Addressing the Embeddability Problem in Transition Rate Estimation

Markov State Models (MSM) and related techniques havegained significanttraction 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 feasibleMD simulations. The MSM analysis often relies on spectral decompositionof empirically generated transition matrices. This work discussesan alternative approach for extracting the thermodynamic and kineticinformation from the so-called rate/generator matrix rather than thetransition matrix. Although the rate matrix itself is built from theempirical transition matrix, it provides an alternative approach forestimating both thermodynamic and kinetic quantities, particularlyin diffusive processes. A fundamental issue with this approach isknown as the embeddability problem. The key contribution of this workis the introduction of a novel method to address the embeddabilityproblem as well as the collection and utilization of existing algorithmspreviously used in the literature. The algorithms are tested on datafrom a one-dimensional toy model to show the workings of these methodsand discuss the robustness of each method in dependence of lag timeand trajectory length.

Automated summary

Markov State Models (MSM) and related techniques are widely used in molecular dynamics (MD) simulations to extract structural, thermodynamic, and kinetic information on proteins. This work proposes an alternative approach using the rate/generator matrix instead of the transition matrix for estimating thermodynamic and kinetic quantities, particularly in diffusive processes. The key contribution of this work is the introduction of a novel method to address the embeddability problem and the utilization of existing algorithms. The algorithms are tested on a one-dimensional toy model to demonstrate their effectiveness and robustness.