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

The importance of orientational order in biological systems is becoming increasingly clear, as it plays a critical role in processes such as tissue morphogenesis, collective cell motion, and cellular extrusion. Orientational order results from the shapes of cells and their alignments with neighboring cells. Disruptions in this order, known as topological defects, are often linked to key biological events. For instance, defects—points or lines where the order breaks down—can drive cell extrusion (Saw et al., 2017; Monfared et al., 2023) or trigger morphological changes in tissues (Maroudas-Sacks et al., 2021; Ravichandran et al., 2025). Orientational order is linked to liquid crystal theories (de Gennes, 1993) and should here be interpreted in a broad sense. Recent evidence extends beyond nematic order, characterized by symmetry under ${180}^{\circ }=2\pi /2$ rotation, to higher-order symmetries such as tetratic order (${90}^{\circ }=2\pi /4$), hexatic order (${60}^{\circ }=2\pi /6$), and even general p-atic orders ($2\pi /p$, $p$ being an integer). In biological contexts, nematic order ($p=2$) has been widely studied in epithelial tissues (Duclos et al., 2017; Saw et al., 2017; Kawaguchi et al., 2017), linking defects to cellular behaviors and tissue organization. More complex orders, such as tetratic order ($p=4$) (Cislo et al., 2023) and hexatic oder ($p=6$) (Li and Ciamarra, 2018; Durand and Heu, 2019; Pasupalak et al., 2020; Armengol-Collado et al., 2023; Eckert et al., 2023), have also been observed in experimental systems, offering new perspectives on how cells self-organize and respond to mechanical cues. Understanding and quantifying orientational order and the corresponding liquid crystal theory are, therefore, essential for unraveling the physical mechanisms that govern biological dynamics.

In physics, p-atic liquid crystals illustrate how particle shape and symmetry influence phase behavior, with seminal works dating back to Onsager’s theories (Onsager, 1949). Most prominently, hexatic order ($p=6$) has been postulated and found by experiments and simulations as an intermediate state between crystalline solid and isotropic liquid in Halperin and Nelson, 1978; Nelson and Halperin, 1979; Murray and van Winkle, 1987; Bladon and Frenkel, 1995; Zahn et al., 1999; Gasser et al., 2010; Bernard and Krauth, 2011. Other examples are colloidal systems for triadic platelets (Bowick et al., 2017) or cubes (Wojciechowski and Frenkel, 2004), which lead to p-atic order with $p=3$ and $p=4$, respectively. Even pentatic ($p=5$) and heptatic ($p=7$) liquid crystals have been engineered (Wang and Mason, 2018; Yu and Mason, 2024). The corresponding liquid crystal theories for p-atic order have only recently been defined (Giomi et al., 2022b; Krommydas et al., 2023) and can also be used in biological contexts. However, while in colloidal systems particle shapes remain fixed, in biological tissues, cells are dynamic: their shapes are irregular, variable, and influenced by internal and external forces. These unique properties make quantifying p-atic order in tissues significantly more challenging, as quantification of the cell shapes is required. We will demonstrate that existing methods, such as bond-orientational order (Nelson and Halperin, 1979) or polygonal shape analysis (Armengol-Collado et al., 2023), might fail to capture the nuances of irregular cell shapes, which has severe consequences on the definition of p-atic order.

To address this, we adapt Minkowski tensors (Mecke, 2000; Schröder-Turk et al., 2013)—rigorous mathematical tools for shape analysis—to quantify p-atic orders in cell monolayers. Minkowski tensors provide robust and sensitive measures of shape anisotropy and orientation, accommodating both smooth and polygonal shapes while remaining resilient to small perturbations. By applying these tools, we re-examine previous conclusions about p-atic orders in epithelial tissues and demonstrate that certain widely accepted results require reconsideration.

