ABSTRACT
Apico-basal cell intercalations (scutoids) optimize packing and energy expenditure in curved epithelia. Further consequences of this new paradigm of tissue packing remain uncharacterized. In particular, how scutoids modify the 3D cellular connectivity is an open question. This property is crucial for understanding epithelial architecture and is instrumental for regulating the biological function of tissues. Here, we address this problem by means of a computational model of epithelial tubes and a biophysical approach that links geometrical descriptors with the energetic cost required to increase the cellular connectivity. Our results predict that epithelial tubes satisfy a novel quantitative principle: the “Flintstones’ law”. In short, cellular connectivity increases with tissue thickness/curvature in a logistic way. We confirm experimentally the existence of this principle using Drosophila’s salivary glands. Our study provides methodological advances to analyze tissue packing in 3D and, more importantly, unveils a morphogenetic principle with key biological consequences.
INTRODUCTION
During the last decades much progress has been achieved in the understanding of the emergence of self-organization in tissues. This problem has been addressed from the viewpoint of energetics drivers (Alt et al, 2017; Nelson et al, 2005; Siedlik et al, 2017; Trepat et al, 2009; Misra et al, 2017; Sugimura et al, 2016; Fletcher et al, 2014; Canela-Xandri et al, 2011), material-like properties (Yang et al, 2017; Bi et al, 2015; Pérez-González et al, 2019; Latorre et al, 2018; Mongera et al, 2018; Campàs et al, 2014), and the analysis of the packing properties. As for the latter, the analysis of epithelial surfaces as tessellations of convex polygons has been successfully used to quantitatively understand different biological aspects such as tissue patterning, cell division, and growth (Mao et al, 2013; Thompson, 1945; Farhadifar et al, 2007; Gibson et al, 2011, 2006; Lewis, 1928; Honda, 1978; Sánchez-Gutiérrez et al, 2016; Curran et al, 2017). Importantly, these studies have also revealed the validity of mathematical principles with biological consequences. One relevant example are the implications of Euler’s formula (Reinhardt, 1918; Wetzel, 1926) about cellular connectivity. This principle states that polygonal cells in packed tissues, on average, have six neighbors (i.e., the average 2D cellular connectivity reads 〈n2D〉 = 6). As for its biological consequences, the degree of cellular connectivity determines, for example, the strength of the cell-cell juxtracrine signaling (Tung et al, 2012; Sharma et al, 2019; Perrimon et al, 2012). Not surprisingly, the validity of this connectivity principle to the third dimension has been taken for granted since the role played by apico-basal cell intercalations has been disregarded and cells have been assumed to have prismatic-like shapes in either planar or bended epithelia.
However, the recent discovery of more complex cellular geometries in epithelial cells, i.e., scutoids, to reach an efficient three-dimensional (3D) tissue packing has set a new paradigm that has not been yet fully explored (Mughal et al, 2018; Gómez-Gálvez et al, 2018; Rupprecht et al, 2017). Scutoids imply spatial changes in the neighboring relationship between cells (Fig. 1a). This phenomenon is then a spatial version of the T1 transitions that produce cell rearrangements with time in numerous developmental processes (Bertet et al, 2004; Spencer et al, 2017; Irvine & Wieschaus, 1994). Thus, the presence of scutoids necessarily modifies the connectivity and the biophysical properties of tissues. Still, the analysis of tissue organization in a 3D context, and the corresponding biological repercussions, have been hindered by the technical difficulties to accurately segment and reconstruct cells from apical to basal surfaces. In addition, very few computational models account for the presence of apico-basal transitions to investigate 3D self-organization in tissues (Okuda et al, 2019; Mughal et al, 2018; Gómez-Gálvez et al, 2018; Rupprecht et al, 2017).
The realistic analysis of 3D packing is in turn utterly relevant in epithelial tubes, where scutoids appear more frequently (Sanchez-Corrales et al, 2018; Iruela-Arispe & Beitel, 2013; Gómez-Gálvez et al, 2018). Epithelial tubes are in fact the primary developmental structures in all organisms with bilateral symmetry (Gilbert & Barresi, 2016) and tubulogenesis is fundamental in a broad variety of key developmental processes, including gastrulation and neurulation (Pilot & Lecuit, 2005; Swanson & Beitel, 2006; Colas & Schoenwolf, 2001; Leptin & Grunewald, 1990; Nelson, 2009; Iruela-Arispe & Beitel, 2013). Furthermore, epithelial tubes are the essential functional unit of many mammalian organs, including glands, components of the digestive apparatus, lungs, and kidney (Huebner & Ewald, 2014). Hence, the faithful formation and function of tubes requires the precise coordination of dynamic changes in the tissue architecture, i.e., packing, during development (Röper, 2018).
Here, we study the packing and the 3D cellular connectivity properties of epithelial tubes. We show that the presence of scutoids implies a breakdown of the principle 〈n3D〉 = 6 and reveal a novel law that quantitatively links the 3D cellular connectivity, geometrical descriptors (e.g., tissue curvature/thickness), and energetics drivers. Our findings are supported by i) a computational model that realistically render the 3D cellular organization of tubular epithelia (including the appearance of scutoids); ii) a biophysical model, supported by mathematical calculations, that connects the tissue energetics with the 3D organization of epithelial tubes; and iii) experimental data of epithelial tubes (Drosophila’s salivary gland) whose 3D cellular structure has been accurately characterized by means of a novel computer-aided image analysis method.
