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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.

A chemical used to produce respiratory uncoupling by carrying protons across the inner mitochondrial membrane.

Respiratory control can be investigated outside an intact biological cell by supplying isolated mitochondria with substrates and ADP. This makes the boundary between oxygen consumption and ATP formation experimentally accessible. The historical mitochondrial oxygen electrode assay describes the 1955 experiment; the following authored protocol adds modern preparation and inhibitor controls.

In Functional Assessment of Isolated Mitochondria In Vitro (2009), Ian R. Lanza and K. Sreekumaran Nair describe human skeletal-muscle biopsy mincing, cold protease treatment, homogenization and differential centrifugation. A low-speed spin removes coarse material; subsequent spins collect and wash mitochondria. A stirred, closed $2\mathrm{mL}$ Oxygraph-2k chamber at $37{}^{\circ }\mathrm{C}$ uses a calibrated polarographic oxygen electrode. Its signal gives dissolved oxygen concentration; the negative time derivative gives consumption. Instrument corrections include electrode consumption and oxygen leakage.^[ref]

Their additions include glutamate plus malate, an ADP pulse, then saturating ADP. Added cytochrome c tests outer-membrane damage: stimulation suggests loss of endogenous cytochrome c. Later oligomycin inhibits ATP synthase; FCCP produces respiratory uncoupling; antimycin A inhibits respiratory complex III to estimate residual background. Figure 6 includes intervening succinate dehydrogenase and respiratory complex I interventions; the modeled traces below simplify that sequence. Isolation can damage or select mitochondria, and saturating assay conditions differ from those in living biological tissue.^[ref]

The control logic matters more than a single steep slope. If ADP accelerates consumption and a finite pulse is followed by slowing, demand is affecting respiratory flux. If oligomycin slows consumption, but subsequent FCCP accelerates it, the electron transport chain can still turn over while ATP synthase remains inhibited. That contrast separates restricted coupled turnover from a completely inactive chain. Failure of FCCP to restore a rate is less specific: substrate limitation, damage, low oxygen or excessive uncoupler can also restrict respiration.

Neither the cytochrome c test alone nor a large respiratory control ratio establishes every aspect of mitochondrial integrity. Likewise, oxygen consumption is not a direct ATP assay: respiratory uncoupling provides a concrete counterexample. The preparation, additions and normalization denominator determine what comparison the experiment can support.

Oxidative phosphorylation couples respiratory redox conversion to ATP formation through chemiosmosis. ATP synthase uses inward proton flow to join ADP and inorganic phosphate.^[ref]

The intermediate electrochemical gradient explains why respiratory uncoupling can preserve oxygen consumption while reducing ATP formation. An Electron ledger alone cannot determine the recovery of useful free energy.

Respiratory uncoupling allows proton return to bypass ATP synthase, dissipating the electrochemical gradient. Electron transport and oxygen consumption can continue while less ATP is recovered.^[ref]

In the thermodynamic example under chemiosmosis, reducing both the activity ratio to one and the electric potential difference to zero removes the driving contribution. The hypothetical ATP-forming conversion then costs $+45\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Continued donor oxidation would not restore coupling unless it rebuilt a usable electrochemical gradient. This distinction makes respiratory uncoupling useful conceptually: abundant oxygen consumption can coexist with poor recovery of ATP.

In 1997, Hiroyuki Noji and colleagues attached a fluorescent actin filament to the central $\gamma$ shaft of isolated $F_1$. With ATP supplied, the marker made sustained rotations. This exposed rotation directly, but used ATP hydrolysis: there was no membrane $F_0$ motor producing ATP from a proton-motive force in this assay.^[ref]

A subsequent primary experiment made the optical readout much faster. Purified Escherichia coli $F_1$ was immobilized on glass in a $30\mu \mathrm{m}$-deep flow cell, with a $60\mathrm{nm}$ gold bead attached to $\gamma$. With $2\mathrm{mM}$ ATP and magnesium supplied, laser illumination and dark-field microscopy recorded bead positions at $4000$ or $8000$ frames per second. The position trajectory resolved turns and intervening dwells. This apparatus follows a shaft-attached marker, rather than imaging the shaft's chemical atoms.^[ref]

Controls addressed perturbation and chemical activity: comparison of $40$–$200\mathrm{nm}$ probes found similar rates for the two smallest sizes, supporting low drag; a separate coupled-enzyme assay followed ATP hydrolysis through NADH absorbance. But the rotational analysis excluded pauses longer than $0.1\mathrm{s}$, so it characterizes continuously active intervals, not total output including long inhibited periods. It neither counts transported protons nor measures simultaneous single-molecule ATP synthesis.^[ref]

The experimental distinction is essential. Sustained directional marker motion tests a rotary mechanism. Demonstrating proton-driven synthesis additionally requires intact $F_0{F}_1$ in a membrane with a controlled proton-motive force and a chemical ATP-production readout; dependence on that driving force and inhibition or respiratory uncoupling controls then test the proposed energy coupling. A rotation movie alone cannot establish the complete material and free energy ledger.