Our study leverages a combination of experimental and computational approaches. Experimentally, we analyze confluent monolayers of MDCK (Madin-Darby Canine Kidney) cells. Figure 1 provide a snapshot of a considered monolayer of 235 wild-type MDCK cells, together with their shape classification by Minkowski tensors. The corresponding statistical data and probability distributions of these quantities are shown in Figure 2. These data indicate the presence of all p-atic orders at once with similar probability distributions, mean values, and standard deviations. In addition to these experiments, we also analyze data of MDCK cells reported in Armengol-Collado et al., 2023. Computationally, we employ two complementary models for cell monolayers: the active vertex model and the multiphase field model. These approaches allow us to systematically vary parameters such as cell activity and mechanical properties, providing a comprehensive view of how p-atic orders emerges in different contexts.

Figure 1

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Figure 2

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By combining these robust mathematical tools, experimental insights, and computational models, we establish a reliable framework for quantifying p-atic orders in biological tissues. This framework not only challenges previous findings, e.g. a proposed hexatic-nematic crossover on larger length scales (Armengol-Collado et al., 2023), but also opens new pathways for understanding the role of orientational symmetries in tissue mechanics and development, which require to consider multiple orientational symmetries.

Quantifying orientational order in biological tissues can be realized by Minkowski tensors. The orientation ${\vartheta }_p$ and the strength of $p$-atic order $q_p$ in Equation 7 can be computed for each cell. Minkowski tensors provide reliable quantities describing how cell shapes align with specific rotational symmetries. As already indicated in Figure 2, situations might occur in which rotational symmetries cannot be associated with one specific $p$, but various symmetries seem to be present at the same time. One might be tempted to compare the probability distribution functions (PDFs) of $q_p$ or the mean values $\overline{q_p}$ for different $p$ in order to identify a dominating p-atic order. This is particularly important for interpreting nematic ($p=2$) and hexatic ($p=6$) orders, which describe distinct symmetries but have been found to coexist in biological systems. However, a direct comparison of these quantities only makes sense if they are comparable to each other. As already mentioned in Figure 4, this is not the case, as, e.g., $q_2=1.0$ cannot be realized for a cell with a given area. Another question one might ask is if these values are independent of each other. To answer this question, we first address statistically if $q_2$ and $q_6$, evaluated for each cell, are independent. Second, we explore how $q_p$ depends on key parameters determining tissue mechanics. In a third step, we coarse-grain these quantities using the measures $Q_2$ and $Q_6$ in Equation 11. With these quantities, we address a proposed hexatic-nematic crossover in epithelial tissue, where hexatic order dominates at small scales and nematic order prevails at larger scales (Eckert et al., 2023; Armengol-Collado et al., 2023; Armengol-Collado et al., 2024). For these three tasks, we analyze simulation data from two computational models (see Appendix 1 including additional references Hakim and Silberzan, 2017; Alert and Trepat, 2020; Moure and Gomez, 2021; Koride et al., 2018; Das et al., 2021; Killeen et al., 2022), the active vertex model (Appendix 1 -Active vertex model, including additional references Honda, 1983; Honda, 2022; Farhadifar et al., 2007; Bi et al., 2015; Tong et al., 2023) and the multiphase field model (Appendix 1 - Multiphase field model, including additional references Vey and Voigt, 2007; Witkowski et al., 2015; Praetorius and Voigt, 2018; Salvalaglio et al., 2021). Although these models differ conceptually, both have been validated in studies of cell monolayer mechanics, including solid-liquid transitions, neighbor exchange dynamics, and stress profiles (Fletcher et al., 2014; Alt et al., 2017; Li et al., 2019; Balasubramaniam et al., 2021; Wenzel and Voigt, 2021; Sknepnek et al., 2023; Melo et al., 2023). Key model parameters are the deformability and the activity strength. They are crucial for capturing the coarse-grained properties of confluent tissues (Jain et al., 2023; Jain et al., 2024). In the active vertex model, deformability is controlled by the target shape index $p_0$, reflecting the balance between cell-cell adhesion and cortical tension. The multiphase field model, by contrast, encodes deformability through the capillary number $Ca$, which directly incorporates cortical tension. We vary activity strength ($v_0$), the shape index ($p_0$), and the capillary number ($Ca$), ensuring all parameter combinations remain in the fluid regime. Fluidity was confirmed using by mean square displacement (MSD) (Loewe et al., 2020), neighbor number variance (Wenzel and Voigt, 2021), and the self-intermediate scattering function (Bi et al., 2016).

