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ATP synthase is a molecular motor that couples downhill proton movement across a cell membrane to formation of ATP from ADP and inorganic phosphate. In oxidative phosphorylation, this is the machinery that makes chemiosmosis productive: an electrochemical gradient becomes reusable chemical driving capacity.^[ref]

Its usefulness is a conversion between currencies. A membrane-spanning proton-motive force can power a local machine, while newly formed ATP can participate in energy coupling elsewhere. The enzyme supplies a route between these forms of free energy; it cannot make an energetically uphill total cycle favourable merely by accelerating its chemistry.

Define side $P$ as the higher proton electrochemical potential reservoir and side $N$ as the synthesis-side reservoir. For transfer $P\to N$ at common temperature $T$, define the positive downhill proton-motive force magnitude $\Delta p_{\mathrm{drive}}$ byHere $a$ is thermodynamic activity, $\psi$ is electric potential, $R$ is the molar gas constant and $F$ is the Faraday constant. Thus both a higher proton thermodynamic activity and a more positive electric potential on side $P$ contribute to the driving magnitude. This definition fixes the sign; conventions for a signed proton-motive force differ. The calculation under chemiosmosis separates these two contributions numerically.

Assume fixed reservoirs, tight mechanical energy coupling, one transported proton per $c$ subunit per revolution, and three ATP produced per revolution. Neglect leaks, slipping and membrane transport of the chemical reagents. Let $g_{\mathrm{ATP}}>0$ be the local molar Gibbs free energy cost of ATP formation at the stipulated composition, pH and magnesium-binding convention. It is not necessarily the standard state value. Adding changes in the state function gives, per mole of complete turns,For this model net synthesis is thermodynamically favoured when $\Delta G_{\mathrm{turn}}<0$. Equality is a reversible balance, not a promise of a useful reaction rate. A negative value establishes an available driving margin; it does not specify how fast the enzyme runs.

Choose $n=10$, $g_{\mathrm{ATP}}=50.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, $F=96485\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$ and $\Delta p_{\mathrm{drive}}=0.180\mathrm{V}$. These are illustrative inputs, not measurements of the preparation in the experiment below. Thenand $\Delta G_{\mathrm{turn}}=-23.673\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Per ATP, the ideal input is $57.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, leaving a driving margin of $7.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The balance point isAt $0.140\mathrm{V}$ the same synthesis cycle instead has $\Delta G_{\mathrm{turn}}=+14.921\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: the chosen reservoirs favour its reverse.

The machine is reversible: sufficient ATP hydrolysis can drive opposite rotation and pump protons uphill. Actual direction also depends on inhibition and available substrates; a weak proton-motive force need not make every intact enzyme immediately run backward. Respiratory uncoupling supplies a different failure mode: a parallel proton leak dissipates the gradient without productive shaft rotation.^[ref]

The motor budget $10/3$ is not the whole cost of ATP delivered outside a mitochondrion. Import of ADP and inorganic phosphate, and export of ATP, also draw on the electrochemical gradient.^[ref] For a deliberately simplified system budget, assign these transports together one additional downhill-proton equivalent per delivered ATP, valued at the same $F\Delta p_{\mathrm{drive}}$. This is a chosen accounting convention, not a claim of one extra literal proton through $F_0$, nor a universal transport stoichiometry. The total becomes $10/3+1=13/3$ equivalents and $75.258\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of gradient expenditure per delivered ATP at $0.180\mathrm{V}$. The ratio of stored chemical free energy to that expenditure is $50/75.258\approx 66.4\%$, compared with $50/57.891\approx 86.4\%$ for the motor alone. This system budget changes the resource expenditure, not the motor's balance point $\Delta p_{\ast }$. Real transport terms require their own electrochemical potentials, and leaks add further cost.

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.^[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.^[ref]

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.^[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.^[ref]