How cells flow in the spreading of cellular aggregates

Significance

Cellular aggregates are in vitro model of tumors. Deposited on adhesive substrates, they spread like liquid droplets with a monolayer expanding from the aggregate. We model spreading dynamics by balancing driving forces at the film periphery and viscous forces associated to the penetration of the cells from the (3D) aggregate into the (2D) film. By confocal microscopy, we observe this mechanism named “permeation” by tracking single cells. Using particle image velocimetry, we characterize the flow field versus substrate rigidity. If cells spread like a viscous liquid on stiff substrate, the flows become irregular, with formation of holes as the rigidity decreases. This work will shed light on the dynamics of tissue spreading occurring during cancer progression and embryonic development.

Abstract

Like liquid droplets, cellular aggregates, also called “living droplets,” spread onto adhesive surfaces. When deposited onto fibronectin-coated glass or polyacrylamide gels, they adhere and spread by protruding a cellular monolayer (precursor film) that expands around the droplet. The dynamics of spreading results from a balance between the pulling forces exerted by the highly motile cells at the periphery of the film, and friction forces associated with two types of cellular flows: (i) permeation, corresponding to the entry of the cells from the aggregates into the film; and (ii) slippage as the film expands. We characterize these flow fields within a spreading aggregate by using fluorescent tracking of individual cells and particle imaging velocimetry of cell populations. We find that permeation is limited to a narrow ring of width ξ (approximately a few cells) at the edge of the aggregate and regulates the dynamics of spreading. Furthermore, we find that the subsequent spreading of the monolayer depends heavily on the substrate rigidity. On rigid substrates, the migration of the cells in the monolayer is similar to the flow of a viscous liquid. By contrast, as the substrate gets softer, the film under tension becomes unstable with nucleation and growth of holes, flows are irregular, and cohesion decreases. Our results demonstrate that the mechanical properties of the environment influence the balance of forces that modulate collective cell migration, and therefore have important implications for the spreading behavior of tissues in both early development and cancer.

Tissue spreading is a fundamental phenomenon in many biological processes. Examples include wound healing (1–4) where the surrounding tissue spreads to close the injury, or the development of the embryo (5–7), which requires the orchestrated movement of cells to specific locations. It is also present in the progression of cancer (8–10). For example, glioblastomas grow and spread aggressively to invade surrounding regions and may lead to dramatic damages (11). The first step of cancer propagation (invasion) is characterized by a loss of cell–cell adhesion associated with an increase in cell motility. Increased cell motility is followed by entry into the blood circulation (intravasation), and subsequent escape from the circulation into distal tissue (extravasation). From this distal site, cell proliferation leads to a secondary tumor (11). Thus, it is crucial to understand how noninvasive tumor cells become metastatic by the loss of cell–cell adhesion and increased migration, which leads to malignancy. Further investigation into this topic requires the design and analysis of model in vitro experimental systems suitable for recapitulating these early stage events.

Model in vitro systems in 3D are essential to recapitulating the early stages of cancer progression. For example, it has been reported that the efficiency of medical drugs tested on 2D cell culture systems is not transposable to 3D in more than 50% cases (12). Recently, 3D cellular aggregates have been identified as a model system that approximates living tissues and tumors (13–15). Cell aggregates in culture consist of hundreds to thousands of cells. Structurally, cell aggregates resemble concentrated emulsions, such as the honeycomb organization of foams (16). However, contrary to foam, which is solid, cell aggregates flow like a liquid when placed onto an adhesive surface or in contact with each other. For example, cell aggregates in solution form spheroids to minimize their surface energy, and fuse when in contact with each other like liquid droplets (15). Since the pioneering work by Steinberg in the 1960s (17), several studies have shown that the mechanical behavior of tissues and cellular aggregates can be characterized through their liquid-like behavior (18, 19). The analogy between tissues and liquids has been very fruitful in describing the spreading of tissues, using the physics of capillarity and wetting (13, 20). The statics of wetting has been previously described in ref. 13, where interfacial energies involved in the wetting of droplets are replaced by intercellular and cell–substrate adhesion strengths. By varying both the intercellular adhesion (W_cc) and the adhesion with the substrate (W_cs), two aggregate spreading regimes have been observed as characterized by the sign of the effective spreading parameter, S = W_cs – W_cc. If S < 0, partial wetting occurs and the aggregate does not spread. If S > 0, complete wetting occurs and a precursor film made of a cellular monolayer flows outwards from the edge of the aggregate. The transition between partial and complete wetting is induced experimentally by varying W_cc, by tuning the level of expression of the cadherins, or W_cs, chemically using PEG–fibronectin substrate coating (21) or soft gels (22–24). A number of observations suggest that the strength of adhesion depends on both the substrate’s chemical receptors and its rigidity (25, 26). On a single-cell level, matrix rigidity has been shown to strongly affect myriad cellular physiological functions, including differentiation (27), spreading (28), and migration (29). On a multicellular level, the transition from complete (S > 0) to partial wetting (S < 0) of cell aggregates can be induced by decreasing the substrate elastic modulus below a critical value E_c ≈ 5–8 kPa (24). Thus, as the rigidity of the substrate is a key parameter that controls the statics of cell aggregate wetting, we focus here on the dynamics of wetting in substrates of varying rigidity (10³ to 10⁹ Pa).

