Past exam of the biological physics course of the University of Oxford / 2017 / 1 / d / Solution

Use steady or initial-rate conditions with effectively clamped substrate concentrations. For either substrate $X\in \{S,R\}$, put $U_X=[AX]$ and $V_X=[A{X}^{\ast }]$. The intermediate balances areThe product flux is thereforeThe free enzyme concentration is common to the two competing pathways, so it cancels even when enzyme sequestration changes its magnitude. ConsequentlyEach post-binding factor is the ratio of probabilities of proceeding rather than dissociating at that stage. The expression assumes positive common forward rates and no significant product back reaction. Without the steady/intermediate-balance or initial-rate assumption, the ratio can depend on intermediate populations and observation time; binding and dissociation constants alone would not determine a unique transient ratio.