3D cell neighbour dynamics in growing pseudostratified epithelia

  1. Harold Fernando Gómez
  2. Mathilde Sabine Dumond
  3. Leonie Hodel
  4. Roman Vetter
  5. Dagmar Iber  Is a corresponding author
  1. Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Switzerland
  2. Swiss Institute of Bioinformatics (SIB), Switzerland

Abstract

During morphogenesis, epithelial sheets remodel into complex geometries. How cells dynamically organise their contact with neighbouring cells in these tightly packed tissues is poorly understood. We have used light-sheet microscopy of growing mouse embryonic lung explants, three-dimensional cell segmentation, and physical theory to unravel the principles behind 3D cell organisation in growing pseudostratified epithelia. We find that cells have highly irregular 3D shapes and exhibit numerous neighbour intercalations along the apical-basal axis as well as over time. Despite the fluidic nature, the cell packing configurations follow fundamental relationships previously described for apical epithelial layers, that is, Euler's polyhedron formula, Lewis’ law, and Aboav-Weaire's law, at all times and across the entire tissue thickness. This arrangement minimises the lateral cell-cell surface energy for a given cross-sectional area variability, generated primarily by the distribution and movement of nuclei. We conclude that the complex 3D cell organisation in growing epithelia emerges from simple physical principles.

Introduction

Common to all animals and plants, epithelia are a fundamental tissue type whose expansion, budding, branching, and folding is key to the morphogenesis of organs and body cavities. Characterised by apical-basal polarity (Figure 1a), epithelial cells adhere tightly to their apical neighbours in a virtually impermeable adhesion belt, form lateral cell-cell junction complexes along the apico-basal axis to provide mechanical stabilisation, and bind tightly to the basal lamina and extracellular matrix (ECM) on the basal side (Drubin and Nelson, 1996; Rodriguez-Boulan and Macara, 2014; Shin and Margolis, 2006). How cell neighbour relationships are organised in these tightly adherent layers, and how these change during tissue and concomitant cell shape changes is poorly understood, despite their importance for cell-cell signalling and the fluidity of the tissue.

Principles of epithelial organisation.

(a) Schematic representation of an epithelial tissue layer. The cells are polarised between an apical and a basal side. Near the apical side, cells adhere tightly via adhesion junctions (green). Nuclei are depicted in blue. (b) Apical surface projection of an embryonic lung bud at E12.5 imaged using light-sheet microscopy. Cell contour segmentations (left) coloured according to neighbour relationships (middle) and area quantifications (right). (c) Current shape representations of 3D epithelial cells: prism, frustum, prismatoid, and scutoid. (d) Planar cell neighbour exchange (T1 transition). (e) 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 (Classen et al., 2005; Escudero et al., 2011; Etournay et al., 2015; Farhadifar et al., 2007; Gibson et al., 2006; Heller et al., 2016; Sánchez-Gutiérrez et al., 2016). Data points for n < 3 were removed as they must present segmentation artefacts. (f) The measured average number of cell neighbours is close to the topological requirement (n¯=6) in all tissues; see panel e for the colour code. (g) Epithelial tissues follow the AW law (black line). The AW law formulates a relationship between the average number of neighbours, n, that a cell has and that its direct neighbours have, mn. The product mnn can be determined by summing over all ni. (h) The relative average apical cell area, An¯/A¯, increases with the number of neighbours, n, and mostly follows the linear Lewis’ law (Equation 2, black line), or the quadratic relationship (Equation 3, yellow line) in case of higher apical area variability. (i) The average internal angle by polygon type is close to that of a regular polygon, θn=(n-2)/n180° (yellow lines). To form a contiguous lattice, the angles at each tricellular junction must add to 360°, and the resulting observed deviation in the angles follows the prediction (red line). (j) The average normalised side length by polygon type. (k) Observed fraction of hexagons versus area coefficient of variation (CV). The curves mark theoretical predictions when polygonal cell layers follow either the linear Lewis’ law (Equation 2, black line) or the quadratic law (Equation 3, yellow line). The colour code in panels g-k is as in panel e, but data is available only for a subset of tissues. The abbreviations in panel (e) are as follows: cNT refers to the Chick neural tube epithelium, EYE to the Drosophila eye disc, dPE to the Drosophila peripodal membrane from the larval eye disc, dWL to the Drosophila larval wing disc, dWP to the Drosophila pre-pupal wing disc, dMWP to the Drosophila mutant wing pre-pupa with reduced expression of myosin II, dPW to the Drosophila pupal wing disc, and dMWL to the wing disc with gigas RNAi clones. TPx indicates subsequent but not further specified pupal time points.

© 2019, Kokic et al. Panels e, f, g and k are reproduced with modifications from Figures 1,5 from Kokic et al., 2019, which are published under the Creative Commons Attribution Non-Commercial 4.0 International License

© 2019, Vetter et al. Panels g and i are reproduced with modifications from Figures 1,5 from Vetter et al., 2019, which are published under the Creative Commons Attribution Non-Commercial 4.0 International License

Cell neighbour relationships can be most easily studied on epithelial surfaces, and the polygonal arrangements of apical surfaces (Figure 1b) have been meticulously analysed (Classen et al., 2005; Escudero et al., 2011; Etournay et al., 2015; Farhadifar et al., 2007; Gibson et al., 2006; Gómez-Gálvez et al., 2018; Heller et al., 2016; Kokic et al., 2019; Ramanathan et al., 2019; Sánchez-Gutiérrez et al., 2016). Widely considered to be a reliable proxy for three-dimensional (3D) cell shape, 3D epithelial cell shapes are often depicted as prisms with polygonal faces that retain the same neighbour relationships along the entire apico-basal axis (Figure 1c). Cells in curved epithelia are pictured as frustra, which have the same number of sides, but different apical and basal areas. If the curvature differs substantially along the principal axes, as is the case in epithelial tubes, neighbour relationships must change along the apical-basal axis. Prismatoids accommodate the neighbour change at the surface, while scutoids undergo the neighbour change somewhere along the apical-basal axis (Figure 1c,d; Gómez-Gálvez et al., 2018). However, even though the curvature is the same in both principal directions of spherically shaped epithelia, the neighbour relationships still differ between the apical and basal sides (Gómez-Gálvez et al., 2018), suggesting that effects other than curvature must determine the 3D neighbour arrangements of cells in epithelia.

Given the challenges in visualising 3D neighbour arrangements, most studies to date have focused on apical cell arrangements, and have revealed striking regularities. First, even though the frequencies of neighbour numbers differ widely between epithelial tissues (Figure 1e), cells have on average (close to) six neighbours (Classen et al., 2005; Escudero et al., 2011; Etournay et al., 2015; Farhadifar et al., 2007; Gibson et al., 2006; Heller et al., 2016; Kokic et al., 2019; Ramanathan et al., 2019; Sánchez-Gutiérrez et al., 2016; Figure 1f). This can be explained with topological constraints in contiguous polygonal lattices, as expressed by Euler's polyhedron formula (Gibson et al., 2006; Rivier and Lissowski, 1982). Thus, if three cells meet at each vertex, the average number of neighbours in infinitely large contiguous polygonal lattices is exactly

(1) n¯=6.

While the average number of neighbours in the entire lattice is (close to) six, the local averages deviate from six, and instead rather closely follow a phenomenological relationship, termed Aboav-Weaire's law (Aboav, 1970). According to Aboav-Weaire's law (Figure 1g), the average number of neighbours of all n cells that border a cell with n neighbours follows as

(2) m(n)=5+8n.

Finally, the average apical area, An¯, of cells with n neighbours is linearly related to the number of cell neighbours, n (Figure 1h, black line), a relation termed Lewis’ law (Lewis, 1928),

(3) An¯A¯=(n2)4.

Here, A¯ refers to the average apical cell area in the tissue.

We have recently shown that Aboav-Weaire's law and Lewis’ law are a direct consequence of a minimisation of the lateral cell-cell contact surface energy (Kokic et al., 2019; Vetter et al., 2019). The lowest lateral cell-cell contact surface energy is obtained in a regular polygonal lattice because regular polygons have the smallest perimeter per polygonal area. The distribution of apical cell sizes that emerges from cell growth and division is, however, such that epithelial tissues cannot organise into perfectly regular polygonal lattices. By adhering to Aboav-Weaire's law and Lewis’ law, cells assume the most regular lattice. In particular, by following Aboav-Weaire's law, the internal angles are closest to that of a regular polygon, while adding up to 360° at each tricellular junction (Figure 1i; Vetter et al., 2019). And by following the relationship between polygon area and polygon type as stipulated by Lewis' law (Equation 2), the difference in side lengths, S¯n/S¯, is minimised between cells (Figure 1j; Kokic et al., 2019). The side lengths would be equal (Figure 1j, yellow line), if cells followed a quadratic relation of the form

(4) AnA¯=n6tan(π6)tan(πn)(n6)2.

This quadratic relation (Figure 1h, yellow line), however, requires a larger area variability than is observed in most epithelia imaged to date. Accordingly, the predicted quadratic relation had not been previously reported, but could be confirmed experimentally by us by increasing the apical area variability (Kokic et al., 2019).

Given the relationship between apical area and neighbour numbers as stipulated by Equations 1,3,4, the apical area variability emerges as the key determinant of apical epithelial organisation, and the theory correctly predicts how the fraction of hexagons in the tissue depends on the apical area variability, as can be quantified by the coefficient of variation (CV = std/mean) (Figure 1k; Kokic et al., 2019). As such, active processes such as growth, cell division, cell death and extrusion, cell intercalation and apical constriction determine the variability of the apical areas and thus determine apical organisation indirectly. Taken together, the apical organisation of epithelia can be understood based on the principles of lateral cell-cell contact surface energy minimisation.

In this work, we leverage these theoretical insights along with light-sheet fluorescence microscopy to study 3D epithelial organisation, both in cleared and growing pseudostratified epithelia. We find that cells have complex 3D shapes with numerous neighbour transitions along their apical-basal axis as well as over time. We show that much as on the apical side, the variation of the cross-sectional areas along the apical-basal axis defines the epithelial organisation at all times and across the entire tissue thickness. The observed neighbour arrangement minimises the lateral cell-cell surface energy for a given cross-sectional area variability. The cross-sectional areas vary as a result of active cell processes, most prominently including interkinetic nuclear migration (IKNM). We conclude that the complex 3D cell organisation in growing epithelia emerges from simple physical principles.

Results

Apical and basal epithelial organisation

We started by exploring the apical and basal cellular organisation in epithelial tubes and buds (Figure 2a). To this end, we imaged CUBIC-cleared mouse embryonic (E12.5) lung rudiments from a ShhGC/+; ROSAmT/mG background using light-sheet microscopy, and segmented the fluorescent membrane boundaries of over 400 cells per dataset in 2.5D (Figure 2b, Figure 2—figure supplement 2). The apical and basal surfaces are both curved and thus differ in their total areas, that is, the total segmented apical area is about 5-fold smaller than the basal area (Figure 2b). We detected less than half as many cells on the apical side, and the mean cross-sectional cell area of apical cells is therefore on average only twofold smaller than that of basal cells, while the area variability, measured as area CV, is higher (Figure 2c). Notably, the frequencies of the different neighbour numbers are not identical on the apical and basal side (Figure 2d), suggesting that the neighbour relationships change along the apical-basal axis, both in the tube and tip datasets. This observation is consistent with previous reports (Gómez-Gálvez et al., 2018; Ramanathan et al., 2019; Rupprecht et al., 2017). The change in neighbour relationships has previously been attributed to a curvature effect in tubes, but the neighbour changes in spherical geometries cannot be explained with curvature alone, since prismatic cells fully accommodate the cell dilation from equal curvature changes in both directions (Gómez-Gálvez et al., 2018; Rupprecht et al., 2017), suggesting that mainly other effects determine epithelial organisation.

