Optimal network sizes for most robust Turing patterns

TL;DR

This paper uses random matrix theory to identify that small, optimally sized molecular networks are most likely to produce robust Turing patterns, challenging previous assumptions about network complexity and diffusion roles.

Contribution

It demonstrates that the most robust Turing networks are surprisingly small and identifies an optimal network size based on stability and diffusion tradeoffs.

Findings

Robust Turing patterns are more likely in small networks.

Optimal network size is surprisingly limited to a few molecular species.

Multiple immobile nodes reduce the importance of differential diffusion.

Abstract

Many cellular patterns exhibit a reaction-diffusion component, suggesting that Turing instability may contribute to pattern formation. However, biological gene-regulatory pathways are more complex than simple Turing activator-inhibitor models and generally do not require fine-tuning of parameters as dictated by the Turing conditions. To address these issues, we employ random matrix theory to analyze the Jacobian matrices of larger networks with robust statistical properties. Our analysis reveals that Turing patterns are more likely to occur by chance than previously thought and that the most robust Turing networks have an optimal size, surprisingly consisting only of a handful of molecular species, thus significantly increasing their identifiability in biological systems. This optimal size emerges from a tradeoff between the highest stability in small networks and the greatest instability with diffusion in large networks. Furthermore, we find that with multiple immobile nodes, differential diffusion ceases to be important for Turing patterns. Our findings may inform future synthetic biology approaches and provide insights into bridging the gap to complex developmental pathways.