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Autocatalysis in Reaction Networks

Bulletin of Mathematical Biology volume 76, pages 2570–2595 (2014)Cite this article

Abstract

The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of “critical siphon.” We explore the combinatorics of critical siphons, with a view toward the persistence conjecture. We introduce the notions of “drainable” and “self-replicable” (or autocatalytic) siphons. We show that: Every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and nonautocatalytic weakly reversible networks are persistent. Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons.

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Acknowledgments

We thank Jaikumar Radhakrishnan for pointing out that our proof of Theorem 4.9 showed a result mildly stronger (\(\mathbb {R}^n_{<0} \subseteq \mathrm{cone }(\mathrm{rows }(A))\)) than what we had asserted in a previous draft (\(\mathrm{cone }(\mathrm{rows }(A)) \cap \mathbb {R}^n_{<0}\) is nonempty). We thank the referees for useful comments.

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Affiliations

  1. Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Gachibowli, Hyderabad, India

    Abhishek Deshpande

  2. School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India

    Manoj Gopalkrishnan

Authors
  1. Abhishek Deshpande
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  2. Manoj Gopalkrishnan
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Correspondence to Manoj Gopalkrishnan.

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Deshpande, A., Gopalkrishnan, M. Autocatalysis in Reaction Networks. Bull Math Biol 76, 2570–2595 (2014). https://doi.org/10.1007/s11538-014-0024-x

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  • DOI: https://doi.org/10.1007/s11538-014-0024-x

Keywords

  • Critical siphons
  • Persistence
  • Reaction network
  • Autocatalysis
  • Petri net