Living cells on the move

Spectacular collective phenomena such as jamming, turbulence, wetting, and waves emerge when living cells migrate in groups.

Much like birds fly in flocks and fish swim in swarms, cells in our body move in groups. Collective cell migration enables embryonic development, wound healing and cancer cell invasion. These phenomena involve complex biochemical regulation, but their dynamics can ultimately be predicted by emerging physical principles of living matter.

When and where do cells migrate in groups?
When seen at the microscale, our body is a busy maze filled with actively moving cells. Cellular movements are slow, rarely exceeding 10 µm/min, but crucial to embryonic development, the immune response, tissue self-renewal and wound healing. Through the same mechanisms that sustain these physiological functions, cell movements drive devastating diseases such as acute inflammation and cancer. These can in fact be considered diseases of cell movement because arresting cells in a controlled manner would be sufficient to prevent their spread. Take cancer as an example; whereas its origin is well-known to be genetic, if we could selectively stop the movement of tumor cells, we would prevent them from escaping primary tumors and reach distant organs to metastasize. Understanding cell migration is therefore crucial to improve current strategies to fight disease.
Cell migration comes in different flavors. Some cell types move as isolated self-propelled particles. For example, to chase and destroy pathogens, immune cells move individually through tissue pores. Similarly, in some types of cancer, single cells dissociate from tumors and travel solo through the surrounding tissue, eventually reaching blood vessels and metastasizing at distant organs. By contrast, in other physiological functions and pathological conditions, cells move as collectives. Collective cell movements are dominant in embryo development, where they enable the precise positioning of tissues and organ progenitors. The later coordination of growth and motion shapes organs well into postnatal life. Collective cell migration also drives wound healing in tissues such as the skin. Here, a continuous sheet of tightly adhered cells moves cohesively over the damaged area to restore a functional tissue. Cancer invasion also involves the migration of cell sheets, strands and clusters (Fig. 1a). Within these groups, cells with different function can self-organize in spatial compartments to behave like an aberrant organ. This added functionality is thought to provide malignant tumors with distinct invasive strategies that improve their chances of spreading and metas-tasizing. Finally, collective cell migration is also involved in maintaining the inner surface of the gut, which is the fastest self-renewing tissue in mammals. It renews entirely every 3-5 days, which implies a daily loss of tens of grams of cells. Self-renewal proceeds thanks to the division of stem cells that reside at the bottom of tissue invaginations called crypts. The progeny of these stem cells then migrates collectively from the crypt to the top of finger-like protrusions called villi, where they are shed into the intestinal lumen and discarded (Fig. 1b).
Collective cell migration is regulated by a myriad of molecular processes, from genetic programs to sensing and signaling pathways. Yet, these molecular processes act upon a limited number of physical quantities to determine cell movement. Therefore, coarse-grained physical approaches may provide crucial insight into biological questions. Furthermore, collective cell behaviors also inspire new physical theories of living systems. In this article, we highlight progress in this direction.

Cell assemblies as living matter
What physical principles underlie collective cell motion? In traditional condensed matter, interactions between electrons and between atomic nuclei give rise to fascinating collective phenomena such as magnetism and superconductivity. Analogously, cell-cell interactions lead to emergent collective phenomena in migrating cell groups. However, when treating cell colonies as materials, we must take into account some key features of living matter, as we illustrate throughout the article.
First, the primary constituents of living tissues are cells and extracellular networks of protein fibers such as collagen. The interactions between these mesoscale constituents are orders of magnitude weaker than interatomic interactions in conventional solids. So, with notable exceptions like bone, most biological tissues are soft materials, which can easily deform and flow.
Second, cells are machines with internal engines. Specialized proteins known as molecular motors harness the energy of chemical reactions to generate forces and produce mechanical work. These energy-transducing molecular processes ultimately power cell migration, allowing cells to move autonomously, without externally applied forces. This continuous supply of energy drives living tissues out of thermodynamic equilibrium. Importantly, the driving is local; it occurs at the level of individual constituents, i.e. single cells. In other words, cells are active constituents, and living tissues are a paradigmatic example of active matter -an exploding new field in nonequilibrium statistical physics. But cells are not only mechanically active: they also sense their environment, process information, and respond by adapting their behavior. For example, stem cells plated on substrates of different stiffness differentiate into distinct cell types, from brain to bone cells. So living tissues are adaptive: they respond in programmed ways to environmental cues such as external forces, the mechanical properties of the extracellular matrix, and concentrations of nutrients and signaling molecules. Consequently, cell-cell and cell-environment interactions are often very complex. Unlike atoms and electrons in conventional condensed matter, cellular interactions cannot in general be fully described via an interaction potential with a fixed functional form. Thus, a key challenge in this field is to find effective ways to capture complex cell behaviors in terms of simple, physically-motivated interactions 1 .