A cellular aggregate deposited on a wettable substrate spreads as a stratified droplet with a single monolayer (30). The dynamics of spreading results from a balance between friction forces and driving forces S (per unit length). In simple liquids, the driving forces are capillary forces. In the spreading aggregate, the driving forces are due to motile cells pulling at the periphery of the film (31). Previously, De Gennes and Cazabat described the dynamics of growth of a stratified precursor film (32). The precursor film has a 2D horizontal velocity field. The aggregate acts as a reservoir to feed the precursor film in a process named permeation, introduced by Helfrich to describe the flows of smectic liquid crystals normal to the layers (33). In our model system, permeation is the process by which the cells from the aggregates enter into the cell monolayer. During spreading, we expect two types of viscous dissipation, due to (i) slippage of the expanding monolayer with the substrate and (ii) permeation normal to the layer. De Gennes and Cazabat (32) predicted that the permeation flow is limited to a narrow ribbon near the contact line, characterized by the permeation width ξ. For simple liquids, they conclude that the friction with the substrate is dominant to permeation. By contrast, with the spreading of cellular aggregates, we will show here that the permeation of cells from the aggregates into the film is the factor limiting the dynamics of spreading (34). For example, aggregates deposited on adhesive stripes spread at constant velocity (21), which can be explained only if permeation is dominant. Thus, to date, the permeation of cells within aggregates has only been introduced as a theoretical hypothesis. Permeation has never been directly visualized for cell aggregates and we propose to directly observe this phenomenon using E-cadherin (Ecad)–expressing cells with a fluorescent nucleus.

Studies on the collective motion of multicellular systems have been mainly restricted to 2D cell culture, confluent monolayers (4, 35), and wound healing assays (1). Our aim in this paper is to assess the collective motion of cells in 3D aggregates as they spread onto a 2D adhesive substrate. Using fluorescence ubiquitination cell cycle indicator (FUCCI) to identify the position of each cell within the aggregate, the migration trajectories of the cells will be monitored over time. Dynamical mapping of the velocity field is performed to fully characterize the collective mode of spreading. To this end, we use particle imaging velocimetry (PIV), an efficient technique already used for cell migration (36–38). PIV is a cross-correlation technique initially developed for characterizing fluid flow (39), by calculating the local displacements of embedded particles imaged in real time. Using PIV for cell migration, we have monitored for the first time, to our knowledge, the evolution of the velocity field within the migrating monolayer that emerges from the aggregate as an analogy to the early stages of tumor invasion. With this analysis, we have observed two regimes: a viscous liquid-like regime with a quasiradial flow of the cells at short times, where the aggregate acts as a reservoir, followed by a chaotic state at long times, once the aggregate has been depleted into the monolayer film. The latter is characterized by the formation of long range rotating velocity fields (“swirls”). In the case of deformable substrates, the cell monolayer is unstable during spreading, with growth of holes in the monolayer leading to rupture.