Altogether, by realistically capturing the organization of cells in tubular epithelia, we shed light on the important issue of how tissues are 3D shaped and we open the door to understand quantitatively key morphogenetic events that ultimately depends on the 3D cellular connectivity.
RESULTS
A computational model unveils the connectivity properties of tubular epithelia
To understand how the geometry of tubular epithelia affects the 3D cellular packing and connectivity, we designed and implemented a computational Voronoi tubular model (Gómez-Gálvez et al, 2018) (Materials and Methods). We analyzed tubes with an increasing surface ratio (radial expansion), sb = Rb/Ra (Fig. 1b). This parameter quantifies the ratio of the non-trivial curvatures of apical and basal tubular surfaces, ka/kb = Rb/Ra = sb, and it is a proxy for the dimensionless tissue thickness, (Rb – Ra)/Ra = (sb – 1). In addition, we explored the cellular organization of tubes by using a Centroidal Voronoi Tessellation (CVT) scale (Fig. 1c). The CVT scale accounts for the number of iterations of the homogenizing Lloyd’s algorithm and makes possible to analyze the effect of the topological order of the tissue (Materials and Methods) (Sánchez-Gutiérrez et al, 2016; Gómez-Gálvez et al, 2018).
Our results showed that the average number of apico-basal intercalations per cell, 〈i(sb)〉, (Fig. 1d), and therefore the percentage of cells adopting the scutoidal shape (Fig. S1), increases with sb and decreases as tubes become more ordered (i.e., as the CVT index increases). To further uncover the 3D organization of tissues, we implemented a benchmark able to reveal simultaneously the existence of apico-basal intercalations (scutoids) and the polygonal distributions of cells. To that end, we computed the probability that cells change their polygonal class between the apical and basal surfaces. Thus, the components (i.e., bins) of this distribution along the diagonal account for prismatic cells (Fig. 2a) whereas the spreading away from the diagonal reveals the existence of scutoids (cells that exchange neighbors due to apico-basal intercalations) and, consequently, changes in the cellular 3D connectivity in the tissue (Fig. 2b). In agreement with the results shown in Fig. 1d, our data indicates that the degree of spreading of the distribution (as quantified by the parameter, η2, Materials and Methods) increases with the surface ratio and decreases when the initial (i.e., apical) Voronoi diagram became more ordered, that is, as the CVT increases (Fig. 2c).
Moreover, we computed the average of the total number of contacts of the cells, 〈n3D〉, as a function of the surface ratio and the initial Voronoi diagram (Fig. 2d). Our data are quantitatively consistent with a mathematical derivation that shows that 〈n3D〉 is linearly proportional to the amount of apico-basal intercalations (Materials and Methods and Fig. S1). Also, these results indicated that the average cellular connectivity grows as the tissue thickness and the randomness of cellular organization increases.
In summary, our computational model suggests a relation between the tissue geometry (i.e., cell thickness/curvature), the cellular planar topological order (CVT index), and the 3D cellular connectivity in epithelial tubes.
The 3D neighbor’s accumulation follows a “poor get richer” principle
In order to shed light on the underlying mechanisms that determine the degree of 3D cellular connectivity in tubular epithelia, we computed the net gain of cellular neighbors as a function of the radial expansion (thickness/curvature of tubes) and the topological properties of cells (CVT index and polygonal class at the apical surface). As a general trend, we observed that, independently of the radial expansion, the smaller the number of neighbors at the apical surface the larger the net gain of 3D cellular contacts (Fig. 3a-b, Fig. S2). Additionally, we also checked that this tendency is satisfied when estimating the net gain of neighbors accumulated from the basal to the apical surface (Fig. S2). These results suggest that in tubular epithelia the 3D cell packing follows a “poor get richer” principle driven by apico-basal intercalations: the less neighbors a cell has in a surface (apical or basal), the larger the net increase of cellular contacts. Interestingly, this result is akin to the behavior found in planar geometries that indicates that the probability of undergoing a T1 transition increases as the number of neighbors decreases (Bi et al, 2014), see Discussion.
An energetics model suggests that cellular connectivity satisfies a logistic-like law
In light of this evidence, and in order to better understand the dependence of the tissue self-organization on the radial expansion, s = R/Ra, we developed a biophysical model (a Kolmogorov rate equation) that accounts for the probability of cells to increase their 3D connectivity driven by energetic cues (Fig. 4a-b and Materials and Methods): where, Pm, is the probability of having m accumulated 3D neighbors (i.e., m = n3D) as the surface ratio (i.e., tissue thickness) changes from s to s + ds, and ri,i+1 accounts for the rate per unit of surface ratio of undergoing an apico-basal intercalation. By drawing parallels between apico-basal intercalations and planar T1 transitions (Sanchez-Corrales et al, 2018; Gómez-Gálvez et al, 2018) we assumed that cells need to overcome an energy barrier to gain a 3D neighbor, that is, ri,i+1~e~ΔEi (Fig. 4a-b). The “poor get richer” principle suggests that Δ/Ei, grows as i increases. In addition, our mathematical calculations proved that the apico-basal intercalation rate becomes null for a finite value of i (Box and Materials and Methods): neighbors’ gaining is necessarily bounded or, energetically speaking, the energy barrier to undergo an apico-basal transition becomes eventually infinite. All these facts led to the following expression for the apico-basal intercalation rate: ri,i+1 = α(Nmax – i)e−iβ, where α is a ‘bare’ transition rate, β is the dimensionless energy (in units of the four-fold vertex energy configuration) per 3D neighbor that a cell needs to increase its connectivity to an additional cell, and Nmax is the maximum 3D cellular connectivity (Materials and Methods).