To flow or not to flow
One way to think about interactions between deformable epithelial cells comes from the physics of foams. In foams, gas bubbles arrange in polygonal packings, with the liquid phase filling the interstitial spaces and providing surface tension to bubble interfaces (Fig. 2a). Similarly, cells in epithelial monolayers also acquire polygonal shapes, with roughly straight edges subject to active tension generated inside cells (Fig. 2b). This description of tissues as polygonal cell packings traces back to work by Hisao Honda and collaborators in 1980, and it was later popularized by work in the group of Frank Jülicher and collaborators 3 . Because cells are deformable, edge lengths vary dynamically. These variations change the energy of the cellular network, which we can write in terms of areas A i and perimeters P i of cells i = 1, . . . , N as This energy assumes that cells resist changes in their area and perimeter around the preferred values A 0 and P 0 with elastic moduli κ and Γ, respectively. The preferred perimeter P 0 depends on cell-cell interactions and cellular activity, with cellcell adhesion promoting longer edges and cellular tension favoring shorter edges. The preferred perimeter and area define a dimensionless parameter p 0 = P 0 / √ A 0 which informs about the preferred cell shape. Higher p 0 corresponds to more elongated cells, and smaller p 0 corresponds to more isotropic shapes, with less perimeter for the same area.
For a given p 0 , edge lengths vary until the system reaches its ground state, minimizing the energy in Eq. (1). In this process, an edge can shrink until it eventually disappears, leading to the formation of a new cell-cell interface (Fig. 2c). These events, known as T1 transitions, allow cells to change neighbors, driving topological rearrangements of the cellular network. The ability to reorganize its constituents determines whether a material is solid or fluid. So, if cell rearrangements are difficult, the cellular network resists shear deformations; the tissue is solid. In contrast, if cells can rearrange easily, the network yields upon shear; the tissue is fluid. At small p 0 , i.e. for roundish cells, Eq. (1) reveals that there is an energy barrier preventing T1 rearrangements (Fig. 2c). However, as we increase p 0 and cells become more elongated, the energy barrier decreases (Fig. 2c). At a critical value of p 0 = p * 0 ≈ 3.81, the barrier vanishes (Fig. 2d); cells can rearrange freely 2 .
This simple model thus predicts a solid-fluid transition in tissues driven by changes in cell shape, encoded in p 0 ormal component indicates the extent to r compressed at a given point, whereas the orces parallel to the surface. enerate active stresses, a key feature that passive soft materials is the self-propulual cell. Cells achieve self-propulsion by f protrusion and adhering them to their in types of protrusion are thin actin-rich extensions called filopodia or lamellipodia, and spherical membrane blisters called blebs. Cells migrate using lamellipodia, filopodia or blebs, depending on a diversity of intrinsic and extrinsic variables, including expression of adhesion proteins, density and composition of the ECM, confinement, topology and cortical contractility 60,61 . To migrate, cells adhere their protrusions to the surrounding matrix either specifically through membrane receptors such as integrins, or unspecifically using frictional interactions. teins (k on , k o in panel f), and the activity that pull on the laments (F in panel f).
successfully described complex features of cell monolayers such as wave propagation in the absence of inertia 76,133 . shear/bulk to describe . However, of external or exchange layer where e rare, cell n processes cells shrinks uring cells, the tissue is n the vertex amount of . Therefore, single-cell all possible y generated s. For each 'sharpness' of the transition, whereas p 0 controls the distance to the transition. This suggests that the trio (r1", r, p 0 p ⇤ 0 ) is analogous to (m, h, T T c ) in the Ising model. Therefore, our scaling ansatz is that the order parameter r1"(r, p 0 ) vanishes at the critical point p 0 = p ⇤ 0 , with fluctuations controlled by r. In that case, near the critical point the order parameter should obey the universal scaling form 33 : Here z = r/|p 0 p ⇤ 0 | is the crossover scaling variable, is the crossover scaling critical exponent, and f , f + are the two branches of the crossover scaling functions for p 0 < p ⇤ 0 and p 0 > p ⇤ 0 , respectively. After re-plotting the data in Fig. 1b using equation (3), we find an excellent scaling collapse onto two branches with = 4.0 ± 0.4, = 1.0 ± 0.2 and a precise location of the critical point p ⇤ 0 = 3.813 ± 0.005, as shown in Fig. 2.
