On the abelian structure of noncompetitive chemical reaction networks

Abstract

Chemical reaction networks (CRNs) are foundational models for describing complex biochemical processes. We study noncompetitive CRNs, a class of networks whose static states, where the CRN is inactive, are rate independent, and that can implement ReLU neural networks. CRNs of interest in biochemistry and systems biology are embedded in complex networks so that CRNs have to respond to internal and environmental cues. We describe the network's response to such perturbations using a new Markov chain that we call CRN sandpile Markov chain, whose state space is the set of static states. The transition mechanism of the CRN sandpile Markov chain is defined by adding a molecule of a randomly chosen species to a static state, and then letting the CRN state evolve toward a new static state. A central contribution of the present work is the observation that one can associate a natural Abelian Network (AN) to each noncompetitive CRN, and use AN theory to get new mathematical results on noncompetitive CRNs. For noncompetitive CRNs on a finite state space, we use AN theory to get that only a fraction of the static states are recurrent for the CRN sandpile Markov chain. We obtain furthermore that the set of recurrent states is in one to one correspondence with the critical group of the AN, which plays a major role in AN theory. Overall, this work establishes a unified algebraic and probabilistic framework for analyzing the long-term behavior of noncompetitive CRNs. We focus on a special class of noncompetitive CRNs called generalized toppling networks, and obtain new mathematical results both for the CRN and AN settings.