Seminar by Meritxell Saez Cornellana

Recovering a reaction network after linear elimination of species

The formalism of chemical reaction network theory (CRNT) puts chemical reaction networks into a mathematical, particularly algebraic, framework. In this framework, the steady states of a chemical reaction network with mass-action kinetics are solutions to a system of polynomial equations. Even for small systems, finding the steady states of the system is a very demanding task and therefore methods that reduce the number of variables are desirable. Elimination of certain species in the network is closely linked to the procedure known as the quasi-steady state approximation, which is often employed to simplify the modeling equations. Under the quasi-steady state approximation, some reactions are assumed to occur at a much faster rate than other reactions, that is, there is a separation of time scales, such that a steady state effectively has been reached for the fast reactions. In this setting, the goal is to study the evolution of the species that have not reached steady state (slow variables), as a new reaction network on their own.

Following the ideas introduced in (Feliu-Wiuf, 2012) for the elimination of so-called non-interacting species, we give a graphical method to find a reaction network on the slow variables as well as their production rates. Our method reinterprets the system of equations obtained after substitution of the eliminated variables (fast variables), as a new reaction network with mass-action kinetics, Michaelis-Menten-like kinetics or similar.