Sequential-enzyme Hill limit ($\rho =E_0{k}_n^{+}{S}^n/({\prod }_i{K}_i+S^n)$)

For sequential enzyme binding, a quasi-steady-state approximation gives occupied-state weights ${\varphi }_j{S}^j$, ${\varphi }_j={\prod }_{i=1}^j{K}_i^{-1}$, relative to free enzyme. If all intermediate weights tend to zero while the terminal weight remains nonzero, enzyme conservation leaves only unbound and fully bound states, and the production rate becomes the displayed Hill equation. One realization holds other rates fixed while the first binding rate tends to zero and the last tends to infinity with positive finite product. A statement merely bounding that product does not by itself specify this nontrivial distinguished limit.