For the mechanically rigid branch in the limit z ! 0, the energy barrier can be rewritten in dimensional units and scales as This indicates that these barriers are completely governed by the perimeter elasticity ⇠ . At the critical point, the two branches of the scaling function merge and f + = f = z / . In this case the dimensional energy barrier scales as   model has a constant-density glass transition governed by singlecell mechanical properties such as cell-cell adhesion and cortical tension encoded in the target shape index p 0 .
Analysing only the ground states of the vertex model, the seminal work of Staple et al. 16 found an ordered-to-disordered transition at p 0 = p hex 0 ⇠ 3.722. However, because almost all biological tissues are strongly disordered, it remained unclear whether this transition was relevant for the observed glass or jamming transitions in multicellular tissues. Therefore, we explicitly study disordered metastable states and transitions between them. In addition ref. 16 uses a linear stability analysis of a single cell to suggest that a rigidity transition also occurs at p hex 0 . In contrast, our analysis explicitly includes multicellular interactions (that is, collective normal modes) and nonlinear e ects (that is, energy barriers). With this more sophisticated analysis, we demonstrate that vertex models exhibit a rigidity transition at a value p 0 = p pent 0 ⇠ 3.81 that is measurably di erent from the prediction p 0 = p hex 0 based on singlecell linear stability.
Importantly, predictions based on this critical rigidity transition have recently been verified in experiments 43 . Specifically, in both simulations and experiments we can measure the shape index p = P/ p A for each cell in a monolayer, where P is the projected cell perimeter and A is the cross-sectional area. In simulations of the vertex model, we find that the median value of the observed shape index p is an order parameter that also exhibits critical scaling: p = p ⇤ 0 ⇠ 3.81 for rigid or jammed tissues and p becomes increasingly larger than p ⇤ 0 as a tissue becomes increasingly unjammed (Fig. 2). This prediction is precisely realized in cultures from primary cells in human patients, with implications for asthma pathobiology 43 .
We expect that this rigidity framework will help experimentalists develop other testable hypotheses about how the mechanical response of tissues depends on single-cell properties. For example, Sadati et al. 44 have proposed a jamming phase diagram where tissues become more solid-like as adhesion increases, based on observations of jamming in adhesive particulate matter at densities far below confluency. Using standard interpretations of the vertex model (equations (1) and (2) ( Fig. 2e). This is a striking prediction, which showcases the bizarre mechanical properties of materials with deformable constituents. In conventional condensed matter, solids melt by either increasing temperature or reducing the packing fraction, i.e. decreasing pressure. Tissues, however, can melt at a fixed temperature and at the maximum packing fraction, φ = 1, i.e. without gaps between cells 2 . Tissues can become fluid by increasing the cell perimeter-to-area ratio, for example by either decreasing intracellular tension or, counterintuitively, increasing cell-cell adhesion (Fig. 2e). The more cells adhere to one another, the more they elongate, and the easier it is for them to rearrange.
Shortly after its prediction 2 , this solid-fluid transition driven by changes in cell shape was experimentally verified in layers of human bronchial epithelial cells 4 . Moreover, this study showed that cells from healthy individuals tended to be caged by their neighbors and form a solid tissue, whereas cells from asthmatic individuals tended to remain unjammed, forming a fluid tissue. Therefore, these experiments suggested that the fluid-solid transition in tissues is involved in disease, opening the door to new treatments based on preventing this phase transition. Fluid-solid transitions also occur during development, enabling tissues to first turn fluid in order to remodel and acquire their shapes, and then solidify and mature. The emerging picture is that, in different biological contexts, cells can tune their shape and use the physical principles governing phase transitions in foams to decide whether to flow or not to flow.

Aligning with neighbors
The action starts once tissues become fluid and cells can move. Cells then engage in collective flows of different types, depending on how cells align with neighbors to coordinate their individual motions. Again, these cell-cell interactions depend strongly on cell shape.
Cells come in many different shapes: some are roughly spherical, some have rod-like shapes, and some cells develop a head-tail asymmetry to migrate persistently in one direction. This direction can be represented by a vector known as cell polarity. In groups, cells can align their individual polarities, forming phases of matter with orientational order. These alignment interactions and the resulting oriented phases can be described using concepts from the physics of magnetism and liquid crystals. For example, cells in a group can spontaneously break symmetry and align in a common direction. This type of alignment is known as polar order. To capture its emergence, one can introduce ferromagnetic-like interactions between individual cell polarities. At a coarse-grained level, collective cell polarity can be thought to result from an effective free energy with the Mexican-hat shape familiar from the Landau theory of phase transitions. This example thus illustrates how tools and concepts from other fields of physics are being borrowed to rationalize cell alignment.
