= Solution
<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.
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