Passivity-based analysis of biochemical networks displaying homeostasis

Abstract

Homeostasis refers to the ability of organisms and cells to maintain a stable internal environment even in the presence of a changing external environment. On the cellular level many compounds such as ions, pH, proteins, and transcription factors have been shown to be tightly regulated, and mathematical models of biochemical networks play a major role in elucidating the mechanisms behind this behaviour. Of particular interest is the control theoretic properties of these models, e.g. stability and robustness. The simplest models consist of two components, a controlled compound and a controller compound. We have previously explored how signalling between these two compounds can be arranged in order for the network to display homeostasis, and have constructed a class of eight two-component reaction kinetic networks with negative feedback that shows set-point tracking and disturbance rejection properties. Here, we take a closer look at the stability and robust control inherent to this class of systems. We show how these systems can be described as negative feedback connections of two nonlinear sub-systems, and show that both sub-systems are output strictly passive and zero-state detectable. Using a passivity-based approach, we show that all eight systems in this class of two-component networks are asymptotically stable.