Boundedness analysis for open Chemical Reaction Networks with mass-action kinetics

Abstract

This paper describes the working principles of an algorithm for boundedness analysis of open Chemical Reaction Networks endowed with mass-action kinetics. Such models can be thought of both as a special class of compartmental systems or a particular type of continuous Petri Nets, in which the firing rates of transitions are not constant or preassigned, but expressed as a function of the continuous marking of the network (function which in chemistry is referred to as the "kinetics"). The algorithm can be applied to a broad class of such open networks, and returns, as an outcome, a classification of the possible dynamical behaviors that are compatible with the network structure, by classifying each variable either as bounded, converging to 0 or diverging to ?. This can be viewed as a qualitative study of Input---Output Stability for chemical networks, or more precisely, as a classification of its possible I---O instability patterns. Our goal is to analyze the system irrespectively of values of kinetic parameters. More precisely, we attempt to analyze it simultaneously for all possible values. Remarkably, tests on non-trivial examples (one of which is discussed in this paper) showed that, as the kinetic constants of the network are varied, all the compatible behaviors could be observed in simulations. Finally, we discuss and illustrate how the results relate to previous works on the qualitative dynamics of closed reaction networks.