Abstract
This paper deals with the analysis of a metabolic network with feedback inhibition. The considered system is an acyclic network of mono-molecular 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.
Highlights
The cellular metabolism is defined as the set of biochemical reactions that occur inside a living cell for growth and reproduction
A special difficulty comes from the existence of negative feedback inhibitions that are known for a long time to potentially induce instabilities and limit cycles
We have shown in [8] that this result is even valid for a larger class of metabolic networks, namely networks that satisfy a weaker version of Assumption 2c: the reactions Xs → Xp are inhibited by metabolites from the-arborescence rooted in Xp
Summary
The cellular metabolism is defined as the set of biochemical reactions that occur inside a living cell for growth and reproduction. In order to perform the stability analysis, we shall use a specific mathematical technique which, to our knowledge, has not been used in previous papers on the stability of metabolic systems This technique provides a sufficient condition under which the considered branched network has a single globally asymptotically stable equilibrium. It is assumed that the velocity of the first reaction X1 −→ X2 is inhibited by the last metabolite with a multiplicative hyperbolic inhibition function of the form: ψα(xn) Under these assumptions and notations, the mass balance dynamical model under study is formulated as:. We have shown that a wide class of models of branched metabolic networks has a single equilibrium and we have proven that, under a small gain condition, this equilibrium is globally asymptotically stable and attractive.
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