= ATP synthase rotary mechanism
{c}
The membrane motor $F_0$ contains a rotating ring of $c$ subunits beside stationary subunit $a$. Two offset water-accessible half-channels connect opposite membrane sides to the ring. A <proton> enters one half-channel and protonates a <glutamic acid residue> on a $c$ subunit; neutralization permits that site to enter the <hydrophobic> <lipid bilayer>. Rotation carries it around to the other half-channel, where it releases the <proton>. A conserved <arginine> in subunit $a$ helps separate the routes and promotes release. Downhill <proton> transfer biases rotation; this is not a freely open transmembrane pore.<Structure of a bacterial ATP synthase by Hui Guo and others>[Guo, Suzuki and Rubinstein, Structure of a bacterial ATP synthase, 2019, Figure 4B–D and Proton translocation through the FO region]{ref}
The central shaft transmits <torque> to $F_1$; a peripheral stalk prevents the catalytic head from simply co-rotating. Its three $\beta$ catalytic sites, within an $\alpha_3\beta_3$ head, cycle through different binding shapes. In the simplified binding-change picture, a loose site captures <ADP> and <inorganic phosphate>, a tight site favours bound <ATP>, and opening releases <ATP>. Rotation changes binding and release as well as chemistry.<Structure of ATP synthase under strain during catalysis by Hui Guo and John L. Rubinstein>[Guo and Rubinstein, Structure of ATP synthase under strain during catalysis, 2022, Introduction and Figure 1: binding-change cycle]{ref}
\Image[/atp-synthase-rotary-mechanism.png]
{title=<ATP synthase>: <proton> transfer, shaft rotation and three catalytic sites}
{description=Original functional schematic, not an atomic structure. The side view separates rotor and stationary support; the inset separates the two half-channels. Ten $c$ subunits are a chosen example.}
One complete, tightly coupled turn cycles each catalytic site once, producing three <ATP> <molecules>. If the ring has $n$ <proton>-binding subunits, the ideal motor budget is $n$ transferred <protons> per turn, or $n/3$ per <ATP>. Ring size varies between organisms; the worked example chooses $n=10$, as in the studied bacterial systems, rather than assigning ten to every <ATP synthase>.<Rotary catalysis of FoF1-ATP synthase by Rikiya Watanabe>[Watanabe, 2013, pp. 51–52: c-ring stoichiometry and H+/ATP ratio]{ref}
For this example the ring's repeat angle is $360^\circ/10=36^\circ$, whereas the three-site catalytic cycle repeats after $120^\circ$. Their ratio is $10/3$, not an integer. Counting complete turns avoids inventing a fixed integer <proton> count for every catalytic event. The schematic does not require both parts to advance simultaneously as rigid gears; <elasticity> allows temporary twisting while their cycles proceed.<Structure of ATP synthase under strain during catalysis by Hui Guo and John L. Rubinstein>[Guo and Rubinstein, 2022, Introduction and Figure 1: symmetry mismatch and elastic coupling]{ref}
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