Random nature of epithelial cancer cell monolayers

Abstract

Although the polygonal shape of epithelial cells has been drawing the attention of scientists for several centuries, only a decade and a half ago it was demonstrated that distributions of polygon types (DOPTs) are similar in proliferative epithelia of many different plant and animal species. In this study, we show that hyper-proliferation of cancer cells disrupts this universal paradigm and results in randomly organized epithelial structures. Examining non-synchronized and synchronized HeLa cervix cells, we suppose that the spread of cell sizes is the main parameter controlling the DOPT in the cancer cell monolayers. To test this hypothesis, we develop a theory of morphologically similar random polygonal packings. By analysing differences between tumoural and normal epithelial cell monolayers, we conclude that the latter have more ordered structures because of their lower proliferation rates and, consequently, more effective relaxation of mechanical stress associated with cell division and growth. To explain the structural features of normal proliferative epithelium, we take into account the spread of cell sizes in the monolayer. The proposed theory also rationalizes some highly ordered unconventional post-mitotic epithelia.

1. Introduction

Symmetry and topology determine the structure and laws of motion for relatively simple abiotic systems studied by physics and chemistry. In living systems, gene expression is usually considered as the fundamental mechanism controlling development and homeostasis [1,2]. Nevertheless, the polygonal (prismatic in three-dimensional) shape of cells, and their highly ordered packing in epithelia, clearly demonstrate that the organization of these cell monolayers directly follows basic physical and topological rules [3–5]. Epithelial growth is achieved by intercalary cell divisions within a constrained volume [6]. Dividing cells change their mediolateral neighbours, thus maintaining the apico-basal architecture and tightness of the intact layer [7], which retains both robustness and plasticity. The most striking phenomenon is the so-called topological invariance observed in almost all proliferative epithelia within phylogenetically distant organisms harbouring different global architectures. In fact, during their formation, different epithelial structures converge to polygonal packings with very similar distributions of polygon types (DOPTs) [8–12]. Therefore, in very different epithelia the probabilities of observing cells with the same number of sides are approximately equal. This topological invariance seems to be closely associated with the physiological invariance of epithelium: in all eumetazoans, epithelial structures form a selective paracellular barrier, which controls fluxes of nutrients, regulates ion and water movements, and limits host contact with antigens and microbes. This universal function of epithelia is achieved by the maintenance of the epithelial tightness throughout various morphogenetic processes, like embryonic development, organogenesis or continuous cell renewal [13–15].

Topology of the cellular borders was first studied back in 1926 [8]. In this pioneer work, the DOPT in the proliferative epithelium of cucumber was investigated: out of 1000 epithelial cells studied, 474 were hexagonal, 251 were pentagonal and 224 were heptagonal. Later, in an excellent article [11], the hypothesis of topological invariance was formulated. It was established that the similar distributions of polygons are observed in epithelia of several more animal species, and a Markov-type theory was proposed to explain such invariance. This theory assumed that the junction that appears during cell division is always located in such a manner that it forms two additional polygonal vertices, which are necessarily on non-adjacent sides of the original polygon. It was also supposed that the probability of the junction formation is independent of the ratio between the parts into which the cell is divided. Nevertheless, the theory [11,12] cannot explain the existence of 4-valent cells, for which the observed fraction varies from 2 to 3% [8,11]. In addition, this approach considers neither the possibility of cellular motility nor mechanical interactions between cells, which are crucial for the epithelium properties [16–18].

These facts, together with new findings regarding the effect of mechanical stress on the reorganization of focal adhesion, adherent junctions and cytoskeleton, as well as cell division and growth [10,16,19–21], motivated development of the microscopic models of epithelia. In the past 15 years, several approaches, which consider cells as polygons and define energy of the system as a function of their areas and the lengths of their edges, were proposed to explain the influence of mechanics on the topology and collective behaviour in cellular monolayers [4,22–27]. One of the major successes of these models is a discovery of the solid–liquid transition controlled by the effective parameter, so-called target shape index [4,23,24], that is defined by the competition between active contractility of actin–myosin subcellular cortex, cortical tension and cell–cell adhesion [24]. It has been demonstrated that decrease of the adhesion between cells leads to the ‘jamming' of the soft liquid-like phase, which is accompanied by the increase of the share of six-valent cells and overall ordering of the structure. Experimental observations confirm that epithelial tissues indeed exhibit glassy behaviour essential for such processes as embryonic development, cancer metastasis and wound healing [24]. Normal development of epithelium involves mesenchymal-to-epithelial transition leading to the dramatic decrease of cell mobility and mitotic rate. At this transition, cells take more regular shapes with smaller average perimeter. The reverse process, leading to the appearance of elongated cells with high mobility, is called the epithelial-to-mesenchymal transition (EMT) [28]; cells undergoing oncogenic EMT are believed to drive metastasis process [29]. In this context, new data on relatively easily measurable geometric parameters of cell monolayers as DOPTs and distribution of cell areas can be helpful to distinguish pathological states and develop theoretical models of cancer and normal epithelia.

