Mesoscale physical principles of collective cell organization

We review recent evidence showing that cell and tissue dynamics are governed by mesoscale physical principles. These principles can be understood in terms of simple state diagrams in which control variables include force, density, shape, adhesion and self-propulsion. An appropriate combination of these physical quantities gives rise to emergent phenomena such as cell jamming, topological defects and underdamped waves. Mesoscale physical properties of cell assemblies are found to precede and instruct biological functions such as cell division, extrusion, invasion and gradient sensing. These properties are related to properties of biomolecules, but cannot be predicted from biochemical principles. Thus, biological function is governed by emergent mesoscale states that can be predicted by a simple set of physical properties. The behaviour of cells and tissues can be understood in terms of emergent mesoscale states that are determined by a set of physical properties. This Review surveys experimental evidence for these states and the physics underpinning them.

M ost organs in the human body consist of sheets of cells termed epithelia 1 . During early development, epithelia are composed by a single-cell layer that adopts a simple shape such as a hollow sphere or an ellipse. As development progresses, epithelial sheets grow, differentiate, fold into complex three-dimensional (3D) shapes and secrete their fibrous microenvironmentthe extracellular matrix (ECM). Epithelial sheets exhibit complex dynamic behaviours that are not observed in single cells, neither can they be simply predicted from the observation of isolated cells [2][3][4][5][6][7][8][9] . Understanding how the properties of subcellular structures relate to the complex physical properties of biological matter presents a compelling challenge [10][11][12] . The advent of new microscopy technologies, such as super-resolution microscopy and correlative live-cell and electron microscopy, is boosting our knowledge of nanoscale structures and interactions in cells 13,14 . Yet the time and length scales at which tissue dynamics emerges remain largely unknown. Here we discuss mesoscale principles of collective cell organization. We begin by summarizing the main stress-bearing elements that form cells and tissues, and the main physical variables that determine their dynamics. We then review emergent phenomena in epithelial cell sheets and discuss theoretical frameworks that capture them in terms of simple integrative principles. Finally, we discuss how such principles might be applicable to more complex living tissues such as 3D assemblies.

Subcellular determinants of collective cell dynamics
In this section, we summarize the main subcellular structures that cells use to generate, transmit and transduce physical forces. These are dynamic structures and organelles such as cytoskeletal filaments, the nucleus, cell-cell junctions and cell-ECM junctions.
The cytoskeleton. The structure of individual cells is governed by the cytoskeleton (Fig. 1). This is a network of three main classes of dynamic polymers. Actin filaments are continuously polymerizing and depolymerizing in a manner that consumes energy in the form of ATP 15 . Individual filaments are directional with actin monomers added at one end and detaching at the other. The rate of polymerization can be regulated by many factors and can lead to rates of filament growth on the order of several micrometres per minute 16 . In most migrating cells, actin monomer addition occurs at the front of the cell and generates a pushing force on the plasma membrane, which has a membrane tension of 200-400 pN μ m −1 . In some cells, all of the force generated by polymerization is used to push the membrane forward, whereas in other cells, actin polymerization at the membrane also leads to rearward movement of the filaments (this is termed retrograde flow). A range of proteins can modulate polymerization rates and nucleate new fibres from the side of existing ones; together with filament crosslinking and bundling proteins, they enable the emergence of actin networks with complex geometries. Myosin motor proteins act as 'molecular machines' that utilize ATP to move along actin filaments and generate a few piconewtons of force per stroke 17 . The dimeric nature of the predominant myosins in epithelial cells enables them to interact with two actin filaments at the same time, thereby generating contractile force 18 . This force can be transmitted to the plasma membrane via linker proteins enabling control of cell shape and, in some cases, internal hydrostatic pressure 19 . Membrane-spanning ion pumps can change the osmotic pressure balance of cells. Changes in cell shape and pressure can combine to change membrane tension, and several molecules exhibit tensiondependent binding to membranes. The curved shape of some membrane-binding linkers confers further complexity by favouring their binding to regions of the membrane with specific curvature 20 .
From a soft-matter physics perspective, the actomyosin network can be treated as an active polar gel whose dynamics differ fundamentally from those of passive polymer gels in equilibrium 21 . Activity arises from the continuous consumption of free energy during polymerization and motor activity, which keeps the material far from thermodynamic equilibrium. Polarity arises from the asymmetric polymerization/depolymerization rates at each end of the filament. The active polar nature of the actomyosin cytoskeleton enables remarkable non-equilibrium phenomena such as oscillations and waves in the absence of inertia 22 . While F-actin networks are highly complex at the nanometre scale, their crosslinking, myosin force generation and perpetual turnover confer viscoelastic properties to the actin network 23,24 . Indeed, different experimental approaches 25 (Box 1) have successfully shown that the actin cytoskeleton behaves as a viscoelastic material at the micrometre scale. Viscoelasticity of the cytoskeleton should not be interpreted in terms of a finite number of timescales represented by a collection of springs and dashpots in series or parallel connection. Instead, extensive data now support that timescales in the cytoskeleton are broadly distributed so that the elastic and loss moduli follow a weak power law in frequency 26,27 . These moduli are influenced positively both by polymerized actin and myosin activity.

NATure Physics
Intermediate filaments are slightly larger in diameter than actin filaments, they are less dynamic and do not rely on ATP 28,29 . Their mechanical response is strongly dependent on loading rate and their extensibility is partly determined by a structural switch from α -helices to β -sheets 30 . These properties suggest that intermediate filaments protect cells from fast, large deformations 31 . Microtubules are the largest class of cytoskeletal filament. Similar to actin, they are polar filaments, but polymerization and depolymerization occur at the same filament end and their dynamics depends on GTP as an energy source 32 . Microtubule motor proteins transport cargo around the cell, including membrane vesicles, but do not generate significant contractile force. By contrast, microtubules can bear significant compressive load and, in some cases, this leads to buckling 33 . In addition to their mechanical properties, microtubules are critical for the polarity of epithelial cells-that is, the asymmetric delivery of cellular components to different of regions of the plasma membrane; either apical or basal in the non-migratory epithelial cells or front or rear in migrating cells. Stable cell polarity depends on motor proteins moving in a directed manner on microtubules delivering or receiving distinct 'cargos' from different regions of the cell 34 . Thus, asymmetry in the plasma membrane is linked to underlying asymmetry in the cytoskeleton. Cytoskeletal state and the movement of membranes is largely controlled by two classes of molecular 'switches' , namely RHO-GTPases and RAB-GTPases 35,36 .
In addition to the cytoskeleton, the physical properties of individual cells are significantly influenced by the nucleus. This organelle is densely packed with DNA wrapped around histone protein complexes and encased by an envelope of intermediate filament proteins called lamins and a bilayered membrane 29 . This nuclear envelope is connected to the cytoskeleton via LINC protein complexes. Lamins play a significant role in determining nuclear stiffness, which can vary in the 0.1-10 kPa range 37,38 . Lamins act to protect the DNA from physical damage, but because the nucleus is the largest organelle in the cell they can also influence the overall cellular stiffness and the ability of cells to move through small gaps 39 .
Interactions with neighbours and substrates. The elements described above are important determinants of the internal physical properties of the cell. However, the behaviour of cells in the context of a tissue depends on their interaction with other cells and the extracellular environment. A multitude of plasma-membrane-spanning molecules can mediate adhesion between cells. Cadherins are critical for epithelial cell cohesion through the formation of adherens junctions. In these junctions, cadherins are coupled to the actin cytoskeleton enabling actomyosin forces to be transmitted between cells 40,41 . This linkage to a contractile network means that junctions are typically under tension, with an individual cadherin junction withstanding 20-50 pN before breaking 42 . Clustering of cadherins can increase the force needed to pull two cells apart into the range of nanonewtons 43 . The linkage of cadherins to the actin network is finely regulated by both internal and external mechanisms. If force is applied to cadherin-mediated junctions, they can enhance coupling to the cytoskeleton and locally stiffen it 44 . Tight junctions perform a barrier function and enable the transport of ions across epithelial layers to be actively regulated. This can play an important role in the control of fluid pressures in tissues. Desmosomes are another class of cell-cell junction. They are coupled with intermediate filaments and the resulting supracellular network confers mechanical resilience on cell layers 45 . Together, adherens junctions, desmosomes and tight junctions are the major mediators of epithelial cell-cell adhesion and their regulation enables emergent behaviours in cell sheets that are not observed in single-cell systems.
Most cells also make contact with the supporting matrix that confers long-lasting form to tissues. This matrix consists of a variety of protein polymers, including collagens and laminins. Like many biological polymers, they exhibit strain-stiffening behaviour and can be extensively crosslinked 46 . The latter process typically increases the bulk modulus and results in elastic behaviour. A subset of matrix proteins are heavily glycosylated and their hydration can provide a gel-like property to the ECM. A large variety of cell surface proteins interact with the matrix and these interactions are critical for both tissue structure and cell migration. The integrin family of heterodimers are central to cell-matrix interactions and are capable of resisting forces in the range 20-100 pN, although most engaged integrins in a cell are subjected to forces < 7 pN 47 . The affinity of the heterodimer can be regulated from the inside of the cell, and, conversely, the engagement of external matrix ligands can alter the intercellular conformation of integrins, leading to changes in cell signalling. Further, integrins frequently cluster into micrometrescale assemblies called focal adhesions that are capable of exerting forces around 25 nN 48 . The intracellular part of integrins is coupled to the actin cytoskeleton via a range of adaptor molecules 49 . This linkage is important for actin polymerization to exert a pushing force on the plasma membrane. The coupling of actin filaments to integrin-mediated adhesions is not constant. Several of the adaptor proteins that link actin and integrins undergo force-induced conformation changes that alter their binding properties. This leads to two important emergent features. First, there is a clutch between actin and integrins that modulates how effectively actin polymerization pushes the cell membrane. Second, when force is applied on integrins, it is transmitted through α -helical domains to the interior of the cell, where it can change the conformation of proteins bound to integrins. For example, talin and p130Cas undergo conformational changes when subjected to forces in the low piconewton range 50 , which influences their ability to interact with other protein partners

Emergent mechanics of cell monolayers
Despite the broad diversity of subcellular components described above, collective cell dynamics is ultimately determined by a limited set of key mechanical properties. In the spirit of soft-matter physics, current efforts attempt to identify these properties and provide a coarse-grained description of the laws that govern collective cell dynamics (Box 2 and Fig. 2). The focus of this type of approach is not so much to identify how one specific molecule might affect one specific physical property, but rather to explain collective tissue dynamics in terms of state diagrams that capture the phenomenology on the basis of a small set of key variables. This set includes, but is not limited to, cell-matrix traction, intercellular stress and self-propulsion.

Key mechanical variables.
A traction is a force per unit area applied at any surface of a cell 52 . In the context of cell mechanics, the traction vector is understood as the force per unit area applied by the cell on its surrounding inert microenvironment. In general, cells exert traction by pulling on focal adhesions using their actomyosin machinery, but cells lacking integrins and even passive liquid drops are able to exert significant traction 53,54 . Several approaches have been described to quantify cell tractions in two and three dimensions by measuring deformations of materials surrounding cells 55 (Box 1). The combination of force measurements with imaging has revealed that the generation of traction is a highly dynamic process in which the force generated by myosin causes continuous binding/unbinding events of integrins and accessory proteins. As a consequence, actin near adhesion sites is usually seen to move retrogradely towards the centre of the cell 56,57 .
Tractions at cell-cell interfaces are usually not called tractions but adhesion forces. In the simple case of a suspended cell doublet in which the cytoskeleton localizes mainly at the cell surface-called the cell cortex-adhesion forces are parallel to the cell-cell interface 58 . In general, force at any point of a cell-cell junction has a component parallel to the cell-cell interface (shear component) and one normal to it (normal component) 59 . Like cell-matrix traction sites, cell-cell adhesion sites are highly dynamic and adhesion proteins undergo frequent turnover.
If the cell is studied at a scale larger than the cytoskeletal mesh, then it can be treated as a continuum (Box 2). At any point of this continuum, the mechanical state of the cell is fully captured by the stress tensor, which defines the force per unit area applied on any surface centred at any point of a tissue. In general, stress has a normal component and Box 1 | Methods for measuring cell mechanics 1. Atomic force microscopy involves placing a cantilever of known deformability against a cell or other material. Known forces are applied and the deflection of the cantilever is measured, usually by reflecting light on it 27 . 2. Bead displacement assays typically involve coating a bead with a biologically adhesive molecule and then placing it against a cell 26 . Depending on the type of bead, force can be applied by the use of magnetic fields or optical tweezers. A modification of the magnetic method allows repeated twisting of the bead, termed magnetic twisting cytometry 124 . This method allows for repeated interrogation of the cell's properties, enabling one to determine whether the cell changes its mechanical properties in response to a mechanical challenge. 3. Insert deformation assays involve the introduction of inserts (typically soft polymer gel spheres or lipid droplets) of known deformability into either a cell or a multicellular tissue 125 . Forces within the cell or tissue can be interpreted from the insert deformation, especially if they are anisotropic and lead to a loss of sphericity. 4. Laser cutting assays involve the use of a high-power focused laser to 'cut' a cellular structure or tissue. The recoil of the surrounding tissue can be used to infer the tension that the cut region was under 126,127 . 5. Time-lapse microscopy involves repeated imaging of the cells to generate a movie of their motion. The movie can be analysed by tracking individual cells, although at high cell density particle imaging velocimetry is often used because the reliable identification and tracking of individual cells may not be possible. Genetically encoded fluorophores are used to label either specific cells or specific proteins within cells. 6. Traction force microscopy involves growing cells on an elastic substrate of known deformability. The spatial deformation of the substrate by the cell is measured with a microscope (usually with the aid of fiducial fluorescent beads embedded in the substrate). Forces can be inferred by comparing the deformed and relaxed states of the substrate 52 .
7. Microplate assays involve plating cells on a deformable substrate that is attached between one fixed plate and another that can be moved. This enables uniaxial compression or stretching to be performed. The response of the cell is typically studied by time-lapse microscopy 31,128 . 8. Wound healing assays rely on damaging an epithelial sheet using a laser or direct mechanical disruption, and the response is frequently monitored using a combination of the techniques listed above, including time-lapse microscopy to obtain dynamics, laser cutting, traction force microscopy, and microplate assays. They are particularly useful because, in conditions of tissue homeostasis, there is relatively little motion of epithelial cells except for that resulting from the birth of new cells or cell death. Perturbation of the homeostatic state therefore reveals regulatory mechanisms 129 . Besides the ability to generate active stresses, a key feature that differentiates tissues from passive soft materials is the self-propulsive ability of each individual cell. Cells achieve self-propulsion by extending distinct types of protrusion and adhering them to their surroundings. The two main 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.

Box 2 | Mechanobiology as a multiscale problem
One the great challenges of mechanobiology is the existence of multiple length scales at which relevant mechanisms are at play. This problem has led to the development of diverse types of modelling approach describing biological tissues at different scales. Examples of mechanical models of cells and tissues include the following. 1. Clutch models describe the dynamics of cell-matrix adhesion at the molecular level (a,e in figure below). They take into account the rigidity of the matrix, the binding/unbinding (k on , k off ) rates of adhesion proteins and the force that myosin motors exert on these proteins through the actin cytoskeleton. Clutch models have been successful at predicting traction forces and dynamics of the leading edge of a protruding cell, sensing of rigidity and matrix ligand spacing, and durotaxis 4,56,57 . These models capture fundamental rheological properties of cells such as strain-stiffening, scale-free elastic modulus, fluidization and ageing 130,131 . 3. Vertex models treat cell monolayers as 2D or 3D polygonal objects that share common vertices (see also Box 3) (c,g in figure below). In three dimensions, these models are governed by tissue geometry and by surface tensions of the apico-basal (γ ab ) and lateral surfaces (γ l ). Vertex models have predicted dynamic aspects of monolayers, such as their ability to deform through intercalation, to jam at constant density and to flock as solids and liquids 82,83,86,87,132 . 4. Continuum models describe tissues as active materials in which the cellular structure is coarse-grained (d,f in figure below). These models are formulated in terms of continuum vectorial and tensorial fields such as velocity, v, polarity p and stress σ. These variables are linked through viscosity η, elasticity k and friction ζ. In three dimensions, they also incorporate luminal pressure ∆ P. Continuum models have successfully described complex features of cell monolayers such as wave propagation in the absence of inertia 76,133 .

NATure Physics
Even at the level of the single isolated cell, the relationship between forces exerted by cells and their velocity is not straightforward. For example, a cell crawling on a flat substrate generates tractions that are orders of magnitude smaller than the force needed to propel their body through the surrounding viscous fluid 62 . Traction forces are thus not generated to achieve the function of migration, but rather to actively probe and adhere to the microenvironment. They may also be crucial for leukocytes to exit the blood and home to areas of tissue damage in the face of high shear forces exerted by blood flow 63 . As discussed above, mechanical sensing of the substrate depends on force-driven conformational changes in proteins within cell-matrix adhesion complexes 64 . The substrate stiffness threshold at which these changes occur is tuned by the binding rates of integrins to the underlying substrate 65 . Nonetheless, traction and migration are not independent because the flow of actin in a protrusion is inversely proportional to the traction generated by that protrusion 56 .
Collective cell dynamics. Despite being genetically identical, cells in monolayers exhibit heterogeneous properties in terms of shape, adhesion and dynamics. For example, cells close to a free edge are usually large, flat, protrusive and able to generate large traction forces 66 . Conversely, cells behind the edge tend to be cuboidal and to generate low traction forces through cryptic lamellipodia protruded underneath their neighbours 67 (Fig. 1). Within these two categories, there exist large variations as well. Some cells at the leading edge are much larger, protrusive and motile than their immediate neighbours, which has led to the idea that collective migration of monolayers is driven by 'leader' cells 68 . In the bulk of the monolayer, some cells also have distinctive dynamic signatures such as the ability to rotate in swirls 69 . The dynamics of these swirls is determined by intrinsic and extrinsic factors such as cell adhesion, division and confinement 70 .
The renewed interest in cell monolayers-an experimental system that dates back more than one century-originates from the development of new technologies to measure not only velocity and deformation fields but also tractions and intercellular stresses (Box 1) 25 . Force mapping has unveiled phenomenological principles of cell organization such as the alignment of the cell body with the direction of maximum stress 5,71,72 . This phenomenon, called plithotaxis, implies that cells organize in sheets so as to minimize intercellular shear stress. Plithotaxis provides a mechanim for cells to migrate collectively in a preferred mechanical direction during wound healing and cancer invasion 73 .
The organization of cells in sheets and clusters has also been shown to enable collective sensing of both chemical and mechanical gradients 4,74 (Fig. 3). When cells are prevented from transmitting forces by disrupting cell-cell junctions or by inhibiting myosin

Box 3 | Vertex models
A variety of models have recently been implemented to explain the rich phenomenology of cell monolayers 79,[121][122][123]136 . Among such models, vertex models treat the cell monolayer as a closepacked mosaic of polygonal objects that represent constituent cells 82,83 . These models study the dynamic evolution of the 2D network formed by cell vertices and by connections between them. The position x k of each vertex k is determined by the equation of motion where F k is the force acting on the vertex and 1/μ is the friction between the cell and its substrate. F k may be specified explicitly as a sum of contractile, adhesive and osmotic contributions, or obtained implicitly by computing the gradient of an energy function of the monolayer. A common approach to compute such energy function is to express the energy of a given cell i in terms of its area A i and its perimeter P i : The first term represents an area elasticity where A i0 is a preferred cell area and K A i is an area elastic modulus. This contribution to the cell energy can be understood in terms of cell incompressibility and resistance to height fluctuations. The second term represents a perimeter elasticity, with a preferred perimeter P i0 and a perimeter elastic modulus K A i . Expansion of this second terms yields a quadratic term in P i , which describes the contractility of the cell cortex, and a linear term in P i , which reflects a competition between effective cell-cell adhesion and cortical tension. In this interpretation, cell-cell adhesion tends to increase the length of the edges, whereas cortical tension tends to shorten them.
To incorporate the self-propulsion, the formulation of vertex models has been combined with terms derived from particle models to give rise to self-propelled Voronoi (SPV) models 86,87 . Rather than focusing on the dynamics of cell vertices, SPV models describe the motion of the centre of each cell obtained by Voronoi tesselation. The energy function describing the interaction between cells is the same as in vertex models but the motion of the cell centre is determined not only by the gradient of the energy function but also by a self-propulsion term v 0 n k . The equation of motion then reads: where n k is a unit vector indicating the polarity of the cell and v 0 is a constant. In the simplest approximation, the polarization dynamics is governed by random rotational diffusion independent of the velocity of the cell and, therefore, of the movement of its neighbours 88 . However, dynamics of monolayers is better captured by imposing an active feedback mechanism inspired by flocking models 137 that tends to align cell polarity with cell velocity. Given that cell velocity is in part due to the force exerted by the neighbours, this feedback mechanism effectively couples the movement of adjacent cells with an alignment rate J (refs 86,87 ). motors, collective gradient sensing is often impaired. Another striking emergent phenomenon in cell monolayers is their ability to propagate mechanical waves. In response to sudden unconfinement, the first row of cells at the monolayer edge spreads and migrates towards the freely available substrate, whereas the cells behind them remain static 66,75 . With time, every cell row becomes progressively engaged in collective motion following a wave of deformation and force generation. Propagation of mechanical waves has now been observed in confined clusters 76,77 , in expanding colonies 66,75 and in colliding monolayers 78 (Fig. 3). Propagation of mechanical waves in inertial matter is a trivial physical phenomenon that can be simply explained by an exchange of potential and kinetic energy. In cell monolayers, which lack inertia, alternative mechanisms must be at play to introduce second derivatives with respect to time and provide an effective inertia 76 . This could be achieved through the interplay between cell mechanics and molecular circuits involving mechanotransduction, such as those involving the mechanosensing protein merlin, which links forces at cell-cell junctions with the regulation of the protrusive and forcegenerating action of RHO-GTPases 6,79 .
A general feature of cell monolayers-and possibly one that is intuitive to any cell biologist who has ever performed cell cultureis that cells growing on a dish slow down their motion as the cell density increases 80 . From a physical perspective, this behaviour is reminiscent of that of granular materials such as sand or coffee beans close to a jamming transition; as the system density increases, each constitutive element becomes trapped by its neighbours, the energy required for structural rearrangements rises, and the system transitions from a fluid to a disordered solid 81 . Careful analysis of cell velocity fields in proliferating cell monolayers showed that the analogy between the behaviour of cell monolayers and granular materials close to a jamming transition is deeper than expected. Like particles in granular materials, cells in dense monolayers move in large groups whose length scale grows with cell density 5 . Moreover, cells in a monolayer exhibit dynamic heterogeneities in cell migration, a non-Arrhenius dependence of relaxation times on cell density, peaks in the vibrational density of states, and a shift in the position of the four-point susceptibility function 80 .
A perhaps less intuitive observation is that fluid-to-solid transitions in cell monolayers can also occur at constant density. This striking result was first predicted by theoretical analysis of vertex models 82 (Box 3). These models aim at capturing the dynamics of tissues in terms of the motion of vertices representing junctions between three or more neighbouring cells 83 . A key ingredient behind the success of vertex models is that they readily enable neighbour exchanges. In the absence of mitosis and extrusion, neighbour exchanges in monolayers occur through T1 transitions, whereby one edge between two cells shrinks until vanishing and a new edge is created at the same point between two different cells. Bi et al. 82 identified that the energy distribution of T1 transitions in a vertex model is a function of a dimensionless geometric factor called the target shape index = ∕ p P A 0 0 0 , where P 0 and A 0 are preferred perimeter and area, respectively. For regular polygons, p 0 increases with the number of sides, with p 0 = 3.72 for a regular hexagon and p 0 = 3.81 for a regular pentagon. More strikingly, they found that below a critical value p 0 = 3.81, energy barriers are finite and the system behaves like a jammed solid. By contrast, above p 0 = 3.81, energy barriers become vanishingly small and the monolayer behaves like an unjammed liquid. This result, which has now been extended to 3D bulk tissues 84 , suggested that cell monolayers exhibit a jamming transition controlled by a geometric parameter independent of tissue density. This prediction was successfully tested using bronchial epithelial cells maturing in an air/liquid interface 85 . As the maturation time increased, p 0 decreased without a noticeable change in the cell density, and the monolayer dynamics slowed down. Remarkably, bronchial epithelial cells from asthmatic donors exhibited higher values of p 0 and remained unjammed for a longer time than cells obtained from healthy donors.
The ability of early 2D vertex models to precisely capture the jamming transition in cell monolayers is remarkable taking into account that they did not include the ability of cells to exert traction forces on their substrate or to self-propel their body. One approach to include self-propulsion in the description of 2D monolayers was the development of SPV models (Box 3) [86][87][88] . These models predict four relevant states in monolayers defined in terms of the shape index p 0 and the characteristic time of alignment between neighbours J (Fig. 3). These states are solid, liquid, solid flock and liquid flock 86,87 . The solid (or jammed) phase emerges at low p 0 and low J, and it is characterized by the absence of cell rearrangements and of global tissue movement. If p 0 is increased at low values of J, the monolayer is predicted to enter a liquid phase in which the spatial correlation length remains low but local rearrangements become frequent. These two phases are analogous to those predicted by 2D vertex models in the absence of self-propulsion or with self-propulsion but in the absence of alignment [86][87][88] . At higher values of J, the SPV model predicts the emergence of flocking (that is, an increase in the spatial correlation In the flocking solid state, the monolayer flows cohesively without neighbour exchanges. By contrast, in the flocking liquid state, the monolayer combines long-range correlation with frequent local rearrangements (Fig. 3). Experimental observations of transitions between solid (jammed) and liquid phases have been reported in maturing bronchial epithelial cells at the air/liquid interface 85 . A transition between non-flocking and flocking liquid was recently observed in response to overexpression of the endocytic GTPase RAB5A 86 . These examples illustrate how differentiation processes or expression changes in one single gene can be rationalized in terms of transitions in a physical phase diagram. Dynamics of cell monolayers has also been interpreted in terms of active nematics 2,3,9 . Like molecules in liquid crystals, cells forming a dense monolayer are polarized and often exhibit an elongated shape. In addition, they are able to self-propel their body, which gives rise to a range of phenomena inaccessible to inert liquid crystals, such as long-range order, giant fluctuations, unattenuated wave propagation and turbulence at low Reynolds numbers 89 . A common feature of liquid crystals is the presence of singular points in the orientational field, known as topological defects. In contrast with passive systems, topological defects in active systems can form and annihilate spontaneously. Dynamic topological defects have been reported in cell monolayers of elongated shape, such as fibroblasts 2 , but also in cell types that are not elongated but have a polarized cytoskeleton such as MDCK cells. In the latter case, cells located at topological defects were shown to display a higher probability to be extruded from the monolayer 3 . The origin of extrusion was proposed to be the compressive stress experienced by cells as a consequence of the topological configuration at the defect site. This phenomenon illustrates how the collective dynamics of a monolayer can precede and instruct biological function.

Scaling up to 3D assemblies
While some biological tissues can be considered essentially as 2D structures, with only low levels of curvature relative to the scale of the cell (for example, the skin excluding hair follicles), the majority of tissues have complex 3D shapes. How the behaviours articulated above in 2D systems determine 3D architecture is poorly understood, but much effort is being directed at this problem. The simplest experimental models involve spheroids of cells that can be either solid or hollow. We will discuss both of these before concluding with remarks on the higher levels of topological complexity found in some tissues.
