Mapping Intrachannel Diffusive Dynamics of Interacting Molecules onto a Two-Site Model: Crossover in Flux Concentration Dependence

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

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.

Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture

Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on Jss-and hence on the MFPT-as the system evolves in time. Here, we show that Jss is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating Jss and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.

Blocker escape kinetics from a membrane channel analyzed by mapping blocker diffusive dynamics onto a two-site model

When a large solute molecule enters a membrane channel from the membrane-bathing electrolyte solution, it blocks the small-ion current flowing through the channel. If the molecule spends in the channel sufficiently long time, individual blockades can be resolved in single-channel experiments. In this paper, we develop an analytical theory of the blocker escape kinetics from the channel, assuming that a charged blocking molecule cannot pass through a constriction region (bottleneck). We focus on the effect of the external voltage bias on the blocker survival probability in the channel. The bias creates a potential well for the charged blocker in the channel with the minimum located near the bottleneck. When the bias is strong, the well is deep, and escape from the channel is a slow process that allows for time-resolved observation of individual blocking events. Our analysis is performed in the framework of a two-site model of the blocker dynamics in the channel. Importantly, the rate constants, fully determining this model, are derived from a more realistic continuum diffusion model. This is done by mapping the latter onto its two-site counterpart which, while being much simpler, captures the main features of the blocker escape kinetics at high biases.

Theoretical insights into mechanisms of channel-facilitated molecular transport in the presence of stochastic gating

Molecular motion through pores plays a crucial role in various natural and industrial processes. One of the most fascinating features of biological channel-facilitated transport is a stochastic gating process, when the channels dynamically fluctuate between several conformations during the translocation. Although this phenomenon has been intensively investigated, many properties of translocation in a dynamically changing environment remain not well understood microscopically. We developed a discrete-state stochastic framework to analyze the molecular mechanisms of transport processes with stochastic gating by explicitly calculating molecular fluxes through the pores. Two scenarios are specifically investigated: (1) symmetry preserving stochastic gating with free-energy changes and (2) stochastic gating with symmetry changes but without modifications in the overall particle-pore interactions. It is found that stochastic gating can either accelerate or slow down the molecular translocation depending on the specific parameters of the system. We argue that biological systems might optimize their performance by utilizing conformational fluctuations of channels. Our theoretical analysis clarifies physical-chemical aspects of the molecular mechanisms of transport with stochastic gating.