These studies demonstrate independence of $q_2$ and $q_6$, a general trend of increasing $\overline{q_2}$ and decreasing $\overline{q_6}$ for higher activity and deformability, and a consistent decrease of $\overline{Q_2}$ and $\overline{Q_6}$ for increasing coarse-graining radius $R$. However, as $q_2$ and $q_6$ are not directly comparable and $q_2$ and $q_6$ (and, therefore, also $Q_2$ and $Q_6$) are independent, the concept of a hexatic-nematic transition - typically requiring a single order parameter - may not be applicable in this context. Even if the proposed hexatic-nematic crossover (Armengol-Collado et al., 2023) is not a formal phase transition, one might expect it to be quantifiable. Yet, its characterization appears to depend strongly on the maximum attainable value of $q_2$, which in turn is influenced by several parameters. To further explore this, we reanalyze the experimental data for MDCK cells from Armengol-Collado et al., 2023 using Minkowski tensors and compute $q_2$ and $q_6$ as defined in Equation 7 (see Appendix 1-Experimental setup for details on the analysis of this experimental data with additional references Stringer et al., 2021; van der Walt et al., 2014). Our analysis also indicates independence of $q_2$ and $q_6$ supporting the same interpretation as above. The statistical properties and probability distributions of these data remain consistent when full cellular boundaries from microscopy images are used. However, an increase in hexatic order ($p=6$) is observed when cell shapes are approximated by polygons. This suggests that the dominant hexatic order reported at the cellular scale in Armengol-Collado et al., 2023 may stem from the geometric simplification of cell boundaries. We also compute $Q_2$ and $Q_6$ (Equation 11) and again observed a consistent decrease with increasing coarse-graining radius $R$, without evidence of a measurable hexatic-nematic crossover. To better understand the differences with the findings in Armengol-Collado et al., 2023, we further analyze both simulation and experimental data using the alternative shape measures ${\gamma }_p$ (Equation 8) and ${\mathrm{\Gamma }}_p$ (Equation 12) considered in Armengol-Collado et al., 2023. These measures reproduce the reported results.

Independence of $q_2$ and $q_6$

We examine the distribution of ($q_2$, $q_6$) values across deformability-activity parameter pairs in both computational models (Figure 7). Both models show consistent trends: In near-solid regimes (Figure 7 lower left), $q_2$ and $q_6$ values cluster tightly due to restricted shape fluctuations. However, even in this regime, small $q_2$ values can correspond to either small or large $q_6$ values, and vice versa. In more fluid-like regimes, with higher activity and higher deformability (Figure 7 upper right), $q_2$ and $q_6$ values become highly scattered. Each $q_2$ value spans a broad range of $q_6$ values, and vice versa, indicating their independence. In order to quantify this, we compute the distance correlation (Székely et al., 2007), which is a statistical measure quantifying linear and non-linear dependency in given data. Thereby, a value of 0.0 corresponds to independence, whereas a value of $1.0$ corresponds to a strong dependence between the datasets. As can be seen in Figure 7—figure supplement 1 the obtained distance correlation for $q_2$ and $q_6$ is quite low, underscoring that these quantities are independent. Furthermore, the corresponding p-values, as shown in Figure 7—figure supplement 2 are mostly larger than $0.1$, indicating that the weak correlation found in Figure 7—figure supplement 1 is not significant. This leads to the conclusion that $q_2$ and $q_6$ measure distinct aspects of cell shape anisotropy. As a consequence, both orders, nematic and hexatic, need to be considered independently. There cannot be a single parameter which describes a crossover between both.

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A full investigation of potential dependencies between $q_p$ for arbitrary combinations of $p$’s resulting, e.g., from symmetry arguments is beyond the scope of this paper.

