Understanding the mechanisms governing and regulating self-organisation in the developing embryo is a key challenge that has puzzled and fascinated scientists for decades. Since its conception in 1952 the Turing model has been a paradigm for pattern formation, motivating numerous theoretical and experimental studies, though its verification at the molecular level in biological systems has remained elusive. In this work, we consider the influence of receptor-mediated dynamics within the framework of Turing models, showing how non-diffusing species impact the conditions for the emergence of self-organisation. We illustrate our results within the framework of hair follicle pre-patterning, showing how receptor interaction structures can be constrained by the requirement for patterning, without the need for detailed knowledge of the network dynamics. Finally, in the light of our results, we discuss the ability of such systems to pattern outside the classical limits of the Turing model, and the inherent dangers involved in model reduction.

Department of Mathematics FNSPE Czech Technical University Prague Prague Czech Republic

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