Boundary conditions and trajectories of diffusion processes

A continuum PNP model of double species ion transport between graphene

In this paper, we employ Poisson-Nernst-Planck equations in conjunction with the Navier-Stokes equations and the mean-field theory to investigate the charge transport of double species monovalent ions in double-layered graphene sheets, driven by an external electric field. Unlike most classical models, we develop a simple mechanical model by incorporating the microscopic effects of the physical systems so that the ionic interactions between ions and the host material, and the steric repulsions among ions are considered. Taking Li⁺-PF6^-1monovalent ions as an example, we find that the transport pattern for the present double species ionic systems and that for the pure lithium ions are dramatically different. Due to the mutual attractive ionic forces between ions and their counter-ions, such double species systems turn out to be more stable than that of the single species systems. In addition, the storage patterns of the former systems are richer, which depend heavily on initial ionic states, however less on the strength of applied electric fields and external temperatures when the graphene separation is not too large. The fluid flows, the electric conductivities and the stability of such double species systems are subsequently scrutinized. The current study could form a theoretical basis to help designing high-performance lithium batteries and explaining ion transport in biological channels.

Fixed-density boundary conditions in overdamped Langevin simulations of diffusion in channels

We consider the numerical integration of Langevin equations for particles in a channel, in the presence of boundary conditions fixing the concentration values at the ends. This kind of boundary condition appears, for instance, when considering the diffusion of ions in molecular channels, between the different concentrations at both sides of the cellular membrane. For this application the overdamped limit of Brownian motion (leading to a first order Langevin equation) is most convenient, but in previous works some difficulties associated with this limit were found for the implementation of the boundary conditions. We derive here an algorithm that, unlike previous attempts, does not require the simulation of particle reservoirs or the consideration of velocity variables or adjustable parameters. Simulations of Brownian particles in simple cases show that results agree perfectly with theory, both for the local concentration values and for the resulting particle flux in nonequilibrium situations. The algorithm is appropriate for the modeling of more complex ionic channels and, in general, for the treatment of analogous boundary conditions in other physical models using first order Langevin equations.

Single-occupancy binding in simple bounded and unbounded systems

The number of substrate molecules that can bind to the active site of an enzyme at one time is constrained. This paper develops boundary conditions that correspond to the constraint of single-occupancy binding. Two simple models of substrate molecules diffusing to a single-occupancy site are considered. In the interval model, a fixed number of substrate molecules diffuse in a bounded domain. In the spherical model, a varying number of molecules diffuse in a domain with boundary conditions that model contact with a reservoir containing a large number of substrate molecules. When the diffusive time scale is much shorter than the time scale for entering the single-occupancy site, the dynamics of binding are accurately described by simple approximations.

Kernel representations for flux and concentration in ion channel models with time-varying concentrations

This paper explores stochastic models for the study of ion transport in biological cells. It considers one-dimensional models with time-varying concentrations at the boundaries. The average concentration and flux in the channel are obtained as kernel representations, where the kernel functions have a probabilistic interpretation which contributes to a better understanding of the models. In particular, the kernel representation is given for the flux at a boundary point, providing a correct version of a representation found in the literature. This requires special attention because one of the kernel functions exhibits a singularity. This kernel representation is feasible due to the linearity of the system that arises from the assumed independence between ions.

Framework models of ion permeation through membrane channels and the generalized King-Altman method

A modern approach to studying the detailed dynamics of biomolecules is to simulate them on computers. Framework models have been developed to incorporate information from these simulations in order to calculate properties of the biomolecules on much longer time scales than can be achieved by the simulations. They also provide a simple way to think about the simulated dynamics. This article develops a method for the solution of framework models, which generalizes the King-Altman method of enzyme kinetics. The generalized method is used to construct solutions of two framework models which have been introduced previously, the single-particle and Grotthuss (proton conduction) models. The solution of the Grotthuss model is greatly simplified in comparison with direct integration. In addition, a new framework model is introduced, generalizing the shaking stack model of ion conduction through the potassium channel.

Langevin trajectories between fixed concentrations

We consider the trajectories of particles diffusing between two infinite baths of fixed concentrations connected by a channel, e.g., a protein channel of a biological membrane. The steady state influx and efflux of Langevin trajectories at the boundaries of a finite volume containing the channel and parts of the two baths is replicated by termination of outgoing trajectories and injection according to a residual phase space density. We present a simulation scheme that maintains averaged fixed concentrations without creating spurious boundary layers, consistent with the assumed physics.