In tubular geometries the 3D cellular connectivity gain decreases as the surface ratio increases and it is bounded.
In this example, the Y axis represents the longitudinal axis of tubes, while the X axis accounts for the Cartesian projection of the transversal axis of radial sections. From left to right different radial sections are represented as s increases (as indicated by the color gradient arrow: from light to dark blue). In a) three Voronoi seeds that correspond to neighboring cells at the apical surface, s = 1, define the triangle ABC. Panels b) and c) track changes in the neighboring relations (accumulated neighbors) of cell A for two increasing values of s: 2 and 4.5 (panels b) and c) respectively). As shown in a), should a new neighboring cell, D, of cell A appear due to an apico-basal intercalation, then its position must lie inside the parabola defined by the points A, B and C, but outside the circle that these points define (white region), (see Lemma 1, Material and Methods). Regions accessible to new neighbors are then coded by the green shading in a)-c). As s increases, see b), and the cells A, B, C, and D become neighbors, then the parabolas and circles defined by ABD and ACD restrict the locations of possible new neighbors. This idea is further reinforced in panel c): winning neighbor E set additional limits to the accessible locations of new neighbors. Thus, the potentiality of a connectivity gain by cell A due to apico-basal intercalations diminishes as the surface ratio increases and eventually becomes null: the number of 3D neighbors of a cell is bounded (Materials and Methods).
The fitting of the in silico data about the average tissue connectivity, 〈n3D(s)〉 = ΣmmPm(s), to this biophysical model showed an excellent agreement and confirmed that 〈n3D〉 >6 as long as the tissue is subjected to some level of anisotropic curvature (Fig. 4c, Fig. S3 and Materials and Methods). We also observed that the energy required per 3D neighbor to undergo an intercalation, β, quickly reached a plateau, β ≃ 5.10-2, as the tissue became more ordered (i.e., as the CVT index increases). Our results also indicate that in Voronoi tubes the scutoidal geometry enables a theoretically increase of the average 3D cellular connectivity up to 〈Nmax〉 ~12 – 15 cells (Table S1). In addition, the plausibility of the Kolmogorov approach was further assessed by predicting the 3D neighbor distribution, Pm(s), thus confirming that a link between geometrical and energetic traits determines the cellular connectivity in tubular epithelia (Fig. 4c).
We also obtained theoretically an analytical formula that characterizes the average 3D cellular connectivity, 〈n3D〉 (Box and Materials and Methods). We concluded that in tubular epithelia 〈n3D〉 can be described by a logistic-like behavior, where b, c, and d are non-independent parameters that are functions of α, β, and 〈Nmax〉 (Table S1 and Materials and Methods). We refer to this logistic-like principle as the “Flintstones’ law” after the cartoon characters.
The analysis of computational tubes revealed the validity of the Flintstones’ law as an effective way to determine the cellular connectivity in tubular epithelia as a function of the radial expansion (Fig. 4c and Fig. S3). More importantly, it provides a straightforward way to estimate/predict the value of the underlying energetic properties regulating apico-basal intercalations and the limiting average 3D cellular connectivity (Table S1).
Experiments confirm that the 3D cellular connectivity in tubular epithelia satisfies the Flintstones’ law
In order to confirm our computational and theoretical predictions, we implemented a novel methodological pipeline that combines several computational image analysis techniques to accurately segment cells of in vivo epithelial tubes (Arganda-Carreras et al, 2017; Machado et al, 2019) (Materials and Methods). We used the Drosophila larval salivary gland as a model due to its ideal characteristics to study complex tubular architectures (Girdler & Röper, 2014) (Fig. 5a-c).
Our methodology allowed to determine the average surface ratio of the salivary glands (sb = 4.0 ± 0.4), 〈n3D(sb)〉 = 6.7 ± 0.2, the average percentage of scutoids (76±11%), and the average number of apico-basal intercalations per cell, 〈i(si)〉 = 1.4 ± 0.4, thus confirming the validity of the formula that relates apico-basal intercalations per cell and the average connectivity: 〈n3D〉 = 6 + 〈i〉/2 (Materials and Methods and Fig. S1). We also calculated the polygonal class distribution in the apical and basal surfaces (Fig. S4). Interestingly, in spite of the prevalence of scutoids, the polygonal organization of apical and basal surfaces was found to be the same and equivalent to that obtained in in silico V8 tubes with a radial expansion sb = 1.75 (Fig. S4). This V8 model (sb = 1.75) also displayed a similar scutoidal prevalence (79 ± 5%), average number of 3D neighbors, average number of apico-basal intercalations per cell, and η2 spreading that in vivo tubes (Fig. 5d and Fig. S4). We additionally confirmed that the apical and basal surfaces of the V8 model and the salivary glands fulfilled, as expected, that 〈n2D〉 ≈ 6 (Reinhardt, 1918; Wetzel, 1926) (Fig. S4). Thus, we concluded that the in silico V8 model with a radial expansion of sb = 1.75 faithfully recapitulates the 3D packing properties of in vivo salivary glands.
As for the 3D cellular connectivity of in vivo tubes, our analyses confirmed that the “poor get richer” principle was satisfied, thus supporting the idea that the smaller the number of neighbors of a cell in a surface, the larger the probability to increase its connectivity (Fig. S4). Additionally, by implementing an un-rolling (i.e., peel-off) algorithm (Yang et al, 2019) (Materials and Methods), we obtained concentric radial sections and quantified the number of 3D neighbors as a function of the radial expansion. The fitting of the data to the Kolmogorov model showed an excellent agreement and allowed to estimate the energetic properties as summarized by the parameter β (Fig. 5e and Table S1). Our results suggested that the energy per 3D cell required to undergo an apico-basal intercalation is larger in in vivo tubes than in the computational V8 model, see Discussion. Importantly, the 3D cellular connectivity data confirmed the applicability of the Flintstones’ law in in vivo tubular epithelia (Fig. 5e and Table S1).
