No abstract

<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>

<p>A zymogen is an inactive precursor of an enzyme, which 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 product is the same enzyme catalyzing the reaction. In this article we investigate the nonlinear chemical dynamics of the intermolecular autocatalytic zymogen activation reaction mechanism, and compare it to that of the Michaelis-Menten reaction mechanism. We show that the intermolecular autocatalytic zymogen activation exhibits significant changes in reaction dynamics relative to the Michaelis-Menten reaction mechanism. These changes include differences in the number of conservation laws, number and stability of equilibrium states, altered structure of the invariant set that influences the long-time rate of the reaction, and qualitative evolution of the reaction depending strictly on the choice of initial conditions. We find a rate law, homologous to the Michaelis-Menten equation, to estimate the kinetic parameters of the intermolecular autocatalytic zymogen activation reaction mechanism, and derive the conditions for the validity for this rate law. Finally, we derive analytical expressions to estimate the timescale for the completion of the zymogen activation, which could have a practical application to calculate the molar enthalpy of the autocatalytic zymogen reaction in calorimetry assays. </p>

<p>A zymogen is an inactive precursor of an enzyme, which 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 product is the same enzyme catalyzing the reaction. In this article we investigate the nonlinear chemical dynamics of the intermolecular autocatalytic zymogen activation reaction mechanism, and compare it to that of the Michaelis-Menten reaction mechanism. We show that the intermolecular autocatalytic zymogen activation exhibits significant changes in reaction dynamics relative to the Michaelis-Menten reaction mechanism. These changes include differences in the number of conservation laws, number and stability of equilibrium states, altered structure of the invariant set that influences the long-time rate of the reaction, and qualitative evolution of the reaction depending strictly on the choice of initial conditions. We find a rate law, homologous to the Michaelis-Menten equation, to estimate the kinetic parameters of the intermolecular autocatalytic zymogen activation reaction mechanism, and derive the conditions for the validity for this rate law. Finally, we derive analytical expressions to estimate the timescale for the completion of the zymogen activation, which could have a practical application to calculate the molar enthalpy of the autocatalytic zymogen reaction in calorimetry assays.</p>

<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>