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

Reviewer #1 (Public review):

Summary:

The authors' stated aim is to introduce so-called Minkowski tensors to characterize and quantify the shape of cells in tissues. The authors introduce Minkowski tensors and then define the p-atic order q_p, where p is an integer, as a cell shape measure. They also introduce a previously defined measure of p-atic order in the form of the parameter γ_p. The authors compute q_pp for data obtained by simulating an active vertex model and a multiphase field model, where they focus on p=2 and p=6 - nematic and hexatic order - as the two values of highest biological relevance. Based on their analysis, the authors claim that q₂ and q₆ are independent, that there is no crossover for the coarse-grained quantities, that the comparison of q_p for different values of p is not meaningful, and determine the dependence of the mean value of q₂ and q₆q₆ on cell activity and deformability. They then apply their method to data from MDCK monolayers and argue that the γ_p "fail to capture the nuances of irregular cell shapes".

Strength:

The work presents a set of parameters that are useful for analyzing cell shape.

Weaknesses:

The main weakness of the manuscript is that the points that the authors make are not sufficiently elaborated or supported by the data. Although they start out with Minkowski tensors, they eventually only consider the parameters q_p, which can be defined without any recourse to Minkowski tensors. Also, I dare to doubt that the average reader will benefit from the introduction to Minkowski tensors as it remains abstract and does not really go beyond repeating definitions. Eventually, for me, the work boils down to the statement that when you want to characterize (2d) cell shape, then it is better to take the whole cell contour instead of only the positions of the vertices of a polygon that approximates the full cell shape. By the way, for polygons, the q_p and γ_p should convey the same information as the vertex positions contain the whole geometric information.

Some statements made about the values of q_p are not supported by the data. For example, an independence of values of q₂ and q₆ cannot be inferred from Figure 7. Actually, Figure 8 points to some dependence between these values as the peaks of the pdfs move in the opposite direction as deformability and activity are changed. Figure 1 suggests that in general, larger cells have lower values of q_p for all p. Some more serious quantification should be obtained here.

The presented experimental data on MDCK cells is anecdotal.

Reviewer #2 (Public review):

Summary:

Orientational symmetries of cells and tissues play an important role in describing processes in development and disease, and the methods used to investigate them rely on the detection of cell shape. In this interesting and very timely manuscript by Lea Happel et al., Minkowski tensors are introduced to study the orientational symmetries of cells and set in comparison to existing shape descriptors, such as the shape function introduced by Armengol-Collado et al., which captures the orientational symmetry by the vertex positions of the polygonal shape of the cell. As an advantage, the Minkowski tensors consider the real cell shape with its arbitrary curvature of the cortex. Using computational models, such as the active vertex model and the multiphase field model, as well as experimental support with MDCK monolayers, the authors find that the orientational symmetries are independent of one another, as well as that they are dependent on the activity and deformability of the cells, resulting in a monotonic trend. A trend that has not been observed for the hexatic symmetry using the shape function. Together with the lack of hexatic-nematic crossover at the tissue scale, the authors suggest a reconsideration of findings from other shape descriptors. Taken together, the Minkowski tensors set a framework to investigate orientational symmetries at a single cell scale and how they may interplay in biological tissues.

Strengths:

The authors introduce the Minkowski tensors, which capture the p-atic orders of cells in tissues, considering their real shape instead of a polygonal approximation as reported for other shape descriptors in the literature. Thus, they do not depend on the vertex positions of the cells nor on the number of neighboring cells. The Minkowski tensors capture the dependence of the p-atic orders on the cell activity and deformability in a monotonic manner, which makes them a robust tool for quantifying p-atic orders at a single-cell scale, especially for rounded cells. The robustness has been tested by comparing the results of two computational model systems that simulate cell monolayers and whose results have been extended with experimental data. The Minkowski tensors have been used to explore the role of cell-cell adhesion and density in epithelial cells and have shown similar results to the shape function, a polygonal shape descriptor.

