Solute translocation probability, lifetime, and "rectification" in membrane channels with localized constriction.

We study the translocation probability and lifetime of a solute molecule in a cylindrical membrane channel that contains a localized constriction at an arbitrary location. Using a one-dimensional continuous diffusion description of solute dynamics in the channel, we explore two models. The first one describes a molecule's interaction with the constriction in terms of a narrow rectangular barrier in the potential of mean force. The second novel model proposed here represents this interaction by introducing an infinitely thin permeable partition. It is shown that when the parameters of the two models are chosen to warrant the same translocation probability, both models predict the same mean lifetime of the molecule in the channel. While the translocation probability is independent of the constriction location, the mean lifetime is a function of the location. The benefit of the thin partition model is that it allows one to lump together the height and length of the potential barrier into a single parameter, which is the partition's permeability. It is shown that in the case of an asymmetric location of the localized constriction and strong repulsion between the solutes, the solute flux through the channel is a function of the direction in which it goes, analogous to the phenomenon known in ion channel electrophysiology as rectification.

Trapping of single diffusing particles by a circular disk on a reflecting flat surface. Absorbing hemisphere approximation.

Recent progress in biophysics (for example, in studies of chemical sensing and spatiotemporal cell-signaling) poses new challenges to statistical theory of trapping of single diffusing particles. Here we deal with one of them, namely, trapping kinetics of single particles diffusing in a half-space bounded by a reflecting flat surface containing an absorbing circular disk. This trapping problem is essentially two-dimensional and the question of the angular dependence of the kinetics on the particle starting point is highly nontrivial. We propose an approximate approach to the problem that replaces the absorbing disk by an absorbing hemisphere of a properly chosen radius. This replacement makes the problem angular-independent and essentially one-dimensional. After the replacement one can find an exact solution for the particle propagator (Green's function) that allows one to completely characterize the kinetics. Extensive testing of the theoretical predictions based on the absorbing hemisphere approximation against three-dimensional Brownian dynamics simulations shows excellent agreement between the analytical and simulation results when the particle starts sufficiently far away from the disk. Our approach fails and the angular dependence of the kinetics is important when the distance of the particle starting point from the disk center is comparable with the disk radius. However, even when the initial distance is only two disk radii, the maximum relative error of the theoretical predictions is about 10%. The relative error rapidly decreases as the initial distance increases.

Permeability and diffusion resistance of porous membranes: Analytical theory and its numerical test.

This study is devoted to the transport of neutral solutes through porous flat membranes, driven by the solute concentration difference in the reservoirs separated by the membrane. Transport occurs through membrane channels, which are assumed to be non-overlapping, identical, straight cylindrical pores connecting the reservoirs. The key quantities characterizing transport are membrane permeability and its diffusion resistance. Such transport problems arising in very different contexts, ranging from plant physiology and cell biology to chemical engineering, have been studied for more than a century. Nevertheless, an expression giving the permeability for a membrane of arbitrary thickness at arbitrary surface densities of the channel openings is still unknown. Here, we fill in the gap and derive such an expression. Since this expression is approximate, we compare its predictions with the permeability obtained from Brownian dynamics simulations and find good agreement between the two.

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.

Counter-Intuitive Features of Particle Dynamics in Nanopores.

Using the framework of a continuous diffusion model based on the Smoluchowski equation, we analyze particle dynamics in the confinement of a transmembrane nanopore. We briefly review existing analytical results to highlight consequences of interactions between the channel nanopore and the translocating particles. These interactions are described within a minimalistic approach by lumping together multiple physical forces acting on the particle in the pore into a one-dimensional potential of mean force. Such radical simplification allows us to obtain transparent analytical results, often in a simple algebraic form. While most of our findings are quite intuitive, some of them may seem unexpected and even surprising at first glance. The focus is on five examples: (i) attractive interactions between the particles and the nanopore create a potential well and thus cause the particles to spend more time in the pore but, nevertheless, increase their net flux; (ii) if the potential well-describing particle-pore interaction occupies only a part of the pore length, the mean translocation time is a non-monotonic function of the well length, first increasing and then decreasing with the length; (iii) when a rectangular potential well occupies the entire nanopore, the mean particle residence time in the pore is independent of the particle diffusivity inside the pore and depends only on its diffusivity in the bulk; (iv) although in the presence of a potential bias applied to the nanopore the "downhill" particle flux is higher than the "uphill" one, the mean translocation times and their distributions are identical, i.e., independent of the translocation direction; and (v) fast spontaneous gating affects nanopore selectivity when its characteristic time is comparable to that of the particle transport through the pore.