Dependence of $q_p$ on activity and deformability

We now explore how tissue properties such as activity and deformability influence cell shape and orientational order. We again focus on nematic ($p=2$) and hexatic ($p=6$) orders, as shown in Figure 8—figure supplements 1 and 2. Additional results for $p=3,4,5$ are provided in Appendix 2-Results for $q_3$, $q_4$, and $q_5$, precisely in Appendix 2—figure 1 - Appendix 2—figure 3 (fixed activity) and Appendix 2—figure 4, Appendix 2—figure 5, Appendix 2—figure 6 (fixed deformability). In all plots, all cells and all time steps are considered. In Figure 8—figure supplement 1, we vary deformability ($p_0$ in the active vertex model and $Ca$ in the multiphase field model) while keeping the activity $v_0$ constant. Results are presented for the active vertex model (left column) and the multiphase field model (right column). Activity increases from bottom to top rows. Both models show qualitatively similar trends in the probability distribution functions (PDFs) of $q_2$ and $q_6$. For $p=6$, increasing deformability shifts the PDF of $q_6$ to the left, indicating lower mean values. In contrast, for $p=2$, higher deformability leads to higher $q_2$ values, reflected in a rightward shift of the PDF. This trend is confirmed by the mean values, shown as a function of deformability ($p_0$ and $Ca$, respectively) in the inlets. Additionally, the PDFs broaden with increasing deformability, and this effect is more pronounced at lower activity levels. A notable difference between the models is the range of $q_p$ values. In the multiphase field model, $q_p$ rarely exceeds $0.8$ due to the smoother, more rounded cell shapes, whereas the active vertex model often produces higher $q_p$ values. In Figure 8—figure supplement 2, deformability is held constant while activity $v_0$ is varied. Results are again shown for the active vertex model (left column) and the multiphase field model (right column), with increasing activity indicated by brighter colors. Deformability increases from bottom to top rows. The trends mirror those observed in Figure 8—figure supplement 1. Increasing activity reduces the mean value of $q_6$ while increasing $q_2$, see inlets. The effects of activity are more pronounced at lower deformabilities; at higher deformabilities, differences between parameter regimes diminish. The overall behavior of $q_2$ and $q_6$ is summarized in Figure 8, which shows the mean values $\overline{q_2}$ and $\overline{q_6}$ as functions of activity and deformability. For both models:

• $\overline{q_2}$ increases with higher activity or deformability

• $\overline{q_6}$ decreases with higher activity or deformability.

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This behavior is consistent with a trend towards nematic order in more dynamic regimes and towards hexatic order in more constrained, less dynamic regimes. The first is further confirmed by recent findings in analysing T1 transitions and their effect on cell shapes (Jain et al., 2025). These studies suggest that cells transiently elongate when they are undergoing T1 transitions. As the number of T1 transitions increases with activity or deformability Jain et al., 2024, this elongation contributes to the observed behavior. The second is consistent with the emergence of hexagonal arrangements in solid-like states.

Corresponding results for $p=3,4,5$ are shown in Appendix 2—figure 7. While $q_3$, $q_4$, $q_5$ also increase with increasing activity or deformability, the dependency is not as pronounced as for $q_2$.

Coarse-grained quantities $Q_2$ and $Q_6$ and potential hexatic-nematic crossover

For every parameter configuration in the active vertex model (Figure 9—figure supplement 1) and in the multiphase field model (Figure 9—figure supplement 2) we compute the coarse-grained quantities $Q_2$ and $Q_6$ for various coarse-graining radii $R$. One might ask the question if the observed trends for $\overline{q_2}$ and $\overline{q_6}$ for higher activity and deformability in Figure 8 are also present on larger scales. We, therefore, investigate the behavior of $\overline{Q_2}$ and $\overline{Q_6}$ upon varying activity and deformability, see Figure 9. This is exemplified for $R/R_{cell}=8.0$. The trends seen for $\overline{q_2}$ and $\overline{q_6}$ - so increasing ($\overline{q_2}$) or decreasing ($\overline{q_6}$) with higher activity or deformability - are lost for ${\overline{Q}}_2$ and less pronounced for ${\overline{Q}}_6$.

