Likelihood-based non-Markovian models from molecular dynamics

Abstract

Significance The analysis of complex systems with many degrees of freedom generally involves the definition of low-dimensional collective variables more amenable to physical understanding. Their dynamics can be modeled by generalized Langevin equations, whose coefficients have to be estimated from simulations of the initial high-dimensional system. These equations feature a memory kernel describing the mutual influence of the low-dimensional variables and their environment. We introduce and implement an approach where the generalized Langevin equation is designed to maximize the statistical likelihood of the observed data. This provides an efficient way to generate reduced models to study dynamical properties of complex processes such as chemical reactions in solution, conformational changes in biomolecules, or phase transitions in condensed matter systems.

Free Energy Landscapes, Diffusion Coefficients, and Kinetic Rates from Transition Paths

“…For chemical reactions in water the memory time scale could be comparable to the transition path time, requiring a non-Markovian equation, whereas for crystal nucleation or protein folding Markovian models are customarily invoked. In the present work, we demonstrated the approach using overdamped Langevin equations: in principle, this can be generalized to other stochastic models by replacing the propagator expression (eq ) with a different suitable approximation, based on ideas in refs and .…”

“…Langevin equations can be obtained from Hamilton’s equations of motion by projecting the high-dimensional deterministic dynamics on a subset of the phase-space variables. , In the limit of strong friction, velocity fluctuations away from the equilibrium distribution are damped quickly (compared to the time resolution τ), yielding the widely employed overdamped Langevin equation: where F ( q ) = − k B T log ρ eq ( q ) is the free-energy landscape, D ( q ) is the position-dependent diffusion coefficient, and η­( t ) is a Gaussian white noise. We remark that a realistic description of the dynamics generally requires a nonconstant D ( q ), and that the use of the overdamped equation can be formally justified (for instance, it provides exact mean first passage times (MFPT)) when q is the optimal CV, that is, any monotonic one-to-one function of the committor …”

“…Hamilton's equations of motion by projecting the highdimensional deterministic dynamics on a subset of the phase-space variables. 7,43 In the limit of strong friction, velocity fluctuations away from the equilibrium distribution are damped quickly (compared to the time resolution τ), yielding the widely employed overdamped Langevin equation:…”

“…A number of algorithms aim at parametrizing Langevin models starting from trajectories of many-particle systems. ,,− More in detail, non-Markovian friction (the so-called memory kernel) can be reconstructed in different ways, for instance based on a set of time-correlation functions and Volterra integral equations (see, e.g., refs , , and ) or via likelihood maximization . In the case of Markovian (overdamped) Langevin equations, the latter approach, ,, together with direct estimation of Kramers-Moyal coefficients, , has been employed.…”

“…11,12,16−43 More in detail, non-Markovian friction (the so-called memory kernel) can be reconstructed in different ways, for instance based on a set of time-correlation functions and Volterra integral equations (see, e.g., refs 11, 12, and 17) or via likelihood maximization. 43 In the case of Markovian (overdamped) Langevin equations, the latter approach, 22,27,29 together with direct estimation of Kramers-Moyal coefficients, 29,35 has been employed. Often, only friction is addressed, assuming that the freeenergy profile is known in advance, or that it can be directly estimated from the histogram of the CV based on long equilibrium trajectories reversibly sampling all relevant states (see, e.g., refs 12, 24, and 27).…”