Intrinsic diffusion resistance of a membrane channel, mean first-passage times between its ends, and equilibrium unidirectional fluxes.

Diffusive flux of solute molecules through a membrane channel driven by the solute concentration difference on the two sides of the membrane is inversely proportional to the channel diffusion resistance. We show that the intrinsic, channel proper, part of this resistance is the ratio of the sum of the mean first-passage times of the molecule between the channel ends and the molecule partition function in the channel. This is derived without appealing to any specific model of the channel and, therefore, is applicable to transport in channels of arbitrary shape and tortuosity and at arbitrary interaction strength of solute molecules with the channel walls.

Capturing single molecules by nanopores: measured times and thermodynamics.

In numerous nanopore sensing applications transient interruptions in ion current through single nanopores induced by capturing solute molecules are a source of information on how solutes interact with the nanopores. We show that the distribution of time spent by a single captured solute molecule in a nanopore is bimodal with the majority of capture events being too fast to be experimentally resolved. As a result, the exact mean durations of the event and inter-event interval are orders of magnitude shorter than their measured values. Moreover, the exact and measured mean durations have qualitatively different dependences on the molecule diffusivity. This leads to a formal contradiction with the thermodynamics of molecule partitioning between the bulk and the nanopore. Here we resolve this controversy. We also demonstrate that, surprisingly, the probability of finding a molecule in the nanopore, obtained from the ratio of the measured mean durations of the capture event and interevent interval, is essentially identical to the exact equilibrium thermodynamic probability.

Crowding breaks the forward/backward symmetry of transition times in biased random walks.

Microscopic mechanisms of natural processes are frequently understood in terms of random walk models by analyzing local particle transitions. This is because these models properly account for dynamic processes at the molecular level and provide a clear physical picture. Recent theoretical studies made a surprising discovery that in complex systems, the symmetry of molecular forward/backward transition times with respect to local bias in the dynamics may be broken and it may take longer to go downhill than uphill. The physical origins of these phenomena remain not fully understood. Here, we explore in more detail the microscopic features of the symmetry breaking in the forward/backward transition times by analyzing exactly solvable discrete-state stochastic models. In particular, we consider a specific case of two random walkers on a four-site periodic lattice as the way to represent the general systems with multiple pathways. It is found that the asymmetry in transition times depends on several factors that include the degree of deviation from equilibrium, the particle crowding, and methods of measurements of dynamic properties. Our theoretical analysis suggests that the asymmetry in transition times can be explored experimentally for determining the important microscopic features of natural processes by quantitatively measuring the local deviations from equilibrium and the degrees of crowding.

Diffusive barrier crossing rates from variationally determined eigenvalues.

Kramers' procedure for calculating the rate of activated processes involves partitioning space into reactant, barrier, and product regions by introducing two dividing surfaces. Then, a nonequilibrium steady state is established by injecting particles on one surface and removing them when they reach the other. The rate is obtained as the ratio of the steady-state flux between the surfaces and the population of the initial well. An alternative procedure that seems less artificial is to estimate the first non-zero eigenvalue of the operator that describes the dynamics and then equate its magnitude to the sum of the forward and backward rate constants. Here, we establish the relationship between these approaches for diffusive dynamics, starting with the variational principle for the eigenvalue of interest and then using a trial function involving two adjustable surfaces. We show how Kramers' flux-over-population expression for the rate constant can be obtained from our variationally determined eigenvalue in the special case where the reactant and product regions are separated by a high barrier. This work exploits the modern theory of activated rate processes where the committor (the probability of reaching one dividing surface before the other) plays a central role. Surprisingly, our upper bound for the eigenvalue can be expressed solely in terms of mean first-passage times and the mean transition-path time between the two dividing surfaces.

Surface-facilitated trapping by active sites: From catalysts to viruses.

