Physical Models of Collective Cell Migration PDF

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CO11CH05_Alert ARjats.cls February 13, 2020 11:16 Annual Review of Condensed Matter Physics Physical Models of Collective Cell Migration Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org Ricard Alert1,2 and Xavier Trepat3,4,5,6 1 Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA; email: [email protected] 2 Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA 3 Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST), Barcelona 08028, Spain; email: [email protected] 4 Facultat de Medicina, University of Barcelona, Barcelona 08036, Spain 5 Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona 08028, Spain 6 Centro de Investigación Biomédica en Red en Bioingeniería, Biomateriales y Nanomedicina, Barcelona 08028, Spain Annu. Rev. Condens. Matter Phys. 2020. 11:77–101 Keywords First published as a Review in Advance on tissue biophysics, cellular Potts models, phase-field models, active network September 27, 2019 models, particle models, continuum models The Annual Review of Condensed Matter Physics is online at conmatphys.annualreviews.org Abstract https://doi.org/10.1146/annurev-conmatphys- Collective cell migration is a key driver of embryonic development, wound 031218-013516 healing, and some types of cancer invasion. Here, we provide a physical per- Copyright © 2020 by Annual Reviews. spective of the mechanisms underlying collective cell migration. We begin All rights reserved with a catalog of the cell–cell and cell–substrate interactions that govern cell migration, which we classify into positional and orientational interactions. We then review the physical models that have been developed to explain how these interactions give rise to collective cellular movement. These mod- els span the subcellular to the supracellular scales, and they include lattice models, phase-field models, active network models, particle models, and con- tinuum models. For each type of model, we discuss its formulation, its limi- tations, and the main emergent phenomena that it has successfully explained. These phenomena include flocking and fluid–solid transitions, as well as wetting, fingering, and mechanical waves in spreading epithelial monolay- ers. We close by outlining remaining challenges and future directions in the physics of collective cell migration. 77 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 1. INTRODUCTION It has long been recognized that cells move as collectives during development, regeneration, and wound healing. Reports from the late nineteenth century already agreed that these processes in- volve collective movements of cells but mechanisms remained controversial (1–4). Some authors proposed that cell movements were driven by pressure, either preexisting in the tissue or generated de novo by cell division (see Reference 3, and references therein). Others claimed that cells would move by spreading their volume to occupy the largest possible surface (1). Still others defended that cell sheets advanced by the active pulling force generated by leader cells at the tissue margin (2). In those early days of cell biology, proposed mechanisms were largely physical in origin but not even the sign of the tissue stress, i.e., tension versus compression, was agreed upon. Later, the discovery of genes and proteins shifted the attention from the whole to the parts, and the search Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org for a global physical understanding of collective migration was largely abandoned. This trend has been reversed in the last decade due to groundbreaking technical (5) and con- ceptual (6–8) advances together with a progressive questioning of reductionist approaches (9). Time-lapse imaging and fluorescence microscopy have become standard tools in life-science lab- oratories, and technologies such as particle imaging velocimetry (borrowed from fluid mechanics) now enable a detailed mapping of velocity fields and strain tensors in the tissue (10, 11). Further- more, a range of new technologies such as traction microscopy have enabled the direct mapping of the forces that cells exert on their surroundings as they migrate (12, 13). All mechanical variables relevant to the problem of collective cell migration have thus become available in time and space (Figure 1). This technological revolution has coincided with the development of the theory of active matter (6–8), which provides an ideal framework to rationalize the collective movement of