The aim of this paper is to relate the macroscopic spreading behavior of cellular populations to the migration of individual cells using simple physical laws, such as the conservation of matter, permeation, and balance between viscous and driving forces. From these laws, we intend to shed light on the dynamics of tissue spreading, as it occurs during the early stages of cancer progression and during tissue development.

Results and Discussion

How Cells Flow from the Aggregate into the Expanding Monolayer.

Aggregates of varying sizes (R₀ radius) are imaged by confocal microscopy for up to 10 h as they spread onto fibronectin-coated glass coverslips (E = 70 GPa) or fibronectin-coated polyacrylamide (PAA) gels (E > 7.4 kPa) (24) (Fig. S1). FUCCI cells are used to form the aggregates and visualize individual cells motion. The dynamics of spreading is then quantified by tracking the fluorescent nuclei of individual cells for aggregates of different sizes. We use these methods to first describe how the cells migrate from the aggregate into the precursor cohesive film on a rigid substrate.

Aggregates on Glass Coverslips.

Large aggregates.

First, we focus on the spreading of large aggregates (R₀ > 150 µm) (Fig. 1A, Upper and Movie S1) onto glass coverslips. As the film extends from the aggregate, we observe that the dynamics of cell migration depend on the spatial localization of the cells within the population (Fig. 1C, Left). Under the aggregate (domain a), cells are immobile and the FUCCI fluorescence is predominantly red (∼56%), indicating that the cells are primarily in the G1 phase of the cell cycle. Around the aggregate (the precursor film; domain c), cells are put into motion, with an average curvilinear velocity (i.e., total distance traveled divided by time) of (7 ± 1) × 10⁻³ µm⋅s⁻¹ (n = 32, results ± SD; Fig. 1B, Left). Moreover, the direction of cell movement is radially away from the center of the aggregate, with an average directionality (defined by the ratio of the Euclidean distance by the accumulated distance of the cells) of 0.9 ± 0.1. The FUCCI fluorescence in domain c is predominantly green (∼69%), indicating that the cells are primarily in the S–G2 phase of the cell cycle or in M entry. Between domains a and c, there is a population of cells which flows from the aggregate into the film in a ring, characterized by a width ξ (domain b) equal to ∼65 µm. Using two-photon microscopy and large aggregates of Ecad–GFP S180 cells, we have been able to visualize the 3D flows in the aggregate and the precursor film (Movie S2). We clearly see a stagnant zone and a permeation zone.

Fig. 1.

Spreading of large and small cell aggregates on glass coverslips (E = 70 GPa) coated with fibronectin and highlighted permeation process. Aggregates are observed in confocal microscopy (GFP and CHFP fluorescence). (A) Spreading of large (Upper, R₀ = 185 µm) and small aggregate (Lower, R₀ = 65 µm) observed at different times (t = 0–10 h). (B) Spreading of the large (Left) and small aggregate (Right) in GFP fluorescence 7 h after their deposition. Dynamics of the cells is followed using manual tracking (ImageJ). (C) The different noticeable areas of the large (Left) and small (Right) aggregates 9 h after their deposition. (D) Flows in a large (Upper) and small (Lower) cellular aggregate spreading on a substrate; the precursor film is fed by permeation (Right). The a, b, and c domains correspond to immobile cells, permeation domains, and flowing monolayers, respectively. All scale bars represent 50 μm in each panel, respectively.

Small aggregates.

Next we ask what happens if the size of the aggregates is smaller than the permeation width ξ. To answer this question, we study the spreading of small aggregates (R < ξ) (Fig. 1A, Lower and Movie S3) onto glass coverslips. Contrary to large aggregates (R >> ξ), where the central zone of contact under the aggregate is almost immobile (domain b), we observe that all cells beneath the aggregate migrate and only domains b and c remain. Similarly to large aggregates, the proportion of cherry fluorescent protein (CHFP⁺) cells under small aggregates is ∼50%, indicating that the cell division is limited inside this domain. To characterize cell migration in domains b and c, each individual cell in both regions is tracked over time and compared. We analyzed the trajectory of cells under (n = 15) or around the aggregate (n = 31), (Fig. 1B, Right). The directionality of the cells and the mean curvilinear velocity under the aggregate are, respectively, (0.7 ± 0.2) × 10⁻³ µm⋅s⁻¹ and (6 ± 1) × 10⁻³ µm⋅s⁻¹ compared with the (0.8 ± 0.2) × 10⁻³ µm⋅s⁻¹ and (7 ± 1) × 10⁻³ µm⋅s⁻¹ values around the aggregate. These directionality values indicate that the trajectories of cells are radial as expected for inert liquids.

