Molecular simulations minimally restrained by experimental data
Huafeng Xu





[Correspondence to ]huafeng.xu@silicontx.com 
Silicon Therapeutics LLC, Boston, USA



  One popular approach to incorporating experimental data into molecular
  simulations is to restrain the ensemble average of observables to
  their experimental values.  Here I derive equations for the
  equilibrium distributions generated by restrained ensemble
  simulations and the corresponding expected values of observables.
  My results suggest a method to restrain simulations so that they
  generate distributions that are minimally perturbed from the
  unbiased distributions while reproducing the experimental values of
  the observables within their measurement uncertainties.





Molecular simulations minimally restrained by experimental data
    Huafeng Xu
    
===============================================================





§ INTRODUCTION


Molecular simulations are routinely used to generate models of
molecular structures and their equilibrium
distributions <cit.>.  They provide structural and dynamic
information at atomistic resolution that are critical to the
interpretation of experimental data <cit.>.  Inaccuracy in
the empirical force fields used to compute the potential energy of the
biomolecules in the simulations, however, can cause the simulations to
generate inaccurate equilibrium distributions of molecular structures,
manifested by the disparity between the experimentally measured values
of observables and the predictions by the
simulations <cit.>.  Biasing the simulations
toward reproducing the experimental values of the observables is a
promising approach to improving the accuracy of the
simulations <cit.>.  How to introduce minimal
bias, however, remains an open question, and no such method exists
that takes into account the measurement uncertainties associated with
the experimental data.

Because many experimental techniques, such as nuclear magnetic
resonance (NMR) and small-angle X-ray scattering (SAXS), measure the
ensemble average of observables, a natural and popular method to
incorporate such experimental data is to simultaneously simulate many
replicas of the molecular system while restraining the average of the
instantaneous values of any observable of interest to be close to its
experimental
value <cit.>.
I will refer to such replica-based restrained simulations simply as
restrained ensemble simulations.  The equilibrium distribution
generated by restrained ensemble simulations depends on the number of
replicas and the strength of the restraints, the latter of which
should reflect the measurement uncertainties of the
observables <cit.>.  There lacks, however, 
a clear theoretical underpinning for the choice of the number of 
replicas <cit.>.

The restrained simulations should be no more biased by the restraints
than is necessary, which is a principle that motivated the alternative
maximum entropy method, which seeks a distribution that reproduces the
experimental values of the observables and is minimally perturbed from
the unbiased distribution
 <cit.>. It
has been established that in some limit of infinite number of replicas
and infinitely strong restraints, restrained ensemble simulations
generate a statistically equivalent distribution as the maximum
entropy
method <cit.>.
In practice, however, the restraining strength is finite, in which case
the two methods are not equivalent. In contrast to restrained
ensemble simulations, the maximum entropy method constrains every
observable to its exact experimental value, thus it does not take into
account the measurement uncertainties, which are ubiquitous in all
experiments.

Here I present theoretical results that enable the determination of
the largest number of replicas to use in restrained ensemble
simulations with fixed strength of restraints so that they generate a
distribution minimally perturbed from the unbiased distribution while
reproducing the experimental values of the observables within their
measurement uncertainties.  As I will demonstrate below, this is
equivalent to the determination of the weakest strength of restraints
to use at fixed number of replicas, provided that the latter is
sufficiently large. The key result is an equation
(Eq. <ref>) that predicts the expected value of
any–restrained and unrestrained–observable in restrained ensemble
simulations for different numbers of replicas, thus permitting the
selection of the appropriate number without computationally expensive
trial and error. In addition, I derive an equation
(Eq. <ref>) for the distribution generated
by restrained ensemble simulations with a finite number of replicas
and a finite restraining strength, which can be used to perform
reweighted simulations that take into account of the experimental data
with their attendant measurement uncertainties.  The validity of these
theoretical results are demonstrated numerically by Monte Carlo
simulations. I illustrate the application of these results using the
molecular dynamics (MD) simulations of the polyalanine peptide Ala_5
restrained by experimental nuclear magnetic resonance (NMR) data.



