document: 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 '' -LRB- Liu , 1999 -RRB- . 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
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