Cell-size distribution in epithelial tissue formation and homeostasis

Abstract

How cell growth and proliferation are orchestrated in living tissues to achieve a given biological function is a central problem in biology. During development, tissue regeneration and homeostasis, cell proliferation must be coordinated by spatial cues in order for cells to attain the correct size and shape. Biological tissues also feature a notable homogeneity of cell size, which, in specific cases, represents a physiological need. Here, we study the temporal evolution of the cell-size distribution by applying the theory of kinetic fragmentation to tissue development and homeostasis. Our theory predicts self-similar probability density function (PDF) of cell size and explains how division times and redistribution ensure cell size homogeneity across the tissue. Theoretical predictions and numerical simulations of confluent non-homeostatic tissue cultures show that cell size distribution is self-similar. Our experimental data confirm predictions and reveal that, as assumed in the theory, cell division times scale like a power-law of the cell size. We find that in homeostatic conditions there is a stationary distribution with lognormal tails, consistently with our experimental data. Our theoretical predictions and numerical simulations show that the shape of the PDF depends on how the space inherited by apoptotic cells is redistributed and that apoptotic cell rates might also depend on size.

1. Introduction

Growth regulation is a key property of living tissues needed to ensure proper functioning. To this aim, strict control of cell proliferation and death, differentiation, metabolism and migration is enforced. All of these cellular processes can be regulated both in space and time in order to give rise to a correctly shaped tissue during development, to regenerate parts of damaged tissues during wound healing and to provide a sufficient degree of self-renewal during homeostasis [1,2].

Cell volume growth regulation and coordination between growth and cell cycle at the single cell level has been the subject of many studies in the context of microbial populations, where cells can either regulate the time between two divisions, the size at which division occurs or the amount of volume added after division (e.g. [3–7]). In metazoans however, cell volume growth is the result of autonomous growth and the mitogenic homo- or hetero-typic paracrine signals coming from neighbouring cells and tissues [1,8].

Growth at the tissue level also results from the spatio-temporal control of cell proliferation and apoptosis, both of which can be driven by neighbouring cells signalling and/or autonomous regulation. Prototypical examples of tissues that must satisfy such constraints are provided by epithelial tissues, which can be formed by fundamental sub-units such as glands, or by layers of cells with different differentiative capacities, depending on the function of the tissue. In these tissues, a precise equilibrium must be maintained in order to keep cell density constant and avoid hyperplastic and/or hypertrophic regions or more severe consequences such as tumour growth. Disruption of spatial ordering of cell division and altered division or apoptotic rates are common initiating steps of many cancer lesions and other pathologies related to defects in morphogenesis [9,10].

Growth disregulations are avoided by cues that can be either cell-autonomous or involving cell–cell interactions, including mechanics [11], paracrine inter-cellular or extracellular signalling. Single cell level mechanisms of growth regulation include controlled apoptosis, controlled cell division/differentiation [1,12–14] and compensatory growth [15,16].

A notable, well-known example of regulation of multicellular growth is contact inhibition, i.e. the arrest of proliferation of non-transformed cells that reach a critical density. In culture, single isolated cells undergo a phase of uncontrolled cell proliferation until they reach a state called confluency, consisting in the coverage of the locally available free surface of the culture dish [17,18]. At this point, cells start to divide in a more controlled fashion, with rates that depend on size [14,19,20]. Larger cells still divide at a relatively fast pace, while smaller cells start to divide slower and slower. At this stage, the total available space (away from boundaries) does not change so that, except for a slight growth in the direction perpendicular to the dish, cells undergo several rounds of purely size reduction divisions, thereby halving their volume at each division. It was shown that during this phase, cell volume between one division event and the next one is also approximately constant [19,21].

As there is no volume exchange, cell-size distribution is uniquely determined by the timing of cell division events. It can be easily shown (see the electronic supplementary material) that if cell division were a Poisson process, with constant rates, cell size heterogeneity would diverge, allowing large cells to coexist in tissues together with very small cells, a condition that is not observed experimentally and instead represents often a hallmark of altered tissue functions. For example, pleomorphisms, i.e. increased variability in cell size and shape, are often associated with tumours [1]. Growth anomalies (either due to hypertrophy or to hyperplasia) are studied in adipose tissues by looking at cell size distribution changes [22–25]. Cell heterogeneity is at the basis of several endotheliopathies in the cornea, where it represents a problem for tissue integrity [26–30].

