The French flag problem revisited: Creating robust and tunable axial patterns without global signaling

Wolperts French flag problem conceptualizes the task of forming axial patterns with broad regions in multicellular systems. Wolpert described two different solutions to his problem, the balancing model and thresholding of a morphogen gradient, both of which require global, long-range signaling between cells. Since global signaling becomes challenging in large multicellular systems, we computationally explore alternative solutions, which use only local cell-cell signaling and are simple enough to potentially be implemented in natural or synthetic systems. We employ cellular automata rules to describe local signal processing logics, and search for rules capable of robust and tunable axial patterning with evolutionary algorithms. This yields large sets of successful rules, which however display only few types of behavior. We introduce a rule alignment and consensus procedure to identify patterning modules that are responsible for the different behaviors. With these modules as building blocks, we then construct local schemes for axial patterning, which function also in the presence of noise and growth, and for patterns with a larger number of different regions. The regulatory logic underlying these modules could therefore serve as the basis for the design of synthetic patterning systems, and as a conceptual framework for the interpretation of biological mechanisms.