New Results

Minimisation of surface energy drives apical epithelial organisation and gives rise to Lewis’ law

Marco Kokic, Antonella Iannini, Gema Villa-Fombuena, Fernando Casares, Dagmar Iber
doi: https://doi.org/10.1101/590729

ABSTRACT

The packing of cells in epithelia exhibits striking regularities, regardless of the organism and organ. One of these regularities is expressed in Lewis’ law, which states that 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. The driving forces behind the almost 100-year old Lewis’ law have remained elusive. We now provide evidence that the observed apical epithelial packing minimizes surface energy at the intercellular apical adhesion belt. Lewis’ law emerges because the apical cell surfaces then assume the most regular polygonal shapes within a contiguous lattice, thus minimising the average perimeter per cell, and thereby surface energy. We predict that the linear Lewis’ law generalizes to a quadratic law if the variability in apical areas is increased beyond what is normally found in epithelia. We confirm this prediction experimentally by generating heterogeneity in cell growth in Drosophila epithelia. Our discovery provides a link between epithelial organisation, cell division and growth and has implications for the general understanding of epithelial dynamics.

INTRODUCTION

The packing of cells in epithelia exhibits striking regularities, regardless of the organism and organ. This suggests that there are fundamental principles to epithelial organization. Epithelial organization has previously been accounted for by topological constraints [1, 2], energy minimization [3–5], and entropy maximization [6]. While each of those theories explains certain aspects of epithelial organisation, a general theory that explains the wide range of epithelial regularities and organizational principles is missing. We will show that epithelial organization can be explained by a minimization of surface energy. Based on our theoretical framework, we explain how cell growth and division as well as mechanical constraints act together to drive epithelial tissue organization.

In epithelia, neighbouring cells adhere tightly to their neighbours via adherence junctions (Fig 1a). As a result, the apical surface of epithelia corresponds to a contiguous, polygonal lattice (Fig. 1a,b, ED Fig. 1a). In epithelia, three edges typically meet in each vertex (Fig 1c). For such a connected planar graph, the total number of cell faces (f), cell edges (e), and cell vertices (v) in the lattice must be related according to Euler’s Formula as v−e+f=2, and in the limit of large tissues, cells must on average have six neighbours Embedded Image [1, 6]. Even though the frequencies of the different polygon types differ widely between different epithelial tissues, all epithelia analysed to date indeed meet this topological constraint in that their cells have on average close to six neighbours (Fig. 1d,e). This particular property of epithelial organisation thus directly follows from topological constraints.

Figure 1:
Figure 1: Introduction to Epithelial Packing

a, A cartoon of epithelial organisation. For details see text. b, A typical epithelial tissue from the Drosophila larval eye disc peripodial epithelium (dPE), coloured according to the number of neighbours per cell. The original image with a segmentation mask is shown in ED Fig. 1a. c, The apical surface can be represented as a finite, connected, planar graph with f faces, v vertices, and e edges without edge intersection. Three edges are incident to each vertex, i.e. the degree of each vertex is three. d, The measured average number of cell neighbours is close to the topological requirement [1, 6], Embedded Image, in all tissues; see panel e for the colour code. e, Tissues differ widely in the relative frequency of neighbour numbers. Topological models only explain a single (dashed and dotted lines) of these many distributions (shaded range). The dashed line marks the hexagon fraction for randomly positioned cell division axes, the dotted line marks the hexagon fraction when cells are divided perpendicular to their longest axis. The legend provides the measured average number of cell neighbours for each tissue and the references to the primary data [1, 3, 11, 15, 43–45]. Data points for n<3 were removed as they must present segmentation artefacts. See ED Fig. 1 for standard deviations and full tissue names. f, Topological changes upon cell division have been suggested to result in the measured relative frequency of neighbour numbers. However, topological models only explain a single (panel e, dashed and dotted lines) of these many distributions (panel e, shaded range). g, Vertex models are based on finite, connected, planar graphs with vertices i,j cell area Aα (grey), edge length lij (blue) and cell perimeter Πα (blue). h, The relative average cell area, Embedded Image, is linearly related to the number of neighbours, n, and follows Lewis’ law [6] (Eq. 1, black line) in all reported tissues [15, 44]. The colour code is as in (c), but data is available only for a subset of those tissues. Few cells with more than 8 neighbours have been measured and estimates of the mean areas are therefore unreliable (shaded part).