The problem of ensuring correct size distribution in a tissue is also very relevant during homeostasis, when rare proliferation is in equilibrium with apoptosis [31,32]. Local, fast, hypertrophic activity is normally observed in response to apoptosis, in order to preserve correct tissue permeability, and slower compensatory proliferation is also observed. However, the details of how homeostatic equilibrium maintains cell size homogeneity are not known.

Recently, it has been observed that cell-size distributions of commonly cultured epithelia and of in vivo tissues [28,33,34] are close to a lognormal with a relatively small variability in cell size. This observation has also been reported for microbial populations [35–38]. How is this distribution preserved across generations? How can cell appearance and loss within tissues be orchestrated in order to maintain size heterogeneity [39]?

In this paper, we apply the theory of fragmentation [40] to the context of tissue growth, both in the case of confluent proliferating tissues and in the case of homeostasis. Our theoretical approach provides a framework to study cell-size distributions in the absence of net growth and allows to theoretical predictions to be obtained that explain the experimentally observed distributions. We show that experimentally observed single cell level regulation of cell proliferation results in self-similar cell-size distributions and therefore, by definition, preserves homogeneity. We also show that equilibrium between proliferation and death yields lognormal distributions and explains the effect of cell-size redistribution among neighbours. Importantly, our theoretical model lends itself to other generalizations such as skin epithelia and tissues with limited self-renewal ability.

2. Results

2.1. Growth regimes in confluent epithelial cultures

When cultured at sub-confluent densities, i.e. when cells do not cover the whole available space in the culture dish, MDCK (Madin–Darby canine kidney) cells proliferate with a constant rate of about 6.7 × 10⁻² h⁻¹. Each time a cell divides, the daughter cells reach up to the same projected area of the mother cell (as shown in figure 1a,b and electronic supplementary material, figure S1A). This holds true when cells form small colonies as well, and have already established cell–cell contacts [19].

Figure 1.

Growth regimes in confluent epithelial cultures. (a) Cartoon illustrating the different regimes of growth and proliferation in epithelial cultures. The red line shows the behaviour of the average cell area over time. Sub-confluent cells, i.e. when cells do not cover all available space in the dish, have constant average area and are free to grow and spread over the dish. Upon cell division, each daughter cell grows back to the same size of the mother cell, and space is set free by neighbouring cells moving away. Cell proliferation occurs every 15 h on average, regardless of size. This regime is shaded in green in the sketch (sub-confluent regime). Whenever space is covered (blue shade, size-reduction regime), cells start dividing with a rate that depends on size, larger cells faster than smaller cells, and each daughter is half the size of the mother and growth is inhibited. Average area in this regime drops of about a factor of 10. Apoptotic events in this regime are extremely rare and negligible with respect to proliferation rates. This ceases to be true when proliferation events become comparably rare due to reduced cell size. In this regime (red shade, homeostatic regime), apoptosis and proliferation balance on average, and average area remains constant. (b,c,d,e) Snapshots illustrating the various regimes of epithelial cell growth in vitro, sub-confluent (b), size-reduction regime (c,d) and homeostatic (e). In each panel two subsequent snapshots are shown to illustrate cell division events. Mother cells are indicated in the left panel, while the corresponding daughter cells are indicated in the right panel. In panel (e) an apoptotic event is also shown, where neighbour cells are indicated with coloured dots, and the apoptotic cell with a black dot. A co-occurring division event in the neighbourhood is also shown. Images are taken using phase contrast microscopy (b) and Ecad-GFP fluorescence, indicating cell edges (c–e). Scale bar for all panels: 20 μm. (Online version in colour.)

As soon as cells become confluent, i.e. when they occupy all the available space, volume growth is substantially arrested and cells only take up approximately half the volume of the mother (figure 1a,c,d). In this regime, proliferation rates become size-dependent and larger cells divide faster than smaller cells [19]. This regulation has been shown to occur during the G0/G1 phase [21], and has been directly linked to the lack of available space for the cells to grow. Indeed, when mechanically stretched in order to artificially increase the size of cells, cells start dividing again [21]. During the post-confluent but not yet homeostatic regime, cells continue to divide, therefore average area drops. For this reason, we shall refer to this regime as ‘size reduction’. Rare apoptotic extrusions can be observed during this regime at a rate of approximately 0.02 per day per cell.

