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Spatial averaging : enhancement of the sampling of the configuration space for atomic clusters and biomolecules

Authors:
  • Qubit Pharmaceuticals
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 MεNεconfigurations.
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=hFiερ(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)=(x22)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
n1
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
Acknowledgements
Pr. Markus Meuwly
Dr. Nuria Plattner (Freie Universität Berlin)
Pr. Jim D. Doll (Brown University, RI, USA)
SNF for funding
Group Members
F.Hédin & M.Meuwly (Uni. Basel, Meuwly Group) SA-MC : enhanced MC sampling SCS Fall Meeting 6th September 2013 14 / 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 MεNεconfigurations.
Evaluate
E(m,n)
old,Boltz =eβE(m,n)
old
and
E(m,n)
new,Boltz =eβE(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(Mε1)
Mε
X(δmδ)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
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