Thus, large aggregates on glass substrates demonstrate three populations (i.e., the domains), which are distinct in their migratory behavior, and their stage of the cell cycle. This includes a nonmotile, stagnant domain beneath the aggregate (domain a); a highly motile precursor film, which extends into free space (domain c); and a ring of width ξ, which connects the two (domain b). For aggregates of size R < ξ, the stagnant domain disappears, and all cells from the aggregate migrate into the precursor film. The spreading in the monolayer is similar for small and large aggregates. The fluorescence of the cells in the film is mainly green (∼70%), indicating that they are mainly in the S–G2 or in M phase of the cycle. We found that the cell cycle does not influence cell migration (SI Cell Cycle Influence on Cell Migration, Fig. S2, and Table S1).

Model of permeation.

In this section we present the model developed to explain why the permeation limits the dynamics of spreading. De Gennes and Cazabat described the spreading of a stratified droplet (32). They showed that two types of flows occur: (i) shear between layers and on the substrate for the first layer; and (ii) permeation normal to the layer in analogy with the process of permeation introduced by Helfrich for the flow in smectics (33). Using these assumptions, we model the spreading of stratified droplets (SI Model, Spreading of Stratified Drops: Ring of Permeation). It shows that the permeation is limited to a ring of width ξ:

$$\frac{1}{{\xi }^2}=\frac{\lambda }{d^3}\left(k_0+4k_1\right)=\frac{1}{{\eta }_p{d}^2}\left({\eta }_0+4{\eta }_1\right),$$ [1]

where k₀ is the friction coefficient with the solid (measured in ref. 40), k₁ is the interlayer friction, and λ is the Helfrich permeation coefficient. We can express k and λ in terms of molecular parameters: k₀ = η₀/d, k₁ = η₁/d, and λ = d²/η_p; where η₀, η₁, and η_p are associated with the friction on the substrate and with intracellular frictions and d with monolayer thickness. We can estimate the permeation width ξ. For simple liquids, η₀ ≈ η₁ ≈ η_p and ξ ≈ d. For ultraviscous pastes, η₀ < η₁ ≈ η_p ≈ η, the tissue viscosity (40), and ξ ≈ d. For smectics, η₀ >> η₁ ≈ η_p and ξ >> d (41).

The size ξ of the ring is of the order of the monolayer thickness for both low and ultraviscous paste. This is in agreement with our data, where we observe a permeation ring of width ξ equal to few cell sizes.

For the dynamics of spreading, we find that if η₀ ≈ η₁ ≈ η_p, the friction on the substrate is dominant and this is the case of simple liquids. On the other hand, if η₀ << η₁ ≈ η_p, permeation is the limiting factor. This is the case of living drops where the permeation regulates the flow of cells.

Role of rigidity.

