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Physical Models of Collective Cell Migration
Ricard Alert, Xavier Trepat
TL;DR
Collective cell migration is widespread, but its physical mechanisms have remained controversial. This review catalogs positional and orientational interactions, surveys models across scales, and summarizes emergent phenomena and remaining challenges.
Problem
The physical mechanisms governing collective cell migration during development, regeneration, and wound healing have remained controversial.
Method
The review classifies cell-cell and cell-substrate interactions and examines lattice, phase-field, active-network, particle, and continuum models across cellular scales.
Results
The reviewed models explain collective phenomena including flocking, solid-fluid transitions, wetting, fingering, and mechanical waves in spreading epithelial monolayers.
Takeaways & Limitations
The field can increasingly make quantitative predictions and test specific models and assumptions through theory-inspired experiments.
Takeaways & Limitations
Polarity remains difficult to distinguish from velocity in tissues because intercellular forces affect cell motion and morphological polarity features may be hidden.
Abstract
from arXiv · showhide
Collective cell migration is a key driver of embryonic development, wound healing, and some types of cancer invasion. Here we provide a physical perspective of the mechanisms underlying collective cell migration. We begin with a catalogue 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 models span the sub-cellular to the supracellular scales, and they include lattice models, phase fields models, active network models, particle models, and continuum models. For each type of model, we discuss its formulation, its limitations, 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 monolayers. We close by outlining remaining challenges and future directions in the physics of collective cell migration.
1 Introduction
Collective cell migration has long been recognized in development, regeneration, and wound healing, but its physical mechanisms remained controversial. Advances in imaging and force-mapping now enable quantitative physical investigation, while this review focuses on experimentally accessible two-dimensional cell sheets.
- Mechanistic background: Historical explanations for collective migration invoked tissue pressure, surface spreading, or pulling by leader cells at the tissue margin.These proposals reflected competing physical interpretations of how cell sheets advance.
- Experimental advances: Time-lapse imaging, fluorescence microscopy, and particle imaging velocimetry now map tissue velocity fields and strain tensors.These tools provide spatially and temporally resolved measurements of tissue mechanics.
- Experimental advances: Traction microscopy directly maps the forces cells exert on their surroundings during migration.Together with kinematic measurements, it makes relevant mechanical variables available across space and time.
- Physical perspective: Collective cell migration is a physical example of collective mechanical phenomena emerging in soft active entities with complex interactions.The topic connects biological systems with questions in condensed matter physics.
- Scope: Migrating tissues occur as two-dimensional sheets or as strands and clusters within complex three-dimensional environments.The review concentrates on two-dimensional sheets because their cellular forces can be accurately mapped in vitro.
- Scope: The review excludes most three-dimensional migration because physical forces in 3D remain difficult to access.Its stated scope is cell sheets migrating on substrates in two dimensions.
2 Forces and interactions of migrating cells
The review classifies migrating-cell interactions by whether they act on cell position or orientation, across cell-cell and cell-substrate interfaces. These interactions transmit active and passive forces, regulate polarity, and shape tissue mechanics and collective motion.
- The framework separates cell-cell and cell-substrate interactions into positional effects and orientational effects.This classification is intended to unify a fragmented literature while acknowledging some oversimplification.
- Cell-substrate interactions: Active traction arises from actomyosin forces transmitted through focal adhesions, while substrate friction balances traction during low-Reynolds-number motion.Cells polarize through front-rear asymmetry and protrusions to define the direction of propulsion.
- Cell-substrate interactions: Cell polarity can be difficult to identify in tissues because intercellular forces alter motion and morphological protrusions may be cryptic.Traction-based polarity estimates require active forces to dominate passive friction, leaving appropriate subcellular polarity markers as an open challenge.
- Cell-cell interactions: Cell-cell adhesion provides cohesion, surface tension, bulk elasticity, and active-stress transmission, while junction remodeling contributes to tissue dissipation.Decreasing adhesion can promote cell-shape changes and jamming into a solid state; junction turnover can support long-time viscous behavior.