In other situations, cells align along one axis but choose no preferred direction of motion. This type of alignment is known as nematic order, taking its name from nematic liquid crystals, used in LCD screens. Prominent features of liquid crystals are singular points known as topological defects, where alignment is locally lost. You can find topological defects literally on your own hands: they are the points where your fingerprint ridges meet (Figs. 3a and 3b). Recently, researchers have discovered topological defects in several cell assemblies, from bacterial colonies to epithelial tissues, confirming that they can be described as liquid crystals (Fig. 3c). Interestingly, topological defects can play important biological roles. For example, in epithelial monolayers, defects pro-according to the cell cycle phase, which is known as the interkinetic nuclear migration 11 . To see whether the direction switching is related to cell cycle, we used the cell-cycle marker Fucci 12 , and found that the switching of direction occurs within the same G1 or G2 cell cycle phase (Supplementary Video 6). This is in contrast to the interkinetic motion seen in vivo, where the motion of the nucleus is directed to the basal end (G1) or the apical end (G2) of the NPC, depending on its cell cycle phase 11 , although there was a difference in the timescale of switching between the two phases (Extended Data Fig. 4a-c) 13 .
Active nematic systems are predicted to have spatio-temporal correlations between densities and strengths of nematic order that are distinct from those of equilibrium systems [3][4][5] . By picking out areas that do not contain defects 14 , we confirmed giant number fluctuation and quasi-long-range order (Extended Data Fig. 5a and b) in the highdensity NPC culture. In the same regions, we calculated the spatial correlations of cell density and alignment fluctuations (Extended Data Figs 6 and 7a-f) and found that the fluctuations of density and alignment were correlated (see Supplementary Information). The correlations were positive in the first and third quadrant when seen as a function of two-dimensional displacements, indicating that the system is an extensile active nematic system 3,5,15 . This feature of the correlation was flipped and weaker in the case of a cultured myoblast cell line C2C12, which also shows a nematic pattern (Extended Data Fig. 8a and b). We further found that the spatial correlation of the cell density was anisotropic when the typical length scale of correlation in the direction parallel to the alignment was larger than that in the direction perpendicular to it (Extended Data Fig. 6).
A clear signature of 'activeness' in a macroscopic system can be found at the topological defects. It has been observed both in experiment and simulation 16-25 that + 1/2 defects typically undergo spontaneous motion in active nematic systems, which is in contrast to defects in equilibrium liquid crystals 26 . Topological defects in the biological context have been observed in whorled grain in wood 17 as well as in tube formation in embryonic development. Examples of defects yielding non-equilibrium collective dynamics include in vitro 21,22,27 and in vivo 28 cytoskeleton dynamics, bacterium flows in liquid crystal mixtures 23,25 , colony growth dynamics 24 , and the cAMP patterning of aggregating social amoeba 29 .
As has been reported in other works on cell cultures 9,30,31 , we clearly observed + 1/2 and − 1/2 defects in the dense NPC culture (Fig. 4a). We found slow motions of defects (3.7 µ m h −1 on average) including the merging of defects with opposite winding numbers (Fig. 4b). In Fig. 4c, we show the local nematic order around the topological defects calculated from the displacement of cells inside small regions. Figure 4d shows the net velocity field calculated from the same data, indicating that there is a flow of cells pointing in the direction of the front of the comet shape in + 1/2 defects (Fig. 4c), whereas no clear directionality exists in the − 1/2 defects. The direction of the defect motion in the NPCs was opposite to that for passive liquid crystals, C2C12 (Fig. 4e)  ase, which is known as the interkinetic hether the direction switching is related ycle marker Fucci 12 , and found that the ithin the same G1 or G2 cell cycle phase s is in contrast to the interkinetic motion n of the nucleus is directed to the basal ) of the NPC, depending on its cell cycle difference in the timescale of switching nded Data Fig. 4a-c) 13 . predicted to have spatio-temporal cornd strengths of nematic order that are rium systems [3][4][5] . By picking out areas we confirmed giant number fluctuation xtended Data Fig. 5a and b) in the highame regions, we calculated the spatial nd alignment fluctuations (Extended nd that the fluctuations of density and see Supplementary Information). The the first and third quadrant when seen onal displacements, indicating that the ematic system 3,5,15 . This feature of the eaker in the case of a cultured myoblast ows a nematic pattern (Extended Data nd that the spatial correlation of the cell the typical length scale of correlation in the direction parallel to the alignment was larger than that in the direction perpendicular to it (Extended Data Fig. 6).