As substrate rigidity has been shown to affect the statics of aggregate wetting, here we investigate whether substrate rigidity affects the dynamics of aggregate wetting. We therefore allow cell aggregates to adhere and spread on deformable PAA gels (E = 16.7 kPa) and reinvestigate the dynamics of cell migration (Fig. S1A, Upper). The thickness of the gel (80 μm) is much larger than the tactile length, below which single cells are sensible to gel thickness (42, 43). When the precursor film spreads, again, we observe differential cell migration that depends on the cell’s spatial location (domain a, b, or c) (Fig. S1C, Left). First, we follow the trajectory of the cells under the aggregate (domain a). We measured an average curvilinear migration velocity of (6 ± 1) × 10⁻³ µm⋅s⁻¹ (n = 20; Fig. S1B, Left). The movement of the cells is random as their average directionality is low with a value of 0.5 ± 0.2. Around the aggregate (domain c), we measured an average curvilinear migration velocity of (8 ± 1) × 10⁻³ µm⋅s⁻¹ (n = 39; Fig. S1B, Left). The cells are moving radially away from the aggregate as their average directionality is elevated in contrast to domain a, with a value of 0.7 ± 0.1. Different domains are further distinguished by their stage in the cell cycle. Beneath the aggregate (domain a), the cells are mainly CHFP⁺, whereas in the precursor film (domain c), they are mainly GFP⁺. Between these two domains, the ring of permeation (domain b) is characterized by a width ξ ≈ 65 µm; the same order that results on glass substrates. We studied small aggregates (Fig. S1A, Lower), and contrary to large aggregates, we observed that all cells migrate similarly, regardless of their spatial location within the population. We analyzed the cell trajectory (n = 33; Fig. S1B, Right) and found that the directionality of the cells is of 0.7 ± 0.1 and the mean curvilinear velocity (6 ± 1) × 10⁻³ µm⋅s⁻¹. These directionality values indicate that the trajectories of all of the cells are radial.

As with the glass substrate, the spreading of large aggregates on PAA gels demonstrates three populations (domains), which are distinct in their migratory behavior and in their stage of the cell cycle. The precursor film and permeation ring are similar in their dimensions and in the dynamics of cell migration. However, in contrast to glass substrates, the cells beneath the aggregate (domain a) are more mobile on fibronectin-coated PAA. Similarly, for small aggregates (R < ξ), there is no stagnant region and all cells from the aggregate migrate into the precursor film. Thus, substrate stiffness affects the migration of cells within different domains in the spreading aggregate.

Flow Field in the Spreading Monolayer.

Before describing the flows of cells, we give a reminder of the theoretical model of the monolayer spreading. As the cell monolayer spreads, the dynamics, described in a previous paper (21), are ruled by a balance between active driving forces due to mobile cells at the film periphery (31, 44) and viscous dissipation due to (i) the permeation of cells entering the monolayer and (ii) the friction of the sliding monolayer.

The dissipation due to permeation and to the sliding film can be written, respectively, as

$$2\pi \eta {\left(\frac{R\dot{R}}{R_c\xi }\right)}^2{R}_c{\xi }^2+k\underset{R_c}{\overset{R}{{\displaystyle \int }}}2\pi rV{\left(r\right)}^{2}dr=2\pi \eta \left(\frac{R^2{\dot{R}}^2}{R_c}\right)+2\pi \frac{{\eta }_0}{d}{R}^2{\dot{R}}^2\ln \frac{R}{R_c},$$ [2]

where $R\dot{R}/R_c$ is the velocity at the contact radius R_c, ξ the width, and R_cξ² the volume of dissipation, and assuming mass conservation (the velocity at distance r from the center of the aggregate) is given by $V\left(r\right)=\left(R/r\right)\dot{R}$.

When we compare the two terms of Eq. 2, we conclude that the spreading is limited by the permeation and not by the slippage on the substrate if η/R_c > k ln(R/R_c). It is the case of our living drops where the bulk viscosity $\eta$ is much higher than the sliding viscosity ${\eta }_0$ (40).

The balance between the friction force F_v deduced from Eq. 2 $2\pi R\dot{R}{F}_v=2\pi \eta {\left(R\dot{R}/R_c\right)}^2$ and the driving force S, describing the competition between active cell forces pushing the cell to spread on the substrate and cell–cell adhesion preventing the spreading, leads to

$$\eta \frac{R\dot{R}}{R_c}=S\Rightarrow \frac{R^2}{R_c}=V\ast t,$$ [3]

