Three-dimensional Brownian Dynamics Simulations Research Articles

Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting.

The transport of particles through channels is of paramount importance in physics, chemistry, and surface science due to its broad real world applications. Much insight can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities, and the total average lifetime in the channel are obtained, and their behavior as a function of the resetting rate is highlighted. It is shown that resetting can expedite the transport through the channel-rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions is supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes-a phenomenon that can spearhead further theoretical and experimental studies.

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.

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.

Fast simulation of Brownian dynamics in a crowded environment.

Brownian dynamics simulations are an increasingly popular tool for understanding spatially extended biochemical reaction systems. Recent improvements in our understanding of the cellular environment show that volume exclusion effects are fundamental to reaction networks inside cells. These systems are frequently studied by incorporating inert hard spheres (crowders) into three-dimensional Brownian dynamics (BD) simulations; however these methods are extremely slow owing to the sheer number of possible collisions between particles. Here we propose a rigorous "crowder-free" method to dramatically increase the simulation speed for crowded biochemical reaction systems by eliminating the need to explicitly simulate the crowders. We consider both the cases where the reactive particles are point particles, and where they themselves occupy a volume. Using simulations of simple chemical reaction networks, we show that the "crowder-free" method is up to three orders of magnitude faster than conventional BD and yet leads to nearly indistinguishable results from the latter.

Discrete-State Representation of Ion Permeation Coupled to Fast Gating in a Model of CLC-Chloride Channels: Analytic Estimation of the State-to-State Rate Constants

Analytical estimation of state-to-state rate constants is carried out for a recently developed discrete state model of chloride ion motion in a CLC chloride channel (Coalson and Cheng, J. Phys. Chem. B 2010, 114, 1424). In the original presentation of this model, the same rate constants were evaluated via three-dimensional Brownian dynamics simulations. The underlying dynamical theory is an appropriate single- or multiparticle three-dimensional Smoluchowski equation. Taking advantage of approximate geometric symmetries (based on the details of the model channel geometry), well-known formulas for state-to-state transition rates are appealed to herein and adapted as necessary to the problem at hand. Rates of ionic influx from a bulk electrolyte reservoir to the nearest binding site within the channel pore are particularly challenging to compute analytically because they reflect multi-ion interactions (as opposed to single-ion dynamics). A simple empirical correction factor is added to the single-ion rate constant formula in this case to account for the saturation of influx rate constants with increasing bulk Cl(-) concentration. Overall, the agreement between all analytically estimated rate constants is within a factor of 2 of those computed via three-dimensional Brownian dynamics simulations, and often better than this. Current-concentration curves obtained using rate constants derived from these two different computational approaches agree to within 25%.

Fluxes of non-interacting and strongly repelling particles through a single conical channel: Analytical results and their numerical tests

Using a diffusion model of particle dynamics in the channel, we study entropic effects in channel-facilitated transport. We derive general expressions for the fluxes of non-interacting particles and particles that strongly repel each other through the channel of varying cross section area, assuming that the transport is driven by the difference in particle concentrations on the two sides of the membrane. For a special case of a right truncated cone expanding in the left-to-right direction, we show how the fluxes depend on the geometric parameters of the channel and on the particle concentrations. For non-interacting particles the flux is direction-independent in the sense that inversion of the concentration difference leads to the inversion of the direction of the flux without changing its magnitude. This symmetry is broken for repelling particles: the flux in the left-to-right direction exceeds its right-to-left counterpart. Our theoretical predictions are supported by three-dimensional Brownian dynamics simulations.

Discrete-State Representation of Ion Permeation Coupled to Fast Gating in a Model of ClC Chloride Channels: Comparison to Multi-ion Continuous Space Brownian Dynamics Simulations

A discrete-state model of chloride ion motion in a ClC chloride channel is constructed, following a previously developed multi-ion continuous space model of the same system (Cheng, M. H.; Mamonov, A. B.; Dukes, J. W.; Coalson, R. D. J. Phys. Chem. B 2007, 111, 5956) that included a simplistic representation of the fast gate in this channel. The reducibility of the many-body continuous space to the eight discrete-state model considered in the present work is examined in detail by performing three-dimensional Brownian dynamics simulations of each allowed state-to-state transition in order to extract the appropriate rate constant for this process, and then inserting the pairwise rate constants thereby obtained into an appropriate set of kinetic master equations. Experimental properties of interest, including the rate of Cl(-) ion permeation through the open channel and the average rate of closing of the fast gate as a function of bulk Cl(-) ion concentrations in the intracellular and extracellular electrolyte reservoirs are computed. Good agreement is found between the results obtained via the eight discrete-state model versus the multi-ion continuous space model, thereby encouraging continued development of the discrete-state model to include more complex behaviors observed experimentally in these channels.

Kinetics and morphology of cluster growth in a model of short-range attractive colloids

We present results from detailed three-dimensional Brownian dynamics simulations of the self-assembly process in quenched short-range attractive colloids. Clusters obtained in the simulations range from dense faceted crystals to fractal aggregates which show ramified morphology on large length scales but close-packed crystalline morphology on short length scales. For low volume fractions of the colloids, the morphology and crystal structure of a nucleating cluster are studied at various times after the quench. As the volume fraction of the colloids is increased, growth of clusters is controlled by cluster diffusion and cluster-cluster interactions. For shallower quenches and low volume fractions, clusters are compact and the growth-law exponent agrees well with Binder-Stauffer predictions and with recent experimental results. As the volume fraction is increased, clusters do not completely coalesce when they meet each other and the kinetics crosses over to diffusion-limited cluster-cluster aggregation (DLCA) limit. For deeper quenches, clusters are fractals even at low volume fractions and the growth kinetics asymptotically reaches the irreversible DLCA case.

