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Hunting \(\varepsilon\): the origin and validity of quasi-steady-state reductions in enzyme kinetics. (English) Zbl 1481.92051

Summary: The estimation of the kinetic parameters that regulate the speed of enzyme catalyzed reactions requires the careful design of experiments under a constrained set of conditions. Many estimates reported in the literature incorporate protocols that leverage simplified mathematical models of the reaction’s time course known as quasi-steady-state (QSS) reductions. Such reductions often – but not always – emerge as the result of a singular perturbation scenario. However, the utilization of the singular perturbation reduction method requires knowledge of a dimensionless parameter, “\(\varepsilon\),” that is proportional to the ratio of the reaction’s fast and slow timescales. To date, no such ratio has been determined for the intermolecular autocatalytic zymogen activation reaction, which means it remains open as to when the experimental protocols described in the literature are even capable of generating accurate estimates of pertinent kinetic parameters. Using techniques from differential equations, Fenichel theory, and center manifold theory, we derive the appropriate “\(\varepsilon\)” whose magnitude regulates the validity of the QSS reduction employed in the reported experimental procedures. Although the model equations are two-dimensional, the fast/slow dynamics are rich. The phase plane exhibits a dynamic transcritical bifurcation point in a particular singular limit. The existence of such a bifurcation is relevant because the critical manifold loses normal hyperbolicity and classical Fenichel theory is inapplicable. We show that while there exists a faux Canard that passes directly through this bifurcation point, trajectories emanating from experimental initial conditions are actually bounded away from the bifurcation point by an asymptotic distance that is proportional to the square root of the linearized system’s eigenvalue ratio. Furthermore, we show that in some cases chemical reversibility can be interpreted dynamically as an imperfection since the presence of reversibility can destroy the bifurcation structure present in the singular limit. By extension, some of these features are also present in the phase-plane dynamics of the famous Michaelis-Menten reaction mechanism. Finally, we show that the reduction method by which QSS reductions are justified can depend on the path taken in parameter space. Specifically, we show that the standard QSS reduction for this reaction is justifiable by center manifold theory in one limit, and via Fenichel theory in a different limit.

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
34D15 Singular perturbations of ordinary differential equations
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