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.
Highlights
A chemical reaction network is usually given by a finite list of reactions that involve a finite set of chemical species
While the results discussed in the previous section apply to general polynomial inequalities, we discuss some examples of how one can use them to study the capacity for multiple equilibria of reaction networks
According to Theorem 2.1, it will follow that r(·, k) is injective on R6>0 for those values of k, and, as we noted in the introduction, this rules out the capacity for multiple equilibria of (2) for those values of k
Summary
A chemical reaction network is usually given by a finite list of reactions that involve a finite set of chemical species. We prove an equivalence between local and global injectivity for quadratic polynomial functions on a convex domain, which will imply that for the class of uni- and bi-molecular reaction networks, the answer to Question 1 is affirmative.
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