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This paper investigates the geometric properties of steady state varieties arising from reaction networks with power-law kinetics, addressing questions about the generic finiteness of steady states, robustness, and nondegenerate multistationarity. The authors provide an ideal-theoretic characterization of generic absolute concentration robustness, a property describing when certain molecular species maintain constant concentrations despite perturbations. Using the framework of vertically parametrized systems, they establish a linear algebra condition that determines when the steady state system admits positive nondegenerate zeros, and identify conditions under which networks capable of multiple steady states also support nondegenerate multistationarity.
Why it matters
Understanding the geometry of steady states in reaction networks has direct implications for the analysis of biochemical systems, including gene regulatory networks and metabolic pathways, where robustness and multistationarity are key properties linked to cellular decision-making and homeostasis. These results provide mathematical tools that could help researchers systematically characterize and predict the behavior of complex biological networks.
arXiv:2412.17798v3 Announce Type: replace
Abstract: We answer several fundamental geometric questions about reaction networks with power-law kinetics, on topics such as generic finiteness of the number of steady states, robustness, and nondegenerate multistationarity. In particular, we give an ideal-theoretic characterization of generic absolute concentration robustness, as well as conditions under which a network that admits multiple steady states also has the capacity for nondegenerate multistationarity. The key tools underlying our results come from the theory of vertically parametrized systems, and include a linear algebra condition that characterizes when the steady state system has positive nondegenerate zeros.