Multiple positive steady states in subnetworks defined by stoichiometric generators

Abstract

In Systems Biology there is a growing interest in the question, whether or not a given mathematical model can admit more than one steady state. As parameter values are often unknown or subject to a very high uncertainty, one is often interested in the question, whether or not a given mathematical model can, for some conceivable parameter vector, exhibit multistationarity at all. A partial answer to this question is given in Feinberg's deficiency one algorithm. This algorithm can decide about multistationarity by analyzing systems of linear inequalities that are independent of parameter values. However, the deficiency one algorithm is limited to what its author calls regular deficiency one networks. Many realistic networks have a deficiency higher than one, thus the algorithm cannot be applied directly. In a previous publication it was suggested to analyze certain well defined subnetworks that are guaranteed to be of deficiency one. Realistic reaction networks, however, often lead to subnetworks that are irregular, especially if metabolic networks are considered. Here the special structure of the subnetworks is used to derive conditions for multistationarity. These conditions are independent of the regularity conditions required by the deficiency one algorithm. Thus, in particular, these conditions are applicable to irregular subnetworks.