where V*t = S/η is a typical spreading velocity. The law of spreading is diffusive with a diffusion coefficient D = V*R_c proportional to the radius of the aggregate and to the velocity V*. Notice that if the sliding friction is dominant (Eq. 2), the law will also be diffusive with a diffusion coefficient $D_s=S\left(d/{\eta }_0\right)\ln \left(R/R_c\right)$, depending only logarithmically on R_c. On the other hand, as shown in ref. 21 for the spreading on stripes, the sliding friction leads to a diffusion law, whereas permeation leads to a spreading at constant velocity V*, which is observed experimentally. We have checked that D depends linearly on the size of the aggregates. By normalizing the monolayer area A by the initial aggregate radius R₀, the data for different aggregate sizes collapse into a straight line, as predicted by Eq. 3. Fig. 3C shows the time evolution of the normalized monolayer area for E = 70 GPa (red markers) and = 16.7 kPa (black markers). The experimental data are well described by a diffusive law according to the model. The mean velocities V* (i.e., the slope of the straight line) are, respectively, 2.2 × 10⁻² µm⋅s⁻¹ and 3.8 × 10⁻² µm⋅s⁻¹. The permeation process observed directly at a cellular scale is the dominant factor regulating the dynamics of spreading.

Fig. 3.

(A) Spreading of an aggregate on a PAA gel (E = 16.7 kPa) observed in bright field at short times (Top). Using the PIV method the direction of velocity fields (Middle) and heat maps show the spatial distribution of velocity fields (Bottom) have been obtained. (B) Bright-field picture of the same aggregate at long times (35 h, Left) and heat map showing spatial distribution of a velocity field (Center Left). The direction of the corresponding velocity field shows the formation of whirls (Center Right). Magnified view of the region delimited by the red box (Right). (C) Spreading of an aggregate on a PAA gel (E = 9 kPa) observed in bright field (Left). The opening and closing of a hole, circled in white on the picture, is represented by the time evolution of its radius R (Right).

Now we describe the flows of cells in the spreading monolayer on fibronectin-coated substrates with rigidities varying from 2 kPa (PAA) to 70 GPa (glass). Thus, the elastic modulus E is varied while a constant chemical environment is maintained. In contrast to our previous methods where we tracked individual cells, we now seek to characterize the flow field of the cell population, in analogy to the flow of viscous liquids. To this end, we measure the position of cells at successive times t and t + Δt. From these positions, we derive the velocity field, the orientation distribution, and the correlations between cell motions.

Aggregates spreading on glass substrates.

Aggregates were placed on a fibronectin glass coverslip and we observed the formation of a cohesive precursor film (Fig. 2A, Top). Migration of the precursor film was reminiscent of the flow of viscous liquids, as their velocity was radial from the center of the aggregate (Fig. 2A, Middle). This result is in agreement with single-cell tracking in the film where we calculated a directionality value of 0.9 ± 0.1. We also observed long-range correlations in velocity with a characteristic length of few cell diameters. The spreading dynamics follows (Eq. 3) with V* = 2.2 × 10⁻² µm⋅s⁻¹. This spreading film phenomenon is observed during the first 30 h. After the precursor film is spread and no aggregate remains (∼30 h), the amplitude of the velocity is reduced and the flow field points in all directions with the formation of swirls and long-range velocity correlations also observed in wound healing (1, 4) and cell monolayers (4). As the driving force of spreading S has been shown to be very sensitive to substrate rigidity and vanishes for a threshold modulus corresponding to the wetting transition (E ≈ 5 kPa) (24), we now describe the flow pattern and the flow dynamics versus substrate rigidity.

Fig. 2.

(A) Spreading of an aggregate on a coverslip (E = 70 GPa) observed in bright field at short times (Top). Using the PIV method, the direction of velocity fields (Middle) and heat maps showing the spatial distribution of velocity fields (Bottom) have been obtained. (B) Bright-field picture of the same aggregate at long times (50 h, Left). Heat map showing the spatial distribution of a velocity field (Center Left). The direction of the corresponding velocity field shows the formation of swirls (Center Right). Magnified view of the region delimited by the red box (Right). (C) Time evolution of the monolayer area of spreading aggregates normalized by the initial aggregates radius R₀. Red markers (n = 8 experiments with R₀ ranging from 69 to 139 µm) correspond to the case of glass coverslips and black markers (N = 8 experiments with R₀ ranging from 146 to 183 µm) correspond to rigid gels (E = 16.7 kPa).

Aggregates spreading on rigid gels: E = 16.7 kPa.

