 Solution

For an outer solution with bounded derivatives as $ϵ\to 0$, the final complex equation gives $r_n=0$. Working backwards in $0=r_i-r_{i+1}$ then gives every $r_i=0$, including $r_1$; setting only their sum to zero would not suffice. This is the quasi-steady-state approximation, valid after the initial complex-formation layer with the relevant dimensionless rate parameters held fixed.

The recurrence is $C_i=S{C}_{i-1}/K_i$, where $K_i=(k_{-i}+k_i^{+})/k_i$. Thus $C_j={\varphi }_j{S}^j{C}_0$, ${\varphi }_j={\prod }_{i=1}^j{K}_i^{-1}$. Substitution into enzyme conservation yieldsEach state $j$ produces reaction product at rate $k_j^{+}{C}_j$, soThe source's uppercase $N$ in the index range is read as the number $n$ of binding sites. This approximation preserves the total enzyme population and the different catalytic rates of the occupied states.