Topological arguments have also been proposed to explain the particular frequencies of polygon types in the different epithelial tissues (Fig. 1e, ED Fig. 1c). According to those topological arguments the observed frequencies of polygon types are the result of sequential cell division (Fig. 1f) [1, 2]. A key prediction from this topological model is that all epithelial tissues would have the same distribution of polygon type frequencies [1, 2]. The exact frequency would depend on the mode of cell division. However, both in case of a random positioning of the cell division axis (marked by a dashed line in Fig. 1e) as well as in case of cell division perpendicular to the longest axis (Hertwig’s rule [7]) (marked by a dotted line in Fig. 1e), about 45% of all cells would have six neighbours. Contrary to this prediction from the topological model, the frequencies of polygon types vary strongly between tissues, ranging from 30% to 80% hexagons in the different epithelia analysed (Fig. 1e, ED Fig. 1b). Other processes, in addition to cell division, must therefore determine epithelial organisation.

Vertex models explain cell organisation in animal and plant epithelia based on physical arguments [2–5, 8]. Here, the apical surface of each cell is approximated by a polygon. Neighbouring cells share edges such that the epithelium is represented as a polygonal tiling (Fig. 1g). Cells can grow and divide. The mechanical behaviour of the cells is represented either via forces or based on an energy functional. In each time step, the vertices of cells are moved based on the equations of motion until the forces balance at each vertex, which corresponds to the energy minimum. As the forces acting at each vertex can be derived from the energy functional, the two approaches are equivalent, even though implementations may differ [9]. Embedded Image represents a well-studied energy functional for epithelial dynamics [3, 5]. Here, the first term serves to limit the area variability in the tissue, the second term describes cell contractility, and the last term represents the net effect of cell adhesion and surface (line) tension [3, 5]. In the first term, Kα ≥ 0 is a constant, the so-called area contractility, A0 is the target apical area, and Aα is the apical area of cell α; the sum goes over all cells in the tissue. In the second term, Γ is the contractility, and Πα is the perimeter of cell α; the sum goes over all cells in the tissue. Finally, in the last term, Λij is the surface (line) tension and lij is the length of the edge between vertex i and j; the sum goes over all edges in the tissue. The wide range of measured polygon type frequencies can be reproduced by adjusting the mechanical parameters Λij and Γ relative to Kα [3]. In addition to the mechanical parameters, also the mode of cell division affects the polygon type frequencies [2]. It has therefore been argued that both topological and mechanical effects determine epithelial organisation in the vertex model [2].

In 1928, F.T. Lewis documented a further intriguing property of epithelia [10]: the average apical area, Embedded Image, of cells with n neighbours is linearly related to the number of neighbours n, such that Embedded Image

Here, Ā refers to the average apical cell area in the tissue. All analysed epithelial tissues follow Lewis’ law (Fig. 1h, ED Fig. 1c) despite the different frequencies of neighbour numbers in the different tissues (Fig. 1e, ED Fig. 1b). Vertex models can reproduce Lewis’ law, but only for a small subset of the parameter space (spanned by the relative contractility, Embedded Image, and relative line tension, Embedded Image) [3, 5, 8], and it is unclear what restricts epithelia to all fall within this narrow parameter range. Similarly, Voronoi tessellations can give rise both to the measured range in the neighbour frequencies and to Lewis’ law, but again not for all Voronoi tessellation paths, and Voronoi tessellation paths do not necessarily recapitulate cell division planes [11]. Finally, entropy maximisation rather than energy minimisation has been proposed as a driving force for Lewis’ law [6], but this has since been ruled out [12, 13]. In conclusion, none of the current theories can explain the robust emergence of Lewis’ law in animal and plant epithelia analysed to date, suggesting that an important driving force of epithelial organisation has so far been overlooked. So, what is the underlying principle that drives all epithelia to organise according to Lewis’ law?

