On the abelian structure of noncompetitive chemical reaction networks

TL;DR

This paper explores the algebraic structure of noncompetitive chemical reaction networks, showing they are a special case of Abelian networks, and analyzes their long-term behavior using probabilistic methods.

Contribution

It establishes that noncompetitive CRNs are instances of Abelian networks and develops a unified framework for their analysis based on algebraic and probabilistic tools.

Findings

Recurrent static states correspond to the critical group of the Abelian network.

Noncompetitive CRNs can implement ReLU neural networks.

The framework links network perturbations to state transitions via sandpile Markov chains.

Abstract

Chemical reaction networks (CRNs) are foundational models for describing complex biochemical processes. We study noncompetitive CRNs, a class of networks whose static states are rate-independent, and that can implement ReLU neural networks. A central contribution of this work is that noncompetitive CRNs are special instances of Abelian networks (ANs) a well-established framework for self-organized criticality. CRNs of interest in biochemistry and systems biology are embedded in complex networks so that local CRNs have to respond to internal and environmental cues. We describe the network's response to such perturbations using a sandpile Markov chain whose state space is the set of CRN's static states, from where no reaction is possible. The addition of molecules to a static state induces reactions that move the system into a new static state. For noncompetitive CRNs of finite state space, we use AN theory to get that the fraction of static states that are recurrent is in one to one correspondence with the critical group of the AN. Overall, this work establishes a unified algebraic and probabilistic framework for analyzing the long-term behavior of noncompetitive CRNs.