Bespoke Turing patterns with specific nonlinear properties

Turing patterns offer a mechanism for understanding self-organization in biological systems. However, due to their flexibility, it is a mechanism that can often be abused. Here, we construct a minimal Turing system defined by just four parameters controlling the: diffusion rate, steady state, linear dynamics and nonlinear dynamics. Using just these four parameters, we can construct a set of kinetics with a number of desirable properties. Firstly, we can turn any homogeneous steady state into a Turing unstable steady state. Secondly, we can ensure that the Turing instability appears within any chosen parameter region. Thirdly, this formulation provides an unbounded patterning parameter space with guaranteed positive solutions. Finally, using weakly nonlinear analysis, we demonstrate that if we have freedom in any two of the parameters, then we can define any required pattern transition (i.e. spots-to-stripes, or stripes-to-spots) under any given changes of one of the parameters. Thus, if a Turing system is going to be applied to understand a specific biological system and, moreover, if it is going to be used to extrapolate predictions for experimental perturbations, then our findings underscore the necessity of heavily restricting the modelling components and parameter values, since any freedom could be exploited to generate potentially contradictory predictions.