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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide

Spatial Averaging : a new Monte Carlo approach

for sampling rare-event problems

Florent Hedin

Laboratory of Physical Chemistry, Team of Prof. M. Meuwly

University of Basel, Switzerland

March 2011 – August 2011

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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide

1Monte Carlo Simulations

History

Goal of MC methods

The Metropolis-Hastings algorithm

2Spatial Averaging

Goal

Theory

Application to Molecular systems

3Lennard-Jones clusters

Goal

LJ7

LJ55

LJ38

4Implementation in CHARMM

MC module

Adding Spatial Averaging

5Conformational study of Alanine dipeptide

Goal

Conﬁgurations

Spatial Averaging

2

Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide

History

The idea of using stochastic processes for solving problems is not

new :

1733 : “Buﬀon’s needle problem” for estimating πusing

randomness.

1946 : Stanislas Ulam suggested to use stochastic methods

for integrals evaluation.

1949 : “The Monte Carlo Method” by S. Ulam, J. von

Neumann and N. Metropolis [1].

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Goal of MC methods

Monte Carlo method : any stochastic algorithm approximating

solutions of quantitative problems by using statistical sampling.

Example : estimating π

Consider a circle inscribed in a square of length 1.

Generate randomly a point Pof coordinates (x,y)with

0≤x≤1 and 0 ≤y≤1.

Check if this point is in the circle, i.e. if (x2+y2)≤1, and

increment a variable iif it is the case.

Repeat ntimes this experiment.

π≈4∗i

n

4

The Metropolis-Hastings algorithm

1953 : Nicholas Metropolis,Arianna & Marshall

Rosenbluth and Augusta & Edward Teller, for Boltzmann

distributions [2].

1970 : W. Keith Hastings extended it to the general case

[3].

Given a conﬁguration A of energy EA, generate a new

conﬁguration B via some MC move and estimate EB.

If EB<EAthe state B is accepted.

Else, check if α < e−(EB−EA)

kB∗T, where αis in [0;1]; if it is the

case, the state B is accepted.

Else, state B is rejected.

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Spatial Averaging – Goal

Goal : increase the sampling of rare-events problems by modifying

the probability density function (pdf).

The modiﬁed pdf has to have two speciﬁc properties:

The integral of the modiﬁed pdf over the whole space is

identical to that of the parent distribution

The modiﬁed pdf is easier to sample than the original one: if

not, there are no beneﬁts for using this modiﬁed pdf.

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Spatial Averaging – Theory

“A spatial averaging approach to rare-event sampling” [4], by

N. Plattner, J. D. Doll and M. Meuwly, 2009 .

For simplicity, a single dimension system is used, but results are

correctly generalisable to multi-dimensions systems.

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,0) = exp(−βV(x))

With β=1

kBT

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Spatial Averaging – Theory

The modiﬁed pdf is deﬁned as :

ρ(x, ) = ZP(y)exp(−βV(x+y))dy

Where P(y)is a Gaussian distribution of standard deviation .

This Gaussian conﬁguration is centred around ρ(x,0)so we have:

Zρ(x,0) = Zρ(x, )

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Spatial Averaging – Application to Molecular systems

N. Plattner, J. D. Doll and M. Meuwly [5], 2010.

Consider a trial conﬁguration ~

x0.

For moving atoms in ~

x0, generate a Gaussian distribution for

Msets of Nconﬁgurations, of standard deviation Wand

centred on ~

x0.

Apply the chosen MC move to all of the M∗N

conﬁgurations.

Evaluate

E(m,n)

old,Boltz =e−β∗E(m,n)

old

and

E(m,n)

new,Boltz =e−β∗E(m,n)

new

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Spatial Averaging – Application to Molecular systems

For each Mset, evaluate:

Sm

old =

N

XE(m,n)

old,Boltz and S m

new =

N

XE(m,n)

new,Boltz

And then:

δm=−ln Sm

new

Sm

old !

Then we deﬁned:

δ=1

M

M

Xδm

and

σ2=1

M∗(M−1)

M

X(δm−δ)2

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Spatial Averaging – Application to Molecular systems

δ+σ2

2will replace the ∆Eof the Metropolis Criterion.

α < exp(−β∗(δ+σ2

2))

Characterised by a triplet [W,M,N]

Classical Metropolis-Hastings simulation : [0.0,1,1]

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Lennard-Jones clusters - Goal

Objective : try Spatial Averaging MC on systems of many

possible conﬁgurations, well studied [6, 7, 8, 9, 10], and see if

our method allows to localise global minima.

