Shape classification of cells in wild-type MDCK cell monolayer. a) Raw experimental data. b) - f) Minkowski tensor, visualized using ϑp and qp, Equation 7 (see Methods and materials) for p = 2, 3, 4, 5, 6, respectively. The brightness and the rotation of the p-atic director indicates the magnitude and the orientation, respectively.

Statistical data for cell shapes identified in Figure 1 (see Methods and materials). a) Mean qp and standard deviation σ(qp) of qp. b) - f) Probability distribution function (PDF) of qp for p = 2, 3, 4, 5, 6, respectively. Normal distributions with the same mean and standard deviations are added to the histograms as a guide to the eye.

Schematic description of a two-dimensional object with contour ∂𝒞. We denote the center of mass with x𝒞 and vectors from x𝒞 to points x on ∂𝒞 with r. The outward-pointing normals are denoted by n, the corresponding angle with the x-axis by θn.

Regular and irregular shapes, adapted from Armengol-Collado et al. (2023) and Schaller et al. (2024), with magnitude and orientation calculated by Equation 7. For regular shapes, the corresponding magnitude of qp is always 1.0 and the detected angle is the minimal angle of the p-atic orientation with respect to the x-axis. Note that no shape with q2 = 1.0 is shown, as this would be a line. The visualization uses rotationally-symmetric direction fields (known as p-RoSy fields in computer graphics (Vaxman et al., 2016)). The brightness scales with the magnitude of the p-atic order, qp, see color bars in Figure 1.

Illustrative description of the definition of q3 for an equilateral triangle. Considering rotational symmetries under a rotation 120° = 2π/3 means that vectors with an angle of 120° or 240° are treated as equal. Applied to the normals n (left), this means that under this rotational symmetry the normals on the three different edges are equal. Mathematically this is expressed through resulting in the triatic director shown instead of the normals n (middle). One leg of the triatic director always points in the direction as the normal. While only shown for 3 points on each edge, we obtain an orientation with the respective symmetry on every point of the contour ∂𝒞. Considering the line integral along the contour provides the dominating triatic director, shown in the center of mass (right). To get a value between 0.0 and 1.0 for q3 we normalize this integral with the length of the contour, which corresponds to q0. As all triatic directors point in the same direction we obtain q3 = 1.0 in this specific example. To be consistent with other approaches we rotate the resulting triatic director by 60° = π/3 leading to the orange triadic director, which is the quantity used for visualization.

Defining p-atic order for deformable objects requires robust shape descriptors. Shown is the strength of p-atic order for a polygon converging to an equilateral triangle. a) using qp and b) using γp. The considered vectors used in the computations, normals n of the contour for the Minkowski tensors and ri for γp, are shown. Note that the choice of the center of mass highly influences the value of γp. We here used the described approach following Armengol-Collado et al. (2023).

a) Cell contour of the active vertex model. Red arrows represent the polarity vectors that set each cell’s instantaneous direction of self-propulsion. b) Zoom in on a vertex surrounded by three cells showing how the direction of self-propulsion on a vertex is calculated.

a) Cell contours of the multiphase field model. b) Corresponding phase field functions along the horizontal line in a). Colours correspond to the once in a).

Nematic (p = 2) and hexatic (p = 6) order are independent of eachother. q6 (y-axis) versus q2 (x-axis) for all cells in the multiphase field model (blue) and active vertex model (red). For each cell and each timestep we plot one point (q2, q6). Each panel corresponds to specific model parameters; Ca and v0 for multiphase field model, and p0 and v0 for the active vertex model, representing deformability and activity, respectively.

Nematic (p = 2) and hexatic (p = 6) order depend on deformability of the cells. Probability distribution functions (PDFs) for q2 (shades of orange) and q6 (shades of blue), using kde-plots, for varying deformability p0 or Ca and fixed activity v0. Inlets show mean values of q2 and q6 as function of deformability. a) - d) active vertex model, e) - h) multiphase field model for decreasing activity.

Nematic (p = 2) and hexatic (p = 6) order depend on activity of the cells. Probability distribution functions (PDFs) for q2 (shades of orange) and q6 (shades of blue), using kde-plots, for varying activity v0 and fixed deformability p0 or Ca. Inlets show mean values of q2 and q6 as function of activity. a) - d) active vertex model and e) - h) multiphase field model for decreasing deformability.

Nematic (p = 2) and hexatic (p = 6) order depend on activity and deformability of the cells. Mean value for p = 2 (left) and p = 6 (right) as function of deformability p0 or Ca and activity v0 for active vertex model (a) and b)) and multiphase field model (c) and d)).