§ THEORY


Consider an ensemble consisting of N replicas of the molecular
system, each with the atomic coordinates _i=1,2,…,N.
Experiments measure the ensemble average of the individual
contributions of the observables: =
N^-1∑_i_i, where _i ≡(_i) =
(D_1(_i), D_2(_i), …, D_M(_i)^t are the
instantaneous values of the vector of M observables for the i'th
replica.  In an unbiased simulation,  may deviate from
experimental values _e.  The posterior distribution given the
experimental data <cit.> is


    p_N({_i=1,2,…,N}| _e)    =    p({_i}|_e,E) 
    
       ∝   
      p(_e|{_i})p({_i}|E)
    
       =    p(_e|{_i}) ∏_i p_0( _i)


where p_0(_i) = Z_0^-1exp( -β E(_i)) is the
Boltzmann distribution of the molecular configurations under the
potential energy function E.  Assuming a normal distribution in each
experimental value, p(_e|{_i}) = (∏_α 2πδ_α^2)^-M/2exp(-1/2( - _e)), where
= diag(δ_1^-2, δ_2^-2, …,
δ_M^-2) is the inverse of the statistical variances of the
experimental measurements, p_N({_i}|_e) becomes


    p_N({_i}; , _e) ∝ 
    e^ -∑_i β E(_i) - 1/2( - _e),


which is the distribution in the restrained ensemble simulations.  The
strength of the restraints, , is thus related to the
measurement uncertainties associated with the experimental data.
In the restrained ensemble simulations, N replicas of the
molecular system are simulated with the same energy function and at
the same temperature, and they interact with one another only
through the quadratic term in the exponent in
Eq. <ref>.

A similar equation was derived by Hummer and Köfinger
(Eq. 26 of Ref. hummer2015), with the difference that
they scaled the restraining term by N and introduced a parameter
(θ) expressing the level of confidence in the unbiased
reference distribution.  My results below show that these two equations 
are equivalent at large N.  

For a fixed , the marginal distribution for a single replica,
p_N(_1; , ) = ∫∏_i=2^N p_N({_i };
, _e), tends to the unbiased distribution p_0(_1)
with increasing N.  Larger N thus corresponds to higher
confidence in the unbiased distribution.  A reasonable choice of N is
to take the largest value such that the expected values of the
observables in the restrained ensemble simulations remain within
the measurement uncertainties of the experimental values: this generates a
distribution minimally perturbed from the unbiased distribution while
still reproducing the experimental data.

I will analyze the behavior of restrained ensemble simulations at
sufficiently large numbers of replicas, i.e. N≫ 1. Let


    p_λ() = Z_λ^-1exp(-β E() +
    λ^t·())


be a biased distribution for an arbitrary vector λ, which
becomes the distribution generated by the maximum entropy method if
λ is the corresponding Lagrangian
multiplier <cit.>.  Dropping the subscript from the
constant vector _e, the partition function for the restrained
ensemble of N replicas is


    Z_N(, ) 
    
       =   ∫∏_i d_i e^ 
      - ∑_i β E(_i)
     - 1/2( N^-1∑_i _i - )
       =    Z_λ^N ⟨ 
     e^ -1/2(N^-1∑_i _i - ) - λ^t·∑_i _i⟩_N, λ
       =    Z_λ^N ∫ d ( ⟨δ( N^-1∑_i _i - ) ⟩_N,λ·
           e^-1/2( - ) - Nλ^t· )
    
       =    Z_λ^N ∫ d 
    e^-1/2( - ) - Nλ^t·ρ_N,λ()


where ⟨⋯⟩_N,λ signifies the ensemble
average according to the probability distribution of N independent
replicas (i.e., = 0), each sampled according to
the biased distribution p_λ(), and
ρ_N,λ() is the corresponding probability
distribution of the random variable = N^-1∑_i
_i.  In this ensemble of N independent replicas, {_i}
are independent and identically distributed (i.i.d.) random variables.
Acccording to the central limit theorem, the random variable
= N^-1∑_i _i, for sufficiently large N,
tends to the normal distribution ρ_N,λ()
≈ (2^Mπ^M|_λ|/N)^-1/2 e^-1/2
    N^-1( - _λ), where _λ =
∫ d () p_λ() are the expected values of
the observables () in the biased distribution (I will use
⟨⋯⟩_λ to denote the
expected values in the biased distribution with parameter λ),
_λ = ⟨ (() - )·(() -
)^t ⟩_λ is the corresponding covariance matrix, and
|_λ| is the determinant of matrix _λ.  If
_λ≡ - N^-1·λ and _λ are close, in the sense that 


    max_ s.t. 
      e^-( - _λ) > ϵN^-1( - _λ) < α


for a sufficiently small ϵ > 0 and some value of α≥ 3,
such that the normal distribution is a good approximation to
ρ_N,λ() for all  around
_λ,  can be integrated out, and the
partition function in Eq. <ref> becomes


