Rare events sampling with spatial averaging: theory, applications

Abstract

Spatial averaging algorithm is an efficient MC method which can be applied to problems where important regions (e.g. transition states) of the energy landscape may be difficult to sample with a standard random walk method, such as Metropolis sampling. At the heart of the method is the realization that from the equilibrium density a related, modified probability density can be constructed through a suitable transformation. This new density is more highly connected which increases the chances for transitions between neighboring states which in turn speeds up the sampling. In order to transform the equilibrium density, Gaussian distributions with variable widths are used. First successful investigations included the diffusion of small molecules in condensed phase environments and localization of lowest energy structures of Lennard-Jones clusters. A more general implementation in CHARMM allowed us to study the conformation space of biomolecules.

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Sampling rare events with spatial averaging: theory and applications
Florent H´
edin, Nuria Plattner and Markus Meuwly
University of Basel – Department of Chemistry
florent.hedin@unibas.ch
Introduction
Random walk procedures, as the Metropolis-Hastings algorithm, are efficient methods for sampling energy surfaces.
But for high potential barriers : random walks may be trapped in one of the minima.
In that case, reaching another importance region may require very long sampling times.
⇒We present a new method whose goal is to improve the sampling process by directly modifying probability densities.
Theory of spatial averaging
Goal: increase the sampling of rare-events problems by modifying the probability density
function (pdf).
For a particle of potential V: the probability density is:
ρ(x,0)∝exp(−βV(x))
The modified pdf is defined as :
ρ(x, ε)∝Zexp(−βV(x+y))dy
Where yis a perturbation following a Gaussian distribution Pεof standard deviation ε.
This Gaussian configuration is centred around xso we have:
Zρ(x,0) = Zρ(x, ε)
Unbiasing
Let hf(x)i0be a thermodynamic property defined by using the densities of states: it is ex-
pressed as following:
hf(x)i0=Rρ(x,0)f(x)dx
Rρ(x,0)dx
In the case of spatial averaging, a bias in induced by the use of modified densities; an unbiased
estimation of the property is obtained as following:
hf(x)i0=Rρ(x, ε)ρ(x,0)
ρ(x, ε)f(x)dx
Rρ(x, ε)dx
For a simple double well potential as V(x)=(x2−1)2, spatial averaging is applied with
ε=0.25.
A reconstructed energy surface is displayed on Figure 1:
Black curve is the reconstruction for a Metropolis simulation.
Red curve is for spatial averaging where ε=0.25: one can easily see that the bias halves the barrier.
Green curve is the unbiased energy surface: it is close to the Metropolis one.
Figure 1: Energy surface built from MC simulations: Metropolis and spatial averaging with ε=0.25
Application to higher dimensional systems
Consider a trial configuration ~
x0, generate Mε∗Nεconfigurations, of standard deviation Wε,
centred on ~
x0.
Apply the chosen MC move to all of the Mε∗Nεconfigurations.
Evaluate E(m,n)
old,Boltz =e−β∗E(m,n)
old and E(m,n)
new,Boltz =e−β∗E(m,n)
new
For each Mεset, evaluate:
Sm
old =
Nε
XE(m,n)
old,Boltz Sm
new =
Nε
XE(m,n)
new,Boltz δm=−ln Sm
new
Sm
old !
Then we defined:
δ=1
Mε
Mε
Xδmσ2=1
Mε∗(Mε−1)
Mε
X(δm−δ)2
δ+σ2
2will replace the ∆Eof the Metropolis Criterion.
α < exp(−β∗(δ+σ2
2))
Lennard-Jones cluster LJ13
Spatial averaging is applied to Lennard-
Jones clusters of rare gases.
VLJ =4
n
X
i=1
n
X
j=i+1
 1
rij !12
− 1
rij !6

13 atoms, 104simulations starting from the
same configuration, each with a maximal
number of steps of 106.
Figure 2: Lennard-Jones cluster of 13 atoms, LJ13
Lennard-Jones cluster LJ13 (ctd.)
Simulation ends when the reduced energy of the best minimum is reached, i.e. E=−44.3268
Spatial averaging sets of parameters (cf. previous section) are written on the form
[Wε;Mε;Nε].
⇒Figure 3 represents cumulated histograms illustrating the required number of steps before
reaching the best minimum :
(a) (b)
Figure 3: Distribution of the required number of steps for reaching E=−44.3268 for (a) Metropolis simulations, and
(b) spatial averaging simulations with different sets of parameters.
Spatial averaging is really efficient here: after 106steps we observe that:
Only 22 % of the simulations converged for Metropolis simulations.
For [0.25;5;5], more than 80 % of convergence is obtained.
For [0.5;5;5], an optimal convergence of ∼98 % is reached in less than 105steps.
For [1.0;5;5]in contrary results are bad as only ∼50 % of the simulations converged: we interpreted this low
efficiency as following: if εis too high, the modified densities will overlap a lot, and so the averaged energy
difference will be lower, leading to a too high acceptance rate: this was proved to be counter-productive for
Metropolis simulations, and it also appears to be the case for spatial averaging.
Alanine dipeptide Free Energy Surface (FES)
Two (φ, ψ)dihedral angles, so FES based on Ramachandran plots easy to build.
CHARMM c37’s MC module, implicit solvent used (ACE), 300 K, 5∗106steps.
⇒Application of unbiasing to molecular systems:
Let Fbe the Helmholtz Free Energy thermodynamic function of state (ensemble NVT): hFi0will
be its unbiased value for a given configuration, estimated from a spatial averaging simulation:
hFi0=hFiε∗ρ(x,0)
ρ(x, ε)and F =−RT ln n
N
Where nis the occurrence of a given configuration when considering Nstates.
When considering application to the alanine dipeptide with CHARMM the densities are written
as following:
ρ(x,0) = exp(−β∗∆E0)and ρ(x, ε) = exp(−β∗∆Eε)
Where ∆E0is the energy difference between the two configurations of a Metropolis Monte
Carlo, and ∆Eεthe average of the energy differences for all the spatial averaging distributions.
(a) (b)
Figure 4: FES for alanine dipeptide , (a) Metropolis MC and (b) spatial averaging.
Conclusion and Outlooks
Spatial averaging increases the probability of sampling rare events, as high energy
states/barriers, rare configurations ...
It is possible to unbias a posteriori the results for obtaining a good approximation of the original
density: this leads to a good estimation of energy surfaces, thermodynamic properties ...
One future application is the study of the folding process of proteins.
References and acknowledgment
J.D. Doll et al., J. Chem. Phys., 131, 104107, (2009)
N. Plattner et al., J. Chem. Phys., 133, 044506, (2010)
F. H´
edin, Master Thesis, University of Basel, Aug. 2011
Work supported by The Swiss National Science Foundation (SNSF).
H´
edin, Plattner and Meuwly (University of Basel) Sampling rare events with spatial averaging florent.hedin@unibas.ch 1 / 1