cells. The traditional view that physics should serve to illuminate biological function is shifting toward the idea that biological systems inspire new physical theories and allow us to test them; the concept of “physics for biology” is now paralleled by the emergent notion of “biology for physics.” In this sense, from the perspective of condensed matter physics, collective cell migration is interesting as a prominent example of the emergence of collective mechanical phenomena in a system of soft active entities with complex interactions. Finally, life scientists have recognized that collective cell migration is not only key to development, regeneration, and wound healing but also to devastating diseases such as cancer (14). The coincidence in time of these different technological and conceptual advances has placed collective cell migration back at the center of research at the interface between life and physical sciences. Collective cell migration comes in different flavors depending on the biological tissue and pro- cess (16). During epithelial morphogenesis, wound healing, and regeneration, cells generally move as sheets adhered on an inert hydrogel called the extracellular matrix (ECM). In some forms of cancer invasion, cells also invade as sheets at the interface between tissues. In general, however, both in development and in tumor invasion, cells invade as strands or clusters within a complex three-dimensional environment composed of ECM and different cell types (16–18). Despite re- cent advances (19), we remain far from accessing physical forces in three dimensions (3D), so we focus this review on cell sheets migrating in two dimensions (2D; Figure 1). In this mode of mi- gration, central to epithelial function, all relevant cellular forces have been accurately mapped in vitro and can therefore be used to test physical models. For other perspectives on this subject, the reader is referred to excellent recent reviews (20–22). 2. FORCES AND INTERACTIONS OF MIGRATING CELLS In this section, we propose a classification of the forces and interactions that a migrating cell exerts on and experiences from other cells and the substrate. In addition to distinguishing cell–cell and 78 Alert Trepat CO11CH05_Alert ARjats.cls February 13, 2020 11:16 a i T┴ (Pa) i 200 100 0 ii –100 ii –200 Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org 20 µm 20 µm b σm ax in σm Veloc it y Figure 1 Mechanical measurements during collective migration of cell monolayers. (a) Vectorial representation of traction forces in LifeAct-GFP MDCK cells closing a wound. Color coding indicates the value of the radial component, with positive forces pointing away from the wound. For clarity, values between 100 and −100 Pa were not plotted. Panels labeled as i and ii show close-ups of the regions indicated by arrows in panel a. (b) Velocity vectors (green) and monolayer stress ellipses (red ) indicating the maximum and minimum principal stresses in an expanding colony of MDCK cells (phase contrast). Abbreviations: GFP, green fluorescent protein; MDCK, Madin–Darby canine kidney. Panel a adapted from Reference 15, and panel b adapted from Reference 11. cell–substrate interactions, we separate the interactions that directly act on a cell’s position from those that affect cell orientation. Even though any categorization may suffer from some degree of oversimplification, we think that it nevertheless provides some unifying principles over a large body of somehow fragmented literature. 2.1. Positional Cell–Substrate Interactions The field of continuum mechanics defines a traction as a force per unit area applied at any point of the surface of a body. In cell mechanics, traction is usually understood as the stress applied by a cell on its underlying inert substrate, typically a polymeric gel known as ECM. Cell–substrate traction can be interpreted as the sum of two contributions: the force that drives cell motion, which we call active traction, and the passive friction between the cell and the substrate. www.annualreviews.org Collective Cell Migration 79 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 a Side view b Top view CRL υ→ Biological structures Variables Focal adhesions Cell velocity υ→ p→ Gap junctions Cell polarity p→ Adherens cell–cell junctions υ→ Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Actomyosin cortex Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org Forces Actomyosin stress fibers Active traction p→ Contact regulation of locomotion (CRL) Cell–substrate friction Contact following of locomotion Cell–cell tension Contact inhibition of locomotion Cell–cell friction Figure 2 Forces and interactions of migrating cells. Schematic representation of a migrating epithelial monolayer. We illustrate a subset of the interactions discussed in Section 2. In particular, we sketch some of the biological structures that generate and transmit cell–substrate and cell–cell forces. We also indicate contact interactions that regulate and coordinate cell migration, as well as physical variables used to describe collective cell migration. 