An allosteric enzyme $E$ reacts with a substrate $S$ to produce a product $P$ according to the mechanism

$$\begin{gathered} S+E \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftharpoons}} C_{1} \stackrel{k_{2}}{\longrightarrow} E+P \\ S+C_{1} \underset{k_{-3}}{\stackrel{k_{3}}{\rightleftharpoons}} C_{2} \stackrel{k_{4}}{\rightarrow} C_{1}+P \end{gathered}$$

where $C_{1}$ and $C_{2}$ are enzyme-substrate complexes. With lowercase letters denoting concentrations, write down a system of differential equations based on the Law of Mass Action which model this reaction mechanism.

The initial conditions are $s=s_{0}, e=e_{0}, c_{1}=c_{2}=p=0 .$ Using $u=s / s_{0}$, $v_{i}=c_{i} / e_{0}, \tau=k_{1} e_{0} t$ and $\epsilon=e_{0} / s_{0}$, show that the nondimensional reaction mechanism reduces to

$$\frac{d u}{d \tau}=f\left(u, v_{1}, v_{2}\right) \quad \text { and } \quad \epsilon \frac{d v_{i}}{d \tau}=g_{i}\left(u, v_{1}, v_{2}\right) \quad \text { for } \quad i=1,2$$

finding expressions for $f, g_{1}$ and $g_{2}$.