The effects of space charge on the ionic currents through biological membranes

In this paper, the ions moving through an ion-selective membrane are regarded as a space charge. The repulsive action among the ions attenuates the ion current through the membrane. For a quantitative description of this effect on the relation between the ion current, the ion concentrations and the membrane voltage, the usual assumption of a constant electric field within the membrane is replaced by that of an inconstant field influenced by the moving ions. This can be done either by inserting the Poisson equation into the Nernst-Planck equation or by directly calculating the Coulomb forces, e.g. between two ions moving through a narrow channel. An attenuation factor is defined and its dependence on membrane thickness, dielectric number, membrane voltage, and ion concentrations is exactly calculated; its dependence on the cross section of an assumed channel is necessary for the current through the channel to be appreciably attenuated by saturation effects. The saturation effect described differs from the Michaelis-Menten kinetics in that no absolute maximum effect is attained.