ABSTRACT
It has long been noted that the cell arrangements in epithelia, regardless of their origin, exhibit some striking regularities: first, the average number of cell neighbours at the apical side is (close to) six. Second, the average apical cell area is linearly related to the number of neighbours, such that cells with larger apical area have on average more neighbours, a relation termed Lewis’ law. Third, Aboav-Weaire’s (AW) law relates the number of neighbours that a cell has to that of its direct neighbours. While the first rule can be explained with topological constraints in contiguous polygonal lattices, and the second rule (Lewis’ law) with the minimisation of the lateral contact surface energy, the driving forces behind the AW law have remained elusive. We now show that also the AW law emerges to minimise the lateral contact surface energy in polygonal lattices by driving cells to the most regular polygonal shape, but while Lewis’ law regulates the side lengths, the AW law controls the angles. We conclude that global apical epithelial organization is the result of energy minimisation under topological constraints.
INTRODUCTION
Epithelia are polarized tissues. This means that cell properties change along the apical-basal axis (Fig. 1a). In particular, cells adhere tightly on the apical side via an adhesion belt composed of Cadherins, while they bind to the extracellular matrix on the basal side. When viewed from the apical side, the tissue appears as a contiguous polygonal lattice (Fig. 1b). The organisation of the polygonal lattice may appear random at first sight, but it has previously been noted to follow certain phenomenological laws. First, even though the fraction of cells with a certain number of neighbours differs hugely between epithelial tissues (Fig. 1c), the average number of neighbours, 
, is always close to six (Fig. 1d) [1–7] because of topological constraints in 2D contiguous polygonal lattices [4, 8]. In fact, in the limit of an infinite number of polygons, the average number of neighbours is exactly
if three cells meet at each vertex; this can be derived from Euler’s formula, v−e+f=2, which formulates a relationship between the number of vertices, v, edges, e, and faces, f. Second, the average apical area is linearly related to the number of cell neighbours, n, a relation termed Lewis’ law (Fig. 1e), i.e.
(a) A cartoon of an epithelial tissue. The tissue is polarized with an apical and basal side. At the apical side cells adhere tightly via adhesion junctions (green). Nuclei are depicted in pink.
(b) The epithelial cells (denoted as faces - f) form a contiguous polygonal lattice on the apical side. Typically, three edges, e, meet in a vertex, v.
(c) Tissues differ widely in the frequency of neighbour numbers. The legend provides the measured average number of cell neighbours for each tissue and the references to the primary data [1–7]. Data points for n < 3 were removed as they must present segmentation artefacts. The tissue samples are: Chick neural tube epithelium (cNT) [2], Drosophila eye disc (EYE) [2], Drosophila peripodal membrane from the larval eye disc (dPE), Drosophila larval wing disc (dWL) [1–6, 19] (Kokic et al, submitted), Drosophila pre-pupal wing disc (dWP) [1, 2], Drosophila mutant wing pre-pupa (dMWP: the expression of myosin II was reduced in the wing disc epithelium by expressing an UAS-zipper-RNAi using the C765-Gal4 line) [2], and Drosophila pupal wing disc (dPW) [1]. In case of the Drosophila peripodal membrane from the larval eye disc (dPE), the numbers indicate the developmental age as the number of ommatidial rows that have formed. In case of the Drosophila pupal wing disc (dPW) [2], TPx indicates subsequent, but not further specified pupal time points [1]; 30h and 48h specifies the time post puparium formation.
(d) The measured average number of cell neighbours is close to the topological requirement 
in all tissues; see panel c for the colour code.
(e) The relative average cell area, 
, is linearly related to the number of neighbours, n, and follows Lewis’ law (Eq. 2, black line) in all reported tissues. The colour code is as in panel c, but data is available only for a subset of those tissues. Few cells with more than 7 neighbours have been measured and estimates of the mean areas are therefore unreliable (shaded part).
(f) The normalised side length of a regular polygon with n vertices for a fixed area.