Trapping by active sites on surfaces plays important roles in various chemical and biological processes, including catalysis, enzymatic reactions, and viral entry into host cells. However, the mechanisms of these processes remain not well understood, mostly because the existing theoretical descriptions are not fully accounting for the role of the surfaces. Here, we present a theoretical investigation on the dynamics of surface-assisted trapping by specific active sites. In our model, a diffusing particle can occasionally reversibly bind to the surface and diffuse on it before reaching the final target site. An approximate theoretical framework is developed, and its predictions are tested by Brownian dynamics computer simulations. It is found that the surface diffusion can be crucial in mediating trapping by active sites. Our theoretical predictions work reasonably well as long as the area of the active site is much smaller than the overall surface area. Potential applications of our approach are discussed.

Localized potential well vs binding site: Mapping solute dynamics in a membrane channel onto one-dimensional description.

In the one-dimensional description, the interaction of a solute molecule with the channel wall is characterized by the potential of mean force U(x), where the x-coordinate is measured along the channel axis. When the molecule can reversibly bind to certain amino acid(s) of the protein forming the channel, this results in a localized well in the potential U(x). Alternatively, this binding can be modeled by introducing a discrete localized site, in addition to the continuum of states along x. Although both models may predict identical equilibrium distributions of the coordinate x, there is a fundamental difference between the two: in the first model, the molecule passing through the channel unavoidably visits the potential well, while in the latter, it may traverse the channel without being trapped at the discrete site. Here, we show that when the two models are parameterized to have the same thermodynamic properties, they automatically yield identical translocation probabilities and mean translocation times, yet they predict qualitatively different shapes of the translocation time distribution. Specifically, the potential well model yields a narrower distribution than the model with a discrete site, a difference that can be quantified by the distribution's coefficient of variation. This coefficient turns out to be always smaller than unity in the potential well model, whereas it may exceed unity when a discrete trapping site is present. Analysis of the translocation time distribution beyond its mean thus offers a way to differentiate between distinct translocation mechanisms.

VDAC Gating Thermodynamics, but Not Gating Kinetics, Are Virtually Temperature Independent.

The voltage-dependent anion channel (VDAC) is the most abundant protein in the mitochondrial outer membrane and an archetypical ß-barrel channel. Here, we study the effects of temperature on VDAC channels reconstituted in planar lipid membranes at the single- and multichannel levels within the 20°C to 40°C range. The temperature dependence of conductance measured on a single channel in 1 M KCl shows an increase characterized by a 10°C temperature coefficient Q10 = 1.22 ± 0.02, which exceeds that of the bathing electrolyte solution conductivity, Q10 = 1.17 ± 0.01. The rates of voltage-induced channel transition between the open and closed states measured on multichannel membranes also show statistically significant increases, with temperatures that are consistent with activation energy barriers of ∼10 ± 3 kcal/mol. At the same time, the gating thermodynamics, as characterized by the gating charge and voltage of equipartitioning, does not display any measurable temperature dependence. The two parameters stay within 3.2 ± 0.2 elementary charges and 30 ± 2 mV, respectively. Thus, whereas the channel kinetics, specifically its conductance and rates of gating response to voltage steps, demonstrates a clear increase with temperature, the conformational voltage-dependent equilibria are virtually insensitive to temperature. These results, which may be a general feature of ß-barrel channel gating, suggest either an entropy-driven gating mechanism or a role for enthalpy-entropy compensation.

Detailed balance for diffusion in a potential with trapping and forward-backward symmetry of trapping time distributions.

For particles diffusing in a potential, detailed balance guarantees the absence of net fluxes at equilibrium. Here, we show that the conventional detailed balance condition is a special case of a more general relation that works when the diffusion occurs in the presence of a distributed sink that eventually traps the particle. We use this relation to study the lifetime distribution of particles that start and are trapped at specified initial and final points. It turns out that when the sink strength at the initial point is nonzero, the initial and final points are interchangeable, i.e., the distribution is independent of which of the two points is initial and which is final. In other words, this conditional trapping time distribution possesses forward-backward symmetry.

Biased diffusion in periodic potentials: Three types of force dependence of effective diffusivity and generalized Lifson-Jackson formula.

Diffusive transport of particles in a biased periodic potential is characterized by the effective drift velocity and diffusivity, which are functions of the biasing force. We derive a simple exact expression for the effective diffusivity and use it to show that the force dependence of this quantity may be a nonmonotonic function with a maximum [as shown in the work of Reimann et al. Phys. Rev. Lett. 87, 010602 (2001) for periodic sinusoidal potential] or with a minimum, or a monotonic function. The shape of the dependence is determined by the shape of the periodic potential.

