Sequential two-site enzyme reaction network

An enzyme may bind one molecule of a substrate to form $C_1$, then a second to form $C_2$, with product-release reactions $C_1\to E+P$ and $C_2\to C_1+P$. The law of mass action gives conservation laws $e+c_1+c_2=e_0$ and $s+c_1+2c_2+p=s_0$. Scaling substrate by $s_0$, complexes by $e_0$ and time by $(k_1{e}_0{)}^{-1}$ exposes the small ratio $ϵ=e_0/s_0$: the substrate evolves on the slow scale while the complex equations have the form $ϵv_i^{\prime }=g_i$. This is a starting point for quasi-steady-state approximation when $ϵ\ll 1$.