To study the spontaneous tissue deformation arising during epithelial growth, we designed an in vitro assay to track symmetry breaking, both spontaneous and driven by external force. We prepared isotropic colonies of Madin Darby Canine Kidney (MDCK) cells, which assume features of epithelial cells in vivo (Reinsch and Karsenti, 1994; Adams et al., 1998; Reffay et al., 2014), such as adherens junctions (Adams et al., 1998), cytoskeletal components, and the Rho signaling pathway regulating cell shapes and dynamics (Reffay et al., 2014; Fodor et al., 2018). The initial size and shape of the colonies were controlled by plating cells in microfabricated circular stencils (Ostuni et al., 2000). When cells reached confluency, the stencil was removed at time t₀. Cell dynamics was followed over time by phase contrast (Video 1) or fluorescence microscopy with strains labeled with GFP cadherin (Figure 1b), that allowed to observe the behavior of individual cells. We observed that large colonies (750 µm in diameter) expanded isotropically (Figure 1—figure supplement 1). In contrast, colonies of 250 µm in diameter (Figure 1b), the typical coherence length of such epithelial tissues (Doxzen et al., 2013), expanded in a non-isotropic manner (Figure 1c). We then further characterized the process of symmetry breaking.

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The Q_xx(t)-Q_xx(0) and Q_xy(t)-Q_xy(0) of individual colonies used in panel (e) and the raw values of cos(2·(θ(t)-θ(t_final))) used in panel (f) of Figure 1.

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In order to compare elongations in each experiment, we quantified the breaking of symmetry by ellipse-fitting the colony shape. Shape change analysis was quantified by a nematic shape elongation tensor Q. It has two independent components defined as Q_xx = ½ ln(a/b)cos(2•(θ − α)) and Q_xy = ½ ln(a/b)sin (2•(θ − α)), where a corresponds to the major axis, b to the minor axis, θ to the orientation of the major axis of the ellipse and α = θ(t_final) (Figure 1d). As can be seen in Figure 1e, MDCK colonies elongated persistently along the main axis of elongation (Q_xx > 0 and Q_xy ≈ 0) for 6 hr (Figure 1e). In addition, we explored if other epithelial cell lines would behave in a similar manner. Circular epithelial colonies of human epithelial colorectal adenocarcinoma cells (Caco2) and human mammary epithelial cells (MCF-10A) also elongated along the main axis of elongation and by the same magnitude that MDCK cells (Figure 1—figure supplement 2). We note that elongation observed during this time for the three epithelial cell lines was similar in magnitude to tissue elongation observed during in vivo morphogenesis, for instance in the wing blade in Drosophila (Etournay et al., 2015). Moreover, the elongation direction (θ_final = θ (t = 6 h)) converges to a constant value within 2 hr after t₀ (Figure 1f). Altogether, large circular epithelial colonies (750 µm in diameter) expand isotropically, whereas small colonies (250 µm in diameter) expand in a anisotropic manner and shape symmetry breaking takes place within the first 2 hr. As a result, we focus here on these first 2 hr during which the elongation axis is established.

It has been previously described for C. elegans embryo elongation (Zhang et al., 2011) and in other organisms (Zhang and Labouesse, 2012) that time periodic stretch can play a role in morphogenesis. Motivated by these observations, we explored whether oscillatory external forces could have an impact on the direction of elongation. We designed an experimental setup where elongating colonies were submitted to cyclic uniaxial stretching (Figure 2a and Video 2). Mechanical cycles of contraction-relaxation can range from 1 s in C. elegans epithelial elongation (Zhang et al., 2011) up to 200 s in dorsal closure (Solon et al., 2009). So, we explored frequencies and extensions around physiological values (Zhang and Labouesse, 2012). We selected three different cycle durations (20, 60, and 120 s period) and three different stretching conditions (5%, 10% and 15% strain). The stretch was applied to a silicon membrane and was transmitted to the colony. We then fitted the colonies shapes with ellipses at successive time and quantified Q_xx and Q_xy with respect to the angle of uniaxial stretching (set as x-axis, α = 0). Figure 2—figure supplement 1 shows the value of the components of the tensor Q along time for the different strains and periods tested. Among different conditions, we observed colony elongation along the direction imposed by the external strain when we stretched cyclically with 60 s timescale and 5% strain (Figure 2b and Video 3). The overall elongation of colonies under cyclic uniaxial stretching was similar to the spontaneous elongation in the absence of externally applied uniaxial stretching during the first 2 hr (Figure 2c). Also, the magnitude of the shape elongation tensor Q under cyclic uniaxial stretching was comparable to the spontaneous elongation of colony when stretch was not applied, but elongation was oriented in the direction of externally applied uniaxial cyclic stretching (Figure 2d). Therefore, application of an external cyclic force can rectify symmetry breaking and set the direction of tissue elongation.

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Raw data used for the panel (c) of Figure 2 and Q_xx(t)-Q_xx(0) and Q_xy(t)-Q_xy(0) for the individual colonies used for panel (d) of Figure 2.

