A bacterial nutrient uptake model is represented by the reaction system

$$\begin{array}{rll} 2 S+E & \stackrel{k_{1}}{\longrightarrow} & C \\ C & \stackrel{k_{2}}{\longrightarrow} 2 S+E \\ C \stackrel{k_{3}}{\longrightarrow} & E+2 P \end{array}$$

where the $k_{i}$ are rate constants. Let $s, e, c$ and $p$ represent the concentrations of $S, E, C$ and $P$ respectively. Initially $s=s_{0}, e=e_{0}, c=0$ and $p=0$. Write down the governing differential equation system for the concentrations.

Either by using the differential equations or directly from the reaction system above, find two invariant quantities. Use these to simplify the system to

$$\begin{aligned} \dot{s} &=-2 k_{1} s^{2}\left(e_{0}-c\right)+2 k_{2} c \\ \dot{c} &=k_{1} s^{2}\left(e_{0}-c\right)-\left(k_{2}+k_{3}\right) c . \end{aligned}$$

By setting $u=s / s_{0}$ and $v=c / e_{0}$ and rescaling time, show that the system can be written as

$$\begin{aligned} u^{\prime} &=-2 u^{2}(1-v)+2(\mu-\lambda) v \\ \epsilon v^{\prime} &=\quad u^{2}(1-v)-\mu v \end{aligned}$$

where $\epsilon=e_{0} / s_{0}$ and $\mu$ and $\lambda$ should be given. Give the initial conditions for $u$ and $v$.

[Hint: Note that $2 X$ is equivalent to $X+X$ in reaction systems.]