DISCUSSION
Here we have shown how mathematical and physical principles underlie the emergence of functionally complex 3D developmental structures, e.g., glands. Our analyses have uncovered for the first time how a 2D organizational trait, i.e., the cellular connectivity, can be extended to the third spatial dimension when the novel paradigm of epithelial cells’ shapes and packing, the scutoid, is considered. In that regard, we have revealed how the 3D cellular connectivity and tissue energetics are coupled, and we have stated a novel principle, the so-called Flintstones’ law. The latter links the activation energy needed to recruit additional neighbors with geometrical descriptors (i.e., tissue thickness/curvature). Our results provide new biological insight into the spatiotemporal regulation of cell-cell connectivity, a property that ultimately regulates juxtracrine signaling and is pivotal for primordia patterning and cell fate determination (Tung et al, 2012; Sharma et al, 2019). In this context, our study points towards an effect of scutoids on the regulation of the physiological properties of tissues. Therefore, our findings, on top of being fundamental to understand self-organization of epithelia in 3D, open new ways to investigate, and draw implications about, primary developmental processes in which epithelial bending is essential such as tubulogenesis, gastrulation, or neurulation.
Our study also provides important methodological advances. Previous software developed to identify the outlines of the epithelial cells does not work on 3D or lacks enough precision to extract the geometrical and topological data needed to quantify tissue packing in 3D (Khan et al, 2014; Heller et al, 2016; Bassel et al, 2014; Gómez-Gálvez et al, 2018). Here we have shown that our methodological pipeline (Materials and Methods) allows to implement a 3D segmentation and the precise reconstruction of cells in epithelia subjected to curvature. We stress that this level of detail is necessary to be able to quantify the apico-basal intercalation phenomenon and, therefore, compare the results with the computational models and extract biological consequences. We then argue that our methodology, by enabling the analysis of 3D packing in a realistic way, will benefit the field of morphogenesis by bringing understanding about the cellular and mechanical basis of self-organization in curved tissues (Inoue et al, 2019; Ambrosini et al, 2017; Hirashima & Adachi, 2019) or even whole embryos (Shahbazi et al, 2019). In addition, our analysis indicates that the Flintstones’ law can be used quantitatively to estimate key connectivity-related parameters, e.g., β and/or 〈Nmax〉 This avoids the burden of solving the optimization problem associated with the Kolmogorov model that is computationally demanding (Material and Methods). Thus, the values obtained by means of fittings to the Flintstones’ law are, at the very least, within the same order of magnitude with respect to the ‘exact’ Kolmogorov calculations (Table S1). This reveals the usability of the Flintstones’ law not just as a principle that is satisfied by tubular epithelia, but as a practical way to connect packing properties, geometrical descriptors, and biophysical traits due to its predictive character.
As a matter of discussion, the connectivity law that we have introduced herein, depends on a prediction obtained from the Voronoi computational model that was confirmed in experiments: the “poor get richer” principle. Roughly speaking, we have shown that the fewer neighbors a cell has on a surface, the larger is the probability of a connectivity increase. Interestingly, a similar idea has been reported in T1 dynamical processes during the remodeling of planar epithelia (Bi et al, 2014). Since the scutoidal geometry can be related to planar T1 transitions by exchanging the concepts of space and time, this result reinforces the idea of the existence of universal principles driving the organization of tissues.
In our study we have found that in real tissues the energy cost per 3D neighbor that a cell requires to increase its connectivity, β, is larger than in Voronoi models. We hypothesize that it is due to the purely geometrical description used in the latter. That is, while in in silico models the apico-basal transitions develop just a result of a topological constraint (Voronoi tessellation), in the salivary glands, on top of geometrical requirements, the cells must actively remodel their cytoskeleton to make the transitions possible. That component would explain the larger effective cost of gaining new neighbors in real tissues. The reduced energetic cost for gaining neighbors in the Voronoi computational approach also explains why in silico tubes led to a larger limiting average 3D cellular connectivity, 〈Nmax〉, and the V8 model with the same surface ratio that the salivary gland, sb = 4, developed more apico-basal transitions than the real samples (Fig. 5e, Fig. S4, and Table S1). These data ultimately explain why it is necessary to rescale appropriately sb to obtain a computational model with packing, topological, and connectivity properties similar to those of the salivary glands. In that regard, our results suggest that salivary glands are optimized to reach a high cellular connectivity. While the in silico V8 model with a radial expansion of sb = 1.75 or sb = 4 are far for reaching their maximum average connectivity (i.e., 〈n3D〉’s are, respectively, ~47% and ~6l% of 〈Nmax〉), in the salivary glands 〈n3D(sb)〉 is ~87% of 〈Nmax〉 (Fig. 5e, Fig. S4, and Table S1). This optimization could be related to a functionality improvement of the gland, similarly to what has been suggested in pituitary growth hormone secretory cells, where the increase of 3D cellular connectivity has been proposed to better coordinate the pulses of hormone secretion (Bonnefont et al, 2005).