2.1.1. Active traction forces. Active traction forces stem from the cell’s actomyosin cytoskele- ton, where the action of myosin molecular motors on actin filaments generates contractile forces. These forces are then transmitted to the substrate through cell–substrate adhesion complexes called focal adhesions, which physically connect the cytoskeleton to the ECM (Figure 2). For traction forces to lead to cell motion, the cell must break symmetry and polarize to define a front and a rear. To do so, the cell often develops frontal actin-based protrusions such as lamellipodia and filopodia. These structures generate an inward-pointing active traction that is most prominent at the cell’s leading edge (Figure 2). The resulting force propels the cell forward in the direction of its polarity, defined by the position of the protrusion with respect to the cell’s center of mass (see the sidebar titled Cell Polarity in Tissues). Thus, in models, the force that drives cell migration is usually assumed proportional to a cell polarity variable, with a coefficient that depends on both cell–substrate adhesion and the active force–generating processes in the cytoskeleton. CELL POLARITY IN TISSUES The front–rear polarity of a cell is a morphological, dynamical, and biochemical asymmetry between the cell’s leading and trailing edges. The polarity direction of an isolated cell can be unambiguously identified from the direction of its migration, regardless of its subcellular origin. However, this may no longer be true in a tissue, where cell motion is also affected by intercellular forces. In this case, to distinguish polarity from velocity, one must identify a subcellular observable that defines cell polarity. Morphological features such as protrusions are often not apparent in cells inside tissues, which may feature cryptic lamellipodia that extend beneath neighboring cells. Thus, an alternative approach is to identify cell polarity from traction forces. This is valid as long as tractions are dominated by active forces, with negligible contributions from passive friction forces. Identifying subcellular features that appropriately account for cell polarity remains an open challenge. 80 Alert Trepat CO11CH05_Alert ARjats.cls February 13, 2020 11:16 2.1.2. Cell–substrate friction forces. Cell motion takes place at very low Reynolds numbers, which implies that inertial forces are negligible and, hence, that the resultant force on the cell’s center of mass must vanish. Indeed, the active traction applied by the cell on the substrate is bal- anced by friction forces. Friction with the surrounding fluid medium is usually negligible in front of cell–substrate friction forces, which are mediated by the attachment and detachment of proteins at focal adhesions (Figure 2). On timescales larger than the inverse binding rates, this protein- mediated friction is expected to be proportional to the velocity of the cell relative to the substrate (23). Thus, in a first approximation, cell–substrate friction is often modeled as a viscous damping force akin to Stokes’s drag, with a coefficient that reflects cell–substrate adhesion. However, the kinetics of focal adhesion proteins under force are extremely nonlinear and involve reinforcement feedbacks that can be accounted for in more detailed models of cellular friction (24). Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org 2.2. Positional Cell–Cell Interactions In addition to cell–substrate forces, cells also exert forces on neighboring cells. Here, we classify and describe different types of intercellular forces. 