Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

Abstract

Dynamical system models of complex biochemical reaction networks are high-dimensional, nonlinear, and contain many unknown parameters.The capacity for multiple equilibria in such systems plays a key rolein important biochemical processes. Examples show that there is a verydelicate relationship between the structure of a reaction network andits capacity to give rise to several positive equilibria. In thispaper we focus on networks of reactions governed by mass-actionkinetics. As is almost always the case in practice, we assume that noreaction involves the collision of three or more molecules at the sameplace and time, which implies that the associated mass-actiondifferential equations contain only linear and quadratic terms. Wedescribe a general injectivity criterion for quadratic functions ofseveral variables, and relate this criterion to a network's capacityfor multiple equilibria. In order to take advantage of this criterionwe look for explicit general conditions that imply non-vanishing ofpolynomial functions on the positive orthant.In particular, we investigate in detail the case of polynomials withonly one negative monomial, and we fully characterize the case ofaffinely independent exponents.We describe several examples, including an example that shows howthese methods may be used for designing multistable chemical systemsin synthetic biology.