Abstract
Rare, but important, transition events between long-lived states are a key feature of many molecular systems. In many cases, the computation of rare event statistics by direct molecular dynamics (MD) simulations is infeasible, even on the most powerful computers, because of the immensely long simulation timescales needed. Recently, a technique for spatial discretization of the molecular state space designed to help overcome such problems, so-called Markov State Models (MSMs), has attracted a lot of attention. We review the theoretical background and algorithmic realization of MSMs and illustrate their use by some numerical examples. Furthermore, we introduce a novel approach to using MSMs for the efficient solution of optimal control problems that appear in applications where one desires to optimize molecular properties by means of external controls.
Highlights
Stochastic processes are widely used to model physical, chemical or biological systems
We have discussed an approach to overcome direct sampling issues of rare events in molecular dynamics based on spatial discretization of the molecular state space
The strategy is to define a discretization by subsets of state space, such that the sampling effort with respect to transitions between the sets is much lower than the direct estimation of the rare events under consideration
Summary
Stochastic processes are widely used to model physical, chemical or biological systems. We will discuss approach (2) to the estimation of rare event statistics via discretization of the state space of the system under consideration. The idea is: (1) to choose the sets such that the sampling effort is much lower than the direct estimation of the rare events under consideration; and (2) to compute all interesting quantities for the MSM from its transition matrix, cf [2,3]. In the algorithmic realization of Markov State Modeling for realistic molecular systems, the transition probabilities and the respective statistical uncertainties are estimated from short molecular dynamics (MD) trajectories only, cf [7]. MSM discretization yields an efficient algorithm for solving the optimal control problem, whose performance we will outline in some numerical examples, including an application to alanine dipeptide
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