Brownian simulations and unidirectional flux in diffusion

The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics. Existing continuum descriptions of ionic permeation fail to capture the rich phenomenology of the permeation process, so it is therefore necessary to resort to particle simulations. Brownian dynamics (BD) simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time 1/gamma of the Langevin equation. We find that the probability of Brownian trajectories that cross an interface in one direction in unit time Deltat equals that of the probability of the corresponding Langevin trajectories if gammaDeltat=2 . That is, we find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirectional flux is proportional to the concentration and inversely proportional to sqrt[Deltat ] to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.

Memoryless control of boundary concentrations of diffusing particles

Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has calculated that flux since the time of Fick (1855), and the flux has been known to arise from the stochastic behavior of Brownian trajectories since the time of Einstein (1905), yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not complete. We consider the trajectories of noninteracting particles diffusing in a finite region connecting two baths of fixed concentrations. Inside the region, the trajectories of diffusing particles are governed by the Langevin equation. To maintain average concentrations at the boundaries of the region at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in Langevin and Brownian simulations of such systems. We analyze models of controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference between the time evolution and the steady state concentrations. While the time evolution of the density is governed by an integral operator, the spatial distribution is governed by the familiar Fokker-Planck operator. The boundary conditions for the time dependent density depend on the model of the controller; however, this dependence disappears in the steady state, if the controller is of a renewal type. Renewal-type controllers, however, produce spurious boundary layers that can be catastrophic in simulations of charged particles, because even a tiny net charge can have global effects. The design of a nonrenewal controller that maintains concentrations of noninteracting particles without creating spurious boundary layers at the interface requires the solution of the time-dependent Fokker-Planck equation with absorption of outgoing trajectories and a source of ingoing trajectories on the boundary (the so called albedo problem).

A framework model based on the Smoluchowski equation in two reaction coordinates

The general form of the Smoluchowski equation in two reaction coordinates is obtained as the diffusion limit of a random walk on an infinite square grid using transition probabilities that satisfy detailed balance at thermodynamic equilibrium. The diffusion limit is then used to construct a generalization of the single-particle model to two reaction coordinates. The state space includes a square on which diffusion takes place and an isolated empty state. Boundary conditions on opposite sides of the square correspond to transitions between the empty state and the square. The two-dimensional (2D) model can be reduced to a 1D single-particle model by adiabatic elimination. A finite element solution of the 2D boundary value problem is described. The method used to construct the 2D model can be adapted to state spaces that have been constructed by other authors to model K+ conduction through gramicidin, proton conduction through dioxolane-linked gramicidin, and chloride conduction through the bacterial H(+)-Cl- antiporter.

Glycerol conductance and physical asymmetry of the Escherichia coli glycerol facilitator GlpF

The aquaglyceroporin GlpF is a transmembrane channel of Escherichia coli that facilitates the uptake of glycerol by the cell. Its high glycerol uptake rate is crucial for the cell to survive in very low glycerol concentrations. Although GlpF allows both influx and outflux of glycerol, its structure, similar to the structure of maltoporin, exhibits a significant degree of asymmetry. The potential of mean force characterizing glycerol in the channel shows a corresponding asymmetry with an attractive vestibule only at the periplasmic side. In this study, we analyze the potential of mean force, showing that a simplified six-step model captures the kinetics and yields a glycerol conduction rate that agrees well with observation. The vestibule improves the conduction rate by 40% and 75% at 10- micro M and 10-mM periplasmic glycerol concentrations, respectively. In addition, neither the conduction rate nor the conduction probability for a single glycerol (efficiency) depends on the orientation of GlpF. GlpF appears to conduct equally well in both directions under physiological conditions.

Self organization of membrane proteins via dimerization

Protein-protein dimerization is ubiquitous in biology, but its role in self-organization remains unexplored. Here we use Monte Carlo simulations to demonstrate that under diffusion-limited conditions, reversible dimerization alone can cause membrane proteins to cluster into oligomer-like structures. When multiple distinct protein species are able to form dimers, then heterodimerization and homodimerization can organize proteins into structured clusters that can affect cellular physiology. As an example, we demonstrate how receptor dimerization could provide a physical mechanism for regulating information flow by controlling receptor-receptor cross talk. These results are physically realistic for some membrane proteins, including members of the G-protein coupled receptor family, and may provide a physiological reason as to why many proteins dimerize.