Microscopic reweighting for nonequilibrium steady-state dynamics

Dynamical Reweighting for Biased Rare Event Simulations

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Reweighting non-equilibrium steady-state dynamics along collective variables

Computer simulations generate microscopic trajectories of complex systems at a single thermodynamic state point. We recently introduced a Maximum Caliber (MaxCal) approach for dynamical reweighting. Our approach mapped these trajectories to a Markovian description on the configurational coordinates and reweighted path probabilities as a function of external forces. Trajectory probabilities can be dynamically reweighted both from and to equilibrium or non-equilibrium steady states. As the system's dimensionality increases, an exhaustive description of the microtrajectories becomes prohibitive-even with a Markovian assumption. Instead, we reduce the dimensionality of the configurational space to collective variables (CVs). Going from configurational to CV space, we define local entropy productions derived from configurationally averaged mean forces. The entropy production is shown to be a suitable constraint on MaxCal for non-equilibrium steady states expressed as a function of CVs. We test the reweighting procedure on two systems: a particle subject to a two-dimensional potential and a coarse-grained peptide. Our CV-based MaxCal approach expands dynamical reweighting to larger systems, for both static and dynamical properties, and across a large range of driving forces.

Path probability ratios for Langevin dynamics-Exact and approximate

Path reweighting is a principally exact method to estimate dynamic properties from biased simulations-provided that the path probability ratio matches the stochastic integrator used in the simulation. Previously reported path probability ratios match the Euler-Maruyama scheme for overdamped Langevin dynamics. Since molecular dynamics simulations use Langevin dynamics rather than overdamped Langevin dynamics, this severely impedes the application of path reweighting methods. Here, we derive the path probability ratio M_L for Langevin dynamics propagated by a variant of the Langevin Leapfrog integrator. This new path probability ratio allows for exact reweighting of Langevin dynamics propagated by this integrator. We also show that a previously derived approximate path probability ratio M_approx differs from the exact M_L only by O(ξ⁴Δt⁴) and thus yields highly accurate dynamic reweighting results. (Δt is the integration time step, and ξ is the collision rate.) The results are tested, and the efficiency of path reweighting is explored using butane as an example.

A method of incorporating rate constants as kinetic constraints in molecular dynamics simulations

From the point of view of statistical mechanics, a full characterization of a molecular system requires an accurate determination of its possible states, their populations, and the respective interconversion rates. Toward this goal, well-established methods increase the accuracy of molecular dynamics simulations by incorporating experimental information about states using structural restraints and about populations using thermodynamics restraints. However, it is still unclear how to include experimental knowledge about interconversion rates. Here, we introduce a method of imposing known rate constants as constraints in molecular dynamics simulations, which is based on a combination of the maximum-entropy and maximum-caliber principles. Starting from an existing ensemble of trajectories, obtained from either molecular dynamics or enhanced trajectory sampling, this method provides a minimally perturbed path distribution consistent with the kinetic constraints, as well as modified free energy and committor landscapes. We illustrate the application of the method to a series of model systems, including all-atom molecular simulations of protein folding. Our results show that by combining experimental rate constants and molecular dynamics simulations, this approach enables the determination of transition states, reaction mechanisms, and free energies. We anticipate that this method will extend the applicability of molecular simulations to kinetic studies in structural biology and that it will assist the development of force fields to reproduce kinetic and thermodynamic observables. Furthermore, this approach is generally applicable to a wide range of systems in biology, physics, chemistry, and material science.

How to learn from inconsistencies: Integrating molecular simulations with experimental data

Molecular simulations and biophysical experiments can be used to provide independent and complementary insights into the molecular origin of biological processes. A particularly useful strategy is to use molecular simulations as a modeling tool to interpret experimental measurements, and to use experimental data to refine our biophysical models. Thus, explicit integration and synergy between molecular simulations and experiments is fundamental for furthering our understanding of biological processes. This is especially true in the case where discrepancies between measured and simulated observables emerge. In this chapter, we provide an overview of some of the core ideas behind methods that were developed to improve the consistency between experimental information and numerical predictions. We distinguish between situations where experiments are used to refine our understanding and models of specific systems, and situations where experiments are used more generally to refine transferable models. We discuss different philosophies and attempt to unify them in a single framework. Until now, such integration between experiments and simulations have mostly been applied to equilibrium data, and we discuss more recent developments aimed to analyze time-dependent or time-resolved data.