Splitting probabilities as a test of reaction coordinate choice in single-molecule experiments

To explain the observed dynamics in equilibrium single-molecule measurements of biomolecules, the experimental observable is often chosen as a putative reaction coordinate along which kinetic behavior is presumed to be governed by diffusive dynamics. Here, we invoke the splitting probability as a test of the suitability of such a proposed reaction coordinate. Comparison of the observed splitting probability with that computed from the kinetic model provides a simple test to reject poor reaction coordinates. We demonstrate this test for a force spectroscopy measurement of a DNA hairpin

Effective dynamics along given reaction coordinates, and reaction rate theory

In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: we first show that if we start with an ergodic diffusion process whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Mori–Zwanzig, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the “equation-free” approach and the “heterogeneous multiscale method” can be seen as special cases of our approach

Model reduction algorithms for optimal control and importance sampling of diffusions

We propose numerical algorithms for solving complex, high- dimensional control and importance sampling problems based on reduced-order models. The algorithms approach the “curse of dimensionality” by a combination of model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either an suitable set of slow collective variables onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behaviour in the small noise regime. The idea is to use the solution of the reduced-order model as a predictor of the exact solution that, in a corrector step, can be used together with the original dynamics, where no explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples

Importance sampling in path space for diffusion processes with slow-fast variables

Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with a simple numerical example

Variational characterization of free energy: theory and algorithms

The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated

Applications of the cross-entropy method to importance sampling and optimal control of diffusions

Abstract. We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method. Key words. importance sampling, optimal control, cross-entropy method, rare events, change of measure. AMS subject classifications. 1 Introduction This article deals with the application of the cross-entropy method to diffusion processes, specifically, with the application to importance sampling for rare events and optimal control. Generally, the cross-entropy method is a Monte-Carlo method that was originally developed for the efficient simulation of rare events in queuing models and that has been extended to, e.g., combinatorial optimization or analysis of networks in the meantim