(g) The AW law formulates a relationship between the number of neighbours, n, that a cell has and the average number of neighbours, mn, that its direct neighbours have. The product mn · n can be determined by summing over all n neighbours.
(h) The epithelial tissues follow the AW law (black line). The colour code is as in panel c, but data is available only for a subset of those tissues.
Panels a-f are reproduced with modifications from (Kokic et al, submitted).
Here, 
refers to the average apical area of cells with n neighbours, and Ā refers to the average apical area of all cells in the tissue. We have recently shown that Lewis’ law can be explained with a minimisation of the lateral cell-cell contact surface energy (Kokic et al, submitted). The lateral contact surface energy is minimal if the cell-cell contact area, and thus the combined cell perimeter is minimised. For a given apical area, the cell perimeter is minimal if the polygons are the most regular. To form a contiguous polygonal lattice from regular polygons, these need to all have the same side/edge length. However, for the same polygon area, side lengths differ between polygon types (Fig. 1f). In epithelial tissues, apical areas vary as a result of several processes, most prominently growth and cell division. By following the relationship between polygon area and polygon type as stipulated by Lewis’ law (Eq. 2), the difference in side lengths is minimised between cells. However, equal side lengths are only achieved if the areas followed a quadratic rather than a linear relationship,
This quadratic relationship only emerges in case of a larger variability in apical areas than previously observed in epithelia. We confirmed this prediction by increasing the area variability in the Drosophila larval wing disc, which led to the predicted quadratic relationship (Kokic et al, submitted).
Finally, Aboav-Weaire’s law (AW law) formulates a relationship between the number of neighbours, n, that a cell has and the average number of neighbours, mn, of its direct neighbours (Fig. 1g). Aboav [9] empirically observed for grains in a polycrystal that
Eq. 4 has previously been found to be a close approximation to the neighbour relationships in epithelial tissue in both plants [10–12] and animals [7] (Fig. 1h). However, Eq. 4 cannot explain regular hexagonal packings, where 
. Weaire suggested a refinement of the form
where 
is the variance of the polygon distribution, and fn is the fraction of cells with n neighbours [13]. Eq. 5 applies to hexagonal lattices as Var(n) = 0, and reduces to Eq. 4 for Var(n) = 2. Efforts to explain the mathematical basis of Eqs. 4 and 5 have led to a wide range of alternative formulations [14], but no physical or mathematical argument has so far been established to explain the experimental observations.
In the following, we will show that the emergence of the AW law in epithelia can be explained with the minimisation of the lateral contact surface energy. We previously showed that Lewis’ law ensures that the side lengths of the different polygon types are the most similar, so that the different polygon types can fit within a polygonal lattice as the most regular polygons. To form a regular polygon also the internal angles of the polygon must match the polygon type, and to form a contiguous lattice, the internal angles around the shared vertex of adjacent cells must add to 360°. We show that this second constraint results in the AW law.
RESULTS
The emergence of the AW law can be explained with the minimisation of lateral surface energy
In the first step, we sought to test whether the AW law could be explained with the minimisation of the lateral contact surface energy. We have previously introduced the simulation framework LBIBCell [15] to simulate epithelial tissues (Kokic et al, submitted). LBIBCell represents tissues in a 2D plane, and can thus be used to simulate the apical tissue dynamics (Fig. 2a). The simulation framework offers high spatial resolution, i.e., it represents the boundaries of all cells separately such that cells can unbind and the spatial resolution is sufficiently high that cell edges can be curved. Cortical tension as well as cell-cell adhesion are implemented via springs between vertices. The fluid inside and outside the cells is represented explicitly and the fluid dynamics are approximated by the Lattice Boltzmann method. The interaction between the elastic cell boundaries and the fluid is realised via an immersed boundary condition.
(a) Representation of the apical epithelial surface as a 2D simulation plane in LBIBCell. The cartoon shows a cubical section of a simple columnar pseudostratified epithelium. LBIBCell simulates the tissue mechanics and morphodynamics of the 2D plane (indicated in pink) that goes through the adherens junctions (marked in green), which are located below the apical surface. Cell boundaries are geometrically highly resolved and cell boundaries and cell-cell junctions are modelled with elastic springs. The cartoon and legend are reproduced with modifications from (Kokic et al, submitted).
