Deficiency of chemical reaction networks: the effect of operations that preserve multistationarity and periodic orbits

Awildo Gutierrez; Elijah Leake; Caelyn Rivas-Sobie; Jordy Lopez Garcia; Anne Shiu

doi:10.3934/dcdsb.2024157

Article Contents

2025, Volume 30, Issue 5: 1745-1761. Doi: 10.3934/dcdsb.2024157

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Deficiency of chemical reaction networks: the effect of operations that preserve multistationarity and periodic orbits

1.
University of Wisconsin-Madison, USA
2.
DePaul University, USA
3.
Johns Hopkins University, USA
4.
Texas A&M University, USA

^*Corresponding author: Awildo Gutierrez
^*Corresponding author: Awildo Gutierrez

Received: December 1, 2023

Revised: September 1, 2024

Early access: October 31, 2024

Published online: October 31, 2024

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Abstract

We investigate six operations on chemical reaction networks, all of which have been proven to preserve important dynamical properties, namely, the capacity for nondegenerate multistationarity (multiple steady states) and periodic orbits. Both multistationarity and periodic orbits are properties that are known to be precluded when the deficiency (a nonnegative integer associated to a network) is zero. It is therefore natural to conjecture that the deficiency never decreases when any of the six aforementioned network operations are performed. We prove that this is indeed the case, and moreover, we characterize the numerical difference in deficiency after performing each network operation.

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Mathematics Subject Classification: Primary: 37N25, 05C90; Secondary: 05C76, 15A03.

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Table 1. Summary of results. Here, $\delta(\mathcal N)$, $\operatorname{rk}(\mathcal N)$, and $s_ \mathcal N$ denote, respectively, the deficiency, rank, and number of species of a network $ \mathcal N$. Also, $\widetilde \ell_{ \mathcal N}$ is the number of linkage classes in $ \mathcal N$ that contain an at-most-unimolecular complex (namely, $0$ or some $X_i$), while $m_{ \mathcal N}$ is the number of at-most-unimolecular complexes that are missing from $ \mathcal N$ (see the Notation following Definition 3.9). Finally, $\operatorname{cyc}(\mathcal N)$ is the cyclomatic number of $ \mathcal N$ (see Definition 2.6), and the rank condition is given in Definition 2.10. For details, see Theorems 3.7, 3.11, 3.13, 3.16, 3.18, and 3.19

Operation	Description	$\delta( \mathcal N')-\delta( \mathcal N)$
E1	Add a new linearly dependent reaction	$0$ or $1$
E2	Add reactions $0 \leftrightarrows X_i$ for all species $X_i$	$\operatorname{rk}( \mathcal N) + m_ \mathcal N + \widetilde \ell_ \mathcal N - s_ \mathcal N - 1$
E3	Add a new linearly dependent species	$\operatorname{cyc}( \mathcal N) - \operatorname{cyc}( \mathcal N')$
E4	Add a new species $Y$ and the pair of reversible reactions $0 \leftrightarrows Y$	$\operatorname{cyc}( \mathcal N) - \operatorname{cyc}( \mathcal N') + 1$
E5	Add reversible reactions with new species such that rank condition holds	$0$
E6	Split reactions and add complexes involving new species with rank condition	$0$

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Table 2. Parameters associated to a network $ \mathcal N$

$s_ \mathcal N$	$c_ \mathcal N$	$r_ \mathcal N$	$\ell_ \mathcal N$	$\operatorname{rk}( \mathcal N)$
Add a new linearly dependent reaction	Add reactions $0 \leftrightarrows X_i$ for all species $X_i$	Add a new linearly dependent species	Add a new species $Y$ and the pair of reversible reactions $0 \leftrightarrows Y$	rank

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