Stochastic collective movement of cells and fingering morphology: no maverick cells

The cellular Potts model on disordered lattices

The cellular Potts model, also known as the Glazier-Graner-Hogeweg model, is a lattice-based approach by which biological tissues at the level of individual cells can be numerically studied. Traditionally, a square or hexagonal underlying lattice structure is assumed for two-dimensional systems, and this is known to introduce artifacts in the structure and dynamics of the model tissues. That is, on regular lattices, cells can assume shapes that are dictated by the symmetries of the underlying lattice. Here, we developed a variant of this method that can be applied to a broad class of (ir)regular lattices. We show that on an irregular lattice deriving from a fluid-like configuration, two types of artifacts can be removed. We further report on the transition between a fluid-like disordered and a solid-like hexagonally ordered phase present for monodisperse confluent cells as a function of their surface tension. This transition shows the hallmarks of a first-order phase transition and is different from the glass/jamming transitions commonly reported for the vertex and active Voronoi models. We emphasize this by analyzing the distribution of shape parameters found in our state space. Our analysis provides a useful reference for the future study of epithelia using the (ir)regular cellular Potts model.

Fingering Instability Accelerates Population Growth of a Proliferating Cell Collective

During the growth of a cell collective, such as proliferating microbial colonies and epithelial tissues, the local cell growth increases the local pressure, which in turn suppresses cell growth. How this pressure-growth coupling affects the growth of a cell collective remains unclear. Here, we answer this question using a continuum model of a cell collective. We find that a fast-growing leading front and a slow-growing interior of the cell collective emerge due to the pressure-dependent growth rate. The leading front can exhibit fingering instability, and we confirm the predicted instability criteria numerically with the leading front explicitly simulated. Intriguingly, we find that fingering instability is not only a consequence of local cell growth but also enhances the entire population's growth rate as positive feedback. Our work unveils the fitness advantage of fingering formation and suggests that the ability to form protrusions can be evolutionarily selected.

Collective rotational motion of freely expanding T84 epithelial cell colonies

Coordinated rotational motion is an intriguing, yet still elusive mode of collective cell migration, which is relevant in pathological and morphogenetic processes. Most of the studies on this topic have been carried out on epithelial cells plated on micropatterned substrates, where cell motion is confined in regions of well-defined shapes coated with extracellular matrix adhesive proteins. The driver of collective rotation in such conditions has not been clearly elucidated, although it has been speculated that spatial confinement can play an essential role in triggering cell rotation. Here, we study the growth of epithelial cell colonies freely expanding (i.e. with no physical constraints) on the surface of cell culture plates and focus on collective cell rotation in such conditions, a case which has received scarce attention in the literature. One of the main findings of our work is that coordinated cell rotation spontaneously occurs in cell clusters in the free growth regime, thus implying that cell confinement is not necessary to elicit collective rotation as previously suggested. The extent of collective rotation was size and shape dependent: a highly coordinated disc-like rotation was found in small cell clusters with a round shape, while collective rotation was suppressed in large irregular cell clusters generated by merging of different clusters in the course of their growth. The angular motion was persistent in the same direction, although clockwise and anticlockwise rotations were equally likely to occur among different cell clusters. Radial cell velocity was quite low as compared to the angular velocity, in agreement with the free expansion regime where cluster growth is essentially governed by cell proliferation. A clear difference in morphology was observed between cells at the periphery and the ones in the core of the clusters, the former being more elongated and spread out as compared to the latter. Overall, our results, to our knowledge, provide the first quantitative and systematic evidence that coordinated cell rotation does not require a spatial confinement and occurs spontaneously in freely expanding epithelial cell colonies, possibly as a mechanism for the system.

Combinatorial mathematical modelling approaches to interrogate rear retraction dynamics in 3D cell migration