A clear signature of 'activeness' in a macroscopic system can be found at the topological defects. It has been observed both in experiment and simulation 16-25 that + 1/2 defects typically undergo spontaneous motion in active nematic systems, which is in contrast to defects in equilibrium liquid crystals 26 . Topological defects in the biological context have been observed in whorled grain in wood 17 as well as in tube formation in embryonic development. Examples of defects yielding non-equilibrium collective dynamics include in vitro 21,22,27 and in vivo 28 cytoskeleton dynamics, bacterium flows in liquid crystal mixtures 23,25 , colony growth dynamics 24 , and the cAMP patterning of aggregating social amoeba 29 .
As has been reported in other works on cell cultures 9,30,31 , we clearly observed + 1/2 and − 1/2 defects in the dense NPC culture (Fig. 4a). We found slow motions of defects (3.7 µ m h −1 on average) including the merging of defects with opposite winding numbers (Fig. 4b). In Fig. 4c, we show the local nematic order around the topological defects calculated from the displacement of cells inside small regions. Figure 4d shows the net velocity field calculated from the same data, indicating that there is a flow of cells pointing in the direction of the front of the comet shape in + 1/2 defects (Fig. 4c), whereas no clear directionality exists in the − 1/2 defects. The direction of the defect motion in the NPCs was opposite to that for passive liquid crystals, C2C12 (Fig. 4e), and the confluent fibroblasts reported in ref. 31.   Fig. 3c-f,h,i). The shear velocity fluctuated in spa a zero mean (Fig. 3e,i).
In summary, for L > L c , the cell population nematic phase with a director at a finite angl direction of the stripe. In parallel, cells sponta complex flows with shear and transverse compo threshold L c , cells orient in the direction of the shear flow develops.
To understand these observations, we modell a confined active nematic fluid 27,28 . Indeed, des practical differences, our confined cells share the tal symmetries as a contractile acto-myosin netw ATP hydrolysis between parallel plates 29 . We the an adapted version of the physical model of ref. 29 active nematics confined in a stripe of width L exh tinuous transition at a critical width L c between a bile state and a flowing state. This transition is Fréedericksz transition of nematic liquid crystals by the intrinsic activity of the system rather than To account for the finite angle of the edge cells   (Fig. 3c-f,h,i). The shear velocity fluctuated in a zero mean (Fig. 3e,i). mote cell death and extrusion 6 . In colonies of the motile soil bacterium M. xanthus, defects promote the formation of multicellular aggregates known as fruiting bodies, which allow the bacterial population to survive starvation 7 .
Flowing on their own Describing orientational order is not enough to account for collective cell flows. To this end, physicists describe cell assemblies as active matter 1,8 . For example, when cells align with polar order, they can start migrating in the direction of alignment 9 . This type of collective motion is known as flocking. The active-matter theory of this phenomenon was developed 25 years ago inspired by the mesmerizing flights of bird flocks 10 . Today, the principles of flocking are applied to many other systems, from synthetic active colloids to bacterial swarms.
To describe nematic cell colonies, researchers use the theory of active liquid crystals, which generalizes the hydrodynamics of liquid crystals to include active (cell-generated) stresses. Among many other phenomena, this theory explains the cell flows observed around topological defects. Again, beyond cell colonies, the theory successfully describes many other systems, from biopolymer gels to shaken granular materials. Employing general theories of active matter to describe cell migration is particularly useful because it reveals connections to apparently-unrelated systems. Progressively, this approach is uncovering a classification of active systems based on symmetries and generic behaviors, in the spirit of universality classes in statistical mechanics.
For example, the theory of active liquid crystals was orig-inally inspired by behaviors in bacterial suspensions and in the cell cytoskeleton. Soon after formulating the theory, researchers predicted that active stresses generate an instability whereby these fluids start flowing spontaneously, without having to apply external forces 14 . To drive flows, active stresses have to overcome alignment forces in the liquid crystal, which happens only at sufficiently large scales. Therefore, the theory predicted that a strip of active fluid would flow only if it was wide enough 14 .
More than a decade later, these predictions were tested in cell monolayers 5 , showcasing the generic character of the theory. Whereas cells confined in narrow stripes did not flow, cells in stripes wider than a critical width developed a collective shear flow as predicted by the theory (Fig. 3d). In very large tissues, cell flows become chaotic, creating disordered patterns of swirls known as active turbulence 15 . Confinement can therefore prevent and organize spontaneous cell flows. This regulatory role of confinement may be relevant in embryonic development and tumor invasion, in which cell groups often migrate in tracks defined by the surrounding tissue (Fig. 1a). Overall, these studies showcase how cells can leverage the physics of active fluids to engage in collective flow patterns, and how confinement controls whether and how cell groups flow on their own.

