Linear Elimination in Chemical Reaction Networks

  • Meritxell Sáez
  • Elisenda Feliu
  • Carsten WiufEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 18)


We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of \({\mathbb R}^n\). The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.


Steady states Elimination Linear system Reaction network Noninteracting Conservation law 



This research was supported by the Lundbeck Foundation (Denmark) and the Danish Research Council.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenKøbenhavnDenmark

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