In the following, we show that the observed epithelial order indeed minimises energy. The key aspect that had been overlooked so far is the impact of the variability in the apical cell area on epithelial organisation. As a result of cell growth and division, apical areas differ in epithelia. In that case, a tiling with a mix of polygon types (rather than hexagonal tiling) has the smallest combined perimeter. Lewis’ law emerges because the side lengths in the apical polygonal tiling are then the most similar in length, resulting in the most regular polygonal shapes within a contiguous lattice of cells with different apical areas. Regular polygons result in the smallest cell perimeter per area, and regular polygonal lattices therefore minimize the contact surface area between cells, and thus the contact surface energy. Interestingly, our theory predicts that the relationship between cell area and neighbor number follows a quadratic law, rather than the linear law found by Lewis. Simulations predict that this quadratic relationship can only be observed in the limit of much higher apical area variability than what is normally found in epithelia. We tested the theory by generating such higher apical area variability in the Drosophila wing disc epithelium, and we indeed confirm the predicted quadratic scaling law. Moreover, we predict and confirm experimentally that the fraction of hexagons in a tissue depends on the area variability in the tissue. We conclude that the observed epithelial organisation is driven by a minimisation of the lateral contact surface energy between epithelial cells. Epithelial organisation and Lewis’ law can thus be understood by considering only two variables: tension and the variability of the apical cell areas in the epithelial tissue. Growth and cell division affect the epithelial organisation via their impact on the apical cell areas.

RESULTS

Epithelial Organisation when all apical areas are equal

We seek to understand the observed distribution of neighbour numbers and the emergence of Lewis’ law in epithelia. Let us first consider a tissue with equal line tension in all edges, Λij = Λ, and equal apical areas, i.e. Aα = A0. The energy E (Eq. 1) then only depends on the combined cell perimeter in the tissue, Π = Σα Πα = 2Σ<i,j> lij the factor 2 comes in as neighbouring cells share edges in the vertex model. For a given area, the perimeter of a polygon depends on its shape and on its number of neighbours: regular polygons have the smallest perimeter per area, and the polygon perimeter Π declines non-linearly as the number of vertices, n, increases (Fig. 2a), Embedded Image

Figure 2:
Figure 2: The impact of apical area variability on Epithelial Order

a, The perimeter Π of regular polygons declines non-linearly with increasing vertex number, n. b, In contiguous lattices composed of polygons with three edges per vertex, the average number of vertices must be six. The combined perimeter of non-hexagonal polygons with on average six vertices, i.e. a square and an octagon, is larger than that of two hexagons. c, The perimeter of regular polygons dependent on the vertex number, n, for different areas, A = 0.1, 1, 1.9. d, The impact of area variability. Two hexagons that differ in their area (A = 0.1 and A = 1.9) have a lower combined perimeter than two hexagons that have the same area (A = 1). Similarly, the combined perimeter of the square and the octagon is smaller when the two polygons differ in their area (A = 0.1 and A = 1.9 instead of A = 1 for both polygons). However, the square and the octagon have a smaller combined perimeter than the two unequally sized hexagons only if the smaller polygon (A=0.1) has fewer vertices (square) and the larger polygon (A=1.9) has more vertices (octagon) (case ii). The combined areas are identical in all cases; the combined perimeter is normalised with the combined perimeter of two hexagons that have the same area.

The perimeter per area is smaller, the larger the number of vertices. However, in contiguous tissues with three edges per vertex, cells must respect the topological constraint Embedded Image (Fig. 1c, d), i.e. cells must have on average six neighbours [1, 6]. Therefore, if some cells have more than six neighbours, other cells must have less than six neighbours. As can be seen in Figure 2a, for a given area, cells with fewer than six vertices gain more in perimeter than what the cells with more than six vertices lose in perimeter. For instance, a square and an octagon, which have on average six vertices, have a larger combined perimeter than two hexagons (Fig. 2a,b). Given the nonlinear dependency of the perimeter on the number of vertices (Fig. 2a), hexagonal tiling always has a smaller combined perimeter than any mixed polygon tiling, and purely hexagonal tiling is then the energetically most favourable configuration. Why would tissues deviate from hexagonal packing and follow Lewis’ law?

