Accuracy Versus Predominance: Reassessing the Validity of the Quasi-Steady-State Approximation

The application of the standard quasi-steady-state approximation to the Michaelis-Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>e</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> , to the Michaelis constant, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>K</mml:mi> <mml:mi>M</mml:mi></mml:msub> </mml:math> , decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>e</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>/</mml:mo> <mml:msub><mml:mi>K</mml:mi> <mml:mi>M</mml:mi></mml:msub> </mml:mrow> </mml:math> provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>e</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> to the Van Slyke-Cullen constant, K. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>e</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi></mml:mrow> </mml:math> offers the most accurate measure of the validity of the standard quasi-steady-state approximation.