2.2.1. Cell–cell adhesion. A characteristic feature of epithelial cells is that they establish sta- ble cell–cell adhesions, whereas mesenchymal cells tend to form transient and weaker adhesions. Cell–cell adhesion is mediated by specific transmembrane protein complexes, which build cell– cell junctions that physically link the actomyosin cortices of the adhering cells, thereby enabling force transmission between cells (Figure 2). Cell–cell junctions endow tissues with cohesion en- ergy and surface tension, as well as with a bulk modulus that quantifies their resistance to rapid isotropic expansions, which is mainly due to cytoskeletal elasticity (22, 25). Thus, different mod- eling frameworks account for cell–cell adhesion by either an interfacial energy contribution, a short-range attraction that opposes cell–cell detachment, or a tissue bulk modulus. In addition, by enabling the transmission of active cytoskeletal tension, cell–cell adhesion is also an implicit factor in tissue-scale active stress terms. The influence of cell–cell adhesion on tissue mechanical properties does not end here. On the one hand, decreasing cell–cell adhesion leads to less elongated cell shapes that can induce a jamming transition whereby the tissue acquires a finite shear modulus, thus becoming a solid material (26, 27). Therefore, cell–cell adhesion may provide not only bulk but also shear elasticity to epithelial tissues. On the other hand, cell–cell adhesion proteins turn over, and hence cell– cell junctions are remodeled. Junction remodeling is a source of dissipation that can relax stress, possibly contributing to the long-time viscous response of fluid tissues (22, 25, 28, 29). 2.2.2. Cell–cell friction. Cell–cell adhesion also entails cell–cell friction forces when cells slide past each other (Figure 2). Similar to cell–substrate friction, cell–cell friction is based on the sliding, turnover, and attachment kinetics of cell–cell junction proteins (30–32). Usually, cell–cell friction is modeled as a shear force proportional to the relative velocity between the cells or, in tissue-level descriptions, as shear viscous stresses. 2.2.3. Cell–cell repulsion. In addition to the attractive interactions mediated by cell–cell adhe- sion, adhered cells also experience a soft repulsion from other cells. At short times, cell compres- sion is resisted by cytoskeletal elasticity (22, 25), which, in epithelial monolayers, gives rise to an area compressibility. At longer times, however, both the cytoskeleton and cell–substrate adhesions may reorganize to enable cell shape changes. If their volume is conserved, cells under compression can lose area and gain height, at least until their nuclei become tightly packed. Furthermore, cells www.annualreviews.org Collective Cell Migration 81 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 can actually change their volume by exchanging fluid with both the surrounding medium through water channels and other cells through cell–cell channels called gap junctions (33, 34) (Figure 2). Finally, under sufficient compression, epithelial cells can be extruded from a monolayer (35–37), thus enabling monolayer area reduction. The opposite process, cell insertion in a monolayer, also occurs in spreading cell aggregates (38). Altogether, this means that epithelial monolayers can change their area via dissipative processes, with an associated viscosity. 2.2.4. Active cell–cell forces. A last type of cell–cell forces are active forces generated by myosin molecular motors in the cytoskeleton and transmitted through cell–cell junctions (Figure 2). Cytoskeletal structures such as the cell cortex and the apical actin belt generate a roughly isotropic tension at the cell scale, thus giving rise to isotropic active stress at the tis- Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. sue level. However, migrating cells are polarized, and hence their cytoskeleton exhibits highly Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org anisotropic structures such as stress fibers (Figure 2). Stress fibers generate anisotropic tension that, in addition to being transmitted to the substrate as traction forces (Section 2.1.1), can also be transmitted to cell–cell junctions through the cell cortex, thus giving rise to anisotropic active stresses at the tissue scale. Given that they are similarly generated, traction forces and anisotropic cell–cell active stresses are interdependent (39). Yet, they are distinct because their respective trans- mission to the ECM and neighboring cells relies on different adhesion complexes (Figure 2). 2.3. Orientational Cell–Cell Interactions Here, we describe cell–cell interactions that affect cell polarity. These orientational interactions need not be effected by direct mechanical forces, but they can also result from biochemical regu- lation of cell migration. 