A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

TL;DR

This paper introduces a category-theoretic framework for modeling biochemical regulatory networks, connecting network motifs, reaction mechanisms, and dynamics with formal mathematical structures to enhance understanding and analysis.

Contribution

It develops a novel categorical formalism for regulatory networks, linking motifs, reaction mechanisms, and dynamics through functorial mappings and structured cospans.

Findings

Functorial mappings between reaction networks and regulatory networks.

Extension of Lotka-Volterra dynamics to open systems.

Formal categorical framework grounded in biological examples.

Abstract

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.