As for the broader implications of our findings, we argue that, while our analyses focus on static tissues from the point of view of tissue architecture, our results can also be relevant to understand active 3D tissue remodeling (e.g., fluidization). Recent studies have revealed that active remodeling involves changes in the material-like properties of tissues that can be connected to an increased activity of neighbor exchanges (Tetley et al, 2019; Mongera et al, 2018). In that regard, here we have shown that the physical basis of 3D self-organization (i.e., 3D cellular packing and connectivity) in tubular epithelia effectively relies on a constant amount of energy, β. Thus, arguably, active 3D tissue remodeling would imply dynamical changes on the value of β that would modify the apico-basal intercalation propensity and therefore the material-like properties: the larger β the more solid-like the tissue would behave. Finally, with respect to the applicability of our results to other areas, we expect that the emerging field of organoids will benefit from our discoveries. A precise quantification of 3D connectivity could then help to understand the lack of reproducibility in organoid production, one of the biggest challenges of the field (Schutgens et al, 2019; Huch et al, 2017; Clevers, 2016). Also, from a medical point of view, it has been recently shown that tissue curvature affects tumor progression due to the imbalance of tensions in apical and basal surfaces of epithelial tubes (Messal et al, 2019). The Flintstones’ law explains how energetic cues affect the 3D packing of these cells and therefore may shed light on the mechanism of tumorigenic morphogenesis in tubular organs.
MATERIALS AND METHODS
Immunohistochemistry and confocal imaging of salivary glands
Flies were grown at 25 °C using standard culture techniques. We dissected the salivary glands from third instar larvae of the wild type Oregon R strain. After PBS dissection, the glands were fixed using 4% paraformaldehyde in PBS for 20 min. The samples were washed three times for 10 min with PBT (PBS, 0.3% Triton) and then incubated for 1 hr 45 minutes at room temperature with Cy3-labeled phalloidin (Sigma) to label the cell contours of the epithelial cells. Stained larval salivary glands were mounted using Fluoromount-G (Southern Biotech). We used two pieces of double-sided adhesive tape (one on top of each other) as a spacer (Aldaz et al, 2013), so the salivary glands preserve their shape. Images were taken using a Nikon Eclipse Ti-E laser scanning confocal microscope. The images were captured using a ×20 dry objective and 2.5 μm steps between slices. The image stacks were exported as 1024 × 1024 pixels TIFF files.
3D glands segmentation
To segment the salivary gland stacks of images and reconstruct (semi-automatically) the shape of cells in three dimensions we used the FIJI (Schindelin et al, 2012) plugin LimeSeg (Machado et al, 2019). We inferred cell outlines by using surface elements (“Surfels”) obtained by placing single ellipsoidal-like seeds on every cell (see https://imagej.net/LimeSeg for details). Once cell outlines were found (Fig. 5b-c), we exported them as point clouds (output). We developed a custom-made Matlab code (2018a MathWorks) to postprocess the output of LimeSeg in order to correct errors and obtain perfectly segmented salivary glands. In addition, we manually segmented the lumen of the glands from the images using Adobe Photoshop CS6 and reconstructed it using a Matlab code. To faithfully represent the gland as a cylinder, we selected a subset of cells: cells that were not ductal, neither located at the tip of the gland. For more information about the processing pipeline: https://osf.io/nd5t6/.
To obtain the cellular neighborhood relations of salivary glands for different values of the radial expansion, we proceeded as follows. We calculated the cell height by estimating the distance between the average voxel positions of the apical surface with respect to the average voxel positions of its basal surface, d(Sa,Sb). Then, to capture a concentric radial section of the gland, we linearly extrapolated the equivalent cell height to the given surface ratio, s: where d(sa,s) is the Euclidean distance between the position of the centroid of the cell in apical and that the position of the centroid at a value s of the radial expansion. Finally, to obtain the gland cylindrical radial section for a given value of the radial expansion, s, we collected all voxels between apical and the upper bound of the calculated distance d(sa,s). Those cylindrical surfaces of the salivary gland were mapped in the Cartesian plane for analysis using a cylindrical coordinates transformation.
Salivary glands measurements
We quantified the following geometrical and topological descriptors of the segmented salivary glands using a custom-made Matlab code:
- Surface ratio expansion (s): Assuming a cylindrical shape for glands, we estimated s by dividing the area of the basal surface of glands by area of the apical surface.
- Polygonal Class. We estimated the number of sides of each cell using the unrolled images (radial cylindrical sections) projected in the Cartesian plane.
Likewise, we carried out the calculations of the percentage of scutoids and the number of apico-basal transitions.
Voronoi tubular model
Using custom-made Matlab code (R2018a) we generated a Voronoi model that simulates the surface of a cylinder unfolded over the Cartesian plane, see details in Gomez-Galvez et al. ((Gómez-Gálvez et al, 2018), Material and Methods). The only difference with the cited methodology, is that in this work the Voronoi diagrams has been constructed by means of the Delaunay triangulation technique. Therefore, we just considered the cells’ vertices information (cartesian coordinates and connections) for a much faster computation. For each realization, we used an initial set of 200 randomly located seeds on a rectangular domain of 512 (X axis; transverse axis of cylinder) per 4096 (Y axis; longitudinal axis of cylinder). In total, we implemented 20 different realizations (i.e., tubes). We performed this procedure for 10 different initial Voronoi diagrams (Voronoi 1 (V1, random seeds) to Voronoi 10 (V10, more ordered and homogeneous cells). These diagrams represent the apical (inner) surfaces of computational tubes, and they were obtained by applying N-1 times the Lloyd’s algorithm (Lloyd, 1982) to the random seeds, where N is then the resulting Voronoi model. For instance, to compute a V1, we use purely random seeds, while to obtain a V4 diagram, it would be required to apply 3 times the Lloyd’s algorithm to random seeds. V8 diagrams provide a polygonal organization in apical surfaces as experimentally observed (Main Text and Fig. S4). Subsequent radial sections that define computational tubes with different surface ratios were obtained by implementing a radial projection of the Voronoi seeds. For each apical surface of the tube, we generated 40 expansions by incrementing the surface ratios (sb) using 0.25 steps: 1 (apical), 1.25, 1.5, …, 10 (maximum basal surface).