The impact of Apical Area Variability on Epithelial Order

The key aspect is the variability in the apical cell areas. As a consequence of growth, cell division, and nuclear movement during the cell cycle, apical cell areas in epithelia differ widely between cells. Changes in the polygon area shift the perimeter curve in a nonlinear fashion (Fig. 2c). As a result, if we again compare the two hexagons with a square and an octagon, we notice that differences in cell area reduce the combined perimeter for all configurations, even though the combined areas of the polygons is the same as before (Fig. 2d). The strongest reduction in the combined perimeter is observed when the smaller polygon has fewer vertices (square) and the larger polygon has more vertices (octagon) (Fig. 2c,d, case ii). Given the relation between polygon perimeter, area, and neighbour number (Eq. 3), this observation holds not only for this particular example, but for any combination of polygons that has on average six vertices (Euler’s formula). While the smaller combined perimeter explains why smaller cells have on average less than six neighbours and larger cells have on average more than six neighbours, it does not yet explain why the linear Lewis’ law emerges.

Lewis’ Law minimizes Surface Energy by driving polygons to be regular

In our mathematical analysis, we have so far assumed that all polygons are regular, which implies that they are equilateral and equiangular. Regular polygons have the smallest possible perimeter at a given apical cell area, and energy minimisation should therefore drive polygonal lattices to the most regular polygonal shapes. However, to fit within a contiguous lattice, neighbouring cells need to have the same side lengths (Fig. 3a). If all cells had the same apical area (Fig. 3b, blue line), then cells with fewer vertices would have larger average side lengths (Fig. 3c, blue line). Cell edges would thus need to be stretched to constitute a contiguous lattice (Fig. 3a). If the apical cell areas follow Lewis’ law, then the differences in side lengths are much smaller between different polygon types (Fig. 3c,d, black lines). In fact, equal side lengths, Embedded Image and thus regular polygons would be obtained (Fig. 3c,d, yellow lines) if Embedded Image where Embedded Image is the variance in the cell neighbour numbers. Here, we solved Eq. 4 for An, i.e. Embedded Image, and estimated the average area as Embedded Image, where N is the highest observed cell neighbour number. Using Euler’s formula, Embedded Image, we then obtain Eq. 5, which is very similar to Lewis’ law, except that the relative areas depend quadratically rather than linearly on the neighbour numbers. So, why do epithelia follow the linear Lewis’ law (Eq. 2) rather than the quadratic relation in Eq. 5?

Figure 3:
Figure 3: Lewis’ law reduces variability in side length

a, To fit into a contiguous lattice, cells must have equal side lengths or be stretched. b,c If all cells have the same apical area (b, blue line), then the edge length changes with the polygon type (c, blue line). If the cell areas follow Lewis’ law (Eq. 2), then the change in edge length is much smaller (black lines). All edges have the same length, if the polygon type-dependency of the apical areas follows Eq. 4 (yellow curves).

LBIBCell Simulations of Epithelial Tissue Dynamics predict a critical role of the apical area variability for Lewis’ law

To address this question, we need to use a computational framework that permits cells to deform within the tissue, and we used the simulation framework LBIBCell [14] to simulate the apical cell dynamics in a 2D plane with high geometric resolution (Fig. 1a, Fig. S1, Movie S1). The opposing effects of cell-cell adhesion and lateral surface tension are represented by springs at cell-cell junctions, and the effect of cortical tension is represented by springs along the cell surface (Fig. 4a). The most quantitative data is available for the Drosophila wing disc epithelium (or disc), and all parameter values could be inferred from available quantitative data (see Supplementary Material for details). Thus, the fraction of hexagons depends on the two mechanical parameters (Fig. 4b), and we adjusted the boundary tension and cell-cell junction strength to reproduce the frequencies of cell neighbours measured by Heller and co-workers in Drosophila wing discs [15] (Fig. 4c). We used the measured declining growth rate [16, 17] to reproduce the measured wildtype wing disc growth kinetics [18] (ED Fig. 2a,b), and we set the cell division threshold (Fig. 4d) such that we reproduce the measured average apical cell area (5.34 μm2) [15] (Fig. 4e) and the fraction of mitotic cells [5] (ED Fig. 2c). Cells were divided perpendicular to their longest axis (Hertwig’s rule [7]), as observed in wing discs [2, 19], but randomised cell division angles give similar results (ED Fig. 3). A fixed cell division threshold (Fig. 4d) results in a much narrower area distribution (CV = std/mean = 0.21) than measured (CV = 0.38, Fig. 4e) and the resulting slope in the Lewis’ plot is shallower than observed (Fig. 4f). Higher area CV values can result from less precise cell division thresholds, asymmetric cell division, and/or cell deformation, i.e. changes in apical area without change in cell volume as occurs in apical constriction. We randomised the critical size for cell division (Fig. 4g, ED Fig. 4, Supplementary Material) to reproduce the measured apical area distribution (Fig. 4h). For the measured apical area CV, Lewis’ law emerges in the simulations (Fig. 4i), thereby confirming the key importance of the apical area variability for epithelial packing. When we further increased the apical area variability (Fig. 4j,k), the simulations diverted from Lewis’ law (Fig. 4l, black line) and matched the predicted quadratic curve (Eq. 4) (Fig. 4l, yellow line). The simulations show and thus predict that the optimal average area for each polygon type is obtained only with wider apical cell area variability than normally observed in epithelia.