When cell size is sufficiently small, division rates become comparable to apoptotic rates and a homeostatic regime is reached. In this regime, there is no variation in average cell size, and cell death and birth balance out. No proliferation events are observed to follow directly an apoptotic event: instead, the space left free by the dying cell is filled up quickly by neighbouring cells, as shown in figure 1e. Proliferation is therefore triggered indirectly by the (moderately) increased cell size of the neighbours.

For the cases of confluent tissue (size-reduction and homeostatic) described above, cell size distribution is characterized by relatively small variability (around 30%). These two distributions however refer to two very distinct dynamical regimes: one of continuously dividing cells, getting smaller and smaller in size, and one where average cell size is constant. The dynamics of cell-size distribution can be described by two different theoretical models, detailed in what follows.

2.2. Size-reduction regime

A two-dimensional epithelial tissue, containing a large number of proliferating cells, can be described as a linear irreversible fragmentation process. Fragmentation is the process by which particles tend to break up forming smaller fragments. Such processes have applications in biology, chemistry, astrophysics and meteorology. Fragmentation processes can either be reversible or irreversible, depending on whether fragments can coagulate upon colliding to form larger objects or not. Linear fragmentation is obtained whenever breakup is produced by a homogeneous external agent, as opposed to when the cause of fragmentation can depend on fragments themselves, as in fragmentation induced by collision [40]. In what follows, we present the application of the theory of fragmentation to the evolution of a tissue due to cell division.

Linear irreversible fragmentation is described by the following integro-differential equation [40]:

2.1

where we have indicated with N(A, t) the number of cells with area A at time t. The breakup of such fragments (cell division) is happening at a rate ρ(A) which can depend on size, and cells of sizes A smaller than B are produced at a rate f(A | B). Motivated by experimental observations presented in [19,21], we choose a homogeneous breakup rate ρ(A) = ρ₀A^α, and f(A | B) = 4δ(B − 2A), i.e. breakup produces two cells of equal size. Under these assumptions, the total area is trivially conserved, and equation (2.1) becomes

2.2

A graphical illustration of the above model is shown in figure 2b. This equation does not have a stationary solution, as cells continue to divide albeit slower and slower. What we want to verify however is how the distribution dynamically change over-time, and in particular, whether variability is decreasing or increasing.

Figure 2.

Fragmentation model for size-reduction and homeostatic regime in confluent epithelial cultures. Graphical illustration of the model described by equation (2.2) and (2.6). (a) The top row sketches the behaviour of a cell size during time, while cells are dividing and undergo apoptosis. (b,c) The middle and bottom rows represent the graphical explanation of the master equation. In (b), cells of area 2A come from the left (larger) area interval and end up in the centre interval. Cells in the central area interval end up in the right (smaller) area interval. In (c), a cell of area A / (1 + 1 / k) undergoes apoptosis and redistributes area to k randomly chosen cells of the same area, which thereby become cells of area A. (Online version in colour.)

To this aim, a scaling solution to equation (2.2) can be found by considering Q(z) = A²N(ρ₀A^αt,t), obeying the following differential equation:

2.3

for which we found an analytical solution, reported in the electronic supplementary material. Importantly, Q(z) has an asymptotic shape which can be regarded as the area occupied by cells that divided n times for large n. Therefore, after a few rounds of division, this non-stationary process has universal properties, i.e. that do not depend on the initial conditions, and these can be studied. As confirmed by direct comparison between numerical simulations of the process (2.1) and theoretical predictions, the function Q(z) describes the temporal evolution of the cell-size distribution, and is a travelling wave with a logarithmic-in-time displacement, as shown in figure 3b (inset). Over time, large cells (sitting on the right side of the PDF) divide and end up on the left side of the distribution, thereby producing a movement of the PDF towards smaller areas.

Figure 3.

