Past exam of the biological physics course of the University of Oxford / 2016 / 3

Use signed ion valence $z$ and define $V={\psi }_1-{\psi }_2$. The molar electrochemical potential in side $i$ of an ideal dilute solution isTransfer of an infinitesimal amount $dn$ from side 1 to side 2 changes the total Gibbs free energy by $({\tilde{\mu }}_2-{\tilde{\mu }}_1)dn$. At equilibrium this derivative is zero, soFor a cell with side 1 inside, this is $E_i=(RT/z_{i}F)\ln (c_{i,\mathrm{out}}/c_{i,\mathrm{in}})$. The concentrations are equilibrium free-ion concentrations, not automatically their exact initial values. For nonideal solutions the ratio must be a ratio of thermodynamic activities. The temperature and solvent must be common to both reservoirs, and an ion must actually be able to exchange for this equilibrium condition to be dynamically established.

Equal compartment volumes are unnecessary for deriving the voltage law, but help specify number conservation if concentration changes are appreciable. For equal volume $\mathcal{V}$ and a monovalent cation transfer $n$ from side 1 to 2, $c_1=c_1^0-n/\mathcal{V}$, $c_2=c_2^0+n/\mathcal{V}$. Combined with the membrane capacitance, this gives an implicit finite-reservoir equilibrium rather than treating both concentrations as externally clamped.

Suppose initially $c_1>c_2$. Positive ions initially move by diffusion from side 1 to side 2. Since negative counterions cannot cross, side 1 acquires a negative interfacial charge and side 2 a positive one. The resulting electrical force opposes further positive-ion transfer. The two contributions balance when the electrochemical potentials are equal; individual ions still cross in both directions, but there is no net electric current of the permeant chemical species.

A simple capacitor model makes the feedback quantitative. If $n$ moles of monovalent positive ions move from initially neutral side 1 to side 2, the charge on side 1 is $-Fn$, hence $V=-Fn/C$. The free-energy stationarity condition isAs $n$ increases, both the concentration imbalance and the voltage drive toward reversal diminish the original diffusion tendency. The equilibrium lies before the bulk concentrations become equal. This demonstrates why a nonzero Nernst potential can coexist with very little change in bulk composition and approximate bulk electroneutrality. A concentration difference alone does not maintain an electrical voltage if both counterions can freely follow each other; selective permeability is essential to this model.

At $T=310.15$ K, $RT/F=26.72666$ mV. Always retain the valence sign. For the two monovalent positive ions,For calcium, $z=2$ and the specified voltage is positive:For chloride, $z=-1$ reverses the sign in the Nernst equation:These values use thermodynamic activities approximated by the given concentrations. Calcium is the free concentration relevant to its electrochemical potential, rather than total bound plus free intracellular calcium. Ratios use matching concentration units, so a ratio of millimolar values is already dimensionless. Failure to include the factor of two for calcium or the negative charge of chloride would change the answer substantially.

At a common membrane voltage, an ion is in electrochemical equilibrium only if its own Nernst potential equals that voltage. The four independently calculated equilibrium voltages areThus at $V_m=-90$ mV only chloride is at its listed equilibrium voltage. Potassium has an outward positive-current driving voltage $V_m-E_K=+7.74$ mV. Sodium and calcium have negative driving voltages, $-156.60$ and $-220$ mV respectively, favoring inward positive-ion movement when their channels are open. Chloride has zero net electrochemical drive under these ideal conditions.

A resting potential is not a claim that every permeant chemical species is in thermodynamic equilibrium. A membrane with several electrical conductances can have zero total electric current even though individual ionic electric currents flow and cancel. Pumps maintain the concentration differences by consuming Gibbs free energy. Conversely, a closed channel can prevent substantial ion flux even when the ion's electrochemical potential difference is nonzero: absence of flux due to impermeability is not the same as electrochemical equilibrium. The listed concentrations could not all be passive equilibrium distributions at one voltage because their Nernst potentials differ.