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To get further insight into the collective nature of the rectification of tissue elongation, we probed the roles of adhesion between cells. First, we stretched single MDCK cells, individually plated. We observed that cells oriented perpendicularly to the externally applied uniaxial cyclic stretching (Figure 3a and b) as previously reported for fibroblasts (Faust et al., 2011). Then we blocked cell-cell junction in circular colonies prior stretching. Briefly, we incubated confluent colonies in medium containing 5 mM EDTA and 10 μg/ml anti-E-cadherin blocking antibody that targeted the extracellular domain of E-cadherin for 30 min (Harris et al., 2014). Then, medium was replaced by normal medium and the evolution of colonies with and without stretch was followed (Figure 3c). In the absence of externally applied uniaxial cyclic stretching, colonies treated with anti E-cadherin antibody expanded more than control colonies. Moreover, this expansion was still along one preferential direction (Figure 3d). Under cyclic uniaxial stretching, elongation was also non-isotropic and along the direction perpendicular to the cyclic stretching direction, in contrast to the parallel elongation observed when cell-cell contacts were intact (Figure 2b–d). This supports the collective nature of colony elongation. It is worth noting that cells inside the colony exhibited a decrease in their mean velocity (Figure 3e) and a large recruitment of myosin within cells similar to reinforcements (Riveline et al., 2001) in stretching conditions, as shown by the appearance of stress fibers (Figure 3f). However, this effect did not appear to affect the overall elongation rate. Altogether these data suggest that the asymmetric expansion of colonies in the direction imposed by cyclic uniaxial stretching is generally associated to a collective effect.

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The Q_xx(t)-Q_xx(0) and Q_xy(t)-Q_xy(0) of individual cells used in panel (b) and individual colonies used in panel (d); we also provide datasets for the velocities of individual cells in panel (e).

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We next sought to identify the source of symmetry breaking in both conditions, with and without application of cyclic uniaxial stretching. It has previously been reported that in MDCK monolayers, cells can migrate tangentially to the monolayer boundary when confined (Doxzen et al., 2013), or perpendicular to the boundary in the form of multicellular groups or fingers during monolayer expansion (Reffay et al., 2011; Reffay et al., 2014). During spontaneous elongation of MDCK colonies in the absence of externally applied cyclic stretching (Video 4), we observed that boundary cells tend to move either perpendicularly or tangentially to the colony boundary (Figure 4a). In most of the experiments, an acto-myosin cable, similar to compartmentalization boundaries in vivo (Monier et al., 2010; Calzolari et al., 2014), was observed in the outer boundary of the colonies at stencil removal (see Figure 4—figure supplement 1). When this supra-cellular structure is intact, cells at the periphery are reported to undergo clockwise and counter clockwise rotations (Doxzen et al., 2013). In contrast, when a local interruption of this cable appeared, the cell at the vicinity could extend a lamellipodia and move away and radially from the center of the colony (Figure 4—figure supplement 2a). Apparently, a local defect in the cable could promote the local protrusion of a cell in the direction normal to the edge as shown in laser ablation experiments previously (Reffay et al., 2014). Several local defects could appear within the same colony, thus providing the opportunity for cells in the vicinity to protrude outwards. This cell has been termed leader cell (Reffay et al., 2011) and the collection of cells protruding from the circular colonies along this cell can be identified as the finger-like structures already reported for MDCK monolayers (Reffay et al., 2011; Reffay et al., 2014).

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Raw data corresponding to panels (c), (d) and (e).

For panels (c) and (d), we list the individual values corresponding to each colony and its associated fingers. For panel (e), we list the values for individual fingers in each condition.

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We performed cell tracking and observed that, on average, these protruding cells are faster than other boundary cells (Figure 4a and Figure 4—figure supplement 2b and c). They are characterized also by radial and directional migrations, in contrast to tangential motion observed in the other cells of the outer region of the colony (Figure 4—figure supplement 2d and e). In general, the motion of these so-called leader cells was directionally persistent and on average the shape of the whole colony followed the same overall direction (Figure 4b and c). To correlate colony elongation with leader cell orientation, we analyzed the evolution of a larger number of colonies for 2 hr after stencil removal (Video 4). We quantified the breaking of symmetry by fitting an ellipse to the shape of each colony. We then tracked the positions where finger-like structures were appearing, as well as the direction and distance performed by each of them. We could observe that the elongation direction of the whole colony correlated on average with the direction of the leader cell migration and associated finger (Figure 4c).

