Mechanics of epithelial closure over non-adherent environments

Initiation of epithelial wound closure by an active instability at the purse string

The ability of biological systems to withstand and recover from various disruptions, such as spontaneous genetic mutations and environmental damage, largely relies on intricate feedback mechanisms. We theoretically study the mechanical response of an epithelial tissue facing damage in the form of a circular wound. Our model describes a feedback loop between the generation of active forces in the actomyosin and tissue mechanics, described by the vertex model. While the exact dynamics of wound closure may be influenced by several biophysical mechanisms that interplay in a nontrivial way, our findings suggest that the closure may initiate as an active instability, triggered by a reduced myosin turnover rate at the wound's perimeter. We explore the interplay between myosin dynamics and the elastic properties of the tissue, elucidating their collective role in determining a wound's loss of stability, leading to the initiation of the closure process.

Interplay of geometry and mechanics in epithelial wound healing

Wound healing is a complex biological process critical for maintaining an organism's structural integrity and tissue repair following an infection or injury. Recent studies have unveiled the mechanisms involving the coordination of biochemical and mechanical responses in the tissue in wound healing. In this article, we focus on the healing property of an epithelial tissue as a material while the effects of biological mechanisms such as cell proliferation, tissue intercalation, cellular migration, cell crawling, and filopodia protrusion is minimal. We present a mathematical framework that predicts the fate of a wounded tissue based on the wound's geometrical features and the tissue's mechanical properties. Precisely, adapting the vertex model of tissue mechanics, we predict whether a wound of a specific size in an epithelial monolayer characterized by certain levels of actomyosin contractility and cell-cell adhesion will heal (i.e., close), shrink in size, or rupture the tissue further. Moreover, we show how tissue-mediated mechanisms such as purse-string tension at the wound boundary facilitate wound healing. Finally, we validate the predictions of our model by designing an experimental setup that enables us to create wounds of specific sizes in kidney epithelial cells (MDCK) monolayers. Altogether, this work sets up a basis for interpreting the interplay of mechanical and geometrical features of a tissue in the process of wound healing.

Substrate stress relaxation regulates monolayer fluidity and leader cell formation for collectively migrating epithelia

Collective migration of epithelial tissues is a critical feature of developmental morphogenesis and tissue homeostasis. Coherent motion of cell collectives requires large scale coordination of motion and force generation and is influenced by mechanical properties of the underlying substrate. While tissue viscoelasticity is a ubiquitous feature of biological tissues, its role in mediating collective cell migration is unclear. Here, we have investigated the impact of substrate stress relaxation on the migration of micropatterned epithelial monolayers. Epithelial monolayers exhibit faster collective migration on viscoelastic alginate substrates with slower relaxation timescales, which are more elastic, relative to substrates with faster stress relaxation, which exhibit more viscous loss. Faster migration on slow-relaxing substrates is associated with reduced substrate deformation, greater monolayer fluidity, and enhanced leader cell formation. In contrast, monolayers on fast-relaxing substrates generate substantial substrate deformations and are more jammed within the bulk, with reduced formation of transient lamellipodial protrusions past the monolayer edge leading to slower overall expansion. This work reveals features of collective epithelial dynamics on soft, viscoelastic materials and adds to our understanding of cell-substrate interactions at the tissue scale.

Significance Statement

Groups of cells must coordinate their movements in order to sculpt organs during development and maintain tissues. The mechanical properties of the underlying substrate on which cells reside are known to influence key aspects of single and collective cell migration. Despite being a nearly universal feature of biological tissues, the role of viscoelasticity (i.e., fluid-like and solid-like behavior) in collective cell migration is unclear. Using tunable engineered biomaterials, we demonstrate that sheets of epithelial cells display enhanced migration on slower-relaxing (more elastic) substrates relative to faster-relaxing (more viscous) substrates. Building our understanding of tissue-substrate interactions and collective cell dynamics provides insights into approaches for tissue engineering and regenerative medicine, and therapeutic interventions to promote health and treat disease.

Physical Forces in Regeneration of Cells and Tissues

The ability to regenerate after the loss of a part is a hallmark of living systems and occurs at both the tissue and organ scales, but also within individual cells. Regeneration entails many processes that are physical and mechanical in nature, including the closure of wounds, the repositioning of material from one place to another, and the restoration of symmetry following perturbations. However, we currently know far more about the genetics and molecular signaling pathways involved in regeneration, and there is a need to investigate the role of physical forces in the process. Here, we will provide an overview of how physical forces may play a role in wound healing and regeneration, in which we compare and contrast regenerative processes at the tissue and cell scales.

Geometric control by active mechanics of epithelial gap closure

Epithelial wound healing is one of the most important biological processes occurring during the lifetime of an organism. It is a self-repair mechanism closing wounds or gaps within tissues to restore their functional integrity. In this work we derive a new diffuse interface approach for modelling the gap closure by means of a variational principle in the framework of non-equilibrium thermodynamics. We investigate the interplay between the crawling with lamellipodia protrusions and the supracellular tension exerted by the actomyosin cable on the closure dynamics. These active features are modeled as Korteweg forces into a generalised chemical potential. From an asymptotic analysis, we derive a pressure jump across the gap edge in the sharp interface limit. Moreover, the chemical potential diffuses as a Mullins-Sekerka system, and its interfacial value is given by a Gibbs-Thompson relation for its local potential driven by the curvature-dependent purse-string tension. The finite element simulations show an excellent quantitative agreement between the closure dynamics and the morphology of the edge with respect to existing biological experiments. The resulting force patterns are also in good qualitative agreement with existing traction force microscopy measurements. Our results shed light on the geometrical control of the gap closure dynamics resulting from the active forces that are chemically activated around the gap edge.