Cell migration in 3D microenvironments is a complex process which depends on the coordinated activity of leading edge protrusive force and rear retraction in a push-pull mechanism. While the potentiation of protrusions has been widely studied, the precise signalling and mechanical events that lead to retraction of the cell rear are much less well understood, particularly in physiological 3D extra-cellular matrix (ECM). We previously discovered that rear retraction in fast moving cells is a highly dynamic process involving the precise spatiotemporal interplay of mechanosensing by caveolae and signalling through RhoA. To further interrogate the dynamics of rear retraction, we have adopted three distinct mathematical modelling approaches here based on (i) Boolean logic, (ii) deterministic kinetic ordinary differential equations (ODEs) and (iii) stochastic simulations. The aims of this multi-faceted approach are twofold: firstly to derive new biological insight into cell rear dynamics via generation of testable hypotheses and predictions; and secondly to compare and contrast the distinct modelling approaches when used to describe the same, relatively under-studied system. Overall, our modelling approaches complement each other, suggesting that such a multi-faceted approach is more informative than methods based on a single modelling technique to interrogate biological systems. Whilst Boolean logic was not able to fully recapitulate the complexity of rear retraction signalling, an ODE model could make plausible population level predictions. Stochastic simulations added a further level of complexity by accurately mimicking previous experimental findings and acting as a single cell simulator. Our approach highlighted the unanticipated role for CDK1 in rear retraction, a prediction we confirmed experimentally. Moreover, our models led to a novel prediction regarding the potential existence of a 'set point' in local stiffness gradients that promotes polarisation and rapid rear retraction.

Mechanism underlying dynamic scaling properties observed in the contour of spreading epithelial monolayer

We found evidence of dynamic scaling in the spreading of Madin-Darby canine kidney (MDCK) cell monolayer, which can be characterized by the Hurst exponent α=0.86 and the growth exponent β=0.73, and theoretically and experimentally clarified the mechanism that governs the contour shape dynamics. Dynamic scaling refers to the roughness of the surface scales, both spatially and temporally. During the spreading of the monolayer, it is known that so-called leader cells generate the driving force and lead the other cells. Our time-lapse observations of cell behavior showed that these leader cells appeared at the early stage of the spreading and formed the monolayer protrusion. Informed by these observations, we developed a simple mathematical model that included differences in cell motility, cell-cell adhesion, and random cell movement. The model reproduced the quantitative characteristics obtained from the experiment, such as the spreading speed, the distribution of the increment, and the dynamic scaling law. Analysis of the model equation shows that the model can reproduce different scaling laws from (α=0.5,β=0.25) to (α=0.9,β=0.75), where the exponents α and β are determined by two dimensionless quantities determined by the microscopic cell behavior. From the analytical result, parameter estimation from the experimental results was achieved. The monolayer on the collagen-coated dishes showed a different scaling law, α=0.74,β=0.68, suggesting that cell motility increased ninefold. This result was consistent with the assay of the single-cell motility. Our study demonstrated that the dynamics of the contour of the monolayer were explained by the simple model, and we propose a mechanism that exhibits the dynamic scaling property.

Tracking collective cell motion by topological data analysis

By modifying and calibrating an active vertex model to experiments, we have simulated numerically a confluent cellular monolayer spreading on an empty space and the collision of two monolayers of different cells in an antagonistic migration assay. Cells are subject to inertial forces and to active forces that try to align their velocities with those of neighboring ones. In agreement with experiments in the literature, the spreading test exhibits formation of fingers in the moving interfaces, there appear swirls in the velocity field, and the polar order parameter and the correlation and swirl lengths increase with time. Numerical simulations show that cells inside the tissue have smaller area than those at the interface, which has been observed in recent experiments. In the antagonistic migration assay, a population of fluidlike Ras cells invades a population of wild type solidlike cells having shape parameters above and below the geometric critical value, respectively. Cell mixing or segregation depends on the junction tensions between different cells. We reproduce the experimentally observed antagonistic migration assays by assuming that a fraction of cells favor mixing, the others segregation, and that these cells are randomly distributed in space. To characterize and compare the structure of interfaces between cell types or of interfaces of spreading cellular monolayers in an automatic manner, we apply topological data analysis to experimental data and to results of our numerical simulations. We use time series of data generated by numerical simulations to automatically group, track and classify the advancing interfaces of cellular aggregates by means of bottleneck or Wasserstein distances of persistent homologies. These techniques of topological data analysis are scalable and could be used in studies involving large amounts of data. Besides applications to wound healing and metastatic cancer, these studies are relevant for tissue engineering, biological effects of materials, tissue and organ regeneration.

Autocrine inhibition of cell motility can drive epithelial branching morphogenesis in the absence of growth

Epithelial branching morphogenesis drives the development of organs such as the lung, salivary gland, kidney and the mammary gland. It involves cell proliferation, cell differentiation and cell migration. An elaborate network of chemical and mechanical signals between the epithelium and the surrounding mesenchymal tissues regulates the formation and growth of branching organs. Surprisingly, when cultured in isolation from mesenchymal tissues, many epithelial tissues retain the ability to exhibit branching morphogenesis even in the absence of proliferation. In this work, we propose a simple, experimentally plausible mechanism that can drive branching morphogenesis in the absence of proliferation and cross-talk with the surrounding mesenchymal tissue. The assumptions of our mathematical model derive from in vitro observations of the behaviour of mammary epithelial cells. These data show that autocrine secretion of the growth factor TGF[Formula: see text]1 inhibits the formation of cell protrusions, leading to curvature-dependent inhibition of sprouting. Our hybrid cellular Potts and partial-differential equation model correctly reproduces the experimentally observed tissue-geometry-dependent determination of the sites of branching, and it suffices for the formation of self-avoiding branching structures in the absence and also in the presence of cell proliferation. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Using a continuum model to decipher the mechanics of embryonic tissue spreading from time-lapse image sequences: An approximate Bayesian computation approach

Advanced imaging techniques generate large datasets capable of describing the structure and kinematics of tissue spreading in embryonic development, wound healing, and the progression of many diseases. These datasets can be integrated with mathematical models to infer biomechanical properties of the system, typically identifying an optimal set of parameters for an individual experiment. However, these methods offer little information on the robustness of the fit and are generally ill-suited for statistical tests of multiple experiments. To overcome this limitation and enable efficient use of large datasets in a rigorous experimental design, we use the approximate Bayesian computation rejection algorithm to construct probability density distributions that estimate model parameters for a defined theoretical model and set of experimental data. Here, we demonstrate this method with a 2D Eulerian continuum mechanical model of spreading embryonic tissue. The model is tightly integrated with quantitative image analysis of different sized embryonic tissue explants spreading on extracellular matrix (ECM) and is regulated by a small set of parameters including forces on the free edge, tissue stiffness, strength of cell-ECM adhesions, and active cell shape changes. We find statistically significant trends in key parameters that vary with initial size of the explant, e.g., for larger explants cell-ECM adhesion forces are weaker and free edge forces are stronger. Furthermore, we demonstrate that estimated parameters for one explant can be used to predict the behavior of other similarly sized explants. These predictive methods can be used to guide further experiments to better understand how collective cell migration is regulated during development.

Colloidal Silver Induces Cytoskeleton Reorganization and E-Cadherin Recruitment at Cell-Cell Contacts in HaCaT Cells

Up until the first half of the 20th century, silver found significant employment in medical applications, particularly in the healing of open wounds, thanks to its antibacterial and antifungal properties. Wound repair is a complex and dynamic biological process regulated by several pathways that cooperate to restore tissue integrity and homeostasis. To facilitate healing, injuries need to be promptly treated. Recently, the interest in alternatives to antibiotics has been raised given the widespread phenomenon of antibiotic resistance. Among these alternatives, the use of silver appears to be a valid option, so a resurgence in its use has been recently observed. In particular, in contrast to ionic silver, colloidal silver, a suspension of metallic silver particles, shows antibacterial activity displaying less or no toxicity. However, the human health risks associated with exposure to silver nanoparticles (NP) appear to be conflicted, and some studies have suggested that it could be toxic in different cellular contexts. These potentially harmful effects of silver NP depend on various parameters including NP size, which commonly range from 1 to 100 nm. In this study, we analyzed the effect of a colloidal silver preparation composed of very small and homogeneous nanoparticles of 0.62 nm size, smaller than those previously tested. We found no adverse effect on the cell proliferation of HaCaT cells, even at high NP concentration. Time-lapse microscopy and indirect immunofluorescence experiments demonstrated that this preparation of colloidal silver strongly increased cell migration, re-modeled the cytoskeleton, and caused recruitment of E-cadherin at cell-cell junctions of human cultured keratinocytes.

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance

For many organisms, shapes emerge from growth, which generates stresses, which in turn can feedback on growth. In this review, theoretical methods to analyse various aspects of morphogenesis are discussed with the aim to determine the most adapted method for tissue mechanics. We discuss the need to work at scales intermediate between cells and tissues and emphasize the use of finite elasticity for this. We detail the application of these ideas to four systems: active cells embedded in tissues, brain cortical convolutions, the cortex of Caenorhabditis elegans during elongation and finally the proliferation of epithelia on extracellular matrix. Numerical models well adapted to inhomogeneities are also presented. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Active Fingering Instability in Tissue Spreading