As for the 3D reconstruction of cells in Voronoi tubes, each set of seeds that characterizes cells on a given cylindrical section defines a unique 2D Voronoi diagram at every surface and hence the corresponding 2D cellular domains. The set of 2D Voronoi regions that belong to the same radially projected seed from the apical to the basal surface then define each 3D cellular shape. Each of the obtained 3D Voronoi cells was further processed using the Matlab function ‘alphaShape’ to transform the set of voxels into a compact, solid, object. This reconstruction pipeline was implemented using Matlab (2018a). Code available at https://osf.io/nd5t6/.
Voronoi tubular model measurements
We measured the following properties of cells in Voronoi tubular models: number of sides of cells for a given radial section, and total number neighbors.Additionally, we computed the percentage of scutoids, the number of apico-basal transitions, the polygon distribution of every surface (radial sections). In these quantifications, we disregarded cells at the boundaries (tips of tubes) to avoid ‘border effects’.
In Voronoi tubes the net gain of 3D neighbors is bounded
Assuming a cylindrical geometry (e.g., epithelial tubes), each point at a given radial surface can be represented into the Cartesian plane; where coordinate x accounts for the cylindrical transversal coordinate and coordinate y for the longitudinal one (see Box). Thus, if the coordinates of a point (e.g., a Voronoi seed) at the apical surface are given by (x,y), the coordinates of that point at a surface with a value of the cylindrical radial expansion s ∈ [1,∞) can be found by defining the function fs: ℝ2→ℝ2 fs(x,y) = (sx,y). Under these conditions, we aim to characterize the seeds that generate scutoids (exchanges in the neighboring relations of seeds) as s changes.
Given three non–colinear points {A,B,C} that define a circle (a nearest-neighbors relation), and another exterior point D, if s > 1 exists such that fs(D) is interior to the circle defined by (fs(A), A(B),A(C)}, then D is inside of the vertical parabola containing {A,B,C} (Box).
If two of the three points {A,B,C} are on the same vertical line, then the parabola considered in Lemma 1 degenerates as a vertical strip. Even in this case, the thesis of the Lemma is true if we replace the interior of the parabola by the inside of the strip.
Proof. Without loss of generality, we can suppose that {A,B,C} are counterclockwise oriented. Thus, the point D(x,y) is outside the circle defined by {A,B,C} if, and only if, the sign of the following determinant is negative:
For the sake of simplicity, we represent the previous equation as:
On the other hand, by considering x and y as variables, the equation corresponds to the circle defined by {A,B,C}, and corresponds to the vertical parabola defined by the same three points. Consequently, the inequality defines the locus of interior points to that parabola.
Now, assuming that s > 1 exists such that fs(D) is interior to the circle defined by 〈fs(A), fs(B), fs(C)}. Then,
Or, equivalently, , so, . If , then and therefore . The latter is in contradiction with . As a result, , and the following inequality holds,
Notice that if the circle defined by {A,B,C} is surrounded by a set of points and we change continuously the parameter s in the interval [1,∞), it is possible to detect the first point touching the circle defined by {fs(A)fs(B),fs(C)}. That point can be obtained by computing all the points at . Hence, the first point contacting the circle will be that with the minimum value of s.
As for proving that the average of the number of neighbours of a cell induced by a seed grows is bounded as a function of the surface ratio, we state the following proposition:
Given a Voronoi seed representing a cell, if n3D(s) is the total number of accumulated cell neighbors as s increases from s = 1 (apical surface) to a given value of s, then 〈n3D(s)〉 is a bounded function for a finite cylinder.
Proof. We model the apical surface as the cylinder 2πr × h, where r representes the inner radius and h the length of the cylinder. Given a seed A in that surface, in the corresponding Delaunay triangulation it appears as a point surrounded by triangles defining the neighbourhood of A. By Lemma 1, each triangle defines a vertical parabola and a circle. So, any other seed touching A in other layer must be inside of one of the parabolas and outside of all circles (see Box). Let’s denote the feasible region for a new neighbour of A in the layer represented by s, i.e., all points inside one of the parabolas and outside all the circles. Thus, if is the number of seeds in that region that are not neighbours of A in the apical surface, obviously, an upper bound to the number of new neighbours to A is given by .
On the other hand, that number of seeds is, in average, proportional to the density of seeds times the area of , therefore, the average number of accumulated neighbours of A, denoted as 〈n3D(A)〉, will be bounded by the change of the density of points when growing s, this is to say, where M represents the total number of seeds (i.e., the total number of cells that is a constant) and the quotient is the area of divided by the area of a given radial layer. In general, it is not possible to integrate equation (4), since the area of is known only in very few, particular, cases.
