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

Abstract

Recent single-molecule experiments have observed transition paths, i.e., brief events where molecules (particularly biomolecules) are caught in the act of surmounting activation barriers. Such measurements offer unprecedented mechanistic insights into the dynamics of biomolecular folding and binding, molecular machines, and biological membrane channels. A key challenge to these studies is to infer the complex details of the multidimensional energy landscape traversed by the transition paths from inherently low-dimensional experimental signals. A common minimalist model attempting to do so is that of one-dimensional diffusion along a reaction coordinate, yet its validity has been called into question. Here, we show that the distribution of the transition path time, which is a common experimental observable, can be used to differentiate between the dynamics described by models of one-dimensional diffusion from the dynamics in which multidimensionality is essential. Specifically, we prove that the coefficient of variation obtained from this distribution cannot possibly exceed 1 for any one-dimensional diffusive model, no matter how rugged its underlying free energy landscape is: In other words, this distribution cannot be broader than the single-exponential one. Thus, a coefficient of variation exceeding 1 is a fingerprint of multidimensional dynamics. Analysis of transition paths in atomistic simulations of proteins shows that this coefficient often exceeds 1, signifying essential multidimensionality of those systems.

Exposure of Von Willebrand Factor Cleavage Site in A1A2A3-Fragment under Extreme Hydrodynamic Shear

“…According to Bell's model [65], when an external force is applied to one terminus of the protein, the barrier-limited unfolding transition is accelerated because of the added mechanical work along the unfolding path, k = k 0 e βFδx , where k 0 is the rate at null force, β = 1/k B T, and δx is the distance between the reactant state and the transition state along the pulling direction. This expression is valid for a two-state, thermally activated reaction happening on a one-dimensional energy landscape, conditions that can be challenged in realistic systems [66]. Moreover, in MD simulations that generally allow to access molecular processes at the ns − µs timescale, computational limitations impose the use of pulling forces higher than those usually employed in experiments.…”

Memory effects and static disorder reduce information in single-molecule signals

“…For the case where the experimental observable x is a continuous variable, one such Markovianity criterion has been established recently 20 based on analyzing spatial intervals [ a, b ] and comparing to theoretical inequalities the frequency with which the trajectory x ( t ) transitions through or loops back when entering the interval, but this approach is inapplicable to processes where the observed experimental quantity takes on discrete values. Other non-Markovianity signatures have been established 21, 22 , and an information-theoretic method based on computing the mutual information of the true dynamics and its Markovian model has been proposed 23 . For discrete-state processes, a solution was already described by Shannon in his classic work 24 where he estimated the information content (i.e.…”

Energy Landscape of Ubiquitin Is Weakly Multidimensional

“…The theory also assumes that FES is onedimensional function of X, which might not always hold. Indeed, experiments and simulations provide evidence that the FES for some biomolecules could be multidimensional, 26,27 involving multiple barriers. 28−30 If the energy landscape were multidimensional, then it could result in the nonlinearity in log k u ( f) vs f plot in force-clamp experiments 19,31,32 or in the most probable rupture force f * vs the logarithm of loading rate log r f .…”