Figure 2 with 4 supplements see all
Apical and basal epithelial organisation.

(a) Epithelium of E12.5 ShhGC/+; ROSAmT/mG mouse embryonic lung imaged using light-sheet microscopy. Scale bar 200 µm. Corresponding 2D sections are shown in Figure 2—figure supplement 1, and Figure 2—video 1. (b) Apical and basal 2.5D cell segmentation overlays on imaged tip and tube sections (dotted boxes in panel a). An illustration of the 2.5D segmentation workflow is presented in Figure 2—figure supplement 2. Number of cells (N) and segmented surface areas (SA) are given. Cells are coloured using random labels. Scale bars 20 µm. (c) Normalised apical and basal cell area distributions in the tip (broken lines) and tube (solid lines) datasets. The colour code in panel c is reused in panels d-f. (d) Frequencies of neighbour numbers on the apical and basal sides in the tip and tube datasets. (e) The apical and basal layers follow the AW law (black line). SEM is smaller than symbols. (f) The normalised average cell area, An¯A¯, increases with the number of neighbours, n. The basal cells (green) follow Lewis’ law (Equation 2, black line), while the apical cells (purple) follow the quadratic relationship (Equation 3, yellow line). SEM as error bars.

So how can we explain the difference in apical and basal epithelial organisation in both datasets? We have previously shown that the apical organisation emerges from the minimisation of the overall lateral cell-cell contact surface energy (Kokic et al., 2019; Vetter et al., 2019). Aboav-Weaire's law (Equation 2, Figure 1g) and the linear Lewis’ law or the quadratic relationship (Equations 3,4, Figure 1h) emerge as global organisation laws from this physical constraint, and ensure that the angles are closest to that of a regular polygon (Figure 1i, yellow lines), and that the side lengths are the most equal (Figure 1j, yellow line). We now find that these hold not only for the apical, but also for the basal datasets (Figure 2e,f). Consistent with our theory, the apical layers, which have a larger area variability than the basal layers (Figure 2c), follow the quadratic law (yellow line) rather than the linear Lewis’ law (black line).

We conclude that basal layers follow the same organisational principles as apical layers, such that their organisation can also be explained with a minimisation of the lateral cell-cell contact surface energy. Accordingly, the observed difference in overall neighbour relationships (Figure 2e,f) is a consequence of the difference in the cross-sectional area distributions (Figure 2c). So, why do the normalised area distributions differ between the apical and basal sides in both the tube segment and the bud, and how do they change along the apical-basal axis?

3D organisation of epithelia

To explore the physical principles behind 3D epithelial cell organisation, we 3D segmented 140 cells from a tube segment and 59 cells from a bud segment in CUBIC-cleared, light-sheet imaged embryonic lung explants (Figure 3a, Figure 2—figure supplement 1, Figure 2—video 1, Figure 3—videos 13). By interpolating between equally spaced sequential contour surfaces (every 1.66 µm in the tube and every 1.72 µm in the bud dataset) along the apical-basal axis, accurate volumetric reconstructions of cell morphology were obtained that allowed for the extraction of morphometric quantifications along the apical-basal axis. In both datasets, the 3D organisation of epithelial cells is highly complex, and cell neighbour relationships change continuously along the apical-basal axis (Figure 3b). As a result, cells are in direct physical contact not only with the cells that are neighbours on the apical side, but also with cells that appear two or even three cell diameters apart (Figure 3c).

Figure 3 with 4 supplements see all
3D epithelial organisation.

(a) 3D iso-surfaces of segmented epithelial tip (N=59) and tube (N=140) cells from a ShhGC/+; ROSAmT/mG E12.5 mouse lung rudiment imaged using light-sheet microscopy. Morphometric quantifications of cell boundary segmentations along the apical-basal axis were used to study spatial T1 transitions. The 3D segmentation workflow is introduced in Figure 3—figure supplement 1 and Figure 3—videos 13 illustrate the rendered epithelial tip, tube and all segmented volumes. Scale bars 10 µm. (b) Frequency of neighbour numbers as quantified along the apical basal axis in the tip (solid lines) and tube (broken lines) datasets. (c) Extent of neighbour contacts (center cell in yellow and neighbours in blue) in 3D as viewed from the apical side. Scale bars 5 µm. (d) Probability distributions of the lateral T1 transitions for tip (total=169, mean=2.86, N=59) and trunk (total=746, mean=5.41, N=140) datasets are consistent with Poisson distributions. (e) Normalised apical-basal distribution of T1 transitions for all cells shows no apical-basal bias, except for fewer transitions close to the basal surface. (f). Schematic of a tubular epithelium with elliptic cross section. The analysed cells are located in the cusp (brown) of the tube, where the local tissue curvature is close to that of the minor ellipse vertex. Subpanel illustrates ellipse fitting to apical (purple) and basal (green) domain boundaries. Lighter outlines correspond to the most proximal segment of the tube in Figure 2b, while darker ones to the most distal section. (g) The predicted impact of a curvature effect on T1 transitions decreases with increasing cell neighbour numbers (blue line). The measured T1 transitions for different neighbour numbers do not support a curvature effect (dots, boxplots, and red line). Boxplots indicate the median, 25% and 75% percentiles of the data.

Remarkably, we record up to 14 cell neighbour changes per cell in the tube and up to eight in the tip, between adjacent cross-sections along the apical-basal cell axis (Figure 3d). We will refer to these neighbour changes as lateral T1 transitions, or T1L. The mean relative apical-basal position for the lateral T1 transitions is 0.489±0.020 (95% CI), and there is no clear apical or basal tendency, though fewer transitions are observed close to the basal surface (Figure 3e). The dispersion index, that is, the ratio of the variance σ2 and the mean number µ of transitions per cell, which equals unity for a Poisson distribution, is close to unity for both samples (Figure 3d). The chi-squared test also confirms that the number of apical-basal T1 transitions per cell is Poisson-distributed (Figure 3d). A Poisson distribution models the probability of a number of independent random events occurring in a given interval at a constant average rate. The consistency with a Poisson distribution, therefore, suggests a stochastic basis to the 3D organisation of epithelial cells.

The large number of observed T1L transitions and their distribution along the apical-basal axis challenges the recently popularised notion of curvature-driven scutoids as cell building blocks for epithelia (Gómez-Gálvez et al., 2018). To further examine the potential influence of tissue curvature on T1L transitions, we measured the apical-basal distance between two consecutive neighbour number changes for each cell in the tube dataset and recorded at which local tissue curvature they occur. For this analysis, we excluded ce ll portions from the apical end to the first transition and from the last transition to the basal end to reduce boundary effects, that is, only interior segments between transitions were considered. The mean apical-basal distance between two transitions is 17.89 ± 0.66 µm (95% CI). Local tissue curvature was approximated by fitting ellipses to the apical and basal surface boundaries of the tubular epithelium in 624 equidistant sections perpendicular to the main tube axis. (Figure 3f). Epithelial tubes in the developing lung are often collapsed (Conrad et al., 2021), making their apical and basal surfaces nearly elliptic in shape. The semi-axes of the fitted ellipses were then averaged over all sections to obtain the semi-axes aapical, abasal, bapical, bbasal. Since our sample of 140 cells was segmented from a region close to the cusp of the nearly elliptical tube, a reasonably close estimate of the local tissue curvature where a T1L transition occurs is given by a linear interpolation between the curvature at the minor vertices of the apical and basal ellipses, according to the relative apical-basal position of the transition. The minor curvature of an ellipse with major and minor semi-axes a and b is given by b/a2. Therefore, we estimate the local radius of curvature R by

Rx=(aapical+x(abasal-aapical))2bapical+x(bbasal-bapical)

where x∈[0,1] is the relative apical-basal location of the T1L transition. The examined tissue exhibits an average curvature fold change of R(1)/R(0)=2.21 from the basal to the apical side. Denoting by R1 and R2 the radii of curvature between two adjacent T1 transitions along the apical-basal axis of a cell, we find that the distribution of curvature fold change R2/R1 shows no significant dependency on the number of neighbours n the cell has along that portion of the cell (Figure 3g). The mean curvature fold change per apical-basal T1L transition per cell is <R2/R1> = 1.142 ± 0.010 (95% CI). By extending the theory of scutoids (Gómez-Gálvez et al., 2018) to multiple T1L transitions per cell, we have derived a quantitative estimate of how tissue curvature would translate into the number of neighbour exchanges within that framework (Supplementary Material). If curvature changes were a main driver of T1L transitions, cells with smaller neighbour numbers n would be expected to change n over a much larger curvature fold change than cells with many neighbours (Figure 3g, blue line). However, we observe no systematic dependency of the curvature fold change on the number of neighbours the cell has along that portion of the cell in the developing mouse lung epithelium (Figure 3g). From this, we conclude that tissue curvature affects cell neighbourhood rearrangements through the tissue thickness at most mildly.

Neighbour changes along the apical-basal axis are driven by changes in cross-sectional area variation

Other than curvature effects, what else could drive the observed changes in neighbour relationships along the apical-basal axis? We notice that much as the apical and basal layers, each layer along the apical-basal axis behaves according to the three relationships previously described for the apical side, that is, Euler's polyhedron formula (Equation 1, Figure 4a), Aboav-Weaire's law (Equation 2, Figure 4b), and Lewis’ law (Equations 3, 4, Figure 4c). As predicted by the theory based on the minimisation of the lateral cell-cell energy (Kokic et al., 2019), the layers with a large area variability (Figure 4a) follow the quadratic law (yellow line) and those with a lower area variability the linear Lewis’ law (black line). The fraction of hexagons also follows the predicted relationship with the cross-sectional area variability (Figure 4d). We conclude that the neighbour relationships of epithelial cells along the entire apical-basal axis can be explained with a minimisation of the lateral cell-cell contact surface energy, as previously revealed for the apical layer.

Neighbour changes along the apical-basal axis are driven by changes in cross-sectional area.

