Aboave-Weaire’s law in epithelia results from an angle constraint in contiguous polygonal lattices

Abstract

It has long been noted that the cell arrangements in epithelia, regardless of their origin, exhibit some striking regularities: first, the average number of cell neighbours at the apical side is (close to) six. Second, the average apical cell area is linearly related to the number of neighbours, such that cells with larger apical area have on average more neighbours, a relation termed Lewis' law. Third, Aboav-Weaire's (AW) law relates the number of neighbours that a cell has to that of its direct neighbours. While the first rule can be explained with topological constraints in contiguous polygonal lattices, and the second rule (Lewis' law) with the minimisation of the lateral contact surface energy, the driving forces behind the AW law have remained elusive. We now show that also the AW law emerges to minimise the lateral contact surface energy in polygonal lattices by driving cells to the most regular polygonal shape, but while Lewis' law regulates the side lengths, the AW law controls the angles. We conclude that global apical epithelial organization is the result of energy minimisation under topological constraints.

Morphological study of embryonicChd8+/-mouse brains using light-sheet microscopy

“…3c). The observed relation between the polygon type of cells n and the average polygon type of their neighbours mn termed Aboav-Weaire's Law (21,22), recapitulated results in both the Drosophila wing disc and chicken neural tube epithelium (23) (Fig. 3d).…”

“…Having quantified epithelial morphology according to the cell area, aspect ratio, neighbour topology, and adherence to empirical laws such as Lewis' and Aboav-Weaire's ( Fig. 3), our data showed no departure in the cortical organization between haploinsufficient brains and controls, suggesting similar mechanical behaviour (20,22).…”

“…a relationship termed Aboav-Weaire's Law (21,22). Lastly, the cell aspect ratio was calculated using an inhouse algorithm that leverages MorphoGraphX's modularity to fit an ellipse and extract major and minor axes for each cell outline.…”

Cell-based Simulations of Biased Epithelial Lung Growth

“…In case of epithelia, quantifications of the apical polygonal lattice are available for a wide range of epithelia from Drosophila and mouse [4,35,36,5,37]. These can be used to adjust the mechanical parameters in the model [37,38]. The additional biological parameters, i.e.…”

Cell-based simulations of Notch-dependent cell differentiation on growing domains

“…Growth of the cells is simulated by adding mass to intracellular grid points and cells that exceed a certain cell area are divided. We have previously defined the parameter ranges that allow us to recapitulate epithelial tissue dynamics Vetter et al 2019;Stopka et al 2019). The set of coupled ordinary differential equations (ODEs) describing the Notch signalling network (Sprinzak et al, 2010;Boareto et al, 2015) can be solved on the cell boundary grid points, using the LB method.…”