Metastability, conformation dynamics, and transition pathways in complex systems

Abstract

Summary. We present a systematic introduction to the basic concepts and techniques for determining transition pathways and transition rates in systems with multiple metastable states. After discussing the classical transition state theory and its limitations, we derive a new set of equations for the optimal dividing surfaces. We then discuss transition path sampling, which is the most general technique currently available for determining transition regions and rates. This is followed by a discussion on minimal energy path for systems with smooth energy landscapes. For systems with rough energy landscapes, our presentation is centered around the notion of reaction coordinates. We discuss the two related notions of free energies associated with a reaction coordinate, and show that at least in the high friction limit, there does exist an optimal reaction coordinate that gives asymptotically the correct prediction for the transition rates. Variational principles associated with the optimal reaction coordinates are exploited under the assumption that the transition paths are restricted to tubes, and this provides a theoretical justification for the finite temperature string method. Blue moon sampling techniques, metadynamics and a new form of accelerated dynamics are also discussed.

Path-sampling strategies for simulating rare events in biomolecular systems

“…In the latter context, the ability to account for directionality and history is critical – particularly tracing back any given trajectory to the most recently occupied state (A or B, “initial” or “target” state), which enables unbiased rate calculation [*11,12,13]; see also [14,15]. This insight from path theory has important practical implications for analyzing ordinary MD simulations and avoiding the Markov assumption [16].…”

Projected transfer operators

“…Recently the interest in MSMs has drastically increased since it could be demonstrated that MSMs can be constructed even for very high dimensional systems [73] and have been especially useful for modeling the interesting slow dynamics of biomolecules [56,57,58,12,11,61] and materials [79] (there under the name "kinetic Monte Carlo"). Their approximation quality on large time scale has been rigorously studied, e.g., for Brownian or Glauber dynamics and Ising models in the limit of vanishing smallness parameters (noise intensity, temperature) where the analysis can be based on large deviation estimates and variational principles [29,84] and/or potential theory and capacities [8,9]. In these cases the effective dynamics is governed by some MSM with exponentially small transition probabilities and its states label the different attractors of the underlying, unperturbed dynamical systems.…”

“…Particularly in molecular dynamics, MSM have become popular as approximations of the conformational dynamics [72,73,84] of large biomolecules, which exhibits various timescales ranging from protein folding to fast vibrations and oscillations within a molecular conformation. There, the subdivision of the conformational state space has been usually achieved by partitioning [56,11,15,35,39,60,64,52,71,80,81].…”

On Markov State Models for Metastable Processes

“…The approximation quality of a MSM on large time scales has been rigorously studied for many different systems, e.g., for diffusion processes, or Glauber dynamics and Ising models in the limit of vanishing smallness parameters (noise itensity, temperature) where the analysis can be based on large deviation estimates and variational principles [21,22] and/or potential theory and capacities [23,24]. In these cases the effective dynamics is governed by some MSM with exponentially small transition probabilities and its states label the different attracting sets of the underlying Markov process.…”