= Solution
<ATP synthase> couples downhill proton motion across a <cell membrane> to synthesis of <ATP> from <ADP> and inorganic phosphate. Its membrane-embedded $F_o$ sector contains an $a$ subunit with offset proton-access half-channels and a rotating $c$-subunit ring. Reversible protonation of sites on the ring permits rotation relative to the stationary $a$ subunit. The central rotor transmits this motion through the $\gamma$ shaft to the soluble $F_1$ head. A peripheral stator holds the head relative to the membrane sector. The common bacterial schematic has an $\alpha_3\beta_3$ head, catalytic sites mainly on the three $\beta$ subunits, and a central $\gamma/\epsilon$ rotor; accessory stator components depend on the organism.
The <binding-change mechanism> cycles the catalytic sites between open, loose-binding and tight-binding conformations. Rotor orientation changes their affinities for <ADP>, phosphate and <ATP>; the sites operate out of phase. Tight binding favors formation of bound <ATP>, and mechanical rotation changes the conformation to permit product release. In the ideal tightly coupled cycle, one revolution gives three catalytic turnovers:
$$
\boxed{3\ \text{ATP per }360^\circ,\qquad 1\ \text{ATP per }120^\circ.}
$$
The proton count is not universally fixed: an ideal ring with $n_c$ proton-carrying subunits gives $n_c$ protons per revolution and $n_c/3$ protons per synthesized <ATP>, before additional transport costs.
For a proton moving from side 1 to side 2, its molar <electrochemical potential> change is
$$
\Delta\mu_{\mathrm H^+}=RT\log\frac{a_{\mathrm H^+,2}}{a_{\mathrm H^+,1}}+F(\psi_2-\psi_1).
$$
The chemical gradient and membrane <electric potential difference> together supply the <proton-motive force>. Synthesis requires the proton free-energy release to meet the positive <Gibbs free energy> of <ATP> synthesis. The coupling is reversible: isolated $F_1$ supplied with <ATP> can hydrolyse it and rotate its shaft, while the intact enzyme can use hydrolysis-driven rotation to pump protons against their gradient. The sketch is structural and schematic; it does not reconstruct the absent stepping trace.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012-s12-atp-synthase-and-dwell-distributions.png]
{title=ATP synthase architecture and the two derived dwell-time distributions}
{description=The structural panel shows a membrane rotor, shaft, stator and three catalytic sites. The distribution panel uses explicitly illustrative dimensionless rates, not values read from the absent source stepping plot.}
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