time scale and we clarify
the geometric meaning of the reduction. In the stochastic description,
by applying a state space decomposition method, we rigorously
obtain the reduced master equation from the full master equation
of the system, which enables us to implement an efficient simulation
algorithm. Lastly, we illustrate the numerical accuracy and efficiency
of the reduction method by simulating several multiple-time-scale
deterministic and stochastic models including a stochastic reaction-diffusion
model of gene expression.
Rare event simulation for T cells recognising foreign antigens
Florian Lipsmeier
Bielefeld University
Coauthors: Ellen Baake (Bielefeld University)
We reconsider the problem of foreign-self distinction in immunobiology,
namely, the discrimination of foreign antigens against a background
of the body's own molecules. As is well known, the precise mechanism,
though one of the major tasks of the immune system, continues
to be a fundamental puzzle. We reconsider it here as a problem
of statistical recognition as recently formulated by van den Berg,
Rand and Burroughs [1], who modelled the probabilistic nature
of the interaction between T cells and antigen-presenting cells
(APC's). Here, the stochasticity is due to the random sample of
antigens present on the surface of every APC, and to the random
receptor type that characterises individual T cells. It has been
shown previously [1, 2] that this model, though highly idealised,
is capable of reproducing important aspects of the recognition
phenomenon, and of explaining them on the basis of stochastic
rare events. The `rare events' come into play here because the
probability that a randomly chosen T cell will be activated by
a randomly chosen APC is very low, whether the APC carries foreign
antigens or not. It is therefore adequate to use large deviations
theory, which characterises tail events. However, the results
obtained so far are asymptotic in nature; simulations have been
restricted to the straightforward simple sampling approach, which
does not allow for sample sizes large enough to address more detailed
questions. Building on the available large deviation results,
we develop an importance sampling technique here that allows for
a convenient exploration of the relevant tail events by means
of simulation [3]. With its help, we investigate the mechanism
of statistical recognition in some depth. In particular, we illustrate
how a foreign antigen can `stand out' against the self background
if it is present in sufficiently many copies, although no a priori
difference between self and nonself is built into the model. This
method will also allow to tackle models that are more realistic
than the basic caricature considered so far.
[1] Van Den Berg, H.A., Rand, D.A., Burroughs, N.J.: A reliable
and safe T cell repertoire based on low-affinity T cell receptors.
J Theor Biol 209(4), 465-486 (2001)
[2] Zint, N., Baake, E., den Hollander, F.: How T-cells use large
deviations to recognize foreign antigens. J Math Biol., in press
[3] Lipsmeier, F. , Baake, E. Rare event simulation for T cells,
in preparation
Spatially-localized scaffold proteins may
simultaneously boost and supress signaling
Xinfeng Liu
University of California at Irvine
Coauthors: Bardwell Lee and Qing Nie
During cell signaling, scaffold proteins are thought to promote
both signal transmission and specificity by binding to multiple
componnents of a given pathway, but the mechanisms by which they
accomplish this are unclear. In this talk, we develop a mathematical
model of generic, spatially localized scaffold protein. The model
indicates that a scaffold protein could boost signaling locally
(i.e. in and near the region where it was localized) while simultaneously
supressing at a distance. Furthermore, localization could switch
a scaffold from a global inhibitor to a local enhancer to distant
supressor. Distant supression was found to be due to reactant
sequestration. Thus, spatial localization increases the versatility
of scaffold proteins, and creates a novel mechanism by which they
can augment signaling specificity.
Elucidating the HPA axis stress response
via computational inverse analysis
James Lu
Johann Radon Institute for Computational and Applied Mathematics
Coauthors: Clemens Zarzer, RICAM; Rainer Machne, Theoretical Biochemistry
Group, University of Vienna; Gottfried Koehler, Max F. Perutz
Laboratories, University of Vienna
The hypothalamic pituitary adrenal (HPA) axis represents a feedback
system that plays an important role in maintaining the body homeostasis
in response to various stresses. When stress is encountered, the
hypothalamus releases the corticotropin releasing hormone (CRH)
as a central neuro-transmitter in the HPA axis. There exist diverse
differential equation models, which account for induction of ACTH
synthesis in the pituitary by CRH, leading to adrenal activation
and release of cortisol, which in turn inhibits the synthesis
of ACTH.
Starting from these basic models, several additional feedback
mechanisms could be included. One is the incorporation of an additional
membrane bound glucocorticoid receptor (GR) in the inhibition
of the ACTH release in the pituitary, responsible for fast feedback
effects. Including such model extensions, computational inverse
analysis is crucial in identifying the possible dynamical behaviors,
such as oscillations,modulated by circadian rhythms and switching
between multiple steady states.
To identify factors controlling the qualitative nature of the
stress responses, we apply the method of inverse bifurcation analysis,
using a hierarchical identification strategy based upon a sparse-promoting
regularization method. In particular, diseased phenotypes as represented
mathematically by the respective bifurcation diagrams are computationally
mapped to the underlying regulation mechanisms. For instance,
the identified mechanisms underlying the delayed activation of
the stress response include the degradation rate of GR as well
as the rate of up-regulation in the GR synthesis via its dimer.
In addition to mapping diseased phenotypes to possible underlying
mechanisms, inverse analysis can also point to mechanistic details
of the model that should be elucidated via experimental studies.
Noise-based rules govern neural circuit
Victor Luria
Columbia University, Department of Genetics and Development
Sensory-motor circuits are assembled by neurons whose axons execute
discrete, binary decisions at sequential trajectory selection
points. Motor axon trajectory selection is controlled by guidance
cues composed of ligands and receptors whose expression levels
are variable. Some cues direct axons to opposite trajectories.
Genetic inactivation of cues results in inaccurate trajectories
that are also variable, suggesting genetic variability is translated
into phenotypic variability. Quantitative modeling shows the total
number of cues is limited by expression noise and energetic cost
constraints. I propose this model applies to trajectory choices
and generally to discrete decisions controlled by noisy and competing
Existence and uniqueness of the total
quasi-steady-state approximation for coupled systems of enzyme
Shev MacNamara
The Institute for Molecular Biosciences, The University Of Queensland,
Coauthors: Alberto M. Bersani, Kevin Burrage, Roger B. Sidje
The total quasi-steady-state approximation (tQSSA) is obtained
merely by introducing a simple change of variable into the conventional
QSSA and has the benefit of being valid over a wider parameter
range. The original work on the tQSSA (Segel et al., Bulletin
of Mathematical Biology, 1996) demonstrated this for the quintessential
example of Michaelis-Menten enzyme kinetics and recently interest
has focused on being able to generalize the approach to more complicated
networks of coupled enzymatic reactions (Ciliberto et al., PLoS
Computational Biology, 2007). In special cases explicit formulae
for the approximation may be derived but for more complicated
systems these are not available. We provide a theorem guaranteeing
the existence and uniqueness of the approximation for networks
of coupled enzymatic reactions, as well as an accompanying numerical
method. These results are applied to the Goldbeter-Koshland switch
and the mitogen-activated-protein kinase cascade. One novel aspect
of this work is the application to the chemical master equation
to understand the dynamics of discrete and stochastic biochemical
kinetics. Previously this has been felt infeasible because of
difficulties involved with the computation of the exponential
of a matrix of very high dimension but by using Krylov methods
and the extra structure present in the reactions arising in enzymatic
networks we show that significant progress can be made.
Parameter sensitivity investigation of
a mathematical model of glioma tumorigenesis mediated by platelet-derived
growth factor
Susan Christine Massey
University of Washington.edu
Coauthors: Peter Canoll, MD, PhD; Kristin R. Swanson, PhD
Gliomas are the most prevalent form of primary brain tumor in
adults. Despite all possible treatment attempts, including aggressive
surgical resection, these tumors are uniformly fatal. Dr. Peter
Canoll at Columbia University has demonstrated that rats develop
brain tumors closely resembling human gliomas when their glial
progenitor cells are injected with a retrovirus expressing platelet-derived
growth factor (PDGF). Most notably, at 17 days post infection
only 30infected progenitor cellsthe other 70progenitor cells,
presumably recruited to the tumor by interactions with PDGF. Using
the empirical data collected by his lab, we have developed a mathematical
model to describe the observed tumor growth in this rat experiment.
We used a sensitivity analysis technique incorporating latin hypercube
sampling (LHS) and partial rank correlation coefficients (PRCC)
to vary parameters against each other and determine which parameters
in the model are most influential upon the ratio of uninfected
progenitor cells to total (infected and uninfected progenitors)
in the tumor at day 17. Our investigation revealed that the two
most influential parameters affecting the observed tumor growth
pattern are the max proliferation rate of infected progenitors
(and thus the amount of extra cellular PDGF available) and the
max rate of consumption of PDGF by nearby uninfected progenitors.
Specifically, even when controlling for variability in the other
unknown model parameters, an increase in the max proliferation
rate of the infected cell population results in an increase in
the percentage of the cells at the core of the tumor that are
recruited (rather than infected). This may suggest that more aggressive
(highly proliferative) gliomas would have the most recruited cells
within the tumor and may benefit most from PDGF targeted therapies
(e.g., Gleevec). This is a novel insight that may help in patient
selection for such targeted therapies. We are exploring the potential
impact of PDGF targeted therapies on tumors with differing affinities
for PDGF within the context of the current model and parallel
experimentation. Our initial results suggest that this model may
lead to a better understanding of what drugs may help glioma patients,
by quantifying the relative importance of PDGF. Future work will
also focus more on analyzing the model to look for PDGF influenced
spatial migration patterns of the respective cells in the growing
tumor to compare with observational studies tracking the migration
of individual cells over several hours.
Estimation of scaling index on space
records of cell proliferation in the developing central nervous
Jorge Mazzeo
Institute of Biomedical Engineering, Buenos Aires University and
Interdisciplinary Group in Theoretical Biology, Favaloro University,
Coauthors: Melina Rapacioli (Interdisciplinary Group in Theoretical
Biology, Favaloro University, Argentina) Santiagoo Duarte (IBCYN,
School of Medicine, Buenos Aires University, Argentina) Carlos
DAttellis (Interdisciplinary Group in Theoretical Biology,
Favaloro University, Argentina) Vladimir Flores (Interdisciplinary
Group in Theoretical Biology, Favaloro University and IBCYN, School
of Medicine, Buenos Aires University, Argentina) firstname.lastname@example.com
The dynamics of neuroepithelial cell proliferation in the chicken
tectum opticum is analyzed using a model within the framework
provided by the theory of stochastic point processes. Spatial
signals of cell proliferation consisting of numerical sequences
of intermitotic intervals were recorded under microscopic observation.
