Past exam of the biological physics course of the University of Oxford / 2017 / 6 / b / Solution

Usable biological free energy has several interconvertible forms. At approximately fixed temperature and pressure, chemical reactions are described by chemical potentials, ${\mu }_i={\mu }_i^{\circ }+RT\log a_i$, and ${\Delta }_{r}G={\sum }_i{\nu }_i{\mu }_i$. ATP hydrolysis, for example, has ${\Delta }_{r}G={\Delta }_r{G}^{\circ \prime }+RT\log (a_{\mathrm{ADP}}{a}_{\mathrm{Pi}}/a_{\mathrm{ATP}})$ under the specified biochemical standard-state convention. The activities and proton/ion convention matter. Breaking a bond alone costs energy; the net free-energy change includes the products, solvent and all bond changes, not a store released simply by snapping one bond.

Redox reactions provide free energy when electrons move between donors and acceptors with a favourable potential difference: $\Delta G=-nF\Delta E$. Electron-transfer chains can couple that drop to ion transport, building an electrochemical gradient. For transfer from an outer to an inner reservoir,Both concentration difference and electrical potential contribute. ATP synthase couples downhill ion flow to nucleotide free energy; driven pumps can perform the reverse conversion and maintain gradients used by transporters and excitability.

Mechanical work appears as $Fdx$ or torque times angle, and can be stored elastically, for example $\kappa x^2/2$. Molecular motors convert chemical free energy into movement or force; mechanically driven reactions can conversely reshape chemical transitions. Electrical energy can also be stored in membrane capacitance, while bulk ionic concentration gradients store additional chemical free energy. Light supplies photon energy $h\nu$, which photosynthetic machinery can convert into redox and chemical potential differences.

Coupling is essential: an overall favourable reaction can drive an unfavourable one only when their processes are linked, rather than merely present in the same cell. At fixed conditions the extracted work is bounded by the consumed free-energy decrease. Dissipation produces heat and entropy, so energy conservation does not mean all energy remains available for work. Thermal fluctuations can drive random motion, but a persistent directional cycle requires a nonequilibrium free-energy source; an equilibrium asymmetric structure alone is not a free motor.