Past exam of the biological physics course of the University of Oxford / 2019 / 2

Take $V=V_{\mathrm{inside}}-V_{\mathrm{outside}}$. Equality of electrochemical potentials for a monovalent cation gives the Nernst equilibrium potentialAt $T=293.15\mathrm{K}$, $RT/F=25.26\mathrm{mV}$. ThereforeThe signs refer to the stated electrode convention, not to the direction of current at every voltage. Activities are approximated by concentrations and the bulk ionic concentrations are treated as fixed during a single spike.

The intracellular recording by Alan Hodgkin and Andrew Huxley shows the membrane voltage rising from a negative resting value to a positive overshoot, with the outside held at zero reference. The action potential therefore does more than transiently abolish a resting polarization: it reverses the sign across the membrane. A simple hypothesis that excitation only makes an otherwise potassium-dominated membrane equally leaky cannot explain that positive overshoot.

The result helped motivate a transient, selective change in membrane permeability, later resolved into rapid inward sodium current followed by outward potassium current. The positive sodium equilibrium potential calculated above makes the overshoot physically plausible, while potassium conductance favours a negative resting voltage and repolarization. The 1939 voltage trace itself did not identify individual channel proteins or establish the full later voltage-dependent gating mechanism; those conclusions required ionic substitution and voltage-clamp evidence.^[ref]

Panel B steps the membrane from approximately $-80$ to $-40\mathrm{mV}$ and holds it there. The sodium current becomes inward, peaks rapidly, and decays almost to zero despite maintained depolarization. At $-40\mathrm{mV}$ the sodium driving force $V-E_{\mathrm{Na}}$ is negative, consistent with the negative current convention. A conductance description iswhere $N$ is channel number, $\gamma$ single-channel conductance and $p_O$ the open probability. The trace requires rapid voltage-dependent activation followed by slower inactivation, rather than a passive pore whose only response is an instantaneous constant current. Recovery normally requires repolarization, helping produce a refractory period and directional propagation of excitation.

A typical eukaryotic voltage-gated sodium channel contains four homologous membrane-spanning domains in one large pore-forming protein. Charged S4 helices sense voltage; pore-associated regions form a selective pathway and activation gate. Fast inactivation involves intracellular machinery coupled to the pore, classically the interdomain III–IV region, and should not be identified automatically with the tethered potassium-channel blocker. The pore permits rapid downhill electrochemical transport while maintaining selectivity against other ions.

Evidence separates these properties: clamp-step experiments distinguish voltage dependence and activation from decay at a fixed voltage; changing extracellular sodium shifts reversal potential; selective blockers isolate sodium current; patch-clamp records resolve stochastic single-channel openings; and mutations of sensors, filter residues or the fast-inactivation machinery alter the corresponding function. A single macroscopic trace alone cannot uniquely determine all microscopic rate constants.

Add an unavailable nonconducting inactivated state to the two-state gating scheme, for exampleThe probabilities obeyDepolarization increases activation, initially raising $p_O$; subsequent transfer to $I$ reduces current while voltage is held constant. Slow recovery at the depolarized voltage permits most channels to become unavailable. Repolarization promotes recovery and deactivation. More closed states and closed-state inactivation may be needed for quantitative data, but this minimal extension already explains the rise followed by decay that fixed-rate $C\leftrightarrow O$ gating cannot produce from a closed population.

A sodium-channel gain-of-function mechanism in Long QT syndrome can reduce the inactivation rate $k_i$ or increase recovery/reopening from inactivated states, leaving a small persistent open population. The resulting late inward sodium current opposes repolarizing outward currents while $V<E_{\mathrm{Na}}$, prolonging the cardiac action potential. Slower entry into $I$ would also extend the typical conducting episode relative to its normal duration. The altered effect is prolonged sodium entry, not reversal of sodium's charge or a pump-driven uphill flux through the channel.

The precise rate change depends on the mutation: the clinical label alone does not uniquely specify one transition constant. Long QT syndromes also include other channel mechanisms; the question here concerns its stated sodium-channel variant.^[ref]

The original Panel A is readable: the baseline is roughly $-45\mathrm{mV}$ and the peak about $+40\mathrm{mV}$, giving a voltage excursion near $85\mathrm{mV}$. Using the specific membrane capacitance and surface area,A coarser $100\mathrm{mV}$ spike estimate gives $6.2\times 10^6$ ions, illustrating the precision appropriate to the graph. These are the net monovalent charges needed to charge the capacitor if sodium supplies the inward charge; simultaneous opposing currents can make the actual sodium influx larger.

The cell contains $0.015\mathrm{mol}{\mathrm{L}}^{-1}\times 3\times 10^{-13}\mathrm{L}\times N_A\simeq 2.71\times 10^9$ intracellular sodium ions. The capacitive estimate adds only about $0.20\%$, or $0.029\mathrm{mM}$. Thus a spike does not equilibrate bulk concentrations. The sodium-potassium ATPase maintains the gradients over many spikes, exporting three sodium ions per two potassium ions imported per ATP; the estimated sodium load alone corresponds to at least about $1.8\times 10^6$ pump cycles. Rapid repolarization itself comes mainly from changing channel conductances, while sustained fast firing requires adequate energy supply and gradient restoration. The small stated volume and large area are used as supplied, without assuming a particular cell geometry.