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

Cell sorting by differential adhesion is one of the basic mechanisms explaining spatial organization of neurons in early stage brain development of fruit flies. The columnar arrangements of neurons determine the large-scale patterns in the fly visual center. Experimental studies indicate that hexagonal configurations regularly appear in the fly compound eye, which is connected to the visual center by photoreceptor axons, while tetragonal configurations can be induced in mutants. We need a mathematical framework to study the mechanisms of such a transition between hexagonal and tetragonal arrangements. Here, we propose a new mathematical model based on macroscopic approximations of agent-based models that produces a similar behavior changing from hexagonal to tetragonal steady configurations when medium-range repulsion and longer-range attraction between individuals are incorporated in previous successful models for cell sorting based on adhesion and volume constraints. We analyze the angular configurations of these patterns based on angle summary statistics and compare between experimental data and parameter fitted ARA (Adhesion-Repulsion-Attraction) models showing that intermediate patterns between hexagonal and tetragonal configuration are common in experimental data as well as in our ARA mathematical model. Our studies indicate an overall qualitative agreement of ARA models in tile patterning and pave the way for their quantitative studies. Our study opens up a new avenue to explore tile pattern transitions, found not only in the column arrangement in the brain, but also in the other related biological processes.Author summary: Biological tile patterns often exhibit hexagonal patterns, such as those seen in insect compound eyes, columnar structures in the brain, auditory epithelial cells, and lobules in the liver, which are thought to be due to physical constraints. Indeed, hexagonal tile patterns minimize the potential energy based on a standard physical potential of particles. In this study, we focus on the arrangement of the columnar structure in the fly brain. Our quantitative method to analyze the symmetric distribution patterns of particles shows that their distribution patterns are not exactly hexagonal or tetragonal. We therefore propose a new mathematical model, the ARA model, which takes into account short-range adhesion, medium-range repulsion and long-range attraction and reproduces hexagonal, tetragonal and intermediate patterns. Finally, the ARA model could reproduce the changes in column arrangement upon decreasing and increasing the expression level of N-cadherin. The current study provides a mathematical basis for studying a wide range of tile patterns found in biological and non-biological systems.