2.3.1. Polarity alignment. One of the most prominent orientational interactions between cells is the tendency to align their polarities. Alignment might simply rely on the elongated and de- formable shape of migrating cells, but it may also involve biochemical regulation of cell migration. Polarity alignment is often explicitly implemented via either Vicsek-like rules or torques on cell polarity in discrete models, and via an orientational stiffness of the polarity field in continuum models. 2.3.2. Contact regulation of locomotion. Here, we propose the term contact regulation of lo- comotion (CRL) to subsume several processes whereby cells tune their migration direction upon contact interactions with other cells (Figure 2). At least three such processes have been described. First, contact inhibition of locomotion (CIL) refers to the process whereby, upon head-to-head collision, many cell types retract their lamellipodia and repolarize in a different direction, thus migrating away from cell–cell contacts (40, 41). Second, contact following of locomotion (CFL) refers to the tendency of cells to follow others upon head-to-tail contact (42, 43). Finally, in ad- dition to altering the migration direction, head-to-tail collisions have also been found to increase the persistence of cell motion, which is a tendency known as contact enhancement of locomotion (CEL) (44). In general, CRL depends strongly on the cell–cell collision angle (45, 46), thus making orientational cell–cell interactions highly anisotropic. The mechanisms underlying CRL may be diverse and, given that cells polarize in response to tension transmitted through cell–cell junctions (47–53), they could rely on mechanotransduction of cell–cell forces (54–57). Although a number of phenomenological models of CRL have been proposed, a coherent theoretical picture of CRL is lacking. 82 Alert Trepat CO11CH05_Alert ARjats.cls February 13, 2020 11:16 2.3.3. Polarity–flow alignment. Inhomogeneous tissue flows may produce shear. Similar to molecules in liquid crystals, elongated cells subject to shear should experience a torque that tends to minimize shear stress. Indeed, shear tissue flows reorient cell polarity in the fly wing (58) as well as the cell division axis in epithelial monolayers (59). Furthermore, cells in epithelial monolayers tend to migrate in the local direction of lowest shear stress, which is a behavior known as plithotaxis (60–63). However, unlike in ordinary liquid crystals, cell reorientation may not entirely stem from cell shape, but it likely involves an active mechanosensitive response. Regardless of its yet-unclear mechanism, this feedback between polarity and flow mediates orientational cell–cell interactions. Even though some recent continuum models have probed the effects of flow–polarity coupling (64, 65), further research is needed to clarify its role in collective cell migration. 2.3.4. Polarity–shape alignment. Almost by definition, cell polarity and cell shape asymmetry Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org are interdependent, and hence they are often assumed to align. Thus, given that cell–cell interac- tions modify cell shape, cell-autonomous polarity–shape alignment can give rise to intercellular alignment interactions (66–68). 2.4. Orientational Cell–Substrate Interactions Cell polarity can also be modified by substrate cues and cell–substrate forces. Here, we briefly summarize how cell-substrate interactions can alter cell polarity. 2.4.1. Polarity–velocity alignment. Through their interaction with the substrate, cells may be able to align their polarity to their velocity, thus tending to align self-propulsion with drag cell– substrate forces (69). Such a polarity–velocity coupling is a generic property of active polar systems interacting with a substrate (70–73). For these systems, the polarity reorients not only in flow gra- dients but also in uniform flows, like a weathercock in the wind. Even though its cellular mecha- nism is not yet well understood, polarity–velocity alignment has been introduced in several models. However, in some situations, polarity and velocity are strongly misaligned in epithelial monolay- ers (74, 75) (Figure 2), possibly due to the dominance of cell–cell interactions (75, 76). How such possibly conflicting polarization cues coexist and cooperate remains poorly understood (77). 