(b) The AW law plotted for LBIBCell simulations with random cell division (blue) or cell division perpendicular to the longest axis (red).
(c) The estimated coefficients a, b when fitting mn · n = a · n + b to the data (Fig. 1h) and the LBIBCell simulations (Fig. 2b).
(d) The estimated coefficient b when fitting mn · n = a · n + b does not positively correlate with 6 + Var(n) in epithelia. ρ is the Pearson correlation coefficient.
(e) The hexagon fraction versus the variance in the polygon distribution Var(n) for all available data.
The biological parameters in the simulation are the growth rate, the cell division threshold (i.e. the apical cell area, at which the cell divides), as well as the spring constants representing cortical tension and cell-cell adhesion. We have previously adjusted these to recapitulate the quantitative data from the Drosophila larval wing disc, a model epithelial tissue, and we have re-used the same setup as described before (Kokic et al, submitted; Supplementary Material, Table S1). When we simulate the Drosophila larval wing disc dynamics while dividing cells either perpendicular to their longest axis (Hertwig’s rule [16]) (Fig. 2b, orange line) or randomly (Fig. 2b, cyan line), the resulting lattice results in polygonal relationships that are close to the AW law (Fig. 2b, black line). The deviation of the simulations from the AW law is similar to that observed for epithelial tissues (Fig. 2c). Thus, if we determine the parameters a and b for which
best fits the simulations and the data, the inferred values differ in a similar way from those of the AW law (a=5, b=8) [9]. We note that the b parameter deviates more than the a parameter. Weaire linked the b parameter to the variance in the observed polygon distribution, i.e. b = 6 + Var(n) (Eq. 5) [13]. However, 6 + Var(n) does not correlate with the parameter b that we infer from the data (Fig. 2d), thus ruling out such a relationship. Given the definition of variance, the hexagon fraction declines with the variance of the polygon distribution (Fig. 2e), and Var(n) = 2 and thus b = 8 as in Aboav’s formulation is observed only for tissues with low hexagon fraction (~30%), which are rare (Fig. 1c).
We conclude that the LBIBCell simulations generate a distribution of cell-cell contacts that lead to similar parameters for the AW law as observed in epithelia, but that differ from those specified by Aboav and Weaire. This suggests that the AW law could be explained with a minimisation of the lateral contact surface energy. But why would energy minimisation result in neighbour arrangements that follow the AW law?
Perfect polygonal lattices do not follow Aboav’s law, but match Weaire’s law
To understand the underlying constraint that leads to the particular form of the AW law, we consider perfect polygonal lattices made of regular polygons. The simplest case is a perfect hexagonal lattice (Fig. 3a, red). Here, all polygons must have the same area. To generate other regular lattices, cells with the appropriate areas must be placed in the correct relative position within the polygonal lattice. The relative areas of the different polygon types must follow the quadratic law (Eq. 3) so that all polygons in the lattice have the same side length. In addition, the internal angles of the cells that meet at a vertex must add up to 360° (Fig. 3b). The internal angles of a regular polygon with n neighbours (Fig. 3c) are given by
and only in few regular configurations, the angles at each vertex will add up to 360° (Fig. 3a). Consider for instance the case of a lattice made of repetitive elements with one square and two octagons: the 90° angle of a square and the 135° angles of two regular octagons (Fig. 3c) add up to 360° at each vertex (Fig. 3a). If in addition, the octagons have about four times the area of the square (Eq. 3), a regular lattice emerges (Fig. 3a, blue). On the contrary, if the areas of neighbouring cells do not match such requirements, irregular lattices emerge (Fig. 3d).
(a) In case of appropriate relative sizes and positions, adherence of circles results in perfect lattices (grey), as confirmed in LBIBCell simulations.
(b) The angles at each tricellular junction have to add up to 360°.