Cancer-associated fibroblasts actively compress cancer cells and modulate mechanotransduction

During tumor progression, cancer-associated fibroblasts (CAFs) accumulate in tumors and produce an excessive extracellular matrix (ECM), forming a capsule that enwraps cancer cells. This capsule acts as a barrier that restricts tumor growth leading to the buildup of intratumoral pressure. Combining genetic and physical manipulations in vivo with microfabrication and force measurements in vitro, we found that the CAFs capsule is not a passive barrier but instead actively compresses cancer cells using actomyosin contractility. Abrogation of CAFs contractility in vivo leads to the dissipation of compressive forces and impairment of capsule formation. By mapping CAF force patterns in 3D, we show that compression is a CAF-intrinsic property independent of cancer cell growth. Supracellular coordination of CAFs is achieved through fibronectin cables that serve as scaffolds allowing force transmission. Cancer cells mechanosense CAF compression, resulting in an altered localization of the transcriptional regulator YAP and a decrease in proliferation. Our study unveils that the contractile capsule actively compresses cancer cells, modulates their mechanical signaling, and reorganizes tumor morphology.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Regeneration in calcareous sponge relies on 'purse-string' mechanism and the rearrangements of actin cytoskeleton

The crucial step in any regeneration process is epithelization, i.e. the restoration of an epithelium structural and functional integrity. Epithelization requires cytoskeletal rearrangements, primarily of actin filaments and microtubules. Sponges (phylum Porifera) are early branching metazoans with pronounced regenerative abilities. Calcareous sponges have a unique step during regeneration: the formation of a temporary structure, called regenerative membrane which initially covers a wound. It forms due to the morphallactic rearrangements of exopinaco- and choanoderm epithelial-like layers. The current study quantitatively evaluates morphological changes and characterises underlying actin cytoskeleton rearrangements during regenerative membrane formation in asconoid calcareous sponge Leucosolenia variabilis through a combination of time-lapse imaging, immunocytochemistry, and confocal laser scanning microscopy. Regenerative membrane formation has non-linear stochastic dynamics with numerous fluctuations. The pinacocytes at the leading edge of regenerative membrane form a contractile actomyosin cable. Regenerative membrane formation either depends on its contraction or being coordinated through it. The cell morphology changes significantly during regenerative membrane formation. Exopinacocytes flatten, their area increases, while circularity decreases. Choanocytes transdifferentiate into endopinacocytes, losing microvillar collar and flagellum. Their area increases and circularity decreases. Subsequent redifferentiation of endopinacocytes into choanocytes is accompanied by inverse changes in cell morphology. All transformations rely on actin filament rearrangements similar to those characteristic of bilaterian animals. Altogether, we provide here a qualitative and quantitative description of cell transformations during reparative epithelial morphogenesis in a calcareous sponge.

Physical forces guide curvature sensing and cell migration mode bifurcating

The ability of cells to sense and adapt to curvy topographical features has been implicated in organ morphogenesis, tissue repair, and tumor metastasis. However, how individual cells or multicellular assemblies sense and differentiate curvatures remains elusive. Here, we reveal a curvature sensing mechanism in which surface tension can selectively activate either actin or integrin flows, leading to bifurcating cell migration modes: focal adhesion formation that enables cell crawling at convex front edges and actin cable assembly that pulls cells forward at concave front edges. The molecular flows and curved front morphogenesis are sustained by coordinated cellular tension generation and transmission. We track the molecular flows and mechanical force transduction pathways by a phase-field model, which predicts that multicellular curvature sensing is more efficient than individual cells, suggesting collective intelligence of cells. The unique ability of cells in curvature sensing and migration mode bifurcating may offer insights into emergent collective patterns and functions of living active systems at different length scales.

A structural bio-chemo-mechanical model for vascular smooth muscle cell traction force microscopy

Altered vascular smooth muscle cell (VSMC) contractility is both a response to and a driver for impaired arterial function, and the leading experimental technique for quantifying VSMC contraction is traction force microscopy (TFM). TFM involves the complex interaction among several chemical, biological, and mechanical mechanisms, making it difficult to translate TFM results into tissue-scale behavior. Here, a computational model capturing each of the major aspects of the cell traction process is presented. The model incorporates four interacting components: a biochemical signaling network, individual actomyosin fiber bundle contraction, a cytoskeletal network of interconnected fibers, and elastic substrate displacement due to cytoskeletal force. The synthesis of these four components leads to a broad, flexible framework for describing TFM and linking biochemical and biomechanical phenomena on the single-cell level. The model recapitulated available data on VSMCs following biochemical, geometric, and mechanical perturbations. The structural bio-chemo-mechanical model offers a tool to interpret TFM data in new, more mechanistic ways, providing a framework for the evaluation of new biological hypotheses, interpolation of new data, and potential translation from single-cell experiments to multi-scale tissue models.

Engineering tools for quantifying and manipulating forces in epithelia

The integrity of epithelia is maintained within dynamic mechanical environments during tissue development and homeostasis. Understanding how epithelial cells mechanosignal and respond collectively or individually is critical to providing insight into developmental and (patho)physiological processes. Yet, inferring or mimicking mechanical forces and downstream mechanical signaling as they occur in epithelia presents unique challenges. A variety of in vitro approaches have been used to dissect the role of mechanics in regulating epithelia organization. Here, we review approaches and results from research into how epithelial cells communicate through mechanical cues to maintain tissue organization and integrity. We summarize the unique advantages and disadvantages of various reduced-order model systems to guide researchers in choosing appropriate experimental systems. These model systems include 3D, 2D, and 1D micromanipulation methods, single cell studies, and noninvasive force inference and measurement techniques. We also highlight a number of in silico biophysical models that are informed by in vitro and in vivo observations. Together, a combination of theoretical and experimental models will aid future experiment designs and provide predictive insight into mechanically driven behaviors of epithelial dynamics.

The analytical solution to the migration of an epithelial monolayer with a circular spreading front and its implications in the gap closure process

The coordinated behaviors of epithelial cells are widely observed in tissue development, such as re-epithelialization, tumor growth, and morphogenesis. In these processes, cells either migrate collectively or organize themselves into specific structures to serve certain purposes. In this work, we study a spreading epithelial monolayer whose migrating front encloses a circular gap in the monolayer center. Such tissue is usually used to mimic the wound healing process in vitro. We model the epithelial sheet as a layer of active viscous polar fluid. With an axisymmetric assumption, the model can be analytically solved under two special conditions, suggesting two possible spreading modes for the epithelial monolayer. Based on these two sets of analytical solutions, we assess the velocity of the spreading front affected by the gap size, the active intercellular contractility, and the purse-string contraction acting on the spreading edge. Several critical values exist in the model parameters for the initiation of the gap closure process, and the purse-string contraction plays a vital role in governing the gap closure kinetics. Finally, the instability of the morphology of the spreading front was studied. Numerical calculations show how the perturbated velocities and the growth rates vary with respect to different model parameters.