2.4.2. Substrate-induced polarization. In addition to polarizing in response to cell–cell forces (Section 2.3.2), cells can also polarize in response to asymmetric forces at the cell–substrate inter- face (52, 78). In particular, given that cells exert larger tractions on more adhesive and/or stiffer substrates, gradients of substrate adhesivity and/or stiffness can polarize cells (79–81). The ensu- ing migrations toward regions of higher adhesivity and/or stiffness are known as haptotaxis and durotaxis, respectively. Furthermore, even changes in uniform substrate properties may lead to cell polarization. Specifically, increasing substrate stiffness triggers an isotropic-nematic transition in the actomyosin cytoskeleton (52, 82–84). This transition results in cell elongation, which, in turn, might promote spontaneous cell polarization (85). 3. PHYSICAL MODELS, FROM SUBCELLULAR TO SUPRACELLULAR SCALES In this section, we review the different physical descriptions that have been used to model collective cell migration. These descriptions cover different levels of coarse-graining; we start from those describing subcellular detail and move up to continuum models that only describe supracellular features. Complementary presentations have been provided in recent reviews (20, www.annualreviews.org Collective Cell Migration 83 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 a Interfacial energy, α b Fluid Solid Snapshot Cell 2 p→ Trajectories p→ Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org Cell 1 Self-propulsion magnitude, P Interfacial energy, α Figure 3 Lattice models: the Cellular Potts Model. (a) Lattice sites corresponding to two different cells are shown in different colors. Cell–cell and cell–medium interfaces have an interfacial energy α (black). Cell migration in the direction of the polarity is favored by a self-propulsion magnitude P. (b) Snapshots of the system and cell trajectories at the fluid (α = 1) and solid (α = 4) regimes. Neighboring cells are colored differently, with arbitrary colors. Cell boundaries are rougher and longer for smaller interfacial energy. Figure adapted from Reference 89 with permission from Europhysics Letters. 86, 87). Here, we emphasize how the cellular forces and interactions reviewed in the previous section can be accounted for by each of the modeling approaches. We focus on two-dimensional models that explicitly include cell migration. 3.1. Lattice Models: The Cellular Potts Model In the spirit of classical models of statistical mechanics, such as the paradigmatic Ising model, lattice models describe individual cells as domains on a lattice, thus resolving subcellular details of cell shape (Figure 3a). In particular, this description is based on the Potts model, and hence it is known as the Cellular Potts Model (CPM) (88). Each lattice site i = 1,... , N is assigned a state variable σi = 1,... , m corresponding to one of m − 1 cells. The state of each lattice site is then updated using a state-exchange Monte Carlo scheme with Metropolis dynamics at a sufficiently low temperature to ensure that cells remain as compact domains. 3.1.1. Effective Hamiltonian. The dynamics minimizes the effective Hamiltonian   m−1  m−1 H= J(σi , σ j ) + λ (Aσ − A0 )2 − P  σ · pσ. R 1. i, j σ =1 σ =1 The original CPM only included the first two terms. The first term, whose sum runs over neigh- boring sites i, j, accounts for the interfacial tension between neighboring cells as well as between cells and the medium (state σ = m), which are encoded in the interaction matrix J. The simplest choice is J(σi , σ j ) = α(1 − δσi ,σ j ), where α is the interfacial energy that controls the amplitude of cell shape fluctuations (Figure 3). This energy captures the combined effects of cell–cell adhesion 84 Alert Trepat CO11CH05_Alert ARjats.cls February 13, 2020 11:16 and cortical tension (Sections 2.2.1 and 2.2.4). The second term penalizes changes in cell area around a preferred value A0 , with an area modulus λ > 0 (Figure 3a and Section 2.2.3). The area  of cell σ , i.e., its number of lattice sites, is simply given by Aσ = N i=1 δσi ,σ. 3.1.2. Cell migration. Later, the third term was added to implement cell motility by decreasing the energy of those configurations in which a cell’s center of mass R  σ = A−1 N (xi , yi ) δσ ,σ has σ i=1 i advanced toward the direction of its polarity pσ (90). This term corresponds to an active polar force of magnitude P > 0 on each cell: Fσ = −∇   H = P pσ (Figure 3a and Section 2.1.1). The model Rσ does not specifically include friction forces. Rather, an effective damping of cell motion arises from the Metropolis dynamics itself, which is dissipative in nature. In fact, the mean cell speed is linear in P/α over a wide range of parameter values (91). Thus, the cell–cell coupling strength α, which Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. controls the diffusion coefficient of a cell in the absence of motility, is proportional to the effective Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org viscous friction coefficient, consistently with the Stokes–Einstein relation. 