- Orientational interactions: Orientational interactions include polarity alignment, contact regulation of locomotion, and feedback between polarity, tissue flow, and substrate-coupled velocity.CRL includes contact inhibition, contact following, and contact enhancement, with effects depending strongly on collision angle.
3 Physical models, from sub-cellular to supracellular scales
The review compares physical descriptions of collective cell migration across coarse-graining levels, progressing from sub-cellular detail to supracellular continuum models.
- Models span different levels of coarse-graining, from descriptions with sub-cellular detail to continuum models of supracellular features.
3.1 Lattice models: The cellular Potts model
The Cellular Potts Model represents cells as deformable lattice domains whose effective Hamiltonian combines interfacial energy, area elasticity, and active polarity-driven motility. It has reproduced several collective phenomena but retains dynamical and interaction-modeling limitations.
- The Cellular Potts Model represents individual cells as domains on a lattice, resolving sub-cellular details of cell shape through state variables and Monte Carlo updates.Each lattice site is assigned a state corresponding to one of the cells.
- Its effective Hamiltonian combines interfacial interactions, area elasticity around a preferred area, and a polarity-directed motility term.Interfacial energy controls shape fluctuations, while the area term penalizes departures from A0.
- Cell polarity can align with velocity or undergo rotational diffusion, and variants incorporate chemoattractant dynamics and additional polarity mechanisms.
- At α = 1 and α = 4, the model illustrates fluid and solid regimes, respectively, with rougher and longer cell boundaries at lower interfacial energy.
- The self-propelled CPM has been used to study velocity correlations, fluid-solid transitions, glassy dynamics, collective rotations, gap closure, tissue spreading, and fingering instabilities.
- The model enables cell-scale analysis of rearrangements but has artificial temperature-dependent fluctuations and unclear treatment of friction and active forces.
3.2 Phase-field models
Phase-field models describe cell shapes continuously and combine free-energy interactions with force-balance dynamics. They support explicit mechanical interactions and have reproduced several forms of collective epithelial behavior.
- Each cell is represented by a continuous phase field that equals 1 inside the cell and 0 outside, without relying on a lattice.Additional phase fields can represent intracellular structures such as the nucleus.
- Cell-cell interactions enter a free-energy functional containing interface stabilization, area regulation, and cell-cell interaction terms.The interface has width ϵ and tension γ, while area deviations from πR^2 are penalized by modulus µ.
- Phase-field dynamics can use a force balance with friction and interaction forces, augmented by active polar forces and stress-tensor contributions.The stress tensor can include cell-cell friction and anisotropic active stresses proportional to the nematic order parameter.
- Polarity dynamics in these models has incorporated CIL, CFL, polarity alignment, polarity-velocity alignment, chemotaxis, and alignment with total interfacial force.
- Phase-field models have explained collective motion from cell-cell interactions, velocity alignment after collisions, extensile nematic behavior, and collective velocity oscillations.
- Compared with CPMs, phase-field models provide physical force-balance dynamics and explicitly represent cell-cell and cell-substrate friction.
3.3 Active network models
Active network models represent epithelial tissues as polygonal-cell networks whose geometry and interactions can generate motility, collective alignment, and solid–fluid transitions. Their main scope is confluent tissues, although extensions address non-confluent systems and missing stress mechanisms.
- Model formulation: Network models represent epithelial tissues as polygonal cells using either vertex degrees of freedom or cell-center-based Voronoi tessellations.The two descriptions differ in degrees of freedom and can therefore produce different mechanical properties.
- Model formulation: Cell areas and perimeters encode tissue properties and interactions through an energy function involving area elasticity, cortical tension, and cell-cell adhesion.When adhesion dominates, the effective line tension becomes negative and interfaces tend to expand.