Weaknesses:

The authors point out the importance of studying the orientational order in biological systems. However, the current version of the manuscript lacks statistical information, a description of analysis methods, and experimental support. This support is needed to strengthen (i) the results of the two computational models and (ii) give weight to the authors' strong claim against other widely accepted shape descriptors capturing p-atic orders. The Minkowski tensors, which consider the real cell shapes, are reported to be a better method to investigate the p-atic orders of cells than the shape function introduced by Armengol-Collado et al. While there may be differences in the reported results coming from the two different approaches, both approaches show similar trends. As it stands, there is substantiated discussion as to why one method would be better than the other. The shape function, γ₆, may not be monotonic for great changes in cell activity and deformability, hinting at a potential weakness. In contrast to the shape function and results by Armengol-Collado et al. and Eckert et al., the coarse-grained Minkowski tensors do not capture the hexatic-nematic crossover at the tissue scale, applied here only to computational models. The cells simulated in the computational models have a similar size and the monolayer has a nearly regular pattern, which does not reflect the density variance in biological tissues. To strengthen the author's claim that there is no crossover at the tissue scale, experimental verification is essential. Further, the robustness of the Minkowski tensors seems to rely on determining the p-atic orders on the shape of individual cells in the tissue. However, when applying the shape descriptor to experimental systems, the p-atic orders are very low, perhaps too low for comparisons between different p-atic orders with meaningful conclusions.

Reviewer #3 (Public review):

Hapel et al. submit an article entitled “Quantifying the shape of cells - from Minkowski tensors to p-atic order”. The paper reports the p-actic quantitative method - established in physics - to extract cell shapes in experiments using phase contrast images of MDCK cells and simulations - vertex model and phase fields. The rationale of the quantification with adaptation of Minkowski tensors, as well as the detailed extraction of distributions of shapes and plots, distributions quantifying shapes are documented, with an emphasis on changes in cell shapes and their importance in epithelial dynamics.

Higher rank tensors are considered as well as representations with intuitive meanings and q_i orders and their potential correlations or absence of correlations. For example, q₂ and q₆, and statements about nematic and hexatic orders. A strong body of evidence is already reported in the papers of Armengol et al., quoted substantially in the paper, and the authors insist on an improvement thanks to the Minkowski tensors approach to challenge the former crossovers correlations statements.

Although the approach seems to present advantages, the paper does not appear sufficiently novel. Beyond the Armengol et al. paper, the advantages of this approach compared to the shear decomposition (from MPI-PKS Dresden) or the links joining centroids and its neighbours approach (MSC/Curie Paris) for example.

Author response:

We thank the editors and the reviewers for their valuable comments. In response to these suggestions, we will add rigorous statistical measures and extend the experimental support of our findings in a revised version. Indeed, as we will show, doing so strengthens all the main claims. Specifically:

Concerning Reviewer 1:

- It is important to emphasise that the advantage of deriving shape measures q_p from Minkowski tensors is their robustness and stability, that is well-established from extensive, rigorous mathematical analyses. Introducing q_p without this connection to revised Minkowski tensors would not allow to claim this stability property for the considered measures.

- Even though for a polygon the vertex positions contain the whole geometric information, using q_p and γ_p lead to different results, see Fig. 6 for an example.

- We wholeheartedly agree that our statement on independence of values of q₂ and q₆ can be extended and more quantitatively established by rigorous statistical measures. This is exactly what we will do in the revised version, not only providing statistical measures on the presented data, but also extending our analyses to the published data from Armengol-Collado JM, Carenza LN, Eckert J, Krommydas D, Giomi L. Epithelia are multiscale active liquid crystals. Nature Physics. 2023; 19:1773–1779. As we shall show these analyses further strengthen this claim, unequivocally establishing the independence of q₂ and q₆ in two different models (active vertex model and multiphase-field model), as well as two different sets of experiments (the ones in the original manuscript, and the published one from Armengol-Collado JM, Carenza LN, Eckert J, Krommydas D, Giomi L. Epithelia are multiscale active liquid crystals. Nature Physics. 2023; 19:1773–1779).

Concerning Reviewer 2:

To fully address this point, we have extended our analyses to explore the published data of Armengol-Collado JM, Carenza LN, Eckert J, Krommydas D, Giomi L. Epithelia are multiscale active liquid crystals. Nature Physics. 2023; 19:1773–1779. As we shall show in the revised manuscript, the crossover between nematic and hexatic is only specific to the use of γ_p for characterizing the shape and coarse-graining of the associated order. Using q_p as the shape measure this crossover disappears. Therefore, this analyses concretely demonstrate that the crossover is not a robust physical feature of the system and is dependent on the method used to define shape characteristics.

Concerning Reviewer 3:

We respectfully note a misunderstanding from the referee: The briefly mentioned approaches of other groups, turn out to be not measuring shape but connections between cells. Conceptually these approaches are therefore related to bond order parameters. We already comment at the end of the section introducing Minkowski tensors that bond order parameters cannot quantify the shape of a cell. The same argumentation also holds for other such approaches. In our revised version we will further clarify this distinction, to avoid any confusion or misinterpretation.