(c) Interior angles of regular polygons.
(d) In case of inappropriate relative sizes and positions, adherence between circles results in imperfect lattices.
(e) The products of mn · n versus neighbour number n for the perfect lattices in panel a deviate from Aboav’s law (Eq. 4, black line), but match Weaire’s law (Eq. 5, coloured lines; colour code as in panel a) rather well. Results from LBIBCell simulations (triangles) are close to analytic results (circles).
As noted above, in a hexagonal lattice the average number of neighbours is six, i.e., 
(Fig. 3e, red triangle), which violates Aboav’s law (Eq. 4, Fig. 3e, black line), but matches Weaire’s law (Eq. 5, Fig. 3e, red line). Also, the other arrangements that result in perfect polygonal lattices (Fig. 3a) deviate from Aboav’s law (Eq. 4, black line), but match Weaire’s law (Eq. 5, coloured lines) rather well (Fig. 3e). These perfect lattices also all emerge in LBIBCell tissue simulations when adhesive circles with the appropriate relative areas are placed at the appropriate relative positions in a grid (Fig. 3a, coloured lattices) and thus represent the configuration with the lowest lateral surface energy. This shows that not all configurations that minimise the lateral contact surface area follow Aboav’s law.
In summary, regular polygonal lattices follow Weaire’s law more closely than Aboav’s law (Fig. 3e), while epithelial tissues follow Aboav’s law more closely than Weaire’s law (Fig. 2c,d). What is special then about epithelial tissues, and what is the underlying physical principle that leads to these regularities?
The AW law in epithelial tissues
In an epithelial tissue, the apical areas are predetermined by several active processes, including cell growth, cell division, and interkinetic nuclear migration. As a result, the position of cells with a certain area does not follow a pattern that would allow the emergence of a contiguous lattice made from regular polygons. But, why would these lattices follow the AW law?
According to Euler’s formula, cells have on average six neighbours [4, 8]. The plot of mn · n with mn = 6 versus n is reasonably close to the AW law (Fig. 4a, black line), but clearly deviates (Fig. 4a, green line). In a contiguous lattice, cells have to meet a further constraint, in that at each vertex point, the combined angle must be 360° (Fig. 3b). At each of the n vertices, the neighbouring cells should have a combined angle of 360° - θn (Fig. 4b, inset). The average angle, 
, of the two neighbouring cells therefore follows from 
, where we used Eq. 6 for θn. We then obtain for the average number of neighbours
(a) The product of mn · n versus neighbour number n for the AW law (black), mn = 6 (green), 
(yellow).
(b) Interpolation between the angles of regular polygons, 
, versus the number of vertices n. Inset: A cartoon of the angles at a tricellular junction between a cell with n neighbours and its two neighbours with angles θ1 and θ2.
(c) Comparison of an analytical approximation for regular polygons (Eq. 8, grey) and its asymptote (a ≈ 4.6 and b = 7.2) (blue) to Aboav–Weaire’s law (black).
(d) Comparison of an analytical approximation for irregular polygons (red) to Aboav–Weaire’s law (black). Inset: A cartoon of the angles at a tricellular junction in case of an irregular central polygon with angle θn + θi.
This relationship approximates the AW law for neighbour numbers close to n = 5 rather well, but deviates for larger and smaller neighbour numbers (Fig. 4a, yellow line). We note that the internal angle of regular polygons, θn, depends non-linearly on the polygon type n (Eq. 6, Fig. 4b). As a consequence, two cells, whose angles at a vertex add up to 2θm yield a different mn depending on whether their angles are very similar or very different. If we take this non-linear effect into account (see Supplementary Information for details), and integrate within sensible physiological limits, i.e. θmin = 60°, the smallest internal angle of a regular polygon (triangle), and 
, the internal angle of a regular dodecagon (12-gon), we obtain
which yields a rather close fit to the AW law for larger n (Fig. 4c, grey line). In the limit of n → ∞, Eq. 8 yields (Fig. 4c, blue line)
which is close to the parameters a = 5 and b = 8 in the AW law as given by Eq. 4 (Fig. 4c, black line), and within the range of parameter values inferred from biological data and the LBIBCell simulations (Fig. 2c). Aboav’s empirical values a = 5 and b = 8 are obtained with θmin = 50.4° and θmax = 160.7°, the lower bound of which is plausible only if trigons become distorted.
