Past exam of the biological physics course of the University of Oxford / 2016 / 1 / c

Write the instantaneous electrical conductance as $g_K(t)={\bar{g}}_{K}n(t{)}^4$. Here ${\bar{g}}_K$ is the maximal electrical conductance of the recorded membrane, $n$ is the activation variable, $V_m$ is inside-minus-outside membrane potential, and $V_K$ is the potassium Nernst equilibrium potential. The factor $V_m-V_K$ is the electrochemical driving voltage; positive $I_K$ conventionally denotes outward positive charge. For an electric current density, ${\bar{g}}_K$ must instead be an electrical conductance per area.

In the Hodgkin-Huxley model, four equivalent independent activation gates must all be permissive, each with probability $n$, so their joint opening probability is $n^4$. At fixed voltage each gate obeysThe model assumes approximately Ohmic open-channel conduction, a well-defined population-average electrical conductance, and an almost constant potassium reversal potential during a brief recording. Four phenomenological gates should not be mistaken for a proof of four physically independent molecular transitions. The simple delayed-rectifier description also omits ion-channel inactivation.

Reducing the available electrical conductance can weaken repolarization, broaden an action potential and alter repetitive firing. Depending on channel class and tissue, excess excitability or impaired recovery can follow; persistent depolarization can also reduce voltage-gated sodium channel availability. Cardiac potassium-current reductions can prolong the action potential and contribute to Long QT syndrome. These are conditional physiological consequences, not a claim that every potassium-current reduction has the same clinical effect.