An equation for the biological transmembrane potential from basic biophysical principles

Biological membranes mediate different physiological processes necessary for life, many of which depend on ion movement. In turn, the difference between the electrical potentials around a biological membrane, called transmembrane potential, or membrane potential for short, is one of the key biophysical variables affecting ion movement. Most of the existing equations that describe the change in membrane potential are based on analogies with resistive-capacitive electrical circuits. These equivalent circuit models assume resistance and capacitance as measures of the permeable and the impermeable properties of the membrane, respectively. These models have increased our understanding of bioelectricity, and were particularly useful at times when the basic structure, biochemistry, and biophysics of biological membrane systems were not well known. However, the parts in the ohmic circuits from which equations are derived, are not quite like the biological elements present in the spaces around and within biological membranes. Using current, basic knowledge about the structure, biophysics, and biochemical properties of biological membrane systems, it is shown here that it is possible to derive a simple equation for the transmembrane potential. Of note, the resulting equation is not based on electrical circuit analogies. Nevertheless, the classical model for the membrane potential based on an equivalent RC-circuit is recovered as a particular case, thus providing a mathematical justification for the classical models. Examples are presented showing the effects of the voltage dependence of charge aggregation around the membrane, on the timing and shape of neuronal action potentials.