The main goal of this work is to determine the possible existence
of some kind of correlation or dependency between proliferating
cells. The central hypothesis is that, if proliferating cells
behave interactively, such interactions should impart some kind
of dependency or memory on the signals representing the spatial
organization of the proliferative activity. Additionally, appropriate
methods of signal analysis should provide information about the
spatial range of such interactions.
The analyses were performed by means of standardized algorithms
designed to characterize the dynamics of numerical sequences by
computing the scaling index of the stochastic processes. Among
these methods, the Hurst index (one of the earliest proposed),
the Detrended Fluctuation Analysis, the Fano Factor, the Power
Spectral Density and the Dispersional Analysis were applied in
this study.
Many authors [1],[2],[3] have pointed out that some of these
methods are sensitive to non-stationarities. To overcome these
pitfalls, elimination (remotion) of global trends is recommended
[4] to transform the signal into a stationary sequence.
Here we analyze inaccuracies in scaling index estimation that,
paradoxically, appear because of global trends removal.
This study was realized on artificial series, synthesized with
defined statistical parameters, and also on spatial records of
cell proliferation. We propose some methodological guidelines
which are relevant to our specific field of research.
[1] Kun Hu, Plamen Ch. Ivanov1, Zhi Chen, Pedro Carpena, H. Eugene
Stanley, Effects of Trends on Detrended Fluctuation Analysis,
arXiv:physics/0103018 v4 14 May 2001
[2] M. Ignaccolo, P. Allegrini, P. Grigolini, P. Hamilton, B.
J. West, Scaling in Non-stationary time series I,
arXiv:physics/0301057 v1 22 Jan 2003
[3] Zhi Chen, Plamen Ch. Ivanov, Kun Hu, H. Eugene Stanley, Effect
of Nonstationarities on Detrended Fuctuation Analysis, arXiv:physics/0111103
v2 15 Apr 2002
[4] Trang Dinh Dang and Sándor Molnár, On
the Effects of Non-Stationarity in Long-Range Dependence Tests,
El. Eng. Vol. 43, No. 4, Pp. 227250, 1999
Modeling Natural Killer Cell Development
and Repertoire Formation
Ramit Mehr
The Mina and Everard Goodman Faculty of Life Sciences, Bar-Ilan
University, Ramat-Gan, Israel
Coauthors: Mali Salmon-Divon, Sofia Johansson, Maria Johansson,
Yishai Pickman, Marjet Elemans, Petter Höglund, and Ramit
NK cells are able to recognize and reject cells lacking expression
of self- MHC class I molecules. Inhibition of lysis is mediated
by inhibitory receptors expressed by NK cells, such as the murine
Ly49 receptors, which bind to MHC class I molecules. NK cells
adapt to the self-MHC environment by a process ensuring that each
cell expresses at least one self-specific inhibitory receptor
but not too many. Two models have been proposed to account for
the development of a useful Ly49 repertoire. The two-step selection
model proposes a stochastic initial receptor expression combined
with selection of cells expressing appropriate receptor compositions.
The sequential model proposes that NK cells sequentially express
Ly49 receptors and continue to do so until an interaction of sufficient
magnitude between a Ly49 receptor and self-class I MHC occurs.
These two models predict different repertoire compositions under
various conditions. The complexity of experimental observations
on NK cell repertoire development necessitates the application
of theoretical techniques in order to elucidate the principles
underlying this development and evaluate the proposed models.
We conducted mathematical modeling and computer simulation studies
of each NK cell education model, fitting them to published (Salmon-Divon
et al, 2003a, b; 2004) and newly generated (Salmon-Divon et al,
in preparation) experimental data. Our results favor the two-step
selection model over the sequential model, and raise several questions,
which will be addressed in our future studies combining experiments,
mathematical modeling and computer simulations.
Antigen-Driven Selection In Germinal Centers
As Reflected By The Shape Characteristics Of Immunoglobulin Gene
Lineage Trees: A Large-Scale Simulation Study
Ramit Mehr
The Mina and Everard Goodman Faculty of Life Sciences, Bar-Ilan
University, Ramat-Gan, Israel
Coauthors: Gitit Shahaf, Michal Barak, Neta S. Zuckerman, Naamah
Swerdlin, Malka Gorfine and Ramit Mehr
Lineage trees of somatically hypermutated immunoglobulin (Ig)
genes from B lymphocytes often serve to qualitatively illustrate
claims concerning the dynamics of affinity maturation in germinal
centers (GC). Using a novel method for graphical quantification
of lineage tree properties, we have in past studies demonstrated
that lineage tree analysis detects fine differences in Ig gene
intraclonal diversity between B cell clones generated under different
conditions. We found age- and tissue-related differences in the
dynamics of the normal humoral immune response in humans, unique
features of Ig gene diversification in B cell malignancies and
autoimmune responses, and B cell diversification in other species
which utilize gene conversion rather than rearrangement as the
main primary diversification mechanism.
In order to test quantitative claims regarding the GC response
and affinity maturation, we created a computer simulation which
combines mathematical models for all mature B cell populations,
stochastic models of hypermutation and selection, and lineage
tree generation and measurement. We ran this program many times
varying the values of dynamical parameters (such as the proliferation,
differentiation and mutation rates, initial affinity of the Ig
to the antigen, and selection thresholds), creating almost a million
simulated lineage trees. We analyzed the data in order to identify
the ranges of dynamic parameters that yield biologically correct
results based on experimental data regarding germinal center responses,
obtaining interesting insights regarding response dynamics. We
found statistically significant correlations between quite a few
tree characteristics and the initial affinity and selection threshold,
which seem to be the main parameters that affect lineage tree
shapes, in both primary and secondary response trees. We found
that GC cells may be divided into a subset possessing low values
of selection threshold and mutation rate, and a second, small
subset with high values of these parameters. The results also
confirmed that recycling from centrocytes back to centroblasts
is highly likely. Finally, analysis of correlations between tree
properties removed redundant properties, improving the statistical
power of this method.
Wavetrain selection following predator
invasions in oscillatory reaction-diffusion systems.
Sandra Merchant
University of British Columbia
Coauthors: Wayne Nagata (University of British Columbia)
Periodic travelling waves, also known as wavetrains, are known
to evolve behind invasion fronts in oscillatory reaction-diffusion
models for predator-prey systems. Mathematical theory predicts
that for a given set of parameter values there is in fact a family
of possible wavetrain solutions and in a particular predator invasion
a single member of this family is somehow selected. Sherratt (1998)
has studied this selection mechanism, using the Normal Form approximation
that is valid for such models near the Hopf bifurcation in the
local system. However, away from this Hopf bifurcation the predictions
from the Normal Form lose accuracy. We conjecture a more general
selection criterion that retrieves the prediction from the Normal
Form system, but that applies to the full (non-reduced) predator-prey
system and that depends on the properties of the wavetrains for
the full system and hence retains accuracy away from the Hopf
bifurcation. We illustrate how to apply this selection criterion
using three sample oscillatory reaction-diffusion models from
the literature on predator invasions. The selection criterion
does indeed provide more accurate predictions for these models
than the criterion based on the Normal Form, but does eventually
lose accuracy as well. We therefore conclude with future directions
for work on this problem.
Neutral Stability Manifolds for Reaction-Diffusion
R. P. Mondaini
Federal University of Rio de Janeiro
Coauthors: Mariano Rodriguez-Ricard (Havana University)
(abstract to be included shortly)
Model identification from noisy data:
solving ill-posed inverse problems using regularization
Stefan Mueller
Radon Institute for Computational and Applied Mathematics, Austrian
Academy of Sciences
Coauthors: James Lu (Radon Institute for Computational and Applied
Mathematics, Austrian Academy of Science) Rainer Machne (Theoretical
Biochemistry Group, University of Vienna) Lukas Endler (Theoretical
Biochemistry Group, University of Vienna)
The quality of a mathematical model for a biological system depends
- aside from its explanatory value - on its consistency with the
data available. From a data-driven viewpoint, modeling is an "inverse
problem": given a certain class of models, one tries to identify
unknown parameters or even functions which give rise to the observed
data or a desired qualitative dynamics. In the presence of data
noise, however, model identification is an ill-posed inverse problem
in the sense that its solution lacks stability properties: a small
amount of data noise can be considerably amplified and may lead
to unreliable solutions. To overcome this problem, we suggest
the use of so-called regularization methods.
One of the systems we study is an ODE model of a metabolic pathway,
which has been used as a benchmark problem for parameter identification.
The ODE model contains 36 parameters all of which are identified
from noisy data. Using simple least squares minimization (without
regularization), a few percent of data noise leads to more than
100% relative error in some of the identified parameters, thus
highlighting the ill-posedness of the inverse problem. Using regularization,
the relative parameter error is comparable with the data noise.
More specifically, we use Tikhonov regularization to counter the
instability of the problem. By choosing the regularization parameter
appropriately (based on the knowledge of the data noise), the
model parameters can be identified in a stable and accurate manner.
A Computational Framework for Simulating
Multiphase Models of Tissue Growth.
Mr James Osborne
Oxford University Computing Laboratory, Oxford, OX1 3QD, UK
Coauthors: Helen M Byrne: Centre for Mathematical Medicine, Division
of Theoretical Mechanics, School of Mathematical Sciences, University
of Nottingham, University Park, Nottingham, NG7 2RD, UK. email:
firstname.lastname@example.com Sarah Waters: Oxford Center
for Industrial & Applied Mathematics, Mathematical Institute,
24-29 St. Giles', Oxford, OX1 3LB, UK, email: firstname.lastname@example.com
Jonathan Whiteley: Oxford University Computing Laboratory, Oxford
University, Oxford, OX1 3QD, UK, email: firstname.lastname@example.com
Multiphase modelling is a natural framework for studying many
biological systems, for example tissue engineering and cancer
development, where different phases represent the constituents
of the tissue of interest (e.g. extracellular matrix, cancer cells
and interstitial fluid when studying solid tumour growth). The
resulting models comprise non-standard mixed systems of nonlinear
PDEs. For example, multiphase models used to describe tissue engineering
applications and solid tumour growth may generate equations that
consist of: (i) viscous fluid flow equations for each phase: (ii)
hyperbolic PDEs for mass conservation; and (iii) elliptic or parabolic
PDEs for chemical concentrations. Analytical progress with such
systems is usually only possible if additional model assumptions
are made (e.g. radial symmetry, or small aspect ratio). A complementary
approach is to seek a numerical solution of the governing equations
without making any such simplifications. The numerical solution
of these equations presents numerous challenges: the numerical
methods for solving fluid flow equations and hyperbolic PDEs are
notoriously prone to complications such as instability and computational
time. Further complexity may be introduced if the problem is posed
on a growing domain (e.g. a growing tumour). Advanced numerical
algorithms are required in order to guarantee an accurate and
efficient solution.