Geometry-mediated bridging drives nonadhesive stripe wound healing

Wound healing through reepithelialization of gaps is of profound importance to the medical community. One critical mechanism identified by researchers for closing non-cell-adhesive gaps is the accumulation of actin cables around concave edges and the resulting purse-string constriction. However, the studies to date have not separated the gap-edge curvature effect from the gap size effect. Here, we fabricate micropatterned hydrogel substrates with long, straight, and wavy non-cell-adhesive stripes of different gap widths to investigate the stripe edge curvature and stripe width effects on the reepithelialization of Madin-Darby canine kidney (MDCK) cells. Our results show that MDCK cell reepithelization is closely regulated by the gap geometry and may occur through different pathways. In addition to purse-string contraction, we identify gap bridging either via cell protrusion or by lamellipodium extension as critical cellular and molecular mechanisms for wavy gap closure. Cell migration in the direction perpendicular to wound front, sufficiently small gap size to allow bridging, and sufficiently high negative curvature at cell bridges for actin cable constriction are necessary/sufficient conditions for gap closure. Our experiments demonstrate that straight stripes rarely induce cell migration perpendicular to wound front, but wavy stripes do; cell protrusion and lamellipodia extension can help establish bridges over gaps of about five times the cell size, but not significantly beyond. Such discoveries deepen our understanding of mechanobiology of cell responses to curvature and help guide development of biophysical strategies for tissue repair, plastic surgery, and better wound management.

Computational modeling and simulation of epithelial wound closure

Wounds in the epithelium may lead to serious injurious events or chronic inflammatory diseases, however, multicellular organisms have the ability to self-repair wounds through the movement of epithelial cell toward the wound area. Despite intensive studies exploring the mechanism of wound closure, the role of mechanics in epithelial wound closure is still not well explained. In order to investigate the role of mechanical properties on wound closure process, a three-dimensional continuum physics-based computational model is presented in this study. The model takes into account the material property of the epithelial cell, intercellular interactions between neighboring cells at cell-cell junctions, and cell-substrate adhesion between epithelial cells and ECM. Through finite element simulation, it is found that the closure efficiency is related to the initial gap size and the intensity of lamellipodial protrusion. It is also shown that cells at the wound edge undergo higher stress compared with other cells in the epithelial monolayer, and the cellular normal stress dominates over the cellular shear stress. The model presented in this study can be employed as a numerical tool to unravel the mechanical principles behind the complex wound closure process. These results might have the potential to improve effective wound management and optimize the treatment.

Purse-string contraction guides mechanical gradient-dictated heterogeneous migration of epithelial monolayer

Mechanical heterogeneity has been recognized as an important role in mediating collective cell migration, yet the related mechanism has not been elucidated. Herein, we fabricate heterogeneous stiffness gradients by leveraging microelastically-patterned hydrogels with varying periodic distance. We observe that a decrease in the periodic distance of the mechanical heterogeneity is accompanied by an overall increase in the velocity and directionality of the migrating monolayer. Moreover, inhibition of ROCK- and myosin ⅡA- but not Rac1-mediated contraction reduces monolayer migration on the mechanically heterogeneous substrates. Furthermore, we find that F-actin and myosin ⅡA form purse-string at the leading edge on the mechanically heterogeneous substrates. Together, these findings not only show that the orientational cell-cell contraction promotes collective cell migration under the mechanical heterogeneity, but also demonstrate that the mechanosensation arising from large-scale cell-cell interactions through purse-string formation mediated cell-cell orientational contraction can feed back to regulate the reorganization of epithelial tissues. STATEMENT OF SIGNIFICANCE: By detecting the links between heterogenous rigidity and collective cell migration behavior at the molecular level, we reveal that collective cell migration in the mechanical heterogeneity is driven by ROCK- and myosin-ⅡA-dependent cytoskeletal tension. We confirm that cytoskeletal tension across the epithelial tissue is holistically linked through F-actin and myosin-ⅡA, which cooperate to form purse-string structures for modulating collective tissue behavior on the exogenous matrix with mechanical heterogeneity. Mechanical heterogeneity initiates tissue growth, remodelling, and morphogenesis by orientating cell contractility. Therefore, tensional homeostasis across large-scale cell interactions appears to be necessary and sufficient to trigger collective tissue behavior. Overall, these findings shed light on the role of mechanical heterogeneity in tissue microenvironment for reorganization and morphogenesis.

Contractility-induced self-organization of smooth muscle cells: from multilayer cell sheets to dynamic three-dimensional clusters

Smooth muscle cells (SMCs) are mural cells that play a vital contractile function in many tissues. Abnormalities in SMC organization are associated with many diseases including atherosclerosis, asthma, and uterine fibroids. Various studies have reported that SMCs cultured on flat surfaces can spontaneously form three-dimensional clusters whose organization resembles that encountered in some of these pathological settings. Remarkably, how these structures form remains unknown. Here we combine in vitro experiments and physical modeling to show that three-dimensional clusters initiate when cellular contractile forces induce a hole in a flat SMC sheet, a process that can be modeled as the brittle fracture of a viscoelastic material. The subsequent evolution of the nascent cluster can be modeled as an active dewetting process with cluster shape evolution driven by a balance between cluster surface tension, arising from both cell contractility and adhesion, and cluster viscous dissipation. The description of the physical mechanisms governing the spontaneous emergence of these intriguing three-dimensional clusters may offer insight into SMC-related disorders.

Emergent collective organization of bone cells in complex curvature fields

Individual cells and multicellular systems respond to cell-scale curvatures in their environments, guiding migration, orientation, and tissue formation. However, it remains largely unclear how cells collectively explore and pattern complex landscapes with curvature gradients across the Euclidean and non-Euclidean spectra. Here, we show that mathematically designed substrates with controlled curvature variations induce multicellular spatiotemporal organization of preosteoblasts. We quantify curvature-induced patterning and find that cells generally prefer regions with at least one negative principal curvature. However, we also show that the developing tissue can eventually cover unfavorably curved territories, can bridge large portions of the substrates, and is often characterized by collectively aligned stress fibers. We demonstrate that this is partly regulated by cellular contractility and extracellular matrix development, underscoring the mechanical nature of curvature guidance. Our findings offer a geometric perspective on cell-environment interactions that could be harnessed in tissue engineering and regenerative medicine applications.