    Z_N(, )    ≈   Z_λ^N e^1/2N^2( + N_λ^-1)^-1λ/√(|I + N^-1·_λ|)
       ·    e^-1/2(^-1 + N^-1)^-1( - _λ)
       ·    e^-Nλ^t·(+N_λ^-1)^-1·(· + N_λ^-1·_λ)


where I is the identity matrix.  Eq. <ref> forms the basis for deriving the key results of this work.

My first key result is that the expected value of any observable,
restrained or unrestrained, in the restrained ensemble simulations can
be predicted from the biased simulation of p_λ().  Let
^∗ = (D_1, D_2, …, D_M, D_M+1, …, D_M+M')^t be
the vector of restrained and unrestrained observables, where =
(D_1, D_2, …, D_M) are the M restrained observables and
_0 = (D_M+1, …, D_M+M')^t are the M' unrestrained
observables. Introducing


    ^∗ = ([           0_M× M';  0_M'× M 0_M'× M' ]),


_e^∗ = (_e^t  0_1× M')^t, and
λ^∗ = (λ^t  0_1× M')^t, the expected
values of the observables ^∗() in the restrained
ensemble simulations are given by


    ⟨^∗⟩_N,,_e
       =   ∫∏_i d_i p_N({_i};, _e) N^-1∑_i ^∗(_i)
    
       =   ∂/∂γln(
      ∫∏_i d_i ( e^-β∑_i E(_i).
    
          ·. . e^-1/2(N^-1∑_i _i - _e). .
    
          ·. . e^γ^t· N^-1∑_i ^∗_i)) |_γ = 0
       =   . ∂/∂γln(Z_N(, _e) e^1/2(^∗ + N_λ^∗ -1)^-1γ. .
    
       ·   . . e^γ^t·(^∗ + N^∗ -1)^-1·(^∗·_e^∗
     + N(_λ^∗ -1·⟨^∗⟩_λ - λ^∗) )|_γ=0
       =   _e^∗ + (I + N^-1^∗_λ^∗)^-1·( ⟨^∗⟩_λ - _e^∗
    - ^∗_λ·λ^∗)


where ^∗_λ = ⟨ (^∗ - ⟨^∗⟩)·(^∗ - ⟨^∗⟩)^t
⟩_λ is the covariance between all the observables in the
biased simulation.
 
Writing ^∗ in the block form


    _λ^∗ = (
    [     _λ _0,λ^t;   _0,λ  _00,λ ])


where _00,λ is the M'× M' covariance matrix for
the unrestrained observables _0 and _0,λ is the
M'× M covariance matrix between the unrestrained and the restrained
observables, the expected values of the restrained observables and the
unrestrained observables are, respectively,


    ⟨(_1) ⟩_N,,_e
       =   _e 
    + (I + N^-1_λ)^-1·(_λ - _e - _λ·λ)


and


    ⟨_0(_1) ⟩_N,,_e
       =   ⟨_0 ⟩_λ  
    + _0,λ·_λ^-1(I + N^-1·_λ^-1)^-1·( _e - _λ)
    
       +   _0,λ·_λ^-1·((I+N^-1·^-1)^-1-I)·λ


Eq. <ref> enables the determination of the
largest N such that the expected values of the restrained
observables remain within the measurement uncertainties of their
experimental values.

My second key result is that the marginal distribution for a single
replica generated by the restrained ensemble simulations is


    p_N(_1; , ) 
    
       =    Z_N^-1(, )e^-β E(_1)∫∏_i=2^N d_i ( e^-β∑_i=2^N E(_i).
    