We have developed a numerical and computational framework based
upon the Galerkin Finite Element Method that allows the numerical
solution of coupled systems of parabolic, elliptic and hyperbolic
PDEs described above in two or three dimensions. This enables
us to investigate the effect of interactions between constitutive
phases. We have used this framework to investigate tissue growth
in a bioreactor and also the development of a solid tumour, under
non-uniform environmental conditions. We illustrate the versatility
of our numerical method by presenting results for these two case
Sensitive dependence of fixation probability
on life history: The lytic virus case.
Zaheerabbas Patwa
University of Western Ontario
Coauthors: Dr. Lindi Wahl (University of Western Ontario)
The fixation probability of a beneficial mutation is the probability
with which the allele takes over the entire population. This probability
is extremely sensitive to assumptions regarding the organism's
life history. We compute the fixation probability using a life-history
model for lytic viruses. The model assumes exponentially distributed
attachment times and a constant time between attachment and host
cell lysis (lysis time). We derive a partial differential equation,
including a delay term, which describes the time evolution of
the probability generating function (p.g.f.) for the number of
individuals in the mutant lineage. By finding the fixed point
of this p.g.f., we compute the fixation probability for mutations
that increase attachment rate, increase burst size, decrease the
lysis time or reduce the probability of clearance. These four
mechanisms of mutation give widely varying fixation probabilities.
It was found that in all cases, the fixation probability of beneficial
mutations was sensitive to the time between population bottlenecks.
Optimal movement and motor capacity
in a short-range aquatic predator
Claire Postlethwaite
University of Houston
Coauthors: Tiffany M. Psemeneki, Jangir Selimkhanov, Mary Silber,
Malcolm A. MacIver
The black ghost knifefish `Apteronotus albifrons' makes fast
reactive strikes at prey that it detects with its weak electric
field and electrosensory system. Prey are detected at short range
throughout an omnidirectional sensory volume around the body.
Following detection, the body is rapidly reoriented to bring the
mouth to the position of the prey. Because of the short range
of detection and need for rapid movements in many directions,
constraints due to the dynamics and mechanics of fish swimming
are likely to play a significant role in determining the behaviour.
Here we examine mechanically optimal trajectories for an idealised
fish body moving in an inviscid fluid, and compare these with
measured prey capture trajectories. We present evidence that the
measured fish trajectories are close to those of the idealised
fish moving to minimise effort. This is particularly notable given
that the idealised fish is able to move with all possible linear
and rotational degrees of freedom while the real fish is not able
to do so.
Mathematical Modeling of Brain Tumor
and Related Therapeutic Strategies
Gibin Powathil
University of Waterloo, Waterloo, Ontario,Canada
Coauthors: M Kohandel, S Sivaloganathan
Gliomas are the most common and aggressive primary brain tumors.
The most common treatment protocols for these brain tumors are
combinations of surgery, chemotherapy and radiotherapy. However,
even with the most aggressive combination of surgery and radiotherapy
and/or chemotherapy schedules, gliomas almost always recur resulting
in a median survival time for patients of not more than 12 months.
This highly diffusive and invasive nature of brain tumors makes
it very important to study the effects of these combined therapeutic
strategies in an effort to improve the survival time of patients.
It is also important to study the tumor micro- environment, since
the complex nature of the cerebral vasculature, including the
blood brain barrier and several other tumors induced conditions
such as hypoxia, high interstitial pressure, and cerebral edema
affect drug delivery as well as the effectiveness of radiotherapy.
Recently, a novel strategy using antiangiogenic therapy has been
studied for the treatment of brain tumors. Antiangiogenic therapy
interferes with the development of tumor vasculature and indirectly
helps in the control of tumor growth. Recent clinical trials suggest
that anti-angiogenic therapy is usually more effective when given
in combination with other therapeutic strategies. In an effort
to study the effects of the above mentioned therapeutic strategies,
we consider a simple spatio-temporal model that incorporates the
tumor cell growth and the effects of radiotherapy and chemotherapy.
We study the effects of different schedules of radiation therapy,
using a generalized linear quadratic model, and compare the results
with published clinical data. The model is then extended to include
the interactions of tumor vasculature and oxygen concentration,
in an effort to explain tumor hypoxia, cerebral edema and high
interstitial pressure (as well as their changes during antiangiogenic
therapy). We also discuss the optimum way of sequencing these
therapeutic strategies so as to maximize patient survival time.
Evolution of asymmetric division: an
in-silico model
Armin Rashidi
Institute for Ageing and Health, Newcastle University, UK
Coauthors: Daryl Shanley
Symmetric reproduction precludes ageing; all individuals would
be affected by any deterioration and the lineage would vanish.
Segregation of damaged macromolecules, by asymmetric division,
to one progeny cell results in an ageing parent and a rejuvenated
daughter. Asymmetric reproduction is also a precursor for germ-soma
specialization, a prerequisite for the evolution of multicellularity.
However, surprisingly little work has been done on the evolution
of asymmetry. Using in-silico experiments, we here determine circumstances
under which selection forces drive the evolution of asymmetry.
Assumptions used in the model are: (i) Limited resource availability
creates a trade-off between investment in growth/reproduction
and in maintenance/repair. (ii) The population is near its carrying
capacity and the optimum investment strategy has already evolved.
(iii) The rate of damage accumulation (r) is inversely related
to the level of investment in maintenance/repair (m). (iv) The
doubling time (T) is shortened by larger reproductive investments
(b). (v) The likelihood of survival up to a certain time decreases
with the time-integrated damage up to that time. (vi) The mode
of damage distribution at division is an evolvable trait. The
damage segregation coefficient, s, was allowed to evolve between
zero (full symmetry) and one (full asymmetry), and was averaged
over the population at any given time. The outcome of each run
was defined as (a) asymmetry: s > 0.8 (more than 90% of damage
segregating to one progeny cell) for at least 50% of the monitoring
period or (b) symmetry: s > 0.8 for less than 25% of the monitoring
period. Two stochastic parameters (random mutations and the likelihood
of survival to next division) and three fundamental constants
(T, C1, C2) are potential determinants of the dynamics of the
system. C1 determines the shape of the relationship between r
and T (and also the one between m and b if r and T are assumed
linear functions of m and b, respectively), and C2 is the level
of time-integrated damage above which the chance of survival is
negligible. For a given T, the selection pressure for asymmetry
is lower for more concave m/b trade-offs and and larger C2 values.
Asymmetry evolves if damage-related mortality makes survival to
the age at reproduction sufficiently unlikely. Of particular note,
convex m/b trade-offs promote the evolution of asymmetry. These
results have important implications to the evolution of multicellularity,
ageing, and division of labour.
Three dimensional simulation of glioma
growth and response to radiation therapy: a case study
Russell Rockne
University of Washington Department of Pathology
Coauthors: Dr. Kristin R. Swanson, Julia L. Moore
Gliomas are human brain tumors that are unique in their extensive
invasion of the surrounding tissue without metastasis outside
the brain. Using tumor volume data collected from a human glioma
patient, we investigate a reaction diffusion model for both untreated
glioma growth (Swanson 1999, Swanson et al 2002), and response
to radiation therapy (Swanson et al 2007, Rockne et al 2008) on
the brainweb 3D virtual brain. Three dimensional spatial analysis
of model predicted disease distribution and that observed on magnetic
resonance imaging reveals a mean point-wise error of 0.51mm, within
measurement error. Using the linear-quadratic model for radiation
efficacy, in conjunction with [18F]-Flouromisonidazole positron
emission tomography, we observe a radiation resistance effect
in regions of the tumor with increased levels of hypoxia and demonstrate
a 3 fold oxygen enhancement ratio for hypoxic cell radio-resistance.
Heterogeneity in radiation dose per cell reveal regions of insufficient
dose and suggests avenues for treatment field modification.
A Mathematical Model of the Human Sleep/Wake
Lisa Rogers
Rensselaer Polytechnic Institute
Coauthors: Mark Holmes (Rensselaer Polytechnic Institute)
Most living organisms exhibit daily biological rhythms. One of
the most curious and difficult to understand is the sleep/wake
cycle. Although there is little discussion about the importance
of the wake phase, the reasons for sleep are not well understood.
In this talk a model of the human sleep/wake cycle will be presented
that consists of a system of nonlinear integro-differential equations
that describe the behavior of inhibitory and excitatory neurons
contributing to sleep and wake. Specifically, the model is based
on recent experimental studies identifying the brain circuitry
and neurotransmitters that regulate the sleep and wake states.
We utilize inherent properties of the REM/NREM cycle as well as
stability theory and computational methods to analyze the system.
A correlation is shown between the models output and experimental
data taken from sleep studies. The consequences of qualitative
and quantitative analysis are utilized to shed light on the various
physiological implications that arise from the mathematical structure
of the human sleep and wake processes.
Network development in biological gels:
Role in lymphatic vessel development.
Dr. Tiina Roose
OCIAM and CMB, Mathematical Institute, University of Oxford
Even though the existence of lymphatic vessels has been known
since the 17th century, until very recently not very much was
known about their functioning and development. This was due to
a failure to understand their importance in the proper functioning
of tissues. However, in last the 10 years lymphatics have come
to the forefront of biomedical research, largely due to findings
highlighting their importance to cancer growth and metastasis
Stacker et al. (2002). Thus, there are now a large number of experimental
studies on the molecular and micromechanical factors that control
lymphatic function and development.
