Past exam of the biological physics course of the University of Oxford / 2015 / 7

ATP synthase couples a membrane-embedded rotary motor to a soluble catalytic head. Its membrane sector contains a rotating ring of c subunits and a stationary a subunit providing offset proton-access pathways. Its catalytic sector contains an ${\alpha }_3{\beta }_3$ head with catalytic sites primarily on the $\beta$ subunits, a central rotating shaft, and a peripheral stator that prevents the head simply co-rotating.

A proton-motive force, combining a membrane potential and a proton chemical potential difference, drives directional proton passage and ring rotation. Rotation of the central shaft changes the conformations and binding affinities of successive catalytic sites, allowing ADP and inorganic phosphate to bind, form ATP, and release it in a coordinated cycle. The ATP synthase rotary mechanism therefore couples ion transport, torque, and chemical work rather than requiring a proton to react directly at every catalytic site.

The motor is reversible: sufficiently favorable ATP hydrolysis can drive proton pumping against its electrochemical potential gradient. In mitochondria, the respiratory chain establishes the gradient used for oxidative ATP production; analogous coupling operates in bacterial membranes and in chloroplast photophosphorylation. The number of protons per revolution depends on the c-ring stoichiometry, whereas a full catalytic rotation normally produces three ATP molecules. It is therefore inappropriate to assign one universal integer proton-to-ATP ratio to all organisms. Transport and leakage can increase the whole-cell energetic cost beyond the motor's ideal ratio.

The lipid cell membrane separates ionic solutions and has selective ion channels, ion pumps, and a finite capacitance. Ionic gradients and unequal permeability allow a small charge separation to generate a membrane potential. The bulk compartments can remain almost electrically neutral even when the narrow interfacial layers carry opposite electric charges.

For one permeant ionic species with valence $z$, its electrochemical potential is $\tilde{\mu }={\mu }^0+k_{B}T\log a+ze\varphi$, where $a$ is its activity. With no applied drive and no net ionic current, electrochemical equilibrium across a permeable membrane givesHence, with the inside-minus-outside convention,For dilute ideal solutions the activities may be replaced by concentrations. At $25^{\circ }\mathrm{C}$ this is $(59.16\mathrm{mV}/z){\log }_{10}(c_{\mathrm{out}}/c_{\mathrm{in}})$. The resulting Nernst equilibrium potential balances diffusive and electrical driving, not necessarily equal concentrations. Physically there must be countercharge, such as impermeant counterions; the single-species idealization concerns the permeant current. With multiple permeant species, zero total current need not mean zero current of each species, and the resting membrane potential is not generally any single species' Nernst equilibrium potential.

Including both flat ends of the cylindrical cell, its membrane surface area isThe capacitance is $C=(0.01\mathrm{F}{\mathrm{m}}^{-2})A=6.34602\mathrm{pF}$. The electric charge needed for the specified depolarization is thereforeIgnoring the ends gives $3.33\times 10^6$, only about one percent smaller. Inward movement of positive ions, or outward movement of negative ions, produces the stated positive change in the inside-minus-outside membrane potential.

The cell volume is $\pi r^2\ell =3.14159\times 10^{-16}{\mathrm{m}}^3$. Distributed over this volume, the transferred number would correspond to only $0.0177955\mathrm{mM}$. Relative to an illustrative, not source-specified, $100\mathrm{mM}$ internal ionic pool, that is a fraction $1.78\times 10^{-4}$. Thus a substantial change in membrane potential requires movement of only a tiny fraction of bulk ions; it does not require wholesale depletion of the intracellular solution. The separated electric charge is concentrated near the two membrane surfaces.