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

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]