A mathematical model of wound healing in bovine corneal endothelium

We developed a mathematical model to describe healing processes in bovine corneal endothelial (BCE) cells in culture, triggered by mechanical wounds with parallel edges. Previous findings from our laboratory show that, in these cases, BCE monolayers exhibit an approximately constant healing velocity. Also, that caspase-dependent apoptosis occurs, with the fraction of apoptotic cells increasing with the distance traveled by the healing edge. In addition, in this study we report the novel findings that, for wound scratch assays performed preserving the basal extracellular matrix: i) the healing cells increase their en face surface area in a characteristic fashion, and ii) the average length of the segments of the cell columns actively participating in the healing process increases linearly with time. These latter observations preclude the utilization of standard traveling wave formalisms to model wound healing in BCE cells. Instead, we developed and studied a simple phenomenological model based on a plausible formula for the spreading dynamics of the individual healing cells, that incorporates original evidence about the process in BCE cells. The model can be simulated to: i) obtain an approximately constant healing velocity; ii) reproduce the profile of the healing cell areas, and iii) obtain approximately linear time dependences of the mean cell area and average length of the front active segments per column. In view of its accuracy to account for the experimental observations, the model can also be acceptably employed to quantify the appearance of apoptotic cells during BCE wound healing. The strategy utilized here could offer a novel formal framework to represent modifications undergone by some epithelial cell lines during wound healing.

Novel Temperature-Sensitive Hydrogel Promotes Wound Healing Through YAP and MEK-Mediated Mechanosensitivity

Wound healing is a significant problem in clinical management. Various functional dressings are studied to promote wound healing through biochemical factors. They are generally expensive, complex to fabricate, and may adversely affect the wound. Mechanical forces are the critical regulators of tissue repair. Although contraction is shown to promote wound healing, the underlying mechanisms are not fully understood. In this study, a novel adhesive temperature-sensitive mechanically active hydrogel with a simple and inexpensive preparation process is developed. The dressing is able to adhere to the wound surface and actively contract the wound in response to body temperature. This mechanical contraction enhances the proliferative activity of basal cells, reduces the inflammatory response of the wound, and promotes wound healing. Furthermore, RNA-seq clarifies how the gene regulatory network is regulated by contraction. Finally, using pharmacological inhibitors, YAP and MEK are identified as the key signaling molecules for contraction-mediated tissue healing in vivo.

Fibroblasts Close a Void in Free Space by a Purse-String Mechanism

The mechanism by which stromal cells fill voids in injured tissue remains a fundamental question in regenerative medicine. While it is well-established that fibroblasts fill voids by depositing extracellular matrix (ECM) proteins as they migrate toward the wound site, little is known about their ability to adopt an epithelial-like purse-string behavior. To investigate fibroblast behavior during gap closure, we created an artificial wound with a large void space. We discovered that fibroblasts could form a free-standing bridge over deep microvoids, closing the void via purse-string contraction, a mechanism previously thought to be unique to epithelial wound closure. The findings also revealed that myosin II mediated contractility and intercellular adherent junctions were required for the closure of the fibroblast gap in our fabricated three-dimensional artificial wound. To fulfill their repair function under the specific microenvironmental conditions of wounds, fibroblasts appeared to acquire the structural features of epithelial cells, namely, contractile actin bundles that span over multiple cells along the boundary. These findings shed light on a novel mechanism by which stromal cells bridge the 3D gap during physiological processes such as morphogenesis and wound healing.

Leading-edge elongation by follower cell interruption in advancing epithelial cell sheets

Collective cell migration is seen in many developmental and pathological processes, such as morphogenesis, wound closure, and cancer metastasis. When a fish scale is detached and adhered to a substrate, epithelial keratocyte sheets crawl out from it, building a semicircular pattern. All the keratocytes at the leading edge of the sheet have a single lamellipodium, and are interconnected with each other via actomyosin cables. The leading edge of the sheet becomes gradually longer as it crawls out from the scale, regardless of the cell-to-cell connections. In this study, we found leading-edge elongation to be realized by the interruption of follower cells into the leading edge. The follower cell and the two adjacent leader cells are first connected by newly emerging actomyosin cables. Then, the contractile forces along the cables bring the follower cell forward to make it a leader cell. Finally, the original cables between the two leader cells are stretched to tear by the interruption and the lamellipodium extension from the new leader cell. This unique actomyosin-cable reconnection between a follower cell and adjacent leaders offers insights into the mechanisms of collective cell migration.

Active chemo-mechanical feedbacks dictate the collective migration of cells on patterned surfaces

Recent evidence has demonstrated that, when cultured on micro-patterned surfaces, living cells can move in a coordinated manner and form distinct migration patterns, including flowing chain, suspended propagating bridge, rotating vortex, etc. However, the fundamental question of exactly how and why cells migrate in these fashions remains elusive. Here, we present a theoretical investigation to show that the tight interplay between internal cellular activities, such as chemo-mechanical feedbacks and polarization, and external geometrical constraints are behind these intriguing experimental observations. In particular, on narrow strip patterns, strongly force-dependent cellular contractility and intercellular adhesion were found to be critical for reinforcing the leading edge of the migrating cell monolayer and eventually result in the formation of suspended cell bridges flying over nonadhesive regions. On the other hand, a weak force-contractility feedback led to the movement of cells like a flowing chain along the adhesive strip. Finally, we also showed that the random polarity forces generated in migrating cells are responsible for driving them into rotating vortices on strips with width above a threshold value (~10 times the size of the cell).

Tissue Engineering with Mechanically Induced Solid-Fluid Transitions

Epithelia are contiguous sheets of cells that stabilize the shape of internal organs and support their structure by covering their surfaces. They acquire diverse morphological forms appropriate for their specific functions during embryonic development, such as the kidney tubules and the complex branching structures found in the lung. The maintenance of epithelial morphogenesis and homeostasis is controlled by their remarkable mechanics-epithelia can become elastic, plastic, and viscous by actively remodeling cell-cell junctions and modulating the distribution of local stresses. Microfabrication, finite element modelling, light-sheet microscopy, and robotic micromanipulation are used to show that collagen gels covered with an epithelial skin serve as shape-programmable soft matter. The process involves solid to fluid transitions induced by mechanical perturbations, generates spatially distributed surface stresses at tissue interfaces, and is amenable to both additive and subtractive manufacturing techniques. The robustness and versatility of this strategy for engineering designer tissues is demonstrated by directing the morphogenesis of a variety of molded, carved, and assembled forms from the base material. The results provide insight into the active mechanical properties of the epithelia and establish methods for engineering tissues with sustainable architectures.

