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Computation 2017, 5(1), 14; doi:10.3390/computation5010014

Simplification of Reaction Networks, Confluence and Elementary Modes

1
CRIStAL—Centre de Recherche en Informatique Signal et Automatique de Lille—CNRS UMR 9189, Université de Lille, F-59000 Lille, France
2
School of Mathematical Sciences, University of Nottingham, NG7 2RD Nottingham, UK
3
INRIA—French Institute for Research in Computer Science and Automation, 59000 Lille, France
4
This paper is an extended version of our paper published in the Proceedings of the International Conference on Computational Methods in Systems Biology 2016, Cambridge, UK, 21
*
Author to whom correspondence should be addressed.
Academic Editors: Gennady Bocharov, Olga Solovyova and Vitaly Volpert
Received: 17 January 2017 / Accepted: 26 February 2017 / Published: 10 March 2017
(This article belongs to the Special Issue Multiscale and Hybrid Modeling of the Living Systems)

Abstract

Reaction networks can be simplified by eliminating linear intermediate species in partial steadystates. Inthispaper,westudythequestionwhetherthisrewriteprocedureisconfluent,so that for any given reaction network with kinetic constraints, a unique normal form will be obtained independently of the elimination order. We first show that confluence fails for the elimination of intermediates even without kinetics, if “dependent reactions” introduced by the simplification are not removed. This leads us to revising the simplification algorithm into a variant of the double description method for computing elementary modes, so that it keeps track of kinetic information. Folklore results on elementary modes imply the confluence of the revised simplification algorithm with respect to the network structure, i.e., the structure of fully simplified networks is unique. We show, however, that the kinetic rates assigned to the reactions may not be unique, and provide a biological example where two different simplified networks can be obtained. Finally, we give a criterion on the structure of the initial network that is sufficient to guarantee the confluence of both the structure and the kinetic rates. View Full-Text
Keywords: simplification; confluence; reaction network; ordinary differential equations; deterministic semantics; elementary modes; system biology; rewriting rules simplification; confluence; reaction network; ordinary differential equations; deterministic semantics; elementary modes; system biology; rewriting rules
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Madelaine, G.; Tonello, E.; Lhoussaine, C.; Niehren, J. Simplification of Reaction Networks, Confluence and Elementary Modes. Computation 2017, 5, 14.

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