          ·.  e^-1/2(N^-1∑_i=1^N_i - ))
    
       =    Z_N^-1(,) Z_N-1((N-1)^2/N^2, N  - _1/N-1) e^-β E(_1)
       ∝    p_0(_1) 
    e^(λ + _λ^-1·( - _λ))^t · (I + N^-1·_λ^-1)^-1·( (_1) - )
       ·    e^-1/2N(N^-1 + _λ)^-1((_1) - ),


where I used Eq. <ref> for both Z_N(,) and
Z_N-1((N-1)^2/N^2, N  - _1/N-1),
and N≈ N-1 for large N.

For N≫ 1, 1/2N(N^-1 +
  _λ)^-1(_1 - )≈ 0, and
Eq. <ref> reduces to


    p_N(; , _e) 
    
       ∝    p_0() 
      e^(λ + _λ^-1·(_e - _λ))^t ·
      (I + N^-1·_λ^-1)^-1·(() - _e),


which depends on N through the product N^-1.  This suggests
that a restrained ensemble simulation with N replicas and the
strength of restraints  is equivalent to another one with N'
replicas and the strength of restraints ' = (N'/N).
This equivalency, to my knowledge, has not been
previously demonstrated except for the harmonic potentials <cit.>, and it
mathematically justifies the scaling by N of the restraining term
proposed by Hummer and Köfinger <cit.>. 

Eq. <ref> suggests that the restrained ensemble
simulations generate the same distribution as the maximum entropy
method if the number of replicas satisfies the following inequality


    1 ≪ N ≪min( eigenvalues of ·_λ).
 

This condition is the generalization of the condition N≪ K derived
for the special case of harmonic potentials <cit.>.
Under this condition, (I + N^-1_λ^-1)^-1≈ I, the
corresponding single-replica distribution becomes


    p_N(_1;,_e) ∝
    p_0(_1) e^(λ + _λ^-1·(_e - _λ))^t· ((_1) - _e),


which is the same as p_λ(_1) if λ takes the value
of the Lagrangian multiplier λ_ME in the maximum
entropy method such that _λ_ME = _e.  Under
this condition, Eq. <ref> becomes ⟨(_1) ⟩_N,,_e≈_e.

The Kullback-Leibler divergence from the distribution in the
restrained ensemble simulations (Eq. <ref>)
to that of p_λ'() is


    Δ_KL[p_λ'||p_N]    =   ln⟨ e^G((), N) - λ'^t·()⟩_λ'
       -   ⟨ G((), N) - λ'^t·()⟩_λ'


where 


    e^G((), N)
       =    e^(λ + _λ^-1·(_e - _λ))^t 
    · (I + N^-1·_λ^-1)^-1·( () - )
       ·    e^-1/2N(N^-1 + _λ)^-1(() - _e)


Eq. <ref> enables the quantitative
estimation of the difference between the distributions generated by
the restrained ensemble simulations and by the maximum entropy method.
Minimization of Δ_KL[p_λ_ME||p_N] with
respect to N yields the number of replicas to use such that the
restrained ensemble simulations generate a distribution closest to
that of the maximum entropy method.



§ RESULTS AND DISCUSSIONS




I demonstrate the validity of Eq. <ref> and
Eq. <ref> using the Monte Carlo simulations of
a 2-dimensional potential that generates a distribution of mixed
Gaussians:

    p(x,y; f_a, f_b) = e^-E(x,y; f_a, f_b)
       =   ∑_s=a,bf_s/√(2^2π^2|σ_s|)e^-1/2σ_s^-1(x-x_s, y-y_s)


where f_a and f_b are the mixing coefficients constrained by f_a
+ f_b = 1. The parameters used in the simulations are σ_a =
((1.5, 0)^t, (0, 1.5)^t), (x_a, y_a) = (-1, 0), and σ_b =
((1.5, 0.1)^t, (0.1, 1.5)^t), (x_b, y_b) = (0.5, 2.5). The
restrained ensemble simulations
(Eq. <ref>) and the maximum
entropy method are applied to a potential with f_a=0.3, restraining
two observables, x and √(x^2 + y^2), to reference values that
are equal to the observables' mean values in a reference distribution
that has f_a = 0.7.  The restraining strength is =
diag( 100, 100 ).

