Past exam of the biological physics course of the University of Oxford / 2015 / 7 / b

The lipid cell membrane separates ionic solutions and has selective ion channels, ion pumps, and a finite capacitance. Ionic gradients and unequal permeability allow a small charge separation to generate a membrane potential. The bulk compartments can remain almost electrically neutral even when the narrow interfacial layers carry opposite electric charges.

For one permeant ionic species with valence $z$, its electrochemical potential is $\tilde{\mu }={\mu }^0+k_{B}T\log a+ze\varphi$, where $a$ is its activity. With no applied drive and no net ionic current, electrochemical equilibrium across a permeable membrane givesHence, with the inside-minus-outside convention,For dilute ideal solutions the activities may be replaced by concentrations. At $25^{\circ }\mathrm{C}$ this is $(59.16\mathrm{mV}/z){\log }_{10}(c_{\mathrm{out}}/c_{\mathrm{in}})$. The resulting Nernst equilibrium potential balances diffusive and electrical driving, not necessarily equal concentrations. Physically there must be countercharge, such as impermeant counterions; the single-species idealization concerns the permeant current. With multiple permeant species, zero total current need not mean zero current of each species, and the resting membrane potential is not generally any single species' Nernst equilibrium potential.