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.

在这项工作中，我们通过维数 Tikhonov-Fenichel 参数及其各自的临界流形，重新审视了标准、逆向和总准稳态近似有效性的标度分析和普遍接受的条件。通过将 Tikhonov-Fenichel 参数与标度分析和能量方法相结合，我们得出了标准、反向和总准稳态近似的近似误差的改进上限。此外，先前的分析表明，反向准稳态近似仅在初始酶浓度大大超过初始底物浓度时才有效。然而，我们的结果表明，当初始酶和底物浓度相等时，这种近似可能是有效的。使用能量方法，我们发现反向准稳态近似的有效性条件远没有之前假设的那么严格，并且我们推导了一个新的“小”参数来确定该近似的有效性。在此过程中，我们扩展了反向准稳态近似的既定有效域。因此，这开辟了利用反向准稳态近似来模拟酶催化反应的可能性，并以比之前认为的低得多的酶与底物比率来估计酶测定中的动力学参数。此外，我们首次证明逆准稳态近似的临界流形包含一个失去正常双曲性的奇点。与这个奇点相关的是跨临界分岔，并且通过标度分析恢复了该分岔相应的标准形式。