Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems

• Published in: Advances in applied mathematics Vol. 44; no. 2; pp. 168 - 184
• Main Authors: Banaji, Murad, Craciun, Gheorghe
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.02.2010; Elsevier
• Subjects: Chemical reactions; Exact sciences and technology; General mathematics; General, history and biography; Injectivity; Mathematics; Multiple equilibria; Network structure; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; SR graph; SR graph; Network structure; 05C50; 34C99; Chemical reactions; 15A15; Multiple equilibria; Injectivity; 05C38; Chemical reaction; Applied mathematics; Implementation
• ISSN: 0196-8858; 1090-2074; 0196-8858
• DOI: 10.1016/j.aam.2009.07.003

In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis–Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with kinetics subject only to some weak natural constraints. The test, which is easy to implement algorithmically, and can often be decided without the need for any computation, rules out the possibility of multiple equilibria for the systems in question.