Cellular Potts model for an epithelial tissue. (a) Cells representation in the cellular Potts model. The cells were colored differently, and a white space represents the medium. (b, b’) An update of a label from b to b’. The panels show an area inside a black line in a. The label on a randomly selected site (yellow arrowhead) was replaced with one on a randomly selected neighboring site (magenta arrowhead). (c) Neighborhood of a site for the contact energy. The panel shows an area inside a blue line in a. A site marked with a circle is adjacent to three sites in a different cell (black star) and a site in the medium (white star). (d) Material labels in the model. Four types of cytosol are colored blue, green, red, and dark gray. Inner and outer body fluid are colored white and light gray, and the apical ECM is colored dark yellow. (e) Cell polarity markings. The apical, basal, lateral, and none-polar surfaces are colored blue, reg, green, and gray respectively. The adherens junction is colored yellow. (f, f’) Tissue and cell representations in our model. The panel f’ shows an area inside a black line in f, and lightens an area around a cell. In f’, the cytosol was drawn with pale blue, the cell apical, lateral, and basal surfaces were drawn with blue, gree, and red, and the apical lateral sites where the adherens junction localized were colored yellow. (g) Algorithm of the cellular Potts model simulation.

Figure 1—figure supplement 1. Epithelial cell surface tension and tissue integrity.

Simulations of epithelial tissue with the increased contractility. (a) The ratio A0/ Ā indicates how much cells were apico-basally compressed by the lateral cell-cell junction contractility Jl. The horizontal axis Jb/Jl indicates a ratio between the basal surface and lateral cell-cell junction contractilities. The vertical axis Ja/Jb indicates a ratio between the apical and basal surface contractilities in center pale blue five cells. The other contractility. (b) Simulations w/o the surface elasticity, where A0/ Ā ∼ 1.01, Jb/Jl = 4, and Ja/Jb = 1.6. The surrounding pale red and green cells were assigned the apical surface contractility equal to the basal surface vertical axis represents time (1 time/1000 updates). (c) Simulations of the supracellular myosin cable. Cells toward the midline. A0/ Ā ∼ 1.01 and Jb/Jl = 4. marked with an asterisk were assigned a potential energy on their adherens junction so that they were pulled were assigned the increased apical surface contractility: A0/ Ā ∼ 1.01, Jb/Jl = 4, and Ja/Jb = 1.6, where A0/Ā

Figure 2—video 1. Simulation of epithelial tissue with the increased contractility. The center pale blue five cells indicates how much cells were apico-basally compressed by the lateral surface contractility Jl, Jb/Jl represents a ratio between the basal and lateral surface contractilities, and Ja/Jb represents a ratio between the apical and basal surface contractilities in center pale blue five cells. There were 1000 updates between frames.

Cell shape and practical force to constrict the apical surface. (a) The edge cell at the early phase of the cellular Potts model simulation with the increased apical surface contractility. (b) Illustration of the edge cell. (c-e) Illustration of surface contractilities around the cell-cell junction. c shows a junction marked by the black arrowhead in b. d shows a junction marked by the orange arrowhead in b. e shows a junction marked by the magenta arrowhead in b. Vectors t1-t9 depict the surface contractilities exerted on the junctions. Pale pink arrows in c are the same vectors with t2 and t3, those in d are the same with t5 and t6, and those in e are the same with t8 and t9. Blue arrow in d depicts a sum of t4, t5, and t6. (f) Phase diagram of cell shapes. For the apical width and a curvature of the right side lateral surface, the energy of the cell is at minimum with the shape. The pressure and the surface contractility were set so that the cell took the columnar shape for the apical width w, apical-basal height h, and 0 curvature. (g) Energy landscape of the cell shapes for the apical width and the lateral curvature. Red line shows a path following a gradient of the energy. (h) Plots of energy with respect to the apical width. Blue plot shows the energy when the lateral surface was restricted to be straight. Red plot shows the energy along the path in e.

Simulations of epithelial tissue with the modi1ed surface elasticity. (a) Results of the simulations. The center pale blue five cells were assigned Pa0 = 0, while the others were assigned Pa0 equivalent to Pa initial value. The ratio A0/Ā indicates how much cells were compressed, Jb/Jl indicates a ratio between the basal surface and lateral cell-cell junction contractility, and Es denotes the surface elastic modulus for the inner pale blue and green 13 cells. The edge pale red cells were assigned 0.1 times smaller surface elastic modulus than the inner cells. (b) Plots showing difference in distance from the apical ECM between the constricting cells and the surrounding cells. Average distances were compared, and a larger difference indicates a deeper invagination. Magenta horizontal lines indicate 0.73, an average difference between the center cells and surrounding cells when all of the cells were assigned Pa0 equivalent to the Pa initial value, as a control. Results of three simulations were averaged.

Figure 4—video 1. Simulation of epithelial tissue with the decreased elastic reference value. The center pale blue five cells were assigned a decreased apical surface elastic reference value Pa0 = 0. The cells were assigned A0/ Ā ∼ 1.01 and Jb/Jl = 2, and the inner pale blue and green cells were assigned Es = 2.5 while the edge pale red cells were assigned Es = 0.25. There were 10000 updates between frames.

