Electric Potential and Concentration of Ion Species in the Proximity of a Cell Membrane:ab initioCalculations

Abstract

The mathematical model of ion transport in the field of electric forces and diffusion gradients is based on particle conservation equations and the Poisson equation which governs the distribution of the electric potential. Designed for evaluating the currents through passive ion channels in cellular membranes, the model is formulated in a two-dimensional approximation assuming rotational symmetry. Three ionic species are considered in the present study: sodium, calcium and chlorine. The program developed for the numerical solution is based on a semi-analytical approximation suggested by Gummel & Scharfetter. The present contribution reports calculations of the steady-state distributions of ionic species induced by an electric field due to a negative potential drop across the cellular membrane, which represent initial conditions for future calculations of the time development of the ion flux through the calcium channel. The numerical calculations were performed from the moment of potential application up to the time 6.2 μs, when a new steady state is established. At this stage, the maximum values of the sodium and calcium concentrations at the membrane surface are about 22% and 47% above the initial values of 145 mM and 1 mM, respectively, while the chlorine concentration is approximately 18% below. The electric field at the membrane surface is about 6.7×104and decreases exponentially with distance. The length of the boundary layer is about 40 Å.Since the model is based on fundamental principles, it can be used for the quantitative solution of any problem possessing rotational symmetry where a continuum approach is applicable and some elementary conditions are fulfilled, such as equilibrium (or at least quasi-equilibrium) of the particle energy distributions with respect to time and electric field, and the assumption that the transport coefficients are defined as functions of the field and their numerical values are known.