Size-reduction regime. (a) Sketch illustrating the theoretical model: large cells are more likely to divide than small cells at rates that depend on cell size as a power law. Cells split perfectly upon division. (b) Numerical simulations (coloured dots) of the theoretical model described by equation (2.2) are compared with theoretical predictions (solid lines) for α = 1. probability density functions (PDFs) are rescaled such to show self-similarity of the distribution. The area is rescaled with the average area and the PDF is rescaled to ensure normalization. (inset) Time evolution of the numerical PDF compared with theoretical predictions. Different times are indicated by different colours as indicated in the colour bar. The series for Q(z) is truncated after the first 20 terms. (c) Average area elevated to the power α for α = 1,2,3 (red triangles, green squares, blue pentagons) for Poissonian (filled symbols), or gamma (empty symbols) distributed division times. Continuous lines are the theoretical predictions of the model for the Poisson case and are in agreement with numerical simulations. The moments are plotted so that they all scale as x⁻¹, as shown by the black triangle. Time is rescaled with the basal proliferation rate. (inset) Average area for the case α = 0 for Poissonian (filled symbols) or gamma (empty symbols) distributed division times. (d) Moments of the area distribution as measured in experimental data compared with theoretical predictions for empirically fitted values of α. Initial discrepancy of data from predictions might be due to a crossover between sub-confluent and confluent regimes. The plateau for large times is due to homeostasis. Colour shades refer to the regimes showed in figure 1a. (inset) Rescaled moments compared with theoretical predictions with α = 3. The latter oscillate in time with logarithmic periodicity within the coloured bands. (e) Experimentally measured PDF collapsed by scaling arguments predicted by our theory. The area is rescaled with the predicted scaling for the moment (with α = 3) and the PDF is rescaled to ensure normalization. (Online version in colour.)

To understand how the shape of the distribution changes, one can calculate its moments which scale as:

2.4

and

2.5

The moments scale as the rescaled time, i.e. measured in units of duplication times, and the slopes depend on the exponent α, the stronger the dependence of duplication time on size, the slower the decay on time as intuitively expected. Importantly, the coefficient of variation of this distribution for α ≠ 0 is independent of time. When proliferation rate is independent of size (α = 0), the moments decay exponentially and in this case the coefficient of variation is diverging over time. Therefore, size-dependent division time might be a mechanism to preserve cell-size homogeneity in confluent epithelial tissues.

Given the time scaling of the moments, the PDF should rescale as , as shown in figure 3b. This has the implication that once properly rescaled, all PDFs measured at different times collapse into a unique function.

In order to verify the experimental validity of such predictions, we used the dataset of [19] and inferred the values of the exponent α and that of the basal rate ρ₀, obtaining α ≃ 2.7, ρ₀ = 4.9×10⁻⁷ μm⁻⁶A days⁻¹ (calculated for α = 3, see the electronic supplementary material). The comparison of the moments of the experimentally measured distribution and theoretical predictions is shown in figure 3d and show perfect agreement. As explained above, for α ≠ 0, the theory also predicts self-similarity of the distribution, which is confirmed with experimental data, as shown in figure 3e. Self-similarity is a property of a series of functions that collapse into only one function upon appropriate rescaling. In the context of cell size distribution, this property has the intuitive implication that upon rescaling with the average cell size, the size distribution will not depend on cell density. Therefore, cell size distributions of tissues taken right after confluency or before homeostasis would be indistinguishable once the size is computed with the rescaled size A/〈A〉.

It can be noted however that the collapsed shape of the experimental distribution is not the one predicted by our theory. The cause of this likely resides in the assumed Poissonian statistics of the division times. It was indeed shown in [19] that, at least in the non-inhibited regime, the distribution of the inter-mitotic times is better described by a gamma distribution with shape parameter κ = 52, and scale parameter θ = τ₂ / κ, where τ₂ is the average time between cell divisions. With this motivation in mind, we performed numerical simulations of a fragmentation process with gamma distributed breakup times, and found that the moments follow the same behaviour of the Poissonian case, as shown in figure 3c, and therefore that the distribution is also self-similar, although with a different shape.

It is worth noting that for gamma distributed times, the asymptotic shape of the cell-size distribution is more sensitive to the initial conditions as it would take longer times to become universal and therefore to be independent of the initial distribution. In the limiting case of purely deterministic division times, the initial distribution would be preserved.

Therefore, in the experimental case where cells undergo only about four rounds of division, if the distribution at confluency were to be lognormal, that initial distribution would be preserved over time by regulating cell division times with cell size.

2.3. Homeostatic equilibrium

Experimental values of apoptotic rates for [19] have been estimated to be smaller than 0.02 per day per cell and can therefore be safely ignored for over a week during the experiments. However, when division and apoptotic rates become comparable, i.e. when cell size becomes small (less than or equal to 20 μm²), apoptosis becomes relevant, and has to be taken into account. To this aim, we extended the fragmentation model to include cell death

2.6

where analogously to the previous case, η(A) is the apoptotic rate per cell, which in general depends on size.