Lennard-Jones clusters (LJN) : ensemble of N non-reactive

atoms in vacuum , interacting only through the Lennard-Jones

potential :

VLJ =4ε

n

X

i=1

n

X

j=i+1

r0

rĳ!12

− r0

rĳ!6

Reduced units are used, i.e. ε=r0=1, energy will be noted

as a multiple of ε.

Number of atoms 4 7 13 19 33

Number of minima 1 4 366 ∼2∗106∼4∗1014

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Lennard-Jones clusters - LJ7

4 minima system, the best one is icosahedral ; parameters : 100

runs of 5000 steps with [0.5;10;10]

LJ7First minima Second minima Third minima Fourth minima

E Theoretic/ε-16.505 -15.935 -15.593 -15.533

E SP-AV MC/ε-16.505 -15.935 -15.593 -15.533

Frequency (%) 32 7 15 46

The 4 minima are easily sampled with Spatial Averaging ; with

classical MC, frequencies of appearance of the minima were

respectively 4%;4%;2%;2%.

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Lennard-Jones clusters - LJ55

System of 55 atoms : number of minima unknown but incredibly

high, the best one is icosahedral.

Parameters : 100 runs of 10000 steps with [2.0;30;30]

LJNN=55

E Th/ε-279.248

E SP-AV MC/ε-279.132

Frequency(%) 2

With this many-minima system, Spatial Averaging proves its

eﬃciency, as classical Metropolis method is not able to ﬁnd the

global minima.

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Lennard-Jones clusters - LJ38

System of 38 atoms with a non-icosahedral minimum really close

in energy to the others.

Parameters : 250 runs of 10000 steps with [2.0;40;40]

LJNN=38

E Th/ε-173.928

E SP-AV MC/ε-173.915

Frequency(%) 0.4

The best minima is found once for 250 runs ; classical Metropolis

was not able to sample it.

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CHARMM – MC module

MC module : mainly written by A. Dinner,J. Hu and A.Ma.

User deﬁnes an ensemble of moves for a given molecular

system.

At each step of the loop, one instance chosen in the allowed

moves.

MOVE ADD MVTP RTRN BYHEavy WEIGht 1.0 DMAX 0.15 LABEL TR -

SELE ALL END

MOVE ADD MVTP RROT BYHEAVY WEIGht 1.0 DMAX 25.0 LABEL ROT -

SELE ALL END

[...]

MC TEMPerature 300.00 NSTEps 10000 IECHeck 100 IACC 0 PICK 0 -

ISEED 1314790643 IUNCrd 2 NSAVc 10

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CHARMM – adding Spatial Averaging

2009–2010, N. Plattner : implementation for RTRN moves,

interesting and promising results published.

Goal : A new implementation for RTRN, RROT, and dihedral

angles torsions (TORS).

Existing subroutines of MC module reused as much as

possible for compatibility.

Nevertheless, several advanced features can not be enabled

when using Spatial Averaging.

4 parameters added to the line launching the MC simulation :

MC TEMPerature 300.00 NSTEps 10000 IECHeck 100 IACC 0 PICK 0 -

ISEED 1314790643 IUNCrd 2 NSAVc 10 -

SPAV WEPS 1.0 MEPS 25 NEPS 25

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Alanine dipeptide

Test system for theoretical studies of backbone conformational

equilibria [11, 12, 13].

Characterised by two (φ, ψ)dihedral angles.

We proposed to apply Spatial Averaging alanine dipeptide, to see if

we can easily sample the diﬀerent conﬁgurations and localise the

transition paths between them.

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Alanine dipeptide – Conﬁgurations

Possible conformations are :

β, also called C5, for (φ, ψ)∼(−140,150)

C7eq for (φ, ψ)∼(−90,80)

αR(Right-handed αhelix) for (φ, ψ)∼(−80,−60)

αL(Left-handed αhelix) for (φ, ψ)∼(60,60)

C7ax for (φ, ψ)∼(60,−60)

In the case of a vacuum study, only the states β,C7eq and C7ax are

observed, as the αones are favoured by water.

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Alanine dipeptide – Spatial Averaging

Methodology : simulations of 10000 steps, with three possible

moves:

RTRN of a maximal distance DMAX =0.15 Å restricted to

heavy atoms.

RROT of DMAX =25◦restricted to heavy atoms.

TORS of DMAX =35◦for the two dihedral angles (φ, ψ).

The starting point is always (180,180) and moves are WEIGHted

1.0.

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Alanine dipeptide – Spatial Averaging

First, classical MC simulation (bottom left picture) : the

system does not quit the zones of βand C7eq , the most stable

ones.

Then, [1.0;10;10] (bottom right) : more conﬁgurations are

sampled in the zones of βand C7eq.