Active Instability and Nonlinear Dynamics of Cell-Cell Junctions

Active cell-junction remodeling is important for tissue morphogenesis, yet its underlying physics is not understood. We study a mechanical model that describes junctions as dynamic active force dipoles. Their instability can trigger cell intercalations by a critical collapse. Nonlinearities in tissue's elastic response can stabilize the collapse either by a limit cycle or condensation of junction lengths at cusps of the energy landscape. Furthermore, active junction networks undergo collective instability to drive active in-plane ordering or develop a limit cycle of collective oscillations, which extends over regions of the energy landscape corresponding to distinct network topologies.

Born to Run? Diverse Modes of Epithelial Migration

Bundled with various kinds of adhesion molecules and anchored to the basement membrane, the epithelium has historically been considered as an immotile tissue and, to migrate, it first needs to undergo epithelial-mesenchymal transition (EMT). Since its initial description more than half a century ago, the EMT process has fascinated generations of developmental biologists and, more recently, cancer biologists as it is believed to be essential for not only embryonic development, organ formation, but cancer metastasis. However, recent progress shows that epithelium is much more motile than previously realized. Here, we examine the emerging themes in epithelial collective migration and how this has impacted our understanding of EMT.

Wound opening in a thin incompressible viscoelastic tissue

We develop a model to investigate analytically and numerically the mechanics of wound opening made in a viscoelastic, isotropic, homogeneous, and incompressive thin tissue. This process occurs just immediately after the wound infliction. Before any active biological action has taken place, the tissue relaxes, and the wound opens mostly due to the initial homeostatic tension of the tissue, its elastic and viscous properties, and the existing friction between the tissue and its substrate. We find that for a circular wound the regimes of deformation are defined by a single adimensional parameter λ, which characterizes the relative importance of viscosity over friction.

Mechanical plasticity in collective cell migration

Collective cell migration is crucial to maintain epithelium integrity during developmental and repair processes. It requires a tight regulation of mechanical coordination between neighboring cells. This coordination embraces different features including mechanical self-propulsion of individual cells within cellular colonies and large-scale force transmission through cell-cell junctions. This review discusses how the plasticity of biomechanical interactions at cell-cell contacts could help cellular systems to perform coordinated motions and adapt to the properties of the external environment.

Extracellular Matrix Alignment Directs Provisional Matrix Assembly and Three Dimensional Fibrous Tissue Closure

Gap closure is a dynamic process in wound healing, in which a wound contracts and a provisional matrix is laid down, to restore structural integrity to injured tissues. The efficiency of wound closure has been found to depend on the shape of a wound, and this shape dependence has been echoed in various in vitro studies. While wound shape itself appears to contribute to this effect, it remains unclear whether the alignment of the surrounding extracellular matrix (ECM) may also contribute. In this study, we investigate the role both wound curvature and ECM alignment have on gap closure in a 3D culture model of fibrous tissue. Using microfabricated flexible micropillars positioned in rectangular and octagonal arrangements, seeded 3T3 fibroblasts embedded in a collagen matrix formed microtissues with different ECM alignments. Wounding these microtissues with a microsurgical knife resulted in wounds with different shapes and curvatures that closed at different rates. Observing different regions around the wounds, we noted local wound curvature did not impact the rate of production of provisional fibronectin matrix assembled by the fibroblasts. Instead, the rate of provisional matrix assembly was lowest emerging from regions of high fibronectin alignment and highest in the areas of low matrix alignment. Our data suggest that the underlying ECM structure affects the shape of the wound as well as the ability of fibroblasts to build provisional matrix, an important step in the process of tissue closure and restoration of tissue architecture. The study highlights an important interplay between ECM alignment, wound shape, and tissue healing that has not been previously recognized and may inform approaches to engineer tissues.

Hindbrain neuropore tissue geometry determines asymmetric cell-mediated closure dynamics in mouse embryos

Gap closure is a common morphogenetic process. In mammals, failure to close the embryonic hindbrain neuropore (HNP) gap causes fatal anencephaly. We observed that surface ectoderm cells surrounding the mouse HNP assemble high-tension actomyosin purse strings at their leading edge and establish the initial contacts across the embryonic midline. Fibronectin and laminin are present, and tensin 1 accumulates in focal adhesion-like puncta at this leading edge. The HNP gap closes asymmetrically, faster from its rostral than caudal end, while maintaining an elongated aspect ratio. Cell-based physical modeling identifies two closure mechanisms sufficient to account for tissue-level HNP closure dynamics: purse-string contraction and directional cell motion implemented through active crawling. Combining both closure mechanisms hastens gap closure and produces a constant rate of gap shortening. Purse-string contraction reduces, whereas crawling increases gap aspect ratio, and their combination maintains it. Closure rate asymmetry can be explained by asymmetric embryo tissue geometry, namely a narrower rostral gap apex, whereas biomechanical tension inferred from laser ablation is equivalent at the gaps' rostral and caudal closure points. At the cellular level, the physical model predicts rearrangements of cells at the HNP rostral and caudal extremes as the gap shortens. These behaviors are reproducibly live imaged in mouse embryos. Thus, mammalian embryos coordinate cellular- and tissue-level mechanics to achieve this critical gap closure event.