We then measured the position and displacement for each finger in control colonies and colonies under cyclic uniaxial stretching (Figure 4d). We observed that, when growing perpendicular to the direction of force application, finger cells performed shorter displacements than when growing parallel to it. In the absence of externally applied cyclic uniaxial stretching, fingers grew a similar amount as when growing parallel the direction of applied uniaxial cyclic stretching and no bias was observed vis-à-vis the nucleation position (Figure 4d and Figure 4—figure supplement 2f). To further explore this effect, we grew MDCK monolayers with straight boundaries either parallel or perpendicular to the external force. Then, we let fingers appear and grow for 2 hr before applying cyclic uniaxial stretching (Figure 4e). When fingers were growing perpendicular to the stretching direction, they shrank upon application of cyclic uniaxial stretching; in contrast, fingers further elongated when parallel to the direction of uniaxial cyclic stretching. Altogether, this suggests that direction of finger-like structures correlates with elongation direction, and that external stretching affects the dynamics of finger growth.

Finger growth correlates with colony elongation. However whether it is a cause or consequence of the symmetry breaking of the shape of the colony remains elusive. We therefore explored the possibility of inducing the growth of fingers and therefore set the direction of elongation of the colonies. Breakage of the acto-myosin cable by laser ablation induces the appearance of leader cells (Reffay et al., 2014). Hence we attempted to trigger the growth of fingers by locally injecting cytochalasin D using a micropipette. The transient injection of this actin polymerization inhibitor was followed by the disruption of the acto-myosin cable (Video 5 and Figure 5—figure supplement 1). However, the cable reformed, and fingers did not appear. This result shows that breakage of the cable alone does not trigger the growth of fingers in our colonies, and suggests that other mechanisms may be involved.

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We observed that when a finger moves outward from the colony, cells in the immediate vicinity elongate and seem to reorient their elongation axis toward the finger (Figure 5a). Recent studies have shown that the nematic field of cell elongation and its topological defects could be involved in the growth of bacterial colonies (Doostmohammadi et al., 2016) and in controlling dynamics, death and extrusion of epithelial cells (Kawaguchi et al., 2017; Mueller et al., 2019; Saw et al., 2017). We wondered if the spontaneous elongation of colonies would also be related to the average cell elongation. We followed the evolution of the cell elongation nematic field in different MDCK and MCF 10A colonies during expansion. We first obtained the spatio-temporal cell elongation nematic orientation field $\mathrm{\varphi }\left(x,y,t\right)$ (see Materials and methods) on the experimental time-lapse images (see Figure 5b, Figure 5—figure supplement 2a-c, Video 6 and Appendix 1C). We could then obtain the angle θ_nematic of the average cell-shape nematic field at t₀ which we compared with final colony orientation θ_colony obtained using the ellipse fitting analysis (Figure 5c, Figure 5—figure supplement 2d and Appendix 1C). Strikingly, we observed a clear average cell elongation even at the time of stencil removal t₀, and the corresponding angle θ_nematic correlated with colony orientation when elongation direction was established for both MDCK and MCF 10A cell lines (Figure 5d, Figure 5—figure supplement 2d and e). The cell elongation orientation field $\varphi \left(x,y,t\right)$ was not homogeneous at t₀ but exhibited a complex pattern with ±½ topological defects (Figure 5b and Appendix 1D). Interestingly, an expression that provides equilibrium orientation of liquid crystals with defects and having one constant Frank free energy,

mimics the experimentally observed orientation field $\varphi \left(x,y\right)$ very well with just one fitting parameter $\alpha$ (see Appendix 1D for details). Here k_i = ±½ and (x_i, y_i) are the strength and location, respectively, of the topological defects obtained from the experimental image. Thus, the defect position and strength can be used to provide an approximate readout for the orientation of the cell-shape nematic field (Figure 5d). Moreover, the location of finger nucleation seemed to be biased toward the position of topological defects. However, some defect locations were not stable in time and in some cases, the nematic field of cell shapes could only be interpreted in terms of virtual defects outside the colonies, thus suggesting that the mean nematic direction is a better readout for the cell-shape nematic field.

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Raw data corresponding to panels (c) and (d).

For panel (c), we list the individual values used to calculate the cumulative distribution function. For panel (d), we list the values for the x-axis, the y-axis and the weight associated to each dot.

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On the one hand, finger nucleation seemed to be correlated with colony elongation direction (Figure 4c). On the other hand, the orientation of tissue elongation correlates with orientation of average cell elongation at t₀ (Figure 5 and Figure 5—figure supplement 2). This suggests that leader cells moving outward from the colony may not be the cause of symmetry breaking in colony shape, but rather follow from the initial cell shape elongation before stencil removal. Moreover, we found no correlation between breaks of the acto-myosin cable surrounding the colony and the mean nematic direction (Figure 5—figure supplement 3), which suggests that breaks are uniformly distributed along the colony border. We have also shown that breakage of acto-myosin cable after stencil removal, which is associated with leader cell formation (Reffay et al., 2014), did not necessarily induce the growth of fingers in our colonies. Altogether, our results could indicate that the orientation of the mean cell-shape nematic of the colony before stencil removal sets the direction of elongation by triggering the growth of fingers, which appear at the discontinuities of the outer acto-myosin cable located along the nematic orientation, while preventing finger growth at discontinuities located in other directions.