We present a model that explains the prepatterning of lymphatic
vessel morphology during development. This model is derived using
the theory of two phase rubber material due to Flory and co-workers
and it consists of two coupled fourth order partial differential
equations describing the evolution of the collagen volume fraction,
and the evolution of the proton concentration in a collagen implant;
as described in experiments of Boardmand&Swartz (2003). Using
linear stability analysis we find that, above a critical level
of proton concentration, spatial patterns form due to small perturbations
in the initially uniform steady state. Using a long wavelength
reduction we can reduce the two coupled partial differential equations
to one fourth order equation that is very similar to the Cahn-Hilliard
equation; however, it has more complex non-linearities and degeneracies.
We present the results of numerical simulations and discuss the
biological implications of our model.
A Coalescent Theory Analysis of
the Population Structure Statistic Fst
Sivan Rottenstreich
Georgetown University
Populations are often divided into subpopulations. Fst is a statistic
used to experimentally test for population subdivision. We consider
a stochastic model of evolution for subdivided populations. We
analyze Fst under different scaling limits for parameters of this
model. We show that the distribution of Fst depends on mutation
rate and we characterize the distribution in the different scaling
limits. Our results are the first to describe the distribution
of Fst and demonstrate its dependence on mutation rate. We use
a coalescent theory approach to derive our results.
A tunable biochemical band-pass filter
robust to internal noise
Marc R. Roussel
Department of Chemistry and Biochemistry, University of Lethbridge
Some biochemical systems respond differently to the same signal
presented at different frequencies, implying a signal filtering
capability. Calcium signals, for instance, can be interpreted
differently in a cell depending on their temporal characteristics.
The ingredients for a minimal, tunable band-pass filter turn out
to be remarkably simple: A competitively inhibited enzyme operating
in the slow, tight-binding inhibition regime is able to accomplish
this function. The central frequency of the filter can be tuned
by varying the concentration of inhibitor. Since biochemical components
are frequently present in very small numbers, we examine the effect
of decreasing the volume of the system holding the concentrations
constant, which corresponds to increasing the level of internal
noise. Remarkably, the band-pass filtering property is retained,
even at very small extensivities corresponding to a few dozen
enzyme molecules.
Clonal expansion determines CD8+ T cell
phenotype in vivo.
Timothy Schlub
University of New South Wales, Australia
Coauthors: Vanessa Venturi (University of New South Wales, Australia)
Katherine Kedzierska (University of Melbourne, Australia) Cameron
Wellard (The Walter and Eliza Hall Institute, Australia) Peter
Doherty (University of Melbourne, Australia) Stephen Turner (University
of Melbourne, Australia) Ruy Ribeiro (Los Alamos National Laboratory,
USA) Philip Hodgkin (The Walter and Eliza Hall Institute, Australia)
Miles Davenport (University of New South Wales, Australia)
The CD8+ killer T cell response to infection involves
extensive T cell division and differentiation. Expression of the
adhesion molecule CD62L is high on naïve cells that have
not seen virus, and rapidly down regulated on the surface of the
majority ( ~ 90%) of cells present in the effector
phase of acute infection. Various models have been proposed to
explain the progression of the cellular differentiation of this
system. We demonstrate that the extent of CD62L down-regulation
is positively correlated with clone size in vivo, suggesting that
the number of divisions a T cell has undergone may determine its
levels of CD62L expression (phenotype). We develop a mathematical
model of division-linked CD62L differentiation that reproduces
the experimental population kinetics and phenotype during the
acute infection. The model is subsequently used to simulate a
heterogeneous clonal population responding to influenza virus
infection and generating a repertoire of responding clonotypes
(T cell receptor sequences that enable recognition of viral peptides).
The model demonstrates that many of the features of the CD62Lhi
and CD62Llo T cell receptor repertoire observed in vivo can be
explained with a simple mechanism of division-linked differentiation.
We further demonstrate that division-linked CD62L differentiation
adequately describes our experimental kinetic and repertoire data.
Moreover, this is robust in terms of the mathematical description
used for cell division, or the parameters used to distinguish
different clonotypes.
A Feline Leukemia Virus Model with
Potential Vector
Matthew Schuette
William Jewell College
Many microparasitic diseases are spread horizontally, i.e. person-to-person
or animal-to-animal. Certain diseases, such as malaria, are spread
by means of a vector, e.g. mosquito.
Feline leukemia is a retroviral disease among cats, caused by
feline leukemia virus (FeLV). Virus is shed in very high quantities
in saliva and nasal secretions, but also in urine, feces, and
milk from infected cats. Cat-to-cat transfer of FeLV may occur
from a bite wound as well as during mutual grooming, and possibly
through the shared use of litter boxes and feeding dishes. Until
recently, no vectors had been reported as possible sources of
transmission. In 2003, Vobis, et al. published a paper suggesting
the cat flea (Ctenocephalides felis) as a potential vector for
this disease.
For this talk, we investigate a deterministic model of FeLV transmission
that includes the spread of the virus by the cat flea. We will
focus primarily on the development of the model and some of the
analytical results.
Population Behaviour towards Voluntary
BCG Vaccination Policies
Schehrazad Selmane
University of Sciences and Technology of Algiers
Mathematical models that take into account the interplay between
human behaviour, economics, and disease ecology may be more useful
in understanding the dynamics and spread of diseases than models
that rely on epidemiology alone. The transmission and control
of infectious disease is strongly influenced by how people make
choices, both individually and collectively, when presented with
opportunities to engage in preventive actions or to use preventive
health care services. The effectiveness of the TB vaccine in preventing
TB is controversial; studies have shown variable efficacy. Game
theory is used to analyze population behaviour towards voluntary
BCG vaccination policies. Such an approach allows quantifying
how risk perception influences expected vaccine uptake and coverage
levels. The threshold in perceived relative risk is computed from
the basic reproduction, computed from a simplified mathematical
model for the dynamics of tuberculosis with vaccination.
Diversity and shape of peripheral
effector and regulatory T cell repertoires: validating the Crossregulation
model with experimental data
Nuno Sepúlveda
Instituto Gulbenkian de Ciência
Coauthors: Jorge Carneiro, Instituto Gulbenkian de Ciência
A healthy immune system involves a fine balance between effector
T cells (Teffs) that mount immune responses, and regulatory T
cells (Tregs) that suppress them. When this balance is perturbed,
immunopathologies arise. Understanding this balance requires to
know how diverse are the repertoires of Teffs and Tregs and how
they relate to each other. A too large intersection between the
repertoires could lead to deleterious inhibition of specific immune
responses against harmfull microorganisms, while a too small overlap
may open the way to autoimmune responses. Here we address this
issue by a Crossregulation model (Immunol. Rev. 216:48-68, 2007)
that describes the peripheral dynamics of a large number of clones
with both Tregs and Teffs competing for antigen-presenting cells.
The model produces different, but testable, predictions for the
shape and diversity of peripheral Teff and Treg repertoires: (1)
a higher diversity of Teffs than of Tregs, (2) a Lognormal distribution
for the clonal size distributions, and (3) a negligible correlation
between clonal size distributions. Here we confront these predictions
with available experimental data.
A "Vuggy" Medium Approach
to Fluid and Drug Transport in Tumours
Rebecca Shipley
Mathematical Institute, University of Oxford
Coauthors: Prof. S. J. Chapman
The neoplastic vasculature of solid tumours comprises a network
of capillaries with highly permeable walls. Understanding the
flow of blood through this vasculature and the surrounding porous
interstitium of the tumour is important for two main reasons.
Firstly, it is a key ingredient when predicting the oxygen distribution
within the tumour, which is crucial for predicting the micro-environment
and growth rate of cancer cells. Secondly, it is vital for predicting
the treatment of cancer by therapeutic drugs administered intraveneously.
For example, the success of treatment by chemotherapy drugs has
been limited by low transport rates across the vasculature into
the main tumour body. Increased convection induced by elevating
the systemic blood pressure or applying intratumoural infusion
has been shown to improve drug delivery by 40% and several orders
of magnitude respectively.
Here we develop a multiscale model of the tumour vasculature,
using a similar approach to that used in the petrochemical industry
to model groundwater aquifiers and petroleum reservoirs. We assume
that the tumour has a locally periodic structure, comprised of
a network of capillaries embedded in the surrounding interstitium
(a porous medium). The capillaries are small compared to the size
of the tumour itself, but much larger than the pore size of the
tissue, and so these disparate length-scales can be exploited
to describe fluid and drug transport in a tumour. On the local
scale, we describe the flow of blood through the capillaries and
across the vascular boundary into the interstitium. A multiple-scales
technique is then used to move from the local to the global descriptions
and so determine the equations describing the effective fluid
transport on the tumour-scale. This approach is extended to describe
drug transport, and finally numerical simulations are presented.
Simulating invasion: Macroscopic and
microscopic approaches
Matthew J Simpson
Department of Mathematics and Statistics, University of Melbourne,
Parkville 3010, Victoria, Australia
Coauthors: Kerry A Landman, firstname.lastname@example.com Barry
D Hughes, firstname.lastname@example.com
Cell invasion is fundamental to many biological processes ranging
from developmental morphogenesis to disease progression. Experimental
data describing invasion is typically collected across a range
of scales encompassing population-level and individual-level data.
Interpretive and predictive tools capable of replicating and connecting
these kinds of data are needed. Continuum population-level and
agent-based individual-level methods are two modelling approaches
that can replicate and connect multiscale experimental data. We
are using agent-based and continuum modelling approaches in parallel
to identify and explore various similarities and inconsistencies
that arise when considering the same problem using different modelling
Propagation of extrinsic perturbation
in a multi-step biochemical pathway
Somdatta Sinha
Centre for Cellular & Molecular Biology, Uppal Road, Hyderabad
500007, AP, India
Coauthors: R. Maithreye
Biochemical pathways underlie cellular processes. These pathways
are inter-connected chemical reactions forming an intricate network
of functional and physical interactions between molecular species
in the cell. Many of the steps in a pathway take place at different
cellular compartments and hence are subjected to different environmental
milieus. It is thus both interesting and surprising as to how
such an interacting dynamical system can faithfully transmit signals
in spite of perturbations of different types acting at different
steps of the multi-step process. Using a simple three-step negatively
auto-regulated model pathway, we show that the effect of perturbation
at different steps of the pathway and its transmission through
the network is dependent on the context (i.e., the position) of
the particular reaction step in relation to the topology of the
regulatory network, stoichiometry of reactions, type of nonlinearity
involved in the reactions and also on the intrinsic dynamical
state of the pathway variables. We delineate the qualitative and
quantitative changes in the pathway dynamics for constant (bias)
and random external perturbations acting on the pathway steps
locally or globally to all steps. We show that constant perturbation
induces qualitative change in dynamics, whereas random fluctuations
cause significant quantitative variations in the concentrations
of the different variables. Thus, the dynamic response of multi-step
biochemical pathways to external perturbation depends on their
biochemical, topological and dynamical features.