To spread or not to spread
What happens if we release confinement and expose a cell monolayer to free space? Cells at the monolayer edge are able to sense that they have neighboring cells on one side but not on the other. In ways that remain to be fully under- With time, these boundary layers became markedly heterogeneous but systematically grew to encompass increasing numbers of cells; cell-cell tension transmission exhibited a growing scale of length (Fig. 2k), and the maximum intercellular shear stress (μ) followed a similar pattern (Fig. 2m,n). Taken together, these findings demonstrate that force transmission from cell-to-cell, and cellular migration across the epithelial sheet, are initiated at the leading edge and progressively penetrate towards the centre (Supplementary Movie S3). Moreover, these stress fields were anisotropic. At each position in the monolayer plane, the maximum ( max ) and minimum ( min ) principal stresses 5 were represented as an ellipse aligned with corresponding principal orientations (Fig. 2p). Throughout epithelial expansion, stress ellipses tended to be spindle-shaped and thus revealed pronounced stress anisotropy. The maximum principal stress orientation tended to be perpendicular to the leading edge and thus roughly parallel to local cell motion (Fig. 2q). As described previously, this mode of local cell guidance defines plithotaxis 2,5 .
Superposed on systematic monolayer spreading were largescale spatio-temporal fluctuations of tractions, monolayer stresses and cellular velocities (Fig. 2f,i,l,o). To better characterize the systematic evolution of mechanical patterns, we averaged these variables over the observable monolayer length (corresponding to the y coordinate), thereby reducing the dimensionality of the system to only one spatial dimension and one temporal dimension. All data could then be represented as kymographs in the x-t plane (Methods). Kymographs of cellular velocity (v x ) revealed motility patterns that were not restricted to the initial phase of inward mobilization (Fig. 3a). To the contrary, after reaching the monolayer midline at ⇠150 min, the two fronts of cell motility coalesced and then continued towards the leading edges. When cells are cohesive and mass is conserved, cellular velocities must be linked to the rate of cell deformation (strain rate," xx ; ref. 11) through the expression" xx = @v x /@x. Remarkably, kymographs oḟ " xx revealed clear evidence of wave-like crests of strain rate that were launched at each leading edge, propagated away from and back to the leading edge at roughly twice the speed of the advancing front edge, and spanned the entire monolayer (Fig. 3b). To distinguish these mechanical waves from other known types of mechanical wave, and because they inscribe an X-shape on the kymograph, we call them X-waves.
To study the physical origin of the X-wave, we next focused on traction generation and stress transmission in the monolayer. Whereas traction kymographs demonstrated extrema at the leading edge, monolayer stresses were highest at the monolayer midline, indicating that local force generation was globally integrated and transmitted through cell-cell junctions to give rise to a stress build-up (Fig. 3c,d). Importantly, monolayer stress at the midline oscillated in time (Fig. 3g,h and Supplementary Movie S4); these oscillations were in phase with fluctuations of cell area (Fig. 3f,h) and demonstrated phase quadrature with strain rate (Fig. 3e). Contrary to long-held assumptions (reviewed in ref. 12), these observations establish that on the ultraslow timescales of cellular migration the dominant cellular stresses in the monolayer are elastic, not viscous.
In the absence of appreciable inertia, there can exist no exchange between kinetic and potential energy storage as is usually associated with propagation of passive mechanical waves, thus suggesting that the mechanism underlying the observed propagation might be active. To investigate this possibility, we inhibited myosin using blebbistatin. Blebbistatin prevented the formation of stress fibres (Supplementary Fig. S2) and had little effect on the velocity of the leading edge, thus confirming previous reports in wound scratch assays 8 . Blebbistatin caused traction forces and intercellular stresses to be abrogated, however (Supplementary Fig. S2 and Movie S5). A well-defined front of strain rate could be clearly identified nonetheless, but this front was stationary, did not propagate It has been reported that traction forces elicit active cell contractile responses (37, 38), which may not arise during compression. A systematic study comparing parallel-plate compression and micropipette aspiration on aggregates of the same cell lines would help to clarify the effect of these differences. It has also been pointed out that whereas in parallel-plate compression the whole aggregate is stressed, micropipette aspiration probes only a subset of the aggregate's cells (39). The number of cells subjected to stress scales as R p 3 . Thus, for the continuum hypothesis to be valid and the technique applicable to characterize aggregate rheology, the micropipette radius should be large enough to probe a few hundreds of cells, which is usually accomplished with R p ≈ 30 mm.
