Consider the enzymatic reaction sequence shown below where $S$ is the substrate, $E$ is the enzyme, $C$ is the enzyme$-$substrate complex and $P$ is the product. The reaction rates are given below in terms of the species concentrations.

$E+S{}^{r_r}C{r}_{r}E+P$

$r_f=k_{f}ES,r_r=k_{r}C,r_c=k_{c}C$

The total free enzyme $(E)$ plus bound enzyme $(C)$ is constant such that: $E+C=E_t.$

$($a$)(10$ pts$)$ Assuming each species has a stoichiometry of one in each reaction, derive the following mass balance equations:

$\frac{dE}{(d)t}=-(k_{f}S+k_r+k_c)E+(k_r+k_c)E_t=f_1(E,S)$

$\frac{dS}{(d)t}=-(k_{f}S+k_r)E+k_r{E}_t=f_2(E,S)$

$($b$)(10$ pts$)$ Determine the single steady$-$state solution $\frac{?}{\text{b}\text{a}\text{r}}(E)$ and $\frac{?}{\text{b}\text{a}\text{r}}(S).$

$($c$)(10pts)$ Linearize the ODEs in part $($a$)$ at the steady$-$state point in part $($b$)$ to obtain a linear ODE system with matrix A that depends on the model parameters.

$($d$)(10pts)$ Given the parameter values $k_f=2,k_r=1,k_c=5$ and $E_t=1,$ compute the eigenvalues of the matrix $A$ in part $($c$)$ to determine the stability of the steadystate point in part $($b$).$

$($e$)(12pts)$ Formulate the forward Euler equations and produce a table showing the first three steps starting from $E_0=S_0=1$ with a step size of $t=0\mathrm{.}1.$