Fig. 2: Theory of bistable crypt morphology switched by lumen volume.
From: Mechanochemical bistability of intestinal organoids enables robust morphogenesis
a, Schematic of a 3D vertex model (top) and the phase diagram of crypt morphology (bottom) as a function of crypt differential tension σc and normalized volume v. In the schematic, Rc and Rv are the radii of curvature in the crypt and villus, respectively; h is the cell height/tissue thickness (Supplementary Note 1.1). b, Three possible energy landscapes that control the crypt morphology (v = 4). Local energy minima are crypt equilibrium states, with θc = 0 indicating the budded (or closed) crypt shape and θc > 0 indicating the bulged (or open) shape. c, Evolution of crypt shape with varying lumen volumes at constant differential tension σc = 0.1 (top) and σc = 0.15 (middle), and evolution with varying crypt differential tension at a constant volume of v = 3 (bottom). The arrows are different paths shown in the phase diagram in a. d, Schematic of the mechanism of curvature–thickness feedback in the crypt epithelium: epithelial bending increases its thickness h and the corresponding active bending moment M ∼ σch, which, in turn, facilitates crypt budding. e, Fold changes of lumen volume before and after villus tissue breakage, in both bulged (N = 11) and budded (N = 9) organoids. Scale bars, 50 µm. f, Tension asymmetry of the crypt apical versus basal surfaces in bulged (N = 11) and budded (N = 28) organoids, and the theoretical threshold (dashed line) for organoids to remain in a budded/bistable state on lumen inflation.