- Active extensions: Active Vertex and Self-Propelled Voronoi models add polar self-propulsion forces to vertices or cell centers, with boundary wetting, surface-tension, and bending forces also modeled.These active forces are combined with interaction forces derived from the tissue energy.
- Orientational interactions: Polarity-velocity alignment is the dominant orientational interaction, alongside polarity-shape alignment, CIL, force-induced polarization, and chemoattractant coupling.A hybrid model combines CIL with polarity-shape alignment.
- Collective phenomena: SPV models predict solid, liquid, solid-flock, and liquid-flock phases; self-propulsion speed and persistence favor fluidity, while alignment produces rotations and flocking.The solid phase supports elastic collective oscillations driven by self-propulsion.
- Limitations: Most network models assume confluent tissues, restricting them primarily to epithelial groups, while they also omit internal dissipation and anisotropic active stresses.Recent work generalizes Voronoi models to non-confluent tissues and adds cell-cell friction or stress-tensor connections.
3.4 Particle models
Particle models describe cells as one or two interacting circular particles, emphasizing positional forces, friction, and polarity dynamics rather than detailed cell geometry. They reproduce collective motion, chemotaxis, tissue stabilization, and density-driven jamming while extending naturally to mesenchymal migration.
- Model formulation: Particle models represent each cell as one or two circular particles, resolving limited shape detail but potentially capturing anisotropy and head-tail asymmetry.Otherwise, cell-shape details are overlooked.
- Forces and motility: Cell-cell interactions use central potentials with short-range repulsion and adhesion, while motility adds active polar forces, substrate friction, and cell-cell friction.The model can also include surface tension and curvature-dependent edge motility.
- Polarity interactions: Particle models implement polarity-velocity alignment, Vicsek-like alignment, CIL, CFL, and other polarity interactions.Short-range forces combined with autonomous polarity-velocity alignment can produce cell-cell alignment and flocking.
- Collective phenomena: CIL can generate collective motion through polarity-density coupling despite anti-aligning cell-pair polarities, and chemoattractant-sensitive CIL enables collective chemotaxis.CIL also stabilizes monolayers against dewetting and maintains tensile intercellular stresses during spreading.
- Collective phenomena: Increasing cell density and friction produces jamming, and jammed self-propelled particles exhibit collective oscillations.These oscillations parallel those predicted for jammed packings in SPV models.
- Scope and limitations: Particle models miss cell-shape and polarity coupling but can describe epithelial and mesenchymal collective migration and compute the tissue stress tensor.Mesenchymal migration involves weak and transient cell-cell contacts.
3.5 Continuum models
Continuum models coarse-grain migrating tissues into multicellular fields such as velocity, polarity, and density. Their equations combine conservation laws, polar-media dynamics, free-energy couplings, and constitutive stress relations to describe tissue-scale behavior.
- Coarse-grained description: Continuum models represent cell colonies using coarse-grained velocity, polarity, and density fields averaged over many cells.Generic field equations follow hydrodynamic principles, symmetries, and conservation laws.
- Free-energy formulation: The free energy penalizes density deviations, expands the polarity field in Landau terms, couples density gradients to polarity, and penalizes polarity and density gradients.The density-polarity coupling yields polarity proportional to the density gradient in equilibrium.
- Field dynamics: Cell-number conservation is imposed through a density continuity equation that includes the net proliferation rate from cell division and death.The polarity field then follows long-wavelength polar-media dynamics.
- Field dynamics: Polarity dynamics include a molecular-field torque, rotational friction, couplings to bulk and shear flows, and a possible coupling to uniform flows.The uniform-flow coupling could represent polarity-velocity alignment, but it had not yet been included in continuum migration models.
3.5.3 Force balance
Continuum force balance relates tissue stress to substrate traction, while constitutive laws specify the deviatoric stresses and traction forces. Viscoelastic fluid models capture frequency-dependent tissue responses and can approach elastic descriptions as relaxation becomes long.