(a) Average number of neighbours (black) and area CV (blue) along the apical basal axis in the tip and tube datasets. (b) All epithelial layers follow the AW law (black line). The colour code in panels c, d follow that in panel b. (c) All epithelial layers follow Lewis’ law (Equation 2, black line) in case of low, and the quadratic relationship (Equation 3, yellow line) in case of high cell area variability. (d) Observed fraction of hexagons versus area CV for segmented cell layers along the apical-basal axis. The lines mark the theoretical prediction if polygonal cell layers follow either the linear Lewis’ law (black line) or the quadratic law (yellow line). (e) 3D iso-surfaces of four segmented epithelial cells in a CUBIC-cleared ShhGC/+; ROSAmT/mG E12.5 distal lung tube, with 140 3D segmented epithelial cells (Figure 3—figure supplement 1c). (f) Cross-sectional area and cell neighbour number along the apical-basal axis for marked cells in panel e. Dotted lines indicate contact with one another. (g) Lateral cross-sections illustrating a T1 transition along the apical-basal axis (0.664 µm in-between frames). Scale bar 6 µm. (h) An increase in the cell cross-sectional area increases the frequency of a neighbour number increasing spatial T1 transitions, and vice versa. (i) Apical-basal distribution of spatial T1 transitions according to neighbour increase or decrease and cross-sectional area variation.

If epithelial cell neighbour relationships are indeed driven by a minimisation of the total lateral cell-cell contact surface energy then the T1L transitions along the apical-basal axes should be driven by changes in the cross-sectional area along the apical-basal axis. If we analyse four 3D segmented cells (Figure 4e) in detail, we indeed see how an increase in the cross-sectional area results in an increase in the neighbour number, and vice versa (Figure 4f) via lateral T1 transitions (Figure 4g). As the cell neighbour arrangements represent global minima, the local analysis does, of course, not provide a perfect correlation. When we consider all 140 segmented cells in the tube segment with their 746 cell neighbour exchanges between adjacent cross-sections (Figure 3e), then we find that the frequency of T1L transitions along the apical-basal axis is indeed higher, the larger the increase in cross-sectional area, and vice versa (Figure 4h,i).

Changes in cross-sectional area as a result of interkinetic nuclear migration (IKNM)

So, what determines the cross-sectional cell areas in each layer? In pseudostratified epithelia, mitosis is restricted to the apical surface (Gundersen and Worman, 2013). Depending on the average diameter of nuclei and the average apical cross-sectional area, there is insufficient space for all nuclei to be accommodated apically. Therefore, as a cell exits mitosis, the nucleus moves from the apical towards the basal side (G1 and S phase) and then back to the apical side (G2 phase) to undergo another round of mitosis, a process referred to as interkinetic nuclear migration (IKNM) (Meyer et al., 2011). Consequently, nuclei are distributed along the entire apical-basal axis, giving the tissue a pseudostratified configuration (Norden, 2017). We wondered to what extent the nuclear distribution, and its effect on the 3D cell shape, explains the observed area distributions and lateral T1 transitions.

To this end, we stained the nuclear envelope with fluorescently tagged antibodies against lamin B1 (Figure 5a, Figure 5—figure supplement 1), and 3D segmented all nuclei within epithelial cells in a tube segment (Figure 5b, Figure 3—figure supplement 1, Figure 3—video 2). The nuclei were distributed along the entire apical-basal axis (Figure 5c), and consistent with the pseudostratified appearance of the epithelium, nuclei in neighbouring cells had different positions along the apical-basal axis (Figure 5b). The nuclear shapes, volumes, and cross-sectional areas (Figure 5d–f) all varied along the apical-basal axis. As expected, nuclei are largest and most spherical at the apical side, where they undergo mitosis (Figure 5d). Thus, a one-sided, two-sample Welch t-test revealed a significantly reduced ellipticity of nuclei located in the first 25% of the apical-basal axis compared to those in the middle 50% (p=0.0002). The nuclear volumes are on average about 50% smaller than the cell volumes and largely correlate (r = 0.79, Figure 5f), likely reflecting parallel expansion during the cell cycle. Where present, the nuclear cross-sectional areas are only slightly smaller than those of the entire cell, and the cross-sectional areas of the cell and the nucleus are strongly correlated (r = 0.94, Figure 5g). The strong correlation can be accounted for by the opposing actions of cells and nuclei in the columnar epithelium. The nuclear volumes are too large to allow for a spherical nucleus to fit into a cylindrical cell of the measured height (Figure 5h). Accordingly, to fit into the cell, the nucleus necessarily has to deform. Nuclei respond to external forces with anisotropic shape changes (Haase et al., 2016; Neelam et al., 2016), which is consistent with the elliptical nuclear shapes that we observe (Figure 5d). However, there is a limit to how much the stiff nucleus can deform (Lammerding, 2011; Shah et al., 2021), resulting in a local widening of the cell where the nucleus is present. Cell sections without nucleus typically have smaller cross-sectional areas, thereby leading to a higher frequency of small cross-sections in cells compared to nuclei. Accordingly, as previously seen for the cell cross-sectional areas, the observed changes in cell neighbour numbers correlates with the observed changes in nuclear cross-sectional areas (Figure 5i) such that most T1L transitions occur where the nucleus starts and ends (Figure 5j).

Figure 5 with 2 supplements see all
Changes in cross-sectional area as a result of interkinetic nuclear migration (IKNM).

(a) Light-sheet microscopy longitudinal sections of an E12.5 CUBIC-cleared lung tube carrying the ShhGC/+; ROSAmT/mG reporter allele (green epithelium) and immunostained for lamin B1 (blue nuclear envelopes). Morphometric quantifications of 3D iso-surfaces (N=140) and cell segmentations along the apical-basal axis were used to study the nature of cross-sectional area variation and the effect of IKNM (Figure 3—figure supplement 1). Scale bar 20 µm. (b) Sequential cell membrane contour surfaces and nuclear iso-surfaces for six epithelial cells. By interpolating between contours and creating iso-surfaces, 3D shapes can be accurately extracted (Figure 3—video 2 and Figure 5—video 1). Scale bar 7 µm. (c) Distribution of nuclei center of mass along the apical-basal axis. (d) Nuclear ellipticity and volume distributions along the apical-basal axis. (e) Average cross-sectional area distribution along the apical-basal axis for all cells (black) and nuclei (blue). (f). Nuclear and cellular volumes of 140 segmented cells are correlated (r=0.79). Line of best fit in red. (g). The cell and corresponding nuclear cross-sectional areas along the apical-basal axis are highly correlated (r=0.94). (h). Diameters of nuclei and cells based on the measured nuclear and cell volumes if nuclei were perfect spheres, and cells were perfect cylinders of the measured height. Given the larger nuclear diameter, nuclei must deform in order to fit within cells. (i) An increase in the nuclear cross-sectional area increases the frequency of a neighbour-number-increasing spatial T1 transitions, and vice versa. (j) The largest number of changes in neighbour relationships occur at the apical and basal limits of the nucleus for all cells, where cross-sectional areas change sharply.

We conclude that the positions of nuclei can explain much of the observed variability in the cross-sectional cell areas. During the cell cycle, nuclei migrate, and the cell volumes first increase, and subsequently halve due to cell division. As all these processes affect the cross-sectional areas of the cells along the apical-basal axis, one would expect continuous spatial-temporal T1L transitions in growing pseudostratified epithelia.

3D cell organisation in growing epithelia

To follow 3D cellular dynamics during epithelial growth and deformation, we cultured embryonic lungs from a ShhGC/+; ROSAmT/mG background and imaged every 20 min for a total of 10 hr using light-sheet microscopy (Figure 6—video 1). We used a subset of this dataset (11 time points, >3 hr) (Figure 6—video 2) to 2.5D segment the apical and basal surfaces, and to explore 3D cell shape dynamics and neighbour relationships in a growing lung bud (Figure 6a, Figure 6—video 3).

Figure 6 with 4 supplements see all
Dynamics of apical and basal epithelial organisation.

(a) Timelapse light-sheet microscopy series of a cultured mouse E12.5 distal lung bud expressing the ShhGC/+; ROSAmT/mG reporter (green epithelium, and red mesenchyme), imaged every 20 min (11 time steps). The white inset denotes the morphology of the lung bud, while the dotted area denotes the segmented cell patch. Corresponding visual provided in Figure 6—video 1. Cells on both the apical and basal domains were 2.5D segmented, and their morphology quantified. Corresponding visual provided in Figure 6—video 3. Scale bars 20 µm. (b) Cell neighbour frequencies for the apical and basal layers over time. (c) Observed average neighbour number and area coefficient of variation (CV) for the apical and basal layers over time. (d) Growing apical and basal layers follow the AW law (black line). Colour code applies to e-f. (e) The relative average apical and basal cell areas are linearly related to the number of neighbours (in all time points) and follow Lewis’ law (black line), or the quadratic relationship in the case of higher area variability (yellow line). (f) Observed fraction of hexagons versus area coefficient of variation (CV) on the apical and basal layers. The lines mark the theoretical prediction if polygonal cell layers follow either the linear Lewis’ law (black line) or the quadratic law (yellow line).

As the explant was growing, we readjusted the 2.5D segmented region such that the segmented surface area and cell numbers remained roughly constant over time (Figure 6—figure supplement 1a). Nonetheless, the segmented bud increased in volume as the thickness of the layer increased with time (Figure 6—figure supplement 1b). Much as in the static dataset, the neighbour number distributions (Figure 6b), and variability of cross-sectional areas (Figure 6c) differ between the apical and basal cell layers in all time points. However, for all time points, both the apical and basal layers conformed to Euler's polyhedron formula (Figure 6c), Aboav-Weaire's law (Figure 6d), and Lewis’ law (Figure 6e). Furthermore, the fraction of hexagons also followed from the variability of the cross-sectional areas, as predicted by the theory (Figure 6f).

We next sought to analyse the 3D dynamics of segmented epithelial cells. As the tracking of packed cells in growing pseudostratified epithelia is challenging, we focused on a small patch with 15 cells in total (Figure 7a). Sequential contour surfaces were drawn to follow cell membrane outlines on several planes along the apical-basal axis and interpolated to reconstruct 3D morphology for each time point (Figure 7—figure supplement 1, Figure 7—video 1). All planar segmentations were then pooled into five groups along the apical-basal axis to enable morphometric analysis in different tissue regions. Over the time course, the volume of individual cells varied between roughly 400 and 800 µm3 (Figure 7b), and the apical-basal length varied between roughly 20 and 30 µm (Figure 7c). We note that all layers conform to Aboav-Weaire's law (Figure 7d) and Lewis’ law (Figure 7e) in all time points. Moreover, consistent with our theory, the fraction of hexagons follows from the variability of the cross-sectional area, though more deviations are observed, given the small number of cells analysed (Figure 7f).

Figure 7 with 2 supplements see all
3D cell organisation in growing epithelia.

(a) 3D segmentation of 15 epithelial cells from timelapse light-sheet microscopy imaging of a mouse E12.5 distal lung bud expressing the ShhGC/+; ROSAmT/mG reporter. The specimen was imaged every 20 min over 3 hr. Planar segmentations along the apical-basal axis were pooled into five groups to enable morphometric analysis in different tissue regions. A full timelapse panel is provided in Figure 7—figure supplement 1 and Figure 7—video 1. Scale bar 30 µm. (b) Epithelial cell volume, and (c) height over time (N=15). (d) Segmented cells in pooled layers along the apical-basal axis follow the AW law (black line) over time (left to right); see panel f for colour code. (e) The relative average cell area in each layer is linearly related to the number of neighbours for all time points (left to right) and follows Lewis’ law (black line), or the quadratic relationship in the case of higher area variability (yellow line); see panel f for colour code. (f) Temporal dynamics of observed fraction of hexagons versus area coefficient of variation (CV) along the apical-basal axis. Dotted lines denote variation per time point. Solid lines mark the theoretical prediction if polygonal cell layers follow either the linear Lewis’ law (black line) or the quadratic law (yellow line).