Toward an understanding of electrically
coupled inhibitory networks in hippocampus
Frances Skinner
Toronto Western Research Institute (TWRI), University Health Network
and University of Toronto
Coauthors: Tariq Zahid (TWRI), Fernanda Saraga (University of
Direct electrical communication is well-established in the hippocampus,
a brain region known to be important for learning and memory.
This form of communication is mediated by gap junctions and it
is known that this coupling is important for brain rhythms such
as gamma (20-80 Hz) which occur during active behavioural states.
It is also known that gap junctions are present at several locations
along the dendrites of inhibitory cells in hippocampus, so that
spatially extended models need to be considered. Weakly coupled
oscillator theory, which uses phase response curves, has been
used to predict network dynamics of electrically coupled networks.
I will describe this work in the context of our use of a quantification
of phase response curves to determine whether synchronous or asynchronous
modes (as predicted from the theory) occur in inhibitory networks
coupled with gap junctions at dendritic locations.
Life or Death: Complement Activation
and Response to a Streptococcus pneumoniae Infection
Amber M. Smith
University of Utah
Coauthors: Frederick R. Adler, University of Utah, Jonathan A.
McCullers, St. Jude Children's Research Hospital, Ruy M. Ribeiro,
Los Alamos National Laboratory, Alan S. Perelson, Los Alamos National
A Streptococcus pneumoniae infection in mice has three possible
outcomes: (i) rapid clearance, (ii) acute infection with clearance,
or (iii) acute infection ending in death. Which of these occurs
depends on the initial dose of bacteria and their interaction
with the complement system, one of the first lines of defense
in the immune system. There are three biochemical pathways of
complement activation: classical, alternative and mannose-binding
lectin (MBL). Initiated by antigen-antibody complexes, bacterial
surfaces and MBL, respectively, the complement cascade involves
more than 30 proteins which act as opsonizers, anaphylatoxins,
and initiators of other components of both the innate and adaptive
immune system. The final product, called the membrane attack complex,
can lyse and kill pathogens. The classical pathway initiates the
response to S. pneumoniae infections, but is amplified by the
positive feedback in the alternative pathway. We use differential
equations to model the kinetics of initiation and amplification
of the cascade, and to identify the factors which determine the
outcome of a bacterial infection.
One and Two Compartment Stochastic Integrate-and-Fire
Neural Models
Charles E. Smith
Biomathematics Program, Dept. of Statistics, North Carolina State
Coauthors: Mamiko Arai, Biomathematics Program, N C State Univ.
One and two compartment stochastic integrate-and-fire neural
models are investigated by simulation and by analytic approximation
methods. The models used are motivated primarily by the papers
of Lansky and Rodriguez (1999 a, b). One main difference is that
the output of our model is a renewal process rather than a correlated
point process. Biophysically this corresponds to antidromic invasion
of the action potential into the dendrite to reset the membrane
voltage following an action potential.
We concentrate on two neurons, both with the same compartment(membrane
electrical properties) at the site of action potential initiation,
however the two compartment model includes the dendritic partition
of the neuron by a second compartment. The two compartment model
is to contrast spatial effects in neurons with longer thinner
dendrites to those with short thicker dendrites that can be modeled
as a one compartment equivalent circuit. Biophysically this means
that the voltage is roughly the same in the spike initiation site
and the proximal dendritic processes.
Euler forward method is used to simulate the Ito version of the
stochastic differential equations corresponding to these equivalent
circuits. A Wiener process is used as the noise term to represent
many smaller synaptic inputs and paralleling the approach of Lansky
and Rodriguez. The approximation methods for first passage times
outlined in Smith(1991) were computed and compared to moments
of simulated output of the neurons.
The shapes of the simulated histograms were fit by a normal,
gamma and third and fourth order Laguerre series approximations
using the method of moments. The corresponding fits and moment
plots (skew vs CV; excess vs. skewness (Pearson plot)) were examined.
The single gamma seemed adequate in most cases.
The simulation was done in a 3 factor experimental design and
blocked on type of model (one vs. two compartment). The factors
were: (1) value of voltage threshold for firing S ;
(2) synaptic input strength u ; and (3) intensity
of the noise input k.
For smaller noise variation and when the mean voltage crosses
the threshold a heuristic explanation explains the systematic
variation in the mean and standard deviation of the first passage
time, namely the mean interval of the output point process and
its standard deviation. The approximation is simply that expected
from the delta method. The standard deviation of the firing time
is approximately the standard deviation of the voltage divided
by the slope the mean voltage trajectory. Both voltage terms are
evaluated at time t* which is when the mean voltage trajectory
reaches the threshold S.
For equal firing rates or output mean intervals, the two compartment
model shows a pronounced reduction of variability in firing times.
Said differently, it can more effectively code input intensity
levels using a mean rate neural coding scheme since it has less
variability about the mean firing time.
Finally some suggestions for further work are presented.
A poroelastic model of transcapillary
Sean Speziale
University of Waterloo
Coauthors: S. Sivaloganathan G. Tenti
Transcapillary exchange is the movement of fluid and molecules
across the porous capillary wall, and plays an important role
in maintaining homeostasis in tissues. To reach the cells of a
given tissue, molecules must traverse a porous matrix known as
the interstitial space, whose main function is to mediate exchange
of oxygen, nutrients and waste products between the vascular and
cellular compartments. The classical picture of transcapillary
exchange was suggested by Starling in 1896, namely that the forces
determining fluid flow were the hydrostatic and osmotic pressure
differences between the capillary and surrounding interstitial
space. However, experimental observations indicated that this
view must be revised, and subsequently Michel and Weinbaum put
forward the idea that the Starling principle should be applied
not across the entire capillary wall but instead across a structure
lining the wall known as the endothelial glycocalyx. Existing
ultrastructural models are quite complicated, so our aim is to
model transcapillary flow using a simpler approach, without losing
the essential characteristics. We adopt the Michel-Weinbaum hypothesis,
but instead of looking at the microstructure we idealize the capillary
wall as a homogenized porous media, and introduce a slight modification
to the theory of Biot. Due to the presence of solutes, a modified
version of Darcy's law is used, in which fluid flow is driven
by both hydrostatic and osmotic gradients. A unique feature of
the present work is to be able to predict the stress and strain
distributions in the capillary wall, which had not been attempted
previously. This work may have implications in understanding edema
formation, as well as in explaining the elevated interstitial
fluid pressure in tumours.
A Computational Model of Cell-Substrate
Interaction in Three Dimensions
Magdalena Stolarska
University of St. Thomas, Saint Paul, MN, 55105
Coauthors: Hans G. Othmer, University of Minnesota, Minneapolis,
MN 55455, firstname.lastname@example.com
Mechanical interactions between a cell and the substrate are
vital for cell migration and are involved in various cellular
processes, such as wound healing, embryonic development, and metastasis
of cancerous tumors. In addition, experiments have shown that
inter-cellular and cell-substrate mechanical interactions affect
signal transduction pathways within the cell (see for example
[1, 2, 3]). As a result, understanding the nature of force generation
by single cells and mechanical interaction of a cell with the
substrate is extremely important.
In this talk, we present a continuum model of single cell motility
in which the stresses that result from the active deformation
of the cell are transmitted to a substrate via controlled adhesion
sites. We propose to use large strain viscoelasticity to describe
this mechanism and study cell-substrate interactions. Both the
cell and the substrate are treated as three-dimensional deformable
continua. A finite element implementation of this model is used
to numerically examine the nature of the stresses generated by
the cell and the resulting traction patterns that occur at the
substrate. The simulations are compared to experimental results
where predictions about the stresses in the cell are based on
measured deformations of the substrate on which the cell is crawling
[4, 5, 6].
[1] P.A. Janmey and D.A. Wietz. Dealing with mechanics: mechanisms
of force transduction in cells. Trends in Biochemical Sciences,
29:364370, 2004.
[2] V. Lecausey and D. Gilmour. Organizing moving groups during
morphogenesis. Current Opinions in Cell Biology, 18:102107,
[3] A. Bershadsky, M. Kozlov, and B. Geiger. Adhesion-mediated
mechanosensitivity: a time to experiment, and a time to theorize.
Current Opinions in Cell Biology, 18:472481, 2006.
[4] J. Lee, M. Leonard, T. Oliver, A. Ishihara, and K. Jacobson.
Traction forces generated by locomoting keratocytes. The Journal
of Cell Biology, 127:19571964, 1994.
[5] S. Munevar, Y.-L. Wang, and M. Dembo. Traction force microscopy
of migrating normal and h-ras transformed 3T3 fibroblasts. Biophysical
Journal, 80:17441757, 2001.
[6] K.S.K Uchida, T. Kitanishi-Yumura, and S. Yumura. Myosin II
contributes to the posterior contraction and the anterior extension
during the retraction phase in migrating dictyostelium cells.
Journal of Cell Science, 116:5160, 2003.
Detecting spill-over: A dynamical systems
modeling approach to glutamatergic synaptic signaling
Emily Stone
Dept. of Mathematical Sciences, University of Montana-Missoula
Coauthors: Greg Leary, Katie Hoffman, Micheal Kavanaugh, Dept.
of Biomedical and Pharmaceutical Science, University of Montana-Missoula
The connectivity of neurons in the hippocampus depends in part
on whether neurotransmitter from one release site can leak out
and activate receptors in another synapse or extrasynaptic patch.
The existence of such "spill-over" is under debate in
the neuroscience community, since direct measurements of neurotranmitter
in such detail cannot, as of yet, be made. Experimental evidence
of spillover is thus indirect, and should be sifted through as
many different filters as possible. In this talk I will present
the contributions of dynamical systems modeling to this effort.
