Fig 1.
Schematics showing possible variations of SC features and illustration of the Cellular Potts Model for simulation.
A Confocal images of ♂ wt (male wildtype) SC (labelled green) at 23 and 36 hours after pupariation. Each scale bar: 20 μm. B Schematic showing the rotation of SC. C Schematics showing three hypotheses for SC rotation: the push model (left) where SC rotates due to the force generated by the expanded distal cells, the pull model (centre) where SC rotates due to the pulling force generated by the contraction of proximal cells, and the push and pull model (right) where SC rotates due to both the pushing and pulling forces from the cells distally and proximally. D Schematics showing possible variations in SC orientations during evolution (top). Images of adult legs of Drosophila species that exemplify these variations (bottom). Each scale bar: 20 μm. E Schematics showing some possible variations in SC shapes during evolution (top). Images of adult legs of Drosophila species that exemplify these variations (bottom). Each scale bar: 20 μm. F Schematics showing possible variations in SC lengths during evolution (top). Images of adult legs of Drosophila species that exemplify these variations (bottom). Each scale bar: 20 μm. G Left: an example configuration of pixels in the Cellular Potts Model. Each square enclosed by dotted lines is a pixel (8 × 8 pixels in this configuration). The number inside the pixel represents the cell index label σ. Each pixel at a single time can only be labelled by one cell index. In this example there are 15 “cells” occupying 64 pixels at the current moment, and the solid lines represent “cell” boundaries. The colours of the cell index labels represent cell types c. Right: illustration of an attempted pixel label flip during a Monte Carlo step (mcs). The circled pixel on the left panel is the randomly chosen “target pixel”, and the pixel with a hexagon is the (also randomly chosen) neighbouring pixel (invading pixel). Whether there is a change to the cell index label of the target pixel is dependent on the relative effective energies of the configuration with and without the flipping. During a single mcs, there can be many such attempted pixel label flips (as specified by the parameters of the simulation–see Table 1). H Illustration of how variables θ and R are calculated for axial preference of epithelial cells. In this example, “cell” 11 is the “invading” cell (since the “invading” pixel belongs to that cell), and the “target” pixel is in cell 9. θ(σ = 11) is the angle subtended between the two vectors: the x axis and the vector that points from the centre of mass (CoM) of the “cell” 11 to the target pixel. R(σ = 11) is the norm of
. In this example only θ(σ = 11) and R(σ = 11) are shown. Similarly, θ(σ = 9) (not labelled in this figure) is the angle subtended between the x axis and the vector
that points from the CoM of cell 9 to the target pixel, while R(σ = 9) (again not labelled in this figure) is the norm of
. I Left: an example initial cell configuration for a 9-tooth SC simulation. As in G and H, we use different colours to differentiate cell types (blue-EP1; magenta-EP2; yellow-EP3; green-SCT; red-BA), but the boundaries (black horizontal and vertical lines) depicted here are cell boundaries, not pixel boundaries. Right: blow-up of a selected rectangular area from the upper panel to illustrate cell types and sizes. In this magnification, there are four types of cells shown: EP1 (blue), EP3 (yellow), SCT (green) and BA (red). Cells of the same type may have different initial areas, as demonstrated by the blue EP1 cells. As an indication of the relative initial areas occupied by different cells, the (square) SC tooth marked with an “X” has an area of 6 × 6 = 36 pixels. “Proximal” refers to the region above the SC and “distal” refers to the region below the SC. Therefore, EP1 and EP2 cells are sometimes called “distal cells” but EP3 cells are sometimes called “proximal cells”.
Table 1.
System-wide simulation parameters.
Fig 2.
Inhomogeneous and differential epithelial cell expansion critical for proper SC rotation.
A,B Approximately homogeneous spatial arrangement of distal epithelial cells. Adhesion parameter J(SCT, SCT) used: 4000. C Inhomogeneous spatial arrangement of distal epithelial cells. Adhesion parameter J(SCT, SCT) used: 4000. Please see S1 Video for a frame-by-frame capture of this simulation.
Table 2.
Mechanical parameters of different cell types for simulations, unless otherwise specified in the main text.
A value of “A” for atarget(c, t = 0) indicates that the initial area target value is the same as the initial area of that cell.
Fig 3.
SC rotation angle is dependent on expansion characteristics of epithelial cells.
A Representative simulation of a minimally rotated SC upon completion of rotation. Adhesion parameter J(SCT, SCT) used: 0. B Representative simulation of an intermediately rotated SC upon completion of rotation. Adhesion parameter J(SCT, SCT) used: 0. C Representative simulation of a maximally rotated SC upon completion of rotation. Adhesion parameter J(SCT, SCT) used: 0. D Summary statistics of the change in areas of distal epithelial cells in each of simulations A, B and C. Vertical bars represent mean areas of distal epithelial cells. Colours correspond to the type of cells depicted in simulations. Non-solid bars represent values calculated at the start of simulations and solid bars represent values calculated at the conclusion of simulation (t = 2000 mcs). Dotted line represents α, final SC angle of each of the simulations. E Graphical illustration of calculation of rotation angle α. A straight line is connected between the CoM of the SC segments that are located at the two extreme ends of the SC. α is the angle between this straight line and the -x-axis.
Fig 4.
