The Michaelis-Menten kinetics reaction is given by

\[E+S \underset{k_{1}}{\stackrel{k_{3}}{\longrightarrow}} E S \underset{k_{2}}{\longrightarrow} E+P\]

The resulting system of equations for the chemical concentrations is

\[\begin{align*} \frac{d[S]}{d t} & =-k_{1}[E][S]+k_{3}[E S] \\ \frac{d[E]}{d t} & =-k_{1}[E][S]+\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[E S]}{d t} & =k_{1}[E][S]-\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[P]}{d t} & =k_{3}[E S] \tag{4.87} \end{align*}\]

In chemical kinetics, one seeks to determine the rate of product formation \(\left(v=d[P] / d t=k_{3}[E S]\right)\). Assuming that \([E S]\) is a constant, find \(v\) as a function of \([S]\) and the total enzyme concentration \(\left[E_{T}\right]=[E]+[E S]\). As a nonlinear dynamical system, what are the equilibrium points?