At short times (t < 25 h), the cells that emerge from an aggregate adhered to deformable PAA display a radial velocity, and thus resemble a viscous liquid. However, the spreading velocity on deformable PAA is higher (V* = 3.8 × 10⁻² µm⋅s⁻¹) than the spreading velocity on glass (2.2 × 10⁻² µm⋅s⁻¹), in agreement with previous results (24). Using PIV, we derived the radial component V_r of the velocity (Fig. S3A, Right) and the flux J(r) = 2πrV_r as a function of the distance r from the center of the aggregate deposited on the PAA gels (E = 16.7 kPa) and glass coverslips (Fig. S3 B and C, respectively). The contact radius R_c of the aggregate with the substrate (the purple inner circle) and the ring of permeation characterized by a thickness ξ (between the two purple circles) have been drawn on the figure. For r < R_c, J(r) is nearly zero. For r > R_c, the flux increases abruptly in the permeation ring, reaches a maximum, and decreases faster than 1/r. This fast, effective decrease in radial velocity is attributed to the noncircular contour of the film, and many zero, values corresponding to empty regions, decrease the mean value of J(r). The maximum value of the flux increases with time until t = 10–25 h with a plateau value J* of 4.2 × 10⁻¹² m²⋅s⁻¹ or 1.5 × 10⁻¹² m²⋅s⁻¹ when the substrate is PAA gel or glass, respectively. These values have to be compared with the theoretical ones J_th = 2πR_cV*. We find that J* and J_th are of the same order of magnitude with J*/J_th = 0.75 for PAA gels and 0.40 for glass substrates. At long times (t > 25 h), the aggregate has been predominantly adsorbed into the precursor film, leaving only a small fraction of the original aggregate (Fig. 3B, Left). At the periphery of the remaining aggregate however, holes in the monolayer nucleate and grow, indicative of the presence of strong cell–cell mechanical tension. The holes have diameters of ∼75 μm (∼5–10 cells in diameter) (Fig. 3B, Left) and last for hours. By contrast, in the monolayer distal to the aggregate, cell density is uniform and large, transient swirls of collective cell motion about 150 µm in diameter (∼10–20 cells in diameter) (red signals in Fig. 3B, Left) are also observed.

Aggregates spreading on soft gels: E = 9 kPa.

On 9-kPa PAA gels, the spreading velocity V* = (1.2 ± 0.2) × 10⁻¹ µm⋅s⁻¹ corresponds to a maximum of S(E). This high cell motility pulls the film against a slow permeation to extract the cells from the aggregate and leads to a stretching of the film. By this description, tension builds within the cell monolayer. As a result of increased tension, transient holes form at the edge of the aggregate and also in the film (white signal in Fig. 3C, Left). Compared with the previous case, these instabilities appear as soon as the film starts to spread. We characterize the formation and growth of holes using PIV. The flow field, as the holes expand, is shown in Fig. S4.

The opening of the holes in cell monolayers resembles the dewetting of a viscous polymer film on a wettable substrate (45), where dry patches open at velocity $\dot{R}\approx \left(\sigma /\eta \right){\left(\eta /{\eta }_0\right)}^{\frac{1}{2}}$, where R is the radius of the hole and σ the film tension. In accordance with this model, we find that the opening radius R(t) can be modeled by a straight line (Fig. 3C, Right) with a mean slope value of (9 ± 5) × 10⁻³ µm⋅s⁻¹ (n = 5). As holes open, the tension of the stretched monolayer σ decreases and the holes close. In few cases, the holes coalesce, leading to a partial dewetting of the precursor film.

Ultrasoft gels: E = 7.6 kPa.

As E decreases, the cohesion of the spreading monolayer decreases, and cells detach from the aggregate individually. We show in Fig. S5 the film for E = 7.6 kPa, which is very similar to the precursor film of E48 cell lines expressing a low level of E-cadherin spreading on fibronectin-coated glass (13). The decrease of W_cc is related to the decrease of W_cs. Maintenance of cellular assemblies is known to be governed by substrate stiffness (46). Indeed, it has been shown that Ecad cells’ adhesion is modulated by external rigidity (25). Isolated Ecad cells are rounder on soft gels, indicating a reduced adhesion to the substrate. Interestingly, when Ecad cells are in contact with neighbors through cadherin-mediated adhesions, they spread more than isolated cells on the substrate and their traction force is increased (25). Reciprocally, when Ecad cells interact with fibronectin, there is a positive regulation of their intercellular adhesion strength (47), illustrating the cross-talk between cadherins and integrins, which stimulates cell–cell adhesion. Accordingly, our observations on the decrease of cellular cohesion on ultrasoft substrates indicate that a decrease in cell adhesion to the substrate impairs cohesivity within the cellular film.