For sufficiently large N, Eq. <ref> predicts the
expected values of both the restrained and unrestrained observables in
quantitative agreement with their mean values in the restrained
ensemble simulations (Fig. <ref>).  As N
increases, the distribution generated by the restrained ensemble
simulations tends to the unbiased distribution
(lim_N→∞Δ_KL[p_0||p_N] = 0).
Notably, Δ_KL[p_0||p_N] is relatively insensitive to
N around the largest value of N for which the expected values of
the restrained observables remain close to their reference values .

My results suggest the following method of setting N so that the
simulations are minimally restrained by experimental data: Iteratively
perform a series of biased simulations of p_λ_j(),
updating λ_j by either 

    λ_j = λ_j-1 +
    _λ_j-1^-1·( _e - _λ_j-1 )

or other update schemes <cit.> such that
lim_j→∞λ_j = λ_ME. For each
iteration, compute ⟨(_1) ⟩_N,,_e
using Eq. <ref> and select the largest N_j
so that the expected values of all the restrained observables are
within the measurement uncertainties from their experimental
values. Continue until N_j no longer changes. 



I illustrate this method of setting N by MD simulations of the
polyalanine peptide Ala_5 restrained by the ^3J couplings
measured by NMR experiments <cit.>.  All MD
simulations were performed using the OpenMM
software <cit.>, with Amber99SB-ILDN force
field <cit.> and a generalized Born implcit
solvent model <cit.>.  Langevin
integrator <cit.> was used to maintain the temperature
of the molecular system at 300 K.  When Ala_5 was simulated without
any restraints, there was substantial discrepancy between the
predictions by the unbiased simulation and the experimental values for the
^3J couplings between H^N and
H^α (denoted as ^3J_H^N,H^α,r for
r=2,3,4,5), which have been used to characterize the four backbone
torsions ϕ_r=2,3,4,5 of Ala_5 <cit.>
(Fig. <ref>).  Below, ^3J_H^N,H^α,r=2,3,4,5 
were used as the restrained observables.

The same molecular system was then simulated with a series of biasing
potentials (Eq. <ref>), in which λ_j
in the j'th simulation was determined by
Eq. <ref>. Eq. <ref> was
used to predict the expected values of the restrained observables
(^3J_H^N,H^α,r=2,3,4,5) in the
restrained ensemble simulations for different values of N, using
the values of λ_j and the corresponding ⟨⟩_λ_j and _λ_j in different biased simulations
as inputs. The largest N (N_max) for which the
predicted values of all the restrained observables are within the
statistical uncertainty from the experimental values is determined for 
each j. In this case, only 1 iteration of biased simulation was
necessary to have a converged estimate of
N_max (Fig. <ref>).

In practice, it is easier to select the number of replicas N based
on the available computing resources (e.g., the number of CPU cores).
Once N_max has been determined by the above procedure, one
can perform the restrained ensemble simulations using N replicas and
the scaled strength of restraints _min =
(N/N_max), because the restrained ensemble simulations with 
the parameters (N, _min) and with the parameters
 (N_max, ) generate equivalent marginal distributions 
(Eq. <ref>).  

My results suggest that the restrained ensemble simulations at large
N are equivalent to a single-replica simulation of the reweighted
distribution in Eq. <ref> (which
becomes Eq. <ref> as N→∞).
In this case, N can be regarded as a parameter that tunes the
strength of the restraints, which are proportional to the
inverse of the covariance of the experimental measurements.  As N
tends to infinity, the resulting distribution becomes unbiased.  To
demonstrate this reweighting approach, the Ala_5 system was
simulated according to the marginal distribution in Eq. 12 for
different values of N, using different values of λ_j and the
corresponding ⟨⟩_λ_j and
_λ_j as inputs. For j≥ 1, the expected values of
the restrained (^3J_H^N,H^α,r=2,3,4,5)
and unrestrained (^3J_H^N,C^β,r=2 and
the radius of gyration R_g) observables in these reweighted
simulations are in good agreement with the corresponding values
predicted by Eq. <ref> (Fig. <ref>).  Such
reweighted simulations may offer a simple alternative to replica-based
restrained ensemble simulations as a way to incorporate into MD
simulations experimental data with their associated measurement
uncertainties.  



