Homeostatic mechanisms in biological systems: Case studies in modeling copper regulation and self-immune recognition

In this paper we investigate homeostatic mechanisms in two biologically motivated models: intracellular copper regulation and self immune recognition. The analysis is based on the notion of infinitesimal homeostasis for systems of ordinary differential equations and the formalism of input–output networks. We show that, unlike the examples of Reed et al. (2017), the obstruction of occurrence of infinitesimal homeostasis in both examples is due to the functional form of the differential equations being to far from the ‘generic form’. More specifically, from a ‘generic’ or ‘model-independent’ point of view the models are expected to exhibit infinitesimal homeostasis, but due to functional form of the differential equations, forced by the modeling assumptions used to define the models, some of the ‘expected generic behaviors’ cannot occur. We first analyze both models from the ‘model-independent’ point of view and then show that in the specific models some ‘expected generic behavior’ cannot occur. We illustrate these phenomena with numerical simulations and give rigorous arguments in some cases.