The Network Basis of Pattern Formation: A Topological Atlas of Multifunctional Turing Networks

Understanding how genetic networks can drive different self-organizing spatial behaviors remains a significant challenge. Here, we use an automated algebraic method to systematically screen for Turing networks capable of generating diverse spatial patterns from noise, including periodic static waves, traveling waves and noise-amplifying patterns. We organize these networks into a topological atlas--a higher-level graph where nodes represent Turing networks linked together when they differ by only one regulatory interaction. In this atlas, Turing networks are arranged into distinct clusters showing a remarkable correspondence between network topology and self-organizing behaviors. Using an analytical approach, we identify the specific regulatory feedbacks that characterize each behavior. Moreover, we discover that different clusters are interconnected by multifunctional networks that can transition between behaviors upon feedback modulation. Among these networks, we find a new class of multiphase Turing networks capable of altering the phase of periodic wave patterns depending on the parameters, and networks that can transition between static and oscillatory Turing behaviors. The atlas further highlights the crucial role of feedback on immobile nodes in regulating pattern formation speed and precision by canalizing system noise. Overall, our study provides a novel framework to study the evolution and development of multicellular self-organization through changes in network topology and feedback modulation. This offers insights into how genetic regulatory networks can be tuned to drive pattern formation in developmental biology and in stem cell systems like embryoids and organoids. Significance StatementBy employing an automated algebraic method, Regueria and Marcon construct a topological atlas that categorizes Turing networks based on their ability to produce periodic patterns, traveling waves, or noise amplifying patterns. The atlas identifies distinct topological clusters linked by multi-functional networks that can transition between behaviors through feedback modulation. Key findings highlight how modulation of regulatory cycle strength in time or space can promote transition between static and oscillatory periodic pattern. The study also reveals the importance of feedback on immobile nodes in managing noise and influencing pattern formation. Overall the topological atlas offers a new framework for examining the evolution and development of multicellular self-organization.