Figure 4—figure supplement 1. Simulation with gradient apical surface contractility

Simulation of apical constriction with the supracellular myosin cable. The center pale blue five cells were assigned Pa0 = 0, and the siding pale green two cells adjacent to the center pale blue cells were assigned the potential energy on their adherens junction so that they were pulled toward the midline. The magnitude Cr indicates a gradient of the potential energy, A0/Ā indicates how much cells were compressed, Jb/Jl indicates a ratio between the basal and lateral cell-cell junction contractility, and Es denotes the surface elastic modulus for the inner pale blue and green 13 cells. The edge pale red cells were assigned 0.1 times smaller surface elastic modulus than the inner cells.

Figure 5—figure supplement 1. Simulation with various cell heights

Figure 5—figure supplement 2. Simulation with cell-ECM adhesion

Change in junctional tension and cell pressure distribution during tracheal pit invagination. (a) Vertices and edges representation of adherens junction inside and around Drosophila embryo tracheal pit. The panels show two tracheal pits from the beginning of the invagination (0 min) and after 15 and 30 minutes. Black bars on top and at left side indicates positions of the y-z and x-z slices. (b) Heat maps showing average junctional tensions in each cell at the three time points. The relative junctional tensions were averaged weighted with the edge lengths for each cell. (c) Heat maps showing relative cell pressure in blue-red and relative junctional tension in purple-orange at the three time points. (d) Change in apical surface area among the invaginated constricting cells (lines) and surrounding cells (dotted lines). Colors indicate different embryos, and the values were averaged in each embryo. Error bars indicate standard deviations. (e) Change in relative cell pressure among the invaginated constricting cells (lines) and surrounding cells (dotted lines). Colors indicate different embryos, and the values were averaged in each embryo. Error bars indicate standard deviations. A scale bar in a represents 10 nm.

Hypothetical model of endocytosis-based apical constriction. (a) Flow diagram of the ratcheting by endocytosis. The cell apical surface was contracted by the pulsed myosin activation (1). Without the endocytosis, the apical surface would be fully relaxed (2). By the endocytosis, the apical surface reference value was decreased (3). Because of the modified reference value, the cell apical surface was partially relaxed (4). (b) Expected deformation by the increased apical surface contractility. (c) Expected deformation by the sporadic apical surface contractility. (d) Expected deformation by the patterned sporadic contractility and endocytosis. (e) Expected deformation by the general sporadic contractility and the patterned endocytosis. (f) Expected deformation with the increased apical surface tension.

Parameters for epitherila cell surface tension and tissue integrity.

Parameters for increased apical contractility.

Parameters for modified surface elasticity.

Parameters for gradient apical contractility.

Parameters for various cell heights.

Parameters for deformation with apical elasticity and cell-ECN adhesion.

Epithelial cell surface tension and tissue integrity. The energy H included the terms of surface contact energy and area constraint. The constant Ā represents a target average area of cells in the simulation. The ratio A0/ Ā indicates how much the cells are compressed, and thus how strong the surface tension is. Here the lateral surface tension was defined by the contact energy between the cytosols in different cells, the basal surface tension was defined by the contact energy between the cytosol and the inner body fluid, and the apical surface tension was defined by the contact energy between the cytosol and the apical ECM or the outer body fluid. The lateral surface tension was determined based on the compression A0/ Ā. The vertical axis Jb/Jl indicates a ratio between the basal and apical surface contact energy Jb and the lateral surface contact energy Jl .

Simulation of epithelial tissue with the gradient contractility. The ratio A0/ Ā indicates how much cells were compressed, Jb/Jl indicates a ratio between the basal and lateral surface contractility, and Ja/Jb indicates a ratio between apical and basal surface contractilities in edge pale blue cells. Darker blue cells were assigned higher apical surface contractility.

Simulation of apical constriction with various cell heights. The cells were 2/3 lower (first and second columns) or 4/3 higher (third and fourth columns) in the apical-basal axis. The ratio A0/ Ā indicates how much cells were compressed, Jb/Jl indicates a ratio between the basal and lateral surface contractility, and Es denotes the surface elastic modulus for the inner pale blue and green 13 cells. The edge pale red cells were assigned 0.1 times smaller surface elastic modulus than the inner cells. The center pale blue five cells were assigned Pa0 = 0, while the others were assigned Pa0 equivalent to Pa initial value.

Simulation of apical constriction by the elasticity remodeling with cell-ECM adhesion. The five center cells were assigned Pa0 = 0, and the other surrounding cells were assigned an affinity to the apical ECM indicated by a ratio between cytosol-ECM and cytosol-outer body fluid contact energies Js/Ja. The ratio A0/ Ā indicates how much cells were compressed, Jb/Jl indicates a ratio between the basal and lateral surface contractility, and Es denotes the surface elastic modulus for the inner pale blue and green 13 cells. The edge pale red cells were assigned 0.1 times smaller surface elastic modulus than the inner cells.