Much as in the static dataset (Figures 3 and 4), we observe up to 14 neighbour number changes (T1L transitions) along the apical-basal axis (Figure 8a,b). The average number of T1L transitions is relatively constant over time (Figure 8b). The mean relative apical-basal position for T1L transitions is again roughly in the middle, but in this small dataset, we now observe more T1L transitions in the center of the cell than at the apical or basal boundaries (Figure 8c). By following a single cell over time, we can appreciate the dynamic cell shape changes, and how a change in the cross-sectional area correlates with a change in neighbour number (Figure 8d). The neighbour relationships are, of course, not determined by the local cell cross-section, but by the overall cross-sectional area distribution in that layer. Accordingly, the correlation between the cross-sectional area and the neighbour number is not perfect for a single cell. By considering a patch of cells, we can, however, see how those T1L transitions occur dynamically in developing tissues (Figure 8e).

3D cell neighbour dynamics in growing epithelia.

(a) Apical, basal and lateral cross-sections from a light-sheet microscopy timelapse of a murine E12.5 distal lung bud expressing the ShhGC/+; ROSAmT/mG reporter. The specimen was imaged every 20 min over 3 hr. Cell membrane outlines illustrate fluid cell neighbour relationships along the apical-basal axis and over time. T1 transitions are marked with white stars; see panel e for cell colour code. Scale bar 14 µm. (b) Number of T1L transitions for all cells (N=15) over time. Diamonds represent the mean. Morphometric quantifications of planar segmentations along the apical-basal axis were used to examine T1L transition dynamics. (c) Spatial distribution along the apical-basal axis of T1L transitions for all cells and time points. (d) Temporal evolution of neighbour relationships along the apical-basal axis for a single cell. Schematic cell width corresponds to cross-sectional area. (e) (top row) 3D iso-surface segmentations of 15 epithelial cells. Scale bar 10 µm. (bottom row) Cross-sectional area and cell neighbour number along the apical-basal axis for a given cell (marked with a white star). Dotted lines indicate contact with a cell that was not segmented.

Discussion

Epithelial tissues remodel into complex geometries during morphogenesis. We used light-sheet microscopy and 3D cell segmentation to unravel the physical principles that define the 3D cell neighbour relationships in pseudostratified epithelial tissues. Our analysis reveals that pseudostratified epithelial layers adopt a far more complex packing solution than previously anticipated: the 3D epithelial cell shapes are highly irregular, and cell neighbour relationships change multiple times along the apical-basal axis, with some cells having up to 14 changes in their neighbour contacts along their apical-basal axis (Figure 3a). Curvature effects can result in neighbour changes, but the data does not show the dependency on cell neighbour numbers that would be expected if curvature effects played a dominating role (Figure 3g). There is also no apical-basal bias (Figure 3e), and the prevalence of contact remodelling is randomly distributed (Figure 4i).

Even though the neighbour relationships are uncorrelated between the apical and basal sides and appear random at first sight, they follow the same fundamental relationships that have previously been described for apical epithelial layers, that is, Euler's polyhedron formula, Lewis' law, and Aboav-Weaire's law across the entire tissue and at all times (Figures 2, 4, 6 and 7). This arrangement minimises the lateral cell-cell surface energy in each plane along the apical-basal axis, given the variability in the cell cross-sectional areas (Figure 1; Kokic et al., 2019; Vetter et al., 2019). Where present, the stiff nucleus determines the cell cross-sectional area, as is apparent from the strong correlation between the cell cross-sectional and the nuclear cross-sectional areas (Figure 5g). Accordingly, most changes in neighbour relationships occur at the apical and basal limits of the nucleus where cross-sectional areas change sharply (Figure 5i). As the nucleus moves along the apical-basal axis during the cell cycle, a process referred to as interkinetic nuclear migration (IKNM) (Meyer et al., 2011), cell neighbour relationships change continuously (Figure 8). We conclude that neighbour relationships in epithelia are fluidic, and the complex, dynamic 3D organisation of cells in growing epithelia follows simple physical principles.

Defining the physical principles behind cell neighbour relationships is only the first step in unravelling the determinants of epithelial 3D cell shapes. The second key aspect is the cell volume distribution along the apical-basal axis, which gives rise to the cell cross-sectional area distribution, which then determines the cell neighbour relationships (Figure 5e). The overall cell volume is determined by cell growth and division, but its distribution along the apical-basal axis depends on the nuclear dynamics (Figure 5g), and the epithelial cell heights. We find that the nucleus occupies, on average, 55% of the cell volume in the embryonic lung epithelia. As the cell nuclei move along the apical-basal axis during the cell cycle (Meyer et al., 2011), the cytoplasm fills the remaining space between the apical and basal surfaces, likely in a way that minimises the total surface area of all cells. The determinants of the epithelial thickness, that is, the distance between the apical and basal surfaces are still unknown, but signalling downstream of Fibroblastic Growth Factor (FGF), Sonic Hedgehog (SHH), Bone Morphogenetic Protein (BMP)/ transforming growth factor-beta (TGF-β), and WNT has been observed to affect cell height, presumably via an effect on cell tension and/or cell-cell adhesion (Gritli-Linde et al., 2002; Hirashima and Matsuda, 2021; Kadzik et al., 2014; Kondo and Hayashi, 2015; Widmann and Dahmann, 2009).

Cell-based modelling frameworks are heavily used to investigate epithelial processes and how they result in morphological changes such as tissue bending, folding, fusion, and anisotropic growth during morphogenesis (Fletcher et al., 2014; Tanaka, 2015). Our data confirms many underlying assumptions of cell-based modelling frameworks and provides quantitative data to calibrate parameters. Once calibrated to reproduce the here identified 3D cell shape distributions, such simulation frameworks will help to reveal the determinants of 3D cell shapes, and will be invaluable in providing insight into how local changes in cell growth, adhesion, tension, or in the basal lamina affect cell shapes locally and within the remaining epithelial layer.

In summary, this study offers a detailed view of 3D cell neighbour relationship dynamics and packing in growing epithelial tissues, and demonstrates that the 3D cell shapes are much more complex than previously anticipated, and that cell neighbour relationships are dynamic and change as result of cell growth and cell cycle-linked IKNM. The complex 3D cell neighbour relationships can nonetheless be understood based on simple physical principles. Although we recognize that tissue architecture is a multifactorial process, our work carries vast implications for the study of cell-cell signalling, epithelial cohesion, and energetic modelling of developing epithelial layers in both healthy and disease contexts.

Materials and methods

Ethical statement

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Permission to use animals was obtained from the veterinary office of the Canton Basel-Stadt (license number 2777/26711). Experimental procedures were performed in accordance with the Guide for the Care and Use of Laboratory Animals and approved by the Ethics Committee for Animal Care of ETH Zurich. All animals were housed at the D-BSSE/UniBasel facility under standard water, chow, enrichment, and 12 hr light/dark cycles.

Animals

To investigate 3D cellular dynamics during mouse embryonic lung development, we used mouse lung rudiments from animals homozygous for the Rosa26mTmG and heterozygous for the ShhGFP-Cre allele (ShhGFP-Cre/+; Rosa26mTmG). The double-fluorescent Shh-controlled Cre reporter mouse expresses membrane-targeted tandem dimer Tomato (mT) before CRE-mediated excision and membrane-targeted green fluorescent protein (mG) after excision (Muzumdar et al., 2007). As a result, only epithelial cell membranes are labelled by GFP, while all adjacent mesenchymal tissue is labelled by tdTomato.

Immunofluorescence

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E12.5 mouse lungs were fixated for 1 hr in 4% paraformaldehyde in PBS, and subsequently incubated with Lamin B1 (Thermo; Material No. 702972; 1:200) at 4°C for 3 days. As a structural component of the nuclear lamina, LaminB1 immunostaining makes crowded nuclei clearly distinguishable and easily segmentable. After washing in D-PBS, lungs were incubated with conjugated fluorescent secondary Alexa Fluor 555 donkey anti-mouse IgG (H+L) (Abcam; Material No. ab150106; 1:250) for 2 days at 4°C.

Optical clearing and lightsheet imaging

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Optical clearing of embryonic lung rudiments enabled the 3D segmentation of numerous epithelial cells from single image stacks. To this extent, the whole-mount clearing of dissected E12.5 lung explants was performed with the Clear Unobstructed Brain/Body Imaging Cocktails and Computational Analysis (CUBIC) protocol (Susaki et al., 2015; Figure 2—figure supplement 3). Reagents for delipidation and refractive index (RI) matching were made as follows: CUBIC-1 [25% (w/w) urea, 25% ethylenediamine, 15% (w/w) Triton X-100 in distilled water], and CUBIC-2 [25% (w/w) urea, 50% (w/w) sucrose, 10% (w/w) nitrilotriethanol in distilled water], respectively. Following fixation and immunostaining, samples were incubated in 1/2 CUBIC-1 (CUBIC-1:H2O=1:1) for four days, and in 1X CUBIC-1 until they became transparent. All explants were subsequently washed several times in PBS and treated with 1/2 CUBIC-2 (CUBIC-2:PBS=1:1) for around four days. Lastly, incubation in 1X CUBIC-2 was done until the desired transparency was achieved. All solutions were changed daily, and CUBIC-1 steps were performed on a shaker at 37°C while CUBIC-2 steps at room temperature. Cleared samples were then embedded in 2% low melting point solid agarose cylinders and immersed in CUBIC-2 for two more days to increase the agarose refractive index. 3D image stacks were acquired on a Zeiss Lightsheet Z.1 microscope using a Zeiss 20x/1.0 clearing objective .

Timelapse light-sheet acquisitions

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Light-sheet acquisitions of live epithelial cell morphology enabled the study of 3D organisation dynamics. Following dissection in DPBS at room temperature, E12.5 lung explants were cultured in sterile Dulbecco's modified Eagle's medium w/o phenol red (DMEM) (Life Technologies Europe BV; 11039021) containing 10% Fetal Bovine Serum (FBS) (Sigma-Aldrich Chemie GmbH; F9665-500ML), 1% Glutamax (Life Technologies Europe BV; A1286001), and 1% penicillin/streptomycin (Life Technologies Europe BV; 10378–016). All specimens were equilibrated at 37°C with 5% CO2 in a humidified incubator for 1 hr.

Following a 1 hr equilibration period, 1.5% LMP hollow agarose cylinders were prepared (Udan et al., 2014). Hollow cylinders, in contrast to solid ones, accommodate unencumbered 3D embryonic growth, provide boundaries to minimise tissue drift, enable imaging from multiple orientations, and allow for better perfusion of gasses and nutrients. All specimens were suspended within each hollow cylinder in undiluted Matrigel (VWR International GmbH; 734–1101), an ECM‐based optically clear hydrogel that provided a near-native 3D environment and supported cell growth and survival. All cylinders were kept at 37°C with 5% CO2 in culture media for 1 hr before mounting.