In this paper, we investigate the structural characteristics of non-synchronized (conventional) and synchronized confluent monolayers obtained from HeLa epithelial cancer cells. The latter monolayers are of particular interest, since most of the cells constituting this epithelial model belong to the second generation, and the Markov-type theory used in [11] goes beyond the scope of its applicability. Moreover, despite much previously published data [30], a putative influence of the cell cycle duration and cell synchronization on the epithelial structure remains unclear [13].

In order to rationalize the observed epithelial structures, we propose a new geometrical model generating polygonal packings that are very similar to the structures observed in non-synchronized and synchronized HeLa monolayers. Testing and applying our approach to several normal proliferative epithelia, we propose the physical mechanism underlying the topological difference between normal and hyperproliferative epithelia. As we demonstrate, in the epithelia with lower proliferation rates the relaxation of mechanical stresses associated with cell division and cell growth results in more ordered structures and maintains the topological invariance.

2. Results

2.1. Structural characterization of cancer cell monolayers

Before carrying out the structural characterization of the epithelial monolayers that we obtained, it is important to discuss some general properties of DOPTs. Note that the cell polygons in the epithelium are generally convex and form a tessellation of the monolayer surface that is similar to the Voronoi one [31], which, in turn, is a dual of Delaunay triangulation [32]. In this case, for an infinite arbitrary flat monolayer, the average number of nearest neighbours equals 6. Indeed, if the Gaussian curvature is absent, then the equality Δ = 0 takes place, where

$$\mathrm{\Delta }=\sum _i{P}_i\cdot (i-6),$$ 2.1

P_i = N_i/Σ_jN_j is the concentration of cells with i nearest neighbours and N_j is the total number of cells with j nearest neighbours. Recall that the quantity $Q={\sum }_i{N}_i\cdot (i-6)$, proportional to Δ, is called the topological charge [33–35]. For a triangulation of the sphere, Q = 12, for a torus, as well as for an infinite plane, Q = 0.

In fact, the non-local equality Δ = 0 or the equivalent statement about the average number of neighbours means that the DOPT must be balanced: the number of cellular n-gons, with n < 6, must balance the number of n-gons with n > 6. In this equilibrium, the weights of 5- and 7-gons are equal to one, the weights of 4- and 8-gons are twice as large, etc.; equation (2.1) can be used to estimate the error in the experimental determination of DOPT. For a finite monolayer, or when averaging over several samples, the obtained value of Δ can deviate from zero, characterizing the error of the experimental method for calculating the DOPT. In particular, due to the relatively large number of ​cells considered in Cucumis epithelium [8], the error (1) for this case is approximately 0.004, which is almost 10 times lower than for the data in [11]. Nevertheless, in the geometrically correct model of cell division [11], with an increase in the number of successive divisions, Δ tends to 0, reaching Δ ≈ 0.001 at 10th division. Since both the left and right sides of the distribution contribute to Δ, it is reasonable to estimate the maximum error in determining the probabilities P_i in the DOPT as Δ/2. Also, the condition Δ = 0 severely restricts the possible shapes of the DOPT. In real epithelia, probabilities P_i, where i > 7 or i < 5, are small. The critical probability is P₆, and other probabilities follow it conserving the condition Δ = 0. For example, if in a hypothetical planar epithelium consisting only of 5-, 6- and 7-valent cells, the percentage of hexagonal cells is P₆, then concentrations of 5- and 7-valent cells should be equal to (1 − P₆)/2.

HeLa cells are human malignant epithelial cells derived from an epidermoid carcinoma of the cervix. The growth of confluent HeLa cell monolayers and synchronization procedure are described in §5 (see also electronic supplementary material, figures S1–S3). In synchronized monolayers, most of the cells belong to the second generation with similar time elapsed after the division process. The characterization of the HeLa and HeLa synchronized epithelia (figure 1) was carried out using nine and seven assembled images obtained by the juxtaposition of contiguous microscope fields. The studied epithelial areas contained from 404 to 933 cells. The first line of figure 1 shows typical non-synchronized and synchronized HeLa epithelial cells (respectively, samples HeLa9 and HeLa_syn5 in table 1). Owing to visualization specificities (see §5), the cell nuclei in micrographs are clearly visible, while the cell boundaries are indistinguishable in most cases. Therefore, in order to determine the number of nearest neighbours and obtain additional structural data, we used the Voronoi tessellation with the nodes located at the centres of the nuclei (see the second line of figure 1 and §5). That is, strictly speaking, we analysed not the valency of cells, but the valency of their nuclei. The areas of the epithelial cells were also calculated as the areas of the cells of the Voronoi tessellations.

Figure 1.

Structural characterization of HeLa cell monolayers. (a) Non-synchronized and (b) synchronized cellular structures. White scale bars are 100 µm. The triangulation with nodes in the centres of cell nuclei is imposed on the monolayers. (c,d) Voronoi tessellations for the monolayers (a) and (b), respectively. Histograms (e,f) show a probability P to observe the certain ranges of cell areas S in the samples (a,b). Polygon types and corresponding contributions to the histograms are colour-coded as shown in the legend. (g) DOPTs for the samples (a,b) and averaged data. Dark blue, blue, red and pink colours correspond to HeLa9, 〈HeLa〉, HeLa_syn5 and 〈HeLa_syn〉 lines from table 1. In all cases, the averaged topological error Δ is less than 0.01.

