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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[1]

Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417
[2]

Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems & Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035
[3]

Marc Bocquet, Alban Farchi, Quentin Malartic. Online learning of both state and dynamics using ensemble Kalman filters. Foundations of Data Science, 2021, 3 (3) : 305-330. doi: 10.3934/fods.2020015
[4]

Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x
[5]

Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021160
[6]

Mark F. Demers, Christopher J. Ianzano, Philip Mayer, Peter Morfe, Elizabeth C. Yoo. Limiting distributions for countable state topological Markov chains with holes. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 105-130. doi: 10.3934/dcds.2017005
[7]

Jiongmin Yong. Remarks on some short rate term structure models. Journal of Industrial & Management Optimization, 2006, 2 (2) : 119-134. doi: 10.3934/jimo.2006.2.119
[8]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193
[9]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813
[10]

C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71
[11]

Dong-Mei Zhu, Wai-Ki Ching, Robert J. Elliott, Tak-Kuen Siu, Lianmin Zhang. Hidden Markov models with threshold effects and their applications to oil price forecasting. Journal of Industrial & Management Optimization, 2017, 13 (2) : 757-773. doi: 10.3934/jimo.2016045
[12]

Roberto C. Alamino, Nestor Caticha. Bayesian online algorithms for learning in discrete hidden Markov models. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 1-10. doi: 10.3934/dcdsb.2008.9.1
[13]

Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2685-2702. doi: 10.3934/jimo.2020089
[14]

Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial & Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165
[15]

Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039
[16]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2022003
[17]

Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195
[18]

Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
[19]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[20]

Badal Joshi. A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1077-1105. doi: 10.3934/dcdsb.2015.20.1077

 Impact Factor: 

Tools

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy