Living Cells on the Move

Living cells on the move

Ricard Alert1, 2 and Xavier Trepat3, 4, 5, 6 1Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544, USA 2Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 3Institute for Bioengineering of Catalonia, The Barcelona Institute for Science and Technology (BIST), Barcelona, Spain, 08028 4Facultat de Medicina, University of Barcelona, Barcelona, Spain, 08036 5Institucio´ Catalana de Recerca i Estudis Avanc¸ats (ICREA), Barcelona, Spain, 08028 6Centro de Investigacion´ Biomedica´ en Red en Bioingenier´ıa, Biomateriales y Nanomedicina, Barcelona, Spain, 08028 (Dated: June 8, 2021) 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 tasizing. Finally, collective cell migration is also involved in in our body move in groups. Collective cell migration enables maintaining the inner surface of the gut, which is the fastest embryonic development, wound healing and cancer cell inva- self-renewing tissue in mammals. It renews entirely every 3- sion. These phenomena involve complex biochemical regula- 5 days, which implies a daily loss of tens of grams of cells. tion, but their dynamics can ultimately be predicted by emerg- Self-renewal proceeds thanks to the division of stem cells that ing physical principles of living matter. reside at the bottom of tissue invaginations called crypts. The progeny of these stem cells then migrates collectively from the When and where do cells migrate in groups? crypt to the top of finger-like protrusions called villi, where When seen at the microscale, our body is a busy maze filled they are shed into the intestinal lumen and discarded (Fig. 1b). with actively moving cells. Cellular movements are slow, Collective cell migration is regulated by a myriad of molec- rarely exceeding 10 µm/min, but crucial to embryonic devel- ular processes, from genetic programs to sensing and signal- opment, the immune response, tissue self-renewal and wound ing pathways. Yet, these molecular processes act upon a lim- healing. Through the same mechanisms that sustain these ited number of physical quantities to determine cell move- physiological functions, cell movements drive devastating dis- ment. Therefore, coarse-grained physical approaches may eases such as acute inflammation and cancer. These can in provide crucial insight into biological questions. Furthermore, fact be considered diseases of cell movement because arrest- collective cell behaviors also inspire new physical theories of ing cells in a controlled manner would be sufficient to prevent living systems. In this article, we highlight progress in this their spread. Take cancer as an example; whereas its origin direction. is well-known to be genetic, if we could selectively stop the movement of tumor cells, we would prevent them from escap- Cell assemblies as living matter ing primary tumors and reach distant organs to metastasize. What physical principles underlie collective cell motion? In Understanding cell migration is therefore crucial to improve traditional condensed matter, interactions between electrons current strategies to fight disease. and between atomic nuclei give rise to fascinating collective Cell migration comes in different flavors. Some cell types phenomena such as magnetism and superconductivity. Anal- move as isolated self-propelled particles. For example, to ogously, cell-cell interactions lead to emergent collective phe- chase and destroy pathogens, immune cells move individually nomena in migrating cell groups. However, when treating cell through tissue pores. Similarly, in some types of cancer, single colonies as materials, we must take into account some key fea- cells dissociate from tumors and travel solo through the sur- tures of living matter, as we illustrate throughout the article. rounding tissue, eventually reaching blood vessels and metas- First, the primary constituents of living tissues are cells and tasizing at distant organs. By contrast, in other physiological extracellular networks of protein fibers such as collagen. The functions and pathological conditions, cells move as collec- interactions between these mesoscale constituents are orders tives. Collective cell movements are dominant in embryo de- of magnitude weaker than interatomic interactions in conven- velopment, where they enable the precise positioning of tis- tional solids. So, with notable exceptions like bone, most bio- arXiv:2106.02752v1 [physics.bio-ph] 4 Jun 2021 sues and organ progenitors. The later coordination of growth logical tissues are soft materials, which can easily deform and and motion shapes organs well into postnatal life. Collective flow. cell migration also drives wound healing in tissues such as the Second, cells are machines with internal engines. Special- skin. Here, a continuous sheet of tightly adhered cells moves ized proteins known as molecular motors harness the energy cohesively over the damaged area to restore a functional tis- of chemical reactions to generate forces and produce mechan- sue. Cancer invasion also involves the migration of cell sheets, ical work. These energy-transducing molecular processes ul- strands and clusters (Fig. 1a). Within these groups, cells with timately power cell migration, allowing cells to move au- different function can self-organize in spatial compartments tonomously, without externally applied forces. This contin- to behave like an aberrant organ. This added functionality uous supply of energy drives living tissues out of thermody- is thought to provide malignant tumors with distinct invasive namic equilibrium. Importantly, the driving is local; it occurs strategies that improve their chances of spreading and metas- at the level of individual constituents, i.e. single cells. In 2

a b 50 μm

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Figure 1 Collective cell migration in the human body. (a) Breast cancer cells from a patient sample of invasive ductal carcinoma migrate collectively into the surrounding extracellular matrix. Colors label cell nuclei (green), the cell-cell adhesion protein E-cadherin (red), and collagen ﬁbers (white). From O. Ilina et al. Dis. Model. Mech. 11, dmm034330, 2018. (b) Intestine cells migrate as indicated by the white arrows to enable fast renewal of the inner surface of the gut. Colors label cell nuclei (blue), goblet cells (pink), the cell-cell adhesion protein p120 (green), and lysosomes (white). The image was kindly shared by Kristen A. Engevik. See also D. Krndija et al. Science 365, 705 (2019).

other words, cells are active constituents, and living tissues in terms of areas Ai and perimeters Pi of cells i = 1,...,N are a paradigmatic example of active matter — an exploding as new field in nonequilibrium statistical physics. N X κ 2 Γ 2 But cells are not only mechanically active: they also sense = (Ai A0) + (Pi P0) . (1) H 2 − 2 − their environment, process information, and respond by adapt- i=1 ing their behavior. For example, stem cells plated on sub- This energy assumes that cells resist changes in their area and strates of different stiffness differentiate into distinct cell perimeter around the preferred values A and P with elastic types, from brain to bone cells. So living tissues are adap- 0 0 moduli κ and Γ, respectively. The preferred perimeter P de- tive: they respond in programmed ways to environmental cues 0 pends on cell-cell interactions and cellular activity, with cell- such as external forces, the mechanical properties of the extra- cell adhesion promoting longer edges and cellular tension fa- cellular matrix, and concentrations of nutrients and signaling voring shorter edges. The preferred perimeter and area de- molecules. Consequently, cell-cell and cell-environment in- fine a dimensionless parameter p = P /√A which informs teractions are often very complex. Unlike atoms and electrons 0 0 0 about the preferred cell shape. Higher p corresponds to more in conventional condensed matter, cellular interactions cannot 0 elongated cells, and smaller p corresponds to more isotropic in general be fully described via an interaction potential with a 0 shapes, with less perimeter for the same area. fixed functional form. Thus, a key challenge in this field is to For a given p , edge lengths vary until the system reaches find effective ways to capture complex cell behaviors in terms 0 its ground state, minimizing the energy in Eq. (1). In this pro- of simple, physically-motivated interactions1. cess, an edge can shrink until it eventually disappears, leading to the formation of a new cell-cell interface (Fig. 2c). These To flow or not to flow events, known as T1 transitions, allow cells to change neigh- One way to think about interactions between deformable bors, driving topological rearrangements of the cellular net- epithelial cells comes from the physics of foams. In foams, work. The ability to reorganize its constituents determines gas bubbles arrange in polygonal packings, with the liquid whether a material is solid or fluid. So, if cell rearrangements phase filling the interstitial spaces and providing surface ten- are difficult, the cellular network resists shear deformations; sion to bubble interfaces (Fig. 2a). Similarly, cells in epithe- the tissue is solid. In contrast, if cells can rearrange easily, the lial monolayers also acquire polygonal shapes, with roughly network yields upon shear; the tissue is fluid. At small p0, i.e. straight edges subject to active tension generated inside cells for roundish cells, Eq. (1) reveals that there is an energy bar- (Fig. 2b). This description of tissues as polygonal cell pack- rier preventing T1 rearrangements (Fig. 2c). However, as we ings traces back to work by Hisao Honda and collaborators increase p0 and cells become more elongated, the energy bar- ∗ in 1980, and it was later popularized by work in the group rier decreases (Fig. 2c). At a critical value of p0 = p0 3.81, of Frank Julicher¨ and collaborators3. Because cells are de- the barrier vanishes (Fig. 2d); cells can rearrange freely≈2. formable, edge lengths vary dynamically. These variations This simple model thus predicts a solid-fluid transition change the energy of the cellular network, which we can write in tissues driven by changes in cell shape, encoded in p0 NATURE PHYSICS DOI: 10.1038/NPHYS3471 ARTICLES

abvalue of p0 and r tested, we obtained the distribution of energy REVIEW1.5 r = A 0.5 RTICLE | INSIGHT⇢(1") REVIEWhttps://doi.org/10.1038/s41567-018-0194-9 ARTICLE | INSIGHTbarrier heights . The functional form of the distributionNATURE PHYSICS r = 0.1 becomes universal (Fig. 1a) when scaled by the mean energy 100 1.0 1"( )

) r = 1 barrier height r,p0 . The rescaled distribution is fitted well by a Box 2 | " Mechanobiology as a multiscale problem k k 1 ! ! −1 k-gamma distribution (k x exp( kx)/(k 1) )withx 1"/1" / 10 ! =

! 0.5 r = 2 ( One the great challenges of mechanobiologyand k is the2.2 existence0.2. The of k-gamma!ese distribution models capture has fundamental been observed rheological properties # −2 10 = ± 30–32 """ multiple lengthr = 10scales at which relevant inmechanisms many non-biological are at play. disorderedof cells such systems as strain-sti, and"ening, generically scale-free elastic modulus, ! 130,131 31,32 10−3 !is problem0.0 has led to the development resultsof diverse from types maximizing of mod- the entropy$uidization subject and to ageing constraints. . This 0123456elling approach3.0 3.2 describing 3.4 3.6 biological3.8 4.0 tissuesconfirms at di that"erent the scales. distribution 3. Vertex of energy models barriers treat cell depends monolayers on theas 2D or 3D polygonal p0 x = !"/!" Examples of mechanical models of cellssingle-cell and tissues properties include thep0 and r onlyobjects through that share its averagecommon1" vertices. (see also Box 3) (c,g following. Figure 1b shows the dependencein #gure of below).1" on Inp0 threefor various dimensions, values these models are Figure 1 | Energy barriers for local cellular rearrangements. a, Illustration 1. Clutch models describe the dynamics of cell–matrixr.Atp0 . 3.8,adhe the- energy barriersgoverned are by always tissue finite—thatgeometry and is, by cells surface tensions of of a T1 transition in a confluent tissue andsion the at normalized the molecular distribution level (a,⇢e inof #guremust below). put ! iney some take into amount of workthe apico-basal to deform (!ab and) and rearrange. lateral surfaces Here (!l). Vertex models normalized energy barrier heights 1"/1"accountfor a large the rigidity range of of parameters the matrix, the binding/unbindingthe tissue behaves (kon like, a solid;have it ispredicted a rigid dynamic material aspects with aof finite monolayers, such as their (r 0.5,1,2 and p0 3.2–3.7). They havek ao" universal) rates of shapeadhesion that proteins is fitted and well the force that myosin motors ability to deform through intercalation, to jam at constant = = yield stress. As p0 is increased, the energy barriers decrease and 82,83,86,87,132 by a k-gamma distribution (solid line), indicatingexert on thatthese1" proteinscompletely through the actinbecome cytoskeleton. vanishingly Clutch small indensity the vicinity and to of $ockp as 3.8,solids so and that liquids cell . models have been successful at predicting traction forces and 4. Continuum models0 ⇡describe tissues as active materials in specifies the distribution and describes the mechanical response. b, 1" as rearrangements cost almost no energy. This suggests that the vertex dynamics of the leading edge of a protruding cell, sensing of which the cellular structure is coarse-grained (d,f in #gure function of the target shape index p0 for various values of the inverse rigidity and matrix ligand spacing, andmodel durotaxis may4,56, undergo57. a critical rigiditybelow). ! transitionese models near arep formulated0 3.8. in terms of continuum perimeter modulus r. ⇡ 2. Polymer network models are based on theTo polymeric test this nature hypothesis, wevectorial searched and tensorial for a scaling #elds such collapse as velocity, v, polarity p of the cytoskeleton (b,f in #gure below).based !ey on take theories into ac for- continuousand stress phase ". transitions !ese variables near are a criticallinked through viscosity and p P /pA is the target shape index or a preferred perimeter- point, such as the Ising model for ferromagnetism. Figure 1b 0 = 0 0 count the properties of single #laments, their spacing (df in #, elasticity k and friction $. In three dimensions, they also to-area ratio. For simplicity, we focuspanel on p f0 below),>0. For thep0 deformability<0 there are anddemonstrates binding rates of that polyr- sets theincorporate overall scale luminal of 1" pressureas well ! P as. Continuum the models have two regimes: a static regime with resultsmer identicalcrosslinking to thoseproteins presented (kon, ko" in panel‘sharpness’ f), and of the the activity transition, whereassuccessfullyp0 controls described the complex distance features to the of cell monolayers 76,133 here for 0

⇤ knownNATURE as T1 transitions PHYSICS,DOI: where10.1038/NPHYS3471 an edge between two cells1 µm shrinks the crossover scaling10 µm functions for p0ARTICLESp0 and p030 µpm0 , respectively.from asthmatic individuals100 µm tended to remain unjammed, form- to a point and a new edge arises between two neighbouring cells, After re-plotting the data in Fig. 1b using equationing a (3 fluid), tissue. Therefore, these experiments suggested that as illustrated3 in Fig. 1a. The mechanicalef response of the tissue isa we find an excellent scalingghc collapse onto two branchesthe with fluid-solid transition in tissues is involved in disease, open- 10 c d 100 governed by the rate100 of cell rearrangements,Myosin II and, within the vertex 4.0 0.4, 1.0 0.2 and a precise location of the criticaling point the door to new treatments based on preventing this phase Actin = ± = ± # ⇤ model, the ratexy of10−1 T1 rearrangements is related to the amount of p0 3.813 0.005, as shown in Fig. 2. transition. Fluid-solid transitions also occur during develop- −2 $ rG v 29 = ± 10 2 b 1.5 kon/koff ment, enabling" tissues to first turn fluid in order to remodel mechanical10 energy−2 required to execute a T1 transitionAdaptors . Therefore, For the mechanically rigid branch in the limit z 0, the 10 Increasing p " !ab !

0 ! Slope = 1 ) df and acquire their shapes, and then solidify and mature. The we first study how these energy barriers change with single-cell 0 energy barrier can be rewritten in dimensional units and scales as l 1.0 −3 −2 −1 0 Crosslinks 2 10 10 10 10 = −4 l 1 1" ⇠ ⇤ 10 emerging picture is that, in different biological contexts, cells xy E KAA A0 p p0 . This indicates that these barriers properties encoded in the model! parameters r and p0. ( 0 0 v P G p0 − p0" " = / ! Integrins − 1 Myosin 0.5II F ⇠ can tune their shape and use the physical principles governing To explore10 the statistics of energy barriers, we test all possible " are completely governed by the perimeter elasticity .Atthe k /k !l on off Substrate phase transitions in foams to decide whether to flow or not to T1 transition paths (see Methods) in ten randomly generated critical0.0 point, the two branches of10 the−6 scaling function merge and Actin−0.6 −0.3/ 0.0 0.3 0.6 3.0 3.5 4.0 flow. disordered samples each consisting of M Fn64 cells. For each f f z . In this case the dimensional energy barrier scales asp = + = = l p0 100 Aligning with neighbors Figuree 5 | A simple four-cell model. a, A four-cell aggregate undergoing a a 2 b 3.72 3.81 −3 10 −2 −1 0 10 10 Mechanical10 structures and models10 of cellsT1 topologicaland tissues swap. at different The thick length (green) scales. edge representsa, A focal adhesion the cell–cell of an unroofed HeLaThe cell action stained starts with fluorescent once tissues become fluid and cells can ! p0" − p0 move. Cells then engage in collective flows of different types, 1 phalloidin (cyan) and imaged with total internalinterface reflection that isSolid contracted fluorescence to a point microscopy/platinum-replica and then resolved Fluid in the transmission electron microscopy (grey) 10 ! p0 < p0 correlative light–electron microscopy134. bperpendicular, An osteosarcoma direction. cellb (U2OS), Energy ofplated a four-cell on fibronectin. aggregate duringActin (red) a T1 was imageddepending using stimulated on how emission cells align with neighbors to coordinate

| #/" ! Figure 4 | Shear0 modulus0 as a functiondepletion of p0⇤↵ microscopyp0, which isand the a focal adhesiontransition, marker which(paxillin, attains green) a maximum was imaged at the using transition confocal point. microscopy.p0 varies from c, A monolayertheir of individual epithelial cells motions. labelled Again, these cell-cell interactions p 10

system-specific− distance to the rigidityusing transition. a membrane-targetedThe value of p0⇤ green↵ is fluorescent1.5 to 3.8protein in equal (green). increments. d, An immunofluorescentc, Energy barrier height image as a functionof a small of intestinalp0 for organoiddepend showing strongly markers on cell of shape. 0 determined fromp 10 the−1 location at whichproliferation floppy modes (EdU, appear red), for differentiation the specific (Krt20,a four-cell green) aggregate and nuclei and a(blue). mean-field e, Clutch estimate model (dashed) of F-actin for the linkage value to of fibronectin (Fn)Cells via come integrins. in manyf, Polymer different shapes: some are roughly /| ! pent $ p = 3.813 ± 0.005 configuration. Here the tissue contains M 100 cells and0 the colours p0 p at which 1" vanishes. " network model. g, 3D vertex model. h, Continuum0 model. Credit: courtesy of J. Taraska and K. Schack (a) and G. Jacquemet and J. Ivaska (b); and r = spherical, some have rod-like shapes, and some cells develop −2 = " = 4.0 ± 0.4 correspond to (top10 to bottom):!r 0.005 (blue), r 0.001135 (green) and p0 > p0 = adapted from= ref.! , Bio-protocol (d). a head-tail asymmetry to migrate persistently in one direc- r 0.02 (red). Inset: the shear modulus can be rescaled# = by 1.0 ± 0.2 = tion. This direction can be represented by a vector known as G (p p )/10r.− Data3 for ten tissues with0.5 random initial configurations model has a constant-density glass transition governed by single-p xy = 0⇤↵ 0 0 are shown. cell mechanical propertiesphex such asp! cell–cell adhesion and cortical cell polarity. In groups, cells can align their individual polari- 10−3 100 103 106 5 109 1012 0 0 a shear component . The normal componenttension indicates encoded inthe the extent target to shape extensions index p0. called filopodia or lamellipodia,ties, forming and spherical phases membrane of matter with orientational order. These ! " Cortical tension Cell−cell adhesion whichz a = material r/|p0 − p0 is| tensed or compressed atAnalysing a given point, only whereas the ground the statesblisters of called the vertex blebs. model,Cells migrate the usingalignment lamellipodia, interactions filopodia and theor resulting oriented phases can 16 the cells are pentagons, whereasshear the component other two indicates remain hexagonal. forces parallelseminal to the surface. work of Staple et al. foundblebs, depending an ordered-to-disordered on a diversity of intrinsic and extrinsic variables, pent 3/4 hex be described using concepts from the physics of magnetism Therefore if p0 < p (7 2p7)/(p2 3 ) 3.812, pentagons transition at p p 3.722. However, because almost all biological 0 = + Besides⇥ the ability⇡ to generate active stresses, a 0key= 0feature⇠ that including expression of adhesion proteins,and liquid density crystals. and composition For example, cells in a group can spon- Figurecost 2 | finite A rigidity energy transition and therefore in confluent the transition tissues. a necessarily, Critical scaling requires collapsetissues of the are average strongly energy disordered, barrier it height remained1", normalized unclear whether by multiplying this 60,61 differentiatespent tissues from passive soft materialsFigure 2 isBiological the self-propul tissues- as foam.of the (a) ECM,A picture confinement, of soap foam. topology andtaneously cortical break contractility symmetry. andTo align in a common direction. r/ p finitep energy., as a function In contrast, of z forr/pp0 p0p , pentagonsfor the data (and shownn-gons in withFig. 1b, confirmingtransition was the relevant scalingansatz for the observedof equation glass (3). orb jamming, The rigidity transitions transition is 0 0⇤ sive0 ability0⇤ of each individual cell. CellsAdapted achieve fromself-propulsion Wikimedia Commons. by migrate,(b) A cells monolayer adhere of their epithelial protrusions to the surrounding matrix | n>5)| cost no energy and= the| cells are| able to remain in the ground in multicellular tissues. Therefore, we explicitly study disordered This type of alignment is known as polar order. To capture its demonstrated in a simple phase diagramextending as a distinct function types of p0, of snapshots protrusion are takenandcells adhering labelled from a typical using them a rigid membrane-targetedto their tissue (peither0 3.7 specifically green) and fluorescentfluid-like through tissue protein membrane (p0 3.96). receptors such as integrins, state throughout the transition, requiring zero energy. The estimate metastable states and transitions between= them. In addition ref. 16 = emergence, one can introduce ferromagnetic-like interactions A rigidity transitionpent occurs at p0 surroundings.p0⇤ 3.813 for disordered The two main metastable types of tissue protrusion(green). configurations. From are thin X. Trepat The actin-rich line and corresponding E. Sahai.or unspecificallyNat. to Phys. the order-to-disorder14 using, 671 (2018).frictional transition interactions. p⇤ p , indicated by a red= dashed⇡ line in Fig. 5c, does identify uses a linear stability analysis of a single cell to suggest that a between individual cell polarities. At a coarse-grained level, reported0 = by0 Staple et al.16 is shown for comparison. Below phex, the ground state(c) For is a a hexagonal group of four lattice, cells and undergoing above phex a T1, the cell ground rearrangement, state is disordered. the the critical target shape index in our mean-field model,0 and is rigidity transition also occurs at phex.0 In contrast, our analysis collective cell polarity can be thought to result from an effec- energy (Eq. (1)) increases as the green0 edge contracts (l < 0), andNAT itURE PHYSICS | www.nature.com/naturephysics consistent with the critical point p⇤ 3.813 0.005 identified by explicitly includes multicellular interactions (that is, collective tive free energy with the Mexican-hat shape familiar from the 0 = ± decreases as the edge expands in the perpendicular direction NATUREthe PHYSICS scaling collapse| VOL 11 | of DECEMBER energy barriers 2015 | www.nature.com/naturephysics in the full vertex model. normal modes) and nonlinear eects (that is, energy barriers). Landau1075 theory of phase transitions. This example thus illus- pent (l > 0). The cell-shape parameter p0 varies from 1.8 to 3.5 in equal Is there an even simpler explanation for p0⇤ p0 ? As in other With this more sophisticated analysis, we demonstrate that vertex trates how tools and concepts from other fields of physics are 6,35,36,40 ⇠© 2015 Macmillan Publishersincrements. Limited. All(d) rightsThe energyreserved barrier for a T1 rearrangementpent rigidity transitions , we expect that the critical shape index models exhibit a rigidity transition at a value p0 p0∗ 3.81 that is decreases with p0, eventually vanishing for p0 == p ≈⇠3.81. (e) being borrowed to rationalize cell alignment. should also be related to isostaticity. In the vertex model with measurably dierent from the prediction p phex based0 on single- Tissues can undergo a fluid-solid transition0 by= tuning0 cell shape via In other situations, cells align along one axis but choose periodic boundary conditions, cells tile the flat two-dimensional cell linear stability. changes in cell-cell adhesion and/or internally-generated cortical plane, and therefore the total number of vertices V , cells M Importantly, predictions based on this critical rigidity transition no preferred direction of motion. This type of alignment is tension. Panels (c)-(e) are from ref.2. and edges E are related through Euler’s formula: 0 V E M. have recently been verified in experiments43. Specifically, in both known as nematic order, taking its name from nematic liq- = + As each edge is shared by two cells, E is also related to the simulations and experiments we can measure the shape index uid crystals, used in LCD screens. Prominent features of liq- average coordination number z of cells or E Mz/2, which yields p P/pA for each cell in a monolayer, where P is the projected uid crystals are singular points known as topological defects, = = V M(z/2 1). The degrees of freedom are simply the motions of cell(Fig. perimeter 2e). This and isA ais striking the cross-sectional prediction, area. which In showcases simulations the of where alignment is locally lost. You can find topological de- = each vertex in two dimensions: ndof 2V . Assuming force balance thebizarre vertex mechanical model, we find properties that the of median materials value with of the deformable observed fects literally on your own hands: they are the points where = (in both directions) and torque balance on each cell generates three shapeconstituents. index p is anIn orderconventional parameter condensed that also exhibits matter, critical solids scaling: melt by your fingerprint ridges meet (Figs. 3a and 3b). Recently, re- constraints per cell: nc 3M. At isostaticity, the number of degrees p p⇤ 3.81 for rigid or jammed tissues and p becomes increasingly = =either0 ⇠ increasing temperature or reducing the packing fraction, searchers have discovered topological defects in several cell of freedom equals the number of constraints: ndof nc, resulting larger than p⇤ as a tissue becomes increasingly unjammed (Fig. 2). = i.e. decreasing0 pressure. Tissues, however, can melt at a fixed assemblies, from bacterial colonies to epithelial tissues, con- in ziso 5 and suggesting a mean-field transition at a shape index This prediction is precisely realized in cultures from primary cells = temperature and at the maximum packing fraction, φ = 143, i.e. firming that they can be described as liquid crystals (Fig. 3c). of p⇤ 3.812. Although it gives a correct prediction, this isostatic in human patients, with implications for asthma pathobiology . 0 ' without gaps between cells2. Tissues can become fluid by in- Interestingly, topological defects can play important biologi- argument makes a strong assumption: that constraints are applied to We expect that this rigidity framework will help experimental- each cell instead of to each vertex. Therefore, an interesting direction istscreasing develop the other cell testable perimeter-to-area hypotheses ratio, about for how example the mechanical by either cal roles. For example, in epithelial monolayers, defects pro- for future research is to understand under what circumstances the response of tissues depends on single-cell properties. For exam- energy functional (equation (2)) eectively groups vertices into ple, Sadati et al.44 have proposed a jamming phase diagram where functional units that are cells. tissues become more solid-like as adhesion increases, based on observations of jamming in adhesive particulate matter at densities Discussion far below confluency. Using standard interpretations of the vertex Although the vertex model has been used extensively to model model (equations (1) and (2)), p0 increases with increasing adhesion, tissues over the past 15 years, there has never been a clear way to and therefore our model predicts that confluent tissues become connect the model parameters to tissue mechanical properties. Here more liquid-like as adhesion increases. This highlights the fact that we show that the vertex model has a new and previously unreported adhesion acts dierently in particulate and confluent materials; in critical rigidity transition that occurs at a critical value of the target particulate matter higher adhesion leads to gelation and solidifica- shape index p⇤ 3.81. This criticality is evident in energy barriers tion, whereas in the vertex model larger adhesion leads to larger 0 ⇠ to local T1 rearrangements, the vibrational spectrum of the linear perimeters, more degrees of freedom, and liquid-like behaviour. response, and the shear modulus of the tissue. Unlike SPP models, These ideas suggest that the role of adhesion in tissue rheology may where the liquid-to-solid transition is governed by density, our be much richer and more interesting than previously thought.

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A clear signature of ‘activeness’ in a macroscopic system can be found found be can system macroscopic a in ‘activeness’ of signature clear A the that found and , Fucci marker cell-cycle the used we cycle, cell to

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in the direction parallel to the alignment was larger than that in the the in that than larger was alignment the to parallel direction the in according to the cell cycle phase, which is known as the interkinetic interkinetic the as known is which phase, cycle cell the to according 180° 0°

C2C12 80

, Large-field image of NPC culture. Colour indicates indicates Colour culture. NPC of image Large-field , b m. 100 bar, Scale µ 75

Angle (deg)

arrows (right bottom panel) indicate velocity reversal of the tracked cells. cells. tracked the of reversal velocity indicate panel) bottom (right arrows 180° 0°

topological defects characterized by the winding numbers. winding the by characterized defects topological mm cells (3,000 high-density show panels bottom The

) cells. The white white The cells. ) y (x)

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− NIH-3T3 , Types of of Types , c mm. 1 bar, Scale indicated. (blue) 1/2 and (red) 1/2 colour). Lines in the right panels are the typical trajectories of single cells. cells. single of trajectories typical the are panels right the in Lines colour). − + –250 0 250

x defects, as shown explicitly in the bottom image with the winding numbers numbers winding the with image bottom the in explicitly shown as defects,

contrast (left) and fluorescent channel (right, nucleus marker in pseudo- in marker nucleus (right, channel fluorescent and (left) contrast Distance (µm)

Singular points where all colours meet in the top figure are the topological topological the are figure top the in meet colours all where points Singular mm cells (1,100 low-density show panels top

) cells observed in phase phase in observed cells ) NATURE PHYSICS

− –250 2500 LETTERS

180° 0° Supplementary Information) from the phase contrast image (bottom). (bottom). image contrast phase the from Information) Supplementary , Imaging at the single-cell level of the two-dimensional culture. The The culture. two-dimensional the of level single-cell the at Imaging ,

a 73 83

µm

the angle of the local alignment calculated using the tensor method (see (see method tensor the using calculated alignment local the of angle the Two-dimensional patterning and collective motion of NPCs. NPCs. of motion collective and patterning Two-dimensional | RESEARCH LETTER 1 Figure 4 Angle (deg)

abc d e f +1 –1/2 +1/2 90° π g aba t = 0 min 180 min c = 0 d hivy vx j RPE1 c 3

1 mm ) –1 0 85 ( µ m h

y 180° 0° v –3 C2C12 80

+1/2 c –1/2 +1 k 0.8

Figure 1 | Two-dimensional patterning and collective motion of NPCs. the angle of the local alignment calculated using the tensor method (see ) 75 Angle (deg) b –1 a, Imaging at the single-cell level of the two-dimensional culture. The Supplementary Information) from the phase contrast image (bottom). 180° 0° −2 y 0.0 top panels show low-density (1,100 cells mm ) cells observed in phase Singular points where all colours meet in the top figure are the topological (x) ( µ m h θ x 70 contrast (left) and fluorescent channel (right, nucleus marker in pseudo- defects, as shown explicitly in the bottom image with the winding numbers v NIH-3T3 –0.8 –250 0 250 colour). Lines in the right panels are the typical trajectories of single cells. + 1/2 (red) and − 1/2 (blue) indicated. Scale bar, 1 mm. cx, Types of − Distance ( m) The bottom panels show high-density (3,000 cells mm 2) cells. The white topological defects characterized by the winding numbers. –250 0 250 µ arrows (right bottom panel) indicate velocity reversal of the tracked cells. x (µm) Scale bar, 100 µ m. b, Large-field image of NPC culture. Colour indicates –250 2500 0.0 2.5 –4.0 0 4.0 –1.4 0 1.4 180° 0° –1 –1 73 –1 83 µm µm h µm h µm h Angle (deg) c Fig. 1 | Confined RPE1 cells align together with a tilt angle and develop a shear flow. a–c, Local angle of the cell bodies when confined in a 500-µ m-wide according to the cell cycle phase, which Figureis known 3 Topological as the interkinetic defects and spontaneous in the flows direction in cell monolayers. parallel to (a) theSchematics alignment of nematic was topological larger than defects. that Theg in local the hi 11 alignment axis (black curves) is undefined at the defect core. Defects are characterized by theirstripe: topological phase contrast charge, indicated (a); line below integral the convolution (b); local directorvy (c). Only a small fractionvx of the directorsj is displayed for clarity. d, Heatmap of the nuclear migration . To see whether the direction switching is related direction perpendicular to it (Extended Data Fig. 6). 3 schematics,12 which is defined as the winding number of the alignment direction around the defectlocal core.angle.(b) e,Topological Histograms defects of the in angles fingerprints. θ of the cells’ bodies (averaged over the width) (NRPE1! !38 FOVs, NC2C12! !60 FOVs, NNIH-3T3! !64 FOVs). f, Profile of +1/2 –1/2 +1 = = = to cell cycle, we used the cell-cycle marker Fucci , and found that the A clear signature of ‘activeness’ in a macroscopic system can be found ) Adapted from Wikimedia Commons, by Frettie. (c) The top panel shows the angle of cell alignmentthe orientation in a cell angle monolayer. across Defects the stripe are the width after averaging along the y direction. , Velocity field within the stripe. The colours code for the speed. Only

g –1 switching of direction occurs withinFigure 1 the| Two-dimensional samepoints G1 where patterning or allG2 colors and cell collective meet.cycle motion The phase bottomof NPCs. panelatthe the showsangle topological of the local defects alignment on defects. acalculated phase-contrast using It thehas tensor image been method of observed the(see cell monolayer. both Panelsin experiment(a) and (c) are from a, Imaging at the single-cell level of the two-dimensional culture. The Supplementary Information)16–25 from the phase contrast image (bottom).a fraction of the vectors is displayed for clarity. Note the shear flow (y!component) near the edges and the relatively smaller x!component of the velocity (Supplementary Video 6). This topis panelsin contrast show low-densityK. Kawaguchi, to the (1,100 interkinetic cells R. mm Kageyama,−2) cells observed motion and in M. phase Sano. andNatureSingular simulation points545, where 327, all 2017. colours(d) thatmeetA in nematic +the 1/2top figure monolayerdefects are the topological typically of epithelial undergo cells confined spontaneous in a stripe 0 contrast (left) and(phase-contrast fluorescent channel (right, image nucleus on the marker left) in exhibitspseudo- a spontaneousdefects, as shown shear explicitly flow in the (right). bottom Fromimage with ref. the5. windingdirected numbers toward the centre. h,i, Heatmaps of the y (h) and x!(i) components of the velocity. j,k, Profiles of the two components of the velocity across the stripe ( µ m h seen in vivo, where the motioncolour). of the Lines nucleus in the right panels is directed are the typical trajectoriesto the ofbasal single cells. motion+ 1/2 (red) andin −active 1/2 (blue) indicated.nematic Scale bar,systems, 1 mm. c, Types which of is in contrast to defects in y − The bottom panels show high-density (3,000 cells mm 2) cells. The white topological defects characterized by the winding26 numbers. after averaging over its length (y direction). For all relevant panels, the solid lines are average values and thev coloured areas are the standard deviations. end (G1) or the apical end (G2)arrows of the (right NPC, bottom panel) depending indicate velocity on reversal its cellof the trackedcycle cells. equilibrium liquid crystals . Topological defects in the biological 11 Scale bar, 100 µ m. b, Large-field image of NPC culture. Colour indicates 17 –3 phase , although there was a difference motein the cell timescale death and of extrusion switching6. In colonies context of the have motile been soil observedinally inspired in whorled by behaviors grain in in bacterial wood suspensions as well as and in between the two phases (Extended Data Fig.bacterium 4a–cM.)13. xanthus, defects promotein the tube formation formation of mul- in embryonicthe cell cytoskeleton. development. Soon after Examples formulating of thedefects theory, re- according to the cell cycle phase, which is known as the interkinetic in the direction parallel to the alignment was larger than that in the 21,22,27 nuclear migrationticellular11. To see whether aggregates the direction known switching as is fruiting related direction bodies, perpendicular which allow to it (Extendedsearchers Data Fig. predicted6). (Figs. that 2a active and stresses3a,b,g) generateand the average an instability shear flow vanished although the For widths largerk than0.8 but close to the critical width (L ≳ Lc), the Active nematic systems are predicted to have spatio-temporal cor- yielding non-equilibrium collective dynamics include in vitro LL to cell cycle, we used the cell-cycle marker Fucci12, and found that the A clear7 signature28 of ‘activeness’ in a macroscopic system can be found θ ()xo~+π−ε c c(sqx) the bacterial population to survive starvation . whereby these fluidscells start remained flowing spontaneously,very motile and without the convergent hav- flow was not affected angle is predicted to) vary across the stripe as θ 2 Lc c relations between densities andswitching strengths of direction of occurs nematic within the same order G1 or G2that cell cycleare phase andat the in topological vivo defects.cytoskeleton It has been observed dynamics, both in experiment bacterium 14 flows in liquid crystal 2π ing to apply external forces . To drive flows, active stresses q –1 3–5 23,16–2525 + (Fig.24 3c–f,h,i). The shear velocity fluctuated in space and time about where c = L and ε is a coefficient that depends on the material prop- distinct from those of equilibrium(Supplementary systems Video 6). This. By is in picking contrast to the out interkinetic areas motion mixturesand simulation , colonythat 1/2 defects growth typically dynamics undergo spontaneous, and the cAMP patterning of c seen in vivo, whereFlowing the motion on oftheir the nucleus own is directed to the basal motion in active nematic systems,have which tois in overcome contrast to defects alignment in forces in the liquid crystal, which 14 26 29 a zero mean (Fig. 3e,i). erties of the system (see Supplementary0.0 Note). Similarly, the y com- that do not contain defects , weend confirmed (G1) or the apical endgiant (G2) ofnumber the NPC, depending fluctuation on its cell cycle aggregatingequilibrium liquid social crystals amoeba. Topological .defects in the biological 2Kqcε LL− 11 Describing orientational order is not enough to account for happens only at17 sufficiently large scales. Therefore, the theory c ( µ m h vx() sin(qx) phase , although there was a difference in the timescale of switching context have been observed in whorled grain in wood as wellIn as summary, for L L , the cell population organizes in a ponent of the velocity is given by y ~ (1) c , where K is the 9,30,31 > c x γν+ Lc 13 and quasi-long-range order (Extendedbetween the twoDatacollective phases Fig. (Extended 5a cell and Data flows. Fig.b) 4a–cin To the) this. high- end, physicistsinAs tube has describe formation been cellin reported embryonic as- development.predicted in other Examples that works a strip of on defects of cell active cultures fluid would, flow we clearly only if it was v

14 nematic phase with a director at a finite angle with the main stiffness associated with splay deformations of the director field, γ is

Active nematic systems are predicted to have spatio-temporal1,8 cor- yielding non-equilibrium collectivewide dynamics enough include .in vitro21,22,27 ab

180 min 180 min 0 = t = 0 = semblies as active matter . For example, when+ cells align − π density NPC culture. In the same regions, we calculated the spatial observed 1/2 and 1/2 defects in the dense NPC culture (Fig. 4a). relations between densities and strengths of nematic order that are and in vivo28 cytoskeleton dynamics, bacterium flows in liquiddirection crystal of the stripe. In parallel, cells spontaneously develop the rotational viscosity and–0.8 ν is the flow alignment parameter. These with polar order, they3–5 can start migrating in the23,25 direction of 24More than a decade later,−1 these predictions were tested in correlations of cell density and distinct alignment from those of equilibriumfluctuations systems (.Extended By picking out areas Wemixtures found, colonyslow growth motions dynamics of, and defects the cAMP (3.7 patterning5 µ m of h on average) including

that do not containalignment defects14, we9. confirmed This type giant of number collective fluctuation motion aggregating is known social as amoeba ﬂock-29. cell monolayers complex, showcasing flows the genericwith shear character and of transverse the the- components. Below the quantities are material parameters–250 of the active0 cells. Therefore,250 it is

Data Figs 6 and 7a–f) and found that the fluctuations of density and the merging of defects with opposite9,30,31 winding numbers (Fig. 4b). In L 2 and quasi-long-range order (Extended Data Fig. 5a and b) in the high- As has been reported in other worksory. on cell Whereas cultures cellsthreshold, we clearly confined L in, cells narrow orient stripes in did the not direction flow, of the stripe and no net expected that the central angle θ (0)2−π∕ , the angle θ ± 2 −π∕ at the RESEARCH ing. The active-matter theory of this phenomenonLETTER was de- c x ( m) () density NPC culture. In the same regions, we calculated the spatial observed + 1/2 and − 1/2 defects in the dense NPC culture (Fig. 4a). L µ alignment were correlated (see Supplementaryveloped 25 Information). years ago inspired The by theFig. mesmerizing 4c, we show flights the of localcells nematic in−1 stripes order wider around than a the critical topological width developed defects a col- vy ± ()LL correlations of cell density and alignment fluctuations (Extended We found slow motions of defects (3.7 µ m h on average)shear including flow develops. 0.0 2.5 –4.0 0 4.0 –1.4edges0 and 1.4 the velocity at the edges ()2 all scale as − c (see correlations were positive in theData first Figs 6 andand bird7a–f third) and flocks found quadrant10 .that Today, the fluctuations when the principles of seen density and of calculated flockingthe merging are of from applied defects thewith to oppositedisplacementlective winding shear numbers of flow cells (Fig. as 4bToinside). predicted Inunderstand small by regions. the these theory Figureobservations, (Fig. 4d 3d ). In we modelled our system as also Supplementary Note). We note that the large fluctuations of the alignment were correlated (see Supplementary Information). The Fig. 4c, we show the local nematic order around the topological defects –1 –1 –1 many other systems, from synthetic active colloids to bacterial very large tissues, cell flows become chaotic, creatingµm h27 disor-,28 µm h µm h as a function of two-dimensionalcorrelations displacements, were positive in the indicating first and third quadrant that whenthe seen showscalculated the from net the displacement velocity of fieldcells inside calculated small regions.a Figure from confined 4d the same active data, nematic indicating fluid15 . Indeed, despite the obvious orientations and velocities (Supplementary Fig. 7) are not described as a function of swarms.two-dimensional3,5,15 displacements, indicating that the shows the net velocity field calculateddered from patternsthe same data, of indicating swirls known as active turbulence . Con- system is an extensile active nematic system . This feature3,5,15 of the that there is a flow of cells pointing inpractical the direction differences, of the ourfront confined of the cells share the same fundamen- by this mean-field model. For this reason, we focus here only on system is an extensileTo active describe nematic nematic system cell. Thiscolonies, feature of the researchers that there is a use flowthe of cells the- pointing finementin the direction can of thereforethe front of the prevent and organize spontaneous cell correlation was flipped and weakercorrelation in was the flipped case and of weaker a cultured in the case of myoblasta cultured myoblast cometcomet shape shape in + 1/2in defects + 1/2 (Fig. defects 4cflows.), whereas Fig.( ThisFig. no clear 1 regulatory4c | directionality Confined),tal whereas symmetries role ofRPEno confinement clear 1 ascells a directionality contractile align may together be relevant acto-myosin with a networktilt angle powered and develop by average a shear quantities. flow. a–c, Local angle of the cell bodies when confined in a 500- m-wide cell line C2C12, orywhich of also active shows a liquid nematiccrystals, pattern (Extended which Data generalizes exists in the the − 1/2 hydrody- defects. The direction of the defect motion in the 29 µ cell line C2C12, which also showsFig. 8a aand nematic b). Wenamics further pattern found ofliquid that the(Extended spatial crystals correlation to Data include of the cell exists activeNPCs was (cell-generated)in opposite the − to 1/2that for defects. passivein liquid embryonic The crystals, direction C2C12 developmentATP (Fig. 4e of hydrolysis), the and defect tumor between invasion,motion parallel inin which the plates cell . We therefore developed Fitting model to experiments for 10 µ m < L < 150 µ m, we find a stripe: phase contrast (a); line integral convolution29 (b); local director (c). Only a small fraction of the directors is displayed for clarity. d, Heatmap of the Fig. 8a and b). We further founddensity that was the anisotropicstresses. spatial when Amongcorrelation the typical many length of other scale the of phenomena, correlationcell NPCsand this the wastheoryconfluent opposite explains fibroblasts reportedto thatgroups in ref.for 31 often .passive migratean liquid adapted in tracks crystals, version defined C2C12 byof thethe ( surrounding Fig.physical 4e), model tis- of ref. that predicts that very good agreement between these theoretical scaling predictions sue (Fig.local 1a). angle. Overall, e these, Histograms studies showcase of the how angles cells canθ of the cells’ bodies (averaged over the width) (N ! !38 FOVs, N ! !60 FOVs, N ! !64 FOVs). f, Profile of density was anisotropic when 328the | NtypicalA TUREthe | VOLlength cell 545 flows | 18scale MAY observed 2017of correlation around topological and the defects. confluent Again, fibroblasts reportedactive in ref. nematics 31. confined in a stripe of width L exhibit a generic con- and the experimentalRPE1 data= yielding LcC2C12 = 46= ± 4 µ m for RPE1NIH-3T3 cells= beyond cell colonies,© 2017 Macmillan the theory Publishers successfullyLimited, part of Springer describes Nature. All rights many reserved. leveragethe the orientation physicstinuous of active transition angle fluids across toat engagea critical the stripe in collectivewidth width L between after averaging an ordered along immo- the y(Fig. direction. 4a,b). Similarg, Velocity results field were within obtained the stripe. for C2C12 The coloursmouse myo- code for the speed. Only flow patterns, and how confinement controls whether and howc 328 | NA TURE | VOL 545 | 18 MAY other2017 systems, from biopolymer gels to shaken granular mate- bile state and a flowing state. This transition is analogous to the blasts. In this case, we measured λ 12 2 m and L 30 4 m. cell groupsa fraction flow on theirof the own. vectors is displayed for clarity. Note the shear flow (y!component) near the edges and the= relatively± µ smallerc = x±!componentµ of the velocity © rials.2017 Macmillan Employing Publishers general theories Limited, ofpart active of Springer matter Nature. to describe All rights reserved. Fréedericksz transition of nematic liquid crystals30 but it is driven The theory thus captures the main features of the biological system cell migration is particularly useful because it reveals connec- directed toward the centre. h,i, Heatmaps of the y (h) and x!(i) components of the velocity. j,k, Profiles of the two components of the velocity across the stripe tions to apparently-unrelated systems. Progressively, this ap- To spread or notby to the spread intrinsic activity of the system rather than an external field. and in particular the existence of the transition that we now iden- proach is uncovering a classification of active systems based Whatafter happens averagingTo if weaccount release over for confinement itsthe length finite andangle (y expose direction). of the a cell edge For cells all (Fig. relevant 2b), we panels, adapted the solidtify as lines a Fréedericksz are average transition values driven and the by colouredcell activity. areas Spontaneous are the standard deviations. on symmetries and generic behaviors, in the spirit of univer- monolayer to freethe space? model Cells by assuming at the monolayer a finite edge anchoring are at the walls. The torque flow transitions are typical of active matter; other examples include sality classes in statistical mechanics. able to sense thatexerted they have on the neighboring cells by cellsthe edges on one is side then balanced by the nematic low-Reynolds-number turbulence or spontaneous topological sin- For example, the theory of active liquid crystals was orig- but not on the other.elastic In torque ways that (see remain Supplementary to be fully under- Note). gularities in translation and rotation27,31. Here, we demonstrate their (Figs. 2a and 3a,b,g) and the average shear flow vanished although the For widths larger than but close to the critical width (L ≳ Lc), the NATURE PHYSICS | VOL 14 | JULY 2018 | 728–732 | www.nature.com/naturephysics 729 ()xoπ−LLc c(sqx) cells remained very motile and the convergent flow was not affected angle θ is predicted to vary across the stripe as θ ~+2 ε L c 2 c q = π (Fig. 3c–f,h,i). The shear velocity fluctuated in space and time about where c Lc and ε is a coefficient that depends on the material prop- a zero mean (Fig. 3e,i). erties of the system (see Supplementary Note). Similarly, the y com- 2Kq cε LL− c vxy()~ sin(qxc ) In summary, for L > Lc, the cell population organizes in a ponent of the velocity is given by γν(1+ ) Lc , where K is the nematic phase with a director at a finite angle with the main stiffness associated with splay deformations of the director field, γ is direction of the stripe. In parallel, cells spontaneously develop the rotational viscosity and ν is the flow alignment parameter. These complex flows with shear and transverse components. Below the quantities are material parameters of the active cells. Therefore, it is L (0)2 θ ± −π∕2 threshold Lc, cells orient in the direction of the stripe and no net expected that the central angle θ −π∕ , the angle ()2 at the L v ± shear flow develops. edges and the velocity at the edges y ()2 all scale as ()LL− c (see To understand these observations, we modelled our system as also Supplementary Note). We note that the large fluctuations of the a confined active nematic fluid27,28. Indeed, despite the obvious orientations and velocities (Supplementary Fig. 7) are not described practical differences, our confined cells share the same fundamen- by this mean-field model. For this reason, we focus here only on tal symmetries as a contractile acto-myosin network powered by average quantities. ATP hydrolysis between parallel plates29. We therefore developed Fitting model to experiments for 10 µ m < L < 150 µ m, we find a an adapted version of the physical model of ref. 29 that predicts that very good agreement between these theoretical scaling predictions active nematics confined in a stripe of width L exhibit a generic con- and the experimental data yielding Lc = 46 ± 4 µ m for RPE1 cells tinuous transition at a critical width Lc between an ordered immo- (Fig. 4a,b). Similar results were obtained for C2C12 mouse myo- bile state and a flowing state. This transition is analogous to the blasts. In this case, we measured λ = 12 ± 2 µ m and Lc = 30 ± 4 µ m. Fréedericksz transition of nematic liquid crystals30 but it is driven The theory thus captures the main features of the biological system by the intrinsic activity of the system rather than an external field. and in particular the existence of the transition that we now iden- To account for the finite angle of the edge cells (Fig. 2b), we adapted tify as a Fréedericksz transition driven by cell activity. Spontaneous the model by assuming a finite anchoring at the walls. The torque flow transitions are typical of active matter; other examples include exerted on the cells by the edges is then balanced by the nematic low-Reynolds-number turbulence or spontaneous topological sin- elastic torque (see Supplementary Note). gularities in translation and rotation27,31. Here, we demonstrate their

NATURE PHYSICS | VOL 14 | JULY 2018 | 728–732 | www.nature.com/naturephysics 729 REVIEWS

even originate far away from the edge and be reveals thick actin bundles (51) as well as stretched brought to the front by flows within the epithelium E-cadherin morphology (48) connecting the lead- (51). Indeed, cells far away from the edge of a ers to the followers, suggesting a strong mechani- monolayer have been shown to extend cryptic la- cal coupling and force transmission within these mellipodia underneath their neighbors (21), sug- finger-like structures. Velocity field analysis and gesting that leader cells are not necessarily mechanical perturbation of leader cells using laser restricted to the edge of a monolayer (FIGURE 3B). ablation strongly support the role of leader cells in These leader cells drag along a small cluster of providing local guidance cues to cells following follower cells, resulting in the formation of “finger- them (53). It was further observed that inhibition like” instabilities of the leading front (51). Staining or expression of a dominant negative form of Rho SPECIALSECTION NATURE PHYSICS DOI: 10.1038/NPHYS2355 ARTICLES

a 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,5 PA gel and cellular migration across the epithelial sheet, are initiated at PDMS membrane cthe leading edge and progressively penetrate towards the centre a (Supplementary Movie S3). b 100 μm 100 μm 20 μm Moreover, these stress fields were anisotropic. At each position t = 0 h in the monolayer plane, the maximum (max) and minimum 5 (min) principal stresses were represented as an ellipse aligned t = 1 h with corresponding principal orientations (Fig. 2p). Throughout Downloaded from epithelial expansion, stress ellipses tended to be spindle-shaped t = 4 h and thus revealed pronounced stress anisotropy. The maximum principal stress orientation tended to be perpendicular to the d 50 c b t = 0 h t = 1 h t = 4 h leading edge and thus roughly parallel to local cell motion (kPa) Intercellular stresses Cell-cell adhesion (Fig. 2q).40 As described previously, this mode of local cell guidance http://science.sciencemag.org/ ⇣ Dewetting

2,5 defines plithotaxis30 .

Actin Superposed20 on systematic monolayer spreading were large- scale spatio-temporal fluctuationsWetting of tractions, monolayer stresses and cellular10 velocities (Fig. 2f,i,l,o). To better characterize the

Traction forces Cell-substrate adhesion systematicContractility 0 evolution of mechanical patterns, we averaged these variables over0 the 50 observable 100 150 monolayer 200 length (corresponding Monolayer radius R (µm)

to the y coordinate), thereby reducing the dimensionality of the on January 28, 2019 E-cadherin system to only one spatial dimension and one temporal dimension. Fig. 2. Wetting analogies of tissue behavior. (A)Cellsorting.Twodifferentcell creased, the dynamics of the precursor film transitions from a liquid (left) to a 2D gas All data could then be represented11 as kymographs in the x–t Figure 4 Tissue wetting and spreading. (a) populations (pink and blue) in a cell aggregate spontaneously sort out(b) according to (right). Reprinted with permission from (3). (E)Dewettingofacellularmonolayeron Transversal view of a cell monolayerplane as it (Methods). starts spreading. Kymographs From ref. of. cellularSchematic velocity of ( thev ) revealed cell-substrate forces and cell-cell stresses involved in tissue spreading. Adaptedtheirfrom surface and S.R.K. interfacial Vedula tensions. et(B)Aggregatefusion.Whenbroughtinto al. Physiology 28, 370, 2013.anonwettablesubstrate,aphenomenonopp(c) Cellx osite to spreading. Reprinted with kind ZO-1 contact, two aggregates of the same cell line fuse to yield a larger, spherical aggre- permission of The European Physical Journal (EPJE) from (6). Scale bars, 100 mm. (F) motility patterns that were12 not restricted to the initial phase of aggregates may either remain as droplet-like spheroids (left) or wet the underlyinggate. (C)Aggregatespreadingonawettablesub substrate (right). From ref.strate. Reprinted. (d) Phase with permission diagramAggregate of active shivering. Active pulsatile contractions are observed during micropipette from (57inward). (D)Long-termspreadingdynamics.As mobilization (Fig.the 3a). aggregate To the cohesivity contrary, is de- aspiration after when reaching the aspiration the pressure is within a certain range. tissue wetting. For a given tissue contractility, which represents cell-cell pulling forces,monolayer only sufﬁcientlymidline at large150 monolayers min, the two wet fronts the substrate. of cell motility 13 ⇠ Adapted from ref. . to a viscoelasticcoalesced model, and the then aggregate continued’selastic may towards not arise during the compression. leading edges.A systematic Whenrequired to break intercellular bonds and induce modulus,cells viscosity, are cohesive and surface tension and aremass de- isstudy conserved, comparing parallel-plate cellular compression velocities and mustcellular reorganization (22). To our knowledge,

Paxillin termined. This technique has the advantage of a micropipette aspiration on aggregates of the same no systematic experimental investigation of the relativelybe simple linked experimental to the setuprate and of image cell deformationcell lines would help (strain to clarify rate, the effect"˙xx of; these ref. 11)existence of a yield stress in tissues has been con- analysis. Using this technique on aggregates of differences. It has also been pointed out that where- ducted. Existing experimental observations such stood, edge cells respond to this asymmetric environment by Recentthrough work the addressed expression this"˙xx limitation@vx /@x. Remarkably, by treating kymographsthe cell of mouse sarcoma cell lines expressing E-cadherin,= as in parallel-plate13 compression the whole aggre- as the spontaneous fusion of two aggregates polarizing towardd free space60 (Fig. 4a). Speciﬁcally, these cells monolayerGuevorkian"˙xx revealedet as al.reportedthattheaggregate a droplet clear evidence of active of’s liquid wave-likegate is stressed,. Using crests micropipette ofthis strain approach, aspiration rate that probes were(40)seemtosuggestthattheyieldstress,ifit surfacelaunched tension increased atwith each the appliedleading pressure edge,only propagated a subset of the away aggregate from’scells( and39). back The exists, to is smaller than the stresses induced by extend protrusions known40 as lamellipodia, with which they one obtains the spreading parameter directly in terms of active 3 (21), which could indicate an active reinforce- number of cells subjected to stress scales as Rp . tissue surface tension, and thus deformations are 6 % the leading edge at roughly twice the speed of the advancing front exert directed and persistent20 traction forces on the underly- cellularment of forces. the aggregate Supported in response to the by applied experiments,Thus, for the continuumthe model hypothesis predicts to be valid relaxed by surface tension over a time scale of 0 stress.edge, Such a reinforcement and spanned has not been the reported entireand monolayer the technique (Fig. applicable 3b). to characterize To distinguish ag- the order of hR/g,whereR is the characteristic )

ing substrate to migrate2 toward open ground. Because cells that the wetting transition results from the competition be- 0 200 400 600 in parallel-platethese compression mechanical experiments. waves Although fromgregate other rheology, known the micropipette types of radius mechanical should aggregate size (typically a few hundred micro- m in parallel-plate compression aggregates are com- be large enough to probe a few hundreds of cells, meters). Moreover, bond formation and dissoci- in the monolayer areµ adheredTime to (min) one another, the migrating tween two types of active forces: cell-substrate traction forces 5 4 wave, and because they inscribe an X-shape on the kymograph, pressed to a constant strain, with the exerted stresses which is usually accomplished with Rp × 30 mm. ation is more accurately described as a dynamic edge cells pull on cells10 in the second row, which then also that promote spreading versus cell-cell pulling forces that pro- × beingwe relaxed call over them a typical X-waves. time scale of about The existence of a yield modulus above which process, where the formation and dissociation polarize, migrate, and pull on inner cells, setting the mono- motean hour, retraction. in micropipette This aspirationactive aggregates wettingaggregatesframework exhibit plastic behavior makes remains an- a mat- rates vary continuously with the applied load. An undergo aTo continuous study traction the at physical constant stress. originter of of controversy. the X-wave, Some authors we have next postulated focusedappealing solution to the debate about the exist-

16Area ( layer under tension 2 (Fig. 4b). At the molecular level, this otherIt has keyon been tractionprediction: reported that generation traction the forces spreading and elicit stressthe parameter existence transmission of a yield depends modulus inthe in ontissues, monolayer. the which ence of tissue plasticity has been suggested by supracellular coordination is mediated by a protein known as dropletactiveWhereas cell radius. contractile traction Cell responses monolayers kymographs(37, 38), which largerphysically demonstrated than woulda arise criticalextrema from the radiusat critical the force leadingMarmottant et al.(41), who proposed that the merlin, which transduces intercellular forces into cell polar- wet theedge, substrate, monolayer whereas stresses smaller weremonolayers highest at the dewet monolayer from midline,it www.sciencemag.org SCIENCE VOL 338 16 NOVEMBER 2012 913 ization. In this mechanically-coordinated0 way, the entire cell (Fig. 4dindicating). This prediction that local force is striking generation because was globallyit has no integrated coun- and monolayer spreads on the0 substrate,100 200 becoming 300 400 progressively 500 600 700 terparttransmitted in the classic through wetting cell–cell picture, junctions in which to the give spreading rise to a stress 11 Time (min) build-up (Fig. 3c,d). Importantly, monolayer stress at the midline thinner (Fig. 4a). Combined with other mechanisms, this parameteroscillated depends in time solely (Fig. on 3g,h surface and tensions. Supplementary For regular Movie liq- S4); these type of collectiveFigure 1 Experimental cell migration model. helpsa, A PDMS to close membrane gaps in is epithe-deposited on auid droplets,oscillations size does were not in phase matter. with In fluctuations contrast, tissue of cell wetting area (Fig. is 3f,h) lial tissues,collagen-coated as in| wound polyacrylamide healing. (PA) gel. Cells are seeded and allowedsize-dependent. to and demonstrated This prediction phase quadrature was verified with in strain experiments, rate (Fig. 3e). But tissuesattachFIGURE only not over always the 3. gapMechanisms spread defined on by the substrates. PDMS of membrane. collective Under When cer- cell confluentproviding migrationContrary evidence from to long-held for a the mechanistic active assumptions nature of perspective (reviewed tissue wetting in ref.13 12),. these tain conditions,cellsA reach: positive a a cellrelatively monolayer (left high) andconfluence, may negative the instead PDMS (right retract membrane) fluorescence from is peeled offBesides imageobservations its of relevance a migrating establish in physics, that monolayer on the the ultraslow existence of MDCK timescales of a cells critical of stably cellular expressing the substrate,andactin-GFP cells eventually start invading showing collapsing the surrounding finger-like into space. a droplet-like instabilitiesb, Transversal cell view capped of size by formigration cells tissue with wetting the mesenchymal dominant might explain cellular morphology drastic stresses changes in thecharacteristic in monolayer tissue are of leader LifeAct MDCK cells at the specified time points after the PDMS membrane elastic, not viscous. aggregate13 cells.(Fig.Right 4c). Tissueimage spreading is superimposed and retraction with are“cartoonized”morphology structures during embryonic (red and development green) to illustrate and cancer the pro- distribution of ac- was removed; scale bar,20 µm. c, Basal actin (LifeAct–GFP), E-cadherin,gression.In For the example, absence of a appreciabledisturbing inertia,possibility there is can that exist a grow- no exchange thereforeZO-1 reminiscenttin and bundles paxillin of immunofluorescence theand wetting lamellipodia and micrographs dewetting that isbefore, of typical liquid 1 h after of and leaders 4 h between and followers kinetic and in potential finger-like energy protrusions. storage as is usuallyB: cryptic associated lamellipodia droplets.after Theextended removing degree the of by membranes. wetting cells far depends Scale away bars on, from15 µ them. d thebalance, Cell leading sheet be- area (fillededge.ing tumorInsetwith might: propagation magnified become of view able passive toof spread the mechanical region onto the waves, bounded surrounding thus bysuggesting the dotted box tween cohesivecircles)showing and forces cell sheetcryptic within area lamellipodia.thedue to liquid proliferation and adhesiveRight (white circles)image forces at different is superimposedtissue oncethat the it reaches with mechanism pseudo-colored a critical underlying size. Overall, the structures observed this work propagation (red exem- and mightgreen) to illus- with thetime substrate.trate points. the The By area distribution analogy, due to proliferation early ofmodels actin was calculatedbundles proposed by and counting that cryptic theplifies lamellipodia.be how active. the Toquest investigateC: to cartoon understand this illustrating possibility, collective we the inhibited cell mechanical migration myosin using interaction of tissue wettingnumbercells was of withcells dictated in the the monolayer by substrate a competition at distinct through time between points focal cell-cell and adhesions then motivates andblebbistatin. with the their development neighborsBlebbistatin of new prevented through physics, the cell-cell leading formation junctions. to theof stress dis- Thefibres former re- multiplying the number of new cells by the average12,17 cell area. Data are (Wcc) and cell-substratesults in traction (Wcs) forces,adhesion and energies the latter, encoded transmitscovery the tension(Supplementary of phenomena across likeFig. the S2) active sheet and wetting. had of cells. little Finally, effectD: cell-cell on this the physics velocity junctions of the are sub- mean s.d. (n 5). Inset: relative contribution of cell proliferation to cell leading edge, thus confirming previous reports in wound scratch in a spreadingjected± parameter to= shearS = andWcs normalWcc. When stressesS < during0, cell- migrationapproach as offers a result clues of on mechanical how cell aggregates coupling. may Cells tune tend active to migrate sheet area. 8 cell adhesionalong dominates the direction and the cell of− maximalaggregate normal retracts from stress andforces minimal toassays control. shear Blebbistatin whether stress. caused to spreadE: particle traction or not forces image to spread. and velocimetry intercellular stresses (PIV) tracking to be abrogated, however (Supplementary Fig. S2 and Movie S5). the substrate (dewetting, Fig. 4c left), whereas when , Withof time, MDCK these cells boundary migrating layers became within markedly wideS channels heterogeneous > 0 (×400A␮ well-definedm) showing front vortex of strain formation rate could (top be) and clearly within identified narrow chan- cell-substratebutnels systematically adhesion (×20 ␮ dominatesm) grew showing to and encompass athe contraction aggregate increasing spreads relaxation numbersMechanical modeof nonetheless, of migration waves but without this (bottom front inertia). was Cartoons stationary, showing did not the propagate distribution of (wetting, Fig.traction 4c right). forces This under simple both conceptual conditions framework are depictedTissue below spreading the respective revealed yetimages another (F) strikingplot of thecollective correlation distance was sufficientNATURE(of to PHYSICS velocities) interpretVOL 8the ofAUGUST behaviorMDCK 2012 cell ofwww.nature.com/naturephysics cell clusters aggregates as a function of phenomenon: of the radius Mechanical of the confining waves start circular spontaneously pattern. at the 629 varying cell-cell and cell-substrate| | adhesion| 12,17. However, leading edge and propagate across the cell monolayer11 this analogy to passive liquids does not explicitly account for (Fig. 5). These waves are slow, with speeds 10 100 ∼ − 374 PHYSIOLOGYthe active• nature Volume of cells. 28 • November 2013 • www.physiologyonline.orgµm/h, and wavelengths that span several cell diameters. Simi- ll Article

A D

ARTICLES NATURE PHYSICS

a c ERK activity f 0.7 1.7 Cell area (µm2) 120 700 (FRET/CFP) Im() µ y 200 µm 200 m Re() 6 q x

a BC E ical waves in tissues implies an active driving mechanism that Stable b Space d compensates damping and generates an effective inertia. 0 200 µm ERK Area Area ERK The quest to understand these waves led to a plethora of 5 0.2

h) physical models, ultimately revealing a palette of possible ( 1

0.1

ime 10 mechanisms . Recently, the situation has been clarified by ex- T q NATURE PHYSICS ERK–area 0 periments revealing that mechanical waves are accompanied 15 cross-correlation ARTICLES –0.1 byUnstable waves of ERK, an extracellular signaling molecule that af- σ$ ERK activity –30 –20 –10 0 10 20 30 oscillatory a (FRET/CFP) 0.8 1.4 Lag time (min) e fects cellular activity (Fig. 5a). Following these observations, Cell migration ERK e g researchers developed a theory based on the feedback between Figure 1.b Cell Deformation Waves Precede ERK Activation Waves during Collective Cell MigrationStress Space (A) Phase contrast images (upper), FRET/CFP ratio indicating ERK activity (middle), and strain rate in the directionPolaritythe of collective biochemical cell migration (x regulator-strain rate; lower) and cell mechanics (Fig. 5b), which are represented at 1 (left), 8 (center), and 14 h (right) after start of time-lapse imaging. In the x-strain rate images, extending and shrinking regions are shown in red produces coupled chemical = 60 andmin mechanical waves. Assum- and blue, respectively. Scale bar, 200 mm. The imaging interval is 1 min. p (B and C) x-t kymographs of ERK activity (B) and x-strain rate (C), from (A).ERK White activity arrows indicate the rightward propagationing that of ERK cells activation polarize wavest (B) and in cell response to stress gradients in the ϕa deformation waves (C). Time Ei (D) Temporal change of ERK activationϕl rate and x-strain rate in a representative cell. monolayer, the theory also explains propagation away from (E) Temporal cross-correlations between x-strain rate and ERK activation rate. The blue line indicates the average temporal cross-correlation coefficients with 18 ϕb Ii the leading edge in spreading tissues (Fig. 5c). standard deviations (SDs). n = 212 cells from three independent experiments. ϕa, b, l Cell length Simulated kymographThese results show that cells can exploit mechanochemical See also Video S1. 100ri µm ri+1 Surface tension p = 40 min li Golgi apparatus Nucleus feedbacks to transmitt local information over long distances. et al.,Fig. 2019 1 | Observation). Earlier and studies theory of have spatiotemporal shown thatpatterns ERK in confluent activation cell monolayers.RESULTS a–c, Confluent MDCK monolayers display isotropic patterns of c b This tissue-scale communication is relevant for wound heal- propagatesERK activity as (a multiple and b, kymograph) waves fromand density leader (c, colour cells code to the indicates follower cell area from Voronoi tesselation) in contrast to directional waves reported in cellsmigrating during collectivemonolayers. See cell also migration SupplementaryERK using wave Videoin vitro 1. d, Cross-correlationcultured cells functionCell of Deformationcell area and ERK activity Waves ing,(defined Precede enabling in the Methods) ERK Activation distant indicates a cells Waves to coordinate their migration toward 25 27 (Handlypositive et correlation al., 2015; between Matsubayashi the two , with et ERK al., trailing 2004; area Nikoli by aroundc et 3 al.,5 min To(ref. investigate) (average of theN = 3 relationship experiments, shaded between areas ERK indicate activation standard and cell deviation). e, Schematic of our mechanochemical model. ERK activation (yellow) causes actomyosin remodelling the(red), differentially wound. affecting Similar apical–basal principles p = 20 min operate in morphogenesis, en- 2006and) and lateralin tensions vivo mouse. The dependence ear skin of (ERKHiratsuka activationet on cell al., length 2015 l completes). deformation a mechanochemical during collectivefeedback loop cell between migration, ERK activation, Madin-Darby cortical canine a,b,l abling coordinated cellt deformations to precisely shape tis- Recently,tensions we and have cell aspect demonstrated ratio. f, LinearTraction that stability forces the of ERK this model activation (equation waves (4)) revealskidney a finite (MDCK) wavelength cells oscillation confluently (Re($) > seeded 0, dashed within line) above compartments a certain of orientstrength the directed %&crit of mechanochemical migration of coupling. a cellcluster g, This instability against is theconfirmed wave in numericala silicone simulations confinement of the model weresues (kymograph). released without for collective requiring cell migration. local genetic control of cellular forces. direction, indicating a critical role of the ERK activation waves ERK activity and cell deformationFrom were a physics evaluated perspective, by FRET imag- this research highlights that the in coordinatedc cell migration (Aoki et al., 2017). However, it re- ing with EKAREV-NLS,f by which ERK activation is quantified as Results Symmetry breaking epithelial tissues, cell–cellERK junctions activitycells’ (FRET/CFP)impose ability mechanical to generate, couplingsTraction sense, force in and x (Pa) respond to signals, both mainsFigureFirstly, largely 5 in Waves agreement unknown during with how previous cells tissue harness studies spreading. of the expanding mechanotrans- (a) MDCKSpace-time betweenan increase cells plots38, in so the that FRET/CFP0.65 the mechanical ratio (stateKomatsu of 1.15each et cell al., influences 2011),–350 and par- 350 duction(knownmonolayers and as the kymographs)25,27 ERK,34, we activation confirmed of waves that the directional activity to coordinate wavesof the collective of signalling cellu- theticle shape, molecule image density velocimetry and velocity (PIV)-basedchemical of its neighbours image450 and 39min. To processing, mechanical, describe this respec- can give450 rise min to emergent phenom- cell migration.lar density and ERK activity propagate away from leading edges wetively. consider, As thein a cellssimple migrated 1D setting,ena toward asa linear counter-intuitive free chain spaces, of coupled sustained cells as ERK mechanical waves without inertia. ERKIntowards this (left) study, theand centre by combiningof of thecolonies strain Fo(Extended¨ rster rate resonance Data (right) Fig. 1a,b). during energy Moreover, trans- tissue with spreading.activation vertexx positions waves were ri and propagated surface tensions from i the (Fig. leader 1e). If cells a given tothe fol- ferWaves, (FRET)-basedwe performed indicated the biosensors same by experiments white (Harvey arrows, at full et two-dimensional al., appear 2008; as Komatsu oblique (2D) cell lines,lower contracts, cells whose (itFigures imposes 1 aA stress and 1B;on itsVideo neighbours, S1), in which agreement is resisted with pre- confluency (Fig. 1a–d and Supplementary Video 1) and found simi- by cellular tensions, as well as by frictional forces with the substrate etslope al., 2011 gives), an the optogenetic wave speed. tool (FromAoki et N. al., Hino 2013), et traction al. Dev.vious Cellϕ reports53, 646 (Aokiϕ et al., 2017; Matsubayashi et al., 2004). The lar patterns of ERKIn phase: and cell no density net polarity (with ERK positivelyϕσ correlated fi = vi (where is a cell–substrateOutlook friction coefficient depen- force microscopy (Trepat et al., 2009), and mathematical cell deformation, i.e., extension and shrinkage, was quantified (2020).with, and(b) slightlySchematic lagging behind, of the 2D feedback cell area), although between in this ERK dent activity, on cell–substrate cell adhesion andThe vi = ϕ physicstri the velocity of ofactive the cell living matter is increasingly success- modeling,case they we displayed show that random each orientation follower cellof propagation. possesses This a mech- sug- vertexby the ri). cell strain rate in the direction of collective cell migration tension,gests that and monolayer cell length, expansion which (mediated produces by leading mechanochemicaledges) is not At the continuum waves. limit and at linear order, force balance in such anochemical feedback system, in which stretch-induced ERK (x-strain rate). Similar toful the ERK at explaining activity,100 µm the x-strain the ratedynamics ex- of100 collective µm cell migration. activation(c) requiredSchematic triggers for wave showing cellformation, contraction.that motivating cells Intercellular us polarize to first study coupling and theoreti migrate of- a hibitedmonolayer against repeated reads the positive and negative values, namely extension cally how isotropic waves of ERK and density arise in the confluent x This core biological process is being understood through con- the ERK-mediated mechanochemical feedback enables sus- and shrinkage, which propagatedDistance ( m) from the leader cells to the fol-Distance ( m) mechanochemicalsetting. wave. In spreading tissues, the wave travels rtr x; t xxr x; t µ xl0 x; t 1 µ ϕσ cepts ϕ$ such asϕ orientationalϕ order, flow, turbulence, jamming, tainedbackwardsFor propagation this, from we first ofthe wrote ERK leading down activation a edge, minimal and directing model contractile of epithelial cell force migration lower cells toward (Figures–300 1A–200 and 1C;–100Video0 S1100), as described–300 previ-–200 –100 0 100 generation,monolayer leading mechanics to multicellular that we highlight alignment here (see of18 Supplementaryfront-rear cell withously all (Serra-Picamal units normalized0 et by al.,wetting the 2012; average Tlili and cell et al.,length, wave 2018 and). propagation.0 Hereinafter,r the The mechanistic origin and polarityfreeSection space. over I for long Panels details).Out distances. of The(b) phase: 3D-(c) shape Thus, netare polarity of our MDCK from study cells, ref. clarifies like. in other a mech- epi- vertexwe refer position to the atpropagation distance x and of thetime x-strain t. The first rate two ascell terms deformation are 5 physical properties5 of these phenomena in cells, however, dif- anismthelial of tissues, intercellular is proposed communication toERK arise from a balance underlyingStress of forces long-range betweenPolairty classicalwaves for (Rodrı overdamped´guez-Franco chainset of al., oscillators, 2017). and represent resp. the active stresses generated by actomyosin cortices at the apical, frictional forces and restoring forces from active cell tensions (with sustained transmission of35– directional37 cues for collective cell To analyze the10 correlationfer between fundamentally the ERK activation10 from and those the in non-living matter. Whereas lateral and basal surfaces (resp. denoted as a, l and b). For a σr = ϕ/k, the ratio of cell–substrate friction to cell stiffness, giving Time (h) migration. cell deformation,Time (h) time series data of the ERK activity and those of single cell on a flat surface, equilibrium of these forces requires that a timescale for the15 spatiotemporalnew diffusion active of matter displacement15 theories r, progressively manage to explain the in-plane length of a cell l is equal to a ratio of active tensions, on length scales of cell size). The third term arises from the fact 1=3 lar to longitudinal sound waves, tissue waves stretch and com-Developmentalthe Cell broad53, 646–660, phenomenology June 22, 2020 647 of collective cell migration in terms such thatd 600l l . However, for confluent, heterogeneous that each cell at position x can in principle have its own preferred a b press cells/ as they travel. Most strikingly, the waveslength l0, are so that self- gradients ofof preferred a small length number result in cellular of physical variables, how cells tune these I ϕ g sustained; they travel long distances ( 1 mm) unattenuated.1.1 NATvariablesURE PHYSICS | www.nature.com/naturephysics through30 thousands of genes and biochemical reac- P)

500 Average F ∼ 1.0 (Pa)

This observation is surprising because cell motionpolarity is so slow tions remains a majorx open question. that inertia is negligible. Therefore, these waves cannot be0.9 400 0.8 0 ty (FRET/C i 0.8 sustained by the common back-and-forth between kinetic0.6 and Acknowledgments 0.7 0.4 potential energy familiar from the harmonic oscillator. More- We apologizeTraction force in to the many colleagues whose work could

300 ERK activ 0.2 0.6 –30 over, there are many sources of dissipation in tissues, includ- 150 200not250 be300 cited350 owing150 to200 space250 300 constraints.350 R.A. acknowl- 0

Period (min) Time (min) Time (min) ing cell-cell200 and cell-substrate friction, which could–0.2 poten-1.0 edges support from1.0 the Human Frontier Science Program n –0.4 0.8 o 0.8 tially damp the waves. Thus, the very existence of mechan-0.6 (LT000475/2018-C).

relati 0.6 r 100 0.4 0.4 0.2 0.2

0 autoco n –0.2 0 –0.2

ERK autocorrelation –0.4 0 ractio 10 20 30 40 50 –0.6 T –0.4 0 100 200 0 100 200 Wavelength (cells) 1. Alert, R. & Trepat, X. Physical Models of Collective Cell Mi- Lag timeairway (min) epithelium. Nat.Lag Mater. time (min) 14, 1040–8 (2015). gration. Annu. Rev. Condens. Matter Phys. 11, 77–101 (2020). 5. Duclos, G. et al. Spontaneous shear flow in confined cellular Fig.2. 3 | Bi,Response D., Lopez, of active J. polar H., traction Schwarz, forces J. to M. an &applied Manning, ERK wave M. and L. comparison A withnematics. traction forceNat. data. Phys. a, 14Migrating, 728–732 MDCK (2018). monolayers 27 polarizedensity-independent towards a free edge as rigidityvisualized transition by the orientation in biological of the Golgi tissues. apparatusNat. (red) 6.relative Saw, to T. the B. nucleuset al. (green)Topological (see ref. defects). b, Schematic in epithelia govern cell of our mechanochemical model extended to include cell polarization and active migration. c, If polarity follows gradients of stresses, monolayers Phys. 11, 1074–1079 (2015). death and extrusion. Nature 544, 212–216 (2017). can exploit phase differences (ϕ) between ERK and stress to produce net polarization from symmetric waves. d, Quantitative phase diagram of the average3. Farhadifar, cell polarity in R., response Roper,¨ to J.-C., applied Aigouy, ERK waves, B., showing Eaton, an S. optimal & Julicher,¨ period and wavelength7. Copenhagen, for inducing K., maximal Alert, average R., Wingreen, polarity (positive N. S. & Shaevitz, J. W. values inF. red The indicate Influence polarity of counter Cell Mechanics, to ERK waves). Cell-Cell Data points Interactions, are previously and reported valuesTopological for wavelength defects and period promote in MDCK layer layers formation17,25,27 in Myxococcus and mouseProliferation skin26 (see onSupplementary Epithelial Section Packing. IC forCurr. details). Biol. Importantly,17, 2095–2104 the waves permissiblexanthus by our colonies. biophysicalNat. model Phys. (black17 ,curve 211–215 for varying (2021).

σE) produce(2007). only positive average polarity, thus providing robustness to the migration response.8. Marchetti, e, Oscillations M. C. inet polarity al. Hydrodynamics and traction forces of are soft active matter. Rev. weakened4. Park, for longer J.-A. ettimescales al. Unjamming of polarization and σp cellwhereas shape their in average the asthmatic (dashed line) is unaffected.Mod. Phys. f, ERK 85activity, 1143–1189 and corresponding (2013). traction forces measured in an expanding monolayer. Dashed lines indicate the leading edge of the monolayer. Kymographs show that the intercellular wave propagation present in ERK activity is absent in traction forces. g, Single-cell traces (six representative cells, colour-coded) of ERK activity show a strong oscillatory component (autocorrelation function), whereas corresponding local traction forces show a much weaker oscillatory component (as predicted in e). Shaded areas, standard deviation.

NATURE PHYSICS | www.nature.com/naturephysics 7

9. Malinverno, C. et al. Endocytic reawakening of motility in 14. Voituriez, R., Joanny, J. F. & Prost, J. Spontaneous ﬂow transi- jammed epithelia. Nat. Mater. 16, 587–596 (2017). tion in active polar gels. Europhys. Lett. 70, 404–410 (2005). 10. Vicsek, T., Czirok,´ A., Ben-Jacob, E., Cohen, I. & Shochet, O. 15. Blanch-Mercader, C. et al. Turbulent Dynamics of Epithelial Novel Type of Phase Transition in a System of Self-Driven Par- Cell Cultures. Phys. Rev. Lett. 120, 208101 (2018). ticles. Phys. Rev. Lett. 75, 1226–1229 (1995). 16. Trepat, X. et al. Physical forces during collective cell migration. 11. Serra-Picamal, X. et al. Mechanical waves during tissue expan- Nat. Phys. 5, 426–430 (2009). sion. Nat. Phys. 8, 628–634 (2012). 17. Douezan, S. et al. Spreading dynamics and wetting transition of 12. Gonzalez-Rodriguez, D., Guevorkian, K., Douezan, S. & cellular aggregates. Proc. Natl. Acad. Sci. U. S. A. 108, 7315– Brochard-Wyart, F. Soft Matter Models of Developing Tissues 7320 (2011). and Tumors. Science 338, 910–917 (2012). 18. Boocock, D., Hino, N., Ruzickova, N., Hirashima, T. & Han- 13.P erez-Gonz´ alez,´ C. et al. Active wetting of epithelial tissues. nezo, E. Theory of mechanochemical patterning and optimal Nat. Phys. 15, 79–88 (2019). migration in cell monolayers. Nat. Phys. 17, 267–274 (2021).