Self-Consistent Calculations of the Current and Access Resistance in Open Ion Channels

Abstract

The problem of calculating the current and access resistance in open ion channels is considered. A self-consistent analytic solution is introduced for an arbitrary number of species within the Poisson-Nernst-Planck (PNP) equations formalism. The model considered is a cylindrical channel of radius a in the protein which allows ions to cross a membrane that is bathed by two solutions of different concentration on its left and right-hand sides. Electro-diffusion in this system is described by the Poisson equation combined with the continuity equations for the mobile ions. The PNP equations are solved in the bulk in the Boltzmann approximation in 3D, assuming spherical symmetry, and in the pore in a 1D approximation. The boundary conditions (BCs) for the potential and concentration are set at infinity. The internal BCs for the current and the gradient of the potential are set at the surfaces of two hemispheres of radius a. The two solutions are matched together at the internal BCs using an iterative procedure in a self-consistent way. The method allows for calculation of the currents for an arbitrary number of ions species that have different diffusion constants in the channel and in the bulk. The sizes of the ions are taken into account by introducing a “filling factor” as an additional fitting parameter. The method is applied to model experimental I-V characteristics of the Gramicidin A channel for various concentrations, yielding qualitative good agreement.