For each overnight culture, the imaging chamber was prepared by sonication at 80°C, followed by ethanol and sterile PBS washes. After assembly, the chamber was filled with culture medium and allowed to equilibrate at 37°C with 5% CO2 for at least 2 hr before a cylinder containing an explant was mounted for imaging. Furthermore, to compensate for evaporation and to maintain a fresh culture media environment, two peristaltic pumps were installed to supply 0.4 mL and extract 0.2 mL of culture medium per hour. Each lung explant was then aligned with the focal plane within the center of a thin light-sheet to enable fine optical sectioning with optimal lateral resolution. For this study, all live imaging was done with a 20x/1.0 Plan-APO water immersion objective.

Image processing

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To efficiently process the resulting volumetric CZI datasets (10s-100s of GBs), all image stacks were transferred to a storage server and subsequently processed in remote workstations (Intel Xeon CPU E5-2650 with 512 GB memory). Deconvolution via Huygens Professional v19.04 (Scientific Volume Imaging, The Netherlands, http://svi.nl) improved overall contrast and resolution while Fiji (ImageJ v1.52t) (Schindelin et al., 2012) aided in accentuating cell membranes, enhancing local contrast, removing background fluorescence, and TIFF conversion.

Cell morphometric quantifications

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Cell morphology on the apical and basal membranes of embryonic lung epithelia was investigated using the open-source software platform MorphoGraphX (MGX) (Barbier de Reuille et al., 2015). By meshing the curved boundaries of input 3D image stacks and projecting nearby signal onto it, MGX builds a curved 2.5D image projection that is distortion-free, unlike planar 2D projections that ignore curvature. We then proceeded to use a suitable implementation of the Watershed transform to extract individual cell geometries, with minimal manual curation, and quantify properties such as surface area and the number of cell neighbours. All border cells were excluded. Apical and basal cell meshes were exported as text files and traversed using the R Programming Environment to extract the neighbour relationships between cells as needed to generate Aboav-Weaire plots (Gómez et al., 2021).

To render time-lapse datasets and extract 3D volumetric surface reconstructions of entire epithelial cells, we employed Imaris v9.1.2 (Bitplane, South Windsor, CT, USA). By computationally interpolating between cell membrane contour surfaces from successive transverse frames into iso-surfaces, faithful cell and nuclear 3D volumes were obtained. Quantified volumetric features included cell and nuclear volume, total surface area, sphericity, and nuclear position along the apical-basal axis. Imaris was also used to generate high-resolution videos, which, despite being strongly downsampled to accommodate vast time-lapse datasets, presented little noticeable loss in image quality. Furthermore, to extract cell areas and the number of neighbours along the apical-basal axis, transverse image frames were imported into ImageJ and processed using the interactive plugin TissueAnalyzer (Aigouy et al., 2016). Like this, cell segmentation masks across layers could be generated, and cell geometry and neighbour topology quantified (Gómez et al., 2021).

Appendix 1

A simple theory for lateral T1 transitions in curved tissues

Drawing inspiration from the notion of ‘scutoids’ (Gómez-Gálvez et al., 2018), we develop here a simple geometrical theory to estimate the effect of tissue curvature, if any, on the occurrence and number of lateral T1 transitions of cells along the apical-basal axis in a pseudostratified epithelium. The purpose of this effort is to verify whether an idea brought forward in the aforementioned article, which is that tissue curvature could be responsible for the occurrence of up to one lateral T1 transition per cell between its apical and basal sides, is consistent with our observations in the developing, pseudostratified mouse lung epithelium.

We start by considering, like the authors of Gómez-Gálvez et al., 2018, the vicinity of an edge shared by two adjacent cells in a two-dimensional cross section of the tissue perpendicular to the apical-basal axis. This vicinity is defined as the quadrilateral (Appendix 1—figure 1A, orange) spanned by the four cell junctions (green) that are directly linked to the edge of interest (blue), covering a fraction of four cells in such a two-dimensional projection. As the projection plane is moved along the apical-basal axis, the edge length can shrink to zero and reappear with different orientation, leaving the cell neighbourhood of all four cells modified by one. It is this process which we refer to as lateral neighbourhood transition of type T1. Each such edge vicinity is characterised by an aspect ratio ϵ=w/h, where w=(w1+w2)/2 is the average width and h=(h1+h2)/2 the average height. We now proceed to quantitatively estimate with a simple geometrical model how a change in tissue curvature translates into a change of aspect ratio of such motifs, and how that in turn would be expected to change the cell neighbourhood if all T1 transitions were effectively governed by the geometric effect of changing tissue curvature.

Appendix 1—figure 1
Definition of the vicinity of a cell edge in a geometric projection on a tubular epithelium.

We consider two different tissue topologies: (i) a cylindrical tube and (ii) a spherical vesicle or hemispherical tube cap. The crucial difference between these two cases is that the former only has a mean curvature but no Gaussian curvature, whereas the latter has both. As the radius r of these surfaces changes, the change of aspect ratio follows by the quotient rule of calculus as

(5) ϵr=(w/h)r=1h(wr-ϵhr).

We now consider the general case where the edge motif is arbitrarily oriented on the surface, and will later average over all possible orientations. In general, the edge of interest spans an angle θ with the first principal curvilinear axis of the surface, which we take as the main cylinder axis for the tubular epithelium (Appendix 1—figure 1), and as any great circle on the spherical epithelium. The change of width and height then read

(6) wr=cosθwxr+sinθwyr,hr=sinθhxr+cosθhyr,

where wx, wy and hx, hy are the components of the width and height vectors, respectively (Appendix 1—figure 1B, gray):

(7) wx=wcosθ,wy=wsinθ,hx=hsinθ,hy=hcosθ.

Their response to a change in tissue radius is given by

(8) wxr=κ1wx,wyr=κ2wy,hxr=κ1hx,hyr=κ2hy,

where κ1 and κ2 are the principal curvatures of the surface in the respective directions. Substituting Equation 8 into Equations 7, 6 and 5 yields

ϵr=ϵ(κ1-κ2)cos(2θ).

A cylinder with radius r has the principal curvatures κ1=0 and κ2=1/r, so

ϵr=-ϵrcos(2θ).

For a sphere with radius r, on the other hand, the two principal curvatures are both equal, κ1=κ2=1/r, and thus the aspect ratio does not change with varying radius,

ϵr=0

independent of the orientation θ.

From here onward, we consider the region along the apical-basal axis between two T1 transitions on a given cell, irrespective of whether these T1 transitions increase or decrease the neighbour number. In our experimental quantifications of lateral T1 transition in the mouse lung epithelium, they were counted in the same way. The sign of the T1 transition is encoded in the sign of ϵ/r, hence we consider only the absolute value |ϵ/r| hereafter, to quantify the change of edge aspect ratio between T1 transitions along the tissue radius.

Carrying on with the tubular geometry, we now average over all orientations θ to calculate the mean change of aspect ratio for the entire cell ensemble in the tissue. The set of possible orientations is θ[0,π/2), because all angles θ[π/2,π) just switch the roles of w and h. Therefore,

|ϵr|=2π0π/2|ϵrcos(2θ)|dθ=2ϵπr.

This is an ordinary differential equation which can easily be solved by separation of variables. The result is

ϵ(r)=ϵ1(R1r)±2/π

and therefore

(9) |ϵr|=2ϵ1πr(R1r)±2/π

where ϵ1 is the aspect ratio at a given cylinder radius R1. For the sphere, on the other hand, we trivially have a constant aspect ratio ϵ(r)=ϵ1.

To link this change of aspect ratio with lateral T1 transitions, a relationship between ϵ and the number of neighbours a cell locally has, n, needs to be established. A simple rough estimate can be found by recalling that cells typically tend toward a regular polygonal shape in a cross-sectional projection (Kokic et al., 2019; Vetter et al., 2019). In a honeycomb lattice, the aspect ratio is

ϵ|n=6=231.155.

For general regular polygons with n4, the relationship reads

(10) ϵ(n)=1+2cos(2π/n)2sin(2π/n)

as can be recognised from Appendix 1—figure 1B. Equation 10 holds locally in the region between two consecutive T1 transitions, where n is constant. From this, we can estimate the change of aspect ratio between two lateral T1 transition as

(11) ϵn=π2+cos(2π/n)n2sin2(2π/n).

On average, for hexagonal cells (n=6), this evaluates to

ϵn|n=6=5π540.291.

So how many T1 transitions can be expected along the apical-basal axis? We can apply the chain rule of calculus to write 

(12) |dndr|=|nϵϵr|=|(ϵn)1ϵr|.

Substituting Equations 9 and 11 into Equation 12, we find

|dndr|=n2sin2(2π/n)2+cos(2π/n)2ϵ1π2r(R1r)±2/π

for a cylindrical epithelium, and dn/dr=0 for a spherical one. A spherical topology is thus entirely compatible with cells being frusta that do not change their neighbour number from their apical to the basal side, whereas a cylindrical topology is not. For tubes, a change of radius R1R2 lets the neighbour number change by

|Δn|=R1R2|dndr|dr=ϵ1πn2sin2(2π/n)2+cos(2π/n)[1(R1R2)±2/π]

per cell edge. How far must R1 and R2 be apart to expect one T1 transition per cell? Over that distance, each of the cell’s n edges will have undergone a change of 1/n on average. The answer can therefore be found by setting |Δn|=1/n with ϵ1=ϵ(n) from Equation 10, and solving for the radius fold change:

(13) R2R1=(1-ϵ(n)nϵ(n))±π/2=(1-2π[2+cos(2π/n)]n2[sin(2π/n)+sin(4π/n)])±π/2

This is the expected fold change of mean curvature inside a tubular epithelium between two consecutive lateral T1 transitions per cell, if the T1 transitions were mainly driven by a change of curvature. Note that the two cases with different signs in the exponent reflect the R1R2 symmetry. As shown in Appendix 1—figure 2, Equation 13 predicts a strong dependency of the required change of curvature between adjacent lateral T1 transitions on the number of neighbours the cell has, n. Cells with a large neighbour number are expected to transition at much smaller changes of curvature on average than cells with few neighbours.

Appendix 1—figure 2
Predicted radius fold change in a tubular epithelium between a pair or consecutive T1 transitions per cell.

Equation 13 relates the fold change of tissue curvature on a tubular sheet between consecutive lateral T1 transitions along the apical-basal axis of a single cell, to the cell’s neighbour number n in any cross section perpendicular to the radius between the two transitions. n here is to be understood locally within the tissue, not as the cell connectivity on either the apical or basal surfaces of the cell. Equation 13 does not directly lend itself to estimating the number of lateral T1 transitions to be expected for cells in a curved tissue, although our theory might be adaptable to this case with some modifications. The sign of T1 transitions is ignored here, leaving the possibility of frequent back-and-forth transitions with zero net effect on n—a phenomenon that we indeed do observe in the developing lung epithelium. Equation 13 quantifies only the net contribution of curvature on the number of T1 transitions a cell undergoes. This net effect strongly depends on the the local neighbour number n, as shown in Appendix 1—figure 2.