Finally, when looking at the evolution of the mean cell elongation nematic field of colonies under uniaxial cyclic stretching, we observed that it did not change over time (Figure 5—figure supplement 4). The initial mean direction of cell elongation, either parallel or perpendicular to the external stretching, was maintained throughout 2 hr of external stretching. This suggests that average cell elongation alone does not determine colony elongation direction when subjected to uniaxial cyclic stretching.

We next sought to evaluate quantitatively the contribution of cellular processes to elongation. We quantified the contributions of each cellular event using image segmentation, cell tracking, and neighbor triangulation (Etournay et al., 2015; Etournay et al., 2016) (see Appendix 1A, Figure 6a and b, Figure 6—figure supplement 1 and Figure 6—figure supplement 2, and Video 7). This procedure decomposes overall tissue elongation, which is quantified in terms of cumulative total pure shear, into contributions from cell elongation and topological changes. Five main events contribute to total shear: cell elongation change, cell division, cell extrusion, T1 transition, and correlation effects (Etournay et al., 2015; Etournay et al., 2016). At the colony scale, shear decomposition plots (Figure 6c and Figure 6—figure supplement 3) revealed that the total pure shear gives values consistent with elongation estimates from ellipse fitting (Figure 6d). Note that various contributions to shear decomposition exhibit significant variability between experiments (Figure 6—figure supplement 3). However, we found that after the first 2 hr, the contribution of cell elongation is generally comparable to the total pure shear, with a smaller contribution from other sources (Figure 6d). When looking at the shear decompositions of colonies under cyclic uniaxial stretching (Figure 6e and Figure 6—figure supplement 3), the cumulative shear values were also similar to the ones obtained by ellipse fitting (Figure 6f). Interestingly, we found however that in that case, shear created by T1 transitions is the main contributor for the total pure shear, while cell elongation does not contribute to tissue elongation (Figure 6f and Figure 6—figure supplement 3). This indicates that applying oscillatory forces to the tissue changes fundamentally the main mode of tissue elongation by favoring topological rearrangements of the cell network.

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We then asked whether a model could reproduce experimental observations of shear decomposition and, in particular, in which conditions tissue elongation would arise from cell elongation or from topological transitions. We developed a vertex model which takes into account mechanical anisotropies such as active stresses and polarized cell bond tension (see Figure 7a and Appendix 2). We generated a confluent colony of circularly confined cells, in which a unit director p that represented the cell polarity was assigned to each cell. Based on orientation of the director, each cell generated an extensile active stress σ_a and bias λ in the base value of its edge contractility to promote cell elongation and active T1 transitions. We assumed that the experimentally measured cell elongation nematic q is a readout of the underlying cell polarity p (Figure 7b). Hence, the initial spatial distribution of p was based on the commonly observed pattern of q (Figure 7c). To evolve p with time, we imposed that p of the exterior cells tended to be parallel to the boundary, whereas the inner cells tended to align their p with those of their neighbors (Appendix 2D).

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Upon removal of confinement, we found that the simulated colony spontaneously elongates, in a direction set by the orientation of the mean cell elongation nematic field, along with the formation of finger-like structures near the +½ defect, as observed experimentally (see Video 8). Our simulations therefore reproduce experimental observations indicating that colony deformation can be understood without forces exerted by a leader cell at the colony boundary (Figure 7c, Video 8, and Appendix 2). To test whether we could also recapitulate different contributions of the total pure shear inside the tissue, we performed a shear decomposition analysis of in silico movies. We found a qualitatively similar cumulative shear-strain decomposition as was observed in experiments (Figure 7d and e), where the main contribution came from cell elongation. Moreover, by changing the relative contributions of the cellular active stress magnitude (σ_a) and the edge tension bias (λ), we could modulate the relative contributions from cell elongations and T1 transitions to the total pure shear (Figure 7—figure supplement 1) as was also observed in experiments with colonies in the absence or presence of cyclic stretching (Figure 6—figure supplement 3). When σ_a was dominant, the colony elongation was primarily due to cell elongation, whereas when λ was the stronger term, T1 transitions were the main cause of colony elongation. These results reveal possible cellular mechanisms that can govern the process of tissue deformation and influence whether cell elongation, or cellular rearrangements, dominate tissue elongation.

