Model reduction permits Turing instability analysis of arbitrary reaction-diffusion models

Synthesising a genetic network which generates stable Turing patterns is one of the great challenges of synthetic biology, but a significant obstacle is the disconnect between the mathematical theory and the biological reality. Current mathematical understanding of patterning is typically restricted to systems of 2 or 3 chemical species, for which equations are tractable, but plausible genetic networks typically consist of dozens of interacting species. In this article, we suggest a method for reducing large biochemical systems to systems with 2 or 3 species which can then be studied analytically. We provide conditions to guarantee that the full system forms patterns if the reduced system does, and vice-versa. We confirm our technique with 3 examples: the Brusselator, an example proposed by Turing, and a biochemically plausible patterning system consisting of 17 species. These examples show that our method significantly simplifies the study of pattern formation in large systems.