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Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution

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10.1063/1.4801325

### Abstract

Direct simulation of biomolecular dynamics in thermal equilibrium is challenging due to the metastable nature of conformation dynamics and the computational cost of molecular dynamics. Biased or enhanced sampling methods may improve the convergence of expectation values of equilibrium probabilities and expectation values of stationary quantities significantly. Unfortunately the convergence of dynamic observables such as correlation functions or timescales of conformational transitions relies on direct equilibrium simulations. Markov state models are well suited to describe both stationary properties and properties of slow dynamical processes of a molecular system, in terms of a transition matrix for a jump process on a suitable discretization of continuous conformation space. Here, we introduce statistical estimation methods that allow *a priori* knowledge of equilibrium probabilities to be incorporated into the estimation of dynamical observables. Both maximum likelihood methods and an improved Monte Carlo sampling method for reversible transition matrices with fixed stationary distribution are given. The sampling approach is applied to a toy example as well as to simulations of the MR121-GSGS-W peptide, and is demonstrated to converge much more rapidly than a previous approach of Noé [J. Chem. Phys.128, 244103 (Year: 2008)10.1063/1.2916718].

© 2013 AIP Publishing LLC

Received 11 January 2013
Accepted 25 March 2013
Published online 24 April 2013

Acknowledgments: The authors would like to thank two anonymous referees for helpful comments and suggestions. One of the authors would like to thank Guillermo Perez, Han Wang, and Ivan Kryven for discussions and helpful suggestions. He thanks Luc Devroye for a suggestion concerning the modified rejection sampling approach. He would especially like to thank Sven Krönke for inspiring discussions. B. Trendelkamp-Schroer acknowledges funding by the DFG fund No. 825/3 and from the “Center of Supramolecular Interactions” at FU-Berlin. Frank Noé acknowledges funding from the research center Matheon.

Article outline:

I. INTRODUCTION

II. PROBABILITY DISTRIBUTIONS FOR TRANSITION MATRICES

III. CONDITIONAL PROBABILITIES

A. Log-concave densities

B. Optimal piecewise approximation

C. Suboptimal piecewise approximation

D. Rejection sampling using the envelope

E. Modified rejection sampling for large *d* values

IV. MAXIMUM LIKELIHOOD ESTIMATION

V. A GIBBS SAMPLER FOR TRANSITION MATRICES WITH FIXED STATIONARY DISTRIBUTION

A. Enforcing sparsity—A prior for metastable dynamics

VI. RESULTS

A. Conditional distributions

B. Convergence of mean values and variances

C. Autocorrelation functions

D. Application to simulation data

E. Computational efficiency

VII. COMPARISON WITH NONREVERSIBLE AND REVERSIBLE ESTIMATION

VIII. DISCUSSION AND CONCLUSION

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/content/aip/journal/jcp/138/16/10.1063/1.4801325

2013-04-24

2014-04-18

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