This model now includes both fragmentation and area redistribution, and allows investigation of the situation where these two phenomena are in equilibrium.

It is assumed that upon cell death the area of the apoptotic cell is redistributed to k cells of equivalent area. In other words, once a cell of area A dies, k cells of area A(1 + 1 /k)⁻¹ will inherit its area, each obtaining a factor A(1 + k)⁻¹, and therefore becoming cells of area A. We will assume that both the division rate and the apoptotic rate depend on cell size as ρ(A) = ρ₀A^α and η(A) = η₀A^−β.

The moments of the PDF can be calculated by recurring to the Mellin transform: , obtaining the exact expression

2.7

where s_n = n(α + β) − α. For the simple case α = β = 0, the moments are either increasing or decreasing exponentially as in the pure division case and the coefficient of variation diverges, therefore a stationary solution does not exist. This observation is consistent with what was found for the size-reduction regime: size independent proliferation (or apoptosis) tends to increase dramatically the size variability within a tissue. In order for the heterogeneity of the distribution to be finite, one needs α + β > 0. In this case, it can be shown (see the electronic supplementary material) that the tails of the distribution are those of a lognormal, with parameters given by

2.8

The right part of the distribution is dominated by redistribution events, i.e. the effect of redistribution to a large number of neighbours k is that of reducing the overall variability in cell size as expected.

In general, equation (2.7) does not give an explicit expression for moments of all orders, but a few case studies can be addressed. In the case , all moments are integers and can be calculated exactly along with the coefficient of variation. Results for experimentally relevant values of α and β are shown in the electronic supplementary material.

To go beyond this simple theoretical model, we implemented a numerical model of fragmentation (figure 4a), which is analogous to the theoretical model presented above, except for the redistribution mechanism of area upon cell death. Indeed, owing to the finiteness of the number of cells in the simulations, the area upon cell death is redistributed among k cells picked at random, instead of selecting among those cells with the same area of the dying cell.

Figure 4.

Homeostatic equilibrium. (a) Sketch of the numerical model: the area left empty by apoptotic cells is inherited by k neighbours picked at random. Mitotic cells inherit half the size of the mother cell each. (b) Simulations of the model for α = 3 and varying β, showing that the effect of increasing β is that of modifying the left tail. (c) Simulations of the model for α = β = 3 and varying k, showing that redistribution changes only the right tail, as predicted theoretically and shown in the electronic supplementary material. (inset) Increasing both β and k lowers the coefficient of variation. Black dotted lines represent the experimentally measured value. Colour scheme as in the main panel. (d) Experimental PDF of the logarithm of the cell size (rescaled to mean) fitted with two normal distributions for left (green) and right (cyan) tails. Fitted parameters yield μ_L = − 0.05 ± 0.01; μ_R = − 0.05 ± 0.03 and σ_L = 0.26 ± 0.01; σ_R = 0.28 ± 0.02. (inset) Original distribution for cell area. Data were obtained from 24 separate experiments, for a total of nearly 40 000 cells. The coefficient of variation (σ / μ) is CV = 0.29. (Online version in colour.)

Numerical results are presented in figure 4b,c and qualitatively reproduce the results of the theory. To mimic the experimental conditions, we performed numerical simulations with α = 3 as obtained in the previous section and varied β and k.

Analogously to theoretical predictions, the mean size is largely determined by the ratio η₀/ρ₀, as for the theoretical model. The left tail depends on the value of β, with larger values of the exponents yielding narrower distributions and vice versa. The effect of varying k is that of varying the right tail, as predicted by the theory, with redistribution to a large number of sister-cells being associated with narrower distributions. Therefore, when apoptosis and proliferation coexist, redistribution to a single neighbour, as opposed to all contacting cells, tends to create larger cells, as expected. The effect of a size-dependent apoptosis is instead that of eliminating small cells, and making the distribution narrower.

The PDF of the numerical model are lognormal as predicted by the theory. It is interesting to note that for a reasonable set of parameters, i.e. β = 0 ÷3 and k = 3÷6, the coefficient of variation is consistent with that observed experimentally. Also note that the case α ≠ 0 and β = 0 is no longer problematic within this model, as redistribution hits only pre-existing cells.

To verify these predictions, we fitted the experimental distribution at the end of the experiment (figure 4d) with a lognormal with two different tails, obtaining fitted values indicated in the caption of figure 4 and good overall agreement with the hypothesis of lognormality.

