Exact continuum solution for a channel that can be occupied by two ions

The classical Nernst-Planck continuum equation is extended to the case where the channel can be occupied simultaneously by two ions. A two-dimensional partial differential equation is derived to describe the steady-state channel. This differential equation is of the form of the generalized Laplace equation, but it has the novel feature that the boundary conditions are periodic. The finite difference solution takes approximately 8 s on a large computer. The equations are solved for the special case of a cylindrical channel with a fixed charge in the center. It is assumed that the forces on the ions result entirely from the sum of the Born image potential, the fixed charge potential, the interaction potential between the two ions, and the applied voltage. Approximate simple analytical expressions are derived for these potential terms, based on the assumption that the electric field perpendicular to the channel wall is zero. The potentials include the contribution from a diffuse charge (Debye-Huckel) reaction field in the bulk solution for the monovalent cation flux was obtained for channels with a radius of 4 A and lengths of 16 and 32 A and a fixed charge valence of -1 and -1.5. For these channels, a significant fraction (up to 90%) of the total resistance is contributed by the bulk solution and results were obtained for the case where the "channel" included 8 A of bulk solution at each channel end. These results for the two-ion channel were compared with the analytical solution for a one-ion channel. The one-ion channel is a fair approximation to the two-ion channel for a fixed charge of -1, underestimating the flux at high concentrations by approximately 30%. However, for a fixed charge of -1.5, the one-ion model is a poor approximation, with the two-ion flux about seven times that of the one-ion model at high concentrations. The absolute conductance and concentration dependence of these channels (with a fixed charge of -1) mimic the behavior of the large conductance K+ channel and the acetylcholine receptor channel.