Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S12 / 3 / c

The observed $90^{\circ }+30^{\circ }$ substeps sum to $120^{\circ }$, consistent with three catalytic cycles per revolution of $F_1$-ATPase. Surface attachment fixes the $\beta$-subunit frame while the gold-labelled $\gamma$ shaft reports relative rotation. A finite dwell before the smaller step indicates hidden chemical transitions rather than a completely smooth motor.

Suppose two irreversible independent first-order transitions, with rates $k_a$ and $k_b$, must occur sequentially before the $30^{\circ }$ step. Their waiting times $X_a$ and $X_b$ have exponential distributions, and $\tau =X_a+X_b$. Convolution givesThis probability density integrates to one and has $⟨\tau ⟩=k_a^{-1}+k_b^{-1}$. For equal rates its continuous limit is $P(\tau )=k^2\tau e^{-k\tau }$, not a singular distribution. In particular $P(0)=0$ distinguishes two resolved sequential waits from a single exponential wait. Hydrolysis and product release are plausible underlying chemical events, but the waiting-time fit alone neither assigns their order nor uniquely identifies which biochemical event causes each substep. Concentration dependence or independent labelling would be needed for that assignment.