Addressing the Embeddability Problem in Transition Rate Estimation
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. Here, we discuss 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. We particularly discuss a fundamental issue with this approach, known as the embeddability problem and offer ways to address this issue. We describe eight different methods to overcome the embeddability problem, including a novel approach developed for this work. The algorithms were tested on data from a one-dimensional toy model to show the workings of these methods and discuss the robustness of each method in terms of its dependence in lag time and trajectory length.