Assessing Position-Dependent Diffusion from Biased Simulations and Markov State Model Analysis

A variety of enhanced statistical and numerical methods are now routinely used to extract comprehensible and relevant thermodynamic information from the vast amount of complex, high-dimensional data obtained from intensive molecular simulations. The characterization of kinetic properties, such as diffusion coefficients, of molecular systems with significantly high energy barriers, on the other hand, has received less attention. Among others, Markov state models, in which the long-time statistical dynamics of a system is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years, with the aim of tackling these fundamental issues. Here, we discuss a general, automatic method to assess multidimensional position-dependent diffusion coefficients within the framework of Markovian stochastic processes and Kramers-Moyal expansion. We apply the formalism to one- and two-dimensional analytic potentials and data from explicit solvent molecular dynamics simulations, including the water-mediated conformations of alanine dipeptide and the transport of drug molecule across three-dimensional heterogeneous porous media.