Transition times in the low-noise limit of stochastic dynamics

On distributions of barrier crossing times as observed in single-molecule studies of biomolecules

Single-molecule experiments that monitor time evolution of molecular observables in real time have expanded beyond measuring transition rates toward measuring distributions of times of various molecular events. Of particular interest is the first-passage time for making a transition from one molecular configuration ( a ) to another ( b ) and conditional first-passage times such as the transition path time, which is the first-passage time from a to b conditional upon not leaving the transition region intervening between a and b . Another experimentally accessible (but not yet studied experimentally) observable is the conditional exit time, i.e., the time to leave the transition region through a specified boundary. The distributions of such times contain a wealth of mechanistic information about the transitions in question. Here, we use the first and the second (and, if desired, higher) moments of these distributions to characterize their relative width for the model in which the experimental observable undergoes Brownian motion in a potential of mean force. We show that although the distributions of transition path times are always narrower than exponential (in that the ratio of the standard deviation to the distribution's mean is always less than 1), distributions of first-passage times and of conditional exit times can be either narrow or broad, in some cases displaying long power-law tails. The conditional exit time studied here provides a generalization of the transition path time that also allows one to characterize the temporal scales of failed barrier crossing attempts.

Kinetics of Loop Closure in Disordered Proteins: Theory vs Simulations vs Experiments

We study intrachain dynamics of intrinsically disordered proteins, as manifested by the time scales of loop formation, using atomistic simulations, experiment-parametrized coarse-grained models, and one-dimensional theories assuming Markov or non-Markov dynamics along the reaction coordinate. Despite the generally non-Markov character of monomer dynamics in polymers, we find that the simplest model of one-dimensional diffusion along the reaction coordinate (equated to the distance between the loop-forming monomers) well captures the mean first passage times to loop closure measured in coarse-grained and atomistic simulations, which, in turn, agree with the experimental values. This justifies use of the one-dimensional diffusion model in interpretation of experimental data. At the same time, the transition path times for loop closure in longer polypeptide chains show significant non-Markov effects; at intermediate times, these effects are better captured by the generalized Langevin equation model. At long times, however, atomistic simulations predict long tails in the distributions of transition path times, which are at odds with both the one-dimensional diffusion model and the generalized Langevin equation model.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

Oscillations in the mean transition time of a particle scattered on a double slit potential

Scattering through a double slit potential is one of the most fundamental problems in quantum mechanics. It is well understood that due to the superposition of amplitudes, one observes a spatial interference pattern in the scattered wavefunction reflecting the superposition of amplitudes coming from both slits. However, the effect of the double slit on the mean time it takes to traverse the slit has not been considered previously. Using a transition path time formalism, we show that when a single Gaussian wavepacket is scattered through a double slit potential, one finds not only oscillations in the scattered density resulting from the spatial interference created by the splitting of the wavepacket but also an oscillatory pattern in the mean scattering time. Long times are associated with low values of a suitably defined momentum, and short times with higher values. The double slit thus serves as a momentum filtering device. We also find an interference pattern in the time averaged momentum weak value profile of the scattered particle implying that the double slit also acts as a weak momentum filter. These results not only demonstrate the value of considering transition path time distributions in their quantum mechanical context but also present a challenge to semiclassical approximations-can they account for temporal interference?

Communication: Transition-path velocity as an experimental measure of barrier crossing dynamics

Experimental observation of transition paths-short events when the system of interest crosses the free energy barrier separating reactants from products-provides an opportunity to probe the dynamics of barrier crossing. Yet limitations in the experimental time resolution usually result in observing trajectories that are smoothed out, recross the transition state fewer times, and exhibit apparent velocities that are much lower than the instantaneous ones. Here we show that it is possible to define (and measure) an effective transition-path velocity which preserves exact information about barrier crossing dynamics in the following sense: the exact transition rate can be written in a form resembling that given by transition-state theory, with the mean thermal velocity replaced by the transition-path velocity. In addition, the transition-path velocity (i) ensures the exact local value of the unidirectional reactive flux at equilibrium and (ii) leads to the exact mean transition-path time required for the system to cross the barrier region separating reactants from products. We discuss the coordinate dependence of the transition path velocity and derive analytical expressions for it in the case of diffusive dynamics. These results can be used to discriminate among models of barrier crossing dynamics in single-molecule force spectroscopy studies.

First passage, looping, and direct transition in expanding and narrowing tubes: Effects of the entropy potential

We study transitions of diffusing particles between the left and right ends of expanding and narrowing conical tubes. In an expanding tube, such transitions occur faster than in the narrowing tube of the same length and radius variation rate. This happens because the entropy potential pushes the particle towards the wide tube end, thus accelerating the transitions in the expanding tube and slowing them down in the narrowing tube. To gain deeper insight into how the transitions occur, we divide each trajectory into the direct-transit and looping segments. The former is the final part of the trajectory, where the particle starting from the left tube end goes to the right end without returning to the left one. The rest of the trajectory is the looping segment, where the particle, starting from the left tube end, returns to this end again and again until the direct transition happens. Our focus is on the durations of the two segments and their sum, which is the duration of the particle first passage between the left and right ends of the tube. We approach the problem using the one-dimensional description of the particle diffusion along the tube axis in terms of the modified Fick-Jacobs equation. This allows us to derive analytical expressions for the Laplace transforms of the probability densities of the first-passage, direct-transit, and looping times, which we use to find the mean values of these random variables. Our results show that the direct transits are independent of the entropy potential and occur as in free diffusion. However, this "free diffusion" occurs with the effective diffusivity entering the modified Fick-Jacobs equation, which is smaller than the particle diffusivity in a cylindrical tube. This is the only way how the varying tube geometry manifests itself in the direct transits. Since direct-transit times are direction-independent, the difference in the first-passage times in the tubes of the two types is due to the difference in the durations of the looping segments in the expanding and narrowing tubes. Obtained analytical results are supported by three-dimensional Brownian dynamics simulations.