Collective migration during a gap closure in a two-dimensional haptotactic model

The ability of cells to respond to substrate-bound protein gradients is crucial for many physiological processes, such as immune response, neurogenesis and cancer cell migration. However, the difficulty to produce well-controlled protein gradients has long been a limitation to our understanding of collective cell migration in response to haptotaxis. Here we use a photopatterning technique to create circular, square and linear fibronectin (FN) gradients on two-dimensional (2D) culture substrates. We observed that epithelial cells spread preferentially on zones of higher FN density, creating rounded or elongated gaps within epithelial tissues over circular or linear FN gradients, respectively. Using time-lapse experiments, we demonstrated that the gap closure mechanism in a 2D haptotaxis model requires a significant increase of the leader cell area. In addition, we found that gap closures are slower on decreasing FN densities than on homogenous FN-coated substrate and that fresh closed gaps are characterized by a lower cell density. Interestingly, our results showed that cell proliferation increases in the closed gap region after maturation to restore the cell density, but that cell–cell adhesive junctions remain weaker in scarred epithelial zones. Taken together, our findings provide a better understanding of the wound healing process over protein gradients, which are reminiscent of haptotaxis.

Microfabricated Gaps Reveal the Effect of Geometrical Control in Wound Healing

The geometry (size and shape) of gaps is a key determinant in controlling gap closure during wound healing. However, conventional methods for creating gaps result in un-defined geometries and poorly characterized conditions (cell death factors and cell debris), which can influence the gap closure process. To overcome these limitations, a novel method to create well-defined geometrical gaps is developed. First, smooth muscle cells (SMCs) are seeded in variously shaped micro-containers made out of hyaluronic acid hydrogels. Cell proliferation and cell tension induce fibrous collagen production by SMCs predominantly around the edges of the micro-containers. Upon removal of SMCs, the selectively deposited collagen results in micro-containers with cell-adhesive regions along the edges and walls. Fibroblasts are seeded in these micro-containers, and upon attaching and spreading, they naturally form gaps with different geometries. The rapid proliferation of fibroblasts from the edge results in filling and closure of the gaps. It is demonstrated that gap closure rate as well as closure mechanism is strongly influenced by geometrical features, which points to an important role for cellular tension and cell proliferation in gap closure.

Adhesion-mediated heterogeneous actin organization governs apoptotic cell extrusion

Apoptotic extrusion is crucial in maintaining epithelial homeostasis. Current literature supports that epithelia respond to extrusion by forming a supracellular actomyosin purse-string in the neighbors. However, whether other actin structures could contribute to extrusion and how forces generated by these structures can be integrated are unknown. Here, we found that during extrusion, a heterogeneous actin network composed of lamellipodia protrusions and discontinuous actomyosin cables, was reorganized in the neighboring cells. The early presence of basal lamellipodia protrusion participated in both basal sealing of the extrusion site and orienting the actomyosin purse-string. The co-existence of these two mechanisms is determined by the interplay between the cell-cell and cell-substrate adhesions. A theoretical model integrates these cellular mechanosensitive components to explain why a dual-mode mechanism, which combines lamellipodia protrusion and purse-string contractility, leads to more efficient extrusion than a single-mode mechanism. In this work, we provide mechanistic insight into extrusion, an essential epithelial homeostasis process.

Actin-ring segment switching drives nonadhesive gap closure

Gap closure to eliminate physical discontinuities and restore tissue integrity is a fundamental process in normal development and repair of damaged tissues and organs. Here, we demonstrate a nonadhesive gap closure model in which collective cell migration, large-scale actin-network fusion, and purse-string contraction orchestrate to restore the gap. Proliferative pressure drives migrating cells to attach onto the gap front at which a pluricellular actin ring is already assembled. An actin-ring segment switching process then occurs by fusion of actin fibers from the newly attached cells into the actin cable and defusion from the previously lined cells, thereby narrowing the gap. Such actin-cable segment switching occurs favorably at high curvature edges of the gap, yielding size-dependent gap closure. Cellular force microscopies evidence that a persistent rise in the radial component of inward traction force signifies successful actin-cable segment switching. A kinetic model that integrates cell proliferation, actin fiber fusion, and purse-string contraction is formulated to quantitatively account for the gap-closure dynamics. Our data reveal a previously unexplored mechanism in which cells exploit multifaceted strategies in a highly cooperative manner to close nonadhesive gaps.

The Research Advance of Cell Bridges in vitro

The microenvironment in which cells reside in vivo dictates their biological and mechanical functioning is associated with morphogenetic and regenerative processes and may find implications in regenerative medicine and tissue engineering. The development of nano- and micro-fabricated technologies, three-dimensional (3D) printing technique, and biomimetic medical materials have enabled researchers to prepare novel advanced substrates mimicking the in vivo microenvironment. Most of the novel morphologies and behaviors of cells, including contact guidance and cell bridges which are observed in vivo but are not perceived in the traditional two-dimensional (2D) culture system, emerged on those novel substrates. Using cell bridges, cell can span over the surface of substrates to maintain mechanical stability and integrity of tissue, as observed in physiological processes, such as wound healing, regeneration and development. Compared to contact guidance, which has received increased attention and is investigated extensively, studies on cell bridges remain scarce. Therefore, in this mini-review, we have comprehensively summarized and classified different kinds of cell bridges formed on various substrates and highlighted possible biophysical mechanisms underlying cell bridge formation for their possible implication in the fields of tissue engineering and regenerative medicine.

Accessible dynamic micropatterns in monolayer cultures via modified desktop xurography

Micropatterned cell cultures provide an important tool to understand dynamic biological processes, but often require specialized equipment and expertise. Here we present subtractive bioscribing (SuBscribe), a readily accessible and inexpensive technique to generate dynamic micropatterns in biomaterial monolayers on-the-fly. We first describe our modifications to a commercially available desktop xurographer and demonstrate the utility and limits of this system in creating micropatterned cultures by mechanically scribing patterns into a brittle, non-adhesive biomaterial layer. Patterns are sufficiently small to influence cell morphology and orientation and can be extended to pattern large areas with complex reproducible shapes. We also demonstrate the use of this system as a dynamic patterning tool for cocultures. Finally, we use this technique to explore and improve upon the well-established epithelial scratch assay, and demonstrate that robotic control of the scratching tool can be used to create custom-shaped wounds in epithelial monolayers, and that the scribing direction leaves trace remnants of matrix molecules that may significantly affect conventional implementations of this common assay.

Transduction of cell and matrix geometric cues by the actin cytoskeleton

Engineered culture substrates have proven invaluable for investigating the role of cell and extracellular matrix geometry in governing cell behavior. While the mechanisms relating geometry to phenotype are complex, it is clear that the actin cytoskeleton plays a key role in integrating geometric inputs and transducing these cues into intracellular signals that drive downstream biology. Here, we review recent progress in elucidating the role of the cell and matrix geometry in regulating actin cytoskeletal architecture and mechanics. We address new developments in traditional two-dimensional culture paradigms and discuss efforts to extend these advances to three-dimensional systems, ranging from nanotextured surfaces to microtopographical systems (e.g. channels) to fully three-dimensional matrices.

