# Ordered Hexagonal Patterns via Notch-Delta Signaling

## TL;DR

This paper presents an exact mathematical analysis of pattern formation via Notch-Delta signaling, explaining the prevalence of hexagonal cellular patterns and proposing mechanisms for defect-free pattern generation.

## Contribution

It introduces a detailed differential equation model for juxtacrine signaling and analyzes bifurcations to explain observed biological patterns and their formation mechanisms.

## Key findings

- Hexagonal patterns with high Delta at centers are prevalent.
- Low cis-coupling leads to novel high Delta and high Notch patterns.
- Biological systems require additional mechanisms for defect-free pattern formation.

## Abstract

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, exploiting this "juxtacrine" property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters 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. We show that outside this range of parameters, in particular for low cis-coupling, a novel kind of pattern is produced, where high Delta cells have high Notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.