Minimisation of surface energy drives apical epithelial organisation and gives rise to Lewis’ law

The packing of cells in epithelia exhibits striking regularities, regardless of the organism and organ. One of these regularities is expressed in Lewis law, which states that 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. The driving forces behind the almost 100-year old Lewis law have remained elusive. We now provide evidence that the observed apical epithelial packing minimizes surface energy at the intercellular apical adhesion belt. Lewis law emerges because the apical cell surfaces then assume the most regular polygonal shapes within a contiguous lattice, thus minimising the average perimeter per cell, and thereby surface energy. We predict that the linear Lewis law generalizes to a quadratic law if the variability in apical areas is increased beyond what is normally found in epithelia. We confirm this prediction experimentally by generating heterogeneity in cell growth in Drosophila epithelia. Our discovery provides a link between epithelial organisation, cell division and growth and has implications for the general understanding of epithelial dynamics.