For small n, Eq. 8 predicts larger values for mn than what is observed. So, what happens for smaller n? Consider a regular triangle. To meet the angle constraint, it requires neighbouring polygons that each have 12 neighbours (Fig. 3a, purple). To maintain a lattice of equilateral polygons, this would require a tissue with a large variability in cell area (Eq. 3), and a strict pattern of large and small cells (Fig. 3a, purple). Such tissues are unlikely to emerge from biological processes. In the case of more physiological variabilities in apical cell areas, a triangular cell will border cells that have a smaller area than what is required for a regular dodecagon (n=12) according to the quadratic law (Eq. 3). As a consequence, such neighbouring dodecagonal cells would have a smaller side length than the triangular cell. To fit into a contiguous lattice, the polygons would then become stretched and their internal angles would no longer correspond to that of a regular polygon. Alternatively, the neighbours can assume a lower neighbour number than 12. Also in this case, the internal angles can no longer reflect that of a regular polygon as they would no longer add to 360°.
Accordingly, we extended the above formalism to allow for an irregular central polygon that deviates from the perfect internal angle θn by an angle θi (Fig. 4d, inset; see Supplementary Information for details). In the limit of n → ∞, we recover the same value for the parameter a as for the regular polygon case (Eq. 9), but b now depends also on the maximal value of the deviatory angle θi. We obtain an excellent fit to the AW law (a=5, b=8, Fig. 4d, black line) when we use θmin = 360° − θn − θi − θmax and θmax = 151.3°, and set the symmetric upper bounds of the angle mismatch to 
, where 
is the upper limit to maintain convexity (Fig. 4d, red line).
In fact, there is sufficient parametric freedom to reproduce all measured values of a and b (Fig. 2c) by adjusting the integration bounds for the angles, θmin and θmax, and the upper bounds 
for the angle deviation θi. Thus, the neighbour relationships in the Drosophila larval wing disc (a=4.76, b=8.34, grey) deviate slightly from the classical AW law (Eq. 4, Fig. 5a, black), but can be reproduced very well by the model (Fig. 5a, red line), and the predicted upper bound, 
, in the angle deviation matches that observed in the tissue very well (Fig. 5b). Importantly, even though we introduced the irregularity assumption to explain the neighbour relationships of cells with few neighbours, it applies to all polygon types. In agreement with this, the great majority of epithelial cells deviate from a regular shape as measured by their ellipticity, and the irregularity of the apical sides of epithelial cells does not dependent on the cells’ neighbour number (Fig. 5c).
(a) The product of mn · n versus neighbour number n for cells in the Drosophila wing disc follows the AW law on average (black line), and is approximated very well by the analytical approximation for irregular polygons (red line) when optimizing the angle bounds 
to match the AW law (a=4.76, b=8.34) that best matches the data (Fig. 2c).
(b) Angle versus polygon type for cells in the Drosophila wing disc. The observed average angles are those of regular polygons (yellow lines). The red solid lines mark 
, the predicted angle range of the irregular central polygon with 
.
(c) Ellipticity of cells in the Drosophila wing disc as measured by the ratio of the axes of fitted ellipses. A ratio of 1 (yellow line) corresponds to regular polygons.
(d) Normalised side length versus polygon type for cells in the Drosophila wing disc. The mean deviates from the ideal curve 
(yellow line).
(e) The relative average cell area, 
, for cells in the Drosophila wing disc. The average is linearly related to the number of neighbours, n, in the Drosophila wing disc, and follows the linear Lewis’ law (Eq. 2, black line) rather than the quadratic law (Eq. 3, yellow line).
