# The standard, reverse and total Quasi-Steady-State Approximations revised: Timescale, small parameters, normal forms and singularities in enzyme kinetics

### Abstract

In this work, we revisit the scaling analysis and commonly accepted conditions for the validity of the standard, reverse and total quasi-steady-state approximations through the lens of dimensional Tikhonov–Fenichel small parameters and their respective critical manifolds. By combining Tikhonov–Fenichel small parameter with scaling analysis, we derive improved upper bounds on the approximation error for the standard, reverse and total quasi-steady-state approximations. We find that the condition for the validity of the reverse quasi-steady-state approximation is far less restrictive than was previously assumed, and we derive a new small” parameter that determines the validity of this approximation. Moreover, we show for the first time that the critical manifold of the reverse quasi-steady-state approximation contains a singular point where normal hyperbolicity is lost. Associated with this singularity is a transcritical bifurcation, and the corresponding normal form of this bifurcation is recovered through scaling analysis. It is commonly accepted that the reverse quasi-steady-state approximation is only valid in regions of parameter space where the influence of the singular point can be ignored. However, by using the method of slowly-varying Lyapunov functions, we provide asymptotic estimates for the validity of the reverse quasi-steady-state approximation in parameter regions where the effect of the singular point cannot be disregarded. In doing so, we extend the established domain of validity for the reverse quasi-steady-state approximation. Previous analysis has suggested that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Consequently, this opens the possibility of utilizing the reverse quasi-steady-state approximation to model enzyme catalyzed reactions and estimate kinetic parameters in enzymatic assays at much lower enzyme to substrate ratios than was previously thought.

Type
Publication
arXiv, 1911.03445