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# Master thesis presentation slides September 2011

Authors:
• Qubit Pharmaceuticals
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
5Conformational study of Alanine dipeptide
Goal
Conﬁgurations
Spatial Averaging
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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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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
0x1 and 0 y1.
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.
π4i
n
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
Goal of MC methods
Monte Carlo method : any stochastic algorithm approximating
solutions of quantitative problems by using statistical sampling.
Example : estimating π
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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(EBEA)
kBT, where αis in [0;1]; if it is the
case, the state B is accepted.
Else, state B is rejected.
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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 MN
conﬁgurations.
Evaluate
E(m,n)
old,Boltz =eβE(m,n)
old
and
E(m,n)
new,Boltz =eβE(m,n)
new
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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(M1)
M
X(δmδ)2
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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 210641014
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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 =25restricted to heavy atoms.
TORS of DMAX =35for the two dihedral angles (φ, ψ).
The starting point is always (180,180) and moves are WEIGHted
1.0.
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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Content Monte Carlo Simulations Spatial Averaging Lennard-Jones clusters Implementation in CHARMM Ala dipeptide
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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Conclusion Acknowledgements Bibliography Questions ?
Questions ?
Thank you for your attention.
Any questions ?
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ResearchGate has not been able to resolve any citations for this publication.
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