Data availability

The source code and plotted data files are available as a git repository at https://git.bsse.ethz.ch/iber/Publications/2021_gomez_3d_cell_neighbour_dynamics.git. The raw data is publicly available as openBIS repository at https://openbis-data-repo.ethz.ch/openbis/webapp/eln-lims/?user=observer&pass=openbis under the name 3D Epithelium.

The following data sets were generated

References

Decision letter

  1. Anna Akhmanova
    Senior and Reviewing Editor; Utrecht University, Netherlands
  2. Matthias Merkel
    Reviewer; Syracuse University, United States

Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.

Acceptance summary:

In this study, the authors measured the three-dimensional organization within an epithelial cell monolayer and found that cell neighbors change frequently along the apicobasal axis. This work constitutes a significant investigation into the drivers of complex three-dimensional cell shapes and tissue structures and will be of interest to scientists studying tissue mechanics.

Decision letter after peer review:

Thank you for submitting your article "3D Cell Neighbour Dynamics in Growing Pseudostratified Epithelia" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, one of whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by Anna Akhmanova as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Matthias Merkel (Reviewer #3).

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1) Claims that imply causation but are based on correlations need to be either substantiated further to justify causation or toned down. Reporting only correlations might not suffice for publication in eLife.

2) Solve the issues spotted by Reviewer #3 in the theory presented in the SM (reviewer's point 4).

3) Estimate uncertainties as requested by Reviewer #3, point 3.

4) Respond to the recommendations for the authors.

Reviewer #1 (Recommendations for the authors):

L 60: Eq. (2) is wrong and should read instead 5+8/n.

Figure 1: I did not understand, why the authors reproduce the previously published data in Figure 1e-k. Figure 1d does not seem to be referred to in the text.

Figure 2f: It is hard to distinguish between linear and quadratic dependences.

Figure 3e: For tips there seems a clear bias for apical transitions. Could you comment on this?

Ll 165: I did not understand this part. Why do you fit ellipses and not circles? Why do you consider two subsequent T1L transitions/consecutive neighbor number changes and not one?

L 211: This statement contradicts later statements that involve the nucleus. Also, here you really write about minimizing a line energy, whereas it should be a surface energy; the lateral and the apico-basal directions are not independent.

L 233: What do you mean by "pseudo-stratification"? Where is it shown?

Ll 241+ Figure 5d: There seems to be only a very weak correlation between nuclear volume and ellipticity or position along the apico-basal axis.

Ll 244 + Figure 5f,g: Since the nuclei are inside the cell, could it be otherwise? Also, the correlation between cell and nuclear volume seems to be rather elevated. Is this correct?

Ll 246: But it could also be the other way around or some independent factor could determine both. I do not think that the consistency you notice has any explanatory power.

Figure 6: Are the results based on one explant?

L 317: "Where present, the stiff nucleus determines the cell cross-sectional area, as is apparent from the strong correlation between the cell cross-sectional and the nuclear cross-sectional areas" Correlations do not imply causation. I do not think that you have data to support the causal relationship.

L 320: Again, this seems to be speculation. Figure 8 does not show the nucleus – and even if: from your the data it is not clear that the motion of the nucleus causes changes in the neighbor relationships.

Reviewer #2 (Recommendations for the authors):

Page 4, line 73, "following, Aboav-Weaire's law": I suggest replacing this with "following Aboav-Weaire's law" (i.e. remove the comma).

Page 4, lines 89-90, "As such, growth and cell division determine the variability of the apical areas" and lines 100-101, "The cross-sectional areas vary as a result of cell growth, division and interkinetic nuclear migration (IKNM)": Maybe I missed it, but can the authors discount any role(s) for 'active' cell intercalations and/or cell death here?

Page 4, lines 97-98, "the variation of […] define the": I suggest replacing this with "the variation of […] axis defines the" (i.e. add an 's').

Page 5, lines 112-113, "We detected less than half as many cells on the apical side": Just to be clear here, do the authors mean that they could only detect half as many cells on the apical side due to their small size and/or lower resolution, or that there are really half as many cells on the apical side and hence over half the cells don't extend fully from the basal side to the apical side?

Page 5, lines 118-119, "the neighbour changes in spherical geometries cannot be explained with such an effect (Gómez-Gálvez et al., 2018)": For clarity, I suggest briefly emphasizing why not (I guess the point is that as one looks along the apical-basal axis in this case, one is just seeing a dilation, so prismatic cells do the job fine).

Page 5, lines 127-130, "Aboav-Weaire's law (Figure 2e) and Lewis' law (Figure 2f) hold not only for the apical, but also for the basal datasets […] the apical layers, which have a larger area variability than the basal layers (Figure 2c), follow the quadratic law (yellow line) rather than the linear Lewis' law (black line)": I may simply have misunderstood things here, but the first sentence seems to indicate that Lewis' law holds for the apical dataset, while the second sentence seems to indicate that the apical layers follow the quadratic law rather than Lewis' law, which seems a contradiction. If I have indeed just misunderstood things here, then perhaps the authors could slightly rephrase these sentences for clarity.

Page 7, lines 232-233, "giving the tissue a pseudostratified configuration": I suppose strictly speaking this is true of pseudostratified epithelia but not ALL epithelia, so it might be worth changing "In epithelia" to "In many epithelia" on line 227, or similar.

Page 9 lines 335-336, "signalling factors that control cellular tension are known to affect cell height (Widmann and Dahmann, 2009)": For clarity, perhaps the authors could name these signalling factor(s) explicitly here?

Page 16, Figure 1, panel (b): The chosen colour scheme makes it very difficult for me to tell the difference between 5, 6, 7 and 8 sided cells, since they're all shades of green. If possible, it would be great if the authors could use more contrasting colours.

Page 16, Figure 1, panel (e): Can the authors briefly explain in the legend what the various abbreviations mean (cNT, EYE, dPE, dWL, dWP, dMWP, dPW, TP, dMWL), or at least broadly which different tissues/species are considered?

Page 19, Figure 3, panel (b): I found this figure difficult to understand partly due to the colour scheme but also because the authors are plotting two datasets on the same image. If there is a way of simplifying this figure at all, that would be great.

Reviewer #3 (Recommendations for the authors):

– Citation of Rupprecht et al., MBoC, 2017 appears justified, as it is the first / one of the first papers to discuss apico-basal T1 transitions in epithelia based on experimental data.

– Recommendations regarding point 3. in the public review:

a) Estimations of uncertainties (e.g. standard error of the mean) would be useful, e.g. in Figure 2 e,f. For instance, in Figure 2f, an indication for the stochastic uncertainty due to the finite sample size would be useful.

b) Aboav-Weaire law: For instance in Figure 2e, everything is close to a diagonal line. However, that there is a line with slope of approximately 6 in the plots of m_n*n over n is already a consequence of the Euler characteristic. To better display the correlation that is the core of the Aboav-Weaire law, I would suggest to directly plot m_n over n (as also one e.g. in Aboav, 1970). Together with error bars indicating uncertainty, this would allow to better scrutinize whether the different data sets fulfill the prediction.

c) Some plots have very similar colors and are therefore hard to distinguish, e.g. the quadratic theory curve vs the basal data in Figure 2f. Also, in Figures 4f and 8e, the polygon type (number of neighbors n) is essential, but the colors in particular for n>5 are very hard to distinguish.

– Suggestions regarding point 4 of the public review (merely ideas):

a) As pointed out by Gómez-Gálvez et al., the curvature-based mechanism to create T1L transitions is due to a change in the aspect ratio of the 2D tissue cross section as one goes from apical to basal in an e.g. cylindrical tissue. One way to estimate the T1L rate that results from this might also be to use existing formalisms to understand how many T1 transitions are needed to create a certain change in 2D tissue aspect ratio (e.g. the texture tensor formalism, Graner et al., EPJE, 2008 and Guirao et al., eLife, 2015, or a triangle formalism, Merkel et al., PRE, 2017).

b) An interesting question could be how oriented the T1L transitions are ("How many of them do (net) contribute to the apico-basal change in 2D tissue aspect ratio?"), which one could address using orientational statistics of the T1 transitions (i.e. how uniform is the T1L orientation angle in the unfolded cylinder mantle).

c) Just a remark: I am happy to be convinced otherwise, but I would be surprised if one could indeed derive a dependence of the T1L transition number on cell neighbor number solely on geometric grounds (i.e. disregarding mechanics etc). This is because the curvature-based mechanism is about T1L transitions creating anisotropic deformation in the 2D cross section, while the cell neighbor number is a scalar (i.e. an "isotropic" quantity). I could be wrong here, but I would presume that any correlation between the two will depend on model details.

https://doi.org/10.7554/eLife.68135.sa1

Author response

Essential revisions:

1) Claims that imply causation but are based on correlations need to be either substantiated further to justify causation or toned down. Reporting only correlations might not suffice for publication in eLife.

As we emphasize also in the title of this paper, we can explain the observed 3D cell neighbour relationships with a minimisation of the lateral cell-cell surface contact energy. It is important to distinguish this from questions regarding the 3D cell shape, which we do not address. The confusion easily arises because our theory explains the polygon type and thus the shape of cross-sectional areas, but not their size. The size of the different cross-sectional areas, however, defines the overall 3D shape of the cell.

We show that, where present, the nuclear cross-sectional area is only slightly smaller than that of the cell, and the two measures correlate strongly (r = 0.94, Figure 5g). Referees 1 and 3 comment that this does not imply that the nucleus affects the cross-sectional area of the cell. We beg to differ here. The measured nuclear volumes are too large to allow a spherical nucleus to fit into a cylindrical cell of the measured height (as we now show explicitly in the new Figure 5h). Accordingly, to fit into the cell, the nucleus has to deform. Nuclei respond to external forces with anisotropic shape changes (Haase et al., 2016; Neelam et al., 2016), which is consistent with the elliptical nuclear shapes that we observe (Figure 5d). However, there is a limit to how much the stiff nucleus can deform (Lammerding, 2011; Shah et al., 2021), necessarily resulting in a local widening of the cell where the nucleus is present. Cell sections without nucleus typically have smaller cross-sectional areas, leading to a higher frequency of small cross-sections in cells compared to nuclei. We have added these additional explanations and references to the manuscript to strengthen the argument.

Nuclei in pseudostratified epithelia are well known to move continuously during the cell cycle, a phenomenon referred to as interkinetic nuclear migration (IKNM). The moving nuclei will continuously change the cross-sectional areas. In the live microscopy, the nuclei are not visible as we lack a live reporter for the nuclei. But given the strong correlation between the cross-sectional areas between nucleus and cell along the entire apical-basal axis and independent of the nuclear position and thus cell cycle phase (Figure 5g), and given the measured cross-sectional size of the nuclei, we believe that it is a safe inference that the nuclei are located at the wide parts of the cells, and these wide parts move during the cell cycle (Figure 8), consistent with IKNM.

In summary, we defined the physical principle behind the 3D cell neighbour relationships, and our data strongly suggest that the shape and movement of the nucleus is a key driver of the cell shape changes that translate into neighbour changes, both along the apical-basal axis and over time. Future work is required to unravel the physical determinants of the 3D shape of the epithelial cells and their nuclei.

