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Spatial averaging : enhancement of the sampling of the

conﬁguration space for atomic clusters and biomolecules.

Florent Hédin, Markus Meuwly

Group Pr. M. Meuwly

Department of Chemistry

Universität Basel, Klingelbergstrasse 80, CH-4056 Basel

SCS Fall Meeting

6th September 2013

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 1 / 15

Table of Contents

1Theoretical background

Sampling rare events

SA-MC Algorithm

2Double well potential model

3Physical and chemical systems

Global minima of LJ clusters

Blocked Alanine dipeptide

4Conclusion and outlooks

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 2 / 15

Sampling rare events

Random walk procedures such as the Metropolis1algorithm provide a suitable

method for sampling free energy surfaces.

Problem: sample disconnected importance regions in a system.

For a system with two importance regions separated by a high barrier, the

random walk is trapped for a long time in the initial region. Replica exchange2

(left) and Umbrella Sampling3(right) are two widely used methods for ad-

dressing this problem.

1N. Metropolis et al. (1953). In: J. Chem. Phys. 21.6.

2R. H. Swendsen et al. (1986). In: Phys. Rev. Lett. 57.21.

3G M Torrie et al. (1977). In: Journal of Computational Physics 23.2.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 3 / 15

SA-MC

Spatial averaging Monte Carlo (SA-MC)4is an eﬃcient algorithm dedicated to the

study of rare-event problems. It increases the sampling of those events by modifying

the probability density function (pdf).

We consider an uni-dimensional particle of potential V: the probability for this

particle of being at a point xwith a potential V(x)is:

ρ(x)∝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)dx =Zρ(x, ε)dx

4J. D. Doll et al. (2009). In: J. Chem. Phys. 131.10

N. Plattner et al. (2010). In: J. Chem. Phys. 133.4.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 4 / 15

Metropolis vs. SA-MC : CHARMM implementation

Consider a trial conﬁguration ~

x0.

For moving atoms in ~

x0, generate a Gaussian distribution for Mεsets of Nε

conﬁgurations, of standard deviation Wεand centred on ~

x0.

Apply the chosen MC move to all of the Mε∗Nεconﬁgurations.

Metropolis

Accept/reject move

tt

Choose move

Evaluate ∆E

jj

Evaluate current energy

**

Evaluate new energy

OO

Apply move

44

SA-MC

Accept/reject move

tt

Choose move

SA-MC acc. crit.

jj

Generate conﬁguration

distribution

Evaluate current energy

Next conﬁguration

OO

oo

Apply move Mε×Nε//Evaluate new energy

OO

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 5 / 15

Thermodynamic properties and unbiasing

Let hf(x)i0be an unbiased thermodynamic property : in the case of SA-MC, a bias

in induced by the use of modiﬁed densities; an unbiased estimation of the property

is obtained as following:

hf(x)i0=ρ(x,0)

ρ(x, ε)f(x)ε

Let Fbe the Helmholtz Free Energy thermodynamic function of state (ensemble

NVT): hFi0will be its unbiased value for a given conﬁguration, estimated from a

SA-MC simulation:

hFi0=hFiε∗ρ(x,0)

ρ(x, ε)and F =−RT ln n

N

Where nis the observed occurrence of a given conﬁguration when considering N

states.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 6 / 15

Double well potential test system

To ﬁrst illustrate the eﬃciency of SA-MC but also the need of unbiasing, we ﬁrst

study a one dimensional problem involving a simple double-well potential, of the

form V(x)=(x2−2)2.

Parameters are : T=0.75;Mε=1;Nε=10;Wε=0.3

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 7 / 15

Lennard-Jones clusters : LJN

LJ clusters are deﬁned as an ensemble of non-

reactive atoms in vacuum interacting only through

Lennard-Jones potentials

VLJ =4

n−1

X

i=1

n

X

j=i+1"σ

rij 12

−σ

rij 6#

For simplifying the study reduced units are em-

ployed, i.e. =σ=1, and the energy will be

reported in units of .

The particular focus for this example is the speed with which a minimum energy

conﬁguration is found and whether or not the global minimum as reported in the

literature5can be found at all.

5D. J. Wales et al. (1997). In: J. Phys. Chem. A 101.28.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 8 / 15

Lennard-Jones clusters : LJN

104independent runs are started from the same initial (random) conﬁguration.

At each step, if the energy diﬀerence with the best conﬁguration is less than

5.0the system is minimised, and if the best conﬁguration is obtained the

calculation is stopped.

If the best minimum is not reached after 106steps it is considered that the

simulation has not converged.

LJNE ref. E MC E SA-MC

13 −44.326 −44.326 −44.326

31 −133.586 –126.081 −133.586

38 −173.928 –160.556 –167.023

55 −279.248 −279.248 −279.248

75 −397.492 –381.173 −397.492

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 9 / 15

FES of Alanine dipeptide

SA-MC is applied to the alanine dipeptide, to see if it can easily sample the

diﬀerent conﬁgurations and localise the transition paths between them.

The solvent is mimicked by using the ACE6implicit solvent model.

The corresponding energy landscape is visualised by using Ramachandran

plots combined to a Helmholtz Free Energy Surface (FES) ∆F.

6M. Schaefer et al. (1996). In: J. Phys. Chem. 100.5.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 10 / 15

FES of Alanine dipeptide

We ﬁrst generate results for Metropolis MC and Molecular dynamics for having a

couple of reference FES.

T=300K

MC (left): 108steps.

MD (right): 1.5µs, timestep 0.5fs.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 11 / 15

FES of Alanine dipeptide

We applied SA-MC with the following parameters :

Wε=0.5 ; Mεand Nε=5

5×106steps at T=300K

Left ﬁg. present biased results ; For right ﬁg. the unbiasing ratio as deﬁned a few

slides previously is used.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 12 / 15

Conclusion and outlooks

SA-MC proved to be eﬃcient for localising rare conﬁgurations such as the best

minima of LJ clusters.

It increases the sampling of the less favourable conﬁgurations of the blocked

alanine dipeptide and allowed a good estimation of the energy of the 4 stable

ones, and the barriers between them.

The unbiasing procedure is a requirement when thermodynamical properties

have to be estimated.

Even if a CPU time penalty of Nε×Mεis observed, the fact that results

are obtained several order of magnitudes faster (in terms of steps) than with

standard methods makes the method interesting to use.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 13 / 15

SA-MC : the acceptance criterion

Consider a trial conﬁguration ~

x0.

For moving atoms in ~

x0, generate a Gaussian distribution for Mεsets of Nε

conﬁgurations, of standard deviation Wεand 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.

F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 15 / 15