Committors, first-passage times, fluxes, Markov states, milestones, and all that.

Milestoning on a one-dimensional potential starts by choosing a set of points, called milestones, and initiating short trajectories from each milestone, which are terminated when they reach an adjacent milestone for the first time. From the average duration of these trajectories and the probabilities of where they terminate, a rate matrix can be constructed and then used to calculate the mean first-passage time (MFPT) between any two milestones. All these MFPT's turn out to be exact. Here we adopt a point of view from which this remarkable result is not unexpected. In addition, we clarify the nature of the "states" whose interconversion is described by the rate matrix constructed using information obtained from short trajectories and provide a microscopic expression for the "equilibrium population" of these states in terms of equilibrium averages of the committors.

Coarse-graining of asymmetric discrete-time random walk on a one-dimensional lattice.

Coarse-graining of an asymmetric nearest-neighbor discrete-time random walk on a one-dimensional lattice allows one to describe this random walk as biased one-dimensional diffusion. The latter is characterized by two parameters: the drift velocity and diffusivity. There is a general expression giving the drift velocity as a function of the parameters determining the random walk. However, a corresponding expression for the diffusivity is known only for the particular case where the random walk escapes from the lattice site at every time step. In this work, we generalize this result and derive an expression for the diffusivity, assuming that the random walk does not necessarily leave the site, and therefore, its mean lifetime on the site can be longer than the time step.

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.

Steady-state flux of diffusing particles to a rough boundary formed by absorbing spikes periodically protruding from a reflecting base.

We study steady-state flux of particles diffusing on a flat surface and trapped by absorbing spikes of arbitrary length periodically protruding from a reflecting base. It is assumed that the particle concentration, far from this comblike boundary, is kept constant. To find the flux, we use a boundary regularization approach that replaces the initial highly rough and heterogeneous boundary by an effective boundary which is smooth and uniform. After such a replacement, the two-dimensional diffusion problem becomes essentially one-dimensional, and the steady-state flux can be readily found. Our main results are simple analytical expressions determining the position of the smooth effective boundary and its uniform trapping rate as functions of the spike length and interspike distance. It is shown that the steady-state flux to the effective boundary is identical to its counterpart to the initial boundary at large distances from this boundary. Our analytical results are corroborated by Brownian dynamics simulations.

Trapping of diffusing particles by small absorbers localized in a spherical region.

We study trapping of particles diffusing in a spherical cavity with an absorbing wall containing small static spherical absorbers localized in a spherical region in the center of the cavity. The focus is on the competition between the absorbers and the cavity wall for diffusing particles. Assuming that the absorbers and, initially, the particles are uniformly distributed in the central region, we derive an expression for the particle trapping probability by the cavity wall. The expression gives this probability as a function of two dimensionless parameters: the transparency parameter, characterizing the efficiency of the particle trapping by the absorbers, and the ratio of the absorber-containing region radius to that of the cavity. This work is a generalization of a recent study by Krapivsky and Redner [J. Chem. Phys. 147, 214903 (2017)] who considered the case where the absorber-containing region occupies the entire cavity. The expression for the particle trapping probability is derived in the framework of a steady-state approach which, in our opinion, is much simpler than the time-dependent approach used in the above-mentioned study.

Stochastic Gating as a Novel Mechanism for Channel Selectivity.

An ideal channel, responsible for metabolite fluxes in and out of the cells and cellular compartments, is supposed to be selective for a particular set of molecules only. However, such a channel has to be wide enough to accommodate relatively large metabolites, and, therefore, it allows passage of smaller solutes, for example, sodium, potassium, and chloride ions, thus compromising membrane's barrier function. Here we show that stochastic gating is able to provide a mechanism for the selectivity of wide channels in favor of large metabolites. Specifically, applying our recent theory of the stochastic gating effect on channel-facilitated transport, we demonstrate that under certain conditions gating hinders translocation of fast-diffusing small solutes to a significantly higher degree than that of large solutes that diffuse much slower. We hypothesize that this can be used by Nature to minimize the shunting effect of wide channels with respect to small solutes.