Addressing the Embeddability Problem in Transition Rate Estimation

Abstract

Markov State Models (MSM) and related techniques have gained significant traction as a tool for analyzing and guiding molecular dynamics (MD) simulations due to their ability to extract structural, thermodynamic, and kinetic information on proteins using computationally feasible MD simulations. The MSM analysis often relies on spectral decomposition of empirically generated transition matrices. This work discusses an alternative approach for extracting the thermodynamic and kinetic information from the so-called rate/generator matrix rather than the transition matrix. Although the rate matrix itself is built from the empirical transition matrix, it provides an alternative approach for estimating both thermodynamic and kinetic quantities, particularly in diffusive processes. A fundamental issue with this approach is known as the embeddability problem. The key contribution of this work is the introduction of a novel method to address the embeddability problem as well as the collection and utilization of existing algorithms previously used in the literature. The algorithms are tested on data from a one-dimensional toy model to show the workings of these methods and discuss the robustness of each method in dependence of lag time and trajectory length.