Chemical Reaction Networks: Comparing Deficiency, Determinant Expansions, and Sign Patterns

This paper will treat three topics motivated by chemical reaction networks with massaction kinetics; they are commonly referred to as deficiency, determinant expansions, and sign patterns. The dynamics of a chemical reaction network are governed by a non-linear system of ordinary differential equations dx dt = f(x), and what is observed in experiments is often equilibria states, {x : f(x) = 0} of this differential equation. A major issue is counting the number of equilibria, and in particular determining if there is a unique equilibrium. There are two main hypotheses which insure that a chemical reaction has a unique equilibrium. The most classical one is known as the deficiency 0 condition, while more recently a condition on the determinant of the Jacobian of f has been fruitful. Since either of these conditions on f separately imply that there is a unique equilibria, it is natural to ask: does one condition imply the other? In this thesis we demonstrate that the answer is no. Indeed we give concrete examples showing that the ”deficiency” and key properties of the Jacobian’s determinant expansion have no simple bearing on each other. The thesis also contains a study of the two standard ways of representing f for a chemical reaction network; we refer to them as the Stoichiometric Representation and the Complexes Representation. We show that a system which has one of these representations also has the other and we layout precisely the correspondence between them. At some level this is in the chemistry literature, but we could not find a reference which did this thoroughly at a high level of generality. The thesis concludes with another topic, that of ”sign patterns” of the Jacobian of f . Through various examples, we show sign patterns also have no relation to the deficiency of a chemical network. Additionally, we extend some known theoretical results on sign patterns of Jacobians and give an approach and results showing how to proceed on networks which fail to have a sign pattern.