3.1.3. Polarity dynamics. The cellular polarity pσ was proposed to align with the velocity over some timescale (91, 92) (Section 2.4.1) or, alternatively, to simply undergo rotational diffusion (89). In a variant of the CPM, cell motion was dictated by the gradient of a self-secreted chemoat- tractant, whose concentration evolves with its own dynamics (93). CPMs with alternative polarity dynamics should be explored in the future. Along these lines, Coburn et al. have recently proposed a hybrid CPM that accounts for CIL (94) (Section 2.3.2). 3.1.4. Collective phenomena. Initially, the self-propelled CPM was primarily used to study ve- locity correlations of complex flows in cell monolayers (30, 91, 92). More recently, it has also been used to study fluid–solid transitions and glassy dynamics in cell monolayers (89, 91) (Figure 3b), collective rotations (95), gap closure (94), and tissue spreading (96), including the fingering insta- bility of the tissue front (93, 96). 3.1.5. Discussion. The CPM is based on an explicit and detailed description of cell shape and cell–cell adhesion, which, by means of intensive simulations, enables close investigation of cell- scale mechanisms of cell rearrangements. However, the Metropolis dynamics yields somewhat artificial cell-shape fluctuations that depend on a temperature parameter not directly related to experimental measurements. Furthermore, the model is not readily suited to incorporate some kinds of cellular interactions relevant for collective cell migration. In particular, how to distinguish cell–cell and cell–substrate friction and how to appropriately capture the active nature of some cellular forces with the relaxational algorithm of the CPM remains unclear. 3.2. Phase-Field Models With their origins in interface dynamics (97), phase-field models also describe cell shape in subcel- lular detail. However, unlike the CPM, they do not rely on a lattice. Rather, each cell i = 1,... , N is described by a phase field φi (r , t ), which is 1 inside the cell and 0 outside (Figure 4a). A similar approach relies on describing cell shape via a contour function (66). Some models even describe intracellular structures, such as the nucleus, using additional phase fields (98). 3.2.1. Phase-field free energy. Cell–cell interactions are built into a free-energy functional of the phase field. Although formulations vary (98–100), a possible form is F = FCH + Farea + Fcell−cell with www.annualreviews.org Collective Cell Migration 85 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 a b.) a.u y( Shape-based Pha ϕ(x,y) 1 anisotropic se fi active stress 0 x (a.u.) el d Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. , Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org c Velocity alignment d Collective motion υ→ Time Time Figure 4  Phase-field models. (a) Phase field of a cell. (b) The overlap between phase fields, i= j φi φ j , identifies cell–cell interfaces (white). The inset shows cell contours, φi = 1/2, along with a sketch of extensile stress along the principal axis of the cell deformation tensor. (c,d ) Collisions between deformable cells lead to velocity alignment (c) and collective motion (d ). Panel a adapted from Reference 99. Panel b adapted with permission from Reference 100; Copyright (2019) by the American Physical Society. Panels c and d adapted from Reference 67. N    γ  i |2 d2r , FCH = 4φi2 (1 − φi )2 +  2 |∇φ 2a. i=1  A  N   2 1 Farea = μ 1− φ 2 2 d  r , 2b. i=1 π R2 A i  N     κ  i |2 |∇φ  j |2 d2r. Fcell−cell = φi2 φ 2j − τ  4 |∇φ 2c. i=1 j=i  A The first contribution is a Cahn–Hilliard free energy that stabilizes the phase-field interface. The first term is a double-well potential with minima at the cell interior (φi = 1) and exterior (φi = 0), which are connected by an interface of width  and tension γ that delineates the cell boundary. Here, we have neglected the bending rigidity of the interface (98). The second con- tribution penalizes departures of cell area from its preferred value π R2 , with area modulus μ (Section 2.2.3). The third contribution accounts for cell–cell interactions. It includes a repulsive term that penalizes cell overlapping (Section 2.2.3), with strength κ, and an attractive interaction between cell boundaries that models cell–cell adhesion (Section 2.2.1), with strength κτ (98). 