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In Armengol-Collado et al., 2023 a similar approach was used to identify a hexatic-nematic crossover (see Armengol-Collado et al., 2023, Figure 3e). This cross-over is considered at the coarse-graining radius at which the two curves for $\overline{Q_2}$ and $\overline{Q_6}$ as a function of $R$ cross. While already conceptually questioned above, we consider these investigations to compare with Armengol-Collado et al., 2023. However, regardless of the model, there is no consistent trend indicative of a potential crossover. These results question the proposed hexatic-nematic crossover reported in Armengol-Collado et al., 2023. To further explore this issue, we next test the existence of such a crossover directly on the data considered in Armengol-Collado et al., 2023.

Analyzing experimental data for MDCK cells

The experimental data for confluent monolayers of MDCK GII cells used in Armengol-Collado et al., 2023 are provided in two different formats, as microscopy images and as polygon data with the calculated vertex points per cell. We consider both formats and all 68 provided configurations. The polygonal data are directly used to compute $q_2$ and $q_6$. The experimental microscopy images are segmented and the extracted cell boundaries are used to compute $q_2$ and $q_6$.

We compute the PDFs of $q_2$ and $q_6$, Figure 10a for both data sets. While they are similar, if the full cellular contour from the microscopy images is considered the PDFs strongly differ if the polygonal shapes are used. The dominating hexatic ($p=6$) order is, therefore, just a consequence of the approximation of the cell boundaries by polygons. This different behavior, which shows larger values for $q_6$ for the polygonal shapes has the same origin as the difference between the regular and rounded shapes in Figure 4. Further differences result from the different accessible parameter range. While $q_6=1.0$ is possible for a perfect hexagon, $q_2=1.0$ cannot be realized, as this would correspond to a line.

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We also test the values for $q_2$ and $q_6$ for independence, see Figure 10b and in the corresponding statistical measures, Figure 10—figure supplements 2 and 3. The indicated independence in the scatter plots Figure 10b is confirmed by the distance correlation and the p-values, as in Figure 7, Figure 7—figure supplement 1 and Figure 7—figure supplement 2. This holds for the polygonal shapes as well as for the more detailed shapes from the microscopy images.

As a consequence, the same arguments as discussed above also hold for the experimental data and thus caution against interpretation of $q_2$ and $q_6$ or their coarse-grained quantities $Q_2$ and $Q_6$ as interdependent order parameters. However, in order to compare with Armengol-Collado et al., 2023 we next compute the averaged coarse-grained quantities $\overline{Q_2}$ and $\overline{Q_6}$ for various coarse-graining radii $R$. In Figure 11 these curves are shown for the polygonal shapes (a) and the microscopy images (b). As in Armengol-Collado et al., 2023, we carried out the coarse-graining until the coarse-graining radius corresponds to half of the domain width. In both plots, the curves for $\overline{Q_2}$ are almost identical, reflecting the similar PDFs in Figure 10a. For $\overline{Q_6}$ the slope is similar but the curves are shifted. A potential crossover, therefore, also depends on the approximation of the cells. In any case, for the considered data, no consistent hexatic-nematic crossover can be observed.

Figure 11

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In order to resolve the discrepancy of these results with Armengol-Collado et al., 2023 we next examine the analysis using the alternative shape measures ${\gamma }_p$ in Equation 8, which have been considered in Armengol-Collado et al., 2023 but are shown to be not stable.

Sensitivity of the results on the considered shape descriptor

We now demonstrate that the alternative shape descriptors ${\gamma }_p$ in Equation 8, which have been used in Armengol-Collado et al., 2023, can lead to qualitatively different and thus misleading results. The corresponding figures to Figures 1 and 2 and Figure 4 are shown in Appendix 2 -Results using polygonal shape analysis, precisely in Appendix 2—figure 9, Appendix 2—figure 10 and Appendix 2—figure 8, respectively. For the experimental data in Figures 1 and 2 we use the Voronoi interface method (Saye and Sethian, 2011; Saye and Sethian, 2012) to calculate the vertices of a polygon approximating the cell shape. The comparison of these figures already indicates differences between the two methods. Such differences can be seen in the approximated cell shapes, the PDFs and more quantitatively also by comparing $\overline{|{\gamma }_p|}$ in Appendix 2—figure 10a with $\overline{q_p}$ in Figure 2a , which, e.g., for $p=6$ almost double.