Inhibition of breast cancer growth
by GM-CSF: A mathematical model
Barbara Szomolay
Mathematical Biosciences Institute, The Ohio State University
Coauthors: Tim D. Eubank, Ryan D. Roberts, Clay B. Marsh, Avner
GM-CSF is a drug that enhances the ability of macrophages to
present antigen and initiate immune response. GM-CSF also stimulates
monocytes to secrete soluble VEGF receptor (sVEGFR-1) which binds
to and inactivates VEGF. Eubank and colleagues in a recent work
discovered that GM-CSF treatment locally in murine breast cancers
reduced tumor growth and metastasis; moreover, GM-CSF lowered
oxygen level and reduced blood vessel density within the tumor.
We developed a mathematical model which addresses the effect of
GM-CSF on the growth of breast cancer in mice. The model takes
into account the experimentally established interactions among
cancer cells, macrophages, endothelial cells, VEGF and M-CSF.
The model simulates the growth of tumor as a function of the local
GM-CSF dose. The model simulations were validated against in vivo
data and show a good fit with experimental results. We used the
model to compare the efficacy of different dosing protocols of
injection of GM-CSF, as well as to suggest new hypotheses for
slowing the progression of breast tumor. For example, the model
suggests that injecting the drug daily, twice or 3 times a week
are comparably effective. In contrast, reducing or over increasing
the frequency of dosing is counterproductive. We hope to further
refine the model in future work, by including the interactions
of macrophages and other immune cells, fibroblasts and cytokines
that communicate between the tumor and its micro-environment.
The Interaction of Swimming Microorganisms
With Flow: Modelling Motile Phytoplankton in Turbulence
Graeme Thorn
Department of Mathematical Sciences, University of Liverpool
The interaction of swimming phytoplankton and typical ocean flow
conditions is important to study in order to understand how the
spatial distribution evolves due to the dispersive properties
of turbulence. Many harmful algal bloom-forming species are motile:
one plausible mechanism for the formation of such a bloom is that
cells swim into a lower-salinity layer at the top of the water
column becoming trapped due to the stratification. Eutrophication
of this layer from pollutants in river run-off can cause a population
explosion leading to a bloom. A study of the interaction of turbulence
with motility will therefore provide insights into how these blooms
develop once they are formed. For a gyrotactic microorganism,
whose preferred motile behaviour is to swim upwards, the interaction
is non-trivial, as turbulence can alter this behaviour due to
relatively rapid (compared to the cells intrinsic reorientation
to the vertical) along its Lagrangian particle path.
As phytoplankton swimming typically occurs on length scales (of
the order 1-100 µm) some orders of magnitude below the Kolmogorov
scale for ocean turbulence (of the order 1 cm), a model for the
swimming behaviour in arbitrarily-oriented simple flows can be
used to parameterise the effects of the smallest eddies on the
mean swimming velocity. This result can then be incorporated into
a population-level equation which then describes the time-evolution
of the spatial distribution of a patch of phytoplankton. This
advection diffusion model extends previous work which has concentrated
on developing models for population dispersal in linear homogeneous
flows using the macroscopic generalised Taylor dispersion method.
This talk will begin with a description of the advection-diffusion
model, by showing how it is built up from the simple flow model,
and show comparisons of this population-level model with simulations
of individuals in turbulent flows. Finally, an application to
the modelling of the effects of turbulence on the recruitment
of gyrotactic cells into a pre-existing stratified fluid.
Spatiotemporal Modelling of Intracellular
Signalling in Bacterial Chemotaxis
Marcus Tindall
Centre for Mathematical Biology, Mathematical Institute, University
of Oxford.
Coauthors: Steven L. Porter, Philip K. Maini and J.P. Armitage
The role that spatial protein localisation plays in altering
the expression of flagellar motor driving proteins in bacterial
chemotaxis has to date largely been ignored. The work presented
here focuses on two spatiotemporal reaction-diffusion models of
signal transduction developed to describe phosphotransfer within
E. coli and R. sphaeroides. R. sphaeroides is a bacterial species
whose phosphotransfer pathway is considerably more complex than
E. coli. The mathematical model developed is used to understand
the role that spatial protein localisation has on affecting the
motor protein expression, both dynamically and in the steady-state.
The model is used to elucidate the role that cytoplasmic and receptor
clusters play in describing the overall bacterial response.
Modeling cancer stem cells: Implications
for novel therapeutic strategies
Colin Turner
University of Waterloo
Coauthors: Mohammad Kohandel, University of Waterloo and Centre
for Mathematical Medicine, email@example.com Siv Sivaloganathan,
University of Waterloo and Centre for Mathematical Medicine, email@example.com
Sheila Singh, McMaster Stem Cell and Cancer Research Institute
and Department of Surgery, McMaster University
In recent years, support has increased for the cancer stem cell
hypothesis, which states that a subpopulation of cancer cells
in possession of properties typically associated with stem cells
is responsible for initiating and maintaining tumour growth. Such
cancer stem cells were first identified in leukaemias,
and have since been been implicated in solid tumours including
those of the breast, brain and colon. Unravelling the details
of the cancer stem cell hierarchy, as well as the interactions
of these cells with various therapies, will be essential in the
design of optimal treatment strategies. We develop a mathematical
model of the cancer stem cell hypothesis that may aid in characterizing
tumours, as well as in predicting effective therapeutic strategies.
This model is stochastic when small numbers of cells are under
consideration; for larger populations of cancer cells, we adopt
a deterministic approach. The importance of certain parameters
in dictating properties of the tumour is discussed, as well as
how the dependence of the tumour on these parameters may be exploited
in therapy.
A mean field Ising model for cortical
rotation in amphibian one cell stage embryos
Jack A. Tuszynski
University of Alberta
Coauthors: Richard Gordon (University of Manitoba, email@example.com)
The fertilized amphibian egg is ideal for mathematical analysis,
with its apparently spherically symmetric cortex (membrane and
a few microns of attached cytoskeleton), and initially axially
symmetric, bottom heavy stratified yolk and cytoplasm inside.
Before the first cell division, the cortex rotates 30deg. It is
believed that the rotation is driven by microtubule motors and/or
polymerization of microtubules attached to the inner surface of
the cortex. While these microtubules are initially randomly oriented,
they take on a common orientation by the end of the cortical rotation.
The basic interaction appears to be two way: the microtubules
drive the cortical rotation, leading to sloshing of the yolk and
cytoplasm, and the fluid motion aligns the microtubules in such
a way as to enhance the rotation. The startup is stochastic in
nature. We model this interaction by a mean field Ising model,
but due to the coupling between the microtubules, the actual field
is equal to the mean field, giving the model an unusual precision.
Here we show what can be gleaned from observed stochastic rotational
trajectories using this model, and how the model is altered if
nearest neighbor microtubule-microtubule interactions are included.
Neurodynamics of epileptiform
activity: advances towards prevention of seizures
Jose Luis Perez Velazquez
University of Toronto and Hospital for Sick Children
Indications of some specific dynamical regimes have been found
in studies of epileptiform activity in brain in vivo and in vitro.
These analyses, in turn, may be of help in the current efforts
to prevent paroxysmal discharges (seizures) from occurring, and
in the solution of the related question of the anticipation, or
prediction, of seizure events. However, so far, a general method
for seizure prediction or control has not been found. In this
presentation, it will be argued that, according to the present
methodologies and concepts, seizure prediction and control is
certainly feasible for individual cases, but a general approach
will be difficult to find considering the multifactorial nature
of brain dynamics.
A simulation approach to understanding
the role of production frequency in the sharing of T cell receptors
between macaques in immune responses to simian immunodeficiency
Vanessa Venturi
Centre For Vascular Research, University of New South Wales, Australia
Coauthors: Hui Yee Chin (University of New South Wales, Australia)
David A Price (Cardiff University School of Medicine, UK) Daniel
C Douek (Vaccine Research Center, NIAID/NIH, USA) Miles P Davenport
(University of New South Wales, Australia)
The effectiveness of T cell immune responses to viruses depends
largely on the diversity of T cell receptors (TCRs) expressed
on the surface of individual T cells that detect viral peptides
(epitopes). These TCRs are produced by a process of germline gene
recombination. The enormous potential diversity of TCRs produced
in the thymus greatly exceeds the number of T cells found in an
individual at any given time (eg. > 1018 vs. 1012 in a human).
Thus, it is considered surprising that T cells expressing identical
TCRs in different individuals are observed in a wide variety of
immune responses. However, variation in the production frequencies
of different TCRs is rarely taken into account. We used simulations
of a random gene recombination process to demonstrate that, even
with unbiased gene recombination, some TCRs can be produced more
frequently than others. For example, in a random generation of
107 in-frame TCR sequences some TCRs are produced 103 times while
others are rarely produced. The simulations also demonstrate that
the variety of different ways that a TCR sequence can be made
from the germline genes plays an important role in how efficiently
it can be produced (as opposed to being produced by one or a few
frequently occurring gene recombination events). We term this
process convergent recombination. To test whether
convergent recombination and TCR production frequency play a role
in vivo, we have investigated the experimentally observed sharing
of TCRs in CD8+ (or killer) T cell responses specific
for simian immunodeficiency virus (SIV) in an outbred population
of rhesus macaques. The in silico TCR production frequency was
found to be a good predictor of the observed extent of sharing
of TCRs between macaques in these immune responses to SIV, suggesting
that the hierarchy of TCR production frequencies in the simulated
repertoire is predictive of that in the observed TCR repertoire
in vivo.
Modeling the relationship between radiotherapy
delay and cancer outcomes
Jon-Paul Voroney
Queen's University Cancer Research Centre, Division of Cancer
Care and Epidemiology
Coauthors: William J Mackillop Sarah J Rauth
Background: Worldwide, delay in radiotherapy for cancer allows
progression of untreated tumours. The current average delay in
Ontario is over 4 weeks. Meta-analyses of retrospective cohorts
relate delay of a one month for radiotherapy to a relative risk
of 1.15 for local control or survival from head & neck cancer.
