Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species. Dynamical systems arising from the law of mass-action kinetics are commonly used to model phenomena described by networks of interacting species. The main examples are found in chemistry and molecular biology but instances are also found in ecology and epidemiology (such as the Lotka-Volterra and SIR models). Typical systems found in molecular biology are high-dimensional and contain many parameters that are unknown or poorly determined. It is therefore of interest to determine properties of these systems that are inherent of the network structure alone and independent of the specific choice of parameters. Of particular relevance for applications in systems biology is the determination of network structures for which the corresponding dynamical systems have multiple positive steady states for some choice of parameters. Multistationarity in cellular systems provides a mechanism for switching between different cellular responses and can be crucial for cellular decision making. Even though different features, such as feedback loops, are known that facilitate multistationarity in systems , it is in general difficult to decide whether a particular system has the capacity to exhibit multiple steady states. Various criteria have therefore been developed to preclude the existence of multiple positive steady states. These criteria typically utilize the structure or qualitative features of the system [1–3] or properties of the class of kinetics that are allowed [4–6]. Additionally, algorithms to positively assert multistationarity have also been developed [7,8]. It is the …