On even softer gels, E < E_c ≈ 5–7 kPa, we observe a wetting transition and no precursor film.

In conclusion, to bridge the macroscopic spreading laws to the flows at a cellular level, we observe the migration of thousands of cells during the spreading of cell aggregates (Ecad) deposited on rigid and soft adhesive substrates. By confocal microscopy, focusing on the first cell layer in contact with the substrate and using fluorescent FUCCI Ecad cells, we follow cell trajectories during cell cycles. By bright-field and 3D two-photon microscopy, we measure the cell velocities using PIV. We apply the concepts of soft matter physics to model the dynamics of spreading of the cellular aggregates. Studies of cell migration have been mainly restricted to 2D cell culture. For 3D aggregates spreading onto a 2D adhesive substrate, we demonstrate the role of permeation for tissue dynamics, which describes how cells migrate from the 3D aggregate into the 2D monolayer. As the cell monolayer extends from the aggregate, we observe three distinct domains: the first corresponding to immobile cells under the aggregate; the second, to a ring of permeation where the cell velocity increases abruptly; and the last, to very motile cells in the expanding monolayer. The ring of permeation has a width of a few cells. We check that for aggregates with a radius smaller than ξ, the central stagnant zone disappears, corresponding to ξ = R. A theoretical model for the dynamics of the spreading film based on the role of permeation leads to a diffusive law R² = V*R_ct, where V* ≈ S/η is a spreading velocity and R_c is the radius of the aggregate. The flows of the expanding film are studied using PIV. At short times cells flow like a viscous liquid, with a radial flow field with long-ranged cell–cell correlations. At long times, on rigid substrates, the aggregates totally spread into a cohesive monolayer of cells. At this stage, we observe the correlated motion of cells pointing in all directions forming swirls such as those observed in wound healing (1) or for confluent monolayers (35). However, as the substrate becomes softer, two main features arise. First, we observe that the precursor film is less and less cohesive, and on very soft substrates, cells escape from the aggregate. It shows that the progression of a noninvasive tumor into a metastatic malignant carcinoma, known as the “epithelial–mesenchymal transition” (EMT) can be induced by substrate rigidity in vitro. Second, the spreading dynamics of the monolayer expanding outward from the aggregates is optimal in a narrow range of rigidity, where the spreading velocity V* presents a maximum. In this regime we observe an additional phenomenon: the fast spreading gives rise to a tension in the monolayer and the film becomes unstable with the formation of holes and swirls at the edge of the aggregates. These mechanisms of collective cell migration play a crucial role in embryonic development and cancer invasion. Moreover, from the spreading laws established in this work, we will be able to test accurately the role of drugs to slow down the cancer propagation and to reduce tumor proliferation induced by the EMT.

Materials and Methods

Detailed experimental methods can be found in SI Materials and Methods. We used FUCCI Ecad and Ecad–GFP— fluorescent murine sarcoma cells expressing E-cadherins at their surface. Aggregates were obtained using the agitation method or cellular capsules technology (48). Fibronectin was either adsorbed on glass subtrates or functionalized using sulfosuccinimidyl-6-(4′-azido-2′-nitrophenylamino) hexanoate on gels. We used spinning-disk confocal microscopy to visualize the dynamics of the cells during the permeation process. Cell tracking was visualized using ImageJ (National Institutes of Health, Bethesda). Cell directionality and curvilinear velocity was obtained using Chemotaxis and Migration Tool (49). The velocity field in the monolayer was mapped by PIV analysis and stacks of images were analyzed by using the MatPIV software package for MATLAB (MathWorks) (50). The MATLAB program used to calculate the radial velocities of the velocity fields was developed by M.P.M.