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Our vertex model assumed that the cell elongation was the main readout for cell polarity, and it did not explicitly account for the effect of substrate stretching. To incorporate uniaxial cyclic stretching, we further developed the model. Our results show that initial cell shape elongation does not have a preferential direction (Figure 5—figure supplement 4), but colony elongation under uniaxial cyclic stretching is along x axis, the direction of stretching, and mainly achieved through T1 transitions (Figure 2c and Figure 6f). Also, we report that the elongation happens along y axis, perpendicular to the direction of stretching, in single cells and cell colonies with lowered Ecadherin levels (Figure 3a–d). These two experimental observations can be implemented in the model. First, we introduced an additional term m_stretch that oriented cell polarization p along x axis, for any given initial condition, upon confinement removal at t₀ – this term is inactive in the absence of uniaxial stretching (Appendix 2D). Then, by using cell active stress σ_a > 0, we mimicked the tendency of single cells to elongate perpendicular to the orientation of polarity, i.e., perpendicular to the uniaxial stretching. In contrast, the bias in the edge tension λ > 0 induced T1 transitions along the polarity of the cell, i.e., parallel to the uniaxial stretching. Thus, in the presence of uniaxial stretching, m_stretch oriented cell polarities along x, while the relative magnitudes of single cell active stress σ_a and edge-contractility bias λ dictated the orientation of the colony elongation. When keeping a lower magnitude of σ_a and a higher value of λ, colonies elongated along x through T1 transitions (Figure 7f and Video 9), mimicking colony elongation under uniaxial cyclic stretching (Figure 2 and Figure 6f). On the contrary, by increasing σ_a and lowering λ (Ecadherin deficient colonies), colonies elongated perpendicular to x (Figure 7g and Video 9), mimicking colonies under uniaxial cyclic stretching treated with E-cadherin antibody (Figure 3c–d). Therefore, we propose that a competition between the strength of active T1 transitions parallel to the external stretching and active cell stress perpendicular to the external stretching dictate overall colony elongation under uniaxial cyclic stretching. When cell-cell junctions are intact, colony elongation is along the direction of stretching and through T1 transitions (Figure 2c and Figure 6f), suggesting that the tendency of single cells to orient along y (Figure 3a–b) is partially screened by cell-cell junctions via T1 transitions. When cell-cell junctions are weakened, active cell stress dominates, and colonies, which could be thought of to be closer to a collection of single cells, elongate perpendicular to the uniaxial stretching direction (Figure 3c and d).

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We showed that upon stretching, cells reduced their speed and myosin structures appeared (Figure 3e and f). These type of cellular responses to external stretching involve the Rho-associated protein kinase (ROCK) (Hart, 2020), which is also involved in cell-cell contacts integrity (Ewald et al., 2012; Nishimura and Takeichi, 2008). We treated MDCK colonies with a ROCK inhibitor (Y-27632 50µM) and followed their behavior for 2 hr, both in the absence and the presence of cyclic uniaxial stretching (Figure 8—figure supplement 1). When cyclic stretching was applied, elongation along the direction of the uniaxial cyclic stretching was absent (Q_xx ≈ Q_xy) (Figure 8—figure supplement 1a and b). However, colonies still elongated anisotropically, similar to colonies in the absence of application of uniaxial cyclic stretching (Figure 8—figure supplement 1c). According to our model, when edge tension bias is sufficiently large, the dominant mechanism for tissue elongation switches from single-cell elongations to T1 transitions (Figure 7—figure supplement 1). We observed that uniaxial cyclic stretching triggers this type of elongation (Figure 6f). Therefore, if the effect that application of uniaxial cyclic stretching has at the cellular level was reduced, colonies subjected to cyclic uniaxial stretching should preferentially elongate as if cellular active stress becomes dominant, that is through single-cell elongation. Strikingly, shear decomposition analysis shows that the elongation mechanism shifts from T1 transition-dominant to cell elongation-dominant when colonies under uniaxial cyclic stretch have ROCK inhibited (Figure 8a-b). In summary, colonies under cyclic uniaxial stretching elongate through T1 transitions rather than through cell elongation (red dot in Figure 8c) similar to what our model predicts for colonies with increased biased edge tension. In contrast, colonies elongate spontaneously largely through single-cell elongation, which the model predicts when the effect of the cellular active stress is more dominant (green dot in Figure 8c). Strikingly, the application of a ROCK inhibitor leads to single-cell elongation dominating over T1 (orange dot in Figure 8c), effectively suppressing the effect of uniaxial cyclic stretching on the mode of colony deformation (Figure 8—figure supplement 1).

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Values of each of the different contributions to cumulative shear as time progresses for the three colonies analyzed.

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Our results show that cell elongation nematic field can have an impact on epithelia morphogenesis. It was already reported that topological defects in the cell elongation nematic field have a key role on epithelial dynamics (Kawaguchi et al., 2017) and on cell death and extrusion (Saw et al., 2017). Now, we showed that circular epithelial colonies when in confinement build up a mean nematic orientation. This symmetry breaking results from the inner activity of cells, and sets the direction for colony elongation. Epithelia changes in shape could be revisited in vivo with this new framework, leading to potential generic rule of morphogenesis in developmental biology.

