Fig. 5: An elastic bilayer model captures growth-induced epithelial morphogenesis.
From: Growth anisotropy of the extracellular matrix shapes a developing organ
a Geometric decomposition for a bilayer structure, composed of DP (purple) and ECMDP (green). In the initial state \({ {\mathcal B} }_{0}\), the disc is unstressed and undeformed. Representative volume elements with unit lateral lengths are shown. The DP and ECM thickness is denoted \({H}_{0,DP}\) and \({H}_{0,ECM}\), respectively. The growth tensor \({\mathbb{G}}\) describes the transformation to the reference state ℬr which is grown and relaxed. Since the DP grows in-plane, its thickness remains \({H}_{0,DP}\). The ECM grows orthogonally to the plane as controlled by the anisotropy parameter ρ, and the relaxed thickness is greater than \({H}_{0,ECM}\). Through the elastic deformation gradient \({\Bbb{A}}\), stress is introduced, leading to the domed observed state ℬt. b Parameter diagram to determine a region in which simulation values are within chosen tolerances of experimental measurements. In the blue, red and orange regions, the simulation results are within tolerance of the measured ECM thickness HECM, DP thickness HDP, and relative increase in ECM thickness upon decellularization (see Supplementary Information for details). In the dark region, all three conditions are satisfied simultaneously. Parameters ρ = 0.45 and μ = 25 were chosen as the best fit, of which the morphology is shown in the central out of 9 insets showing cross-sections of the simulated wing discs. The result ρ = 0.45 suggests that the bottom ECMDP growth anisotropy lies midway between planar (ρ = 0) and isotropic growth (ρ = 1). c Model results (ρ = 0.45, μ = 25) compared to experimental data. Left two plots: the insets show linear profiles of γDP and γECM. Right two plots: comparison between simulated and experimental values for DP thickness HDP, and ECM thickness HECM. The coloured bands show the region within 22.4% of the mean of experimental values, in which the simulation values are contained. d The shape of the simulated wing disc, using the best fit parameters, is shown compared to cross-sections of representative wing discs. The simulated discs are shown with a quarter of each disc removed for illustration purposes (simulations are axisymmetric).
