A new paradigm considering multicellular adhesion, repulsion and attraction represent diverse cellular tile patterns.

Area of Science:

• Developmental Biology
• Computational Neuroscience
• Mathematical Modeling

Background:

• Differential adhesion drives neuronal spatial organization in early fruit fly brain development.
• Columnar neuron arrangements form large-scale patterns in the fly visual center.
• Hexagonal configurations are observed in the fly compound eye, with tetragonal patterns appearing in mutants.

Purpose of the Study:

• To develop a mathematical framework for studying transitions between hexagonal and tetragonal cell arrangements.
• To model the mechanisms underlying pattern changes in neural development.

Main Methods:

• Proposed a new mathematical model based on macroscopic approximations of agent-based models.
• Incorporated medium-range repulsion and longer-range attraction into existing cell sorting models.
• Analyzed angular configurations using angle summary statistics and compared with experimental data.

Main Results:

• The Adhesion-Repulsion-Attraction (ARA) model successfully reproduces the transition from hexagonal to tetragonal configurations.
• Intermediate patterns between hexagonal and tetragonal are common in both experimental data and the ARA model.
• The model shows qualitative agreement with experimental tile patterning.

Conclusions:

• The ARA mathematical model provides a framework for understanding cell pattern transitions in biological systems.
• The study opens new avenues for exploring pattern dynamics in neural development and other biological processes.
• Further quantitative studies of ARA models are warranted.