The existence of a yield modulus above which aggregates exhibit plastic behavior remains a matter of controversy. Some authors have postulated the existence of a yield modulus in tissues, which physically would arise from the critical force required to bre cellular reorga no systematic existence of a y ducted. Existin as the sponta (40) seem to exists, is sma tissue surface relaxed by sur the order of h aggregate size meters). More ation is more a process, wher rates vary cont appealing solu ence of tissue Marmottant et stood, edge cells respond to this asymmetric environment by polarizing toward free space (Fig. 4a). Specifically, these cells extend protrusions known as lamellipodia, with which they exert directed and persistent traction forces on the underlying substrate to migrate toward open ground. Because cells in the monolayer are adhered to one another, the migrating edge cells pull on cells in the second row, which then also polarize, migrate, and pull on inner cells, setting the monolayer under tension 16 (Fig. 4b). At the molecular level, this supracellular coordination is mediated by a protein known as merlin, which transduces intercellular forces into cell polarization. In this mechanically-coordinated way, the entire cell monolayer spreads on the substrate, becoming progressively thinner 11 (Fig. 4a). Combined with other mechanisms, this type of collective cell migration helps to close gaps in epithelial tissues, as in wound healing.
But tissues not always spread on substrates. Under certain conditions, a cell monolayer may instead retract from the substrate, eventually collapsing into a droplet-like cell aggregate 13 (Fig. 4c). Tissue spreading and retraction are therefore reminiscent of the wetting and dewetting of liquid droplets. The degree of wetting depends on the balance between cohesive forces within the liquid and adhesive forces with the substrate. By analogy, early models proposed that tissue wetting was dictated by a competition between cell-cell (W cc ) and cell-substrate (W cs ) adhesion energies 12,17 , encoded in a spreading parameter S = W cs − W cc . When S < 0, cellcell adhesion dominates and the cell aggregate retracts from the substrate (dewetting, Fig. 4c left), whereas when S > 0, cell-substrate adhesion dominates and the aggregate spreads (wetting, Fig. 4c right). This simple conceptual framework was sufficient to interpret the behavior of cell aggregates of varying cell-cell and cell-substrate adhesion 12,17 . However, this analogy to passive liquids does not explicitly account for the active nature of cells.
Recent work addressed this limitation by treating the cell monolayer as a droplet of active liquid 13 . Using this approach, one obtains the spreading parameter directly in terms of active cellular forces. Supported by experiments, the model predicts that the wetting transition results from the competition between two types of active forces: cell-substrate traction forces that promote spreading versus cell-cell pulling forces that promote retraction. This active wetting framework makes another key prediction: the spreading parameter depends on the droplet radius. Cell monolayers larger than a critical radius wet the substrate, whereas smaller monolayers dewet from it (Fig. 4d). This prediction is striking because it has no counterpart in the classic wetting picture, in which the spreading parameter depends solely on surface tensions. For regular liquid droplets, size does not matter. In contrast, tissue wetting is size-dependent. This prediction was verified in experiments, providing evidence for the active nature of tissue wetting 13 .
Besides its relevance in physics, the existence of a critical size for tissue wetting might explain drastic changes in tissue morphology during embryonic development and cancer progression. For example, a disturbing possibility is that a growing tumor might become able to spread onto the surrounding tissue once it reaches a critical size. Overall, this work exemplifies how the quest to understand collective cell migration motivates the development of new physics, leading to the discovery of phenomena like active wetting. Finally, this physics approach offers clues on how cell aggregates may tune active forces to control whether to spread or not to spread.

Mechanical waves without inertia
Tissue spreading revealed yet another striking collective phenomenon: Mechanical waves start spontaneously at the leading edge and propagate across the cell monolayer 11 (Fig. 5). These waves are slow, with speeds ∼ 10 − 100 µm/h, and wavelengths that span several cell diameters. Simi-  (Trepat et al., 2009), and mathematical modeling, we show that each follower cell possesses a mechanochemical feedback system, in which stretch-induced ERK activation triggers cell contraction. Intercellular coupling of the ERK-mediated mechanochemical feedback enables sustained propagation of ERK activation and contractile force generation, leading to multicellular alignment of front-rear cell polarity over long distances. Thus, our study clarifies a mechanism of intercellular communication underlying long-range sustained transmission of directional cues for collective cell migration.

Cell Deformation Waves Precede ERK Activation Waves
To investigate the relationship between ERK activation and cell deformation during collective cell migration, Madin-Darby canine kidney (MDCK) cells confluently seeded within compartments of a silicone confinement were released for collective cell migration.