Solid spheroids. The simplest 3D form that a group of cells can adopt is a sphere, but even this form has additional complexity compared to simple monolayers. The distance of a cell from the sphere's surface affects its access to nutrients and oxygen. This imposes a biochemical gradient not observed in 2D systems and, in many cases, can limit the size to which the sphere can grow 90,91 . Cell movement within a spheroid could mitigate against this by enabling cells to move in and out of nutrient/oxygen-limited areas, but this has not been explored in any detail. Indeed, much less is known about the migratory behaviour of cells in spheroids. Unlike 2D systems where all cells have contact with the underlying matrix substrate, cells in the interior of the spheroid contact only other cells or the small amounts of matrix proteins on the surface of other cells. Consistent with this, both adherens junctions and desmosomal connections have been shown to play a key role in cells migrating in 3D multicellular structures 92 . Frictional mechanisms of traction are also likely to play a more significant role in migration in confined environments. In such contexts, frictional traction stresses have been measured in the pascal range 93 . However, frictional mechanisms can, in principle, generate significantly higher stresses. The movement of one cell requires the displacement of its neighbours, and the strength of cell-cell adhesion, shape and deformability of the surrounding cells all influence the ability of a cell to move and its coordination with neighbours 94,95 . Strong cell-cell adhesion and low deformability typically lead to low levels of movement within spheroids or jamming. The transitions from jammed to un-jammed states in three dimensions are less well understood than in 2D systems, but the same principles are likely to apply with both a shape factor p 0 and alignment rate J being critical. The outer layer of cells does contact the matrix that supports the spheroid and can exert traction forces in a similar manner to 2D cell systems. Experimental observation reveals that in some cases there is sufficient coordination to generate rotational behaviours 96 . The mechanisms underlying the emergence of these behaviours remain incompletely understood, but the analysis of the flocking liquid and solid states in 2D systems is likely to be relevant. Rotational behaviour is not just a curiosity of simple 3D cell systems but is linked to the development of more complex shapes. In Drosophila, the egg chamber is initially spherical, but the rotational movement of the outer cell layer influences the underlying matrix structure and is required for the prolate shape of the egg to develop 97,98 .
Pressure can also have an important influence on cell behaviour in 3D systems. Unless spheroids are growing in aqueous suspension, in which case the fluid pressure is negligible, the matrix that supports the spheroid will exhibit some resistance to deformation. The conversion of soluble nutrients to biomass as a result of cell growth then requires some deformation of the surrounding matrix to accommodate the change in volume. This is not uniform throughout the spheroid, but is greater in the centre and can feedback on the growth rate of cells 99 . This can ultimately limit the growth of spheroids or require biological remodelling of the surrounding matrix to generate space for cell growth. Pressure in spheroids and tumours has been estimated in the range 1-40 kPa (refs 100,101 ).
Hollow spheroids. Many epithelial cells have the ability to form fluid-filled spheres and more complex branched structures. Key to this behaviour is cell-cell adhesions mediated by cadherins and cell polarity that is stabilized by microtubules. Cells in the nascent sphere in contact with the supporting matrix receive signals that polarize the cytoskeleton. Microtubules and RAB-GTPases coordinate the delivery of specialized membrane to the part of the cell away from the basal surface 102 . This generates a specialized apical membrane in the interior, and the action of ion pumps and the death of cells not in contact with the substrate leads to a hollow sphere. Cells in a hollow sphere can exhibit jammed or unjammed behaviours similar to 2D systems. Cell-cell junctions ensure that fluid flow between cells in an epithelium is tightly regulated. Ion pumps can regulate the osmotic pressure of the fluid within a spheroid and thereby modulate its rate of expansion 103 . Consideration of the bulk tissue properties therefore requires one to take into account the inter-luminal or interstitial pressure. In developed organs with a lumen, such as the mammary gland, the pressure can exceed 10 4 Pa 104 , whereas in vitro estimations in monolayered semi-spheres are on the order of 100 Pa 105 . Luminal pressure (Δ P in Fig. 2h) depends on coordinated cell polarization to ensure that solutes are transported in a consistent direction by all of the cells in the structure, and it is linked to interfacial tension and epithelial curvature through Laplace's law. In line with this idea, numerous studies have shown that the loss of key cell polarity genes, many of which are tumour-suppressor genes, prevents the formation of ordered hollow spheroids. Loss of polarity therefore contributes causally to the disordered nature of tumours and the recurrent observation that tumour cells form solid, not hollow, spheroids. distinctive properties, such as altered cell shape. Even relatively few elongated cells, such as fibroblasts, are predicted to perturb jamming. Cells with different cell-cell adhesion properties may also sort themselves into distinct regions. Simple models predict that cells with low cell-cell adhesion are preferentially sorted to the exterior of spheroids 106 . However, more detailed analysis suggests that this mechanism may not be sufficient and different tension at the boundaries between cell types and their jamming state may also play a role 58,107,108 . Experimental studies have confirmed the importance of myosin for the formation of sharp tissue boundaries 109 . Differences in substrate traction forces and actin protrusion dynamics will also add complexity. Highly protrusive fibroblasts with strong traction forces are better able to move into the matrix that surrounds spheroids, thus acting as leader cells 110 . These lead to the emergence of cell strands leaving the spheroid and generate complex 'stellate' 3D shapes 111 . Leaders and followers impact each other's function continuously 112 ; leaders pull on followers and open gaps in the ECM to enable collective invasion 111 , whereas followers contribute to polarize leaders and to steer their migration 111,113 .
