Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements

We extend previous work on injectivity in chemical reaction networks to general interaction networks. Matrix- and graph-theoretic conditions for injectivity of these systems are presented. A particular signed, directed, labelled, bipartite multigraph, termed the ``DSR graph'', is shown to be a useful representation of an interaction network when discussing questions of injectivity. A graph-theoretic condition, developed previously in the context of chemical reaction networks, is shown to be sufficient to guarantee injectivity for a large class of systems. The graph-theoretic condition is simple to state and often easy to check. Examples are presented to illustrate the wide applicability of the theory developed.Comment: 34 pages, minor corrections and clarifications on previous versio

Toric dynamical systems

Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded.Comment: We include the proof of our Conjecture 5 (now Lemma 5) and add some reference

The structure of the moduli spaces of toric dynamical systems

We consider complex balanced mass-action systems, also called toric dynamical systems. They are polynomial dynamical systems arising from reaction networks and have remarkable dynamical properties. We study the topological structure of their moduli spaces (i.e., the toric locus). First we show that the complex balanced equilibria depend continuously on the parameter values. Using this result, we prove that the moduli space of any toric dynamical system is connected. In particular, we emphasize the product structure of the moduli space: it is homeomorphic to the product of the set of complex balanced flux vectors and the affine invariant polyhedron.Comment: 16 pages, 4 figure

On the Connectivity of the Disguised Toric Locus of a Reaction Network

Complex-balanced mass-action systems are some of the most important types of mathematical models of reaction networks, due to their widespread use in applications, as well as their remarkable stability properties. We study the set of positive parameter values (i.e., reaction rate constants) of a reaction network $G$ that, according to mass-action kinetics, generate dynamical systems that can be realized as complex-balanced systems, possibly by using a different graph $G'$. This set of parameter values is called the disguised toric locus of $G$. The $\mathbb{R}$-disguised toric locus of $G$ is defined analogously, except that the parameter values are allowed to take on any real values. We prove that the disguised toric locus of $G$ is path-connected, and the $\mathbb{R}$-disguised toric locus of $G$ is also path-connected. We also show that the closure of the disguised toric locus of a reaction network contains the union of the disguised toric loci of all its subnetworks.Comment: 18 pages, 2 figure