= Solution
Use enzyme conservation to write
$$
e=e_0(1-v_1-v_2).
$$
Since $\tau=k_1e_0t$, $u=s/s_0$, $v_i=c_i/e_0$, and $\epsilon=e_0/s_0$, substitution into the mass-action equations gives
$$
\boxed{
f(u,v_1,v_2)
=-u(1-v_1-v_2)+\alpha v_1-\beta uv_1
},
$$
and
$$
\boxed{
\begin{aligned}
g_1(u,v_1,v_2)
&=u(1-v_1-v_2)-\alpha v_1-\beta uv_1+\gamma v_2,\\
g_2(u,v_1,v_2)
&=\beta uv_1-\gamma v_2,
\end{aligned}}
$$
where
$$
\alpha=\frac{k_{-1}}{k_1s_0},
\qquad
\beta=\frac{k_2}{k_1},
\qquad
\gamma=\frac{k_3}{k_1s_0}.
$$
The factors of $\epsilon$ multiplying $dv_i/d\tau$ identify the complex concentrations as fast variables in a <singular perturbation>.
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