Confocal images of SC experiments for ♀wt, ♂babPR72 and ♂wt. Each scale bar: 10 μm. Initial = 23 hours AP; Final = 36 hours AP.
Please see S2 Video. for a frame-by-frame capture of the entire SC rotation process of three flies with identical genotypes as the three here. A Minimally rotated SC in female wildtype. B Intermediately rotated SC in mutant babPR72. C Maximally rotated SC in male wildtype. D Summary statistics of the change in areas of distal epithelial cell in experiments with the above three fly genotpyes: ♀wt, ♂babPR72 and ♂wt. Conventions as in Fig 3D. The values displayed here are the average values of the 5 pupae for each genotype.
Fig 5.
Simulation of rotation of SCs of different lengths and adhesion parameters.
A Illustration of ABASCT (“angle between adjacent sex comb tooth”) and how it is related to the curvature of SC. B Initial conditions and example SC rotation simulations of different lengths. Left panel of each row: initial spatial configuration for SC rotation simulations. Second from left: rotated SC with low ABASCT SD. Third from left: rotated SC with higher ABASCT SD. Second from right: broken rotated SC. Right: Simulated SC length in number of SC teeth. Adhesion parameter J(SCT, SCT) used in the examples shown: 8000 for 5-tooth and 7-tooth SCs, 6000 for 9-tooth SCs, 4000 for 11-tooth SCs.
Fig 6.
SC breaking statistics for different SC lengths and adhesion parameters.
A Graph of intact ratio vs. adhesion parameter between SC teeth for rotated SC simulations of various SC lengths. B Two different parameter regimes for SC breaking. Top: SC breaking in the parameter regime where mutual adhesion between SC teeth is strong. This 9-tooth example is from the adhesion parameter regime marked with “#” in A. Bottom: SC breaking in the parameter regime where mutual adhesion between SC teeth is weak. This 9-tooth example is from the adhesion parameter regime marked with “##” in A.
Table 3.
Adhesion parameter J between two cell types (see Eq 2), unless otherwise specified.
A value of “S” indicates that the parameter is dependent on specific simulations. J is symmetric against interchange of the two cell types.
Fig 7.
Statistics on ABASCT standard deviation for both experiment and simulated SC rotations.
A Graph of ABASCT SD vs. SC length in experiments. Please see Table 4 for details on the mutant genotypes studied. B Graph of aggregate ABASCT SD statistics of simulated and intact rotated SCs, grouped by adhesion parameters, of all SC lengths examined. C Example outputs showing interplay between adhesion and expansion of epithelial cells on ABASCT SD. Upper panel. Higher ABASCT SD is generally obtained in rotated SCs with stronger adhesion between SC teeth. This 5-tooth example is drawn from the adhesion parameter regime marked with “#” in B. Lower panel. Lower ABASCT SD is generally obtained in rotated SCs with weaker adhesion between SC teeth. This 5-tooth example is drawn from the adhesion parameter regimes marked with “##” in B. D Example outputs showing “slowed down” rotation of 5-tooth SCs. With the exception of a larger τ for epithelial cell expansion, the example from the left panel shares identical parameters with the top one in C (adhesion parameter regime marked with “#”), while the example from the right panel has the same parameters as the bottom one in C (adhesion parameter regime marked with “##”).
Table 4.
Summary of mutant fly genotypes studied.
Fig 8.
A suitable temporal sequence of expansion of distal epithelial cells improves breaking statistics for longer SCs.
A Four graphs of intact ratio vs adhesion between SC teeth, with different line colours representing the effect of different magnitudes (in mcs) of delayed expansion of the distal epithelial cells closer to the base of the SC (blue coloured EP1 cells in Fig 5B) relative to the epithelial cells closer to the tip of the SC (magenta EP2 cells in Fig 5B) on intact ratio. Each graph shows the results for one particular SC length (5, 7, 9 or 11-tooth SC). B Two example 11-tooth SC simulations showing that delayed expansion of distal epithelial cells closer to the base of SC reduces incidences of SC breaking during rotation. Left: all distal epithelial cells expand at the same time. Right: delayed expansion of distal epithelial cells closer to the base of SC (i.e. EP1) by 120 mcs. Both example simulations share the same random seed and other parameters (with SC adhesion in the parameter regime labelled with “#” in A) except the cell expansion sequence. S3 and S4 Videos show the frame-by-frame capture of the two simulations, respectively.
Fig 9.
Experiments showing rotation of longer SCs exhibits delayed expansion of distal epithelial cells closer to the base of comb.
A Left: confocal image of short SC and its surrounding cells at the start of area measurement (23 hours AP). Shaded areas (magenta and blue) are the cells selected for area measurement. Middle: same confocal image with identical shaded cells at 36 hours AP (end of area measurement at 41.5 hours AP). Right: graph of percentage change of apical areas of the shaded cells vs. time and the fitted logistic curves. Each coloured straight line is the tangent at the inflexion point of the same coloured logistic curve. Its intercept with the time axis is mathematically constructed as the start of the expansion phase of the cells. Each scale bar: 20 μm. B Same as A but for wildtype comb. Start of area measurement at 22 hours AP. C Same as B but for longer comb. Start of area measurement at 22 hours AP.
Table 5.
Variables of the simulations.