Markov State Models Based on Milestoning

Markov State Models (MSMs) have become the tool of choice to analyze large amounts of molec- ular dynamics data by approximating them as a Markov jump process between suitably predefined states. Here we investigate ”Core Set MSMs”, a new type of MSMs that builds on metastable core sets acting as milestones for tracing the rare event kinetics. We present a thorough analysis of Core Set MSMs based on the existing milestoning framework, Bayesian estimation methods and Transi- tion Path Theory (TPT). As a result, Core Set MSMs can now be used to extract phenomenological rate constants between the metastable sets of the system and to approximate the evolution of certain key observables. The performance of Core Set MSMs in comparison to standard MSMs is analyzed and illustrated on a model potential and the torsion angle dynamics of Alanine dipeptide

On Markov State Models for Metastable Processes

We consider Markov processes on large state spaces and want to find low-dimensional structure-preserving approximations of the process in the sense that the longest timescales of the dynamics of the original process are reproduced well. Recent years have seen the advance of so-called Markov state models (MSM) for processes on very large state spaces exhibiting metastable dynamics. It has been demonstrated that MSMs are especially useful for modelling the interesting slow dynamics of biomolecules (cf. Noe et al, PNAS(106) 2009) and materials. From the mathematical perspective, MSMs result from Galerkin projection of the transfer operator underlying the original process onto some low-dimensional subspace which leads to an approximation of the dominant eigenvalues of the transfer operators and thus of the longest timescales of the original dynamics. Until now, most articles on MSMs have been based on full subdivisions of state space, i.e., Galerkin projections onto subspaces spanned by indicator functions. We show how to generalize MSMs to alternative low-dimensional subspaces with superior approximation properties, and how to analyse the approximation quality (dominant eigenvalues, propagation of functions) of the resulting MSMs. To this end, we give an overview of the construction of MSMs, the associated stochastics and functional-analysis background, and its algorithmic consequences. Furthermore, we illustrate the mathematical construction with numerical examples

Markov state models of biomolecular conformational dynamics

It has recently become practical to construct Markov state models (MSMs) that reproduce the long-time statistical conformational dynamics of biomolecules using data from molecular dynamics simulations. MSMs can predict both stationary and kinetic quantities on long timescales (e.g. milliseconds) using a set of atomistic molecular dynamics simulations that are individually much shorter, thus addressing the well-known sampling problem in molecular dynamics simulation. In addition to providing predictive quantitative models, MSMs greatly facilitate both the extraction of insight into biomolecular mechanism (such as folding and functional dynamics) and quantitative comparison with single-molecule and ensemble kinetics experiments. A variety of methodological advances and software packages now bring the construction of these models closer to routine practice. Here, we review recent progress in this field, considering theoretical and methodological advances, new software tools, and recent applications of these approaches in several domains of biochemistry and biophysics, commenting on remaining challenges

MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations

Molecular dynamics (MD) simulations can model the interactions between macromolecules with high spatiotemporal resolution but at a high computational cost. By combining high-throughput MD with Markov state models (MSMs), it is now possible to obtain long-timescale behavior of small to intermediate biomolecules and complexes. To model the interactions of many molecules at large lengthscales, particle-based reaction-diffusion (RD) simulations are more suitable but lack molecular detail. Thus, coupling MSMs and RD simulations (MSM/RD) would be highly desirable, as they could efficiently produce simulations at large time- and lengthscales, while still conserving the characteristic features of the interactions observed at atomic detail. While such a coupling seems straightforward, fundamental questions are still open: Which definition of MSM states is suitable? Which protocol to merge and split RD particles in an association/dissociation reaction will conserve the correct bimolecular kinetics and thermodynamics? In this paper, we make the first step towards MSM/RD by laying out a general theory of coupling and proposing a first implementation for association/dissociation of a protein with a small ligand (A + B C). Applications on a toy model and CO diffusion into the heme cavity of myoglobin are reported

Variational cross-validation of slow dynamical modes in molecular kinetics

Markov state models (MSMs) are a widely used method for approximating the eigenspectrum of the molecular dynamics propagator, yielding insight into the long-timescale statistical kinetics and slow dynamical modes of biomolecular systems. However, the lack of a unified theoretical framework for choosing between alternative models has hampered progress, especially for non-experts applying these methods to novel biological systems. Here, we consider cross-validation with a new objective function for estimators of these slow dynamical modes, a generalized matrix Rayleigh quotient (GMRQ), which measures the ability of a rank-$m$ projection operator to capture the slow subspace of the system. It is shown that a variational theorem bounds the GMRQ from above by the sum of the first $m$ eigenvalues of the system's propagator, but that this bound can be violated when the requisite matrix elements are estimated subject to statistical uncertainty. This overfitting can be detected and avoided through cross-validation. These result make it possible to construct Markov state models for protein dynamics in a way that appropriately captures the tradeoff between systematic and statistical errors