Figure 4:
Figure 4: Simulations in LBIBCell confirm the impact of the cell size distribution on epithelial packing

a, Representation of tissue and cells in LBIBCell [14]. Cortical tension and cell-cell adhesion are incorporated via springs (red). b, The fraction of hexagons increases with higher membrane tension and stronger cell adhesion in the simulations. c, The measured hexagon frequency in the wing disc [15] (blue line) can be reproduced. d-f, Cell division at a fixed cell size (d) results in a narrow cell size distribution (e), and fails to recapitulate Lewis’ law (Eq. 2, black line) (f). g-i, With a normally distributed cell division threshold (with mean μ and standard deviation c) (g) the simulations (red) recapitulate the measured (blue) cell size distribution [15] (h) and Lewis’ law (black line) (i). The blue symbols in f,i represent three independent measurements [15]. j-l, With an even wider distribution of cell division thresholds (j) the cell size distribution widens further (k) and the simulations deviate from Lewis’ law (black line) and approach the curve for which all side lengths are equal (yellow line, Fig. 3b) (l).

Experimental Test: higher apical area variability results in the predicted quadratic law

We sought to test this prediction by artificially creating epithelia with wider apical cell area distributions. To this end, we induced the expression of an RNAi targeting the growth regulator gigas/TSC2 [20] in cell patches randomly located within Drosophila wing discs. gigas attenuation results in larger cells because cells repeat S phase without entering M phase [20]. Indeed, in mosaic gigas/TCS-RNAi wing discs we observe cells with substantially larger apical areas than in the control, scattered throughout the epithelium (Fig. 5a). Importantly, most gigas/TSC2 cells still have an apical area in the wt range, but the largest apical areas are substantially larger than in the control such that the area distribution is considerably wider (Fig. 5b).

Figure 5:
Figure 5: Experimental Confirmation of the generalised Lewis’ law

a, Representative epithelial tissue from a wing disc with gigas RNAi clones. b, The apical cell area distribution, c, the relative frequency of neighbour numbers, and d, the relative average apical area, Embedded Image for different neighbour numbers in control (blue) and mutant (“gigas”, black) wing discs. The black line in panel d represents Lewis’ law (Eq. 2); the yellow line the quadratic relation (Eq. 5).

Based on our theory, we make two predictions: first, that the largest cells will on average have more neighbours than any of the wt cells – and indeed in the gigas/TCS-RNAi wing discs the polygon distribution is wider and cells with more than 10 neighbours are observed (Fig. 5c). Consistent with theoretical considerations (Fig. 2c,d) and our simulations (ED Fig. 4c), not only the polygon distribution is wider, but also the hexagon frequency is lower (Fig. 5c). The second prediction from the model and the simulations is that the tissue with the wider area distribution will follow a quadratic relationship rather than the linear Lewis’ law (Fig. 5d). Indeed, the average apical cell area for each neighbour number now corresponds to the predicted optimum (Eq. 4) – that is, a quadratic relationship, rather than the Lewis’ law (Fig. 5d).

