Nonlinear Electrical Effects in Lipid Bilayer Membranes. II. Integration of the Generalized Nernst-Planck Equations

In this paper the ion transport across a thin lipid membrane is treated using a generalized form of the Nernst-Planck equations. An additional term is introduced into the flux equations to account for the image force acting on the ion. As the membrane thickness is of the same order of magnitude as the range of the image forces, the potential energy of the ion in the membrane is strongly dependent on position. The integration of the flux equations leads to a general expression for the integral membrane conductance lambda as a function of the voltage u. The ratio lambda(u)/lambda(0) (lambda(0) = membrane conductance in the limit u --> 0) depends on the dielectric constant and the thickness of the membrane, but is independent of the ionic radius. When the numerical values of the potential energy function, as calculated by the method of electrical images, are inserted into the expression for lambda(u)/lambda(0), a strongly non-linear current-voltage characteristic is obtained. The theoretical current-voltage curve agrees satisfactorily with the experimental data at a low ionic strength and at low voltages; at higher voltages the observed membrane conductance exceeds the predicted value.