Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics

Daniele Cappelletti; Badal Joshi; Daniele Cappelletti; Badal Joshi

doi:10.3934/mbe.2022359

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 7649-7668. doi: 10.3934/mbe.2022359

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Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics

Daniele Cappelletti ^{1
,
,},
Badal Joshi ²

1.
DISMA-Dipartimento di Scienze Matematiche "G.L. Lagrange", Politecnico di Torino, Torino, Italy
2.
Department of Mathematics, California State University San Marcos, San Marcos, USA

Academic Editor: Grzegorz A. Rempala

Received: 19 February 2022 Revised: 14 April 2022 Accepted: 18 April 2022 Published: 24 May 2022

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.
- reaction networks,
- stochastic models,
- state space truncation,
- stationary distribution
Citation: Daniele Cappelletti, Badal Joshi. Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7649-7668. doi: 10.3934/mbe.2022359

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Abstract

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.

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