The apical area distribution defines the hexagon fraction

Finally, based on our theory, we can predict and experimentally test a novel relationship. Lewis’ law (Eq. 2; Fig. 6a, black) and the quadratic law (Eq. 5; Fig. 6a, yellow) relate the apical area to a polygon type (Fig. 6a, dotted lines). Based on these laws, we expect that the wider the area distribution, the smaller the fraction of cells that falls within the area range that is predicted to form hexagons (Fig. 6b, shaded green). This trend is indeed observed in our experiments in the Drosophila wing disc (Fig. 5b,c). We can make this relationship more precise and quantitatively test it with the available data.

Figure 6:
Figure 6: The apical area distribution defines the hexagon fraction

a, Lewis’ law (Eq. 2, black line) and the quadratic law (Eq. 5, yellow line) relate the relative apical area to a polygon type n (dotted lines). b, Weibull distribution (Eq. 6) with λ = 1.12, k = 2 to approximate the normalised area distribution. The white lines indicate the limits of the polygon types, such that the average area for each polygon type n is closest to Embedded Image. The area shaded in green indicates the part of the area distribution that would correspond to hexagons. c,d, The fraction of hexagons for (c) different Weibull shape parameters k, and (d) different area CV values (Gaussian distribution) as predicted by the linear Lewis’ law (Eq. 2, black line) or the quadratic law (Eq. 3, yellow line) and for the data (dots). The abbreviations are as in ED Figure 1. dMWL refers to the wing disc with gigas RNAi clones from Figure 5.

To predict how the fraction of hexagons depends on the width of the area distribution, we approximate the normalised area distributions either by a Gaussian distribution with variance σ2, Embedded Image or by a Weibull distribution, which is positive for positive values of A/Ā, but is zero otherwise, Embedded Image

Here, λ is the scale parameter that predominantly affects the position of the maximum, and k > 1 is the shape parameter of the Weibull distribution that predominantly affects the width of the distribution; as k increases, the distribution sharpens (ED Figure 5). The expected fraction of hexagons can then be derived by first assigning each polygon type n to a part of the normalised area distribution such that the average area for each polygon type n is closest to Embedded Image (Fig. 6b, white lines), and by subsequently calculating the value of the cumulative distribution function for hexagons (Fig. 6b, shaded green). The predicted fraction of hexagons for the different Weibull shape parameters k and for different area CV values (Gaussian distribution) (Fig. 6c,d, lines) fit the data (Fig. 6c,d, dots) very well, both when we use the linear Lewis’ law (Eq. 2, black line) or the quadratic law (Eq. 3, yellow line). A hexagon fraction below these predicted values arises when cells seek to maximise rather than to minimise their lateral cell-cell contact surface, as is the case when the adhesion strength is increased in the LBIBCell simulations (ED Figure 6).

In summary, the theory, computational modelling, and experimentation indicate that the minimisation of surface energy drives the emergence of Lewis’ law, and that a generalization of this law, that applies in the limit of a broader variability in apical cell areas than normally found, is described as a quadratic relationship between average cells area and neighbour number.

DISCUSSION

Different theories have been proposed to explain epithelial order [1–6, 11]. We have now shown that epithelial organisation can be understood in terms of energy minimisation (Fig. 7). Lewis’ law emerges in all epithelial tissues studied to date because of the particular role played by apical cell area variability, which we describe here. Growth, cell division, and nuclear migration affect epithelial order indirectly via their impact on the variability in the apical cell area.

Figure 7:
Figure 7: Summary Figure

(Left column) If all cells have the same apical area, then hexagonal packing results in the smallest average perimeter per cell, and thus in the lowest possible surface energy. (middle column) For an intermediate width of the apical area distribution, lowest surface energy is achieved for non-hexagonal packing and if the average areas per polygon type follow Lewis’ law (Eq. 2). (right column) For wide apical area distributions, the lowest surface energy is achieved for non-hexagonal packing and if the average areas per polygon type follow a quadratic dependency on the number of neighbours (polygon type) (Eq. 4). In this case, the side lengths of all polygons are the most similar, and cells thus do not need to be stretched to fit into a contiguous lattice.