Table 1.

Characterization of the studied samples. The columns contain: sample name, average area of Voronoi cells S_av, dimensionless effective spread of areas ΔS/S_av, probability P₆, number of identified Voronoi cells N_vor and total number of cells N_tot. The last two lines in italics correspond to the averaged structural data (including standard deviations) for HeLa and HeLa synchronized samples.

code	Sav (μm2)	$\mathrm{\Delta }\mathit{S}/{\mathit{S}}_{\mathit{a}v}$	P6	Nvor	Ntot
HeLa1	839.0	0.444	0.318	402	578
HeLa2	1024.3	0.388	0.373	263	404
HeLa3bis	951.2	0.431	0.349	373	561
HeLa4	942.8	0.501	0.336	318	532
HeLa5	831.0	0.462	0.348	399	623
HeLa6bis	1057.9	0.388	0.343	539	742
HeLa8	987.9	0.411	0.371	582	780
HeLa9	813.1	0.441	0.352	691	915
HeLa10	923.9	0.444	0.343	577	828
HeLasyn1	499.8	0.452	0.379	688	933
HeLasyn3	628.9	0.418	0.337	591	826
HeLasyn4	853.9	0.391	0.325	409	593
HeLasyn5	696.5	0.386	0.355	512	714
HeLasyn6	970.3	0.367	0.371	353	525
HeLasyn8	815.9	0.378	0.316	396	624
HeLasyn9	693.9	0.391	0.372	521	739
〈HeLa〉	924 ± 82	0.43 ± 0.03	0.35 ± 0.02	460.4	662.6
〈Helasyn〉	705 ± 147	0.40 ± 0.03	0.35 ± 0.02	495.7	707.7

To study correlation between DOPTs and the spread of cell sizes, we analysed histograms of cell area distribution for the considered samples (figure 1e,f). All the histograms (including the samples not shown in figure 1) are wide and similar to the Gauss type, which can be explained by the spread of the cell sizes before the mitosis, and the possibility of cell division into substantially unequal parts. Asymmetry of distributions, in our opinion, is due to the fact that in the considered cellular structures, the minimum cell area is limited, not by zero, but by a specific positive value. We have also noticed that the maximum spread of areas strongly fluctuates and can notably differ in structures with similar morphology. Therefore, it is reasonable to characterize a cellular structure with the average area S_av of its cells and the average spread ΔS, which we define as the difference between the smallest and the largest cell areas among the half of the cells with the areas closest to the S_av value. We then introduce a dimensionless effective spread ΔS/S_av.

Structural data on the investigated epithelia are presented in table 1. The last two lines contain data that are averaged for all the non-synchronized (〈HeLa〉) and synchronized (〈Hela_syn〉) samples studied, respectively. Note that in 2nd–4th columns, the values are weighted arithmetic means. Namely, the S_av, ΔS/S_av and P₆ values are weighted by the numbers of N_vor in each line.

The average cell area in the HeLa non-synchronized epithelium is larger than in the synchronized one (table 1), since most of the cells in the latter belong to the second generation with similar time elapsed after the division process. In all the examined specimens of both types, the ΔS/S_av value was substantial. The spread in the S_av value is apparently due to the growth of samples upon coverslips with an uneven surface. Therefore, the cellular monolayer undergoes a strain, which we consider to be homogeneous in the image size scale. The spread in the average cell areas between the samples of the same type can be also related to the following experimental feature. To prevent the formation of multilayers, the growth is stopped just before the total confluence of the cells. As a result, small and relatively sparse empty areas appear (see Methods). Note also that the averaged value 〈ΔS/S_av〉 is 7% larger in the HeLa non-synchronized dataset (hereinafter, angle brackets denote averaging over all monolayers of the corresponding type). However, the averaged DOPTs remain very close, with the differences in probabilities $\mathrm{\Delta }{P}_i\hspace{0.17em}\lesssim \hspace{0.17em}0.01$ (figure 1g).

2.2. Model of random polygon packing

As we have already mentioned, synchronized cell monolayers are beyond the scope of applicability of the Markov-type theory [11]. Its inapplicability for the non-synchronized case is evidenced by our finding that DOPTs in both types of monolayers are very similar. Differences in structural parameters of different samples, substantial spread of the cell sizes, and the specific asymmetry of the cell area distributions lead to the hypothesis that in the hyperproliferative epithelia the cell packings are close to random but, nevertheless, satisfy the geometric constraint associated with the existence of minimal cell size. Below we develop the theory of random polygonal packings and then test our hypothesis on epithelial cancer cell monolayers.

To construct the random polygonal packings, we start from random distributions of points with minimal allowed distance between them, d_min. We use periodic boundary conditions (which ensure Δ = 0) and follow the random sequential adsorption algorithm [36]. Points are randomly and sequentially inserted into the fundamental region with the area A_r. If the distance between the point that is currently being placed and any of the points placed earlier is less than the distance d_min, then this point is deleted and the random insertion is repeated. When the desired N number is achieved, the insertion is stopped. The subsequent Voronoi tessellation yields polygons with the average area S_av = A_r/N.

