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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