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

TL;DR

This paper introduces Bayesian estimation techniques for Markov model transition matrices that incorporate known stationary distributions, improving convergence speed for dynamical property calculations in biomolecular systems.

Contribution

It presents novel maximum likelihood and Monte Carlo methods for estimating reversible transition matrices with fixed stationary distributions, enhancing efficiency over previous approaches.

Findings

Faster convergence in estimating transition matrices.

Effective incorporation of known stationary distributions.

Successful application to peptide simulation data.

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 discretiza- tion 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 ma- trices 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 in [F. Noe, J. Chem. Phys. 128, 244103 (2008)]