Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S12 / 3

ATP synthase combines a membrane $F_o$ proton-conducting rotor with an $F_1$ catalytic head. The c-ring and central gamma shaft rotate relative to a stator and the alpha/beta head. Proton flow down its electrochemical potential can drive conformational changes in three beta catalytic sites to bind substrates, form and release ATP. Reverse operation uses ATP hydrolysis to turn the shaft and pump protons. The immobilized $F_1$ assay isolates this latter rotation; it does not require the membrane proton motor to be present.

An ATP-binding step with pseudo-first-order rate $k_{1}c$ has survival probability $S_1=e^{-k_{1}ct}$ and waiting density $P_1=-S_1^{\prime }=k_{1}c{e}^{-k_{1}ct}$. It decreases from a finite value at zero, integrates to one and has mean $1/(k_{1}c)$. Its concentration dependence suggests binding-limited waiting before the large 80-degree substep.

Two independent sequential exponential steps of rates $k_a,k_b$ give the convolutionIt begins at zero, rises to a maximum at $\log (k_a/k_b)/(k_a-k_b)$ and decays; its mean is $k_a^{-1}+k_b^{-1}$. At equal rates it becomes $k^{2}t{e}^{-kt}$. This supports at least two kinetic events during the waiting associated with the 40-degree substep, plausibly chemical transformation/product release and conformational changes. It does not uniquely identify which rate belongs to hydrolysis or a particular product: different hidden-state networks can have the same waiting distribution. Three 120-degree catalytic cycles per revolution, partitioned into 80 and 40 degrees in the assay, are consistent with three catalytic sites.