Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2010 / B / 4 / a

Use $S,C_i,P$ for concentrations and define net complex-formation rates $R_i=k_i{C}_{i-1}S-(k_{-i}+k_i^{+})C_i$, with $R_{n+1}=0$. The law of mass action givesEach 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 $i\ge 1$. All rates are nonnegative on the boundary of the nonnegative concentration region in the direction required to preserve it. These equations also conserve $S+P+{\sum }_{i=1}^{n}i{C}_i=S_0$, accounting for bound substrate as well as free substrate.