- Force balance: Force balance equates the divergence of the tissue stress tensor with cell-substrate traction forces.The stress is decomposed into pressure, symmetric deviatoric stress, and antisymmetric polarity-related stress.
- Constitutive closure: Pressure follows the Gibbs-Duhem relation, while constitutive equations specify deviatoric stress and traction in terms of tissue variables.This constitutive specification is the key modeling step for closing the continuum description.
- Rheology: Migrating tissues can be modeled as elastic or fluid media, and both descriptions have reproduced experimental observations.The viscoelastic-fluid formalism approaches an elastic description in the limit τ →∞.
- Rheology: The Maxwell model describes aggregates as viscoelastic fluids whose stress relaxes over a characteristic time-scale τ.Such tissues respond elastically at high frequencies and viscously at low frequencies.
- Rheology: The relaxation time reflects intracellular cytoskeletal reorganization and intercellular sliding, with relevant protein-turnover processes occurring over at most tens of minutes.Cell division, death, and extrusion are additional processes affecting stress relaxation.
3.5.5 Constitutive equation for the traction forces
Continuum traction models combine constitutive force laws with boundary conditions and simplifying assumptions to describe tissue migration. Their formulations variously retain polarity, density, viscosity, friction, and interfacial effects.
- The traction equation combines cell-substrate viscous friction, an active polar force driving migration, and polarity-velocity alignment forces.
- Models impose edge conditions such as vanishing density and stress, line tension, or polarity anchoring toward free space.
- Homeotropic anchoring produces a polarized boundary layer with width related to K/a, while planar or tilted anchoring represents alternative edge alignment.
- Many models omit density, neglect flow-polarity coupling, or treat polarity dynamics as quasi-static; others take the small-correlation-length limit and localize traction at the edge.
- Models commonly retain either internal viscosity or substrate friction, although both may be needed when the screening length is comparable to tissue size.
- Two-dimensional incompressibility is not generally valid because monolayer area is not conserved.
3.5.8 Collective phenomena
Continuum models explain collective phenomena in spreading epithelia, including fingering, wetting transitions, and mechanical-wave behavior. Their analytical tractability enables predictions, but coarse-graining leaves cellular interactions phenomenological and may require additional fields near jamming.
- An active instability can generate multicellular fingers with an intrinsic wavelength even without motility regulation at the monolayer edge.
- Wetting occurs only for monolayers larger than a critical size, because spreading versus retraction depends on the balance between active tractions and contractile stresses.
- Continuum models address whether tissue spreading is better represented by liquid or elastic rheology, reflecting liquid-like behavior alongside long-time effective elasticity.
- Toner-Tu models describe compressible polar dry fluids without hydrodynamic interactions or a separate distinction between polarity and velocity fields.
- The analytical tractability of continuum models supports predictions without exhaustive simulations, but cellular interactions are encoded through phenomenological couplings whose relationship to cell processes may be unclear.
- Near jamming, cell-shape fluctuations become a slow field, motivating models that couple cell-shape anisotropy to polarity.
4 Conclusions and outlook
The review identifies progress toward quantitative, experimentally testable theories of collective cell migration while emphasizing major unresolved problems. These include understanding migration in mechanically complex 3D environments, connecting scales, and unifying cell-cell contact mechanics.
- The field can increasingly formulate quantitative predictions and test specific models and assumptions in theory-inspired experiments.
- The review is restricted to two-dimensional cell sheets, while collective migration in remodeled three-dimensional environments remains unknown and force-probing techniques are limited.
- A central challenge is deriving continuum descriptions from cell-scale models and linking molecular actin and adhesion mechanisms to modeled self-propulsion and friction forces.
- The field lacks a unifying picture of the mechanical consequences of cell-cell contacts, which may involve adhesion, repulsion, or no interaction with biological nuances.