Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 6C / d / Solution

On times $\tau \gg ϵ$, the quasi-steady-state approximation setsThe second equation givesIn the first equation the terms $-\beta u{v}_1+\gamma v_2$ then cancel, leavingSolving these two equations givesSince $\dot{p}=k_3{c}_2=k_3{e}_0{v}_2$,

At small substrate concentration this allosteric rate is quadratic, $dp/dt\sim (A/\alpha )u^2$, so its graph leaves the origin with zero slope. At large $u$ it saturates at $A\gamma /\beta =k_3{e}_0$. A Michaelis-Menten equation also saturates, but its rate $V_{\max }u/(K_M+u)$ is linear near the origin. The allosteric curve is therefore more sigmoidal:This is the quasi-steady rate law for a two-substrate allosteric enzyme.