A General Relation Between Membrane Potential, Ion Activities, and Pump Fluxes for Symmetric Cells in a Steady State

For steady states of cells that have uniform cell membranes, there must be zero net fluxes of each ion. This provides a set of conditions more restrictive than the condition of zero net current flow that is used to derive the Goldman equation. We show that this leads to a set of general relations between the membrane potential, ion activities, ion permeabihties, and pump fluxes and that the Goldman equation is but one of this set. Further, we demonstrate that the transmembrane potential is uniquely defined by the intracellular and extracellular activities, the permeability coefficients, and the ratio of the net mediated (nondiffusional) fluxes of any two ions of the same valence in a steady state, regardless of the behavior of other ions and without assumptions with respect to the electrical potential profile across the membrane. In the course of this, we also give an exact and general derivation of the Mullins-Noda relation. INTRODUCTION The Goldman equation as originally derived [ 1,2] depended on the assumptions of a constant field across the membrane, zero net current flow across the membrane, and zero net current due to pumps. Over the years efforts to generalize the Goldman equation have been directed to relaxing the constant field assumption [3-71 and to incorporating the effects of electrogenic pumps [6-121. General derivations that do both are now available [6,7]. On reviewing the applicability of the Goldman equation, one of us (S.G.S.) noted that for a steady state distribution of ions in symmetric cells Q American Elsevier Publishing Company, Inc., 1974 20 J. JACQUEZ AND S. SCHULTZ that have uniform membrane properties, many more restrictions on fluxes hold than are required for the derivation of the generalized Goldman equation [6,7]. This raised the possibility that relations other than the Goldman equation connecting ion fluxes and the membrane potential might be found. The basic assumption used to derive the generalized Goldman equation is that there is a zero net rate of charge transfer across the membrane. This is given by in which zk, J,“, and Jkp are the valence, net diffusive, and net mediated fluxes of the kth ion. We use the convention that fluxes are positive when directed into the cell. We also note that the derivation of the generalized Goldman equation depends on the assumption that only univalent ions are present. Equation (1) introduces no assumption of a steady state for any ion; all it does is to impose a particular restriction on total current flow. However, for a symmetric cell that has uniform membrane properties, the steady state condition implies there is a zero net flux for each ion, as given by Eq. (2) which is much more restrictive than Eq. (1).