Boolean analysis of lateral inhibition
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We study Boolean networks which are simple spatial models of the highly conserved Delta-Notch system. The models assume the inhibition of Delta in each cell by Notch in the same cell, and the activation of Notch in presence of Delta in surrounding cells. We consider fully asynchronous dynamics over undirected graphs representing the neighbour relation between cells. We show that all attractors are fixed points for the system, independently of the neighbour relation. The fixed points correspond to the so-called fine-grained "patterns" that emerge in discrete and continuous modelling of lateral inhibition. These patterns are shown to have a simple characterisation in terms of vertex covers of the graph encoding the neighbour relation. The reachability of fixed points is studied, and a description of the trap spaces is used to investigate the robustness of patterns to perturbations. The results of this qualitative analysis can complement and guide simulation-based approaches, and serve as a basis for the investigation of more complex mechanisms.
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