Obviously, there exists an averaged maximum limit on the number of points N_max that can be randomly placed in the region with area A_r for the given minimal distance d_min. Thanks to the discovery of new types of disordered structures and the continuous increase of available computing power, this limit, also known as the jamming limit [36], has been extensively studied in the literature over the past 30 years for various shapes and dimensions of adsorbed particles [36–39]. In the case of the random sequential adsorption of equivalent discs, the average ratio of the area occupied by them to the total surface area tends to L ≈ 0.547 [36]. Naturally, in our model the average ratio $\pi N_{\max }{d}_{\min }^2/A_r$ tends to the same value.

If a packing is not random, its surface coverage can differ significantly. For example, for the densest hexagonal packing of equivalent discs the surface coverage equals $\pi /2\sqrt{3}\approx 0.9064$. Let us consider the densest hexagonal packing with N discs per area A_r. In this packing d_min = d_hex, where $d_{\mathrm{hex}}=\sqrt{4A_r/N\sqrt{3}}$ is the distance between the disc centres. Then, in our packing algorithm the jamming limit corresponds to $d_{\min }=\sqrt{2L\sqrt{3}/\pi }\hspace{0.17em}{d}_{\mathrm{hex}}\approx 0.776$ d_hex.

Note that to model the hyperproliferative epithelia we use the unsaturated structures, where the ratio η = d_min/d_hex is smaller than the above critical value 0.776. Figure 2a–f shows examples and area distribution histograms of random polygonal packings with different ratios η and different degrees of hexagonality. The plots in figure 2g are calculated up to η = 0.75, since at larger η the calculation time increases sharply. Note also, that packings with the same η can have slightly different morphology, so the averaging is needed. In particular, when N ∼ 5000, in the region where η > 0.4, the ratio ΔS/S_av and probabilities P_i are reproduced with standard deviations smaller than 0.01 (figure 2g).

Figure 2.

Examples of random packings and their geometric characteristics. (a–d) Packings obtained at η values equal to 0.1, 0.4, 0.6 and 0.758, respectively. It can be observed how the fraction of hexagonal cells grows with increasing η (these cells are shown in yellow; the colouring is the same as in figures 1c–f). (e,f) Area distribution histograms obtained for η = 0.1 and 0.758. For both cases, the value of S_av is renormalized to 1. (g) Dependences ΔS/S_av and P_i on η. Plots of ΔS/S_av, P₆, P₅, P₇, P₄, P₈, P₉, calculated with the step Δη = 0.05, are coloured with black, yellow, orange, green, red, light blue and dark blue colours, respectively. Probabilities P₃, P₁₀ and P₁₁ are too small to be shown in the chosen scale. The centre of each vertical bar represents the averaging of 10 calculations at N = 5000. The bar sizes denote the standard deviations obtained for the calculations.

The proposed random packing model is in perfect agreement with the averaged structural data for the non-synchronized and synchronized HeLa epithelial cell monolayers (see the last two lines of table 1 (〈Hela〉 and 〈Hela_syn〉) and the histograms in figure 1g). Indeed, as presented in figure 2g, the change in ΔS/S_av from 0.40 to 0.43 corresponds to the variation of η from 0.50 to 0.47, respectively. In this region, the slope of the plot P₆(η) is small, and the values of η correspond to very close DOPTs with P₆ ≈ 0.35 − 0.37. The deviations of other averaged probabilities (figure 1g) from their theoretical values also do not exceed 0.02, which is within the spread of model calculations at N ∼ 5000.

The random packing model can also explain the scatter of ΔS/S_av and P₆ values between different samples presented in table 1. Note that these monolayers contain, on average, slightly fewer than 500 cells, and such a small value of N increases the morphological inhomogeneity of the generated packings. To justify this, one can perform a series of computations with the appropriate inputs: the number of calculations should not be less than the total number of monolayers, and N is equal to N_vor in the sample. As a result, it is highly probable that a packing that deviates from the mean by about (or even more than) the considered sample will be generated.

As we already mentioned, small and relatively sparse empty domains are present in non-synchronized HeLa monolayers. We decided to evaluate the impact of these domains on the obtained results. For this purpose, on the assembled images, we selected 40 smaller regions without empty areas. These confluent rectangular regions contained from 36 to 76 cells. When averaging over these regions, the values of S_av were found separately, and cells for which it was impossible to determine the number of nearest neighbours were not considered. This treatment resulted in ΔS/S_av ≈ 0.394 and P₆ ≈ 0.38. Comparing these values with those for averaged 〈HeLa〉 and using the graphs shown in figure 2g, we observe that the decrease in ΔS/S_av corresponds to the small increase in P₆, and the order is still properly described by the developed theory of random polygonal packings. So, we conclude that both types of considered hyperproliferative epithelia represent the random packings, while the main structural difference between the non-synchronized and synchronized HeLa monolayers consists in different values of S_av.