A new insight into diffusional escape from a biased cylindrical trap

Recent experiments with single biological nanopores, as well as single-molecule fluorescence spectroscopy and pulling studies of protein and nucleic acid folding raised a number of questions that stimulated theoretical and computational investigations of barrier crossing dynamics. The present paper addresses a closely related problem focusing on trajectories of Brownian particles that escape from a cylindrical trap in the presence of a force F parallel to the cylinder axis. To gain new insights into the escape dynamics, we analyze the "fine structure" of these trajectories. Specifically, we divide trajectories into two segments: a looping segment, when a particle unsuccessfully tries to escape returning to the trap bottom again and again, and a direct-transit segment, when it finally escapes moving without touching the bottom. Analytical expressions are derived for the Laplace transforms of the probability densities of the durations of the two segments. These expressions are used to find the mean looping and direct-transit times as functions of the biasing force F. It turns out that the force-dependences of the two mean times are qualitatively different. The mean looping time monotonically increases as F decreases, approaching exponential F-dependence at large negative forces pushing the particle towards the trap bottom. In contrast to this intuitively appealing behavior, the mean direct-transit time shows rather counterintuitive behavior: it decreases as the force magnitude, |F|, increases independently of whether the force pushes the particles to the trap bottom or to the exit from the trap, having a maximum at F = 0.

Mean Direct-Transit and Looping Times as Functions of the Potential Shape

Any trajectory of a diffusing particle making a transition between two end points of an interval can be divided into two segments, which we call direct-transit and looping parts. The former is the final segment of the trajectory, when the particle goes from one end point of the interval to the opposite end point, without retouching the starting point. The rest of the trajectory is the looping part. We study mean durations of the two parts in the presence of a symmetric linear cusp potential which, depending on the parameter values, forms either a barrier or a well between the end points. For the cusp barrier, we find that the mean direct-transit time decreases as the barrier height increases at a fixed interval length. This happens because the increase in the barrier height results in the increase of the magnitude of the force acting on the particle on both sides of the barrier. Interestingly, though the mean looping and direct-transit times are different in the case of the barrier and well potentials with equal height and depth, respectively, the mean first-passage times for the two cases are identical.

Transition path time distribution and the transition path free energy barrier

Free energy profile, showing why the transition path barrier is lower than the free energy of activation.

Computation of transit times using the milestoning method with applications to polymer translocation

Milestoning is an efficient approximation for computing long-time kinetics and thermodynamics of large molecular systems, which are inaccessible to brute-force molecular dynamics simulations. A common use of milestoning is to compute the mean first passage time (MFPT) for a conformational transition of interest. However, the MFPT is not always the experimentally observed timescale. In particular, the duration of the transition path, or the mean transit time, can be measured in single-molecule experiments, such as studies of polymers translocating through pores and fluorescence resonance energy transfer studies of protein folding. Here we show how to use milestoning to compute transit times and illustrate our approach by applying it to the translocation of a polymer through a narrow pore.

Transition paths, diffusive processes, and preequilibria of protein folding

Fundamental relationships between the thermodynamics and kinetics of protein folding were investigated using chain models of natural proteins with diverse folding rates by extensive comparisons between the distribution of conformations in thermodynamic equilibrium and the distribution of conformations sampled along folding trajectories. Consistent with theory and single-molecule experiment, duration of the folding transition paths exhibits only a weak correlation with overall folding time. Conformational distributions of folding trajectories near the overall thermodynamic folding/unfolding barrier show significant deviations from preequilibrium. These deviations, the distribution of transition path times, and the variation of mean transition path time for different proteins can all be rationalized by a diffusive process that we modeled using simple Monte Carlo algorithms with an effective coordinate-independent diffusion coefficient. Conformations in the initial stages of transition paths tend to form more nonlocal contacts than typical conformations with the same number of native contacts. This statistical bias, which is indicative of preferred folding pathways, should be amenable to future single-molecule measurements. We found that the preexponential factor defined in the transition state theory of folding varies from protein to protein and that this variation can be rationalized by our Monte Carlo diffusion model. Thus, protein folding physics is different in certain fundamental respects from the physics envisioned by a simple transition-state picture. Nonetheless, transition state theory can be a useful approximate predictor of cooperative folding speed, because the height of the overall folding barrier is apparently a proxy for related rate-determining physical properties.

Protein folding kinetics and thermodynamics from atomistic simulation

Advances in simulation techniques and computing hardware have created a substantial overlap between the timescales accessible to atomic-level simulations and those on which the fastest-folding proteins fold. Here we demonstrate, using simulations of four variants of the human villin headpiece, how simulations of spontaneous folding and unfolding can provide direct access to thermodynamic and kinetic quantities such as folding rates, free energies, folding enthalpies, heat capacities, Φ-values, and temperature-jump relaxation profiles. The quantitative comparison of simulation results with various forms of experimental data probing different aspects of the folding process can facilitate robust assessment of the accuracy of the calculations while providing a detailed structural interpretation for the experimental observations. In the example studied here, the analysis of folding rates, Φ-values, and folding pathways provides support for the notion that a norleucine double mutant of villin folds five times faster than the wild-type sequence, but following a slightly different pathway. This work showcases how computer simulation has now developed into a mature tool for the quantitative computational study of protein folding and dynamics that can provide a valuable complement to experimental techniques.