Overcoming the Embeddability Problem: A More Robust Calculation of Kinetic Information from Sparsely Sampled Molecular Dynamics Simulations

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 embeddability-corrected 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