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# The Emergence of Geometric Order

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Supplementary Material The Emergence of Geometric Order in Proliferating Metazoan Epithelia Matthew C Gibson Ankit B Patel Radhika Nagpal Norbert Perrimon 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 Table of Contents 1 Graph Model Derivation in the Presence of Boundary Conditions 2 Extended Markov Model Error in Interface Formation 3 Extended Markov Model Alternative Cleavage Plane Models 4 Imaginal Disc Polygonal Cell Counts 5 Detailed Methods 1 Graph Model Derivation in the Presence of Boundary Conditions In the paper we approximate st the average number of sides per cell at generation t as st 2et ft 2 et 1 3ft 1 2ft 1 st 1 2 3 1 We solve this recurrence to get st 6 2 t s0 6 2 However this equation for st 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 st is st 2et etbd ft 6 2 t s0 6 etbd ft 3 where etbd is the number of boundary edges and ft 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 etbd O r and ft O r2 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 ft O 2t and etbd O ft Note that this would not be true for a strip of cells that proliferate in a single direction since then etbd O ft Thus the original approximate formula for st in 2 is still

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