Past exam of the biological physics course of the University of Oxford / 2022 / 3

In the continuum cylindrical-pore estimate, the electrical resistance is $R_p=\rho \ell /(\pi a^2)$, with radius $a=0.075\mathrm{nm}$, length $\ell =0.5\mathrm{nm}$ and electrical resistivity $\rho =80\Omega \mathrm{cm}=0.80\Omega \mathrm{m}$. ThereforeUnrestricted approach from the reservoirs removes access barriers, but still permits a continuum spreading or access resistance of a circular pore. Including $\rho /(4a)$ on each side givesThus the usual estimate neglecting the access contribution is about $44\mathrm{pS}$, or about $36\mathrm{pS}$ with spreading resistance. A $1.5$-ångström diameter is at the molecular scale, smaller than hydrated ions; bulk electrical resistivity cannot quantitatively describe dehydration, selectivity or discrete-ion transport. These values are the requested continuum estimates, not evidence that such a pore freely passes hydrated potassium ions.

The rising sigmoid shows activation by depolarization: $P_o$ is almost zero at very negative membrane potential, rises steeply around $-20\mathrm{mV}$, and approaches about $0.8$ at positive membrane potential. The graph describes the fraction of time a single ion channel is open, not a continuously variable opening amplitude. A convenient phenomenological fit isHere $V_{1/2}$ refers to half the maximum, about $0.4$, not absolute $P_o=0.5$. The effective gating charge describes the steepness; a sigmoid alone does not identify the microscopic gating sequence or explain why the observed maximum is below one.

A voltage-gated potassium channel has four pore-forming subunits. Each canonical subunit contains an S1–S4 voltage-sensing domain and S5–S6 pore elements. Positive charges in S4 respond to the electric field; their displacement during depolarization changes the energetic preference of sensor conformations. Coupling through the sensor–pore interface favors opening of the intracellular pore gate. The potassium selectivity filter is structurally distinct from this voltage sensor.

Several independent observations support this account. Voltage clamp detects transient gating electric currents before pore conduction, even when ionic conduction is blocked: charge movement precedes opening. Charge-neutralization studies change total gating charge and voltage dependence, with contributions from charged S4 residues and an S2 residue. Accessibility and voltage-sensitive optical measurements support conformational motion rather than permeating potassium itself acting as the sensor. Structural measurements locate the charged sensor beside the pore. These observations constrain a coupled moving-sensor mechanism, while the exact displacement cannot be deduced from this $P_o$ curve alone.^[ref][ref]

An open ion channel carries the electric currentThe macroscopic mean is $I=N{P}_{o}i$, so $I/i\simeq 204$ is the mean number open, not the total number present. Reading $P_o(-20\mathrm{mV})\simeq 0.40$ from the original graph givesThe graph reading is approximate, so a precision of hundreds is appropriate. Only a few hundred active ion channels generate a nanoampere-scale cellular electric current; molecular discreteness and stochastic gating therefore matter. For independent identical ion channels, the open-count standard deviation is $\sqrt{N{P}_o(1-P_o)}\simeq 11$, corresponding to a coefficient of variation in electric current of about $5.4\%$. This is an instantaneous ideal independent-gating estimate, before temporal filtering and other noise sources, and requires the single-channel recording $P_o$ to apply to the cellular recording conditions.

In ball-and-chain inactivation, also called N-type inactivation, a flexible intracellular N-terminal region carries an inactivation particle. Once a voltage-gated potassium channel opens, this tethered particle can enter and occlude the inner pore, stopping electric current even while the activation machinery favors an open configuration. Recovery requires the particle to leave, after which the activation gate can return to its resting conformation. This supplies an additional unavailable state, not merely reversal of activation.

The decisive evidence separates necessity from sufficiency. Removing the relevant N-terminal region greatly reduces fast ion-channel inactivation while preserving activation. Applying a peptide corresponding to the missing N-terminal region on the cytoplasmic side restores blocking with similar properties. Changes in the blocking region alter blocking kinetics, and altering the tether affects the rate at which the particle reaches the pore. Open-pore access and competition with intracellular pore blockers support occlusion of the inner cavity. A tether makes the particle's local effective concentration high, explaining rapid intramolecular block.^[ref][ref]

This mechanism is distinct from slower filter-associated C-type inactivation. An N-terminal particle is a mechanism for certain potassium ion channels, not a universal explanation of all ion-channel inactivation.

In a classical action potential, delayed activation of voltage-gated potassium channels supplies outward electric current that drives repolarization. In the Hodgkin-Huxley model, for example, $I_K={\bar{g}}_K{n}^4(V-E_K)$ and $\dot{n}={\alpha }_n(V)(1-n)-{\beta }_n(V)n$. Increasing the opening rate at depolarized membrane potential, other properties fixed, recruits outward electric current earlier: the action potential generally becomes narrower and may have a smaller peak. Slower opening delays repolarization and broadens the action potential.

Slower closing after repolarization retains excess potassium electrical conductance, lengthening the after-hyperpolarization and often the interval of reduced excitability. Faster closing shortens this tail and permits earlier recovery. If activation or the available electrical conductance is insufficient, normal repolarization may fail; changing rates can also change the steady open fraction. Duration is therefore determined by the whole balance of inward and outward electric currents, not by a closing rate in isolation.