Additive Eigenvector Analysis of Diffusion: Wave-function Phase as Optimal Reaction Coordinate

Complex multidimensional stochastic dynamics can be approximately described as a diffusion along reaction coordinates (RCs). If the RCs are optimally selected important properties of dynamics can be computed exactly. The committor is a primary example of an optimal RC. Recently, additive eigenvector (addevs) have been introduced in order to extend the formalism to non-equilibrium dynamics. An addev describes a sub-ensemble of trajectories together with an optimal RC. The sub-ensemble is conditioned to have a single RC optimal for both the forward and time-reversed non-equilibrium dynamics of the sub-ensemble. Here we continue the fundamental development of the framework and illustrate it by analysing diffusion. In particular, we show that the forward and time-reversed committors are functions of the addev, meaning a diffusive model along an addev RC can be used to compute important properties of non-equilibrium dynamics exactly. While the addev conditioning is a reasonable requirement for a RC, it is rather peculiar for physics in general. Nevertheless, we show that one family of addev solutions for diffusion is described by equations of classical mechanics, while another family can be approximated by quantum mechanical equations. We show how the potential can be introduced in to the formalism self-consistently. The developments are illustrated on simple examples. Deep fundamental connections between quantum mechanics and stochastic processes, in particular, between the Schr\"odinger and diffusion equations, are well established. Here we demonstrate yet another one, emphasizing the role of the phase of a wave-function as an optimal RC

Optimal reaction coordinate as a biomarker for the dynamics of recovery from kidney transplant.

The evolution of disease or the progress of recovery of a patient is a complex process, which depends on many factors. A quantitative description of this process in real-time by a single, clinically measurable parameter (biomarker) would be helpful for early, informed and targeted treatment. Organ transplantation is an eminent case in which the evolution of the post-operative clinical condition is highly dependent on the individual case. The quality of management and monitoring of patients after kidney transplant often determines the long-term outcome of the graft. Using NMR spectra of blood samples, taken at different time points from just before to a week after surgery, we have shown that a biomarker can be found that quantitatively monitors the evolution of a clinical condition. We demonstrate that this is possible if the dynamics of the process is considered explicitly: the biomarker is defined and determined as an optimal reaction coordinate that provides a quantitatively accurate description of the stochastic recovery dynamics. The method, originally developed for the analysis of protein folding dynamics, is rigorous, robust and general, i.e., it can be applied in principle to analyze any type of biological dynamics. Such predictive biomarkers will promote improvement of long-term graft survival after renal transplantation, and have potentially unlimited applications as diagnostic tools

Modulation of a protein free-energy landscape by circular permutation

Circular permutations usually retain the native structure and function of a protein while inevitably perturb its folding dynamics. By using simulations with a structure-based model and a rigorous methodology to determine free-energy surfaces from trajectories we evaluate the effect of a circular permutation on the free-energy landscape of the protein T4 lysozyme. We observe changes which, while subtle, largely affect the cooperativity between the two subdomains. Such a change in cooperativity has been previously experimentally observed and recently also characterized using single molecule optical tweezers and the Crooks relation. The free-energy landscapes show that both the wild type and circular permutant have an on-pathway intermediate, previously experimentally characterized, where one of the subdomains is completely formed. The landscapes, however, differ in the position of the rate-limiting step for folding, which occurs before the intermediate in the wild-type and after in the circular permutant. This shift of transition state explains the observed change in the cooperativity. The underlying free-energy landscape thus provides a microscopic description of the folding dynamics and the connection between circular permutation and the loss of cooperativity experimentally observed

Lorentz Covariant Theory of Light Propagation in Gravitational Fields of Arbitrary-Moving Bodies

The Lorentz covariant theory of propagation of light in the (weak) gravitational fields of N-body systems consisting of arbitrarily moving point-like bodies with constant masses is constructed. The theory is based on the Lienard-Wiechert presentation of the metric tensor. A new approach for integrating the equations of motion of light particles depending on the retarded time argument is applied. In an approximation which is linear with respect to the universal gravitational constant, G, the equations of light propagation are integrated by quadratures and, moreover, an expression for the tangent vector to the perturbed trajectory of light ray is found in terms of instanteneous functions of the retarded time. General expressions for the relativistic time delay, the angle of light deflection, and gravitational red shift are derived. They generalize previously known results for the case of static or uniformly moving bodies. The most important applications of the theory are given. They include a discussion of the velocity dependent terms in the gravitational lens equation, the Shapiro time delay in binary pulsars, and a precise theoretical formulation of the general relativistic algorithm of data processing of radio and optical astrometric measurements in the non-stationary gravitational field of the solar system. Finally, proposals for future theoretical work being important for astrophysical applications are formulated.Comment: 77 pages, 7 figures, list of references is updated, to be published in Phys. Rev. D6

Is Protein Folding Sub-Diffusive?

Protein folding dynamics is often described as diffusion on a free energy surface considered as a function of one or few reaction coordinates. However, a growing number of experiments and models show that, when projected onto a reaction coordinate, protein dynamics is sub-diffusive. This raises the question as to whether the conventionally used diffusive description of the dynamics is adequate. Here, we numerically construct the optimum reaction coordinate for a long equilibrium folding trajectory of a Go model of a l-repressor protein. The trajectory projected onto this coordinate exhibits diffusive dynamics, while the dynamics of the same trajectory projected onto a sub-optimal reaction coordinate is subdiffusive. We show that the higher the (cut-based) free energy profile for the putative reaction coordinate, the more diffusive the dynamics become when projected on this coordinate. The results suggest that whether the projected dynamics is diffusive or sub-diffusive depends on the chosen reaction coordinate. Protein folding can be described as diffusion on the free energy surface as function of the optimum reaction coordinate. And conversely, the conventional reaction coordinates, even though they might be based on physical intuition, are often sub-optimal and, hence, show subdiffusive dynamics

Single biomolecules -in silico, in vitro and in vivo High-resolution free energy landscape analysis of protein folding

Abstract The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics