Physics of Living Systems

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

Lea Happel
Griseldis Oberschelp
Valeriia Grudtsyna
Harish P Jain
Rastko Sknepnek
Amin Doostmohammadi
Axel Voigt author has email address

Institute of Scientific Computing, TU Dresden, Dresden, Germany
Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark
Njord Centre, Physics Department, University of Oslo, Oslo, Norway
School of Life Sciences, University of Dundee, Dundee, United Kingdom
School of Science and Engineering, University of Dundee, Dundee, United Kingdom
Center of Systems Biology Dresden, Dresden, Germany
Cluster of Excellence, Physics of Life, TU Dresden, Dresden, Germany

https://doi.org/10.7554/eLife.105680.2

Open access
Copyright information

Figures and data

Shape classification of cells in wild-type MDCK cell monolayer.
a) Raw experimental data. b) - f) Minkowski tensor, visualized using ϑ_p and q_p, 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. The visualization uses rotationally-symmetric direction fields (known as p-RoSy fields in computer graphics (Vaxman et al., 2016)).

Statistical data for cell shapes identified in Figure 1 (see Methods and materials).
a) Mean 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.

Statistical data for cell shapes identified in Figure 1 (see Methods and materials).
a) Mean 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.

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

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 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 q₂ = 1.0 is shown, as this would be a line. The visualization is according to 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.

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 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 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 each other.
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.
Figure 7—figure supplement 1. Distance correlation dCor(q₂, q6) between q₂ and q₆
Figure 7—figure supplement 2. P-values of the distance correlation between q₂ and q₆

Nematic (p = 2) and hexatic (p = 6) order are independent of each other.
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.
Figure 7—figure supplement 1. Distance correlation dCor(q₂, q6) between q₂ and q₆
Figure 7—figure supplement 2. P-values of the distance correlation between q₂ and q₆

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)).
Figure 8—figure supplement 1. Nematic (p = 2) and hexatic (p = 6) order depend on deformability of the cells
Figure 8—figure supplement 2. Nematic (p = 2) and hexatic (p = 6) order depend on activity of the cells

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)).
Figure 8—figure supplement 1. Nematic (p = 2) and hexatic (p = 6) order depend on deformability of the cells
Figure 8—figure supplement 2. 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_cell = 8 depend on activity and deformability of the cells.
Mean value for p = 2 (left) and p = 6 (rigth) as function of deformability p₀ or Ca and activity v₀ for active vertex model (a) and b)) and multiphase field model (c) and d)).
Figure 9—figure supplement 1. versus for different coarse graining radii in the active vertex model.
Figure 9—figure supplement 2. versus for different coarse graining radii in the multiphase field model.

Coarse-gained nematic (p = 2) and hexatic (p = 6) order for R/R_cell = 8 depend on activity and deformability of the cells.
Mean value for p = 2 (left) and p = 6 (rigth) as function of deformability p₀ or Ca and activity v₀ for active vertex model (a) and b)) and multiphase field model (c) and d)).
Figure 9—figure supplement 1. versus for different coarse graining radii in the active vertex model.
Figure 9—figure supplement 2. versus 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).
a): Probability distribution functions (PDFs) using kde-plots, for q₂ (yellow) and q₆ (blue), once using the polygonal approximation of the cell shape and once using the detailed cell outline obtained from the microscopy pictures. b): q₆ (y-axis) versus q₂ (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 (q₂, q₆).
Figure 10—figure supplement 1. Experimental image, segmented cell outline and polygonal shape
Figure 10—figure supplement 2. Distance correlation dCor(q₂, q₆) between q₂ and q₆
Figure 10—figure supplement 3. P-values of the distance correlation

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 q₂ (yellow) and q₆ (blue), once using the polygonal approximation of the cell shape and once using the detailed cell outline obtained from the microscopy pictures. b): q₆ (y-axis) versus q₂ (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 (q₂, q₆).
Figure 10—figure supplement 1. Experimental image, segmented cell outline and polygonal shape
Figure 10—figure supplement 2. Distance correlation dCor(q₂, q₆) between q₂ and q₆
Figure 10—figure supplement 3. P-values of the distance correlation

versus 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 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 axis. Error bars are obtained as s.e.m..

versus 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 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 axis. Error bars are obtained as s.e.m..

Mean value as function of deformability p₀ and activity v₀ for active vertex model.
a) nematic order (p = 2), b) hexatic order (p = 6).

Mean value as function of deformability p₀ and activity v₀ for active vertex model.
a) nematic order (p = 2), b) hexatic order (p = 6).

versus 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 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 axis. Error bars are obtained as s.e.m..

versus 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 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 axis. Error bars are obtained as s.e.m..

Values of the dimensionless parameters used in the active vertex model

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

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 decreasing 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 deformability 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 deformability 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 deformability 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)).

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)).

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|.

Shape classification of cells in wild-type 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.

Statistical data for cell shapes identified in Appendix 3 Figure 22.
(a) Mean 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.

Statistical data for cell shapes identified in Appendix 3 Figure 22.
(a) Mean 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.

|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.
Appendix 3—figure 24—figure supplement 1. Distance correlation dCor(| γ₂|, |γ₆|)
Appendix 3—figure 24—figure supplement 2. P-values of the distance correlation

|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.
Appendix 3—figure 24—figure supplement 1. Distance correlation dCor(| γ₂|, |γ₆|)
Appendix 3—figure 24—figure supplement 2. P-values of the distance correlation

Coarse-Grained nematic (p = 2) and hexatic (p = 6) order for R/R_cell = 8 depending on activity and deformability of the cells.
as function of deformability p and activity v for active vertex model. a) nematic order (p = 2), b) hexatic order (p = 6).
Appendix 3—figure 25—figure supplement 1. versus for different coarse graining radii in the active vertex model.

Coarse-Grained nematic (p = 2) and hexatic (p = 6) order for R/R_cell = 8 depending on activity and deformability of the cells.
as function of deformability p and activity v for active vertex model. a) nematic order (p = 2), b) hexatic order (p = 6).
Appendix 3—figure 25—figure supplement 1. versus 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).
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 |γ₂| (orange) and |γ₆| (blue). b): Nematic (p = 2) and hexatic (p = 6) order are independent of eachother. |γ₆| (y-axis) versus |γ₂| (x-axis) for all cells. For each cell and each timestep we plot one point (|γ₂|, |γ₆|).
Appendix 3—figure 26—figure supplement 1. Distance correlation dCor(|γ₂|, |γ₆|)
Appendix 3—figure 26—figure supplement 2. P-value of the distance correlation

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 |γ₂| (orange) and |γ₆| (blue). b): Nematic (p = 2) and hexatic (p = 6) order are independent of eachother. |γ₆| (y-axis) versus |γ₂| (x-axis) for all cells. For each cell and each timestep we plot one point (|γ₂|, |γ₆|).
Appendix 3—figure 26—figure supplement 1. Distance correlation dCor(|γ₂|, |γ₆|)
Appendix 3—figure 26—figure supplement 2. P-value of the distance correlation

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).
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. 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.

P-values of the distance correlation between q₂ and q₆ 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; Ca and v₀ for multiphase field model, and p₀ and v₀ 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 grey dotted/grey dashed line to guide the eye.

P-values of the distance correlation between q₂ and q₆ 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; Ca and v₀ for multiphase field model, and p₀ and v₀ 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 grey dotted/grey dashed line to guide the eye.

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.

versus 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 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; p₀ and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

versus 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 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; p₀ and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

versus 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 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; Ca and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

versus 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 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; Ca and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

Cell shapes in the experiments from Armengol-Collado et al. (2023).
a) and the 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.

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

P-values of the distance correlation between q₂ and q₆.
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 grey dotted/grey dashed line to guide the eye.

P-values of the distance correlation between q₂ and q₆.
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 grey dotted/grey dashed line to guide the eye.

Distance correlation dCor(|γ₂|, | γ₆|) for the simulation data of the active vertex model.
Each panel corresponds to specific model parameters p₀ and v₀, 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.

P-values of the distance correlation for the simulation data of the active vertex model.
Each panel corresponds to specific model parameters p₀ and v₀, 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 grey dotted/grey dashed line to guide the eye.

P-values of the distance correlation for the simulation data of the active vertex model.
Each panel corresponds to specific model parameters p₀ and v₀, 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 grey dotted/grey dashed line to guide the eye.

versus 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 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; p₀ and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

versus 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 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; p₀ and v₀, representing deformability and activity. A logarithmic scaling was used for both axis. Error bars are obtained as s.e.m..

Distance correlation dCor(|γ₂|, |γ₆|) 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.

P-value of the distance correlation 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.

P-value of the distance correlation 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.

Sign up for email alerts