# A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems

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## Abstract

We introduce a mixed-integer linear programming (MILP) framework capable of determining whether a chemical reaction network possesses the property of being endotactic or strongly endotactic. The network property of being strongly endotactic is known to lead to persistence and permanence of chemical species under genetic kinetic assumptions, while the same result is conjectured but as yet unproved for general endotactic networks. The algorithms we present are the first capable of verifying endotacticity of chemical reaction networks for systems with greater than two constituent species. We implement the algorithms in the open-source online package CoNtRol and apply them to a large sample of networks from the European Bioinformatics Institute’s BioModels Database. We use strong endotacticity to establish for the first time the permanence of a well-studied circadian clock mechanism.

## Keywords

Chemical reaction network Chemical kinetics Persistence Permanence Endotactic## Mathematics Subject Classification

92C42 90C35## Notes

### Acknowledgments

M. J. is supported by grant Army Research Office grant W911NF-14-1-0401. P.D. is supported by EPSRC grant EP/J00826/1. The authors thank Gheorghe Craciun for helpful discussions and suggestions, and Murad Banaji for organizing the workshop “Combinatorial and algebraic approaches to chemical reaction networks” at the University of Portsmouth (UK) at which this collaboration began. The authors also thank the anonymous referees whose comments have significantly improved the readability of the paper.

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