If the case of a finite cylinder, leads, suming up to all the seeds and dividing by M, to the upper bound thus, 〈n3D(s)〉 is necessarily a bounded function. This expression indicates that the number of new neighbours when increasing s exhausts since the number of cells is a resource shared by all the layers. It is possible to obtain an upper bound to 〈Nmax〉 = lims→∞〈n3D(s)〉 since, after a flip in the Delaunay triangulation, the edge disappearing (i.e., a cell contact loss) can never be recovered in a cylindrical geometry. Thus, M · (〈Nmax〉 – n3D(1)) is bounded by the number of edges that complement the original Delaunay triangulation on the apical surface, that is, leading to
Where we have assumed that 〈n3D(1)〉 = 6. The simulations of the computational Voronoi model and the data of the salivary gland show that 〈Nmax〉 is in fact much smaller that the theoretical bound .
Relation between total accumulated 3D neighbors and the number of intercalation events
Scutoids have a Euler characteristic χ = 2 such that V – E + F = 2, where V, E, and F accounts for the number of vertexes, edges, and faces respectively. We assumed that the apical, a, and basal, b, faces of scutoids tesellating a cylindrical space have radial coordinates Ra and Rb respectively. Then, for any value of the surface ratio expansion, s = R/Ra, these solids can be mapped into a connected plane graph with the same Euler characteristic (a sort of projection of the vertexes and connectors into the plane, see Fig. S5. Thus, as a function of s, the accumulated number of 3D neighbors reads n3D(s) = E(s) – V(s). Since in tubular geometries the centroids of tessellating scutoids always separate from each other as s increases (i.e., apico-basal intercalations are not reversible) for a single scutoid, where {V(s)} = {V(1),V(1 + ds), …, V(sb)} and i(s) denotes the number of intercalation points in the interval s ∈ [1,sb]. In the case of a 3D tessellation with N cells, where M of them do not show any intercalation, the total number of accumulated neighbors reads,
Given that each intercalation point is shared by four cells, two of them necessarily increase their number of vertices in a given s-plane and two of them decrease their number of vertices (see Fig. 1a). Thus, in the case of a decrease max({V(j)(s)}) = V(j)(1) and in the case of an increase . Consequently, where we used the fact that for every intercalation event that increases by one the number of neighbors there is one that decreases the number of neighbors in the same amount; consequently, we can add up all intercalation events and divide by two. Hence the average number of accumulated 3D neighbors, 〈n3D(s)〉 = n3D(s)/N reads being the average number of apico-basal intercalations per cell. Finally, by considering that any s-surface, and in particular the apical surface s = 1, corresponds to a 2D tessellation of convex polygons, 〈V(1)〉 = 6 we conclude that,
A Kolmogorov rate equation for the 3D cellular connectivity
The probability, P, of having m accumulated 3D neighbors (i.e., m = n3D) as the surface ratio increases from s to s + ds can be described by the following Markov equation (Fig. 4b), where Ti,j is the probability of incrementing the number of neighbors from i to j due to an apico-basal intercalation. Since ΣjTi,j = 1 (normalization of the transition probabilities) and (each intercalation can only possibly induce to win one neighbor) then Tm,m = 1 – Tm,m+1 and the above Markov equation can be written as a Kolmogorov equation (a.k.a. Master equation): where ri,j accounts for the probability of apico-basal intercalations per unit of surface ratio, i.e., Ti,j = ri,jds.
If we assume an Arrhenius-like kinetics (Bi et al, 2014) then , where is the so-called pre-exponential factor that modulates the “bare” frequency of intercalations (per unit of surface ratio expansion) and ΔEi is the activation energy in some energy units, e.g., in units of Eo (the value of the energetic barrier of the four-fold vertex configuration). The observed “poor get richer” behavior suggests that the activation energy, ΔEi increases with i. For the sake of simplicity, up to first order in i: ΔEi = i · β (β being the dimensionless activation energy of a cell per 3D neighbor in units of E0). On the other hand, the mathematical calculations (see Eq. (9)) indicate that the intercalation rate ri,i+1 becomes null for a finite value of i or, alternatively, that the activation energy becomes infinite for a finite value of i. Otherwise, the net gain of new neighbors is not bounded. This fact can be accounted for by assuming that the bare frequency is a function of the number of neighbors, , such that and becomes null for a finite value of i. Again, for the sake of simplicity, we assume that up to first order in , where Nmax is the asymptotic, maximum, number of 3D neighbors a cell can possibly have. Summarizing, we assume that the apico-basal intercalation rate ri,i+1 reads,
Under these conditions, the Kolmogorov equation reads,
On the other hand, the equation satisfied by the average number of accumulated 3D neighbors, 〈n3D〉 = 〈m〉, reads,
Alternatively, in order to obtain an analytical expression able to recapitulate, effectively, the mathematical principle that govern the net gain of 3D neighbors, we perform the following approximations. First, we perform a mean-field-like approximation, i.e., 〈F(m)〉 ≈ F(〈m〉),
Where 〈Nmax〉 is the limiting average cellular connectivity. Second, since β < 1,
Equation (21) is formally a logistic-like growth equation, that has as solution,
Thus, if c > 0 then . The parameters b, c, and d are further constrained by the following facts: (3D neighbors can only accumulate) and (“poor get richer” principle). Moreover, if we impose the condition 〈m(1)〉 = 6 (the average number of neighbors in the apical surface is 6) these parameters are not independent since,
All the above implies that the logistic-like fitting function, Eq. (24), describes, approximately but effectively, the analytical mathematical law (“Flintstone’s law”) underlying the 3D average connectivity if the following conditions hold, either −1<d<0 or c ln(–d)< 1 if d < –1.
