准稳态近似再论：酶动力学中的时标、小参数、奇点和normal forms
The Quasi-Steady-State Approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetic
摘要：
在这项工作中，我们重温尺度分析和普遍接受的标准，反向和总的准稳态近似的有效性条件，通过镜头的维Tikhonov-Fenichel参数和它们各自的临界流形。通过结合Tikhonov-Fenichel参数、标度分析和能量方法，我们得到了标准、逆和全准稳态近似的近似误差的改进的上界.此外，以前的分析表明，反向准稳态近似是有效的，只有当初始酶浓度大大超过初始底物浓度。然而，我们的研究结果表明，这种近似可以是有效的，当初始酶和底物浓度是相同的幅度。使用能量的方法，我们发现，反向准稳态近似的有效性的条件是远远低于以前假设的限制，我们推导出一个新的“小”参数，确定这种近似的有效性。在这样做时，我们扩展了反向准稳态近似的有效性域。因此，这开辟了利用反向准稳态近似模型酶催化反应和估计动力学参数的酶测定在比以前认为的低得多的酶底物比的可能性。此外，我们第一次表明，临界流形的反向准稳态近似包含一个奇点正常双曲性丢失。与此奇异性相关联的是一个跨临界分岔，并通过标度分析恢复了该分岔的相应规范形。
英文摘要：
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.
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