Fig. 4.

Geometry and mechanics of a catenary-curved dome during LRP tissue growth. (A) Longitudinal confocal sections from time-lapse imaging of a developing Arabidopsis LRP visualized using 35S:Lti6b-GFP (a plasma membrane marker). The elapsed time (h) after gravistimulation for inducing LRP development is indicated in each panel. Red lines indicate LRP dome outlines. (B) A graph of catenary parameter a (y-axis) plotted against dome height h (x-axis) of growing LRP outlines quantified by the NLS method. (C) Cross validation test (Eqn 12) of in vivo and in silico LRP outlines fitted with catenary function. Averaged MSE (y-axis) against the dome height h (x-axis) in vivo [blue, n=10 (h<10), n=10 (10≤h<30), n=10 (30≤h<50), n=10 (50≤h)] and in silico (red, n=5 for each dome height range, h<10, 10≤h<30 and 30≤h<50) are shown. (D) Rules of cell divisions (white, no division; light blue, single division; deep blue, three consecutive divisions) and anisotropic cell expansion in the proximodistal direction (magenta, presence; other colors, absence) observed in the LRP development in vivo. (E) Tissue-mechanical simulation from a flat primordium to dome formation during LRP development with a mass of overlaying cells (gray). Cell division and expansion rules (color-coded as in D) were incorporated into the simulation. See also Movie 1. (F) Catenary curves of different parameter values a with its width (x-axis) approximately equal to that of an actual LRP (Fig. 4A). (G) Catenary curves formed by chains of increasing length with fixed ends under gravity. (H) The mechanics of the catenary curve; the gravity works as vertically uniform force W on the chain, and is balanced with the tangential tension T at the mechanical equilibrium. s, a, x, ρ, g and θ denote chain outline length, catenary parameter, x-coordinate of the catenary chain, mass density, the gravitational acceleration and the angle from horizontal x-axis, respectively. ρg represents the gravitational (vertical) force per unit length. (I) Distribution of vertical force (red arrows) and tangential force (black arrows) on dome outlines after cell expansion in five representative simulations (bottom panel shows a representative outcome; dashed black lines indicate the x- and y-axes). The magnitude of vertical force normalized by its spatial average over the dividing zone (dark blue and light blue cells in the bottom panel) plotted as a function of x-coordinate along the dome width (upper panel). Data are mean±s.d. of five independent simulations. The upper and lower hinges, and the middle lines of box plots in C and I represent the 25th, 75th and 50th percentiles, respectively. Scale bars: 50 µm.