Figures and data in Quantifying the shape of cells, from Minkowski tensors to p-atic orders

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Additional files

29 figures, 2 tables and 1 additional file

Figures

Figure 1

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Shape classification of cells in wild-type Madin-Darby canine kidney (MDCK) cell monolayer.

(a) Raw experimental data. (**b-f**) Minkowski tensor, visualized using $ϑ_{p}$ and $q_{p}$ , Equation 7 (see Methods) 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. The visualization uses rotationally symmetric direction fields known as $p$ -RoSy fields in computer graphics (Vaxman et al., 2016). See Appendix 1-Experimental setup for details on the experimental data.

Figure 2

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Statistical data for cell shapes identified in Figure 1 (see Methods).

(a) Mean $\bar{q_{p}}$ and standard deviation $σ (q_{p})$ of $q_{p}$ . (**b- f**) Probability distribution function (PDF) of $q_{p}$ for $p = 2, 3, 4, 5, 6$ , respectively. Kde-plots are used to show the probability distribution.

Figure 3

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Schematic description of a two-dimensional object $C$ with contour $\partial C$ .

We denote the center of mass with $x_{c}$ and vectors from $x_{c}$ to points $x$ on $\partial C$ with $r$ . The outward-pointing normals are denoted by $n$ , the corresponding angle with the $x$ -axis by $θ_{n}$ .

Figure 4

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Regular and irregular shapes, adapted from Armengol-Collado et al., 2023 and Schaller et al., 2024, by Equation 7.

For regular shapes, the corresponding magnitude of $q_{p}$ 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 $q_{2} = 1.0$ is shown, as this would be a line. The visualization is according to Figure 1.

Figure 5

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Illustrative description of the definition of $q_{3}$ for an equilateral triangle.

Considering rotational symmetries under a rotation $120^{\circ} = 2 π / 3$ means that vectors with an angle of $120^{\circ}$ or $240^{\circ}$ 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 $e^{3 i θ_{n}}$ 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 three points on each edge, we obtain an orientation with the respective symmetry on every point of the contour $\partial C$ . Considering the line integral along the contour provides the dominating triadic director, shown in the center of mass (right). To get a value between $0.0$ and $1.0$ for $q_{3}$ , we normalize this integral with the length of the contour, which corresponds to $q_{0}$ . As all triatic directors point in the same direction, we obtain $q_{3} = 1.0$ in this specific example. To be consistent with other approaches, we rotate the resulting triadic director by $60^{\circ} = π / 3$ leading to the orange triadic director, which is the quantity used for visualization.

Figure 6

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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 $q_{p}$ and (b) using $γ_{p}$ . The considered vectors used in the computations, normals $n$ of the contour for the Minkowski tensors and $r_{i}$ for $γ_{p}$ , are shown. Note that the removal of the forth vertex highly influences the value of $γ_{p}$ . How $x_{c}$ is calculated - as the mean of the vertex coordinates or as the center of mass of the polygon - can also slightly alter the results. We used the described approach following Armengol-Collado et al., 2023.

Figure 7 with 2 supplements

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) orders are independent of each other.

$q_{6}$ (y-axis) versus $q_{2}$ (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 $(q_{2}, q_{6})$ . Each panel corresponds to specific model parameters; $C a$ and $v_{0}$ for multiphase field model, and $p_{0}$ and $v_{0}$ for the active vertex model, representing deformability and activity, respectively.

Figure 7—figure supplement 1

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Distance correlation $d C o r (q_{2}, q_{6})$ between $q_{2}$ and $q_{6}$ for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

Each panel corresponds to specific model parameters; $C a$ and $v_{0}$ for multiphase field model, and $p_{0}$ and $v_{0}$ for the active vertex model, representing deformability and activity, respectively. We compute one value for every timestep and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles.

Figure 7—figure supplement 2

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P-values of the distance correlation $P_{d C o r (q_{2}, q_{6})}$ between $q_{2}$ and $q_{6}$ for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

Each panel corresponds to specific model parameters; $C a$ and $v_{0}$ for multiphase field model, and $p_{0}$ and $v_{0}$ for the active vertex model, representing deformability and activity, respectively. We compute one value for every timestep and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles. 0.05 and 0.1 are marked with a gray dotted/gray dashed line to guide the eye.

Figure 8 with 2 supplements

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on activity and deformability of the cells.

Mean value $\bar{q_{p}}$ for $p = 2$ (left) and $p = 6$ (right) as function of deformability $p_{0}$ or $C a$ and activity $v_{0}$ for active vertex model (a and b) and multiphase field model (c and d).

Figure 8—figure supplement 1

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on deformability of the cells.

Probability distribution functions (PDFs) for $q_{2}$ (shades of orange) and (shades of blue), using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ . Inlets show mean values of $q_{2}$ and $q_{6}$ as function of deformability. (**a-d**) Active vertex model, (**e-h**) Multiphase field model for decreasing activity.

Figure 8—figure supplement 2

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on activity of the cells.

Probability distribution functions (PDFs) for $q_{2}$ (shades of orange) and $q_{6}$ (shades of blue), using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ . Inlets show mean values of $q_{2}$ and $q_{6}$ as function of activity. (**a-d**) active vertex model and (**e-h**) multiphase field model for decreasing deformability.

Figure 9 with 2 supplements

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Coarse-gained nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for $R / R_{c e l l} = 8$ depend on activity and deformability of the cells.

Mean value $\bar{Q_{p}}$ for $p = 2$ (left) and $p = 6$ (right) as function of deformability $p_{0}$ or $C a$ and activity $v_{0}$ for active vertex model ( $a$ and $b$ ) and multiphase field model ( $c$ and $d$ ).

Figure 9—figure supplement 1

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$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii in the active vertex model.

$Q_{p}$ was calculated according to Equation 11, the averaging of this and the choice of $R_{c e l l}$ follow the description in Coarse-grained quantities. The maximal coarse-graining radius corresponds to half the domain width. Each panel corresponds to specific model parameters; $p_{0}$ and $v_{0}$ , representing deformability and activity. A logarithmic scaling was used for both axes. Error bars are obtained as s.e.m.

Figure 9—figure supplement 2

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$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii in the multiphase field model.

$Q_{p}$ was calculated according to Equation 11, the averaging of this and the choice of $R_{c e l l}$ follow the description in Coarse-grained quantities. The maximal coarse-graining radius corresponds to half the domain width. Each panel corresponds to specific model parameters; $C a$ and $v_{0}$ , representing deformability and activity. A logarithmic scaling was used for both axes. Error bars are obtained as s.e.m.

Figure 10 with 3 supplements

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for the cells in the experiments from Armengol-Collado et al., 2023.

(a) Probability distribution functions (PDFs) using kde-plots, for $q_{2}$ (yellow) and $q_{6}$ (blue), once using the polygonal approximation of the cell shape and once using the detailed cell outline obtained from the microscopy pictures. (b) $q_{6}$ (y-axis) versus $q_{2}$ (x-axis) for all cells from the experimental data in Armengol-Collado et al., 2023, once using the polygonal approximation of the cell shape (blue) and once using the detailed cell outline obtained from the microscopy pictures (red). For each cell and each timestep, we plot one point $(q_{2}, q_{6})$ .

Figure 10—figure supplement 1

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Experimental image, segmented cell outline and polygonal shape.

Cell shapes in the experiments from Armengol-Collado et al., 2023. (a) Cell shapes used for the shape quantification, (b) Detailed cell outline obtained from the microscopy image, (c) Polygonal approximation of the cell shape. Shown for frame number 1.

Figure 10—figure supplement 2

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Distance correlation $d C o r (q_{2}, q_{6})$ between $q_{2}$ and $q_{6}$ .

On the left side (a) we use the polygonal approximation of the cell shape, on the right side (b) we use the detailed cell outline obtained from the microscopy pictures. We compute one value for every frame and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles.

Figure 10—figure supplement 3

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P-values of the distance correlation $P_{d C o r (q_{2}, q_{6})}$ between $q_{2}$ and $q_{6}$ .

On the left side (a) we use the polygonal approximation of the cell shape, on the right side (b) we use the detailed cell outline obtained from the microscopy pictures. We compute one value for every frame and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles. 0.05 and 0.1 are marked with a gray dotted/gray dashed line to guide the eye.

Figure 11

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$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

On the left side (a) we use the polygonal approximation of the cell shape, on the right side (b) we use the detailed cell outline obtained from the microscopy pictures. $Q_{p}$ was calculated according to Equation 11, the averaging of this and the choice of $R_{c e l l}$ follow the description in Coarse-grained quantities. The maximal coarse-graining radius corresponds to half the domain width. A logarithmic scaling was used for both axes. Error bars are obtained as s.e.m.

Figure 12

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Mean value $\bar{| γ_{p} |}$ as function of deformability $p_{0}$ and activity $v_{0}$ for active vertex model.

(a) nematic order ( $p = 2$ ), (b) hexatic order ( $p = 6$ ).

Figure 13

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$\bar{| Γ_{6} |}$ versus $\bar{| Γ_{2} |}$ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

We use only the polygonal approximation of the cell shape as $γ_{p}$ can only work with polygons. $Q_{p}$ was calculated according to Equation 12, the averaging of this and the choice of $R_{c e l l}$ follow the description in Coarse-grained quantities. The maximal coarse-graining radius corresponds to half the domain width. A logarithmic scaling was used for both axes. Error bars are obtained as s.e.m.

Appendix 1—figure 1

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Illustration of the active vertex model.

(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.

Appendix 1—figure 2

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Illustration of the multiphase field model.

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

Appendix 2—figure 1

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PDFs for $q_{3}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

Inlets show mean values of $q_{3}$ as function of deformability. (**a-d**) Active vertex model, (**e-h**) Multiphase field model for decreasing activity.

Appendix 2—figure 2

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PDFs for $q_{4}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

Inlets show mean values of $q_{4}$ as function of deformability. (**a-d**) Active vertex model, (**e-h**) Multiphase field model for decreasing activity.

Appendix 2—figure 3

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PDFs for $q_{5}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

Inlets show mean values of $q_{5}$ as function of deformability. (**a-d**) Active vertex model, (**e-h**) Multiphase field model for decreasing activity.

Appendix 2—figure 4

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PDFs for $q_{3}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

Inlets show mean values of $q_{3}$ as function of activity. (**a-d**) Active vertex model and (**e-h**) Multiphase field model for decreasing deformability.

Appendix 2—figure 5

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PDFs for $q_{4}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

Inlets show mean values of $q_{4}$ as function of activity. (**a-d**) Active vertex model and (**e-h**) Multiphase field model for decreasing deformability.

Appendix 2—figure 6

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PDFs for $q_{5}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

Inlets show mean values of $q_{5}$ as function of activity. (**a-d**) Active vertex model and (**e-h**) Multiphase field model for decreasing deformability.

Appendix 2—figure 7

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Mean value $\bar{q_{p}}$ for $p = 3$ (left), $p = 4$ (middle) and $p = 5$ (right) as a function of deformability $p_{0}$ or $C a$ and activity $v_{0}$ for active vertex model (**a-c**) and multiphase field model (**d-f**).

Appendix 2—figure 8

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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 $| γ_{p} |$ .

Appendix 2—figure 9

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Shape classification of cells in wild-type Madin-Darby canine kidney (MDCK) cell monolayer.

(a) Raw experimental data. (**b-f**) Polygonal shape classification, visualized using $γ_{p}$ 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. See Appendix 1-Experimental setup for details on the experimental data.

Appendix 2—figure 10

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Statistical data for cell shapes identified in Appendix 2—figure 9.

(a) Mean $\bar{| γ_{p} |}$ and standard deviation $σ (| γ_{p} |)$ of $| γ_{p} |$ . (**b- f**) Probability distribution function (PDF) of $| γ_{p} |$ for $p = 2, 3, 4, 5, 6$ , respectively. Kde-plots are used to show the probability distribution. For this first analysis, we regard only one frame with 235 cells.

