Section: New Results
Reaction Models and Influence Graphs
Biologists use diagrams to represent interactions between molecular species, and on the computer, diagrammatic notations are also more and more employed in interactive maps. These diagrams are fundamentally of two types: reaction graphs and positive/negative influence graphs. The analysis of circuits in the influence graphs has been introduced by René Thomas in the late 70's with some conjectures on necessary conditions for oscillations (homeostasie) and multistability (cell differentiation). These conjectures have been proven under various conditions in the setting of ordinary differential setting.equations, and more recently in  by Rémy, Ruet and Thieffry in the setting of boolean logical models.
In  , we study the formal relationship between reaction graphs and influence graphs. We consider systems of biochemical reactions with kinetic expressions, as written in the Systems Biology Markup Language SBML, and interpreted by a system of Ordinary Differential Equations over molecular concentrations. We show that under a general condition of increasing monotonicity of the kinetic expressions, and in absence of both positive and negative influences between a pair of molecules, the influence graph inferred from the stoichiometric coefficients of the reactions is equal to the one defined by the signs of the coefficients of the Jacobian matrix. Under these conditions, satisfied by mass action law, Michaelis-Menten and Hill kinetics, the influence graph is thus independent of the precise kinetic expressions, and is computable in linear time in the number of reactions. We apply these results to Kohn's map of the mammalian cell cycle (500 variables and 800 rules) and to the MAPK signalling cascade. Then we propose a syntax for denoting antagonists in reaction rules and generalize our results to this setting.