Flux through a time-periodic gate: Monte Carlo test of a homogenization result

We investigate via Monte Carlo numerical simulations and theoretical considerations the outflux of random walkers moving in an interval bounded by an interface exhibiting channels (pores, doors) which undergo an open/close cycle according to a periodic schedule. We examine the onset of a limiting boundary behavior characterized by a constant ratio between the outflux and the local density, in the thermodynamic limit. We compare such a limit with the predictions of a theoretical model already obtained in the literature as the homogenization limit of a suitable diffusion problem

Homogenization of an alternating Robin–Neumann boundary condition via time-periodic unfolding

We consider the homogenization of a parabolic problem in a perforated domain with Robin–Neumann boundary conditions oscillating in time. Such oscillations must compensate the blow up of the boundary measure of the holes. We use the technique of time-periodic unfolding in order to obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition

Monte Carlo study of gating and selection in potassium channels

The study of selection and gating in potassium channels is a very important issue in modern biology. Indeed such structures are known in all types of cells in all organisms where they play many important functional roles. The mechanism of gating and selection of ionic species is not clearly understood. In this paper we study a model in which gating is obtained via an affinity-switching selectivity filter. We discuss the dependence of selectivity and efficiency on the cytosolic ionic concentration and on the typical pore open state duration. We demonstrate that a simple modification of the way in which the selectivity filter is modeled yields larger channel efficiency

Permeability of interfaces with alternating pores in parabolic problems

We study a parabolic problem set in a domain divided by a perforated interface. The pores alternate between an open and a closed state, periodically in time. We consider the asymptotics of the solution for vanishingly small size of the pores and time period. The interface condition prevailing in the limit is a linear relation between the flux (on either side) and the jump of the limiting solution across the interface. More exactly this behaviour only takes place when the relative sizes of the relevant geometrical and temporal parameters are connected by suitable relations. With respect to the stationary version of this problem, which is known in the literature, we demonstrate the appearance of a new admissible asymptotic standard. More in general, we describe the precise interplay between the geometrical and temporal parameters leading to the quoted interface condition. This work represents a preliminary mathematical investigation of a model of selective transport of chemical species through biological membranes

Homogenization of a parabolic problem with alternating boundary conditions.

We consider the homogenization of a parabolic problem in a perforated domain with Robin–Neumann boundary conditions oscillating in time. Boundary conditions alternating in time appear in biological applications, for example in the modeling of ion channels, see [1]. Our approach relies upon a generalization of the unfolding technique, see e.g., [2], to the time-periodic case. To this end we show how the method of periodic unfolding can be applied to classical homogenization problem for a parabolic equation with diffusion and capacity-like coefficients in the diffusion equation oscillating both in space and time, with general independent scales. From an analytical point of view, in the present case such oscillations must compensate the blow up of the boundary measure of the holes.We obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition; this term keeps memory of the underlying temporal and spatial microstructures. [1] D. Andreucci, D. Bellaveglia. Permeability of interfaces with alternating pores in parabolic problems. Asymptotic Analysis. 79 (2012), 189–227. [2] D. Cioranescu, A. Damlamian and G. Griso. The periodic unfolding method in homogenization. SIAM Journal on Mathematical Analysis. 40(4) (2008), 1585–1620

Alternating Robin-Neumann boundary value problem as a model for transport through biological membranes

It is known that transport of chemical species through biological membranes often can not be modelled by standard diffusion through openings (pores, or channels) in the membrane. Actually, it has been observed that in many physical situations, pores alternate between two states (open and closed), either periodically or according to a random scheme. As shown by Andreucci--Bellaveglia in a previous paper through homogenization techniques, the limiting behavior of problems of this kind sharply depends on the relative scalings of the time and space variables. Here, having in mind a model of cell absorption of a selected protein or drug, we consider the homogenization of a parabolic problem in a perforated domain with Robin--Neumann boundary conditions oscillating in time. Such oscillations must compensate the blow up of the boundary measure of the holes. We use the technique of time--periodic unfolding in order to obtain a macroscopic parabolic problem containing an extra linear term due to the absorption determined by the Robin condition. Our approach is based on the results obtained in a previous paper of the same authors, where the time-periodic unfolding operator is introduced, inspired by the operators of space-periodic unfolding introduced and applied by Cioranescu, Damlamian, Donato, Griso, Onofrei, Zaki in some quite recent researches. Finally, we identify two possible limiting behaviors depending on the relative magnitude of the time-period of the oscillations and the diameter of the holes and spatial period of the lattice

Time-periodic unfolding operator in parabolic homogenization

We extend the recent technique of the unfolding operator to the case of a time-depending setting. We apply this technique to the homogenization of parabolic problems with time-depending oscillating coefficients

Effect of Intracellular Diffusion on Current–Voltage Curves in Potassium Channels

We study the effect of intracellular ion diffusion on ionic currents permeating through the cell membrane. Ion flux across the cell membrane is mediated by specific channels, which have been widely studied in recent years with remarkable results: very precise measurements of the true current across a single channel are now available. Nevertheless, a complete understanding of this phenomenon is still lacking, though molecular dynamics and kinetic models have provided partial insights. In this paper we demonstrate, by analyzing the KcsA current-voltage currents via a suitable lattice model, that intracellular diffusion plays a crucial role in the permeation phenomenon. We believe that the interplay between the channel behavior and the ion diffusion in the cell is a key ingredient for a full explanation of the current-voltage curves

A Mathematical Model for Alternating Pores in Biological Membranes

We consider a mathematical model for selective permeation of chemical species through cell membranes. The mechanism relies on gating, that is on the alternate closing and opening of the pores. Firstly, we perform a preliminary analytical study of a parabolic problem set in a domain divided by a perforated interface, in the presence of periodic gating. We prove that, for vanishingly small size of the pores and time period, the interface condition prevailing in the limit is a linear relation between the flux (on either side) and the jump of the limiting solution across the interface. Note that such an interface condition only appears when the relative sizes of the relevant geometrical and temporal parameters are suitably connected. We demonstrate the appearance of a new admissible asymptotic standard with respect to the stationary version of this problem. Secondly, we investigate the issue of selection in the framework of a random walk model based on the same concepts. This study is performed through Monte Carlo numerical techniques, and is aimed at investigating how selective transport, and gating as well, can be obtained by stochastically switching the affinity of the pore for the target species. © 2011 American Institute of Physics