Global stability of the attracting set of an enzyme-catalysed reaction system

The essential feature of enzymatic reactions is a nonlinear dependency of reaction rate on metabolite concentration taking the form of saturation kinetics. Recently, it has been shown that this feature is associated with the phenomenon of "loss of System coordination" [1]. In this paper, we study a system of ordinary differential equations representing a branched biochemical system of enzyme-mediated reactions. We show that this system can become very sensitive to changes in certain maximum enzyme activities. In particular, we show that the system exhibits three distinct responses: a unique, globally-stable steady-state, large amplitude oscillations, and asymptotically unbounded solutions, with the transition between these states being almost instantaneous. It is shown that the appearance of large amplitude, stable limit cycles occurs due to a "false" bifurcation or canard explosion. The subsequent disappearance of limit cycles corresponds to the collapse of the domain of attraction of the attracting set for the system and occurs due to a global bifurcation in the flow, namely, a saddle connection. Subsequently, almost all nonnegative data become unbounded under the action of the dynamical system and correspond exactly to loss of system coordination. We discuss the relevance of these results to the possible consequences of modulating such systems.