3. Discussion

In the literature, one can find many biophysical approaches describing the correlation between epithelial properties and averaged cell area and perimeter (e.g. [4,16,23,24,26,27,40]). The spread in these geometrical parameters is often considered as a ‘biological noise' and its importance for the epithelium is discussed in only handful of papers [34,40–42]. In particular, the recent paper [41] demonstrates that the relation between average aspect ratio of the cells comprising epithelium and its standard deviation governs processes as diverse as maturation of the pseudostratified bronchial epithelial layer cultured from non-asthmatic or asthmatic donors, and formation of the ventral furrow in the Drosophila embryo. Interestingly, in the paper [40] a possibility of a similar connection between cell area variability and DOPT was only briefly mentioned. In our paper, as far as we know, we have for the first time proposed a model that establishes the relationship between the cell size spread ΔS/S_av and DOPT in the cancer epithelial monolayers. Examining non-synchronized and synchronized HeLa cervix cells, we have shown that this spread is the main parameter controlling the DOPT in both types of the monolayers that represent a real-life example of random packings.

Below, we discuss how a healthy proliferative epithelium distinguishes from random cancer monolayers and point out a physical mechanism underlying this difference. As one can see from figure 2g, at η ∼ 0.62−0.7, the model of random polygon packing reproduces perfectly the DOPTs [8–11] typical of many plant and animal proliferative epithelia: P₄ ≈ 0.02 − 0.03, P₅ ≈ 0.25 − 0.27, P₆ ≈ 0.41− 0.47, P₇ ≈ 0.22, P₈ ≈ 0.03 − 0.05 and P₉ ≈ 0.001 − 0.006. A more detailed analysis, however, shows that the proposed approach leads to a value of ΔS/S_av that is slightly smaller than the one observed experimentally. In particular, after analysing the images of the Cucumis proliferative epithelium [8], we have estimated the value of ΔS/S_av as 0.29, which in the random polygon packing model corresponds to η ∼ 0.63 and P₆ ≈ 0.43 instead of the observed value P₆ ≈ 0.47 [8].

Let us consider the situation in more detail using our experimental data on healthy proliferative Human Cervical Epithelial Cells (HCerEpiC). Figure 3a,b demonstrates a typical sample of HCerEpiC confluent monolayer with the corresponding Delaunay triangulation and Voronoi tessellation. Forty analogous experimental images, previously obtained in our laboratory, capture relatively small separate fragments showing simultaneously from 30 to 54 cells.

Figure 3.

Structural characterization of HCerEpiC epithelium. (a) Typical sample of the epithelium with the superimposed triangulation; white scale bar is 50 µm. Vertices of the Delaunay triangulation coincide with the cell nuclei. The borders between the cells are shown in green. (b) Dual Voronoi tessellation for the same sample. A cell was taken into account in the statistical analysis if its shape could be determined unambiguously and the cell vertices were not too close to the border of the image (see §5). Such cells are coloured. (c) Averaged histogram of cell areas in forty samples. (d) Voronoi tiling for a random polygonal structure with the same ΔS/S_av value and very similar to (c) histogram of the cell area distribution. The polygon colouring is the same as in figures 1c–f. (e,f) More ordered structures obtained from (d) by minimizing the model energies of elastic intercellular interaction (see main text). (e) Minimization of energy (3.2) at βS_av/ζ = 1 and q = 3.81 results in a highly ordered structure with P₆ ≈ 0.7. (f) Minimization of energy (3.3) at βS_av/ζ = 10 and q = 3.81 yields the packing structurally similar to HCerEpiC epithelium: ΔS/S_av ≈ 0.45 and P₆ ≈ 0.41.

Note that borders of the HCerEpiC cells are more clearly visible than those of HeLa cells, and thus can be determined directly, without using Voronoi tessellation. First, we analysed the images following the same procedure that we used for HeLa cells. In this way, we have obtained the following structural data: S_av ≈ 3.0 × 10³ µm², ΔS/S_av ≈ 0.43, P₄ ≈ 0.04, P₅ ≈ 0.30, P₆ ≈ 0.41, P₇ ≈ 0.20, P₈ ≈ 0.04 and P₉ ≈ 0.01. It is interesting to note that these P_i values differ from those obtained for the Xenopus frog [11] by less than 2% and the HCerEpiC epithelium, as expected, has a structure typical of other normal proliferative epithelia [8–11]. We then analysed the same forty images directly, basing on the cell borders. Although cells with common borders are not necessarily linked in the Delaunay triangulation (discrepancy ≈15% of cases), the resulting difference in the calculated average structural parameters is several times smaller, indicating compatibility of these two methods. Therefore, below we refer only to the structural data obtained using Voronoi tessellation.

We have also considered the fact that the S_av value can vary from one HCerEpiC monolayer fragment to another because of the influence of the substrate it was grown on. With the averaging method taking this difference into account, the ratio ΔS/S_av decreases from 0.43 down to 0.36, corresponding in the random packing model to P₆ ≈ 0.39 (instead of 0.37). Nevertheless, this slightly increased probability is still smaller than the observed value P₆ ≈ 0.41. Consequently, based on the Cucumis and HCerEpiC examples, we can conclude that in normal proliferative epithelia the P₆ value is greater than the one predicted by the random packing model for the same ΔS/S_av ratio.