§ ACKNOWLEDGMENTS

H. X. thanks Woody Sherman for helpful suggestions on the manuscript.



§ NOTES

The author declares no competing financial interest.



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@bib@innerbibempty

[Dror et al.(2012)Dror,
  Dirks, Grossman, Xu, and  Shaw]dror2012
  author author R. O. Dror, author R. M. Dirks,
  author J. Grossman, author H. Xu,  and author
  D. E. Shaw, title title Biomolecular Simulation: A Computational Microscope for
  Molecular Biology, 10.1146/annurev-biophys-042910-155245 journal journal Annual Review of Biophysics volume 41,  pages 429–452 (year 2012)NoStop
[Bottaro and Lindorff-Larsen(2018)]bottaro2018
  author author S. Bottaro and author K. Lindorff-Larsen, title title
  Biophysical experiments and biomolecular simulations: A perfect match?  10.1126/science.aat4010 journal journal Science volume 361, pages
  355–360 (year 2018)NoStop
[Gunsteren, Dolenc, and  Mark(2008)]gunsteren2008molecular
  author author W. F. v.  Gunsteren, author J. Dolenc,  and author A. E. Mark, title title Molecular
  simulation as an aid to experimentalists, 10.1016/j.sbi.2007.12.007 journal journal
  Current Opinion in Structural Biology volume 18,  pages 149–153 (year 2008)NoStop
[Boomsma et al.(2014)Boomsma, Tian, Frellsen, Ferkinghoff-Borg, Hamelryck, Lindorff-Larsen, and Vendruscolo]Boomsma:2014b
  author author W. Boomsma, author P. Tian,
  author J. Frellsen, author J. Ferkinghoff-Borg, author T. Hamelryck, author
  K. Lindorff-Larsen,  and author
  M. Vendruscolo, title
  title Equilibrium simulations of proteins using
  molecular fragment replacement and NMR chemical shifts, 10.1073/pnas.1404948111 journal journal
  Proceedings of the National Academy of Sciences volume 111, pages 13852–13857 (year
  2014)NoStop
[Perez et al.(2016)Perez,
  Morrone, Brini, MacCallum,  and Dill]Perez:2016
  author author A. Perez, author J. A. Morrone,
  author E. Brini, author J. L. MacCallum,  and author K. A. Dill, title
  title Blind protein structure prediction using
  accelerated free-energy simulations, 10.1126/sciadv.1601274 journal journal Science
  Advances volume 2, pages e1601274
  (year 2016)NoStop
[Lindorff-Larsen et al.(2005)Lindorff-Larsen, Best, DePristo,
  Dobson, and Vendruscolo]Lindorff-Larsen:20056f8
  author author K. Lindorff-Larsen, author R. B. Best, author M. A. DePristo,
  author C. M. Dobson,  and  author M. Vendruscolo,  title title Simultaneous determination
  of protein structure and dynamics, 10.1038/nature03199
  journal journal Nature volume 433, pages 128 (year
  2005)NoStop
[Vendruscolo(2007)]Vendruscolo:20076f8
  author author M. Vendruscolo, title title
  Determination of conformationally heterogeneous states of proteins,  10.1016/j.sbi.2007.01.002 journal journal Current Opinion in Structural Biology volume 17, pages 15–20 (year
  2007)NoStop
[Roux and Islam(2013)]Roux:2013a
  author author B. Roux and author S. M. Islam, title title
  Restrained-Ensemble Molecular Dynamics Simulations Based on Distance
  Histograms from Double Electron–Electron Resonance Spectroscopy, 10.1021/jp3110369 journal journal The
  Journal of Physical Chemistry B volume 117,  pages 4733–4739 (year 2013)NoStop
[Hummer and Köfinger(2015)]hummer2015
  author author G. Hummer and author J. Köfinger, title title Bayesian
  ensemble refinement by replica simulations and reweighting, 10.1063/1.4937786 journal journal The
  Journal of Chemical Physics volume 143, pages 243150 (year 2015)NoStop
[Olsson and Cavalli(2015)]olsson2015
  author author S. Olsson and author A. Cavalli, title title Quantification