Dewetting: From Physics to the Biology of Intoxicated Cells

Pathogenic bacteria colonize or disseminate into cells and tissues by inducing large-scale remodeling of host membranes. The physical phenomena underpinning these massive membrane extension and deformation are poorly understood. Invasive strategies of pathogens have been recently enriched by the description of a spectacular mode of opening of large transendothelial cell macroaperture (TEM) tunnels correlated to the dissemination of EDIN-producing strains of Staphylococcus aureus via a hematogenous route or to the induction of gelatinous edema triggered by the edema toxin from Bacillus anthracis. Remarkably, these highly dynamic tunnels close rapidly after they reach a maximal size. Opening and closure of TEMs in cells lasts for hours without inducing endothelial cell death. Multidisciplinary studies have started to provide a broader perspective of both the molecular determinants controlling cytoskeleton organization at newly curved membranes generated by the opening of TEMs and the physical processes controlling the dynamics of these tunnels. Here we discuss the analogy between the opening of TEM tunnels and the physical principles of dewetting, stemming from a parallel between membrane tension and surface tension. This analogy provides a broad framework to investigate biophysical constraints in cell membrane dynamics and their diversion by certain invasive microbial agents.

Functional reservoir microcapsules generated via microfluidic fabrication for long-term cardiovascular therapeutics

Cardiovascular disease is a chronic disease that leads to impaired cardiac function and requires long-term management to control its progression. Despite the importance of hydrogels for therapeutic applications, a contradiction between the size of a hydrogel and the amount of loaded drug has been encountered when using conventional fabrication methods. In this study, biocompatible reservoir microcapsules (diameter ∼100 μm) with a large liquid core and polymeric shell were fabricated via a one-step phase separation of poly(ethylene glycol)diacrylate (PEGDA) and dextran within pre-gel droplets through microfluidics. By controlling the process of phase separation, high drug-loading efficiency (∼80%) for long-term release (30 days) of vascular endothelial growth factor (VEGF) and platelet-derived growth factor (PDGF) was achieved. Drug molecules were dispersed within the liquid core at a concentration above saturation solubility for sustained delivery via regulation of the shells. Effective therapeutic enhancement of human umbilical vein endothelial cell (HUVEC) and umbilical artery smooth muscle cell (SMC) proliferation and tube formation in vitro promoted rapid cell proliferation and increased the number of migrated cells by ∼1.7 times. Moreover, in vivo blood vessel regeneration for cardiovascular control induced by sustained dual-drug (VEGF and PDGF) delivery to the rat heart was achieved, showing the effectiveness of long-term protein delivery in improving cardiac function and significantly reducing ventricular wall thickness and fibrosis of the infarct region. The ratio of heart tissue scarring was reduced to 11.2% after microcapsule treatment compared with 21.4% after saline treatment in the rat model. By using these reservoir microcapsules, similar sustained delivery of proteins, mRNAs and biologic drugs could be developed for the treatment of a range of long-term chronic diseases and regenerative medicine.

The role of single cell mechanical behavior and polarity in driving collective cell migration

The directed migration of cell collectives is essential in various physiological processes, such as epiboly, intestinal epithelial turnover, and convergent extension during morphogenesis as well as during pathological events like wound healing and cancer metastasis. Collective cell migration leads to the emergence of coordinated movements over multiple cells. Our current understanding emphasizes that these movements are mainly driven by large-scale transmission of signals through adherens junctions. In this study, we show that collective movements of epithelial cells can be triggered by polarity signals at the single cell level through the establishment of coordinated lamellipodial protrusions. We designed a minimalistic model system to generate one-dimensional epithelial trains confined in ring shaped patterns that recapitulate rotational movements observed in vitro in cellular monolayers and in vivo in genitalia or follicular cell rotation. Using our system, we demonstrated that cells follow coordinated rotational movements after the establishment of directed Rac1-dependent polarity over the entire monolayer. Our experimental and numerical approaches show that the maintenance of coordinated migration requires the acquisition of a front-back polarity within each single cell but does not require the maintenance of cell-cell junctions. Taken together, these unexpected findings demonstrate that collective cell dynamics in closed environments as observed in multiple in vitro and in vivo situations can arise from single cell behavior through a sustained memory of cell polarity.

Effects of Microstripe Geometry on Guided Cell Migration

Cell migration on material surfaces is a fundament issue in the fields of biomaterials, cell biology, tissue engineering, regenerative medicine, etc. Herein, we aim to guide cell migration by flat microstripes with significant contrast of cell adhesion and varied geometric features of the adhesive stripes. To this end, we designed and fabricated cell-adhesive arginine-glycine-aspartate (RGD) microstripes on the nonfouling poly(ethylene glycol) (PEG) background and examined the microstripe-guided adhesion and migration of a few cell types. The migration of cell clusters adhering on the RGD regions was found to be significantly affected by the widths and arc radiuses of the guided microstripes. The cells migrated fastest on the straight microstripes with width of about 20 μm, which we defined as single file confined migration (SFCM). We also checked the possible left-right asymmetric bias of cell migration guided by combinatory microstripes with alternative wavy and quasi-straight stripes under a given width, and found that the velocity of CCW (counter clockwise) migration was higher than that of CW (clockwise) migration for primary rat mesenchymal stem cells (rMSCs), whereas no left-right asymmetric bias was observed for NIH3T3 (mouse embryonic fibroblast cell line) and Hela (human cervix epithelial carcinoma cell line) cells. Comparison of migration of cells on the nanotopological stripe and smooth surfaces further confirmed the importance of cell orientation coherence for guided cell migration and strengthened the superiority of SFCM.