These results lead us to speculate that apoptotic rates might be dependent on size as well, albeit not as markedly as for division. However, statistically solid experimental data for division rates and apoptotic rates in this regime are much harder to obtain as events are extremely rare.

3. Discussion

We have applied the theory of fragmentation to the study of the dynamics of epithelial growth and we have shown that a dividing confluent epithelial sheet can be modelled by linear irreversible fragmentation with perfect splitting.

We have demonstrated that power-law dependence of the division rate on cell size leads to the emergence of self-similarity in the area distribution of cells and that this is sufficient to constrain cell-size heterogeneity. Therefore, single cell level size-control of proliferation has an important impact on the heterogeneity of the distribution: that of maintaining cell size homogeneity during the formation of the tissue. Theoretical predictions have been compared with experimental data which indeed confirm the predicted power-law relation between cell size and division rates, and consistent time behaviour of the moments of the distribution. Furthermore, as expected, experimentally measured cell distribution is self similar.

An interesting observation per se is that cell division rates depend on cell size as a power-law, with exponent between 2 and 3. The physico-chemical mechanism producing that scaling is not known although it is reasonable to assume that adhesion to the substrate might be one of the important players.

For the regime where apoptotic rates cannot be neglected, we propose a generalization of the previously presented fragmentation model including apoptosis and redistribution of cell size by assignment of the area to other cells. As expected, in the limit where both cell division and apoptosis do not depend on size, the coefficient of variation of the distribution diverges, calling for a more controlled dynamics. Instead, by assuming both phenomena to be dependent on cell size, this model predicts that tails of the stationary distribution are lognormal, and allows analytical calculations of the moments of the distribution. Experimental PDFs are indeed lognormal with coefficient of variation consistent with the choice of experimentally relevant parameters. While the determination of the parameters calls for more specific experiments to be performed, we can conclude that lognormality is determined by the equilibrium between cell death and proliferation and that cell loss might also be cell size dependent.

In this paper, we systematically neglected area/volume growth of cells. This assumption is motivated by experimental observations that total area is conserved [19] and that single cells do not actually show any net growth. When describing sub-confluent growth, inclusion of a growth term—as in [33,41]—is necessary to better describe space filling dynamics. It is however experimentally irrelevant in the cases of confluent cultures considered here, in both size-reduction and homeostatic regimes.

One related aspect is the transfer of area from cell to cell, which is not taken into account in this model but could be included in certain approximations. For example, random underlying movement of cells might induce exchanges of areas/volumes [34] and might be modelled by a diffusive-like term in the fragmentation equation. More spatially local exchanges of volume cannot be included in the mean-field formulation and call for a single cell level model such as the vertex model or Potts model.

The mean-field nature of our model implies the neglect of spatial correlations. The redistribution of cell size following an apoptotic event is inherently a local event, and geometry and mechanical interactions are ultimately what determine the redistribution. Therefore, a very detailed description of the role of geometry would be better achieved by other models. Yet this model has important advantages such as requiring relatively few parameters that can be inferred from experiments.

Last, this model lends itself to other interesting generalizations: in the case of the epidermis cells are shed from the surface at a rate independent of size [42], and are brought to the surface upon differentiation in the layer underneath. However, whether single cell level regulation of cell appearance and loss is required to ensure the correct size distribution is not known and statistical properties and single cell level division time statistics have never been measured to the best of our knowledge.

Another context where this model can be applied is that of the corneal endothelium, which is known to be a very ordered tissue, undergoing contact inhibition [29,43] with few topological defects and coefficients of variation around 0.25 [44]. While the coefficient of variation of cell size remains constant throughout life, despite the decrease in cell density [45,46], other conditions, such as prolonged wear of contact lenses, is associated with increased cell loss and slightly increasing cell heterogeneity [44]. These processes can be easily described within the fragmentation framework and will be the subject of future investigations.¹

The effect of a soluble factor or drug could also be potentially studied by our model [47]. In this context, the measure of size distribution upon treatment (provided that the tissue stays confluent) can be much more informative than just looking at the mean size. One could indeed see directly whether a new homeostatic condition is established with an increased apoptotic rate and therefore a smaller overall density. Also, it would be possible to study whether the drug acts selectively on larger rather than smaller cells or vice versa. Higher apoptotic rates in our model would cause higher proliferation rates and might indeed increase the turnover within the tissue, a condition that is highly relevant in prolonged therapies, as it is potentially prone to resistance [48]. The effects of the drug/factor within this framework could be described in terms of alteration of division rates in specific cell populations (discriminated by size) and might give information on the mechanism of action of the drug within a more realistic model of a tissue than single cells.