Paradoxically, in studies that do not adjust for confounding,
delay can be associated with better prognosis: more advanced tumours
are often preferentially treated first. Treating poor-prognosis
patients first is a triaging strategy appropriate for an emergency
department, and does not provide a population with as high a cure
rate as would a triaging strategy based on priority for those
cancer patients who will be most harmed by delay. Objectives:
To model the effect of delay in radiotherapy on cancer outcomes,
including local control and overall survival, with particular
emphasis on data from published series in head & neck cancer.
Methods: (1) Meta-analysis of the head & neck cancer literature
on the prognostic effect of tumour volume on local control and
overall survival in patients treated with radiotherapy, including
modeling the functional relationship between volume and radiotherapy
outcome. (2) Meta-analysis of radiologic and serologic markers
of tumour volume and their time-dependence in untreated patients.
(3) Mathematical modeling of the effect of radiotherapy delay
on outcomes through combining the prognostic effect of tumour
volume and tumour volume doubling times. Results: (1) We pooled
results from 55 studies in head & neck cancer, to demonstrate
relationships between initial tumour volume, V, and radiotherapy
outcomes. The relationship is exponential, exp(-kV), with the
rate constant k depending on site, whether primary tumour volume
or total tumour volume (including lymph node spread) was measured,
and choice of radiotherapy outcome (local control or survival).
This is consistent with a Poisson distribution for the fraction
of surviving cells after treatment depending linearly on the initial
tumour volume. The rate k is also dependent on treatment, surrogates
of radioresistance, and the choice of the model relating volume
to outcomes e.g. a one-parameter model exp(-kV) versus two-parameter
model Aexp(-kV). The prognostic effect of volume is robust- it
persists in studies when additional therapy is given to patients
with higher tumour volumes. (2) We abstracted data from 100 studies
(5702 tumours) on tumour volume doubling times. Doubling time
distribution is lognormal within each site. Summary statistics
show considerable variation of medians and inter-quartile ranges
for doubling times, depending on the primary tumour site, the
histologic subtype, and whether the tumour is metastatic or recurrent.
For example, primary prostate cancer has a median tumour volume
doubling time of 7 years, while a head & neck cancer recurrence
has a doubling time of 1 week. (3) Calculation of the effect of
treatment delay, using the relationship between outcomes and volume,
and tumour volume doubling times is thus possible: for example,
for an average head and neck tumour with a volume of 12 cc, volume
doubling time of 8 weeks, and dependence of volume on prognosis
of exp(-0.03V), the relative risk of 1 month delay is 1.16. An
initial cancer local control rate of 70% drops to 60%. Incorporating
distributions in population values of model parameters and incorporating
a Gompertz or logistic function into modelling tumour growth are
refinements that improve modelling for large tumours and long
delays. Conclusions: The increased risk of poor outcomes with
RT delay is predictable using tumour growth and knowledge of the
association between outcomes and tumour volume. We have validated
our model of delay in radiotherapy for head & neck cancer,
and this model can be tailored to data from particular treatment
centres, particular populations, and particular tumours.
Rigid and Nonrigid Registration Methods
for Medical Images
Justin Wan
University of Waterloo
Coauthors: Lin Xu and Zhao Yi
In image guided procedures such as radiation therapies and computer-assisted
surgeries, physicians often need to align images that are taken
at different times or by different modalities. Typically, a rigid
registration is performed first, followed by a nonrigid registration.
We are interested in 2D-3D registration which align digitally
reconstructed radiographs generated from 3D datasets (e.g. CT
volume from pre-operative planning) with 2D portal image slices
(e.g. X-ray image collected in real-time). It is a very computationally
intensive procedure. In this talk, we propse fast models that
can achieve real-time performance. Afterwards, nonrigid registration
needs to be used to enhance the results. Elastic and fluid models
were usually used but edges and small details often appear smeared
in the transformed templates. In this talk, we also propose a
new inviscid model formulated in a particle framework. We will
derive the corresponding nonlinear partial differential equations
for computing the spatial transformation. Our idea is to simulate
the template image as a set of free particles moving toward the
target positions under applied forces. Our model can accommodate
both small and large deformations, with sharper edges and clear
texture achieved at less computational cost. We demonstrate the
performance of our model on a variety of images including 2D and
3D, mono-modal and multi-modal, synthetic and clinical data.
Incorporating prior knowledge to the dynamic
Bayesian networks modeling of pancreas development gene expression
Xujing Wang
Max McGee National Research Center for Juvenile Diabetes &
Human and Molecular Genetics Center, Medical College of Wisconsin,
8701 Watertown Plank Road, Milwaukee, WI, 53226, USA
Coauthors: Shouguo Gao
The importance of the network structure underlies genes and proteins
is gaining increasing appreciation. This is not only fundamental
to the understanding of genetic regulation and its functional
structure, but also critical to dissect complex diseases. Time
series gene expression data offer a rich source for network inference.
We have adopted the dynamic Bayesian network (DBYN) approach to
model transcription regulatory and co-expression networks, and
developed new algorithms to incorporate existing biological information
(co-citation, GO (gene ontology) similarity, positional and binding
information, etc) in public databases as prior knowledge. We introduced,
for the first time, fuzzy theory-based rules to the MCMC learning
of DBYN in order to efficiently incorporate the prior biological
knowledge, which are often incomplete and plagued with quality
issues. Further we defined gene expression (phase) synchronization
module and utilized it to assist initial network structure construction.
We show that these lead to significantly improved performance.
We then applied the algorithm to investigate the pancreatic development.
We first compiled a list of pancreas-specific genes by: (1) Manually
collecting curated genes that are known to be involved in pancreas
development from the literature; (2) Tissue specific gene expression
data was downloaded from http://www.t1dbase.org. We then determined
for each gene the Z score of its expression in pancreas versus
the mean in all tissues. We focus on those with pancreas Z>0.2,
and being annotated to GO:0032502 (developmental process) or its
descendant categories. Together we obtained a total of 45 genes.
Two data sets were obtained from RNA Abundance Database (www.cbil.upenn.edu/RAD)
to perform network reconstruction: (1) study id 2, expression
of mouse pancreas development at 7 time points: E14.5, E16.5,
E18.5, birth, postnatal day 7, and at adulthood; (2) study id
1790, mouse pancreas at 12, 24 and 48 hrs after 50% Ppx or sham
operation, which also received Ex-4 or vehicle every 24 hours.
We found that with GO and co-citation information our DBYN predicted
number of experimental established relationships were improved
1.5 to 2 fold. The improvement is more when we used the experimentally
confirmed gene interaction as an initial structure to train the
Bayesian network.
Mathematical modelling of the
gastrointestinal epithelium stem cell niche.
Sarah Waters (on behalf of Sarah Eastburn)
University of Oxford
Coauthors: Dr Sarah Waters, Dr James Oliver, Prof. Helen Byrne
and Dr Felicity Rose
The gastrointestinal tract is lined by a mono-layered epithelium
that contains invaginations called the crypts of Lieberkühn.
These crypts contain stem cells which are responsible for the
regeneration and maintenance of the epithelial lining. The stem
cells are thought to reside in a niche at the crypt base. The
crypts have a natural hierarchy and transit cells produced by
the stem cells, divide, differentiate and migrate towards the
intestinal lumen.
Using a number of complementary modelling approaches we investigate
the mechanics of the intestinal epithelium and the population
kinetics, resulting from, e.g., cell proliferation and cell death.
We apply these models to the in vitro experiments carried out
in the Tissue Engineering Group at Nottingham, with the aim of
determining the spatiotemporal distribution of cell density and
stress, and consider the role of different substrate geometries
on the morphology of a growing cell aggregate.
We model the tissue mechanics of a growing aggregate of cells
using a lattice-free spatial framework. Cells are unrestricted
in their position and move in a continuous fashion in response
to the resultant force exerted on each cell by its neighbours,
with drag balancing the cell-cell interactions which are modelled
as springs between neighbouring cells. The model behaviour is
captured in one dimensionless parameter, a, the ratio of a typical
cell cycle time to the spring relaxation time, with cell cycle
times modelled by a simple uniform distribution.
The cell kinetics of the intestinal crypt are modelled via a
stochastic branching process model for the evolution of a spatially-homogeneous
mixed population of stem, transit and fully differentiated cells.
From this model we derive analytic expressions for the expected
number and variance of each cell type, and for the total number
of crypt cells which are compared with experimental data.
These two models are then coupled together, the kinetic model
replacing the original simple cell proliferation assumption in
the mechanical model. We investigate how this modified kinetic
approach changes the resulting cell densities and stresses. We
will present an overview of the modelling techniques used, as
well as our key findings from these models.
Crystal aggregation and deposition in
the catheterised lower urinary tract
Sarah Waters
University of Oxford
Coauthors: Leah Band, Linda Cummings, Jonathan Wattis
Urethral catheters often become blocked with crystals of magnesium
struvite and calcium phosphate. The encrustation can block the
catheter, which can cause urine retention in the bladder and reflux
of urine into the kidneys. We develop a mathematical model to
investigate crystal deposition on the catheter surface, modelling
the bladder as a reservoir of fluid and the urethral catheter
as a rigid channel. At a constant rate, fluid containing crystal
particles of unit size enters the reservoir, and fluid flows from
the reservoir through the channel and out of the system. The crystal
particles aggregate, which we model using Becker-Doring coagulation
theory, and are advected through the channel, where they continue
to aggregate and can deposit on the channel's walls. Inhibitor
particles also enter the reservoir, and can bind to the crystals,
preventing further aggregation and deposition. In the reservoir,
we assume the crystal concentrations are spatially homogeneous,
whereas in the channel we consider concentrations that vary as
a result of advection, diffusion and deposition. We investigate
how the aggregation rate and the influx of inhibitor particles
affect the amount of deposition. For all parameter values, we
find that crystals deposit along the full length of the channel,
with maximum deposition at the channel's entrance.
Mathematical modelling of LDL and VLDL
endocytosis by HepG2 cells.
Jonathan Wattis
University of Nottingham
Coauthors: Brendan O'Malley (Unilever Corporate Research, UK),
Marcus Tindall (University of Oxford, UK), Laura Pickersgill (Unilever
Corporate Research, UK), Hannah Blackburn (Unilever Corporate
Research, UK), Helen Byrne (University of Nottingham, UK), Kim
Jackson (University of Reading, UK), Jasmina Panovska (Unilever
Corporate Research, UK).
