= Solution
\b[Animal cells conserve part of the free energy from nutrient oxidation in <ATP> and electrochemical gradients, then couple their consumption to synthesis, transport and movement.] The relevant quantity is the actual <Gibbs free energy> change,
$$
\Delta G=\Delta G^{\circ\prime}+RT\ln Q,
$$
where $Q$ is the appropriate <reaction quotient>, $R$ the <molar gas constant> and $T$ the absolute <temperature>. A reaction with a positive free-energy change can proceed when physically coupled to one whose negative change makes the combined process favourable. Merely placing an exergonic reaction nearby does not accomplish coupling: an enzyme, shared intermediate or molecular machine must link the processes.
Oxidation of carbohydrates and fats supplies <NADH> and other reducing equivalents. In the mitochondrial <electron transport chain>, transfer towards <oxygen> releases free energy. Complexes I, III and IV couple electron transfer to proton translocation from the matrix to the intermembrane space; complex II transfers electrons without directly pumping protons. The inner mitochondrial membrane retains the resulting <proton-motive force>. For movement of one mole of protons from outside to inside, define
$$
\Delta G_{\mathrm H^+}=F(\psi_{\mathrm{in}}-\psi_{\mathrm{out}})+RT\ln\frac{[\mathrm H^+]_{\mathrm{in}}}{[\mathrm H^+]_{\mathrm{out}}}.
$$
Here $F$ is the <Faraday constant> and $\psi$ is the electrical potential. Both the relatively negative matrix potential and its relatively alkaline pH favour inward movement. An inward driving voltage of magnitude $0.18\,\mathrm V$ corresponds to approximately $17.4\,\mathrm{kJ\,mol^{-1}}$ per mole of protons. This is an illustrative calculation, not a universal mitochondrial measurement.
Proton movement through <ATP synthase> drives rotation and conformational changes that couple <ADP> and phosphate to <ATP> formation. Several protons are required per <ATP>, and transport of phosphate and adenine nucleotides adds energetic costs; there is no universal whole-number <ATP> yield for every animal cell. Reconstitution of a light-driven proton pump with ATP-synthesizing machinery in lipid vesicles showed that an independently generated gradient can support <ATP> formation. This experiment used an artificial mixture of components, providing a mechanistic test rather than a model of animal-cell photosynthesis.<Reconstitution of Purple Membrane Vesicles Catalyzing Light-driven Proton Uptake and Adenosine Triphosphate Formation by Efraim Racker and others>[Racker and Stoeckenius, 1974, pp. 662–663, reconstituted light-driven proton uptake and ATP formation]{ref} In mitochondria, respiratory inhibitors, an ATP-synthase inhibitor and a <protonophore> give distinct predictions for <oxygen> consumption, <ATP> production and the gradient, helping distinguish electron transfer from coupling.
<ATP> then supplies a mobile chemical coupling intermediate. Transfer of a phosphate group can make an otherwise unfavourable biosynthetic step favourable through a common intermediate. <sodium-potassium ATPase> couples <ATP> hydrolysis to ion pumping; sodium-dependent cotransport uses the stored gradient for <secondary active transport>. <Myosin> couples its ATPase cycle to force and displacement on <actin>. <Substrate-level phosphorylation> also generates <ATP> directly during <glycolysis> and other reactions without an obligatory respiratory gradient. <ATP> hydrolysis is favourable because of the free-energy difference between the full reactant and product states under cellular conditions, not because breaking a bond alone releases energy.
Conversion is never perfectly efficient: heat is produced, proton leak can bypass <ATP> synthase, and controlled uncoupling can favour heat production. <ATP> concentration and the <ATP>/<ADP>/phosphate ratio reflect a maintained nonequilibrium state. \b[Energy transduction is therefore a chain of coupled chemical, electrical, osmotic and mechanical changes, sustained by metabolism and constrained by thermodynamics.]
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