In conclusion, a physical model of fragmentation has proven itself effective in the description of living tissue dynamics and provides a tool to explore the impact of single cell alterations in proliferation and shedding on large-scale quantities such as size distribution.

4. Material and methods

4.1. Experimental methods

Experimental data presented in this paper are part of the same dataset presented in [19]. Briefly, MDCK-II and MDCK-Ecad-GFP were cultured in MEM (GIBCO, 11095-098) supplemented with penicillin-streptomycin and 5% FBS (Cellgro, 35-010-CV) at 37°C and 5% CO₂. Cells were seeded at uniform density (around 600 cells mm⁻²) on a fibronectin (Sigma-Aldrich, F1141-2MG) coated PDMS membrane (McMaster-Carr, 87315K62) and imaged in phenol red free IMEM (Cellgro, 10-26-CV) supplemented with penicillin–streptomycin and 5% FBS. The media was replaced daily and the culture conditions were kept at 37°C and 5% CO₂. All imaging was performed on an inverted microscope (Olympus IX-70) equipped with a 20×/0.7NA/Ph2 objective.

Images were analysed by means of several custom written Matlab codes. Quantitative data on cell area were obtained with the help of contrast enhancement by Gabor-filtering and successive removal of low contrast regions. Frames taken 10 min apart were compared to remove poorly segmented cells.

To aggregate different experiments, cell sizes were normalized with mean cell area. For each experiment, a total of at least 1000 cells was considered. Mean overall area was (54 ± 12) μm² (mean, s.d.).

4.2. Numerical simulations

Numerical simulations of both models (size-reduction and homeostatic regimes) were performed by means of a custom written C code based on Gillespie's algorithm, available upon request. Equation (2.1) and (2.6) were simulated by stochastic simulations. A given initial number of cells was considered, each with its own area and division or apoptotic rate assigned according to cell size ρ(A) = ρ₀A^α and η(A) = η₀A^−β . Simulations consisted of finding the next occurring cell undergoing either division or apoptosis, generating the new cells in the case of division or destroying the dying cell and reassigning area in the case of apoptosis. At this point, rates were updated and the process started over. We implemented Gillespie's algorithm as follows [49]:

• (1) Update division and apoptosis rates for all cells as τ⁻¹_d = ρ₀A^α and τ⁻¹_a = η₀A^−β, respectively.

• (2) Generate random number r₁ and compute the next reaction time by computing

  4.1

  where i runs over the cells, while μ = d, a.
• (3) Generate a random number r₂ and determine which reaction (division or apoptosis of a given cell) will occur, by picking a couple j, δ such that

  4.2

  where δ + 1 is the following rate (either division for cell j + 1 if δ = a, or apoptosis for cell j if δ = d).

• (4) Perform reaction j, δ and go back to 1.

The algorithm described is the one for the homeostatic equilibrium, the one relative to size-reduction regime can be easily obtained by eliminating the sums over μ. Note that following this procedure, division (apoptotic) times are Poissonian by construction, as prescribed by the master equations presented in the text.

Redistribution was numerically implemented by assigning the area of the apoptotic cells among k cells picked at random (instead of looking for k cells with area A), to avoid numerical artefacts due to finiteness of the number of cells.

Gamma distribution for division times were simulated by executing division after m Poisson processes with rate ρ(A)/m which is gamma distributed, with mean rate ρ(A). Values of parameters were chosen to fit time distributions found in [19], with m = 53.

Numerical data presented in the paper are the result of several realizations.

Authors' contributions

A.P. and A.C. conceived the study, developed the theoretical model and analysed data. A.P. developed and performed numerical simulations. A.P. and A.C. wrote the manuscript. L.P. participated in the design of the study and helped drafting and revising the manuscript. All authors discussed the project, shared ideas and gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

This study was supported by University of Torino - Compagnia di San Paolo - GeneRNet to L.P. and by Fondo Investimenti per la Ricerca di Base (FIRB) RBAP11BYNP-Newton to L.P. A.P. was supported by ‘Fondazione Umberto Veronesi’ through a post-doctoral fellowship.