Note that the mitosis rate in HeLa cells is approximately 5.5 times higher than in HCerEpiC cells, while the rate of apoptosis is the same in both cases (see electronic supplementary material, figure S2). Owing to this difference, for the same number of seeded cells, the considered monolayers are at confluence after 2 days for HeLa cells compared to 4 days for HCerEpiC ones. Thus, we can assume that the P₆ increase in healthy epithelium is associated with the lower rate of cell division, although intercellular interactions are also very important for this phenomenon.

To justify the latter statement, we recall that ordering of equivalent particles, retained on a planar surface and interacting with each other via very different pair potentials, readily leads to the formation of a simple hexagonal order [35]. In our opinion, the mechanism of the P₆ increase in healthy proliferative epithelia is similar and caused by minimization of the elastic energy associated with the mechanical interaction between cells. Namely, an internal local stress caused by the cell division and nonuniform growth can relax through cell motility, which increases, on average, the hexagonal coordination.

Below, we consider a simple way to demonstrate that mechanical interactions between cells can order the epithelial structure by increasing the number of cells with six neighbours. Following the works [23,24], the elastic deformation energy E of a monolayer containing N cells can be written as:

$$E=\sum _{i=1}^N\hspace{0.17em}{\beta }_i{(A_i-A_i^0)}^2+{\zeta }_i{(P_i-P_i^0)}^2,$$ 3.1

where the subscript i labels each cell; A_i and P_i are the cell area and perimeter, respectively. The origin of the elastic moduli β_i is associated with a combination of the cell volume incompressibility and resistance to height differences between nearest cells [24]. The second term including the cell perimeters P_i results from active contractility of the actomyosin subcellular cortex (quadratic in P_i) and effective cell membrane tension due to cell–cell adhesion and cortical tension, which are linear in perimeter. Usually, different cells of the same type are assumed to have the same elastic moduli β_i and ζ_i and thus index ‘i’ is omitted. The same goes for the parameters $P_i^0$ and $A_i^0$, which are usually substituted with P_eff and S_av, correspondingly. The resulting energy reads:

$$E=\sum _{i=1}^N\hspace{0.17em}\beta {(A_i-S_{av})}^2+\zeta {(P_i-P_{eff})}^2.\hspace{0.17em}$$ 3.2

In [24], using the Surface Evolver software [43], the energy (3.2) was minimized with respect to the position and shape of the intercellular boundaries, while in [23] an overdamped dynamic model was built. The latter model takes into account the effective temperature of the monolayer, which corresponds to the value of additional (not related to intercellular interaction (3.2)) random forces simulating active cell migration. In the model [23], the shape of the cells coincides with the shape of Voronoi cells, therefore the energy (3.2) depends only on positions of the Voronoi cells centers. The results of the works [23,24] are similar, and the properties of both models depend on the so-called target shape index $q=P_{\mathrm{eff}}/\sqrt{S_{\mathrm{av}}}$ [23,24]. To clarify the meaning of this quantity, note that for the regular honeycomb packing the quantity $q=6/\sqrt{1.5\sqrt{3}}\approx 3.72$, while q = 4 and q ≈ 4.56 minimize the energy (3.2) for regular square and triangular lattices, respectively. As shown in the works [23,24] at q > q₀, where q₀ = 3.81−3.813, the cells acquire the possibility of relative movement without overcoming the potential barrier, and the monolayer demonstrates the fluid-like behaviour [23,24]. At the same time, as can be seen from the model monolayers presented in these works, at q > q₀ the number of 6-valent cells decreases, and many cells acquire an elongated shape.

Since our analysis of experimental data is also based on the Voronoi tiling, let us discuss the results [23] in more detail. Image analysis (fig. 2b in [23]) reveals that ΔS/S_av are 0.1 and 0.12, respectively, for the solid and liquid states of the model monolayers shown. However, even in Cucumis epithelium, the most ordered proliferative monolayer considered in our work, ΔS/S_av ≈ 0.29, which is almost three times more than in the model structures [23]. Also note that the probability of P₆ in the solid phase [23] (its comparison with normal epithelium is reasonable) is 0.595, whereas for ordinary proliferative epithelia, the value of P₆ lies in the range of 0.41–0.47 [8–11].

To assess the influence of cell-to-cell interactions on the DOPT in the model [23], we performed a series of numerical experiments, where the energy (3.2) was minimized using the ordinary coordinate descent method. As the initial positions of N = 400 cell centres, we used random coordinates obtained within the framework of the above algorithm for constructing random polygonal packings. The analysis showed that mechanical intercellular interactions can strongly order the system. In the region q < q₀, depending on the initial positions of the particles and the ratio of the coefficients 0.01 < βS_av/ζ < 100, minimization of (3.2) led to the probabilities P₆ = 0.6−0.8 (figure 3e). The scatter of the obtained P₆ values is associated not only with the change in the model parameters, but also with the existence of numerous local energy minima. Note that the anomalous epithelia with P₆ ≈ 0.6−0.9 are also observed in nature. For example, after the mitosis in the wings of a Drosophila stops (and before the start of hair growth), the equilibrium changes and a sharp increase in the proportion of 6-valent cells occurs [40]. As our preliminary estimates show, the use of Lennard–Jones potential [44] to describe the intercellular interaction and assumption of the cell size dispersion [34] allow us to obtain epithelium-like structures with P₆ up to the level of 0.9.