A traction force threshold signifies metastatic phenotypic change in multicellular epithelia

Cancer metastasis has been believed as a genetically programmed process that is commonly marked by biochemical signals. Here using extracellular matrix control of cellular mechanics, we establish that cellular force threshold can also mark in vitro metastatic phenotypic change and malignant transformation in HCT-8 cell colonies. We observe that for prolonged culture time the HCT-8 cell colonies disperse into individual malignant cells, and the metastatic-like dispersion depends on both cell-seeding gel stiffness and colony size. Cellular force microscopies show that gel stiffness and colony size are also two key parameters that modulate cellular forces, suggesting the correlations between the cellular forces and the metastatic phenotypic change. Using our recently developed biophysical model, we construct an extracellular traction phase diagram in the stiffness-size space, filled with experimental data on the colony behavior. From the phase diagram we identify a phase boundary as a traction force threshold above which the metastatic phenotypic transition occurs and below which the cell colonies remain cohesive. Our finding suggests that the traction threshold can be regarded as an effective mechano-marker for the onset of the metastatic-like dispersion and malignant transformation.

Tissue Fluidity Promotes Epithelial Wound Healing

The collective behaviour of cells in epithelial tissues is dependent on their mechanical properties. However, the contribution of tissue mechanics to wound healing in vivo remains poorly understood. Here we investigate the relationship between tissue mechanics and wound healing in live Drosophila wing imaginal discs and show that by tuning epithelial cell junctional tension, we can systematically alter the rate of wound healing. Coincident with the contraction of an actomyosin purse string, we observe cells flowing past each other at the wound edge by intercalating, reminiscent of molecules in a fluid, resulting in seamless wound closure. Using a cell-based physical model, we predict that a reduction in junctional tension fluidises the tissue through an increase in intercalation rate and corresponding reduction in bulk viscosity, in the manner of an unjamming transition. The resultant fluidisation of the tissue accelerates wound healing. Accordingly, when we experimentally reduce tissue tension in wing discs, intercalation rate increases and wounds repair in less time.

Sustained Oscillations of Epithelial Cell Sheets

Morphological changes during development, tissue repair, and disease largely rely on coordinated cell movements and are controlled by the tissue environment. Epithelial cell sheets are often subjected to large-scale deformation during tissue formation. The active mechanical environment in which epithelial cells operate have the ability to promote collective oscillations, but how these cellular movements are generated and relate to collective migration remains unclear. Here, combining in vitro experiments and computational modeling, we describe a form of collective oscillations in confined epithelial tissues in which the oscillatory motion is the dominant contribution to the cellular movements. We show that epithelial cells exhibit large-scale coherent oscillations when constrained within micropatterns of varying shapes and sizes and that their period and amplitude are set by the smallest confinement dimension. Using molecular perturbations, we then demonstrate that force transmission at cell-cell junctions and its coupling to cell polarity are pivotal for the generation of these collective movements. We find that the resulting tissue deformations are sufficient to trigger osillatory mechanotransduction of YAP within cells, potentially affecting a wide range of cellular processes.

Forceful closure: cytoskeletal networks in embryonic wound repair

Embryonic tissues heal wounds rapidly and without scarring, in a process conserved across species and driven by collective cell movements. The mechanisms of coordinated cell movement during embryonic wound closure also drive tissue development and cancer metastasis; therefore, embryonic wound repair has received considerable attention as a model of collective cell migration. During wound closure, a supracellular actomyosin cable at the wound edge coordinates cells, while actin-based protrusions contribute to cell crawling and seamless wound healing. Other cytoskeletal networks are reorganized during wound repair: microtubules extend into protrusions and along cell-cell boundaries as cells stretch into damaged regions, septins accumulate at the wound margin, and intermediate filaments become polarized in the cells adjacent to the wound. Thus, diverse cytoskeletal networks work in concert to maintain tissue structure, while also driving and organizing cell movements to promote rapid repair. Understanding the signals that coordinate the dynamics of different cytoskeletal networks, and how adhesions between cells or with the extracellular matrix integrate forces across cells, will be important to elucidate the mechanisms of efficient embryonic wound healing and may have far-reaching implications for developmental and cancer cell biology.

Large-scale curvature sensing by directional actin flow drives cellular migration mode switching

Cell migration over heterogeneous substrates during wound healing or morphogenetic processes leads to shape changes driven by different organizations of the actin cytoskeleton and by functional changes including lamellipodial protrusions and contractile actin cables. Cells distinguish between cell-sized positive and negative curvatures in their physical environment by forming protrusions at positive ones and actin cables at negative ones; however, the cellular mechanisms remain unclear. Here, we report that concave edges promote polarized actin structures with actin flow directed towards the cell edge, in contrast to well-documented retrograde flow at convex edges. Anterograde flow and contractility induce a tension anisotropy gradient. A polarized actin network is formed, accompanied by a local polymerization-depolymerization gradient, together with leading-edge contractile actin cables in the front. These cables extend onto non-adherent regions while still maintaining contact with the substrate through focal adhesions. The contraction and dynamic reorganization of this actin structure allows forward movements enabling cell migration over non-adherent regions on the substrate. These versatile functional structures may help cells sense and navigate their environment by adapting to external geometric and mechanical cues.

Continuum Models of Collective Cell Migration

Collective cell migration plays a central role in tissue development, morphogenesis, wound repair and cancer progression. With the growing realization that physical forces mediate cell motility in development and physiology, a key biological question is how cells integrate molecular activities for force generation on multicellular scales. In this review we discuss recent advances in modeling collective cell migration using quantitative tools and approaches rooted in soft matter physics. We focus on theoretical models of cell aggregates as continuous active media, where the feedback between mechanical forces and regulatory biochemistry gives rise to rich collective dynamical behavior. This class of models provides a powerful predictive framework for the physiological dynamics that underlies many developmental processes, where cells need to collectively migrate like a viscous fluid to reach a target region, and then stiffen to support mechanical stresses and maintain tissue cohesion.

Computational Modeling of Collective Cell Migration: Mechanical and Biochemical Aspects

Collective cell migration plays key roles in various physiological and pathological processes in multicellular organisms, including embryonic development, wound healing, and formation of cancer metastases. Such collective migration involves complex crosstalk among cells and their environment at both biochemical and mechanical levels. Here, we review various computational modeling strategies that have been helpful in decoding the dynamics of collective cell migration. Most of such attempts have focused either aspect - mechanical or biochemical regulation of collective cell migration, and have yielded complementary insights. Finally, we suggest some possible ways to integrate these models to gain a more comprehensive understanding of collective cell migration.