= Solution
A major source of <ATP> is <chemiosmosis>: an <electron transport chain> uses downhill electron transfer to establish a <proton-motive force>, and <ATP synthase> converts that <electrochemical gradient> into <ATP> formation. In aerobic <oxidative phosphorylation>, reduced carriers supply electrons, ultimately reducing oxygen to <water>; selected respiratory complexes couple the favorable <oxidation-reduction reaction> to uphill <proton> translocation. In a <mitochondrion>, this takes place across the inner membrane, creating a relatively positive, acidic exterior side and a relatively negative, alkaline matrix. <Bacteria> use their <cell membrane>. In photosynthetic systems, light-driven electron transfer can supply the gradient instead. <Substrate-level phosphorylation> is another <ATP>-producing route, so the description is not universal to every organism and condition.
The membrane $F_O$ region of <ATP synthase> contains a rotating ring of c-subunits next to subunit a. Two offset aqueous half-channels expose the ring's proton-binding sites to opposite membrane sides. <Proton> binding and release bias ring rotation down the <electrochemical gradient>. The central shaft transfers <torque> to the $F_1$ catalytic head, while a peripheral stator prevents the head from simply co-rotating. This is the <ATP synthase rotary mechanism>, rather than unrestricted <proton> leakage through a pore.
The three catalytic beta subunits cycle through different binding states as the shaft turns: <ADP> and <inorganic phosphate> bind, bound <ATP> is favored, then <ATP> is released. The binding-change mechanism uses mechanical <work> to alter binding affinities and product release as well as the chemical step. A full coupled revolution produces three <ATP> molecules; a ring with $n_c$ c-subunits transfers approximately $n_c$ <protons> per revolution in the tightly coupled model, or $n_c/3$ per <ATP>.<Structure of a bacterial ATP synthase by Hui Guo and others>[Proton-translocation half-channels and rotor architecture]{ref}<Rotary catalysis of FoF1-ATP synthase by Rikiya Watanabe>[Rotary coupling and c-ring stoichiometry]{ref}
The overall cycle must have a favorable <Gibbs free energy> change. If the <proton-motive force> becomes too small, synthesis stops; under suitable conditions the same <molecular motor> can reverse, using <ATP hydrolysis> to pump <protons> uphill. A catalyst changes kinetics, not the thermodynamic condition for synthesis.
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