MDCK cells (GFP-E-cadherin strain [Adams et al., 1998], GFP-Actin strain, mCherry-actin / GFP-myosin strain [Klingner et al., 2014]) were cultured in Dulbecco’s modified Eagle medium (D-MEM) 1 g/l glucose (Invitrogen), supplemented with 10% fetal bovine serum (FBS) (Invitrogen) and 1% penicillin-streptomycin (and the corresponding resistance for each strain). Cells were passaged every 2–3 days using standard procedures. Caco-2 cells (ATCC) were cultured in minimum essential media (MEM) supplemented with Earle’s salts (Life Technologies), 20% fetal calf serum (FCS) (Invitrogen), 0.1 mM non-essential amino acids, 1 mM sodium pyruvate, and 40 µg/ml Gentamicin at 37°C and 5% CO₂. Culture was passaged every 3–4 days. MCF 10A cells (ATCC) were cultured in Dulbecco’s modified Eagle medium (D-MEM) 1 g/l glucose (Invitrogen), supplemented with 10% horse serum (Invitrogen), 5 µg/ml insulin, 40 µg/ml Gentamicin, 2 mM L-Glutamine, 0.5 µg/ml Hydrocortisone, and 2 ng/ml human epidermal growth factor (hEGF). Cells were passaged every 2–3 days. Cells tested negative for mycoplasma.

Poly(dimethylsiloxane) (PDMS) (Sylgard) was prepared by mixing the pre-polymer and the cross-linker at a 10:1 ratio. To prepare stretchable membranes, uncured PDMS was first centrifuged (3000 rpm for 5 min) to remove air bubbles. Afterwards, the PDMS was spin-coated on a flat polystyrene (PS) surface (500 rpm for 1 min) and cured at 65°C overnight. PDMS stencils were prepared as described previously (Ostuni et al., 2000). Briefly, SU-8 2050 molds containing circles of 250 µm in diameter were prepared by standard photolithography. Uncured PDMS was then spin-coated on molds to a thickness lower than the height of the microstructures (50 µm) and cured overnight at 65°C. Stencils for the finger experiments were prepared by spin-coating uncured PDMS on a flat surface.

The PDMS stencils were cut, peeled off the mold, and placed in a 2% Pluronic F-127 (Sigma-Aldrich) in PBS for 1 hr. The stencils were then kept in PBS for 2 hr and dried under the hood flow. PDMS stretchable membranes were cut and then activated using O₂ plasma. The membranes and the stencils were exposed to UV light in the cell culture hood for 10 min. Afterwards, stretchable membranes were incubated with fibronectin 20 µg/ml for 1 hr, rinsed with PBS and dried. PDMS membranes were placed on a PS holder, and the PDMS stencils were deposited on top of the membrane right after. A rectangular PDMS chamber was attached onto the membrane using vacuum grease, and cells were seeded at a density of 20,000 cells/mm² (Serra-Picamal et al., 2012) for 4 hr. When cells were attached, the medium was changed and the membrane with the cells was kept in the incubator. Local cell density could vary within each colony. We followed the dynamics of assembly of the colony prior removal of the stencil and we could see that cellular clusters size distribution and respective location within the cavity at plating could contribute to these variations. Once they formed confluent circular colonies, the stencils were removed with tweezers carefully before starting the experiment. Some of the colonies exhibited elongation in the short time window between stencil removal and the start of image acquisition.

After stencil removal, the medium was replaced by L-15 (Leibovitz’s L-15 medium, Invitrogen) supplemented with 10% FBS. Cells were then observed under a Nikon Ti inverted microscope using either a x10 or a x20 objective for 6 hr at 37°C. Images were acquired every 5 min.

The stretching device was designed in the lab. Briefly, a Thorlabs motor (Thorlabs z812 motor, Thorlabs) was controlling the motion of a PDMS membrane, and everything was mounted on a custom-made microscope stage. Circular colonies were plated on PDMS membranes, and after removal of the stencils, samples were placed on the microscope. Cyclic uniaxial stretches were applied and images were taken every 30 min typically shortly to prevent interfering with the time course of the experiments. We probed three times for cycles, 20, 60, 120 s, and three extensions, 5, 10, and 20%. The shape of the cycles was triangular. We checked that the PDMS membranes were elastic at all extension and frequency ranges.

To prevent the formation of E-cadherin-mediated adhesion, cells were incubated for 30 min with L-15 medium containing 5 mM EDTA (Sigma-Aldrich) and 10 µg/ml anti-E-cadherin blocking antibody that targeted the extracellular domain of E-cadherin (Gumbiner et al., 1988) (uvomorulin, monoclonal anti-E- cadherin antibody, Sigma); after incubation, the medium was replaced by normal L-15 and the experiment started. The inhibition of ROCK was done by incubating cells with Y-27632 50 µM solution (Sigma-Aldrich) from 30 min before the experiment started until the end of the experiment.

For the finger test after growth, we let finger grow for 2 hr, and we subsequently applied the cyclic stretch.

Shape change analysis was performed using ImageJ (http://rsb.info.nih.gov/ij, NIH). The outline of the colony on phase contrast images was ellipse fitted at every time point. Major axis a, minor axis b, and ellipse orientation θ were obtained. We computed Q, defined as Q_xx(t) = ½ ln(a(t)/b(t))·cos2·(θ(t) − α) and Q_xy(t) = ½ln(a(t)/b(t))·sin2·(θ(t) − α ) being α = θ(t_final) to quantify cell colony elongation. In uniaxial stretching experiments, the x axis corresponds to the direction of the external stretch and Q components are defined as Q_xx(t) = ½ ln(a(t)/b(t))·cos2·(θ(t) − α) and Q_xy(t) = ½ln(a(t)/b(t))·sin2·(θ(t) − α) being α = 0.

The centroid trajectories of cells were tracked using the manual tracking plug-in in ImageJ. Data analysis was performed using a custom-made code in MATLAB (The MathWorks). Cell positions were characterized by a vector r(t), with t denoting time and r position in space (bold letter refers to a vector). Every recorded cell position during the time-lapse experiment was defined as r_i = r(t_i), where t_i = iΔt are the times of recording and Δt denotes the duration of time-lapses. The average velocity of each cell was then defined as v = (1/T)·∑_ir_i, being r_i the module of the vector r_i and T the total duration of the trajectory.

Movies acquired using an MDCK GFP-E-cadherin strain were first pre-processed with FIJI. The subtract background function was applied to remove noise. Images were then loaded to Tissue Analyzer (TA) v8.5 (Aigouy et al., 2010) for edge detection and cell tracking.

First, the background noise of the time-lapse images of the elongating MDCK colonies was reduced with the subtract background function by using a rolling ball radius of 40 px. The resulting images were then subjected to band-pass filter with upper and lower limits of 40 px and 3 px, respectively. The background noise from this output was reduced by using the subtract background command again with a rolling ball radius of 40 px. The processed images from each experiment were analyzed with the OrientationJ plugin of FIJI to quantify their local spatial orientation that reflects the underlying cell elongation. Within this plugin, we used a local smoothing window of 20 px (approximately of the size of the cells) to obtain the structure tensor at discrete points on a grid of 20 px × 20 px. The plugin provides the dominant direction ${\mathrm{\varphi }}_i$ of the structure tensor at each point $\left(x_i,y_i\right)$ that represents the local orientation field ${\displaystyle {\mathbf{q}}_i=\cos {\varphi }_i{\hat{\mathbf{e}}}_1+\sin {\varphi }_i{\hat{\mathbf{e}}}_2}$ for cell elongation. The OrientationJ plugin also provides the coherence C of the structure tensor to quantify the strength of the orientation; $C\approx 0$ and $C\approx 1$ would approximately correspond to rounded and elongated cells, respectively. The orientation or the director field q thus obtained was further quantified by studying the spatiotemporal evolution of ±1/2 topological defects that were obtained by calculating the winding number over unit-cells of the underlying grid. The local smoothing window of 20 px for obtaining the structure tensor, which is approximately of the size of cells, ensured that the most robust defects were observed. The validity of this procedure was cross-verified with the smoothed cell-shape nematic field and the corresponding ±½ defects from the segmented and triangulated data of the experiments processed in Tissue Miner (TM) (Figure 5—figure supplement 2 and Appendix 1). Finally, the orientation of mean cell-shape nematic calculated at 0 hr was compared with shape orientation of the colony at t = 2 hr. For obtaining the cell-shape nematic field for MCF 10A colonies, the procedure was the same as for MDCK control but the numerical parameters used were, rolling ball radius for subtract background 50 px, no band pass filter, and local smoothing window of 15 px and grid size of 30 px for OrientationJ. Similarly, for obtaining the cell-shape nematic field for stretched colonies, the procedure was the same as for MDCK control but the numerical parameters used were, rolling ball radius for subtract background 50 px, no band pass filter, and local smoothing window of 40 px and grid size of 30 px for OrientationJ. See Appendix 1 for more details (also see Video 6).

In order to image acto-myosin supracellular cables at the boundary of colonies either cells expressing mCherry-actin / GFP-myosin strain or cell immunostaining were used. To disrupt acto-myosin cables, local injection of 4 µM Cytochalasin D using a micropipette was performed. Glass micropipettes were connected to a microinjection system (CellTram vario, Eppendorf). The position of the pipette tip was controlled in x, y, z by using a micromanipulator. The system was mounted on an epifluorescence inverted microscope to record the process. Cytochalasin D was released locally for about 10 min. To detect discontinuities in the cable, fluorescent images of actin and myosin were treated using Fiji as follows. First, a median filter was applied and the background subtracted. Then, a contrast-limited adaptive histogram equalization (CLAHE) was used and the background was again removed using an exponential function. Images corresponding to actin and myosin were added. Finally, a Laplacian of Gaussian filter was applied to the resulting image. Once the cable structure was revealed, the positions of the defects were identified.

After obtaining the geometrical and topological information of the colonies from the series tracked images generated using TA, TM was used to extract, triangulate and store the data with the help of an automated workflow. The database obtained after this stage of analysis was used to quantify various state properties such as cell area, neighbor number, cell elongation and the contribution of different cellular processes to deformation using scripts written both in R and in Python. TM was further used to quantify the contributions of various cellular events such as cell elongation and topological transitions to the colony deformation. More details about this analysis can be found in Appendix 1 (also see Video 7).

