Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / B / 5 / iii / Solution

Set $T_f=(k_1{S}_0{)}^{-1}$, $ϵ=E_0/S_0$, $K=(k_{-1}+k_2)/(k_1{S}_0)$ and $\lambda =k_2/(k_1{S}_0)$. On fast time ${\tau }_f=t/T_f$,For ${\tau }_f=O(1)$, leading substrate remains $s=1$. The initial complex buildup therefore solves $c^{\prime }=1-(1+K)c$, $c(0)=0$, givingTime belongs inside the exponential. Its placement outside the exponential bracket in the printed expression is a typographical error; the derived form satisfies both the kinetic equation and zero initial complex concentration.