  of Entropy-Loss in Replica-Averaged Modeling, 10.1021/acs.jctc.5b00579 journal journal Journal
  of Chemical Theory and Computation volume 11,  pages 3973–3977 (year 2015)NoStop
[Boomsma, Ferkinghoff-Borg, and  Lindorff-Larsen(2014)]Boomsma:2014
  author author W. Boomsma, author J. Ferkinghoff-Borg,  and author K. Lindorff-Larsen, title title Combining Experiments and Simulations Using the Maximum
  Entropy Principle, 10.1371/journal.pcbi.1003406
  journal journal PLoS Computational Biology  volume 10, pages e1003406 (year 2014)NoStop
[Różycki, Kim, and  Hummer(2011)]Rozycki:20116f8
  author author B. Różycki, author Y. C. Kim,  and author G. Hummer,  title title SAXS Ensemble Refinement of
  ESCRT-III CHMP3 Conformational Transitions, 10.1016/j.str.2010.10.006 journal journal
  Structure volume 19, pages 109–116
  (year 2011)NoStop
[Groth et al.(1999)Groth,
  Malicka, Czaplewski, Ołdziej, Łankiewicz, Wiczk, and  Liwo]Groth:19996f8
  author author M. Groth, author J. Malicka,
  author C. Czaplewski, author S. Ołdziej, author
  L. Łankiewicz, author
  W. Wiczk,  and author
  A. Liwo, title title Maximum entropy approach to the determination of solution
  conformation of flexible polypeptides by global conformational analysis and
  NMR spectroscopy – Application to DNS1-c-[d-A2bu2, Trp4,Leu5]- enkephalin
  and DNS1-c-[d-A2bu2, Trp4, d-Leu5]enkephalin, 10.1023/a:1008349424452 journal journal Journal
  of Biomolecular NMR volume 15, pages
  315–330 (year 1999)NoStop
[Massad et al.(2007)Massad,
  Jarvet, Tanner, Tomson,
  Smirnova, Palumaa, Sugai,
  Kohno, Vanatalu, and Damberg]Massad:20076f8
  author author T. Massad, author J. Jarvet,
  author R. Tanner, author K. Tomson, author
  J. Smirnova, author
  P. Palumaa, author M. Sugai, author T. Kohno, author K. Vanatalu,  and author P. Damberg, title title Maximum entropy reconstruction of joint ϕ, ψ-distribution
  with a coil-library prior: the backbone conformation of the peptide hormone
  motilin in aqueous solution from ϕ and ψ-dependent J-couplings,  10.1007/s10858-007-9150-1 journal journal Journal of Biomolecular NMR volume 38,  pages 107–123 (year 2007)NoStop
[Pitera and Chodera(2012)]pitera2012
  author author J. W. Pitera and author J. D. Chodera, title title On the Use of
  Experimental Observations to Bias Simulated Ensembles, 10.1021/ct300112v journal journal Journal of
  Chemical Theory and Computation volume 8, pages 3445–3451 (year 2012)NoStop
[Roux and Weare(2013)]Roux:20136f8
  author author B. Roux and author J. Weare,  title title On the statistical
  equivalence of restrained-ensemble simulations with the maximum entropy
  method, 10.1063/1.4792208 journal
  journal The Journal of Chemical Physics volume 138, pages 084107 (year
  2013)NoStop
[Cavalli, Camilloni, and  Vendruscolo(2013)]Cavalli:20136f8
  author author A. Cavalli, author C. Camilloni,
   and author M. Vendruscolo,  title title Molecular dynamics
  simulations with replica-averaged structural restraints generate structural
  ensembles according to the maximum entropy principle, 10.1063/1.4793625 journal journal The Journal of
  Chemical Physics volume 138, pages
  094112 (year 2013)NoStop
[Rieping, Habeck, and Nilges(2005)]Rieping2005
  author author W. Rieping, author M. Habeck,   and author M. Nilges,  title title Inferential structure
  determination, @noop  journal journal
  Science volume 309, pages 303 – 306
  (year 2005)NoStop
[Wang, Kulkarni, and Verdú(2005)]wang2005divergence
  author author Q. Wang, author S. R. Kulkarni,
   and author S. Verdú,  title title Divergence estimation of
