Journal of Mathematical Chemistry

, Volume 57, Issue 4, pp 1001–1018 | Cite as

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

  • Alessandro Ceccato
  • Paolo Nicolini
  • Diego FrezzatoEmail author
Original Paper


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.


Chemical kinetics Mass-action law Open reaction networks Polynomial ordinary differential equations Embedding into Lotka–Volterra Slow manifolds 

Mathematics Subject Classification

80A30 34A05 34A34 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Chemical SciencesUniversity of PadovaPaduaItaly
  2. 2.Department of Control Engineering - K335, Faculty of Electrical EngineeringCzech Technical University in PraguePrague 2Czech Republic

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