Appendix 2—figure 11 with 2 supplements

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$| γ_{6} |$ (y-axis) versus $| γ_{2} |$ (x-axis) for all cells in the active vertex model.

For each cell and each timestep, we plot one point $(| γ_{2} |, | γ_{6} |)$ . Each panel corresponds to specific model parameters $p_{0}$ and $v_{0}$ , representing deformability and activity.

Appendix 2—figure 11—figure supplement 1

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Distance correlation $d C o r (| γ_{2} |, | γ_{6} |)$ for the simulation data of the active vertex model.

Each panel corresponds to specific model parameters $p_{0}$ and $v_{0}$ , representing deformability and activity. We compute one value for every timestep and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles.

Appendix 2—figure 11—figure supplement 2

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P-values of the distance correlation $P_{d C o r (| γ_{2} |, | γ_{6} |)}$ for the simulation data of the active vertex model.

Each panel corresponds to specific model parameters $p_{0}$ and $v_{0}$ , representing deformability and activity. We compute one value for every timestep and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles. 0.05 and 0.1 are marked with a gray dotted/gray dashed line to guide the eye.

Appendix 2—figure 12 with 1 supplement

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Coarse-Grained nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for $R / R_{c e l l} = 8$ depending on activity and deformability of the cells.

$\bar{| Γ_{p} |}$ as function of deformability $p_{0}$ and activity $v_{0}$ for active vertex model. (a) nematic order ( $p = 2$ ), (b) hexatic order ( $p = 6$ ).

Appendix 2—figure 12—figure supplement 1

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$\bar{Γ_{6}}$ versus $\bar{Γ_{2}}$ for different coarse-graining radii in the active vertex model.

$Γ_{p}$ was calculated according to Equation 12, the averaging of this and the choice of $R_{c e l l}$ follow the description in Coarse-grained quantities. The maximal coarse-graining radius corresponds to half the domain width. Each panel corresponds to specific model parameters: $p_{0}$ and $v_{0}$ , representing deformability and activity. A logarithmic scaling was used for both axes. Error bars are obtained as s.e.m.

Appendix 2—figure 13 with 2 supplements

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Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for the cells in the experiments from Armengol-Collado et al., 2023.

We use the polygonal approximation of the cell shape as $γ_{p}$ can only work with polygons. (a): Probability distribution functions (PDFs) using kde-plots, for $| γ_{2} |$ (orange) and $| γ_{6} |$ (blue). (b): Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order are independent of eachother. $| γ_{6} |$ (y-axis) versus $| γ_{2} |$ (x-axis) for all cells. For each cell and each timestep, we plot one point $(| γ_{2} |, | γ_{6} |)$ .

Appendix 2—figure 13—figure supplement 1

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Distance correlation $d C o r (| γ_{2} |, | γ_{6} |)$ for all cells from the experimental data in Armengol-Collado et al., 2023.

We use the polygonal approximation of the cell shape as $γ_{p}$ can only work with polygons. We compute one value for every frame and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles.

Appendix 2—figure 13—figure supplement 2

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P-value of the distance correlation $P_{d C o r (| γ_{2} |, | γ_{6} |)}$ for all cells from the experimental data in Armengol-Collado et al., 2023.

We use the polygonal approximation of the cell shape as $γ_{p}$ can only work with polygons. We compute one value for every frame and present the resulting distribution with a box plot. In the box plots, the orange line corresponds to the median of the data and the box ranges from the first to the third quartile of the data. The whiskers go from the lowest data point greater than (value of the first quartile) − 1.5 × (Interquartile range) to the highest data point below (value of the third quartile) + 1.5 × (Interquartile range). Outliers are shown with circles.

Author response image 1

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Tables

Appendix 1—table 1

Values of the dimensionless parameters used in the active vertex model.

Parameter	Description	Numerical value
$N$	Number of cells	100
$L$	Simulation box size	10
$T$	Total simulation time	300
$k_{P}$	Perimeter elastic modulus	1.0
$τ$	Simulation time step	0.01
$v_{0}$	Self-propulsion strength (i.e. activity)	0.1–0.4
$D_{r}$	Rotation diffusion coefficient	0.05
$p_{0}$	Shape index of active cells	3.5–3.875

Appendix 1—table 2

Values of the dimensionless parameters used in the multiphase field model.