Rationalization of normal proliferative epithelia within the framework [23] is problematic not only due to the significant effective temperature and monolayer disordering required to reduce the probability P₆ from 0.6−0.8 to 0.4−0.47. The assumption of the same equilibrium area S_av for all cells (made in equation (3.2)) limits the ability of the model to generate structures with realistic values of ΔS/S_av. In this context, transition from energy (3.1) to (3.2) by replacing $A_i^0,\hspace{0.17em}{P}_i^0$ with corresponding averages seems to be an oversimplification; thus, we use the following energy instead:

$$E=\sum _{i=1}^N\hspace{0.17em}\beta {(A_i-A_i^0)}^2+\zeta {\left(P_i-q\sqrt{A_i^0}\right)}^2,$$ 3.3

where the equilibrium areas $A_i^0$ are defined by our random polygonal packing model.

Methods to analyze an energy landscape are known [42,45], nevertheless, for energy (3.3) it is a separate complex problem, since the energy (3.3) has a huge number of local minima with different DOPTs. We have only verified that minimization of this energy approximately preserves the initial value of ΔS/S_av, and in the region q < q₀, leads to an increase in P₆ that is smaller than the one occurring during the minimization of energy (3.2).

Using the energy (3.3), one can more realistically simulate normal proliferative epithelia, in particular HCerEpiC epithelium. The DOPT in HCerEpiC epithelium corresponds to the random polygonal packing with η ≈ 0.62, while the observed ΔS/S_av corresponds to η ≈ 0.53. To correct the DOPT, we generated a random polygonal structure with η ≈ 0.53 (figure 3d), which yields a distribution of cell areas very close to that of the HCerEpiC epithelium (figure 3c). At q = 3.81, we minimized energy (3.3) at different ratios βS_av/ζ in the region 0.01−100. This gave rise to structures with P₆ = 0.38−0.43. One of these structures, closest to the HCerEpiC epithelium, is shown in figure 3f. Finally, we note that in epithelia there are multiple processes bringing the system out of equilibrium such as mitosis, apoptosis and varying cell growth rate. Their influence on DOPT cannot be taken in account by the simple minimization of energy (3.3). The construction of such a more complex dynamic model is clearly beyond the scope of this work and is a task for following studies.

4. Conclusion

In this paper, we have demonstrated the significant difference in topology between cancerous and normal epithelial monolayers. In epithelial cancer monolayers, the cell arrangement is close to random but, nevertheless, satisfies the geometric constraint associated with the existence of minimal cell size. This type of the cell order can be described with good accuracy by a single dimensionless control parameter, namely the normalized halfwidth ΔS/S_av of the cell area distribution. Cancer cells divide so quickly that the relaxation processes associated with the minimization of free energy are unable to effectively rearrange cells and order the monolayer by increasing the number of 6-valent cells. Consequently, the growing disorder predominates over the intercellular interactions, and this yields structures very similar to random polygonal packings. By contrast, normal proliferative and non-proliferative epithelia are morphologically different from them. Overall, our study of epithelial monolayers provides novel mathematical tools for a topological analysis of epithelial morphogenesis in a developmental and/or pathological context. Elaborated tools could be used to analyze the change in cell topology during the two-/three-dimensional transition observed throughout epithelial tumourigenesis. In the near future, we plan to apply our approach to studing the development of zebrafish embryonic epidermis before and after tumour cell injection in vivo.

5. Methods

5.1. Cell line growth and synchronization procedure conditions

Human cervical cancer cell line HeLa was purchased from the American Type Culture Collection (Manassas, VA, USA) and maintained in Dulbecco's Modified Eagle Medium (DMEM), high glucose containing 5% heat-inactivated fetal bovine serum and supplemented with GlutaMAX™ (Gibco Life Technologies), penicillin (100 units ml⁻¹) and streptomycin (100 µg ml⁻¹). Normal primary cervical epithelial cells (HCerEpiC) isolated from human uterus were purchased from ScienCell Research Laboratories (Clinisciences S.A.S., Nanterre, France). HCerEpic cells were grown in Cervical Epithelial Cell Medium according to the manufacturer's instructions. Cells were incubated in a humidified incubator at 37°C in 5% CO₂.

For confocal microscopy analysis of non-synchronized cells, HeLa or HCerEpiC cells were seeded on glass coverslips (12 mm diameter round) coated with 10 µg ml⁻¹ of poly-l-Lysine (P4707, Sigma) at 7 × 10⁴ cells coverslip⁻¹ in a 24-well culture plate. Confluency of the monolayer was achieved 48 h and 4 days later for HeLa and HCerEpiC cells, respectively. HeLa cells were synchronized in G0/G1 phase of the cell cycle using the double thymidine block procedure. The day before the first thymidine block, cells were seeded on poly-l-Lysine treated glass coverslip at a density of 7.5 × 10⁴ cells coverslip⁻¹. The next day, 2.5 mM thymidine was added for 16 h (first block). Then cells were washed with phosphate-buffered saline (PBS) and incubated for 8 h without thymidine. Lastly, the second thymidine block was applied for 16 h. At the end of the second thymidine block, cells were washed and incubated for two additional hours, after which cells were treated with antibodies and analysed by confocal microscopy. Following this procedure, cells are fully confluent and cell synchronisation in G0/G1 is about 92%. This was confirmed by cytofluorometry analysis (FACS) (see electronic supplementary material, figure S3).

