= Solution
<ATP synthase> has a membrane-embedded $F_o$ sector, a catalytic $F_1$ head and mechanical connections between them. The $F_o$ sector contains an ion-conducting subunit $a$ adjoining a rotating ring of $c$ subunits. A central shaft including the $\gamma$ subunit connects that ring to $F_1$. A peripheral stator holds the head fixed relative to the membrane. The $F_1$ head has three catalytic $\beta$ subunits alternating with three $\alpha$ subunits, surrounding the asymmetric central shaft; the middle panel of <past-exam-of-the-short-options-paper-shared-by-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2019/s12/2/c/solution> labels these relationships.
A transmembrane <proton-motive force> drives proton movement through offset half-channels and protonation changes around the ring, producing <torque>. Shaft rotation changes the conformations of the three catalytic sites, cycling their affinities for <ADP>, inorganic phosphate and <ATP>. In the <binding-change mechanism>, mechanical work particularly promotes release of tightly bound <ATP>; the threefold head couples a full revolution to three catalytic turnovers. The number of protons per revolution depends on the ring stoichiometry. Reversing the available <free energy> can reverse the motor: isolated $F_1$ hydrolyses <ATP> and rotates, and the coupled enzyme can pump protons. This structural separation of rotor, stator and catalytic head converts electrochemical <free energy> into chemical <free energy>.<Energy, Life, and ATP by Paul D. Boyer>[Nobel lecture, asymmetric catalytic sites and coupling of rotation to nucleotide binding changes]{ref}
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