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 Incoming links: Past exam of the biological physics course of the University of Oxford / 2018 / 2 / c / Solution

Past exam of the biological physics course of the University of Oxford / 2018 / 3 / a / Solution 2026-10-25

Represent one permissive voltage-gated potassium channel gate by an available closed state and an activated state, with voltage-dependent transition rates

α_{n} (V)

and

β_{n} (V)

. Gain minus loss gives

\frac{d n}{d t} = α_{n} (1 - n) - β_{n} n = (α_{n} + β_{n}) (n_{\infty} - n), n_{\infty} = \frac{α _{n}}{α _{n} + β _{n}}, τ_{n} = \frac{1}{α _{n} + β _{n}} .

(1)

During a voltage clamp step the imposed voltage and hence these rates are constant. Integration with

n (0) = n_{0}

gives

n (t) = n_{\infty} - (n_{\infty} - n_{0}) e^{- t / τ_{n}} .

(2)

The fourth power in the electrical conductance describes four permissive activation variables in the Hodgkin-Huxley model phenomenology. If all must be permissive and are statistically independent, their joint probability is

n^{4}

. Even a simple exponential rise in

n

then gives a delayed-looking macroscopic potassium electric current. The second panel of the model figure in solution illustrates this. A continuously changing voltage makes the rates time dependent and requires solving the differential equation along that voltage trajectory; one fixed exponential is not the general action-potential solution.

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Past exam of the biological physics course of the University of Oxford / 2018 / 3 / d / Solution 2026-10-25

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The equivalent circuit has membrane capacitance in parallel with electrical conductance–battery branches: voltage-dependent sodium and potassium conductances in series with their reversal potentials, a chloride electrical conductance in series with

E_{Cl}

, and other leak conductances. For externally supplied electric current,

C_{m} \dot{V} = I_{ext} - g_{N a} (V, t) (V - E_{N a}) - g_{K} (V, t) (V - E_{K}) - g_{Cl} (V - E_{Cl}) - g_{L} (V - E_{L}) .

(1)

The model figure in solution includes this circuit and an illustrative passive response. Although chloride electric current is small when

V ≃ E_{Cl}

, a perturbation produces the restoring electric current

δ I_{Cl} = g_{Cl} δ V

. A large chloride electrical conductance therefore shunts perturbations, lowers input resistance and shortens the passive time constant,

R_{in} ≃ 1/ \sum g_{j}

and

τ_{m} ≃ C_{m} / \sum g_{j}

Reducing that electrical conductance increases the voltage response to a given electric current or extracellular-potassium disturbance and weakens stabilization near rest, making repeated excitation easier. There need not be an instantaneous shift of resting voltage if

E_{Cl}

equals the old rest exactly. In skeletal muscle, loss of ClC-1-mediated stabilization is associated with myotonic hyperexcitability.^[ref]

Later cessation of contraction can be explained by sustained depolarization causing voltage-gated sodium channel ion-channel inactivation and failure to recover: fewer available channels support progressively smaller or less frequent spikes until depolarization block occurs. Accumulated extracellular potassium in restricted spaces and incomplete recovery between spikes can promote that depolarization. This requires the nonlinear voltage-gating dynamics; a passive chloride resistor and capacitor alone cannot generate or terminate an action-potential train. The stated membrane permeability change does not uniquely determine the time to block or prove one particular ionic accumulation history.

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