For a more detailed comparison of $|{\gamma }_p|$ and $q_p$ we investigate the data from the active vertex model and the polygonal approximation of the cells in Armengol-Collado et al., 2023. We restrict ourselves to this data, as for multiphase field data, the usage of ${\gamma }_p$ first requires the approximation of a cell by a polygon and we have already seen that this approximation strongly influences the results.

We focus on Figure 8 and the corresponding results in Figure 12. Instead of the monotonic trend for activity and deformability, which was found for $\overline{q_6}$, $\overline{|{\gamma }_6|}$ exhibits non-monotonic trends, peaking at intermediate values of activity or deformability. For completeness, we also provide the corresponding figures to Figures 7 and 9 in Appendix 2—figures 11 and 12, respectively.

Figure 12

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For the polygonal data of the MDCK cells considered in Armengol-Collado et al., 2023, we compare the coarse-grained quantities $\overline{Q_2}$ and $\overline{Q_6}$, already considered in Figure 11a, with $\overline{|{\mathrm{\Gamma }}_2|}$ and $\overline{|{\mathrm{\Gamma }}_6|}$ computed from ${\gamma }_2$ and ${\gamma }_6$ using Equation 12, see Figure 13. For completeness, we also provide the corresponding figure to Figure 10 in Appendix 2—figure 13. Besides minor differences regarding the calculation of $R_{cell}$ Figure 13 corresponds to Armengol-Collado et al., 2023, Figure 3e. Comparing Figure 11a and Figure 13 one might come to the conclusion that there is no hexatic-nematic crossover using $\overline{Q_2}$ and $\overline{Q_6}$, but there is a hexatic-nematic crossover using $\overline{|{\mathrm{\Gamma }}_2|}$ and $\overline{|{\mathrm{\Gamma }}_6|}$. As the only difference between these two evaluations is the considered shape characterization, this analysis adds another argument that the proposed crossover is not a robust physical feature of the system.

Figure 13

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These results also demonstrate that not only the approximation of the cell boundaries by polygonal shapes heavily influences the characterization of p-atic order but also the considered method to classify the shape might lead to qualitative different results. This confirms the argumentation in Methods that Minkowski tensors should be preferred because of their stability properties.

In this study, we introduced Minkowski tensors as a robust and versatile tool for quantifying p-atic order in multicellular systems, particularly in scenarios involving rounded or irregular cell shapes. By applying this framework to extensive datasets from two distinct computational models—the active vertex model and the multiphase field model—we identified universal trends: increasing activity and deformability of the cells enhance nematic order ($p=2$) while diminishing hexatic order ($p=6$). The consistency of these findings across two models, despite their inherent differences, underscores the generality of our results.

While various shape characterization methods, such as the bond order parameter (Loewe et al., 2020; Monfared et al., 2023) and the shape function ${\gamma }_p$ (Armengol-Collado et al., 2023), have been explored in the literature, we demonstrated that the choice of shape descriptor significantly impacts the conclusions drawn. Such divergences, together with limited mathematical foundations, highlight the limitations of these alternative shape measures in capturing consistent patterns and emphasize the need for stable, reliable shape measures like the Minkowski tensors. As the stability of Minkowski tensors - in contrast to the bond order parameter or the shape function ${\gamma }_p$ - can be mathematically justified Minkowski tensors should be the preferred shape descriptor. This finding is not merely a technical nuance, it leads to qualitative differences. Analyzing experimental data for MDCK cells, e.g., has demonstrated that a strong hexatic order on the cellular scale has no physical origin but is a consequence of the approximation of the cell boundaries by polygonal shapes, which is a requirement to use the shape function ${\gamma }_6$. Considering the full cellular boundaries and $q_6$, which is derived from the Minkowski tensors, leads to a different picture, with weaker hexatic order. A critical question in the literature has been whether shape measures for different p-atic orders can be directly compared. While some studies have suggested relationships between ${\gamma }_2$ and ${\gamma }_6$ (Armengol-Collado et al., 2023), our results refute this notion. We demonstrated that measures like $q_2$ and $q_6$ are independent and capture fundamentally distinct aspects of cell shape and alignment. Comparing them directly is mathematically but also physically and biologically misleading. We further tested the hypothesis of a hexatic-nematic crossover at larger length scales by coarse-graining $q_2$ and $q_6$. To discuss such a crossover requires direct comparison of $q_2$ and $q_6$ or their coarse-grain quantities $Q_2$ and $Q_6$, which is conceptually questionable. However, the results showed no consistent trends indicative of a crossover, regardless of the considered model or the experimental data. This leads to the conclusion that the proposed hexatic-nematic crossover in Armengol-Collado et al., 2023 is not a physical phenomena but specific to the considered method.

