  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 parameters and their respective critical manifolds. By
combining Tikhonov-Fenichel parameters with scaling analysis and energy
methods, we derive improved upper bounds on the approximation error for the
standard, reverse and total quasi-steady-state approximations. Furthermore,
previous analyses suggest 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.
Using energy methods, 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. In doing so, we extend the established domain
of validity for the reverse quasi-steady-state approximation. Consequently,
this opens up 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. 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.
