Turing’s theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry, and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and the mechanisms of pattern selection, are well understood in small networks. However, a general set of rules explaining how network topology determines fundamental system properties and constraints has not been found. Here we provide a first general theory of Turing network topology, which proves why three key features of a Turing system are directly determined by the topology: the type of restrictions that apply to the diffusion rates, the robustness of the system, and the phase relations of the molecular species.

This research was supported by the ERC advanced grant SIMBIONT (670555) and the Ministerio de Economía y Competitividad (through Centro de Excelencia Severo Ochoa 2013-2017, SEV-2012-0208). X. D. acknowledges support by the ERC-FP7 Grant Swarmorgan (601062). J. S. ackowledges support from ICREA. P. M. and L. M. were supported by ERC Starting Grant QUANTPATTERN (637840).