The majority of epithelial tissues are not spherical but highly folded as this benefits fitting a large surface area (functionality) into a small volume. Our understanding of the emergence of the geometries found in tissues such as the lungs, intestine, brain or mammary gland is still in its infancy; however, there are two areas that are being intensively studied at the moment. The first is tissue folding. Physically, tissue folding can be understood as a mechanical instability originating from excess compressive stress in a specific layer of a tissue 114 . One possible mechanism for this process involves differential contraction or expansion of apical and basal membranes in 2D systems leading to curvature. Two-dimensional vertex modelling is being adapted to 3D geometries to provide a theoretical framework for this process (Box 2). The tractability of these models is appealing and they are likely to be relevant when folding mechanisms lead to the generation of hollow tubes 7,8 . However, the emergence of complex tissues typically happens early in embryogenesis when the small scale of the organism means that curvature is typically high. Thus, while these models could be powerful for explaining some of the early folding events in embryogenesis (when the surface curvature of the whole embryo is relatively low), they may be less suited to explaining how a complex tissue arises from a very small number of progenitor cells. Insights into how a complex structure emerges from a low number of cells in a highly curved structure are coming from organoid cultures 115 . In these systems, a small number of progenitor cells initially form a hollow spheroid and this then becomes branched or 'multi-lobed' . Non-uniformity in the ability of cells to expand or proliferate could lead to the preferential expansion of some regions of the spheroid that then become the nascent branches. Differential levels of proliferation and differentiation of specific tissue layers with respect to their surrounding has also been shown to trigger mechanical instabilities that give rise to the characteristic folded structures of diverse tissues such as brain 116 , lung and gut 117,118 . High pressure of the luminal fluid could further assist expansion and branching 103 . The origin and biophysical nature of the non-uniformity is not clear; it could relate to regions of jamming or increased contractility. Consistent with the latter idea, the growing ends of branches in mammary organoids are associated with a lack of contractile myoepithelial cells 119 . Folding and patterning associated with the initiation of hair follicles arise from compressive stresses caused by contraction of the epidermis by the dermis 120 . Imaging has also revealed that mammary cells in expanding branching regions exchange position very frequently and are not jammed. Further analysis has revealed that stochastic branching events followed by linear branch growth, and branch termination in the event of collision can explain the complex higher-order patterns of branching observed in adult tissue collision 115 .

concluding remarks
Understanding the transition from a single-cell fertilized egg into the complex 3D forms of multicellular organisms remains one of the great problems in all science. Principles from soft-matter physics such as cell jamming and active nematics have yielded many new insights into the generation of organized multicellular structures. The rich phenomenology of cohesive tissues is increasingly well captured by theoretical frameworks such as those based on vertex models 82,83 , particle models 121,122 or Potts models 123 . Moreover, several new approaches now enable the accurate quantification of the key physical variables that govern tissue dynamics. In relatively simple tissues such as cell monolayers, we are currently at a point where theory and experiments can be confronted and feedback on each other closely. Despite these exciting advances, important challenges remain. Further work is needed to understand how the molecular structure and dynamics of biomolecules confers distinct physical properties on cell-cell and cell-matrix interactions. It will also be crucial to enhance our understanding of 3D systems and the feedback between cell-driven changes in the ECM, cell adhesions and cell state. To achieve this goal, models will need to incorporate tissue-matrix interactions. Finally, new tools are needed to probe tissue mechanics in systems of increasing complexity such as embryos or regenerating tissues.
Physicists and biologists have long collaborated to develop quantitative tools to probe living systems, but they have also regarded each other with skepticism. Physicists have traditionally considered that living systems are far too messy to be understood in terms of a simple set of unifying laws. Biologists, by contrast, have considered that physical approaches are far too abstract to capture cells and tissues in the broadest diversity and specificity. These views are rapidly changing. Physicists now understand that there is new physics in living matter, and that there is no better laboratory than the living cell to study the physics of systems far from thermodynamic equilibrium. Conversely, biologists are increasingly regarding physics not only as a toolbox to extract much-needed quantitative information, but also as a source of new conceptual frameworks that explain and predict the behaviour of living systems at multiple length scales. Physics and biology are thus merging into a unified discipline to provide an integrative understanding of active living matter that captures its diversity and specificity as well as its fundamental laws.