Fundamental limits on symmetry breaking by Turing-like activator-inhibitor mechanisms.

Turing's longstanding reaction-diffusion hypothesis explains how molecular patterns can self-organise de novo in otherwise homogeneous tissues. However, whilst Turing-like activator-inhibitor models can qualitatively recapitulate patterning in silico, they are often highly simplified approximations of the molecular complexity operating in vivo. Here, we investigate significantly more complex reaction-diffusion systems that seek to more directly capture the mechanisms involved in intercellular signalling. By combining large-scale simulations with formal mathematical proofs, we show, rather generally, that symmetry breaking is strongly constrained by the extracellular interactions in the system but is relatively insensitive to the intracellular dynamics assumed. When applied to the activator-inhibitor paradigm, we find a broader repertoire of self-organising circuits than previously recognised, including some which are unexpectedly robust to parameters. Beyond these examples, we have packaged our highly performant numerical methods into a freely available and easy-to-use software pipeline, ReactionDiffusion.jl, that allows arbitrarily complex reaction-diffusion systems to be simulated at scale.