5.2. Immunocytochemistry and fluorescence microscopy

Cells cultured on glass coverslips were fixed with 4% paraformaldehyde in PBS for 20 min and washed in Tris-buffered saline (25 mM Tris pH7.4, 150 mM NaCl) (TS) for 10 min. After permeabilization with 0.2% Triton X-100 in TS for 4 min, non-specific binding was blocked with 0.2% gelatin from cold water fish skin (#G7765 Sigma-Aldrich Chimie, Lyon) in TS for 30 min. Cells were incubated with primary antibodies in blocking buffer for 1 h and then washed three times with 0.008% TritonX-100 in TS for 10 min. Rabbit anti-ezrin antibody [46] was used to visualize cell body and membrane. Cells were incubated for 30 min with Alexa-Fluor 488-labelled secondary antibodies (P36934-Molecular Probes, InVitrogen) in blocking buffer. After rinsing in washing buffer, cell nuclei were stained with 1 µg ml⁻¹ Hoechst 33342 (62249-Thermo Scientific Pierce) in TS for 5 min. Finally, coverslips were mounted with Prolong™ Gold Antifade (P36934-Molecular Probes, InVitrogen) and examined under a Leica TCS SPE confocal microscope equipped with a 25X/0.75 PL FLUOTAR oil objective (HCerEpiC) and a 40X/1.15 ACS APO oil objective (HeLa) (figure 3 and electronic supplementary material, figure S2). In the case of HeLa cells (figure 1a) and synchronized HeLa cells (figure 1b), the analysis was performed with a Zeiss LSM880 FastAiryScan confocal microscope equipped with a 40X/1.4 Oil Plan-apochromat DIC objective.

Between 7 and 9 ‘z-stacks’ (0.457 µm thickness each) were acquired per field and a two-dimensional image was generated by applying maximum intensity projection processing. For each coverslip, the acquisition pattern was six neighbouring images per row for a total of two or three rows. The resulting images (12 or 18 images) were adjusted for brightness, contrast and colour balance by using ImageJ and assembled side by side in PowerPoint to reconstruct a cell monolayer consisting of N > 500 cells.

5.3. Image analysis

After determining the geometric centres of the cell nuclei, triangulation was performed by the Delaunay method [31]. Next, Voronoi tiling was constructed, and the areas of the epithelial cells were calculated as the areas of Voronoi cells. Obviously, for a correct statistical analysis, it is necessary to discard the cells located too close to the image border, for which the number of neighbours cannot be determined. Note that even if it is possible to construct a closed Voronoi cell, then it is also necessary to check whether the cell polygon boundary can be changed by additional hypothetical nuclei lying directly outside the image border. Therefore, the centre of a reliably constructed Voronoi cell should be located at least twice as far from the image border as any of the vertices of this cell. However, this method leads to the appearance of an excessive total positive topological charge, which is localized at the image border. On the one hand, 4- and 5-valent cells have a smaller area [8], while on the other hand, the smaller the cell located near the image border, the more chances its nucleus has to satisfy the selection criterion formulated above. This fact, when processing images with a small number of nuclei (about 40), leads on average to the formation of a 5% preponderance of the total positive topological charge (which is carried by 4- and 5-valent cells) over the total negative topological charge. This, in turn, leads to errors when constructing DOPTs diagrams and determining the values of P₆ and ΔS/S_av. To avoid preferential selection of small cells, we used additional cutting of the image borders. In the statistical analysis, we took into account only cells whose nuclei centres fall within the rectangle that has maximum possible size and does not contain any nucleus with an uncertain coordination. Thus, it is possible to significantly reduce the total topological charge of the images and, accordingly, the error in the values of ΔS/S_av.

Data accessibility

This article has no additional data.

The data are provided in electronic supplementary material [47].

Authors' contributions

D.S.R.: data curation, formal analysis, investigation, software, visualization; M.M.: data curation, formal analysis, investigation, methodology, resources, writing—review and editing; K.F.: software, visualization; I.G.: formal analysis, software, validation, writing—review and editing; V.M.: methodology, resources, writing—review and editing; S.B.: conceptualization, data curation, formal analysis, investigation, project administration, resources, supervision, writing—original draft, writing—review and editing; S.B.R.: conceptualization, investigation, methodology, project administration, supervision, validation, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

The authors declare no competing interests.

Funding

S.R. and D.S.R. were financially supported by the Russian Foundation for Basic Research (grant no. 19-32-90134). M.M. and V.M. were supported by the French National Research Agency (grant no. ANR-10-INBS-04, ‘Investments for the future’). S.B. was financially supported by ‘Mission pour les initiatives transverses et interdisciplinaires du CNRS’. I.G. was financially supported by Russian Science Foundation (grant no. 20-72-00164).