Recasting the mass-action rate equations of open chemical reaction networks into a universal quadratic format

Abstract

Recasting the rate equations of mass-action chemical kinetics into universal formats is a potentially useful strategy to rationalize typical features that are observed in the space of the species concentrations. For example, a remarkable feature is the appearance of the so-called slow manifolds (subregions of the concentration space where the trajectories bundle), whose detection can be exploited to simplify the description of the slow part of the kinetics via model reduction and to understand how the chemical network approaches the stationary state. Here we focus on generally open chemical reaction networks with continuous injection of species at constant rates, that is, the situation of idealized biochemical networks and microreactors under well-mixing conditions and externally controllable input of chemicals. We show that a unique format of pure quadratic ordinary differential equations can be achieved, regardless of the nonlinearity of the kinetic scheme, by means of a suitable change and extension of the set of dynamical variables. Then we outline some possible employments of such a format, with special emphasis on a low-computational-cost strategy to localize the slow manifolds which are indeed observed also for open systems.