2) Solve the issues spotted by Reviewer #3 in the theory presented in the SM (reviewer's point 4).

We would like to thank the referee for their careful checking of our theory. The comments helped us clarify the theoretical derivations. In the revised version, all issues have been fixed (for details, see point-by-point response). As they concerned only the formal derivation, but not the final outcome as shown in Figure 3g, the theoretical result and its interpretation are unaffected by these changes.

3) Estimate uncertainties as requested by Reviewer #3, point 3.

We have added the requested estimates of uncertainty by adding error bars showing the standard error of the mean.

4) Respond to the recommendations for the authors.

We thank the referees for their thoughtful comments, which we address below.

Reviewer #1 (Recommendations for the authors):

L 60: Eq. (2) is wrong and should read instead 5+8/n.

Thanks for spotting the typo – we have corrected this.

Figure 1: I did not understand, why the authors reproduce the previously published data in Figure 1e-k. Figure 1d does not seem to be referred to in the text.

We reproduced the data to provide an introduction and context for the reader. The previous studies in planar epithelial surfaces included many more cells than we can 3D segment in the curved mouse lung epithelia. In the former, the phenomenological laws are therefore much better visible. Also, in light of the different referees’ comments, we believe that introducing Aboav-Weaire’s and Lewis’ laws clearly at the beginning is important to make our rather involved work accessible to a broad audience. We would therefore like to keep Figure 1 as it is.

Figure 2f: It is hard to distinguish between linear and quadratic dependences.

We are plotting the linear Lewis’ law and the quadratic relationship in comparison to the data. Given the small number of cells, the relationships are not as perfectly matched as for the planar apical sheets that we show in Figure 1. To improve visibility, we have adjusted the colors for basal and apical data in 2f.

Figure 3e: For tips there seems a clear bias for apical transitions. Could you comment on this?

The effect alluded to by the reviewer is likely to be an indirect consequence of the small number of cells considered for this dataset. Although nuclei are distributed along the entire apical-basal axis, consistent with pseudostratification, a few dividing cells in the tip dataset were very large, round, and apically located. This preferentially introduces apical cell geometry heterogeneity and cell neighbour rearrangements as dividing cells make room in an already crowded apical domain. This effect is likely to be more pronounced in a small dataset like this one (N=59) as this trend was no longer visible when we considered a dataset with many more cells (tube N=140). In the tube dataset with more cells, the fraction of T1 transitions appears uniform, except for the apical-most, and basal-most slice.

Ll 165: I did not understand this part. Why do you fit ellipses and not circles? Why do you consider two subsequent T1L transitions/consecutive neighbor number changes and not one?

Epithelial tubes in the developing lung have mostly collapsed rather than open and perfectly round lumens [Conrad et al., Development dev194209, 2021]. Consequently, their cross-sectional shape is not well approximated by circles. Instead, we find that ellipses typically approximate both their apical and basal shapes very well, which is why we fitted ellipses and not circles. To make this clearer to the readership, we have added this information to the paragraph in question. We have also extended Figure 3f to showcase the elliptic shape of the cross sections.

The reason why we considered the apicobasal distance between two consecutive T1L transitions rather than analysing the locations of individual transitions is that the former approach allows us to study the effect a change of tissue curvature has on them. Considering only individual T1L transitions and the tissue curvature they occur at would not provide any insight into the differential effect, i.e., how the transitions are affected by how quickly the tissue curvature changes. Our theory that we developed in the supplementary material quantifies the fold-change of tissue curvature between two T1L transitions. To allow for a fair comparison between theory and data, we analysed the data in the same way.

L 211: This statement contradicts later statements that involve the nucleus. Also, here you really write about minimizing a line energy, whereas it should be a surface energy; the lateral and the apico-basal directions are not independent.

Thanks for highlighting this important point so that we can further clarify. As we emphasize also in the title of this paper, our theory explains the 3D cell neighbour relationships, NOT the 3D cell shape. To be even more precise: while our theory does not attempt to explain the cross-sectional areas of the cells along their apical-basal axis (and thus the cell shape), it does explain the polygonal shape of each cross-section, and this then determines the number of neighbours that a cell has. Differently put: other processes (including the effect of the nucleus) define the cross-sectional areas of the cells, and, given the area distribution, the minimisation of the lateral cell-cell contact surface energy results in the observed polygon distribution, and thus cell neighbour relationship. To make this clearer, we have now rephrased the statement to read:

“We conclude that the neighbour relationships of epithelial cells along the entire apical-basal axis can be explained with a minimisation of the lateral cell-cell contact surface energy, as previously revealed for the apical layer.”

L 233: What do you mean by "pseudo-stratification"? Where is it shown?

The term pseudostratified epithelia is well established in the scientific literature on the subject and refers to epithelia which comprise only a single layer of cells, but have their cell nuclei positioned in a manner suggestive of stratified epithelia. The developing proximal mouse lung epithelium is well known to be pseudostratified (e.g. Que et al., Development, 2007, https://doi.org/10.1242/dev.003855), and we confirm in our measurements that the nuclei are indeed found along the entire apical-basal axis (Figure 5c). We are now citing a review by Caren Norden for readers not familiar with pseudostratified epithelia (https://doi.org/10.1242/jcs.192997).

Ll 241+ Figure 5d: There seems to be only a very weak correlation between nuclear volume and ellipticity or position along the apico-basal axis.

We agree with the referee, and we make no claim toward strong correlation. All we say is that they vary along the apical-basal axis, but there is at most a weak correlation. The only exception is close to the apical surface, where nuclei tend to be larger and more spherical, and we confirm this with a one-sided, two-sample Welch t-test.

Ll 244 + Figure 5f,g: Since the nuclei are inside the cell, could it be otherwise? Also, the correlation between cell and nuclear volume seems to be rather elevated. Is this correct?

Being inside the cell, the cross-sectional area of nuclei can, of course, not be larger than that of the cell, but it could be (substantially) smaller. As such, a strong correlation between the nuclear cross-section and cell cross-section is not an obvious necessity. As indicated in the figure panel, we find a sample correlation coefficient of 0.79 between nuclear and cell volume, which suggests strong association, likely because the nuclear volume expands in parallel to the cell volume during the cell cycle. We are now stating this explicitly in the text:

“The nuclear volumes are on average about 50% smaller than the cell volumes, and largely correlate (r = 0.79, Figure 5f), likely reflecting parallel expansion during the cell cycle.”

Ll 246: But it could also be the other way around or some independent factor could determine both. I do not think that the consistency you notice has any explanatory power.

We have now expanded on this statement to provide more context:

“Where present, the nuclear cross-sectional areas are only slightly smaller than those of the entire cell, and the cross-sectional areas of the cell and the nucleus are strongly correlated (r = 0.94, Figure 5g). The strong correlation can be accounted for by the opposing actions of cells and nuclei in the columnar epithelium. The nuclear volumes are too large to allow for a spherical nucleus to fit into a cylindrical cell of the measured height (Figure 5h). Accordingly, to fit into the cell, the nucleus necessarily has to deform. Nuclei respond to external forces with anisotropic shape changes (Haase et al., 2016; Neelam et al., 2016), which is consistent with the elliptical nuclear shapes that we observe (Figure 5d). However, there is a limit to how much the stiff nucleus can deform (Lammerding, 2011; Shah et al., 2021), resulting in a local widening of the cell where the nucleus is present. Cell sections without nucleus typically have smaller cross-sectional areas, thereby leading to a higher frequency of small cross-sections in cells compared to nuclei.”

As such, the sharp transition in cross-sectional area at the limits of the nuclei, and the concomitant T1 transitions, reflect the impact of the nucleus in locally widening the epithelial cell. This is what we meant to say, but we appreciate that this statement warranted a better explanation.

Figure 6: Are the results based on one explant?

This is correct. The analysis of dynamic cellular organization on the apical and basal cell surfaces was based on a single tissue explant. Lamentably, obtaining suitable specimens for morphometric analysis remains very challenging. Not only is the culturing and timelapse light-sheet microscopy imaging of mammalian rudiments complex, but target lung buds (such as the one in Figure 6) frequently present a high degree of heterogeneity in their shape – and the apical surfaces come so close that the tube lumen appears closed in many cases [Conrad et al., Development dev194209, 2021]. Naturally, inaccessible geometries like these did not allow for the analysis of both the basal and apical domains through 2.5D segmentation techniques, and as such, limited the overall number of samples included.

L 317: "Where present, the stiff nucleus determines the cell cross-sectional area, as is apparent from the strong correlation between the cell cross-sectional and the nuclear cross-sectional areas" Correlations do not imply causation. I do not think that you have data to support the causal relationship.

We absolutely agree of course; correlation does not imply causation. However, as explained above, we believe that our various data taken together strongly indicate that anything but causation is unlikely (see also answer to next point).

L 320: Again, this seems to be speculation. Figure 8 does not show the nucleus – and even if: from your the data it is not clear that the motion of the nucleus causes changes in the neighbor relationships.

As explained above, the nucleus cannot deform as much as would be required to fit into a columnar cell shape with perfectly straight lateral sides. The presence of nuclei therefore necessarily widens the cells locally. Thus, during nuclear migration, the location where the cell is locally widened by the nuclei moves with the nuclei. This is evidenced by the strong correlation in Figure 5g independent of nuclear position and thus cell cycle phase, which would not be expected to be found if it were otherwise. This in combination with the observation that T1L transitions are found with overwhelming predominance in the vicinity of the nuclei’s ends (Figure 5j) leaves us with the conclusion that the distinctly most plausible explanation is that nuclear migration alters the cell area, which in turn alters cell connectivity.

Reviewer #2 (Recommendations for the authors):

Page 4, line 73, "following, Aboav-Weaire's law": I suggest replacing this with "following Aboav-Weaire's law" (i.e. remove the comma).

Thanks for spotting this typo.

Page 4, lines 89-90, "As such, growth and cell division determine the variability of the apical areas" and lines 100-101, "The cross-sectional areas vary as a result of cell growth, division and interkinetic nuclear migration (IKNM)": Maybe I missed it, but can the authors discount any role(s) for 'active' cell intercalations and/or cell death here?

We did not mean to exclude any such processes and have rewritten the two sentences to read:

“As such, active processes such as growth, cell division, cell death and extrusion, cell intercalation and apical constriction determine the variability of the apical areas and thus determine apical organisation indirectly.”

“The cross-sectional areas vary as a result of active cell processes, including interkinetic nuclear migration (IKNM).”

Page 4, lines 97-98, "the variation of […] define the": I suggest replacing this with "the variation of […] axis defines the" (i.e. add an 's').

Thanks for spotting this typo.

Page 5, lines 112-113, "We detected less than half as many cells on the apical side": Just to be clear here, do the authors mean that they could only detect half as many cells on the apical side due to their small size and/or lower resolution, or that there are really half as many cells on the apical side and hence over half the cells don't extend fully from the basal side to the apical side?

As rightfully speculated by the reviewer, the difference in detections between layers stems from the small size of cells on the apical side coupled with resolution limitations. Our analyzed explants had highly curved spherical/tubular morphologies with cells that fully extended from the apical to the basal layer – as is the case in pseudostratified tissues. However, this extreme curvature difference between tissue layers meant that while basal domains were large and easy to segment, cell geometries in constricted apical domains were small and hard to detect given the resolution of our Z.1 light sheet microscope.

