Cycle Structure in SR and DSR Graphs: Implications for Multiple Equilibria and Stable Oscillation in Chemical Reaction Networks

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. 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 in chemical reaction networks 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, 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 what CRN behaviours are determined primarily by reaction network structure, with limited assumptions about the kinetics. A variety of representations of CRN structure are possible, for example via matrices or generalised graphs. Of these, a signed, labelled, bipartite multigraph, termed an SR graph, and its directed version, the DSR graph, are formally similar to Petri nets. This relationship is discussed further below. It is now well established that graphical representations can tell us a great deal about asymptotic behaviours in the associated dynamical systems. Pioneering early work on CRNs with mass-action kinetics ([2, 3] for example), had a graph-theoretic component (using graphs somewhat different from those to be presented here). More recently, graph-theoretic approaches have been used to draw conclusions about multistationarity and oscillation in CRNs with restricted classes of kinetics [4, 5]. Recent Advances in Petri Nets and Concurrency, S. Donatelli, J. Kleijn, R.J. Machado, J.M. Fernandes (eds.), CEUR Workshop Proceedings, ISSN 1613-0073, Jan/2012, pp. 7–22. The applicability of such work, particularly in biological contexts, is greatly increased if only weak assumptions are made about kinetics. Consequently, there is a growing body of recent work on CRNs with essentially arbitrary kinetics. It has been shown that examination of Petri nets associated with a CRN allows conclusions about persistence, that is, whether ω-limit sets of interior points of R≥0 can intersect the boundary of R≥0 [6]. Work on multistationarity has been extended beyond the mass-action setting [7, 8]: some conclusions of this work will be outlined below. Finally, recent work applying the theory of monotone dynamical systems [9, 10] in innovative ways to CRNs [11] has close links with some of the new material presented below. Outline. After some preliminaries, the construction of SR and DSR graphs is presented, and their relationship to Petri nets is discussed. Some recent results about multistationarity based on cycle structure in these objects are described. Subsequently, a new result on monotonicity in CRNs is proved. This result, Proposition 4, is a graph-theoretic corollary of results in [12]. It bears an interesting relationship to results in [11], which provide stronger conclusions about convergence, but make different assumptions, and a somewhat different claim. Finally, several examples, some raising interesting open questions, are presented. At various points, in order to simplify the exposition, the results are presented in less generality than possible, with more technical results being referenced.