We derived Lewis’ law as the configuration that minimises the overall cell perimeter in a contiguous tissue. Cells in a contiguous tissue seek to minimise their perimeter when the effects of cell contractility and lateral surface tension dominate the effects of cell-cell adhesion. In a tissue with equal apical areas, hexagonal tiling results in the smallest combined perimeter (Fig. 2a,b) and is therefore the configuration with the lowest energy (Fig. 7, left column). This configuration has previously been referred to as ground state [3]. In real tissue, apical areas differ as a result of growth, cell division, and interkinetic nuclear movements. If the variability is sufficiently high, cell lattices composed of mixed polygon types have a smaller combined perimeter if the smaller polygon has fewer than six and the larger polygon has more than six neighbours (Fig. 2c,d). Overall, the average number of neighbours must be six due to topological constraints (Euler’s formula) [1, 6]. Lewis’ law (Eq. 2) emerges because cells then have a more similar side length (Fig. 7, middle column), and regular, equilateral polygons have the smallest perimeter at a given area. Equal side lengths of all cells require a quadratic (Eq. 5) rather than a linear (Eq. 2) Lewis’ law. We predict and confirm experimentally that such quadratic law only emerges at higher apical area variability than previously observed in epithelia (Fig. 7, right column).

The role of apical area variability and the regularity of polygons is also observed in other simulation frameworks. Thus, Voronoi tessellations result in epithelia-like packing [11, 21] if the employed algorithm produces nearly equilateral polygons. In vertex models, Lewis’ law has previously been observed only when the mechanical parameters are sufficiently high compared to the area contractility [3]. This can now be understood in terms of our results: a small area contractility corresponds to a weak constraint on the area variability (Eq. 1), which allows a sufficiently wide area distribution for Lewis’ law to emerge. If the mechanical parameter values are further increased compared to the area contractility, then the linear Lewis’ law disappears in the vertex models [3]. Based on our analysis, we expect that a quadratic relationship should be observed in the vertex model when the area contractility is sufficiently small compared to the mechanical parameters.

According to Lewis’ law and the quadratic law, the area distribution determines the width of the neighbour number distribution and thus the fraction of hexagons (Fig. 6). So how do the mechanical properties of cells affect epithelial organisation? The critical aspect here is that in Eq. 1 the line tension term is linear while the contractility term is quadratic in the perimeter. If both constants are positive, then cells will always seek to minimise their combined perimeter, and the area distribution will determine epithelial packing independent of the exact value of the mechanical parameters. Likewise, if both constants are negative, cells will seek to maximise their combined perimeter and will assume deformed cell shapes; these latter tissues are predicted not to follow Lewis’ law. Here, we note that, despite their shape, puzzle cells in plants do not fall into the latter category. This is because puzzle cells do not emerge to maximise their cell perimeter, but to minimize stress in large cells [22]. Importantly, the puzzle shape emerges only after cells have stopped dividing and a contiguous lattice with defined neighbour numbers has already formed, which can no longer change because of rigid cell walls [22]. Accordingly, puzzle cells in plants still follow Lewis’ law [23].

Epithelial organisation reflects the mechanical properties of cells only if the line tension constant, Λij, and contractility, Γ, have different signs. The line tension constant will typically be negative because the effects of cell-cell adhesion can be expected to be stronger than those of lateral surface tension; contractility will be positive due to the effects of the acto-myosin ring. Given that the linear term is then negative and the quadratic term is positive, a minimal cell perimeter will minimise the energy (Eq. 1) only for those cells that have a cell perimeter above a critical value, and this critical value depends on the relative value of the negative line (surface) tension constant, Λij, and the positive contractility, Γ. Accordingly, an increase in the cell-cell adhesion strength relative to cell contractility results in deformed cells (ED Fig. 6a-d), a smaller fraction of hexagons (ED Fig. 6e), and a deviation from Lewis’ law (ED Fig. 6f). Likewise, external forces that stretch the entire tissue result in a deviation from Lewis’ law (ED Fig. 3d). As a practical consequence, relative forces (and thus stresses) in an epithelium can be inferred from the observed cell shapes [24, 25].

