Ordered Hexagonal Patterns via Notch-Delta Signaling

Many developmental processes in biology utilize Notch-Delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, we show that this "juxtracrine" property allows for an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. Furthermore, we show that the possible patterns that are realized can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high Delta at their center, as opposed to those with central high Notch levels. Also, our theory suggests a simple strategy for producing defect-free patterns.