  continuous distributions based on data-dependent partitions, 10.1109/tit.2005.853314 journal journal
  IEEE Transactions on Information Theory volume
  51, pages 3064–3074 (year 2005)NoStop
[Stinis(2005)]stinis2005a
  author author P. Stinis, title title A maximum
  likelihood algorithm for the estimation and renormalization of exponential
  densities, 10.1016/j.jcp.2005.03.001 journal journal Journal of Computational Physics volume 208, pages 691–703 (year
  2005)NoStop
[Graf et al.(2007)Graf,
  Nguyen, Stock, and Schwalbe]Graf:2007
  author author J. Graf, author P. H. Nguyen,
  author G. Stock,  and author H. Schwalbe, title
  title Structure and Dynamics of the Homologous Series
  of Alanine Peptides: A Joint Molecular Dynamics/NMR Study, 10.1021/ja0660406 journal journal
  Journal of the American Chemical Society volume
  129, pages 1179–1189 (year 2007)NoStop
[Wickstrom, Okur, and Simmerling(2009)]Wickstrom:2009
  author author L. Wickstrom, author A. Okur,   and author C. Simmerling,  title title Evaluating the Performance
  of the ff99SB Force Field Based on NMR Scalar Coupling Data, 10.1016/j.bpj.2009.04.063 journal journal Biophysical Journal volume 97, pages 853–856 (year 2009)NoStop
[Eastman et al.(2013)Eastman, Friedrichs, Chodera, Radmer, Bruns, Ku, Beauchamp, Lane, Wang, Shukla, Tye, Houston, Stich, Klein, Shirts, and Pande]eastman2013
  author author P. Eastman, author M. S. Friedrichs, author J. D. Chodera, author R. J. Radmer,
  author C. M. Bruns, author J. P. Ku, author
  K. A. Beauchamp, author
  T. J. Lane, author
  L.-P. Wang, author
  D. Shukla, author T. Tye, author M. Houston, author T. Stich,
  author C. Klein, author M. R. Shirts,  and author V. S. Pande, title
  title OpenMM 4: A Reusable, Extensible, Hardware
  Independent Library for High Performance Molecular Simulation, 10.1021/ct300857j journal journal
  Journal of Chemical Theory and Computation volume
  9, pages 461–469 (year 2013)NoStop
[Lindorff-Larsen et al.(2010)Lindorff-Larsen, Piana, Palmo,
  Maragakis, Klepeis, Dror,  and Shaw]LindorffLarsen:2010
  author author K. Lindorff-Larsen, author S. Piana, author K. Palmo,
  author P. Maragakis, author J. L. Klepeis, author
  R. O. Dror,  and author
  D. E. Shaw, title title Improved side-chain torsion potentials for the Amber
  ff99SB protein force field, 10.1002/prot.22711
  journal journal Proteins: Structure, Function,
  and Bioinformatics volume 78, pages
  1950–1958 (year 2010)NoStop
[Onufriev, Bashford, and  Case(2004)]onufriev2004exploring
  author author A. Onufriev, author D. Bashford,
   and author D. A. Case,  title title Exploring protein native
  states and large-scale conformational changes with a modified generalized
  born model, 10.1002/prot.20033 journal
  journal Proteins: Structure, Function, and Bioinformatics  volume 55, pages 383–394 (year 2004)NoStop
[Leimkuhler and Matthews(2015)]Leimkuhler2015
  author author B. Leimkuhler and author C. Matthews, title Numerical methods for stochastic
  molecular dynamics, in 10.1007/978-3-319-16375-8_7
  booktitle Molecular Dynamics: With Deterministic and
  Stochastic Numerical Methods (publisher Springer
  International Publishing, address Cham, year
  2015) pp. pages 261–328NoStop
[Vögeli et al.(2007)Vögeli, Ying, Grishaev, and  Bax]vogeli2007limits
  author author B. Vögeli, author J. Ying,
  author A. Grishaev,  and  author A. Bax, title title Limits on variations in protein backbone
  dynamics from precise measurements of scalar couplings, 10.1021/ja070324o journal journal Journal of the
  American Chemical Society volume 129, pages 9377–9385 (year 2007)NoStop


