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On the connectivity of the disguised toric locus of a reaction network

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Abstract

Complex-balanced mass-action systems are some of the most important types of mathematical models of reaction networks, due to their widespread use in applications, as well as their remarkable stability properties. We study the set of positive parameter values (i.e., reaction rate constants) of a reaction network G that, according to mass-action kinetics, generate dynamical systems that can be realized as complex-balanced systems, possibly by using a different graph \(G'\). This set of parameter values is called the disguised toric locus of G. The \({\mathbb {R}}\)-disguised toric locus of G is defined analogously, except that the parameter values are allowed to take on any real values. We prove that the disguised toric locus of G is path-connected, and the \({\mathbb {R}}\)-disguised toric locus of G is also path-connected. We also show that the closure of the disguised toric locus of a reaction network contains the union of the disguised toric loci of all its subnetworks.

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Notes

  1. Note that when \(\varvec{y}_0 \not \in V\) or \(\varvec{y}_0 \not \in \tilde{V}\), that side is considered as an empty sum, which is zero.

  2. For simplicity, in the rest of this paper, we abuse the following notation: Given \(\varvec{k}\in {\mathbb {R}}_{>0}^{E}\), we will refer to the mass-action system generated by G and \(\varvec{k}\) as in Eq. (2) as “the mass-action system \((G, \varvec{k})\)”. Moreover, we will still refer to this system as “the mass-action system \((G, \varvec{k})\)” even if we have \(\varvec{k}\in {\mathbb {R}}^{E}\) instead of \(\varvec{k}\in {\mathbb {R}}_{>0}^{E}\).

  3. Some authors exclude the empty set from being path-connected, but we do not follow this convention here.

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Acknowledgements

This work was supported in part by the National Science Foundation grant DMS-2051568.

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Authors and Affiliations

  1. Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, USA

    Gheorghe Craciun

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

    Abhishek Deshpande

  3. Department of Mathematics, The Ohio State University, Columbus, USA

    Jiaxin Jin

Authors
  1. Gheorghe Craciun
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  2. Abhishek Deshpande
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  3. Jiaxin Jin
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Correspondence to Abhishek Deshpande.

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Craciun, G., Deshpande, A. & Jin, J. On the connectivity of the disguised toric locus of a reaction network. J Math Chem 62, 386–405 (2024). https://doi.org/10.1007/s10910-023-01533-0

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