Reweighting Techniques for Monte Carlo and Molecular Dynamics - - PowerPoint PPT Presentation

Reweighting Techniques for Monte Carlo and Molecular Dynamics Simulations I

The 13th KIAS Protein Folding Winter School High 1 Resort, Korea January 19-24, 2014

Yuko OKAMOTO （岡本 祐幸） Department of Physics and Structural Biology Research Center Graduate School of Science and Center for Computational Science Graduate School of Engineering and Information Technology Center NAGOYA UNIVERSITY （名古屋大学） e-mail: okamoto{a}phys.nagoya-u.ac.jp URL: http://www.tb.phys.nagoya-u.ac.jp/

Papers

• 1. A.M. Ferrenberg and R.H. Swendsen, “New Monte Carlo

Technique for Studying Phase Transitions,” Physical Review Letters 61, 2635-2638 (1988).

• -- Single-Histogram Reweighting Techniques
• 2. S. Kumar, D. Bouzida, R.H. Swendsen, P.A. Kollman, and

J.M. Rosenberg, “The Weighted Histogram Analysis Method for Free-Energy Calculations on Biomolecules.

• I. The Method,” Journal of Computational Chemistry 13,

1011-1021 (1992).

• -- Multiple-Histogram Reweighting Techniques (WHAM)
• 3. M.R. Shirts and J.D. Chodera, “Statistically Optimal

Analysis of Samples from Multiple Equilibrium States,” Journal of Chemical Physics 129, 124105 (10 pages) (2008).

• -- A Variant of WHAM (Multistate Bennett Acceptance

Ratio Estimator: MBAR)

Contents

• 1. Metropolis Monte Carlo Method
• 2. Simulated Annealing and Related Methods
• 3. Single-Histogram Reweighting Techniques
• 4. Generalized-Ensemble Algorithms I (MUCA)
• 5. Multiple-Histogram Reweighting Techniques (WHAM)
• 6. Multistate Bennett Acceptance Ratio Estimator

(MBAR)

• 7. Generalized-Ensemble Algorithms II (REM)

• 1. Metropolis Monte Carlo Method

Microcanonical Ensemble ミクロカノニカルアンサンブル Isolated System： E tot = const 孤立系： E tot = 一定

Canonical Ensemble カノニカルアンサンブル

System in Heat Bath (Exchange Energy w/ Heat Bath)

熱浴中の系（熱浴とエネルギーをやりとり）： T = const = 一定

cano

E PB(E) = n(E)WB(E) Canonical Probability Distribution

E WB(E) = exp(-  E ) Boltzmann Factor E n(E) Density of States

Canonical Ensemble at Temperature T

 

1 2 3 1 2 3

, , , , ; , , , ,    x

N N

q q q q p p p p

         

1 2 3 1 v v j k

x x x x x x x



       

         

 

 

1 1 2 1

; , , ,

v v v j k j k

w x x x x x x x w x x

 

      MONTE CARLO

Generate states one after another. Suppose at the -th step the state was xj and the candidate for the (+1)-th step is xk . Suppose also that the transition probability w satisfies the following: Markov Chain State x

   

  



eq j j k eq k k j

P x w x x P x w x x   

  



  

  

1 v v k j j k j

P x P x w x x



 



 

   

 



eq k eq j j k j

P x P x w x x

By definition, the transition probability w satisfies where P()(x) is the probability distribution of state x at the -th step. The equilibrium probability distribution Peq(x) should satisfy A sufficient condition for this to be satisfied is to impose a detailed balance condition:

One solution to this detailed balance condition is (Metropolis method):

• N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller & E. Teller,
• J. Chem. Phys. 21, 1087 (1953).

In Canonical Ensemble, where the equilibrium probability distribution is ∝ the Boltzmann weight factor, we have

Suppose at the -th step the state was xj and the candidate for the (+1)-th step is xk .

z

0.5 0.5

x y

       

12 6

4

LJ i j ij ij

E r r   



                         



ij i j

r   q q

 

, ,

1 2 N

x  ,  q q q

i i i

   q q q

i

   q 

 

z x y

      

where

Trial Move: qold → qnew

Example of MC simulations: Lennard-Jones fluid where

random numbers

Met-Enkephalin: Global Minimum Structure in Gas Phase

E = -12 kcal/mol potential energy: ECEPP/2 degrees of freedom: dihedral angles

Tyr-Gly-Gly-Phe-Met (N = 5)

30 20 10

• 10

E 200000 150000 100000 50000 MC Sweeps

Canonical 1000K

30 20 10

• 10

E 200000 150000 100000 50000 MC Sweeps

Canonical 600K

30 20 10

• 10

E 200000 150000 100000 50000 MC Sweeps

Canonical 300K

30 20 10

• 10

E 200000 150000 100000 50000 MC Sweeps

Canonical 50K

• Cf. Molecular Dynamics Method

2 2

3                       



       

i i i i i i B i

E s s m m m s s s Qs s m Nk T Q s q q f q q q

     

i i i

E mq f q

MOLECULAR DYNAMICS

Newton’s equations of motion: Microcanonical Ensemble Nose’s method: Canonical Ensemble at temperature T

• S. Nose, Mol. Phys. 52, 255 (1984); J. Chem. Phys. 81, 511 (1984).

• 2. Simulated Annealing and Related

Methods

SA-2

徐冷法 (Simulated Annealing)

• S. Kirkpatrick, C. Gelatt, Jr. & M. Vecchi, Science 220, 671 (1983).

Reproduce a Crystal-Making Process on a Computer

• H. Kawai, T. Kikuchi & Y.O., Protein Eng. 3, 85 (1989).

See also:

• S. Wilson, W. Cui, J. Moskovitz & K. Schmidt, Tetrahedron Lett. 29, 4373 (1988).
• C. Wilson & S. Doniach, Proteins 6, 193 (1989).
• A. Brunger, J. Mol. Biol. 203, 803 (1988).
• M. Nilges, G. Clore & A. Gronenborn, FEBS Lett. 229, 317 (1988).

For a review see:

• Y.O., Recent Res. Devel. In Pure & Applied Chem. 2, 1 (1998).

Application of Simulated Annealing to Systems of Biopolymers

SA-2

PB(E) E E PB(E) = n(E)W

B (E)

High T PB(E) E Emin Low T

Simulated Annealing（徐冷法）

30 20 10

• 10

E 200000 150000 100000 50000 MC Sweeps

Simulated Annealing: T = 1000 K  50 K Canonical 1000K Canonical 50K

C-peptide-1

C-Peptide (N = 13): X-Ray C-Peptide of Ribonuclease A

Amino-Acid Sequence: Lys-Glu-Thr-Ala-Ala-Ala-Lys-Phe-Glu-Arg- Gln-His-Met

Cp-first

• M. Masuya & Y.O., unpublished.

Cp-last

• M. Masuya & Y.O., unpublished.

C-Peptide (N = 13): X-Ray Level 0 Solvation (Gas Phase)

RMSD = 1.9 A

Level 1 Solvation (distant- dependent dielectric) Level 2 Solvation (SASA)

RMSD = 1.4 A RMSD = 0.8 A

• M. Masuya & Y.O., unpublished.

Because we lower the temperature during the simulation, we are always breaking thermal equilibrium (and detailed balance conditions). Hence, thermodynamic quantities obtained from SA are not reliable.

Problems of Simulated Annealing (SA)