The Dimension of the Disguised Toric Locus of a Reaction Network

Mathematical models of reaction networks are ubiquitous in applications, especially in chemistry, biochemistry, chemical engineering, ecology, and population dynamics. Under the standard assumption of mass-action kinetics, reaction networks give rise to general dynamical systems with polynomial right-hand side. These depend on many parameters that are difficult to estimate and can give rise to complex dynamics, including multistability, oscillations, and chaos. On the other hand, a special class of reaction systems called complex-balanced systems are known to exhibit remarkably stable dynamics; in particular, they have unique positive fixed points and no oscillations or chaotic dynamics. One difficulty, when trying to take advantage of the remarkable properties of complex-balanced systems, is that the set of parameters where a network satisfies complex balance may have positive codimension and therefore zero measure. To remedy this we are studying disguised complex balanced systems (also known as disguised toric systems), which may fail to be complex balanced with respect to an original reaction network $G$, but are actually complex balanced with respect to some other network $G^{\prime }$, and therefore enjoy all the stability properties of complex-balanced systems. This notion is especially useful when the set of parameter values for which the network $G$ gives rise to disguised toric systems (i.e., the disguised toric locus of $G$) has codimension zero. Our primary focus is to compute the exact dimension (and therefore the codimension) of this locus. We illustrate the use of our results by applying them to Thomas-type and circadian clock models.