Quasi-steady enzyme kinetics at low substrate saturation ($\epsilon =e_0/(s_0+K_m)\ll 1$)

A quasi-steady-state approximation for enzyme-substrate complexes can remain useful when total enzyme and initial substrate concentrations are comparable. The relevant parameter can involve the sum $s_0+K_m$, rather than $s_0$ alone. In a mechanism-based inhibition model with $s_0/K_m=O(\epsilon )$, bounded intermediate decay rates and an inactivation rate of order $\epsilon$, the fast complexes adjust in a layer of scaled width $O(\epsilon )$. On bounded slow-time intervals the main complex follows the substrate, the later complex and accumulated inactivated enzyme remain $O(\epsilon )$, and the substrate has leading exponential depletion. An outer algebraic complex concentration does not satisfy the unbound initial condition inside that initial layer.