Markov State Models (MSM) and related techniques have gained significant traction as a tool for analyzing and guiding molecular dynamics simulations due to their ability to extract structural, thermodynamic, and kinetic information on proteins using computationally feasible simulations. Here, we revisit the common practice in extracting the thermodynamic and kinetic information from the empirical transition matrix. We propose to build a rate/generator matrix from the empirical transition matrix to provide an alternative approach for estimating both thermodynamic and kinetic quantities. We also discuss a fundamental issue with this approach, known as the embeddability problem and the ways to address this issue. We use a one-dimensional toy model to show the workings of the proposed methods. We also show that the common method of estimating the rates based on the eigendecomposition and our proposed method based on the rate matrix are complementary in that they can potentially provide an upper and a lower limit for transition rates.

We have formulated a Riemannian framework for describing the geometry of collective variable spaces of biomolecules within the context of collective variable based molecular dynamics simulations. The formalism provides a theoretical framework to develop enhanced sampling techniques, path-finding algorithms, and transition rate estimators consistent with a Riemannian treatment of the collective variable space, where the quantities of interest such as the potential of the mean force, minimum free energy path, the diffusion constant, and the transition rate remain invariant under coordinate transformation due to the Riemannian treatment of the collective variable space.Specific algorithms within this framework are discussed such as the Riemannian umbrella sampling, the Riemannian string method, and a Riemannian-Bayesian estimator of free energy and diffusion constant, which can be used to estimate the transition rate along a minimum free energy path. *

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.

We have formulated a Riemannian framework for describing the geometry of collective variable spaces of biomolecules within the context of molecular dynamics (MD) simulations. The formalism provides a theoretical framework to develop enhanced sampling techniques, path-finding algorithms, and transition rate estimators consistent with a Riemannian treatment of the collective variable space, where the quantities of interest such as the potential of mean force (PMF) and minimum free energy path (MFEP) remain invariant under coordinate transformation. Specific algorithms within this framework are discussed such as the Riemannian umbrella sampling, the Riemannian string method, and a Riemannian-Bayesian estimator of free energy and diffusion constant, which can be used to estimate the transition rate along an MFEP.

Markovian models provide a convenient way for estimating kinetic properties of biomolecular systems from molecular dynamics (MD) trajectories. These formalisms present attractive characteristics which potentially can overcome sampling problems by constructing long-term kinetic and thermodynamic information from short trajectories. Markov State Models (MSM) traditionally use the time-dependent transition probability matrix (TPM) to estimate different properties of the system as opposed to the time-independent transition rate matrix (TRM) or the generator matrix, which is formally the matrix logarithm of TPM divided by the lag time, used to construct the TPM. This is partly done because of the possibility of the matrix logarithm of sparse TPMs not being a valid TRM, e.g. non-physical/complex rate predictions. This problem, known as the embeddability problem is what our work addresses. We present a comparative study of MSM results using the standard TPM approach to results using an embeddabilitycorrected TRM in situations of increasingly sparse sampling. Our work attempts to overcome this problem and devise a more robust method for kinetic predictions from MD trajectories. As a means to testing our work, we characterize the thermodynamics and kinetics of three systems, a simple bistable potential, a more complex multi-state toy model, and finally, as a means of presenting the general applicability of our methods, a realistic membrane protein in explicit membrane simulated using all-atom MD. We use this study to compare the effectiveness of 8 algorithms, ranging from simple deterministic methods to stochastic Monte Carlo methods, in situations of progressively insufficient sampling.