The Design Principles of Discrete Turing Patterning Systems

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-node networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find the Turing topologies to be substantially more robust to changes in the parameters of the model. We also find that Turing instabilities are a much weaker predictor for emerging patterns in simulations in our discrete modelling framework. We propose a modification of the definition of a Turing instability for cellular automata models as a better predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.