Some Results on Injectivity and Multistationarity in Chemical Reaction Networks

Abstract

Abstract. The goal of this paper is to gather and develop some necessary and sufficient criteria for injectivity and multistationarity in vector fields associated with a chemical reaction network under a variety of more or less general assumptions on the nature of the network and the reaction rates. The results are primarily linear algebraic or matrix-theoretic, with some graph-theoretic results also mentioned. Several results appear in, or are close to, results in the literature. Here, we emphasise the connections between the results, and where possible, present elementary proofs which rely solely on basic linear algebra and calculus. A number of examples are provided to illustrate the variety of subtly different conclusions which can be reached via different computations. In addition, many of the computations are implemented in a web-based open source platform, allowing the reader to test examples including and beyond those analysed in the paper.

Bistability and Bifurcation in Minimal Self‐Replication and Nonenzymatic Catalytic Networks

“…In real systems, the free energies of the various solutions may not be identical, in which case only the minimum value represents thermodynamic equilibrium and the other points will be local minima, and over time, the system will tend to approach its equilibrium. However, many cases of bistability and multistability that have been studied, both theoretically and experimentally, are kinetic steady‐state solutions in open, dissipative systems that also display various types of switching among these states ,,,,,,,,,,,,,. We note that our solutions are obtained by using the same mathematical methods, that is, by using mass‐action kinetics to find the points at which the rates of production are zero and are, therefore, closed‐system analogues to the previous solutions.…”

Mathematical Analysis of Chemical Reaction Systems

“…More details and examples about the use of the Jacobian criterion can be found in . Further results and generalizations for closed or “semi‐open” systems can be found in .…”

Mathematical Methods for Modeling Chemical Reaction Networks

“…1. biology, and the development of new and powerful mathematical techniques. A very fruitful area of research has been in identifying conditions for multiple positive steady states in reaction networks [1,6,7,8,9]. In particular, the cycle structure of the SR graph [1] or DSR graph [7] associated to a network may rule out bistability for any choice of rate constants.…”