Parameter	Description	Numerical value
$N$	Number of cells	100
$L$	Simulation box size	100
$T$	Total simulation time	150
$ϵ$	Interface width	0.15
$τ$	Simulation time step	0.005
$v_{0}$	Self-propulsion strength (i.e. activity)	0.4–1.0
$D_{r}$	Rotation diffusion coefficient	0.01
$α$	Alignment parameter	0.1
$C a$	Capillary number	0.05–0.2
$I n$	Interaction number	0.1
$a_{a}$	Cell-cell attraction strength	1.0
$a_{r}$	Cell-cell repulsion strength	1.0
$h$	Mesh size	$5 h \approx ϵ$

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Lea Happel
Griseldis Oberschelp
Valeriia Grudtsyna
Harish P Jain
Rastko Sknepnek
Amin Doostmohammadi
Axel Voigt

(2025)

Quantifying the shape of cells, from Minkowski tensors to p-atic orders

eLife 14:RP105680.

https://doi.org/10.7554/eLife.105680.3

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Shape classification of cells in wild-type Madin-Darby canine kidney (MDCK) cell monolayer.

Statistical data for cell shapes identified in Figure 1 (see Methods).

Schematic description of a two-dimensional object C with contour ∂C.

Regular and irregular shapes, adapted from Armengol-Collado et al., 2023 and Schaller et al., 2024, by Equation 7.

Illustrative description of the definition of q3 for an equilateral triangle.

Defining p-atic order for deformable objects requires robust shape descriptors.

Nematic (p=2) and hexatic (p=6) orders are independent of each other.

Distance correlation dCor(q2,q6) between q2 and q6 for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

P-values of the distance correlation PdCor(q2,q6) between q2 and q6 for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

Nematic (p=2) and hexatic (p=6) order depend on activity and deformability of the cells.

Nematic (p=2) and hexatic (p=6) order depend on deformability of the cells.

Nematic (p=2) and hexatic (p=6) order depend on activity of the cells.

Coarse-gained nematic (p=2) and hexatic (p=6) order for R/Rcell=8 depend on activity and deformability of the cells.

Q6¯ versus Q2¯ for different coarse-graining radii in the active vertex model.

Q6¯ versus Q2¯ for different coarse-graining radii in the multiphase field model.

Nematic (p=2) and hexatic (p=6) order for the cells in the experiments from Armengol-Collado et al., 2023.

Experimental image, segmented cell outline and polygonal shape.

Distance correlation dCor(q2,q6) between q2 and q6.

P-values of the distance correlation PdCor(q2,q6) between q2 and q6.

Q6¯ versus Q2¯ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

Mean value |γp|¯ as function of deformability p0 and activity v0 for active vertex model.

|Γ6|¯ versus |Γ2|¯ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

Illustration of the active vertex model.

Illustration of the multiphase field model.

PDFs for q3 using kde-plots, for varying deformability p0 or Ca and fixed activity v0.

PDFs for q4 using kde-plots, for varying deformability p0 or Ca and fixed activity v0.

PDFs for q5 using kde-plots, for varying deformability p0 or Ca and fixed activity v0.

PDFs for q3 using kde-plots, for varying activity v0 and fixed deformability p0 or Ca.

PDFs for q4 using kde-plots, for varying activity v0 and fixed deformability p0 or Ca.

PDFs for q5 using kde-plots, for varying activity v0 and fixed deformability p0 or Ca.

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

Regular and irregular shapes, adapted from Armengol-Collado et al., 2023, with magnitude and orientation calculated by Equation 8 and Equation 9.

Shape classification of cells in wild-type Madin-Darby canine kidney (MDCK) cell monolayer.

Statistical data for cell shapes identified in Appendix 2—figure 9.

|γ6| (y-axis) versus |γ2| (x-axis) for all cells in the active vertex model.

Distance correlation dCor(|γ2|,|γ6|) for the simulation data of the active vertex model.

