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Weakly reversible deficiency zero realizations of reaction networks

Neal Buxton, Gheorghe Craciun, Abhishek Deshpande, Casian Pantea

TL;DR

The paper addresses when a given mass-action reaction network admits a ${\\textrm{WR}_0}$ realization for all rate constants and proves the existence of a unique target network ${\\mathcal{N}}'$ with ${\\mathcal{N}}$ realizable by ${\\mathcal{N}}'$. It develops a framework based on polyhedral cones and network realizations to decide and construct this WR0 realization, establishing independence from rate constants and providing a constructive algorithm. A key contribution is the proof that, whenever a WR0 realization exists for a network, it is unique and universal across rate constants, enabling a canonical surrogate network for robust dynamics analysis; this is complemented by a practical software implementation in CoNtRol using GLPK. The work enhances understanding of dynamical equivalence and robustness in reaction networks and offers tools for model reduction by substituting a WR0 surrogate while preserving the ODE dynamics.

Abstract

We prove that if a given reaction network $\mathcal{N}$ has a weakly reversible deficiency zero realization for all choice of rate constants, then there exists a $\textit{unique}$ weakly reversible deficiency zero network $\mathcal{N}'$ such that $\mathcal{N}$ is realizable by $\mathcal{N}'$. Additionally, we propose an algorithm to find this weakly reversible deficiency zero network $\mathcal{N}'$ when it exists.

Weakly reversible deficiency zero realizations of reaction networks

TL;DR

The paper addresses when a given mass-action reaction network admits a realization for all rate constants and proves the existence of a unique target network with realizable by . It develops a framework based on polyhedral cones and network realizations to decide and construct this WR0 realization, establishing independence from rate constants and providing a constructive algorithm. A key contribution is the proof that, whenever a WR0 realization exists for a network, it is unique and universal across rate constants, enabling a canonical surrogate network for robust dynamics analysis; this is complemented by a practical software implementation in CoNtRol using GLPK. The work enhances understanding of dynamical equivalence and robustness in reaction networks and offers tools for model reduction by substituting a WR0 surrogate while preserving the ODE dynamics.

Abstract

We prove that if a given reaction network has a weakly reversible deficiency zero realization for all choice of rate constants, then there exists a weakly reversible deficiency zero network such that is realizable by . Additionally, we propose an algorithm to find this weakly reversible deficiency zero network when it exists.

Paper Structure

This paper contains 11 sections, 8 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A few remarks on polyhedral cones
  4. Reaction networks
  5. Weakly reversible deficiency zero networks
  6. Realizations of mass-action systems
  7. $\textrm{WR}_0$ realizations of mass-action systems
  8. $\textrm{WR}_0$ realizations of mass-action networks
  9. Main results
  10. Software implementation
  11. Discussion and Future Work

Key Result

Theorem 2.4

If $\mathcal{N}$ is a $\textrm{WR}_0$ network, then for any rate constants $\kappa$ the mass-action system $(\mathcal{N},\kappa)$ has a unique positive equilibrium point in each compatibility class, and this equilibrium is asymptotically stable.

Figures (3)

  • Figure 1: Network $\mathcal{N}$ in (a) and $\mathcal{N}_0$ in (b) give rise to the same ODE system if we choose all rate constants equal to 3 for $\mathcal{N}$ and all rates equal to $1$ for ${\mathcal{N}_0}$.
  • Figure 2: Mass-action system $\mathcal{N}$ in (a) has $\textrm{WR}_0$ realization (b) when $\kappa_2=\kappa_3$
  • Figure 3: Mass-action system $\mathcal{N}$ in (a) has $\textrm{WR}_0$ realization (b) when $\kappa_1>\kappa_4$

Theorems & Definitions (24)

  • Definition 2.1: Reaction network
  • Definition 2.2: Weakly reversible networks
  • Definition 2.3: Deficiency
  • Theorem 2.4: Deficiency Zero Theorem feinberg1979lectures
  • Example 2.5
  • Definition 2.6: Realizations of mass-action systems
  • Definition 2.7: Dynamically realizable networks
  • Theorem 2.8
  • Definition 3.1: $\textrm{WR}_0$ realization of mass-action systems
  • Example 3.2
  • ...and 14 more