Hunting $\varepsilon$: The origin and validity of quasi-steady-state reductions in enzyme kinetics

TL;DR

This paper investigates the mathematical foundations of quasi-steady-state reductions in enzyme kinetics, revealing the conditions under which these simplifications are valid and exploring complex dynamical behaviors like bifurcations.

Contribution

It derives the key dimensionless parameter $oldsymbol{ ext{ extepsilon}}$ governing reduction validity and analyzes the rich fast-slow dynamics, including bifurcations and the impact of reversibility.

Findings

The validity of QSS reductions depends on the path in parameter space.

A dynamic transcritical bifurcation influences the model's behavior.

Reversibility can act as an imperfection, affecting bifurcation structure.

Abstract

The estimation of the kinetic parameters 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 known as quasi-steady-state 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. Using techniques from differential equations, Fenichel theory, and center manifold theory, we derive the appropriate $\varepsilon$ whose magnitude regulates the validity of the quasi-steady-state reduction employed in the reported experimental procedures for intermolecular autocatalytic zymogen activation reaction. 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 losses normal hyperbolicity and classical Fenichel theory is inapplicable. 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. 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 quasi-steady-state reduction for this reaction is justifiable by center manifold theory in one limit, and via Fenichel theory in a different limit.