We conclude that Lewis’ law emerges as cells seek to minimize the overall cell perimeter in a tissue, which is the case when contractility and surface tension dominate the effects of cell-cell adhesion. The mechanical properties of the cells determine to what extent cells seek to minimise their perimeter – and thus to what extent Lewis’ law applies in a tissue. The fact that Lewis’ law is observed throughout epithelia shows that the combined effects of contractility and surface tension typically dominate the effects of cell-cell adhesion and cells thus seek to minimise their perimeter while forming a contiguous lattice. Whether the linear (Eq. 2) or quadratic (Eq. 5)

Lewis’ law is observed, depends on the area variability.

Epithelial cells interact not only at the apical side, but also along the apical-basal axis. Accordingly, similar physical constraints are likely to apply. Further quantitative studies are required to elucidate the 3D organisation of epithelia.

METHODS

Imaging of Tissue Samples

Drosophila (wild type Oregon-R strain) eye discs and wing discs were dissected from third instar larvae or pupae. Eye discs were staged according to their number of ommatidial rows (as this number increases as development proceeds). Pupal wing discs were dissected at 30 and 48 hours post puparium formation. Staining and confocal imaging was carried out as described[26]. Primary antibodies used were mouse anti-Dlg (4F3 Hybridoma bank, 1:500; eye discs) and rabbit anti-aPKC (Abcam, AB5813, 1:500; pupal wing discs).

Mosaic induction of gigas RNAi[27, 28] in wing discs was achieved using the flip-out system[29]. Larvae from crossing flip122; tub>stop>GAL4, UAS-GFP females to either the wild type strain Or-R (“wild type”) or UAS-gigasRNAi (“gigas”; VDRC stock# 103417) males were heat-shocked at 37°C for 30min at 72-96 hours after egg laying (AEL). Wing discs from crawling third larval stage (≈120h AEL) were dissected, fixed and stained with mouse anti-Armadillo antibody (N27A1, Developmental Studies Hybridoma Bank, University of Iowa; 1/100), which labels the subapical region of epithelial cells, and rabbit anti-GFP (A11122, Molecular Probes; 1/1000). Alexa Fluor 488 and 568 (Molecular Probes) secondary antibodies were used at 1/500. Samples were imaged on a Leica SPE confocal setup, with an oil-immersion 40X objective (N.A.=1,15), as z-stacks of 12 bits, 1024X1024 pixel images, with z-step= 0.3 microns.

Image Analysis

3D image stacks used in morphometric analyses were first pre-processed using Fiji [30]. We selected image planes that only contained apical cell boundaries as marked by the anti-armadillo antibody staining, removed frames containing non-apical and non-pouch signal, used a maximum intensity projection to project all substacks to 2D, and enhanced their contrast using the Contrast Limited Adaptive Histogram Equalization (CLAHE) [31]. Using Ilastik [32], we then built a random forest classifier that was used for discriminating cell boundaries from background and noise in problematic image regions. The resulting probability maps were binarised and used to build a Voronoi diagram using Fiji. All diagrams (masks) were manually curated using Tissue Analyzer [33] until they exactly mapped to all apical cell boundaries. Pertinent morphometric measurements were derived and exported by using the EpiTools [15] plugin for Icy [34]. The parameters of the Weibull distribution were fitted using the R package fitdistrplus [35]. All plotting was done using several R packages, including ggplot2, gdata, gtools, and data.table [36–42].

Set-Up of the LBIBCell simulations

The details of how the LBIBCell simulations were set up are provided in the Supplementary Text.

Author Contributions

DI developed the theoretical framework, MK carried out all simulations and created all figures, AI, GV-F, FC contributed the data. DI, 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. We are grateful to Anna Stopka and Harold Gómez for advice on LBIBCell and image processing. This work has been supported through an SNF Sinergia grant to DI, and grants BFU2012-34324 and BFU2015-66040 (MINECO, Spain) to FC.

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Supplementary Material

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Minimisation of surface energy drives apical epithelial organisation and gives rise to Lewis’ law
Marco Kokic, Antonella Iannini, Gema Villa-Fombuena, Fernando Casares, Dagmar Iber
bioRxiv 590729; doi: https://doi.org/10.1101/590729
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