SUPPLEMENTARY FILE S1: 3D AIRWAY TUBE RECONSTRUCTION AND CELL-BASED MECHANICAL MODEL. RELATED TO FIGURE 1, FIGURE 7, AND STAR METHODS. 1. 3D AIRWAY TUBE RECONSTRUCTION. RELATED TO FIGURE 1 AND STAR METHODS Obtaining accurate information about the geometry of the airway tube is crucial for analyzing and modeling of the spindle behaviors. We developed a computer program to reconstruct the threedimensional (3D) airway tube. The raw image data is composed of 512 X 512 single-color bitmap Z- stack files (Supplementary file S1-Fig. 1.1), which were obtained via two-photon confocal microscopy. From the basis of these Z-stack images, we sought to reconstruct 3D airway tubes by identifying the inner surface, the outer surface, and the central axis of the airway tube. Supplementary file S1-Fig. 1.1. An example of raw image from two-photon confocal microscopy. The image is composed of grey-scale pixels. The green curves mark both the inner and outer surfaces of the airway tube. The first step was to automatically identify the inner and outer boundaries of the airway epithelium from the image data. We anticipated three main design challenges as we developed our automatic live-image-processing program. First, the pixel intensity of some cells was almost as dark as the
background noise, which made it hard for the computer program to recognize the true signals. Second, there were some unrelated bright spots that needed to be eliminated. Third, the right lobe of a lung was sometimes appeared in a given image field (for example see the lower left corner of Supplementary file S1-Fig. 1.1); recall that we analyzed the left lobes of lungs in this study. To overcome the first two problems, we employed a sticking method in which we randomly threw a huge number (~10 7 ) of short sticks (25-30 pixel units in length) on the image. If both ends of the stick hit white pixels (pixels whose brightness was above a certain threshold), the stick was accepted and retained, and every pixel along the stick was kept as valid pixel. If at least one end of the stick hit a black pixel (pixels whose brightness was lower than a threshold), the stick was rejected and discarded. The resulting image composed of all of the retained sticks (Supplementary file S1-Fig. 1.2) was smoothed using a gradient-flow method. Note that after this sticking process, isolated small white spots were removed. Supplementary file S1-Fig. 1.2. Black and white pixels of an image composed of the retained sticks following our sticking procedure. The right lobes of lungs were still present in in the lower-left corner of the image (Supplementary file S1-Fig. 1.2). To remove these unwanted objects (the third challenge), which were usually located at the corners and were isolated from the main object in the center, we used four red spots
(Supplementary file S1-Fig. 1.3) to march towards the center of the figure. During the marching process, if a line hits a white pixel, we selected all of the white pixels connecting to it.. Supplementary file S1-Fig. 1.3. Selecting the object of interest. The red spots correspond to the starting points of the marching process, and the green lines represent the marching paths. Regions hit by marching paths are selected in this procedure. Next, the contour lines that separate the black and white regions are drawn, and these identified the boundaries. The program can distinguish the inner (apical side) and outer (basal side) boundaries based on the geometrical information. Several processed images at different z from the same lung are shown in Supplementary file S1-Fig. 1.4. Supplementary file S1-Fig. 1.4. Processed 2D images with inner (orange) and outer boundaries (blue) are automatically identified. The situation in the third panel, where there are two orange loops, occurs when the cutting plane is close to the tips of a banana-shaped airway tube.
By stacking the 2D contour lines obtained from each Z-stack imaging series, we were able to obtain the 3D airway shape (Supplementary file S1-Fig. 1.5). Supplementary file S1-Fig. 1.5. An automatically reconstructed 3D airway tube. Orange lines represent the apical surface and blue lines represent the basal surface of an airway tube. The green line in the center of the tube represents the estimated central-axis of the tube. From the reconstructed 3D geometry of an airway tube, we could determine the central-axis of the tube; this is essential for the spindle orientation analysis. We used a simple approach to determine the central axis. The approach is illustrated in Supplementary file S1-Fig. 1.6. First, we projected the inner boundaries to the xy-plane. Then, the program identified the direction parallel to the tube; this direction was defined as the one with the minimal distance between a pair of parallel lines that fully contain the tube (two black solid-lines in Supplementary file S1-Fig. 1.6). After locking this direction, we used a set of lines to cut the tube in the direction that was perpendicular to the solid lines (black dashed-lines in Supplementary file S1-Fig. 1.6). Each cutting-line may intersect with multiple projected boundaries, and the average xy-coordinates of all of these intersecting points were obtained as an approximation of the central axis position projected on the xy-plane (pink asterisks). To determine the z-axis of the central axis, we averaged the z-coordinates of all intersecting points between each cutting-line that was perpendicular to the tube direction and the projected boundaries. Thus, we were
able to obtain a curved line representing the central axis of the airway tube. Supplementary file S1-Fig. 1.6. Determining the central axis of the tube. The blue lines are the inner boundaries of the airway tube projected onto the xy-plane. The red circles are the point-pairs that give the longest distance in different directions, while the direction of the two solid lines indicates the shortest distance among them. The dashed lines are perpendicular to the solid lines. The pink asterisks are the averaged intersecting points between each dashed-line and the blue curves. Their location corresponds to the xy-coordinates of the estimated central axis. 2. CELL-BASED MECHANICAL MODEL. RELATED TO FIGURE 7 AND STAR METHODS Geometry setup: One section of the airway tube is unfolded into an epithelial sheet that is modeled as a two-dimensional rectangular area bounded between L and min L in the longitudinal direction max and H and min H in the circumferential direction. The initial simulation domain is set to be a max rectangular area of 200µm in length and 180µm in height. A total number of 360 cells (not including ghost cells at the boundaries) were placed in the simulation domain. Cells are represented by polygons obtained from Voronoi tessellation. The generator of the Voronoi tessellation is not a set of