During the spreading of epithelial tissues, the advancing tissue front often develops fingerlike protrusions. Their resemblance to traditional viscous fingering patterns in driven fluids suggests that epithelial fingers could arise from an interfacial instability. However, the existence and physical mechanism of such a putative instability remain unclear. Here, based on an active polar fluid model for epithelial spreading, we analytically predict a generic instability of the tissue front. On the one hand, active cellular traction forces impose a velocity gradient that leads to an accelerated front, which is, thus, unstable to long-wavelength perturbations. On the other hand, contractile intercellular stresses typically dominate over surface tension in stabilizing short-wavelength perturbations. Finally, the finite range of hydrodynamic interactions in the tissue selects a wavelength for the fingering pattern, which is, thus, given by the smallest between the tissue size and the hydrodynamic screening length. Overall, we show that spreading epithelia experience an active fingering instability based on a simple kinematic mechanism. Moreover, our results underscore the crucial role of long-range hydrodynamic interactions in the dynamics of tissue morphology.

Fingering instabilities in tissue invasion: an active fluid model

Metastatic tumours often invade healthy neighbouring tissues by forming multicellular finger-like protrusions emerging from the cancer mass. To understand the mechanical context behind this phenomenon, we here develop a minimalist fluid model of a self-propelled, growing biological tissue. The theory involves only four mechanical parameters and remains analytically trackable in various settings. As an application of the model, we study the evolution of a two-dimensional circular droplet made of our active and expanding fluid, and embedded in a passive non-growing tissue. This system could be used to model the evolution of a carcinoma in an epithelial layer. We find that our description can explain the propensity of tumour tissues to fingering instabilities, as conditioned by the magnitude of active traction and the growth kinetics. We are also able to derive predictions for the tumour size at the onset of metastasis, and for the number of subsequent invasive fingers. Our active fluid model may help describe a wider range of biological processes, including wound healing and developmental patterning.

Growth of adult spinal cord in knifefish: Development and parametrization of a distributed model

The study of indeterminate-growing organisms such as teleost fish presents a unique opportunity for improving our understanding of central nervous tissue growth during adulthood. Integrating the existing experimental data associated with this process into a theoretical framework through mathematical or computational modeling provides further research avenues through sensitivity analysis and optimization. While this type of approach has been used extensively in investigations of tumor growth, wound healing, and bone regeneration, the development of nervous tissue has been rarely studied within a modeling framework. To address this gap, the present work introduces a distributed model of spinal cord growth in the knifefish Apteronotus leptorhynchus, an established teleostean model of adult growth in the central nervous system. The proposed model incorporates two mechanisms, cell proliferation by active stem/progenitor cells and cell drift due to population pressure, both of which are subject to global constraints. A coupled reaction-diffusion equation approach was adopted to represent the densities of actively-proliferating and non-proliferating cells along the longitudinal axis of the spinal cord. Computer simulations using this model yielded biologically-feasible growth trajectories. Subsequent comparisons with whole-organism growth curves allowed the estimation of previously-unknown parameters, such as relative growth rates.

Collective cell migration: a physics perspective

Cells have traditionally been viewed either as independently moving entities or as somewhat static parts of tissues. However, it is now clear that in many cases, multiple cells coordinate their motions and move as collective entities. Well-studied examples comprise development events, as well as physiological and pathological situations. Different ex vivo model systems have also been investigated. Several recent advances have taken place at the interface between biology and physics, and have benefitted from progress in imaging and microscopy, from the use of microfabrication techniques, as well as from the introduction of quantitative tools and models. We review these interesting developments in quantitative cell biology that also provide rich examples of collective out-of-equilibrium motion.

Phenomenological approaches to collective behavior in epithelial cell migration

Collective cell migration in epithelial tissues resembles fluid-like behavior in time-lapse recordings. In the last years, hydrodynamic velocity fields in living matter have been studied intensely. The emergent properties were remarkably similar to phenomena known from active soft matter systems. Here, we review migration experiments of large cellular ensembles as well as of mesoscopic cohorts in micro-structured environments. Concepts such as diffusion, velocity correlations, swirl strength and polarization are metrics to quantify the cellular dynamics both in experiments as well as in computational simulations. We discuss challenges relating collective migration to single cell and oligocellular behavior as well as linking the phenotypic parameters to the underlying cytoskeleton dynamics and signaling networks. This article is part of a Special Issue entitled: Mechanobiology.