Individuals with elevated levels of low-density-lipoprotein cholesterol
(LDL-C) in their plasma are considered to at risk of developing
coronary heart disease. LDL particles are removed from the body
mainly by hepatocytes through receptor-mediated endocytosis. Apolipoprotein
B-100 present on the surface of LDL particles binds to receptors
in pits on the surface of hepatocytes. Upon internalisation of
a pit, the bound complex of LDL and receptor is degraded into
its constituent parts (cholesterol, fatty acids and amino acids),
which are released for use by the cell or are excreted. Very low-density
lipoprotein particles (VLDL) are known to inhibit the take-up
of LDL [Jackson et al J Lipid Res 47, 393, (2006)].
We formulate a mathematical model of the binding, internalisation
and processing of LDL and VLDL particles by cells. The model is
calibrated to experimental data of Brown & Goldstein [PNAS
76, 3330, (1979)], and Harwood & Pellarin [Biochem J 323,
649, (1997)]. We find good agreement with in vitro data on LDL-take
up, and inhibition of take-up by VLDL. As well as the 'single
bolus' scenario where a large dose of lipoprotein is delivered
at the start of the experiment, we consider the more realistic
(in vivo) case where lipoprotein is continually delivered to the
system. We show how the the average occupancy of pits, and the
proportions of receptors which are free, bound, or internalised
changes over time. We analyse how the cell's flexibility in adapting
to changes in its environment depends on the efficiency of receptor-recycling,
how different types of VLDL influence LDL takeup, and differences
in behaviour exhibited by the single-bolus and continual-delivery
Epidemiological interactions between the
local and the mean-field
Steven Webb
University of Strathclyde
Coauthors: Mike Boots (Sheffield University) Matt Keeling (Warwick
The assumption that populations are completely mixed is reasonable
for many populations, but there is likely to be some degree of
local interaction whether spatially or socially in many systems.
An important question is therefore how strong these local interactions
need to be before there are significant effects on the dynamics
of the system. Our approach is to use correlation models, namely
pair-wise models, to capture the spatial relationships of contacts
and interactions between individuals. We first extend previous
pairwise models to include immunity and reproduction from infecteds
dynamics that have been largely ignored in such systems.
We then derive a multi-scale pair-approximation to move between
completely local and completely mixed host-parasite interactions,
thereby extending the reproduction processes of hosts and infection,
and examine the long term effects of these differing spatial scales
on the disease characteristics. Possible evolutionary traits of
the pathogen within this multi-scale framework are then derived
using adaptive dynamics.
Examining Cerebrospinal Fluid Pulsations
as a Causative Mechanism for Hydrocephalus
Kathleen Wilkie
University of Waterloo
Coauthors: Prof. S. Sivaloganathan
Hydrocephalus is a condition characterized by dilated ventricles
and compressed brain tissue in the cranial cavity. The ventricles
expand due to an increase in cerebrospinal fluid (CSF) with the
result that the brain parenchyma is compressed against the skull,
often causing neurological impairment. Although the pathophysiology
of hydrocephalus has been, and currently is, the focus of much
clinical and experimental research, it's etiology is still largely
unknown. In the literature, two different causative mechanisms
have been proposed: (1) increased intracranial pressure gradient
from the ventricles to the subarachnoid space (this fails to explain
both communicating hydrocephalus and normal pressure hydrocephalus)
and (2) intracranial CSF pulsatility (this begs the question "why
don't all subjects develop hydrocephalus?", since CSF flow
is always pulsative - even in normal subjects). In this talk,
I will present a simple poroelastic model of the brain which allows
for space- and time-dependent analytic solutions. Numerical simulations
allow us to determine the mechanical effects of the intracranial
pulsations on brain parenchyma and whether or not the pulsations
have the potential to induce ventricular dilation.
Modelling combined phototaxis and gyrotaxis
in suspensions of swimming microorganisms
Rosie Williams
University of Glasgow
Coauthors: Martin Bees
Hydrodynamic instabilities associated with cell aggregations
in suspensions of swimming, typically negatively buoyant, microorganisms
can lead to the formation of intricate patterns, termed bioconvection.
Phototaxis, a response to light, and gyrotaxis, defined as the
balance between gravitational (due to bottom heavy cells) and
viscous torques, bias the cell swimming direction, cause aggregations
and influence the resulting instabilities. We construct a general
model to predict stability of suspensions of photo-gyrotactic
unicellular microorganisms, specifically looking for the wavenumber
and growth rate of the first most unstable mode that grows from
an equilibrium state. We extend recent models for purely gyrotactic
cells, which couple the Navier-Stokes equation with a cell conservation
equation and model cell swimming directions probabilistically
using the Fokker-Planck equation. We include the effect of phototaxis
in three separate ways. First we include the effect of light in
the cell swimming speed, secondly we model the cells as though
they change their center of mass offset depending on available
light and thirdly we consider torque-like effects due to gradients
in light. Equilibrium solutions are found for each model and then
perturbed to assess stability for a range of realistic parameter
values for the green alga Chlamydomonas nivalis. Results of the
stability analysis are presented and the differences between each
model are highlighted and their relative viability discussed.
A systematic model of bacterial chemotaxis:
from signal transduction to cell motility in Escherichia coli.
Xiangrong Xin
Department of Biomedical Engineering and School of Mathematics,
University of Minnesota
Coauthors: David J. Odde, Department of Biomedical Engineering,
University of Minnesota, Email: firstname.lastname@example.com; Hans G. Othmer,
School of Mathematics and Digital Technology Center, University
of Minnesota, Email: firstname.lastname@example.com
The movement of bacteria in response to environmental changes
of specific metabolites and signaling molecules is called bacterial
chemotaxis. Chemotaxis in Escherichia coli (E. coli) is a best
studied system. The authors will present a systematic model of
E. coli chemotaxis that can capture many features of the system
and reproduce a full range of experimental observations from signaling
(excitation, perfect adaptation, robustness, high sensitivity,
wide dynamic range, etc.) to motor behavior and cellular motility.
A remarkable feature of the signaling pathway is its high sensitivity
to small relative changes in concentrations of chemical stimuli
over a broad range of ambient concentrations. To account for it,
the signaling part of the model is based on the structural and
functional unit of receptor clusters, trimer of chemoreceptor
dimers, which has been solidly experimentally established
but not well quantitatively modeled, so the theoretical work includes
more molecular mechanism in modeling and provides a more mechanistically
based description of the origin of high sensitivity than the existing
models in the field.
A multi-scale model of HIV-1 transmission
Lilit Yeghiazarian
Department of Biostatistics, UCLA
Coauthors: William G. Cumberland (UCLA) Otto O. Yang (UCLA)
Interactions of HIV-1 with the immune system of the host have
been extensively studied experimentally and theoretically using
mathematical models. On the other end of the spectrum, epidemiological
studies provided insight on HIV-1 dynamics within human populations.
The importance of the topology of human social networks in disease
epidemics became apparent as research on the underlying structure
of a variety of technological networks such as the Internet progressed.
We have developed a mathematical model that bridges the scales
between in-host processes and HIV-1 transmission in human sexual
networks. Each individual goes through a sequence of health states
that reflect his/her HIV-1 status, treatment stage, and vital
information. Associated with each health state is an in-host model
describing the interactions between HIV-1 and the immune system.
At the same time, each individual is modeled as a node in a sexual
network within which the disease spreads as sexual encounters
take place. We investigate the effect of a variety of medical
care decisions such as timing and type of therapy, on the spread
of disease within populations. We have found that early therapy
initiation, namely during the acute infection phase, may substantially
decrease the spread of disease.
Modeling the evolution of insect phenology:
Can insect populations adapt to climate change?
Brian Yurk
Department of Mathematics and Statistics, Utah State University,
Logan, Utah
Coauthors: James Powell (Department of Mathematics and Statistics,
Utah State University)
Warming temperatures are likely to disrupt insect phenology (the
timing of developmental events) to such an extent that some species
will face local extinction while others will erupt in new habitats.
It is unknown whether phenology can evolve rapidly enough to moderate
these effects. Since phenology is a critical determinant of fitness
in insect populations, there are strong selective pressures on
maintaining appropriate phenology. For example, it is important
that development is timed to avoid the coincidence of sensitive
life stages with extreme weather. An individual's fitness may
also be highly dependent on synchrony between its phenology and
the phenology of its biotic resources, as in the case of plant-pollinator
interactions. At low population densities, developmental synchrony
within a population can also be an important determinant of fitness;
the probability of finding mates increases when a large portion
of the population reaches reproductive age within a short time
period. Synchronized emergence within a population of herbivorous
insects may also be necessary to overwhelm resource defenses,
as is the case of mountain pine beetles attacking pine trees.
Temperature plays a major role in determining the phenology of
insects, since the time it takes for an insect to develop through
a life stage is highly dependent on the temperature that it experiences.
Previous phenology models have described this plastic response
of development time to temperature without considering genetic
evolution of the response. Without evolution, these models predict
that insect populations may lose developmental synchrony or synchrony
with biotic resources. We will present a modeling approach that
extends previous phenology models to allow for evolution of the
dependence of development time on temperature. Our model results
show that evolution may allow populations to adapt to warming
temperatures and changing resource phenology, but there are limits
to this adaptation. We will also discuss the existence of steady
distributions of the evolution model, in which the temporal structure
of the mean phenotype and phenotypic variance are invariant under
the evolution map with periodic temperatures. Both long term and
short term dynamics are controlled by the presence of phenotypes
that allow for individuals and their offspring to be oviposited
at the same time of year in consecutive years.
A new discrete distribution induced by
the Luria-Delbruck mutation model
Qi Zheng
Texas A&M School of Rural Public Health, College Station,
Texas 77843
The Luria-Delbruck mutation model has been a subject of mathematical
investigation for over six decades. A recent investigation of
this celebrated model led to the discovery of a new discrete distribution
that can potentially be applied to model data generated by other
biological processes. This two-parameter distribution arises as
a limiting form of the probability generating function discovered
by M.S. Bartlett. We first show that an obvious extension of the
limiting form is a valid probability generating function and then
present an algorithm for computing the probability mass function.
The asymptotic behavior of the probability mass function is revealed
by employing the technique of singularity analysis of generating
functions. We also suggest likelihood based algorithms for estimating
the parameters. The new distribution is found to be infinitely
divisible and possess divergent moments.
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