Transition Path Times in Non-Markovian Activated Rate Processes

Abstract

Transition paths are brief excursions taken by molecules when they cross barriers separating stable molecular conformations. When observed in single-molecule experiments, they offer insights into the underlying reaction dynamics and mechanisms. A common model used to analyze transition paths assumes that the dynamics along the reaction coordinate is a memoryless, diffusive process. Recent work, however, suggests that memory effects are often important in the dynamics of the reaction coordinates that can be accessed experimentally. Here we study how memory affects the temporal duration of transition paths using the simple model of dynamics governed by a generalized Langevin equation with an exponential memory kernel. We discuss several approximate theories for the distribution and the mean of the transition path times and test them against numerical simulations. We find that the extreme case of long memory is particularly interesting in that it cannot be described by the existing approximations; yet it can be explained using the view where the non-Markov effects arise as a result of coupling of the reaction coordinate to an auxiliary degree of freedom.

Measuring the average shape of transition paths during the folding of a single biological molecule

“…However, there is an intriguing connection between the anomalously low barrier height implied by the flatter-than-expected shape of 〈t(x)〉 and previous work finding that transition path time distributions also implied barrier heights significantly lower than measured directly (10). Recent theoretical work has shown that distortions in the transition path time distribution implying anomalously low barriers can be induced by memory effects in the folding dynamics (37). Such memory effects would be expected to arise naturally in force spectroscopy measurements from the compliant linker coupling the molecule to the force probe, because of the finite response time for propagation of molecular motions through the linker to the probe.…”

“…Other factors involving deviations from the assumptions underlying Eq. 1 may also contribute to the overestimate, however, including the presence of anharmonicity in the measured barrier profiles (9,37), the possibility of position dependence in the diffusivity (34,38), the presence of memory effects (36,37), and the finite barrier size (14). Fig.…”

Broad distributions of transition-path times are fingerprints of multidimensionality of the underlying free energy landscapes

“…A value of C that is below 1, however, does not necessarily mean that the system is 1D, nor does it imply that a 1D description is a good approximation. Indeed, a common reason requiring one to invoke multidimensionality to describe single-molecule force spectroscopy experiments is the coupling of the molecular coordinate x to the observable experimental degrees of freedom such as the position y of the force probe (23,(41)(42)(43)(44).This coupling leads to memory effects in the dynamics of both x and y even if the intrinsic dynamics of these degrees of freedom is Markovian (19,48), and thus a 1D diffusion model may no longer be an adequate description of the observed experimental signal even when the intrinsic dynamics along x is diffusive. However, we find here that elastic coupling of the molecule to the experimental observables is not likely to lead to a broad distribution of the transition path time (with C > 1).…”

“…Although the coupling of the molecular coordinates to those of the experimental setup has been reported to broaden the distributions of the transition path time as compared to the 1D case, both experimental observations and simulations often report still narrower-than-singleexponential distributions (10,19,44). Here, we have studied this case in more detail by considering the standard model where the coupling between the two degrees of freedom is described as a harmonic spring with a free energy of the form k(x − y) 2 =2, where k represents the stiffness of the polymeric handle connecting the molecule and the force probe (SI Appendix).…”

“…In particular, the observed distributions are much broader than those predicted by the 1D diffusion model: The latter, when the diffusivity is adjusted to give the correct mean transition path time, significantly underestimates the contribution of both short and long transition paths (13). Several explanations to this discrepancy have been proposed, including roughness of the free energy landscape (15), entropic barriers (16,17), heterogeneity of the pathways (14,18), and memory effects (19)(20)(21), but so far the…”