points representing the cell center; rather it is a set of ellipses. In addition to the position of cell centers (coincident with the center of the ellipses), these ellipses also contain information about cell shapes (reflected by the length of the semi major axis a and semi minor axis b). In our case, we choose a=6.9µm and b=4.7µm. Hence the average area of a cell is about A0 = πab= 100µm 2. The semi major axis is always in parallel with the longitudinal direction to ensure the shape of the cells obtained from Voronoi tessellation based on these ellipses tend to be elongated in the longitudinal direction. One possible reason for the elongated cell shape could be the different curvature in the circumferential and longitudinal directions of a cylinder. Boundary conditions: Periodic boundary conditions are applied on the upper and bottom boundaries (circumferential direction), and free boundary conditions are applied on the left and right borders (longitudinal direction). To implement the free boundary condition, we introduced an extra layer of cells (which we call ghost cells ) at both the left and right borders of the tissue. The ghost cells form two vertical barriers function to bound the normal cells inside the simulation domain. We also used ghost cells to apply external forces to the tissue. For example, we can simultaneously apply longitudinal forces in opposite directions to ghost cells located at the two borders. Contact force: Cells interact with other cells through contact forces. The contact forces considered in our model come from two sources. One is called the bulk effect, in which the pressure difference between two neighboring cells leads to a force normal to their contact edge. The pressure p i on cell i is given by Avor A0 pi = k A0 and similarly for cell j. Here, A vor is the area of the cell obtained from the Voronoi tessellation, which can be larger or smaller than the area of the corresponding ellipse A 0. k is a coefficient measuring how stiff the cell is. The bulk effect force acting on cell i is (Supplementary file S1-Fig. 2.1),
bulk 1 Fij = ( pi + pj ) lijn ij. 2 Here l ij is the length of the edge shared by cell i and j. n ij is the unit normal vector to the interface of cell i and j, pointing from cell i. Supplementary file S1-Fig. 2.1. Bulk force between cell i and j. The other source of contact forces considered in the model is referred to as local configuration effects. To compute the local configuration force, we measure i A Δ, which is the area of the section of the polygon associated with the edge between cell i and cell j (Supplementary file -Fig. 2.2), and which is the area of the part of the ellipse spanned by the edge between cell i and cell j (Supplementary file S1-Fig. 2.2), and similarly for cell j. We have i A sec, i i j j config A A sec A A Δ Δ sec Fij = C C i + j. Avor A0 Avor A0 Here, C is called the jamming contact coefficient. The local configuration force depends on the mismatch between the area of polygon and ellipse. In the original model (Jennings J, A new computational model for multicellular biological systems, 2014), all forces are acting on the cell edge. For simplicity, we here treat all forces as cell-centered because we do not consider cell slippage in our model (we did not monitor cell rearrangements during tissue growth in our experiments).
Supplementary file S1-Fig. 2.2. Local configuration force between cell i and j. Tissue growth: During the simulation, both the length and the width of the rectangular domain change as the tissue grows (because of cell division) or changes under an external force (as in the lungexplant-stretching experiment). For the left and right borders with free boundary conditions, L and min L are set as the average position of the corresponding ghost cells. For the bottom and top max boundaries with periodic boundary conditions, we let H min = 0 and the change of H to be max proportional to the sum of the absolute value of the contact forces projected in the vertical direction. In the absence of both cell division and external force, the contact forces will be zero at the equilibrium state. Computing the aspect ratio: The aspect ratio of a cell is determined by the shape of the polygon obtained from the Voronoi tessellation. Given a polygon in a plane defined by its n vertices, vi = ( xi, yi) for i =1, 2... n, in a counter-clockwise manner ( v1, v2, L v n ), we derive the moment of inertia tensor for the polygon, 2 y xy I = ( r.r) Id r r da= dxdy 2 A A xy x where Id is the identity matrix. The 2-by-2 matrix I has two eigenvalues: λ 1 and λ 2 ( λ1 λ ). The 2
aspect ratio of the cell is defined as q = λ1/ λ2 1. The long axis of the cell is defined as the eigenvector corresponding to eigenvalue λ 1. Orientated cell divisions: The occurrence of cell divisions over time is modeled as a Poisson process with a constant rate of 0.1 per cell division per time step. When a division event is triggered, one cell is randomly chosen among the existing cells (excluding the ghost cells). The dividing cell splits into two daughter cells, each assigned with an ellipse with the semi major axis a=0.64µm and semi minor axis b=0.5µm, the same as other cells. The centers of the two daughter cells are placed near the center of the dividing cell, along the orientation of the cell division. The orientation of cell division depends on the aspect ratio of the dividing cell. Based on our experimental results, we have here specified a threshold q 0 (q 0 =1.53) for the aspect ratio. Cells whose aspect ratio < q 0 are termed round cells, while cells whose aspect ratio > q 0 are termed elongated cells. When a round cell divides, the division angle is random (uniform distributed from 0 to π ). If an elongated cell divides, it behaves as a fixed-spindle, and its dividing angle θ is given by θ = ˆ θ + η where ˆ θ is the angle between the long-axis of the dividing cell and the longitudinal direction of the airway tube, and η is a Gaussian random variable with mean 0 and standard deviation σ =15 o (the value of σ is estimated from the experimental data shown in Figure 5C).