Transactions on Petri Nets and Other Models of Concurrency V

Associated with a chemical reaction network is a natural labelled bipartite multigraph termed an SR graph, and its directed version, the DSR graph. These objects are closely related to Petri nets, but encode weak assumptions on the reaction kinetics, and are more generally associated with continuous-time, continuous-state models rather than discrete-event systems. The construction of SR and DSR graphs for chemical reaction networks is presented. Conclusions about asymptotic behaviour of the associated dynamical systems which can be drawn easily from the graphs are discussed. In particular, theorems on ruling out the possibility of multiple equilibria or stable oscillation based on computations on SR/DSR graphs are presented. These include both published and new results. The power and limitations of such results are illustrated via several examples. 1 Chemical Reaction Networks: Structure and Kinetics Models of chemical reaction networks (CRNs) are able to display a rich variety of dynamical behaviours [1]. In this paper, a spatially homogeneous setting is assumed, and the state of a CRN is defined to be the set of concentrations of the reactants involved, each a nonnegative real number. In addition, continuoustime models are treated, so that CRNs involving n chemicals give rise to local semiflows on R≥0, the nonnegative orthant in R . These local semiflows are fully determined if we know 1) the CRN structure, that is, which chemicals react with each other and in what proportions, and 2) the CRN kinetics, that is, how the reaction rates depend on the chemical concentrations. An important question is which CRN behaviours are determined primarily by reaction network structure, with limited assumptions about the kinetics. As will be seen below, a variety of representations of CRN structure are possible, for example via matrices or generalised graphs. Some of these representations encode weak assumptions on the reaction kinetics. Of these, a signed, labelled, bipartite multigraph, termed an SR graph, and its directed version, 1 The term derives from “species-reaction graph” [2]. However, while CRNs provided the original motivation for their construction, such graphs have since proved useful in more general contexts. The construction here follows [3]. K. Jensen, S. Donatelli, and J. Kleijn (Eds.): ToPNoC V, LNCS 6900, pp. 1–21, 2012. c © Springer-Verlag Berlin Heidelberg 2012