Equilibria and stability analysis of a branched metabolic network with feedback inhibition

This paper deals with the analysis of a metabolic network with feedback inhibition. The considered system is an acyclic network of monomolecular enzymatic reactions in which metabolites can act as feedback regulators on enzymes located "at the beginning" of their own pathway, and in which one metabolite is the root of the whole network. We show, under mild assumptions, the uniqueness of the equilibrium. We then show that this equilibrium is globally attractive if we impose conditions on the kinetic parameters of the metabolic reactions. Finally, when these conditions are not satisfied, we show, with a specific fourth-order example, that the equilibrium may become unstable with an attracting limit cycle.