(f) Normalised area distribution for cells in the Drosophila wing disc. The yellow vertical lines mark the areas for each polygon type required to form a regular polygon lattice (Eq. 3).
(g-i) Ellipticity, as measured by the ratio of the axes of fitted ellipses, (f), variation of side length (g), and variation in the angles (h) of single cells in the Drosophila wing disc versus the fold-deviation of their apical area, Ai, from that appropriate for their polygon type, 
, according to either the linear Lewis’ law (Eq. 2, black line) or the quadratic law (Eq. 3, yellow line). Variability in side length and angles was calculated for each cell as 
, where xj refers to side length or angle, respectively, of each cell j, and < x > indicates the expected value for the particular polygon type of the cell.
Much as the internal angles, we expect the side lengths to deviate from that of a regular polygon. In a regular polygonal lattice, all side lengths should be equal (Eq. 3, Fig. 5d, yellow line), and the average normalised side lengths are indeed close to one (Fig. 5d). Single cells mostly deviate from these averages, much as only the average area per polygon type follows Lewis’ law (Eq. 2, Fig. 5e, black line). There are two sources for the observed irregularity. For one, the cellular processes result in a spatial distribution of apical areas in epithelia that are not consistent with the requirements of regular polygonal lattices (Fig. 3a). Secondly, apical areas in epithelia follow a continuous area distribution (Fig. 5f, black line) while the quadratic law (Eq. 3) specifies discrete optimal areas for each polygon type (Fig. 5f, yellow lines). With regard to the second point, we indeed observe that cell shape, side lengths, and angles deviate the least from that of a regular polygon when a cell’s neighbour number corresponds to the apical area according to the linear Lewis’ law (Eq. 2, Fig. 5g-i, black line) or the quadratic law (Eq. 3, Fig. 5g-i, yellow line).
CONCLUSION
We conclude that the AW law, like Lewis’ law, emerges in epithelial tissues because cells minimise the overall lateral surface energy and thus the combined cell perimeter. By following the relationship between polygon area and polygon type as stipulated by Lewis’ law (Eq. 2) or the quadratic law (Eq. 3), the difference in side lengths is minimised between cells. By following the neighbour relationships that are described by the phenomenological AW law, the internal angles are closest to that of a regular polygon while adding to 360° at each vertex. Both the linear Lewis’ law and the AW law deviate from the curve predicted for a regular polygonal lattice. Regular polygonal lattices, however, require strict patterns of correctly sized cells. Given that cell growth and cell division processes result in a more variable spatial distribution of apical cell sizes, these cannot give rise to a regular polygonal lattice. The observed lattices therefore represent a trade-off to achieve the most regular cell shapes in a contiguous lattice with differently-sized apical areas.
METHODS
Image Processing
Cell boundaries and cell properties (cell area, polygon type, ellipse fit axes) were obtained in (Kokic et al., submitted). In addition, the side lengths and the internal angles of all cells were extracted using EpiTools [5] and the Fiji [17] plugin Tissue Analyzer [18]. To this end, the EpiTools source code was modified to also export the IDs of the ni neighbours of a cell to permit the calculation of mn. R scripts were used to extract and analyse data, calculate statistics, and plot the results.
Set-Up of the LBIBCell simulations
The LBIBCell simulations were set up as described in detail before (Kokic et al, submitted) and is summarised in the Supplementary Material.
Data and Software availability
All source codes are freely available at:
https://git.bsse.ethz.ch/iber/Publications/2019_vetter_kokic_aboav-weaire-law.
Author Contributions
DI, RV developed the theoretical framework, MK carried out all image processing, simulations and created all figures, HG, LH, BG contributed preliminary image processing, modelling and data analysis, AI, GV-F, FC contributed the data. DI, RV, FC, MK wrote the manuscript. All authors approved the final manuscript.
Acknowledgements
We thank Davide Heller and Luis M. Escudero for providing access to their raw data. This work has been supported through an SNF Sinergia grant to DI, and grants BFU2012-34324 and BFU2015-66040 (MINECO, Spain) to FC.
REFERENCES
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