Our findings suggest that p-atic orders should be studied independently, also across length scales, as they describe complementary aspects of cellular organization. The coexistence of distinct orientational orders emphasized in different studies—such as nematic ($p=2$) (Duclos et al., 2017; Saw et al., 2017; Kawaguchi et al., 2017), tetratic ($p=4$) (Cislo et al., 2023), and hexatic ($p=6$) (Li and Ciamarra, 2018)—is not contradictory but highlights the rich, multifaceted nature of cellular organization. Rather than searching for a single dominant order, future research should focus on the interplay of different p-atic orders and their associated defects. This suggests to not only consider p-atic liquid crystal theories (Giomi et al., 2022b) for one specific $p$, but combinations of these models for various $p$’s. Understanding how these orders interact may reveal how they collectively regulate morphogenetic processes.

Connecting p-atic orders to biological function remains a critical avenue for exploration. While the mathematical independence of $q_2$, $q_6$, and other shape measures precludes the identification of a universal dominant order, biological systems may exhibit context-dependent preferences. For example, a specific p-atic order might correlate with or drive a key morphogenetic event. Investigating these connections could yield insights into how tissues achieve functional organization and adapt to environmental cues. As such, while it is difficult to speak of dominating orders from a mathematical point of view, there could be a dominating order from a biological point of view, meaning the p-atic order connected to the governing biological process.

A code illustrating the extraction of the contour and the calculation from q_p and ϑ_p for grayscale images can be found on Zenodo at https://doi.org/10.5281/zenodo.15430268. Simulation code for the vertex and multiphase field models can be found at https://github.com/sknepneklab/RheoVM (Sknepnek, 2023) and https://gitlab.mn.tu-dresden.de/iwr/amdis (Vey et al., 2022), respectively. The data from Armengol-Collado et al., 2023 to which we compare in the result section was made publicly available by the authors of said paper under GitHub at https://github.com/hexanematic/orientation_tracker.

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Author details

1. Lea Happel

   Institute of Scientific Computing, Dresden, Germany

   Contribution

   Software, Formal analysis, Validation, Investigation, Visualization, Writing – original draft

   Contributed equally with

   Griseldis Oberschelp

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-4525-9185

2. Griseldis Oberschelp

   Institute of Scientific Computing, Dresden, Germany

   Contribution

   Software, Formal analysis, Validation, Visualization, Methodology, Writing – original draft

   Contributed equally with

   Lea Happel

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0009-0000-1735-7493

3. Valeriia Grudtsyna

   Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark

   Contribution

   Data curation, Investigation

   Competing interests

   No competing interests declared

4. Harish P Jain

   Njord Centre, Physics Department, University of Oslo, Oslo, Norway

   Contribution

   Software, Formal analysis

   Competing interests

   No competing interests declared

5. Rastko Sknepnek

   1. School of Life Sciences, University of Dundee, Dundee, United Kingdom
   2. School of Science and Engineering, University of Dundee, Dundee, United Kingdom

   Contribution

   Data curation, Software, Funding acquisition

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-0144-9921

6. Amin Doostmohammadi

   Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark

   Contribution

   Funding acquisition, Investigation, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1116-4268

7. Axel Voigt

   1. Institute of Scientific Computing, Dresden, Germany
   2. Center of Systems Biology Dresden, Dresden, Germany
   3. Cluster of Excellence, Physics of Life, TU Dresden, Dresden, Germany

   Contribution

   Conceptualization, Supervision, Funding acquisition, Investigation, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   axel.voigt@tu-dresden.de

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-2564-3697

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