Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property

Abstract

Abstract. For mass action kinetics, the capacity for multiple equilibria in an isothermal homogeneous continuous flow stirred tank reactor is determined by the structure of the underlying network of chemical reactions. We suggest a new graph-theoretical method for discriminating between complex reaction networks that can admit multiple equilibria and those that cannot. In particular, we associate with each network a species-reaction graph, which is similar to reaction network representations drawn by biochemists, and we show that, if the graph satisfies certain weak conditions, the differential equations corresponding to the network cannot admit multiple equilibria no matter what values the rate constants take. Because these conditions are very mild, they amount to powerful (and quite delicate) necessary conditions that a network must satisfy if it is to have the capacity to engender multiple equilibria. Broad qualitative results of this kind are especially apt, for individual reaction rate constants are rarely known fully for complex reaction networks (if they are known at all). Some concluding remarks address connections to biology. 1]). Models in cell biology sometimes invoke pictures and mathematics reminiscent of CFSTRs [9,12,17,14], so it not unreasonable to expect that theory presented here might ultimately provide insight that is useful in biological applications. Indeed, in biology one rarely has detailed knowledge of reaction rate constants; at the outset, then, it is especially appropriate to seek a qualitative understanding of the relationship between reaction network structure and the capacity for various kinds of behavior (e.g., bistability). As we indicated in the first article of this series [4], the connection between the two is quite delicate. The theory offered here is intended to render the relationship between reaction network structure and behavior more concrete.Our principal results will serve to describe very large classes of networks, including highly complex ones, that cannot give rise to multiple steady states regardless of parameter values. These results provide very strong necessary conditions that a network must satisfy if it is to have the capacity to give rise, for example, to bistable behavior.

Understanding bistability in complex enzyme-driven reaction networks

“…The following theorem (19)(20)(21) extends considerably an earlier theorem of Schlosser and Feinberg (17 The theorem indicates that, for a mass-action network to have the capacity for more than one steady state, its SR graph must satisfy quite stringent conditions. In particular, when every stoichiometric coefficient is a 1 (the common situation), two even-cycles must split a c-pair.…”

Parameter-free methods distinguish Wnt pathway models and guide design of experiments

“…We test the injectivity of each model following graph-theoretic or Jacobian-based approaches used in CRNT (36)(37)(38)(39)(40). We find that only the Schmitz et al (7) and the shuttle model fail injectivity and exhibit multistationarity.…”

Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models

“…As explained in [4], the added outflow reactions A → 0, B → 0, and C → 0 serve to model the contributions of the outflow stream to the CFSTR differential equations (taking each rate constant to be g/V ), while the feed reactions 0 → A, 0 → B, and 0 → C serve to model the contributions of the feed stream (taking the rate constants to be, respectively, gc f A /V , gc f B /V , and gc f C /V ). In general, to obtain the augmented network, one adds to the network of true chemical reactions an outflow reaction s → 0 for each s ∈ S , and a feed reaction 0 → s for each species s deemed to be in the feed stream.…”

“…, c f Z the (fixed) concentrations of the species in the feed stream. We assume hereafter that the rates of the chemical reactions are governed by mass action kinetics [4,6,7,10,11]. In this case, the system of differential equations associated with network (1.1) is the following:…”