Algebraic methods for the study of biochemical reaction networks

Abstract

We will concentrate on biochemical reaction networks, of interest in systems biology, in particular enzymatic networks, consisting of different types of multisite phosphorylation networks. One source of inspiration for our study with algebro-geometric tools is the following quote from the abstract of the paper [44]:"Multisite phosphorylation cycles are ubiquitous in cell regulation systems andare studied at multiple levels of complexity, from molecules to organisms, withthe ultimate goal of establishing predictive understanding of the effects of geneticand pharmacological perturbations of protein phosphorylation in vivo. Achievingthis goal is essentially impossible without mathematical models, which providea systematic framework for exploring dynamic interactions of multiple networkcomponents."We will mainly concentrate on recent advances on the determination of multistationarity for these networks, whose dynamics are usually modeled with mass-action kinetics. For many classes of chemical networks, as the complex balanced networks, monostationarity is an important property. Instead, for biochemical reaction networks, that is, chemical reaction networks modeling pathways in systems biology, multistationarity is a general feature and it is important because it is intepreted as a way for the cell to take different decisions. Indeed, differential systems with mass-action kinetics are deterministic. But the occurrence of multiple stable steady states in the same stoichiometric compatibility class implies that trajectories starting from different initial conditions with the same conserved quantities can converge to steady states with different properties. We will end the chapter with some open questions.