= Solution
Use $S,C_i,P$ for <concentrations> and define net complex-formation rates $R_i=k_iC_{i-1}S-(k_{-i}+k_i^+)C_i$, with $R_{n+1}=0$. The <law of mass action> gives
$$
\boxed{\dot C_0=-R_1,\quad \dot C_i=R_i-R_{i+1}\ (1\leq i<n),\quad\dot C_n=R_n,}
$$
$$
\boxed{\dot S=-\sum_{i=1}^n(k_iC_{i-1}S-k_{-i}C_i)=-\sum_{i=1}^n(R_i+k_i^+C_i),\quad \dot P=\sum_{i=1}^nk_i^+C_i.}
$$
Each binding event removes one free <substrate> molecule; each unbinding returns one; each catalytic event releases one <reaction product> and reduces the number of occupied sites by one. Suitable initially unbound <enzyme> and <substrate> data are $S(0)=S_0$, $P(0)=0$, $C_0(0)=E_0$, $C_i(0)=0$ for $i\geq1$. All rates are nonnegative on the boundary of the nonnegative <concentration> region in the direction required to preserve it. These equations also conserve $S+P+\sum_{i=1}^niC_i=S_0$, accounting for bound <substrate> as well as free <substrate>.
Back to article page