Entropic effects in channel-facilitated transport: interparticle interactions break the flux symmetry.

We analyze transport through conical channels that is driven by the difference in particle concentrations on the two sides of the membrane. Because of the detailed balance, fluxes of noninteracting particles through the same channel, inserted into the membrane in two opposite orientations, are equal. We show that this flux symmetry is broken by particle-particle interactions so that one of the orientations can be much more efficient for transport under the same external conditions. The results are obtained analytically using a one-dimensional diffusion model and confirmed by three-dimensional Brownian dynamics simulations.

Diffusion-limited binding to a site on the wall of a membrane channel

The authors develop a theory of diffusion-controlled reactions with a site located on the wall of a cylindrical membrane channel that connects two reservoirs containing diffusing particles which are trapped by the site at the first contact. An expression for the Laplace transform of the rate coefficient, kt, is derived assuming that the size of the site is small compared to the channel radius. The expression is used to find the stationary value of the rate coefficient, kinfinity, as a function of the length and radius of the channel, the radius of the site, and its position inside the channel (distances from the two ends of the channel) as well as the particle diffusion constants in the bulk and in the channel. Their derivation is based on the one-dimensional description of the particle motion in the channel, which is generalized to include binding to the site into consideration. The validity of the approximate one-dimensional description of diffusion and binding was checked by three-dimensional Brownian dynamics simulations. They found that the one-dimensional description works reasonably well when the size of the site does not exceed 0.2 of the channel radius.

Conduction Mechanisms of Chloride Ions in ClC-Type Channels

The conduction properties of ClC-0 and ClC-1 chloride channels are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We create an open-state configuration of the prokaryotic ClC Cl − channel using its known crystallographic structure as a basis. Two residues that are occluding the channel are slowly pushed outward with molecular dynamics to create a continuous ion-conducting path with the minimum radius of 2.5 Å. Then, retaining the same pore shape, the prokaryotic ClC channel is converted to either ClC-0 or ClC-1 by replacing all the nonconserved dipole-containing and charged amino acid residues. Employing open-state ClC-0 and ClC-1 channel models, current-voltage curves consistent with experimental measurements are obtained. We find that conduction in these pores involves three ions. We locate the binding sites, as well as pinpointing the rate-limiting steps in conduction, and make testable predictions about how the single channel current across ClC-0 and ClC-1 will vary as the ionic concentrations are increased. Finally, we demonstrate that a ClC-0 homology model created from an alternative sequence alignment fails to replicate any of the experimental observations.

Channel-facilitated membrane transport: Average lifetimes in the channel

The transport of many solutes across biological membranes happens with the help of specialized proteins that form water-filled channels traversing the membranes. Recent experimental and theoretical work demonstrates that solute translocation can be facilitated by attractive interactions between the channel and penetrating particle. Here we consider an important aspect of channel-facilitated passive transport, the average lifetimes in the channel for those particles that traverse the channel and those that return, as well as the total average lifetime of the particle in the channel. Exact expressions for the average lifetimes are derived in the framework of a one-dimensional diffusion model. The validity of our one-dimensional analysis is verified by good agreement of the theoretical predictions with the average lifetimes found in three-dimensional Brownian dynamics simulations.

Conducting-State Properties of the KcsA Potassium Channel from Molecular and Brownian Dynamics Simulations

The mechanisms underlying transport of ions across the potassium channel are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We first build open-state configurations of the channel with molecular dynamics simulations, by pulling the transmembrane helices outward until the channel attains the desired interior radius. To gain insights into ion permeation, we construct potential energy profiles experienced by an ion traversing the channel in the presence of other resident ions. These profiles reveal that in the absence of an applied field the channel accommodates three potassium ions in a stable equilibrium, two in the selectivity filter and one in the central cavity. In the presence of a driving potential, this three-ion state becomes unstable, and ion permeation across the channel is observed. These qualitative explanations are confirmed by the results of three-dimensional Brownian dynamics simulations. We find that the channel conducts when the ionizable residues near the extracellular entrance are fully charged and those near the intracellular side are partially charged. The conductance increases steeply as the radius of the intracellular mouth of the channel is increased from 2 Å to 5 Å. Our simulation results reproduce several experimental observations, including the current-voltage curves, conductance-concentration relationships, and outward rectification of currents.

Brownian dynamics study of an open-state KcsA potassium channel

Three-dimensional Brownian dynamics simulations are used to study conductance of the KcsA potassium channel using the known crystallographic structure. Employing an open-state channel created by molecular dynamics simulations, current–voltage and current–concentration curves broadly consistent with experimental measurements are obtained. In the absence of an applied potential, the channel houses three potassium ions at positions that are in close agreement with X-ray diffraction maps.

Particle number fluctuations in a membrane channel

Channel-facilitated transport of metabolites across biological membranes results in excess noise in the current carried by small ions. This noise originates from fluctuations of the number of metabolite molecules in the channel due to their diffusion. We have carried out a theoretical study of particle number fluctuations in a cylindrical pore. First, we obtain the power spectral density of these fluctuations as a function of pore length and radius, as well as the diffusion constants of the particle in the pore and in the bulk, in the absence of particle–pore interactions. We then perform three-dimensional Brownian dynamics simulations that show excellent agreement with the analytical result. Finally, we demonstrate that explicit expressions for the low-frequency limit of the spectral density can be found even when the particle interacts with the pore.