Page 5, lines 118-119, "the neighbour changes in spherical geometries cannot be explained with such an effect (Gómez-Gálvez et al., 2018)": For clarity, I suggest briefly emphasizing why not (I guess the point is that as one looks along the apical-basal axis in this case, one is just seeing a dilation, so prismatic cells do the job fine).

As the referee correctly suggests, spherical shapes can be achieved with prismatic cells because the curvature change is the same in all directions, i.e., there is only a dilation. We have added a brief explanation at this position in the manuscript. We also discuss this in more detail in the next section in the context of figure 3f,g, and in the supplementary material.

Page 5, lines 127-130, "Aboav-Weaire's law (Figure 2e) and Lewis' law (Figure 2f) hold not only for the apical, but also for the basal datasets […] the apical layers, which have a larger area variability than the basal layers (Figure 2c), follow the quadratic law (yellow line) rather than the linear Lewis' law (black line)": I may simply have misunderstood things here, but the first sentence seems to indicate that Lewis' law holds for the apical dataset, while the second sentence seems to indicate that the apical layers follow the quadratic law rather than Lewis' law, which seems a contradiction. If I have indeed just misunderstood things here, then perhaps the authors could slightly rephrase these sentences for clarity.

As we explain in the Introduction, the linear Lewis’ Law and the quadratic law result from the same constraint for different area variabilities; hence the difference between the apical and basal surface. But we accept that it is confusing the way it was written, and we have rephrased this.

Page 7, lines 232-233, "giving the tissue a pseudostratified configuration": I suppose strictly speaking this is true of pseudostratified epithelia but not ALL epithelia, so it might be worth changing "In epithelia" to "In many epithelia" on line 227, or similar.

Thanks for catching this. We have added “pseudostratified” to the text, and we have added a reference that summarises nuclear positioning in different cells and tissues.

Page 9 lines 335-336, "signalling factors that control cellular tension are known to affect cell height (Widmann and Dahmann, 2009)": For clarity, perhaps the authors could name these signalling factor(s) explicitly here?

We have now added examples to the text:

“The determinants of the epithelial thickness, i.e., the distance between the apical and basal surfaces are still unknown, but signalling downstream of Fibroblastic Growth Factor (FGF), Sonic Hedgehog (SHH), Bone Morphogenetic Protein (BMP)/ transforming growth factor-β (TGF-β) and WNT has been observed to affect cell height, presumably via an effect on cell tension and/or cell-cell adhesion (Gritli-Linde et al., 2002; Hirashima and Matsuda, 2021; Kadzik et al., 2014; Kondo and Hayashi, 2015; Widmann and Dahmann, 2009).”

Page 16, Figure 1, panel (b): The chosen colour scheme makes it very difficult for me to tell the difference between 5, 6, 7 and 8 sided cells, since they're all shades of green. If possible, it would be great if the authors could use more contrasting colours.

We thank the reviewer for helping add clarity to our figures. The color palette has been adapted to increase the contrast and visibility of different cell types.

Page 16, Figure 1, panel (e): Can the authors briefly explain in the legend what the various abbreviations mean (cNT, EYE, dPE, dWL, dWP, dMWP, dPW, TP, dMWL), or at least broadly which different tissues/species are considered?

Thank you for pointing this out. We have amended the legend for Figure 1 to elaborate on these abbreviations.

Page 19, Figure 3, panel (b): I found this figure difficult to understand partly due to the colour scheme but also because the authors are plotting two datasets on the same image. If there is a way of simplifying this figure at all, that would be great.

We thank the reviewer for helping add clarity to our figures. The color palette has been adapted to increase the contrast and visibility of different cell types.

Reviewer #3 (Recommendations for the authors):

– Citation of Rupprecht et al., MBoC, 2017 appears justified, as it is the first / one of the first papers to discuss apico-basal T1 transitions in epithelia based on experimental data.

We are now citing Rupprecht et al., MBoC, 2017 in the revised version. We apologise to the authors for the oversight of not including this paper in the original version.

– Recommendations regarding point 3. in the public review:

a) Estimations of uncertainties (e.g. standard error of the mean) would be useful, e.g. in Figure 2 e,f. For instance, in Figure 2f, an indication for the stochastic uncertainty due to the finite sample size would be useful.

We have added error bars showing the SEM to all data shown in Figure 2f. In Figure 2e, we omitted the error bars because they are smaller than the symbols.

b) Aboav-Weaire law: For instance in Figure 2e, everything is close to a diagonal line. However, that there is a line with slope of approximately 6 in the plots of m_n*n over n is already a consequence of the Euler characteristic. To better display the correlation that is the core of the Aboav-Weaire law, I would suggest to directly plot m_n over n (as also one e.g. in Aboav, 1970). Together with error bars indicating uncertainty, this would allow to better scrutinize whether the different data sets fulfill the prediction.

Aboav-Weaire’s law is phenomenological and holds only approximately – the constants 5 for the slope and 8 for the intercept in Aboav’s original relationship are merely nearest-integer numbers. Deviations of the actual data from the line are therefore expected. We have described the deviation from the original simple straight line in much detail in an earlier publication [Vetter et al., bioRxiv 2019]. In Figure 2e, we plot the phenomenological relationship only for reference; the data is not expected to approach it with shrinking uncertainty bounds. Nevertheless, we agree that error estimates help judge the statistical significance with which we find the data to deviate from the simple phenomenological laws. In Figure 2e, the error bars are smaller than the symbols. But we have added error bars (SEM) to all data points in Figure 2f.

c) Some plots have very similar colors and are therefore hard to distinguish, e.g. the quadratic theory curve vs the basal data in Figure 2f. Also, in Figures 4f and 8e, the polygon type (number of neighbors n) is essential, but the colors in particular for n>5 are very hard to distinguish.

We thank the reviewer for helping add clarity to our figures. The color palette has been adapted to increase the contrast and visibility of different cell types.

– Suggestions regarding point 4 of the public review (merely ideas):

a) As pointed out by Gómez-Gálvez et al., the curvature-based mechanism to create T1L transitions is due to a change in the aspect ratio of the 2D tissue cross section as one goes from apical to basal in an e.g. cylindrical tissue. One way to estimate the T1L rate that results from this might also be to use existing formalisms to understand how many T1 transitions are needed to create a certain change in 2D tissue aspect ratio (e.g. the texture tensor formalism, Graner et al., EPJE, 2008 and Guirao et al., eLife, 2015, or a triangle formalism, Merkel et al., PRE, 2017).

Thank you for these interesting references. Indeed, it appears that these other theoretical frameworks could offer a means of deriving similar (potentially equivalent) estimations of curvature effects on T1 transitions. As we were able to resolve all issues raised regarding our theory elsewise, we did not look into possible analogies here. We shall keep these references in mind for potential future considerations related to this aspect.

b) An interesting question could be how oriented the T1L transitions are ("How many of them do (net) contribute to the apico-basal change in 2D tissue aspect ratio?"), which one could address using orientational statistics of the T1 transitions (i.e. how uniform is the T1L orientation angle in the unfolded cylinder mantle).

We agree that this is an interesting aspect that could be worth investigating; thank you for this idea. A potential difficulty in interpreting the data might be that stress relaxation (visco-elastic and/or plastic) on the cellular level might affect the distribution of T1L orientations in a fashion that could be difficult to quantify. Nevertheless, this is an interesting future avenue that might be relatable to the cell orientation bias in the developing lung epithelium [Conrad et al., Development 148 (2021) dev194209].

c) Just a remark: I am happy to be convinced otherwise, but I would be surprised if one could indeed derive a dependence of the T1L transition number on cell neighbor number solely on geometric grounds (i.e. disregarding mechanics etc). This is because the curvature-based mechanism is about T1L transitions creating anisotropic deformation in the 2D cross section, while the cell neighbor number is a scalar (i.e. an "isotropic" quantity). I could be wrong here, but I would presume that any correlation between the two will depend on model details.

Thanks for the remark – we absolutely agree. Indeed, our data confirms this to some degree, in that we find that the observed number of T1L transitions does not agree with the geometrical curvature effect (Figure 3g), but is rather correlated with nuclear arrangement (Figure 5j), which in turn is undoubtedly strongly affected by various mechanical properties.

https://doi.org/10.7554/eLife.68135.sa2

Article and author information

Author details

  1. Harold Fernando Gómez

    1. Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Basel, Switzerland
    2. Swiss Institute of Bioinformatics (SIB), Basel, Switzerland
    Contribution
    Data curation, Software, Formal analysis, Supervision, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-5200-4933
  2. Mathilde Sabine Dumond

    1. Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Basel, Switzerland
    2. Swiss Institute of Bioinformatics (SIB), Basel, Switzerland
    Contribution
    Data curation, Software, Formal analysis, Funding acquisition, Validation, Investigation, Visualization, Methodology
    Competing interests
    No competing interests declared
  3. Leonie Hodel

    Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Basel, Switzerland
    Contribution
    Formal analysis, Investigation
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-2541-4133
  4. Roman Vetter

    1. Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Basel, Switzerland
    2. Swiss Institute of Bioinformatics (SIB), Basel, Switzerland
    Contribution
    Software, Formal analysis, Supervision, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0003-2901-7036
  5. Dagmar Iber

    1. Department of Biosystems Science and Engineering (D-BSSE), ETH Zürich, Basel, Switzerland
    2. Swiss Institute of Bioinformatics (SIB), Basel, Switzerland
    Contribution
    Conceptualization, Formal analysis, Supervision, Funding acquisition, Investigation, Methodology, Writing - original draft, Project administration, Writing - review and editing
    For correspondence
    dagmar.iber@bsse.ethz.ch
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-8051-1035

Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (CRSII5_170930)

  • Dagmar Iber

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

This work has been supported through an SNF Sinergia grant to DI. We acknowledge the many discussions and varied input of Marco Kokic, Anđela Markovic, Odyssé Michos, and Erand Smakaj, and we thank Richard Smith for help with MorphographX.

Ethics

Animal experimentation: Permission to use animals was obtained from the veterinary office of the Canton Basel-Stadt (license number 2777/26711). Experimental procedures were performed in accordance with the Guide for the Care and Use of Laboratory Animals and approved by the Ethics Committee for Animal Care of ETH Zurich.

Senior and Reviewing Editor

  1. Anna Akhmanova, Utrecht University, Netherlands

Reviewer

  1. Matthias Merkel, Syracuse University, United States

Publication history

  1. Received: March 5, 2021
  2. Preprint posted: March 7, 2021 (view preprint)
  3. Accepted: September 27, 2021
  4. Accepted Manuscript published: October 5, 2021 (version 1)
  5. Version of Record published: November 5, 2021 (version 2)

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© 2021, Gómez et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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  1. Harold Fernando Gómez
  2. Mathilde Sabine Dumond
  3. Leonie Hodel
  4. Roman Vetter
  5. Dagmar Iber
(2021)
3D cell neighbour dynamics in growing pseudostratified epithelia
eLife 10:e68135.
https://doi.org/10.7554/eLife.68135

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