Current-voltage curves of porous membranes in the presence of pore-blocking ions. I. Narrow pores containing no more than one moving ion.

We propose a physical model for voltage-dependent conductance changes of excitable cell membranes. It is based on competition of uni- and bivalent ions for chains of stable sites extending through the membrane. These one-dimensional pathways (pores) have different profiles of chemical potential for the two ionic species so that bivalent ions can block the passage of univalent ions at large membrane potentials. We treat the special case that each pore is either empty or, because of electrostatic repulsion, contains no more than one uni- or bivalent ion at a time. A system of linear differential equations describes the time-dependent probabilities of the various possible pore states. The states are limited by transition rate constants involving the profile of the chemical potential, the membrane voltage, the ionic concentrations in the adjacent baths, and electrostatic interactions between the ions. The steady-state solutions (Kirchhoff-Hill theorem) yield expressions for the relationship between the small signal conductance of univalent ions and the concentration of these ions in the external bathing medium (a saturation curve) and for the ionic currents and the steady-state current-voltage curve (N-shaped). From the latter curve we compute the shift of theshold potential caused by concentration changes of the external bathing medium. The model yields a number of predictions which can be tested experimentally.