The relation between the fitting parameters of the logistic fitting with α and β are,
For finding the parameters α and β in in silico tubes and salivary glands we then implemented two possible approaches. On the one hand, we implemented an error minimization algorithm that recursively solved, numerically, Eq. (20) to obtain 〈m(s)〉 = ΣmmPm(s) taking also into account the normalization condition (code available at https://osf.io/nd5t6).
On the other hand, we obtained values using the fitting logistic function Eq. (24). We notice that the values obtained through the first method are exact as compared to the values obtained from the fitting that are based on a series of approximations as explained above (see Table S1).
The values of α and β are obtained from the exact methodology were further used to compare the predicted probability distribution of having m accumulated neighbors for a given value of s: Pm(s). We evaluated the relative error of this prediction with respect to the actual distribution from data, , by computing . This quantity is normalized such that in case of the following situation of full disagreement between the distributions, and Pm(s) = δm,j with i ≠ j, provides ε2 = 1 (i.e., 100% error).
Quantitative characterization of spreading in neighbor exchange distributions between apical and basal surfaces
In order to characterize the spreading away from the diagonal in neighbor exchange distributions between apical and basal surfaces, Fig. 2a-b, we followed the same approach used to quantify intrinsic noise during gene expression processes (see (Elowitz, 2002). Thus, where representing any function of na and nb and p(na,nb) being the probability of neighbor exchange events.
Logistic data fitting
To obtain the logistic function that fit best our data points, we analyzed all the possible parameters combination and achieved the global minimum solution. This ‘fit’ was based on the ‘least squares’ method and minimizes the residual where, yi and yi′ stand for the observed values and the fitted ones, respectively. The logistic equation (Eq. (24)): was then fitted to find the average 3D cell connectivity, but with a series of constraints on the parameters (as explained above): and if d < –1 then cln(–d) < 1. The goodness of fitting was estimated by means of the coefficient of determination, R2.
Data availability
All the necessary material to reproduce this study is available at the Center for Open Science repository: https://osf.io/nd5t6.
Figure S1. Scutoidal prevalence and relation between 〈n3D〉 and 〈i〉.
a) The density plot shows the percentage of scutoidal cells in the tissue (scutoidal prevalence) as a function of the surface ratio (from s = 1 to s = 4 with steps of 0.25) and the Voronoi class (same samples that in Fig. 1d). We notice that from s = 4 onward, the scutoidal prevalence in all Voronoi diagrams is 100%. b) Average number of 3D neighbors, 〈n3D〉, as a function of the average number of apico-basal intercalations per cell, 〈i〉, in Voronoi tubes (from s = 1 to s = 10 with steps of 0.5, 20 samples) and in salivary glands (from s = 1 to s = 3.5 with steps of 0.5,20 samples).
Figure S2. Poor get richer principle in Voronoi tubes.
Average net gain of neighbors (density plot) with respect to the apical surface in Voronoi tubes with surface ratios sb = 1.5 and sb = 2 (a), and with respect to the basal surface for sb = 5 (b) as a function of the Voronoi and the polygonal class (n = 20).
Figure S3. Comparison between results obtained in the Kolmogorov model and simulations of V1 and V10 in silico tubes.
As in Fig. 3b, the left/center density plots represent the connectivity distribution (i.e., the fraction of cells with a given number of 3D neighbors) as a function of the radial expansion obtained in the Voronoi simulation (left) and as predicted by the Kolmogorov model (center); the purple open circles (left/right) indicate the average number of 3D neighbors per cell 〈n3D〉; the solid red line and the dashed white line (center/right) shows 〈n3D〉 as obtained by the Kolmogorov model and the Flinstones’ law respectively. The density plot on the right shows the difference between the predicted and the actual connectivity distributions and the corresponding error, ε2 (magenta lines).
Figure S4. Polygon distribution in apical and basal surfaces and “poor get richer” principle: salivary gland and V8 tubes.
a) Polygon distribution of salivary glands and Voronoi 8 (V8) in silico tubes: the error bar accounts for the standard deviation (n = 20). b) Average net gain of neighbors (density plot) with respect to the apical surface in salivary glands (top) and V8 tubes with a surface ratio sb = 1.75 (middle) and sb = 4 (bottom) as a function of the apical polygonal class (n = 20). c) Absolute net gain of 3D neighbors as a function of the polygonal class in the apical surface (green: salivary gland; blue: V8 tubes). The size of the circle accounts for the relative data count within the apical polygon class (integer numbers indicate the number of cells that gained 3D neighbors). The boxes indicate the 25% – 75% percentile interval, black lines the mean values, gray lines the standard deviation, and the red dotted lines the statistical median. Cells with a smaller polygonal class are more prone to gain neighbors.
Figure S5. Euler characteristic in scutoids.
A scutoid solid can be simplified as a collection of vertexes (circles) that can be connected by straight edges. In the figure the green circle accounts of an apico-basal intercalation point. Alternatively, we can represent the solid as a connected plane graph. In both cases the Euler characteristic is 2.
Table S1. Fitting parameters (Flintstones’ law and Kolmogorov model) and spreading quantification in Voronoi tubes and salivary glands.
Tab 1 (Flintstones) shows the independent fitting parameter values (c,d,〈Nmax)〉, the values of b, 〈n3D〉 for different, relevant, values of the surface ratio, and the estimated values of α and β (Materials and Methods). Tab 2 (Kolmogorov) shows the independent fitting parameter values (α,β,〈Nmax〉). Tab 3 (Spreading) shows the spreading values of the 3D histograms of neighbor exchange between basal and apical surfaces (Materials and Methods).