American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 171-181. doi: 10.3934/proc.2013.2013.171

Analysis of the accelerated weighted ensemble methodology

Ronan Costaouec 1, , Haoyun Feng 2, , Jesús Izaguirre 2, and  Eric Darve 3,

1. 

Mechanical Engineering Department, Stanford University, CA, United States
2. 

Computer Science and Engineering, University of Notre Dame, IN, United States, United States
3. 

Mechanical Engineering Department and Institute for Computational and Mathematical Engineering, Stanford University, CA, United States

Received  September 2012 Revised  September 2013 Published  November 2013

The main issue addressed in this note is the study of an algorithm to accelerate the computation of kinetic rates in the context of molecular dynamics (MD). It is based on parallel simulations of short-time trajectories and its main components are: a decomposition of phase space into macrostates or cells, a resampling procedure that ensures that the number of parallel replicas (MD simulations) in each macro-state remains constant, the use of multiple populations (colored replicas) to compute multiple rates (e.g., forward and backward rates) in one simulation. The method leads to enhancing the sampling of macro-states associated to the transition states, since in traditional MD these are likely to be depleted even after short-time simulations. By allowing parallel replicas to carry different probabilistic weights, the number of replicas within each macro-state can be maintained constant without introducing any bias. The empirical variance of the estimated reaction rate, defined as a probability flux, is expectedly diminished. This note is a first attempt towards a better mathematical and numerical understanding of this method. It provides first a mathematical formalization of the notion of colors. Then, the link between this algorithm and a set of closely related methods having been proposed in the literature within the past few years is discussed. Lastly, numerical results are provided that illustrate the efficiency of the method.
Keywords: colors, Markov State Models., resampling, weighted ensemble, Reaction rate.
Mathematics Subject Classification: Primary: 65C35, 68U20; Secondary: 65N7.
Citation: Ronan Costaouec, Haoyun Feng, Jesús Izaguirre, Eric Darve. Analysis of the accelerated weighted ensemble methodology. Conference Publications, 2013, 2013 (special) : 171-181. doi: 10.3934/proc.2013.2013.171
References:
[1]

D. Bhatt and D. M. Zuckerman, Heterogeneous path ensembles for conformational transitions in semiatomistic models of adenylate kinase, Journal of Chemical Theory and Computation, 6 (2010), 3527-3539.

[2]

D. Bhatt, B. W. Zhang, and D. M. Zuckerman, Steady-state simulations using weighted ensemble path sampling, The Journal of Chemical Physics, 133 (2010), 014110.

[3]

D. Bhatt and D. M. Zuckerman, Symmetry of forward and reverse path populations, arXiv:1002.2402 (physics.comp-ph), 2010.

[4]

E. Darve and E. Ryu, Computing reaction rates in bio-molecular systems using discrete macro-states, Innovations in Biomolecular Modeling and Simulations,1 (2012), 138-206.

[5]

A. Dickson and A. R. Dinner., Enhanced Sampling of Nonequilibrium Steady States, Annu. Rev. Phys. Chem., 61 (2010), 441-459.

[6]

A. Dickson, A. Warmflash and A. R. Dinner., Separating forward and backward pathways in nonequilibrium umbrella sampling, J. Chem. Phys., 131 (2009), 154104.

[7]

A. Dickson, A. Warmflash and A. R. Dinner., Nonequilibrium umbrella sampling in spaces of many order parameters, J. Chem. Phys., 130 (2009), 074104.

[8]

W. E and E. Vanden-Eijnden, Towards a Theory of Transition Paths, Journal of Statistical Physics, 123 (2006), 503-523.

[9]

G. A. Huber and S. Kim, Weighted ensemble Brownian dynamics simulations for protein association reactions, Biophysical journal, 70 (1996), 97110.

[10]

P. Metzner, C. Schütte and E. Vanden-Eijnden, Illustration of transition path theory on a collection of simple examples, The Journal of Chemical Physics, 125 (2006), 084110.

[11]

V. S. Pande, K. Beauchamp and G. R. Bowman, Everything you wanted to know about Markov state models but were afraid to ask, Methods, 52 (2010), 99105.

[12]

N. Singhal and V. S. Pande, Error analysis and efficient sampling in Markovian state models for molecular dynamics, The Journal of Chemical Physics, 123 (2005), 204909.