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Alanine dipeptide – Spatial Averaging

For [1.0;25;25] (bottom left) : the C7ax is sampled by a great

number of points, and we can clearly see a path C7eq →C7ax .

For [2.0;10;10] (bottom right) : results are similar to

[1.0;25;25], with only 10*10 conﬁgurations, which is much

more faster than 25*25.

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Conclusion Acknowledgements Bibliography Questions ?

Conclusion and outlook

Spatial Averaging seems to be eﬃcient for sampling really rare

conﬁgurations, as we saw with LJ clusters.

Implementation for RTRN, RROT and TORS in CHARMM.

The method can easily and quickly sample the diﬀerent

conﬁgurations of Alanine dipeptide in vacuum, and ﬁnd a

transition path between them.

Work is in progress for an implicit solvation model, which may

allow to use this method with big systems in solution :

cyclic-di-GMP, insulin dimer ...

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Conclusion Acknowledgements Bibliography Questions ?

Acknowledgements

I want to thanks Professor Markus Meuwly, who welcomed me in

his team for those six months of work, and all the members of the

team, who considered me as a full member : Lixian Zhang, Dr.

Pierre-André Cazade, Franziska Hofmann, Maksym Soloviov,

Juvenal Yosa Reyes, Dr. Stephan Lutz, Prashant Gupta, Dr. Jing

Huang, Dr. Myung Won Lee, Dr. Yonggang Yang, Dr. Jaroslaw

Szymczak, Manuella Utzinger, and Andi Meier.

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Conclusion Acknowledgements Bibliography Questions ?

[1] Nicholas Metropolis and S. Ulam. The monte carlo method. Journal of the

American Statistical Association, 44(247):335–341, 1949. ArticleType:

research-article / Full publication date: Sep., 1949 / Copyright Âľ 1949

American Statistical Association.

[2] Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth,

Augusta H. Teller, and Edward Teller. Equation of state calculations by fast

computing machines. The Journal of Chemical Physics, 21(6):1087, 1953.

[3] W. K. HASTINGS. Monte carlo sampling methods using markov chains and their

applications. Biometrika, 57(1):97 –109, April 1970.

[4] JD Doll, JE Gubernatis, N Plattner, M Meuwly, P Dupuis, and H Wang. A

spatial averaging approach to rare-event sampling. JOURNAL OF CHEMICAL

PHYSICS, 131(10), September 2009.

[5] N Plattner, JD Doll, and M Meuwly. Spatial averaging for small molecule

diﬀusion in condensed phase environments. JOURNAL OF CHEMICAL

PHYSICS, 133(4), July 2010.

[6] David J. Wales and Jonathan P. K. Doye. Global optimization by Basin-Hopping

and the lowest energy structures of Lennard-Jones clusters containing up to 110

atoms. The Journal of Physical Chemistry A, 101(28):5111–5116, July 1997.

[7] Sigurd Schelstraete and Henri Verschelde. Finding Minimum-Energy

conﬁgurations of Lennard-Jones clusters using an eﬀective potential. The Journal

of Physical Chemistry A, 101(3):310–315, January 1997.

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Conclusion Acknowledgements Bibliography Questions ?

[8] JPK Doye, MA Miller, and DJ Wales. The double-funnel energy landscape of the

38-atom Lennard-Jones cluster. JOURNAL OF CHEMICAL PHYSICS,

110(14):6896–6906, April 1999.

[9] JPK Doye, MA Miller, and DJ Wales. Evolution of the potential energy surface

with size for Lennard-Jones clusters. JOURNAL OF CHEMICAL PHYSICS,

111(18):8417–8428, November 1999.

[10] Xiang, Cheng, Cai, and Shao. Structural distribution of Lennard-Jones clusters

containing 562 to 1000 atoms. The Journal of Physical Chemistry A,

108(44):9516–9520, November 2004.

[11] Douglas J. Tobias and Charles L. Brooks. Conformational equilibrium in the

alanine dipeptide in the gas phase and aqueous solution: a comparison of

theoretical results. The Journal of Physical Chemistry, 96(9):3864–3870, April

1992.

[12] Dmitriy S. Chekmarev, Tateki Ishida, and Ronald M. Levy. Long-Time

conformational transitions of alanine dipeptide in aqueous solution:âĂĽ

continuous and Discrete-State kinetic models. The Journal of Physical Chemistry

B, 108(50):19487–19495, December 2004.

[13] Ao Ma and Aaron R. Dinner. Automatic method for identifying reaction

coordinates in complex systemsâĂă. The Journal of Physical Chemistry B,

109(14):6769–6779, April 2005.

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