A vertex model was developed with an addition of a unit nematic director p to every cell. The orientation of the boundary cell p was maintained parallel to the boundary, whereas p for internal cells were modeled to tend to align with the p of its neighbors. In these simulations, an extensile active stress ${\displaystyle {\sigma }_a(\mathbf{p}\mathbf{p}-\frac{1}{2}\mathbf{I})}$ with ${\sigma }_a$<0 acts to increase cell elongation along p. In addition, a bias λ was also applied on the basic edge tension with respect to the director p of its neighboring cells. For positive λ, this bias reduces (increases) the tension of the edge along (perpendicular to) p. Consequently, the closure (formation) of edges is enhanced in the direction perpendicular (parallel) to p. Hence, the T1 transitions in the region around the cells are oriented to cause shear elongation (contraction) along (perpendicular) to p. The colony is provided with an initial condition for p that mimics the initial configuration of experimentally frequently observed cell shape nematic fields q with two +½ defects that are separated by a distance. The initial polar vector orientation along the nematic axis are chosen at random such that the total polarity is zero, and the dynamics of polarity reorientation is independent on a p → -p flipping of the polarity axis. To begin with, the cell positions and director orientations were evolved under colony confinement until cell position and p do not change significantly. The confinement is then removed to see how the colony breaks symmetry in its shape. In another set of simulations, a small motility v₀ was also provided to the internal cells (Figure 7—figure supplement 1). Similar to the experiments, the output of these simulations was also processed in TM and analyzed for topological defects and pure shear decomposition. For colonies subjected to uniaxial cyclic stretching, p for any cell was provided with an additional tendency to align along the stretching orientation (x-axis). Moreover, the active stress in this case ${\displaystyle {\sigma }_a>0}$ was contractile, that is the cell tended to elongate perpendicular to the orientation of p. More details of the simulations are provided in Appendix 2 (also see Videos 8 and 9).

No statistical methods were used to predetermine sample size. Measurements were performed on different colonies (n) obtained in different independent experiments (N). The exact values are defined at the caption of each figure. Data presentation (as Mean value ± standard deviation (SD) or as Mean value ± standard error of the mean (SE)) is defined at the corresponding figure caption. D’Agostino normality test, Mann-Whitney test and Pearson’s correlation were performed using GraphPad Prism 8. Specific values are noted at the corresponding figure captions.

In the first step of analysis, time-lapse images of MDCK colonies are segmented to partition the tissue into individual cells and then tracked for their movements using Tissue-Analyser (TA) plugin of FIJI (Aigouy et al., 2010; Aigouy et al., 2016) (see Figure 6—figure supplement 1a-i). This information of the tracked cell colony is then passed to Tissue-Miner (TM), which extracts data and stores the cellular geometry and connectivity in the form of cell vertices and bonds using a snakemake based automated workflow (see Figure 6—figure supplement 1j-l). As described in Merkel et al., 2017; Etournay et al., 2016, TM has R or Python API that can be used to quantify fine-grained (cell-level) and coarse-grained (colony-level) deformation properties of epithelial colonies. Even though the scripts to calculate the basic kinematic quantities are available, users have the freedom to add new computational routines to calculate the quantities of their interest.

A planar vertex model is one of the most commonly used computational tool to understand mechanics of epithelial monolayers (Farhadifar et al., 2007). In this description, the tissue is assumed to be a 2-D sheet composed of cells which are represented by planar polygons sharing vertices and edges with their neighbors (Figure 7a). Governing equations are formulated to study the dynamics of evolution of each vertex of a cell. The total effective energy or work-function W of the tissue arises from area deformation of the cells and a combination of acto-myosin contractility and membrane adhesion energy at the cell interfaces (Figure 7a).

where for a particular cell $\alpha$, $A_{\alpha }$ is the actual area and $A_{\alpha }^0$ is the preferred cell area. $l_{\alpha \beta }$ is the length of the bond connecting shared between the cells ${\displaystyle W}$ and $\beta$ and ${\mathrm{\Lambda }}_{\alpha \beta }$ is the effective contractility of that bond (Alt et al., 2017). Note that the energy W depends both on the position ${\mathbf{𝐫}}_i$ of the vertices and cell connectivity.

For this model, the force ${\mathbf{𝐅}}_{\mathrm{vm}}^i$ acting on a particular vertex i can be obtained by taking a derivative of ${\displaystyle W}$ (Fletcher et al., 2014; Farhadifar et al., 2007) with respect to the vertex position ${\mathbf{𝐫}}_i$ as,

and is balanced by the friction force on the vertex from the substrate. Here, $\eta$ represents effective substrate viscosity experienced by the vertex via cell-substrate friction (Fletcher et al., 2014). A mathematically equivalent description of the vertex model could also be made by using mechanical equilibrium or principle of virtual work (Chen and Brodland, 2000; Alt et al., 2017).