Fundamental physical cellular constraints drive self-organization of tissues

Abstract

Morphogenesis is driven by small cell shape changes that modulate tissue organization. Apical surfaces of proliferating epithelial sheets have been particularly well studied. Currently, it is accepted that a stereotyped distribution of cellular polygons is conserved in proliferating tissues among metazoans. In this work, we challenge these previous findings showing that diverse natural packed tissues have very different polygon distributions. We use Voronoi tessellations as a mathematical framework that predicts this diversity. We demonstrate that Voronoi tessellations and the very different tissues analysed share an overriding restriction: the frequency of polygon types correlates with the distribution of cell areas. By altering the balance of tensions and pressures within the packed tissues using disease, genetic or computer model perturbations, we show that as long as packed cells present a balance of forces within tissue, they will be under a physical constraint that limits its organization. Our discoveries establish a new framework to understand tissue architecture in development and disease.

Keywords: Voronoi diagrams; epithelial organization; morphogenesis; neuromuscular diseases.

Figure 1. Polygon distribution of biological tissues compared with Voronoi diagrams

1. Segmentation of a Drosophila prepupal wing imaginal disc epithelium (dWP).

2. Biceps control adult (BCA) biopsy stained with anti‐collagen VI (green) and anti‐myosin heavy chain (red) antibodies.

3. Segmentation of the image showed in (B).

4. Segmented image from a chicken embryo neural tube epithelium (cNT).

5. Polygon distribution of the different biological tissues analysed. Two types of distribution can be distinguished: the Poisson‐Voronoi polygon distribution for Diagram 1 (D1) and cNT, and the conserved polygon distribution for the rest of image categories. Data are represented as mean ± SEM. Diagrams 1, 5 and 6: 20 replicates. cNT: 16 replicates. dWP: 16 replicates. Gibson et al.: 12 replicates. BCA: 29 replicates.

6. Diagram with Voronoi seed distributed randomly in the plane.

7. Poisson‐Voronoi tessellation of the seed showed in (F).

8. Resulting Voronoi diagram after performing four Lloyd's algorithm iterations to the Diagram 1 (Diagram 5, D5).

Data information: Scale bar, 10 μm in (A) and (D); 150 mm in (B).

Figure 2. The CVT path matches the polygon distribution of natural tissues

1. A

   Polygon distribution Voronoi diagrams resulting from the iterative application of Lloyd's algorithm. Data are represented as mean ± SEM. All diagrams: 20 replicates.

2. B–E

   Density plot showing the frequency of a value for four‐, five‐, seven‐ and eight‐sided cells, respectively, depending of the percentage of hexagons (P6) in the CVT diagrams. The darker region reflects the higher probability of a determined value. Coloured circles represent the average values of each polygon class for the different sets of images analysed in this study. dWL, Drosophila larva wing disc, light green. dMWP, Drosophila mutant wing disc, violet. EYE, Drosophila eye disc, orange. BCA, biceps control adult, dark green. BNA, biceps neurogenic atrophy, dark blue.

Figure EV1. Polygon distribution for CVT diagrams from 1 to 200, dWL and Volvox images

1. Comparison of the different polygon distributions along the CVT path. The number of hexagons never reached more than 70% of the total number of cells. Data are represented as mean ± SEM. All diagrams: 20 replicates.

2. Volvox is a green alga that forms spherical colonies. We compared the data obtained from Korn and Spalding (1973) with the different CVT diagrams. The Volvox polygon distribution was very similar to a Voronoi Diagram 12. Drosophila larva wing disc epithelium (dWL) polygon distribution was very similar to the one for Diagram 4.

Figure 3. Relationship between area and polygon distribution

1. Area distribution of several Voronoi diagrams along the CVT path. Cell areas became progressively more homogeneous after each Lloyd iteration.

2. Comparison of area distribution for proliferating epithelia cNT (blue), dWL (green) and dWP (red). At the right of the image biceps (green) and Diagram 5 (grey) images.

Figure EV2. Analysed packed tissues largely hold to several geometrical laws

1. Lewis' law. All samples show a linear relationship between cell area and the number of cell sides (z). Each solid point represents an average of cell areas across all samples. Blue lines show a linear fit to the data.

2. Aboav–Weaire law. All samples show a linear relationship between the number of cell sides (z) and the product of this quantity with the average number of sides of a cell's neighbour (m). This demonstrates that all samples follow the predicted trend.

3. To see whether there are any minor deviations from the Aboav–Weaire law, we plotted the inverse of the number of cell sides (1/z) against m. The slight curvature in the profiles demonstrates that there is in fact a minor deviation for most samples, which is consistent with previous studies of random Voronoi tessellations (Hilhorst, 2006).

