# Graphical reduction of reaction networks by linear elimination of species

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## Abstract

The quasi-steady state approximation and time-scale separation are commonly applied methods to simplify models of biochemical reaction networks based on ordinary differential equations (ODEs). The concentrations of the “fast” species are assumed effectively to be at steady state with respect to the “slow” species. Under this assumption the steady state equations can be used to eliminate the “fast” variables and a new ODE system with only the slow species can be obtained. We interpret a reduced system obtained by time-scale separation as the ODE system arising from a unique reaction network, by identification of a set of reactions and the corresponding rate functions. The procedure is graphically based and can easily be worked out by hand for small networks. For larger networks, we provide a pseudo-algorithm. We study properties of the reduced network, its kinetics and conservation laws, and show that the kinetics of the reduced network fulfil realistic assumptions, provided the original network does. We illustrate our results using biological examples such as substrate mechanisms, post-translational modification systems and networks with intermediates (transient) steps.

### Keywords

Reduced network Quasi-steady-state Species graph Noninteracting Dynamical system Positivity### Mathematics Subject Classification

92C42 80A30## Notes

### Acknowledgments

We thank E. Tonello for pointing out a mistake in a previous version of the paper. We thank the reviewers for useful suggestions that improved the presentation of the manuscript. MS, EF, CW are supported by The Lundbeck Foundation (Denmark). EF and CW acknowledge funding from the Danish Research Council of Independent Research. MS has been supported by the project MTM2012-38122-C03-02/FEDER from the Ministerio de Economía y Competitividad, Spain.

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