Applying the quasi-steady-state approximation to the autocatalytic zymogen activation: A novel general methodology for scaling analysis

<p><br></p><table><tr><td>A zymogen is an inactive precursor of an enzyme that needs to go through a chemical change to become an active enzyme. The general intermolecular mechanism for the autocatalytic activation of zymogens is governed by the single-enzyme, single-substrate catalyzed reaction following the Michaelis-Menten mechanism of enzyme action, where the substrate is the zymogen and the product is the same enzyme that is catalyzing the reaction. In this article we investigate the nonlinear chemical dynamics of the intermolecular autocatalytic zymogen activation reaction mechanism. In so doing, we develop a general strategy for obtaining dimensionless parameters that, when sufficiently small, legitimize the application of the quasi-steady-state approximation. Our methodology combines energy methods and exploits the phase-plane geometry of the mathematical model, and we obtain sufficient conditions that support the validity of the standard, reverse and total quasi-steady-state approximations for the intermolecular autocatalytic zymogen activation reaction mechanism. The utility of the procedure we develop is that it circumnavigates the direct need for a priori timescale estimation, scaling, and non-dimensionalization. At the same time, a novel result emerges from our analysis: the discovery of a dynamic transcritical bifurcation that exists in the singular limit of the model equations. Moreover, associated with the dynamic transcritical bifurcation is an imperfect term. We prove that when the imperfect term vanishes and the singular vector field is perturbed, there exists a canard that follows a repulsive slow invariant manifold over timescales of <i>O</i>(1). This is the first report of such a solution for the intermolecular and autocatalytic zymogen activation reaction. By extension, our results illustrate that canards also exist in the classic single enzyme, single-substrate reversible Michaelis-Menten reaction mechanism.</td></tr></table>