[13]

W. C. Swope, J. W. Pitera and F. Suits, Describing Protein Folding Kinetics by Molecular Dynamics Simulations. 1. Theory, The Journal of Physical Chemistry B, 108 (2004), 65716581.

[14]

E. Vanden-Eijnden, M. Venturoli, G. Ciccotti and R. Elber, On the assumptions underlying milestoning, The Journal of Chemical Physics, 129 (2008), 174102.

[15]

E. Vanden-Eijnden and M. Venturoli., Exact rate calculations by trajectory parallelization and tilting, J. Chem. Phys., 131 (2009), 044120.

[16]

A. Warmflash, P. Bhimalapuram and A. R. Dinner., Umbrella sampling for nonequilibrium processes, J. Chem. Phys., 127 (2007), 154112.

[17]

B. W Zhang, D. Jasnow and D. M. Zuckerman, "Weighted Ensemble Path Sampling for Multiple Reaction Channels," arXiv.org, physics.bio-ph, February 2009.

[18]

B. W. Zhang, D. Jasnow and D. M. Zuckerman, The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures, The Journal of Chemical Physics, 132 (2010), 054107.

show all references

References:
[1]

D. Bhatt and D. M. Zuckerman, Heterogeneous path ensembles for conformational transitions in semiatomistic models of adenylate kinase, Journal of Chemical Theory and Computation, 6 (2010), 3527-3539.

[2]

D. Bhatt, B. W. Zhang, and D. M. Zuckerman, Steady-state simulations using weighted ensemble path sampling, The Journal of Chemical Physics, 133 (2010), 014110.

[3]

D. Bhatt and D. M. Zuckerman, Symmetry of forward and reverse path populations, arXiv:1002.2402 (physics.comp-ph), 2010.

[4]

E. Darve and E. Ryu, Computing reaction rates in bio-molecular systems using discrete macro-states, Innovations in Biomolecular Modeling and Simulations,1 (2012), 138-206.

[5]

A. Dickson and A. R. Dinner., Enhanced Sampling of Nonequilibrium Steady States, Annu. Rev. Phys. Chem., 61 (2010), 441-459.

[6]

A. Dickson, A. Warmflash and A. R. Dinner., Separating forward and backward pathways in nonequilibrium umbrella sampling, J. Chem. Phys., 131 (2009), 154104.

[7]

A. Dickson, A. Warmflash and A. R. Dinner., Nonequilibrium umbrella sampling in spaces of many order parameters, J. Chem. Phys., 130 (2009), 074104.

[8]

W. E and E. Vanden-Eijnden, Towards a Theory of Transition Paths, Journal of Statistical Physics, 123 (2006), 503-523.

[9]

G. A. Huber and S. Kim, Weighted ensemble Brownian dynamics simulations for protein association reactions, Biophysical journal, 70 (1996), 97110.

[10]

P. Metzner, C. Schütte and E. Vanden-Eijnden, Illustration of transition path theory on a collection of simple examples, The Journal of Chemical Physics, 125 (2006), 084110.

[11]

V. S. Pande, K. Beauchamp and G. R. Bowman, Everything you wanted to know about Markov state models but were afraid to ask, Methods, 52 (2010), 99105.

[12]

N. Singhal and V. S. Pande, Error analysis and efficient sampling in Markovian state models for molecular dynamics, The Journal of Chemical Physics, 123 (2005), 204909.

[13]

W. C. Swope, J. W. Pitera and F. Suits, Describing Protein Folding Kinetics by Molecular Dynamics Simulations. 1. Theory, The Journal of Physical Chemistry B, 108 (2004), 65716581.

[14]

E. Vanden-Eijnden, M. Venturoli, G. Ciccotti and R. Elber, On the assumptions underlying milestoning, The Journal of Chemical Physics, 129 (2008), 174102.

[15]

E. Vanden-Eijnden and M. Venturoli., Exact rate calculations by trajectory parallelization and tilting, J. Chem. Phys., 131 (2009), 044120.

[16]

A. Warmflash, P. Bhimalapuram and A. R. Dinner., Umbrella sampling for nonequilibrium processes, J. Chem. Phys., 127 (2007), 154112.

[17]

B. W Zhang, D. Jasnow and D. M. Zuckerman, "Weighted Ensemble Path Sampling for Multiple Reaction Channels," arXiv.org, physics.bio-ph, February 2009.

[18]

B. W. Zhang, D. Jasnow and D. M. Zuckerman, The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures, The Journal of Chemical Physics, 132 (2010), 054107.

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