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Dynamical reweighting methods permit to estimate kinetic observables of a stochastic process governed by a target potential U(x) from trajectories that have been generated at a different potential V(x). In this article, we present Girsanov reweighting and Square Root Approximation (SqRA): the first method reweights path probabilities exploiting the Girsanov theorem and can be applied to Markov State Models (MSMs) to reweight transition probabilities; the second method was originally developed to discretize the Fokker-Planck operator into a transition rate matrix, but here we implement it into a reweighting scheme for transition rates. We begin by reviewing the theoretical background of the methods, then present two applications relevant to Molecular Dynamics (MD), highlighting their strengths and weaknesses.

Metadynamics is a computational method to explore the phase space of a molecular system. Gaussian functions are added along relevant coordinates on the fly during a molecular-dynamics simulation to force the system to escape from minima in the potential energy function. The dynamics in the resulting trajectory are however unphysical and cannot be used directly to estimate dynamical properties of the system. Girsanov reweighting is a recent method used to construct the Markov State Model (MSM) of a system subjected to an external perturbation. With the combination of these two techniques—metadynamics/Girsanov-reweighting—the unphysical dynamics in a metadynamics simulation can be reweighted to obtain the MSM of the unbiased system. We demonstrate the method on a one-dimensional diffusion process, alanine dipeptide, and the hexapeptide Val-Gly-Val-Ala-Pro-Gly (VGVAPG). The results are in excellent agreement with the MSMs obtained from direct unbiased simulations of these systems. We also apply metadynamics/Girsanov-reweighting to a β-hairpin peptide, whose dynamics is too slow to efficiently explore its phase space by direct simulation

In recent years, for the analysis of molecular processes, the estimation of time-scales and transition rates has become fundamental. Estimating the transition rates between molecular conformations is—from a mathematical point of view—an invariant subspace projection problem. We present a method to project the infinitesimal generator acting on function space to a low-dimensional rate matrix. This projection can be performed in two steps. First, we discretize the conformational space in a Voronoi tessellation, then the transition rates between adjacent cells is approximated by the geometric average of the Boltzmann weights of the Voronoi cells. This method demonstrates that there is a direct relation between the potential energy surface of molecular structures and the transition rates of conformational changes. We will show also that this approximation is correct and converges to the generator of the Smoluchowski equation in the limit of infinitely small Voronoi cells. We present results for a two dimensional diffusion process and alanine dipeptide as a high-dimensional system.

This chapter explains the different ways in which Markov State Models (MSMs) can be helpful in structure-based drug design. MSMs are constructed from the time series of molecular dynamics (MD), which can be generated by classical MD simulations. Several features of the MSMs can be utilized for rational drug design. The discretization of a validated MSM is particularly suited to extract meaningful representatives from the conformational ensemble, because the discretization yields a small number of microstates and mirrors the features of the free energy landscape. Long-lived conformations consist of a set of microstates which show high transition rates within the set and low transition rates to microstates outside of the set. The Bayesian agglomerative clustering engine (BACE) algorithm uses the observed transition counts to extract long-lived conformations from an MSM. By iteratively merging microstates according to the Bayes factor and recalculating the Bayes-factor matrix, the algorithm yields an aggregation of the microstates into long-lived conformations.

Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker–Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker–Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to $\mathcal{O}\left(1{0}^{6}\right)$ states can be obtained within seconds on a single high-performance compute server if regular grids are used.

Conformational dynamics is essential to biomolecular processes. Markov State Models (MSMs) are widely used to elucidate dynamic properties of molecular systems from unbiased Molecular Dynamics (MD). However, the implementation of reweighting schemes for MSMs to analyze biased simulations is still at an early stage of development. Several dynamical reweighing approaches have been proposed, which can be classified as approaches based on (i) Kramers rate theory, (ii) rescaling of the probability density flux, (iii) reweighting by formulating a likelihood function, (iv) path reweighting. We present the state-of-the-art and discuss the methodological differences of these methods, their limitations and recent applications.

The sensitivity of molecular dynamics on changes in the potential energy function plays an important role in understanding the dynamics and function of complex molecules. We present a method to obtain path ensemble averages of a perturbed dynamics from a set of paths generated by a reference dynamics. It is based on the concept of path probability measure and the Girsanov theorem, a result from stochastic analysis to estimate a change of measure of a path ensemble. Since Markov state models (MSMs) of the molecular dynamics can be formulated as a combined phase-space and path ensemble average, the method can be extended to reweight MSMs by combining it with a reweighting of the Boltzmann distribution. We demonstrate how to efficiently implement the Girsanov reweighting in a molecular dynamics simulation program by calculating parts of the reweighting factor “on the fly” during the simulation, and we benchmark the method on test systems ranging from a two-dimensional diffusion process and an artificial many-body system to alanine dipeptide and valine dipeptide in implicit and explicit water. The method can be used to study the sensitivity of molecular dynamics on external perturbations as well as to reweight trajectories generated by enhanced sampling schemes to the original dynamics.

Modern potential energy surfaces have shifted attention to molecular simulations of chemical reactions. While various methods can estimate rate constants for conformational transitions in molecular dynamics simulations, their applicability to studying chemical reactions remains uncertain due to the high and sharp energy barriers and complex reaction coordinates involved. This study focuses on the thermal cis-trans isomerization in retinal, employing molecular simulations and comparing rate constant estimates based on one-dimensional rate theories with those based on sampling transitions and grid-based models for low-dimensional collective variable spaces. Even though each individual method to estimate the rate passes its quality tests, the rate constant estimates exhibit disparities of up to four orders of magnitude. Rate constant estimates based on one-dimensional reaction coordinates prove challenging to converge, even if the reaction coordinate is optimized. However, consistent estimates of the rate constant are achieved by sampling transitions and by multi-dimensional grid-based models.

We have combined infrared (IR) experiments with molecular dynamics (MD) simulations in solution at finite temperature to analyse the vibrational signature of the small floppy peptide Alanine-Leucine. IR spectra computed from first-principles MD simulations exhibit no distinct differences between conformational clusters of -helix or -sheet-like folds with different orientations of the bulky leucine side chain. All computed spectra show two prominent bands, in good agreement with the experiment, that are assigned to the stretch vibrations of the carbonyl and carboxyl group, respectively. Variations in band widths and exact maxima are likely due to small fluctuations in the backbone torsion angles.