Supplementary Information for "The Emergence of Geometric Order in Proliferating Metazoan Epithelia"

This supplementary document presents two extensions of the mathematical model, as well as a more detailed description of the Drosophila experimental data. In addition, we present some clarifications on the graph model and a detailed methods section. In the paper we approximate s t , the average number of sides per cell at generation t as: s t ≈ 2e t / f t = 2(e t-1 + 3f t-1) / 2f t-1 = (s t-1 / 2) + 3 (1) We solve this recurrence to get: s t ≈ 6 + 2-t (s 0 – 6) (2) However, this equation for s t is only an upper bound approximation because it double counts edges on the boundary of the epithelium. If we account for boundary cells, then the exact equation for s t is: s t = (2e t-e t bd) / f t = 6 + 2-t (s 0 – 6)-e t bd / f t (3) where e t bd is the number of boundary edges and f t is the number of cells at generation t. This basically is the same as equation (1) but corrects for the over counted edges. In a 2D epithelium comprised of similarly-sized cells, the number of boundary edges approximately measures the perimeter and the number of cells approximately measures area. Hence e t bd ~ O(r) and f t ~ O(r 2), where r is the radius of the epithelium (in cells). The third term in equation (3) is therefore O(1/r) and this approaches zero as t gets large and the epithelium gets large. This implies that the average number of neighbours per cell will approach six from below. In fact, the convergence rate is exponential since f t ~