Data information: In both (A) and (B), solid points represent averaged quantities across all samples, while blue lines show a linear fit to the data.

Figure 4. Polygon and area distribution of altered tissues

1. Segmentation of a prepupal eye imaginal disc from Drosophila (anterior to the left). The analysed developing photoreceptors are labelled in green. The cells adjacent to the photoreceptors are labelled in blue (these are “cone cells” in the most posterior clusters).

2. Segmentation of a prepupal wing imaginal disc epithelium where myosin II levels have been reduced.

3. Adult biceps biopsy from a patient with neurogenic atrophy pathology. Fibres outlined with collagen VI antibody (green).

4. Segmentation of the image showed in (C) highlighting two atrophic elongated fibres that present seven neighbours.

5. Comparison of polygon distributions of different tissues with altered contractibility, tension or pressure balance within the tissue. Data are represented as mean ± SEM. Diagram 1 and 2: 20 replicates. EYE: 3 replicates. dMWP: 10 replicates. BNA: 12 replicates.

Data information: Scale bars, 10 μm in (A) and (B); 75 mm in (C).

Figure EV3. Comparison of area distribution of diverse natural tissues

1. Comparison of the area distribution of Diagram 1 and EYE. The eye presents a left‐skewed bimodal distribution of areas.

2. Comparison of the area distribution of dWP and dMWP. The mutant wing presents a more irregular frequency of sizes and slight left skewness.

3. Comparison of the area distribution of BCA and BNA. The high number of small atrophic fibres characteristic of neurogenic atrophies produces a left‐skewed distribution shape in the case of BNA.

Figure 5. Biophysical alterations in proliferating tissues using computer simulations

1. A

   Diagram resulting from a vertex model simulation that includes cell proliferation.

2. B

   Diagram resulting from a vertex model simulation that includes cell proliferation and heterogeneous reduction of line tension among the tissue cells.

3. C

   Diagram resulting from a vertex model simulation with heterogeneous reduction of line tension and an impairment of the cell division when tension value is under the 30 percentage of the initial value.

4. D

   Similar simulation than in (C) with a threshold of 40 percentage.

5. E–H

   Density plot showing the frequency of a value for four‐, five‐, seven‐ and eight‐sided cells, respectively, depending of the percentage of hexagons (P6) in the CVT diagrams. The values for the individual myosin mutant images (violet) and the individual diagrams of the four simulations (light grey, grey, light pink and pink) are shown in each plot. The average value for each simulation is shown with a black circumference.

Figure 6. Computer simulations indicate that changes to homogeneous resting areas are able to alter cell topology

1. Diagram resulting from a vertex model simulation with homogeneous parameters for contractility, line tension and ideal area.

2. Diagram resulting from a vertex model simulation where in ten percentage of the cells (grey), the line tension and ideal area parameters were reduced.

3. Control diagram where in the ten percentage of the cells (blue) only line tension parameter was reduced.

4. Comparison of polygon distribution for the 10% smallest cells from BCA (green bar) and control simulation (grey bar) with the sick cells from BNA (light blue bar) and atrophy simulation (grey cells from (B), purple bar).

Figure EV4. Polygon distribution of different subset of cells from muscle samples and simulations

Comparison of the polygon distribution of different subsets of cells: the atrophic cells of the BNA images (see Materials and Methods), the 10% of the smallest cells from BCA and the three simulations, and the selected “sick” cells with different parameter values from the atrophy simulation and ideal area = 1 simulation (see Materials and Methods). “control simulation 10%”, “ideal area = 1 simulation 10%”, “BCA 10%”, “BNA 10%” and “atrophy simulation 10%” presented a higher percentage of pentagons than hexagons. On the other hand, when analysing the “atrophic cells” from “BNA sick” and “atrophy simulation sick,” an increase of hexagons and heptagons was observed. “Atrophy simulation sick”, “atrophy simulation 10%” and “atrophy simulation sick and 10%” presented a similar polygon distribution since in this simulation most of the sick cells are included in the 10% smaller cells (Table EV2). The small difference between “BNA 10%” and “BNA sick” reflected the presence of small rounded cells inside the smaller 10% of BNA cells. The polygon distribution of the “ideal area = 1 sick” cells was due to the higher area presented by this subset of cells that gives them higher number of neighbours (Fig 6C and Table EV2).

Figure EV5. Original images processed to obtain segmented data

1. A

   Projection of the original confocal stack used to segment dWP3 sample.

2. A'

   dWP3 final segmented image.

3. B

   Projection of the original confocal stack used to segment dWP3 sample. Some confocal sections included signal from the peripodial membrane of the wing imaginal disc. Therefore, part of the processing was manual.

4. B'

   dMWP3 segmented final image. The signal from the peripodial membrane has been removed.

Data information: Scale bars: 10 μm.