P-values of the distance correlation PdCor(|γ2|,|γ6|) for the simulation data of the active vertex model.

Coarse-Grained nematic (p=2) and hexatic (p=6) order for R/Rcell=8 depending on activity and deformability of the cells.

Γ6¯ versus Γ2¯ for different coarse-graining radii in the active vertex model.

Nematic (p=2) and hexatic (p=6) order for the cells in the experiments from Armengol-Collado et al., 2023.

Distance correlation dCor(|γ2|,|γ6|) for all cells from the experimental data in Armengol-Collado et al., 2023.

P-value of the distance correlation PdCor(|γ2|,|γ6|) for all cells from the experimental data in Armengol-Collado et al., 2023.

Screenshot from p6 = 3.

Values of the dimensionless parameters used in the active vertex model.

Values of the dimensionless parameters used in the multiphase field model.

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Schematic description of a two-dimensional object $C$ with contour $\partial C$ .

Illustrative description of the definition of $q_{3}$ for an equilateral triangle.

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) orders are independent of each other.

Distance correlation $d C o r (q_{2}, q_{6})$ between $q_{2}$ and $q_{6}$ for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

P-values of the distance correlation $P_{d C o r (q_{2}, q_{6})}$ between $q_{2}$ and $q_{6}$ for all cells in the multiphase field model (MPF - purple box) and active vertex model (AV - green box).

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on activity and deformability of the cells.

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on deformability of the cells.

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order depend on activity of the cells.

Coarse-gained nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for $R / R_{c e l l} = 8$ depend on activity and deformability of the cells.

$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii in the active vertex model.

$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii in the multiphase field model.

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for the cells in the experiments from Armengol-Collado et al., 2023.

Distance correlation $d C o r (q_{2}, q_{6})$ between $q_{2}$ and $q_{6}$ .

P-values of the distance correlation $P_{d C o r (q_{2}, q_{6})}$ between $q_{2}$ and $q_{6}$ .

$\bar{Q_{6}}$ versus $\bar{Q_{2}}$ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

Mean value $\bar{| γ_{p} |}$ as function of deformability $p_{0}$ and activity $v_{0}$ for active vertex model.

$\bar{| Γ_{6} |}$ versus $\bar{| Γ_{2} |}$ for different coarse-graining radii for the experimental data from Armengol-Collado et al., 2023.

PDFs for $q_{3}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

PDFs for $q_{4}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

PDFs for $q_{5}$ using kde-plots, for varying deformability $p_{0}$ or $C a$ and fixed activity $v_{0}$ .

PDFs for $q_{3}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

PDFs for $q_{4}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

PDFs for $q_{5}$ using kde-plots, for varying activity $v_{0}$ and fixed deformability $p_{0}$ or $C a$ .

Mean value $\bar{q_{p}}$ for $p = 3$ (left), $p = 4$ (middle) and $p = 5$ (right) as a function of deformability $p_{0}$ or $C a$ and activity $v_{0}$ for active vertex model (a-c) and multiphase field model (d-f).

$| γ_{6} |$ (y-axis) versus $| γ_{2} |$ (x-axis) for all cells in the active vertex model.

Distance correlation $d C o r (| γ_{2} |, | γ_{6} |)$ for the simulation data of the active vertex model.

P-values of the distance correlation $P_{d C o r (| γ_{2} |, | γ_{6} |)}$ for the simulation data of the active vertex model.

Coarse-Grained nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for $R / R_{c e l l} = 8$ depending on activity and deformability of the cells.

$\bar{Γ_{6}}$ versus $\bar{Γ_{2}}$ for different coarse-graining radii in the active vertex model.

Nematic ( $p = 2$ ) and hexatic ( $p = 6$ ) order for the cells in the experiments from Armengol-Collado et al., 2023.

Distance correlation $d C o r (| γ_{2} |, | γ_{6} |)$ for all cells from the experimental data in Armengol-Collado et al., 2023.

P-value of the distance correlation $P_{d C o r (| γ_{2} |, | γ_{6} |)}$ for all cells from the experimental data in Armengol-Collado et al., 2023.

Screenshot from p₆ = 3.