86 Alert Trepat CO11CH05_Alert ARjats.cls February 13, 2020 11:16 3.2.2. Phase-field dynamics and force balance. The dynamics of cell shape reads as  i = − δF. ∂t φi + vi · ∇φ 3. δφi Here, vi is a cell velocity defined as (99)   1 δF  1 δFcell−cell  1 vi = ∇φi d2r = ∇φi d2r = Fiint , 4. ξ A δφi ξ A δφi ξ where ξ is a friction coefficient (Section 2.1.2), and Fiint is an interaction force on the interface of cell i due to overlaps with neighboring cells. This relationship can be generalized to include cell motility in the form of an active polar contribution Ta pi to the force balance (69) (Section 2.1.1): Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org ξvi = Fiint + Ta pi. 5. The interaction force Fiint can also be generalized to account for additional interactions. In con- tinuum mechanics, short-range interaction forces are described in terms of the stress tensor field σ (r , t ). In the phase-field formulation, this corresponds to (100)   Fiint =  · σ d2r = − σ · ∇φ φi ∇  i d2r. 6. A A In addition to the phase-field interactions, which give a pressure term, the stress tensor may also include other contributions such as viscous and active stresses. In this case, combining the ap- proaches of References 69 and 100, the stress tensor could read as  σ (r , t ) = −P(r , t ) I + ξc  j (r , t ) − ζ Q(r , t ), (vi − v j ) ∇φ 7. j=i where the second term accounts for cell–cell friction with coefficient ξc (Section 2.2.2), and the third term describes anisotropic active stresses proportional to the nematic order parameter ten-  sor field, Q(r , t ) = Ni=1 φi ( r , t ) Si (Section 2.2.4). Here, Si is the orientation tensor of cell i, which may be based on either cells’ polarities, Si = pi pi − 1/2 |pi |2 I, or cells’ shapes as proposed in Ref- erence 100, Si = − A [(∇φ  i )T ∇φ  i − |∇φ i |2 I]d2r (Figure 4b). 3.2.3. Polarity dynamics. Regarding the polarity dynamics, interactions such as CIL and CFL (Section 2.3.2), polarity alignment (Section 2.3.1), and polarity–velocity alignment (Section 2.4.1) have been explored (98), as well as couplings to chemotactic fields (101). More recently, an align- ment of cell polarity toward the direction of the total interfacial force has also been implemented (69) (Section 2.3.2). 3.2.4. Collective phenomena. Phase-field models have primarily addressed the emergence of collective motion from cell–cell interactions. Whereas some works focused on explicit orienta- tional interactions (98), other studies showed that, when cell polarity is coupled to cell-shape asymmetry (Section 2.3.4), collisions between deformable cells lead to cell–cell velocity alignment and collective motion (66, 67) (Figure 4c,d ). Recently, the phase-field model has been employed to explain the emergence of extensile nematic behavior (100) and to recapitulate collective velocity oscillations (69) in epithelial monolayers. 3.2.5. Discussion. The phase-field formalism provides a detailed description of cell shape while tackling some of the issues of the CPM. Foremost, it introduces a force balance (Equation 5) that www.annualreviews.org Collective Cell Migration 87 CO11CH05_Alert ARjats.cls February 13, 2020 11:16 provides physical dynamics, thus going beyond the energy minimization process of the CPM, which imposes static mechanical equilibrium at each step. Furthermore, the phase-field model is currently better connected to tissue mechanics (Equation 6), and it can explicitly account for cell–cell and cell–substrate friction as well as for active stresses (Equations 5–7) (69, 100). 3.3. Active Network Models With precedents in the physics of foams (102), network models describe epithelial tissues as net- works of polygonal cells (103). Thus, albeit in less detail than lattice and phase-field models, these models still describe subcellular features of cell shape. They encompass two subtypes of models: vertex and Voronoi models. Access provided by 2601:85:c300:1af0:d430:e710:22d4:8b9a on 03/17/20. For personal use only. Annu. Rev. Condens. Matter Phys. 2020.11:77-101. Downloaded from www.annualreviews.org 3.3.1. Vertex and Voronoi models. In vertex models, the degrees of freedom are the vertices of the polygons. Alternatively, the network can be described by the cell centers, and this reduces the number of degrees of freedom. These descriptions are known as Voronoi models because, given the positions of the cell centers, the cell–cell boundaries are delineated by the Voronoi tessellation