Results
Firstly, in agreement with previous studies of expanding MDCK monolayers 25,27,34 , we confirmed that directional waves of cellular density and ERK activity propagate away from leading edges towards the centre of colonies (Extended Data Fig. 1a,b). Moreover, we performed the same experiments at full two-dimensional (2D) confluency (Fig. 1a-d and Supplementary Video 1) and found similar patterns of ERK and cell density (with ERK positively correlated with, and slightly lagging behind, 2D cell area), although in this case they displayed random orientation of propagation. This suggests that monolayer expansion (mediated by leading edges) is not required for wave formation, motivating us to first study theoretically how isotropic waves of ERK and density arise in the confluent setting.
For this, we first wrote down a minimal model of epithelial monolayer mechanics that we highlight here (see Supplementary Section I for details). The 3D shape of MDCK cells, like in other epithelial tissues, is proposed to arise from a balance of forces between the active stresses generated by actomyosin cortices at the apical, lateral and basal surfaces 35-37 (resp. denoted as γ a , γ l and γ b ). For a single cell on a flat surface, equilibrium of these forces requires that the in-plane length of a cell l is equal to a ratio of active tensions, such that l / γ l γ a þγ b 1=3 I . However, for confluent, heterogeneous epithelial tissues, cell-cell junctions impose mechanical couplings between cells 38 , so that the mechanical state of each cell influences the shape, density and velocity of its neighbours 39 . To describe this we consider, in a simple 1D setting, a linear chain of coupled cells with vertex positions r i and surface tensions γ i (Fig. 1e). If a given cell contracts, it imposes a stress on its neighbours, which is resisted by cellular tensions, as well as by frictional forces with the substrate f i = −ζv i (where ζ is a cell-substrate friction coefficient dependent on cell-substrate adhesion and v i = ∂ t r i the velocity of the cell vertex r i ). At the continuum limit and at linear order, force balance in such a monolayer reads τr∂trðx; tÞ ¼ ∂xxrðx; tÞ À ∂xl0ðx; tÞ ð 1Þ with all units normalized by the average cell length, and r the vertex position at distance x and time t. The first two terms are classical for overdamped chains of oscillators, and represent resp. frictional forces and restoring forces from active cell tensions (with τ r = ζ/k, the ratio of cell-substrate friction to cell stiffness, giving a timescale for the spatiotemporal diffusion of displacement r, on length scales of cell size). The third term arises from the fact that each cell at position x can in principle have its own preferred length l 0 , so that gradients of preferred length result in cellular (c) Schematic showing that cells polarize and migrate against the mechanochemical wave. In spreading tissues, the wave travels backwards from the leading edge, directing cell migration toward free space. Panels (b)-(c) are from ref. 18 . lar to longitudinal sound waves, tissue waves stretch and compress cells as they travel. Most strikingly, the waves are selfsustained; they travel long distances (∼ 1 mm) unattenuated. This observation is surprising because cell motion is so slow that inertia is negligible. Therefore, these waves cannot be sustained by the common back-and-forth between kinetic and potential energy familiar from the harmonic oscillator. Moreover, there are many sources of dissipation in tissues, including cell-cell and cell-substrate friction, which could potentially damp the waves. Thus, the very existence of mechan-ical waves in tissues implies an active driving mechanism that compensates damping and generates an effective inertia.
The quest to understand these waves led to a plethora of physical models, ultimately revealing a palette of possible mechanisms 1 . Recently, the situation has been clarified by experiments revealing that mechanical waves are accompanied by waves of ERK, an extracellular signaling molecule that affects cellular activity (Fig. 5a). Following these observations, researchers developed a theory based on the feedback between the biochemical regulator and cell mechanics (Fig. 5b), which produces coupled chemical and mechanical waves. Assuming that cells polarize in response to stress gradients in the monolayer, the theory also explains propagation away from the leading edge in spreading tissues 18 (Fig. 5c).
These results show that cells can exploit mechanochemical feedbacks to transmit local information over long distances. This tissue-scale communication is relevant for wound healing, enabling distant cells to coordinate their migration toward the wound. Similar principles operate in morphogenesis, enabling coordinated cell deformations to precisely shape tissues without requiring local genetic control of cellular forces. From a physics perspective, this research highlights that the cells' ability to generate, sense, and respond to signals, both chemical and mechanical, can give rise to emergent phenomena as counter-intuitive as mechanical waves without inertia.

Outlook
The physics of active living matter is increasingly successful at explaining the dynamics of collective cell migration. This core biological process is being understood through concepts such as orientational order, flow, turbulence, jamming, wetting and wave propagation. The mechanistic origin and physical properties of these phenomena in cells, however, differ fundamentally from those in non-living matter. Whereas new active matter theories progressively manage to explain the broad phenomenology of collective cell migration in terms of a small number of physical variables, how cells tune these variables through thousands of genes and biochemical reactions remains a major open question.