In experimental system of MDCK cells, nematic (p = 2) and hexatic (p = 6) order respond differently to change in cell density. Probability distribution functions (PDFs) using kde-plots, for q2 (orange shaded) and q6 (blue shaded) for different densities in a) wildtype MDCK cell monolayer and b) E-cadherin knockout cell monolayer.

In experimental system of MDCK cells nematic (p = 2) and hexatic (p = 6) order are independent of eachother for different cell densities and different cell-cell adhesion strengths. q6 (y-axis) versus q2 (x-axis) for different densities a) low density and b) high density, for wildtype MDCK cell monolayer and E-cadherin knockout cell monolayer. For each cell and each timestep we plot one point (q2, q6).The E-cadherin cells lack an adhesion protein and have therefore weaker cell-cell adhesions.

Values of the dimensionless parameters used in the active vertex model

Values of the dimensionless parameters used in the multiphase-field model simulations

Q6 versus Q2 for different coarse graining radii in the active vertex model, calculated according to Equation 18. A logarithmic scaling was used for both axis. qp and ϑp follow from Equation 7, R 𝒞 is chosen as .

Q6 versus Q2 for different coarse graining radii in the multiphase field model, calculated according to Equation 18. A logarithmic scaling was used for both axis. qp and ϑp follow from Equation 7, R is chosen as .

Regular and irregular shapes, adapted from Armengol-Collado et al. (2023), with magnitude and orientation calculated by Equation 8 and Equation 9. The brightness scales with the magnitude |γp|.

Shape classification of cells in wild-type MDCK cell monolayer. a) Raw experimental data. b) - f) Polygonal shape classification, visualized using γp calculated by Equation 8 and Equation 9 for p = 2, 3, 4, 5, 6, respectively. The brightness and the rotation of the p-atic director indicates the magnitude and the orientation, respectively.

Statistical data for cell shapes identified in Appendix 3 Figure 16 (. a) Mean and standard deviation σ(|γp|) of |γp|. b) - f) Probability distribution function (PDF) of |γp| for p = 2, 3, 4, 5, 6, respectively. Normal distributions with the same mean and standard deviations are added to the histograms as a guide to the eye.

Mean value as function of deformability p0 and activity v0 for active vertex model. a) nematic order (p = 2), b) hexatic order (p = 6). While the behaviour of |γ2| qualitatively agrees with the behaviour of q2, the behaviour of |γ6| is quite different to the one of q6.

γ6 (y-axis) versus γ2 (x-axis) for all cells in the active vertex model. For each cell and each timestep we plot one point (γ2, γ6). Each panel corresponds to specific model parameters p0 and v0, representing deformability and activity.

Γ6 versus Γ2 for different coarse graining radii in the active vertex model, calculated according to Equation 19. A logarithmic scaling was used for both axis. γp follows from Equation 8, R𝒞 is chosen as .

PDFs for q3 using kde-plots, for varying deformability p0 or Ca and fixed activity v0. Inlets show mean values of q3 as function of deformability. a) - d) active vertex model, e) - h) multiphase field model for descreasing activity.

PDFs for q4 using kde-plots, for varying deformability p0 or Ca and fixed activity v0. Inlets show mean values of q4 as function of deformability. a) - d) active vertex model, e) - h) multiphase field model for decreasing activity.

PDFs for q5 using kde-plots, for varying deformability p0 or Ca and fixed activity v0. Inlets show mean values of q5 as function of deformability. a) - d) active vertex model, e) - h) multiphase field model for decreasing activity.

PDFs for q3 using kde-plots, for varying activity v0 and fixed de- formability p0 or Ca. Inlets show mean values of q3 as function of activity. a) - d) active vertex model and e) - h) multiphase field model for decreasing deformability.

PDFs for q4 using kde-plots, for varying activity v0 and fixed de- formability p0 or Ca. Inlets show mean values of q4 as function of activity. a) - d) active vertex model and e) - h) multiphase field model for decreasing deformability.

PDFs for q5 using kde-plots, for varying activity v0 and fixed de- formability p0 or Ca. Inlets show mean values of q5 as function of activity. a) - d) active vertex model and e) - h) multiphase field model for decreasing deformability.

Mean value for p = 3 (left), p = 4 (middle) and p = 5 (right) as function of deformability p0 or Ca and activity v0 for active vertex model (a) - c)) and multiphase field model (d) - f)).