A modeling approach to study the effect of cell polarization on keratinocyte migration

The skin forms an efficient barrier against the environment, and rapid cutaneous wound healing after injury is therefore essential. Healing of the uppermost layer of the skin, the epidermis, involves collective migration of keratinocytes, which requires coordinated polarization of the cells. To study this process, we developed a model that allows analysis of live-cell images of migrating keratinocytes in culture based on a small number of parameters, including the radius of the cells, their mass and their polarization. This computational approach allowed the analysis of cell migration at the front of the wound and a reliable identification and quantification of the impaired polarization and migration of keratinocytes from mice lacking fibroblast growth factors 1 and 2 – an established model of impaired healing. Therefore, our modeling approach is suitable for large-scale analysis of migration phenotypes of cells with specific genetic defects or upon treatment with different pharmacological agents.

Modeling the finger instability in an expanding cell monolayer

Curvature-controlled cellular forces at the edge of an expanding monolayer are sufficient for the initiation and growth of finger-like instability.

Probabilistic Voxel-Fe model for single cell motility in 3D

BACKGROUND

Cells respond to a variety of external stimuli regulated by the environment conditions. Mechanical, chemical and biological factors are of great interest and have been deeply studied. Furthermore, mathematical and computational models have been rapidly growing over the past few years, permitting researches to run complex scenarios saving time and resources. Usually these models focus on specific features of cell migration, making them only suitable to study restricted phenomena.

METHODS

Here we present a versatile finite element (FE) cell-scale 3D migration model based on probabilities depending in turn on ECM mechanical properties, chemical, fluid and boundary conditions.

RESULTS

With this approach we are able to capture important outcomes of cell migration such as: velocities, trajectories, cell shape and aspect ratio, cell stress or ECM displacements.

CONCLUSIONS

The modular form of the model will allow us to constantly update and redefine it as advancements are made in clarifying how cellular events take place.

Cell migration

Cell migration is fundamental to establishing and maintaining the proper organization of multicellular organisms. Morphogenesis can be viewed as a consequence, in part, of cell locomotion, from large-scale migrations of epithelial sheets during gastrulation, to the movement of individual cells during development of the nervous system. In an adult organism, cell migration is essential for proper immune response, wound repair, and tissue homeostasis, while aberrant cell migration is found in various pathologies. Indeed, as our knowledge of migration increases, we can look forward to, for example, abating the spread of highly malignant cancer cells, retarding the invasion of white cells in the inflammatory process, or enhancing the healing of wounds. This article is organized in two main sections. The first section is devoted to the single-cell migrating in isolation such as occurs when leukocytes migrate during the immune response or when fibroblasts squeeze through connective tissue. The second section is devoted to cells collectively migrating as part of multicellular clusters or sheets. This second type of migration is prevalent in development, wound healing, and in some forms of cancer metastasis.

A computational model for collective cellular motion in three dimensions: general framework and case study for cell pair dynamics

Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.

Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model

Modelling the displacement of thousands of cells that move in a collective way is required for the simulation and the theoretical analysis of various biological processes. Here, we tackle this question in the controlled setting where the motion of Madin-Darby Canine Kidney (MDCK) cells in a confluent epithelium is triggered by the unmasking of free surface. We develop a simple model in which cells are described as point particles with a dynamic based on the two premises that, first, cells move in a stochastic manner and, second, tend to adapt their motion to that of their neighbors. Detailed comparison to experimental data show that the model provides a quantitatively accurate description of cell motion in the epithelium bulk at early times. In addition, inclusion of model "leader" cells with modified characteristics, accounts for the digitated shape of the interface which develops over the subsequent hours, providing that leader cells invade free surface more easily than other cells and coordinate their motion with their followers. The previously-described progression of the epithelium border is reproduced by the model and quantitatively explained.

Cell migration

Cell migration is fundamental to establishing and maintaining the proper organization of multicellular organisms. Morphogenesis can be viewed as a consequence, in part, of cell locomotion, from large-scale migrations of epithelial sheets during gastrulation, to the movement of individual cells during development of the nervous system. In an adult organism, cell migration is essential for proper immune response, wound repair, and tissue homeostasis, while aberrant cell migration is found in various pathologies. Indeed, as our knowledge of migration increases, we can look forward to, for example, abating the spread of highly malignant cancer cells, retarding the invasion of white cells in the inflammatory process, or enhancing the healing of wounds. This article is organized in two main sections. The first section is devoted to the single-cell migrating in isolation such as occurs when leukocytes migrate during the immune response or when fibroblasts squeeze through connective tissue. The second section is devoted to cells collectively migrating as part of multicellular clusters or sheets. This second type of migration is prevalent in development, wound healing, and in some forms of cancer metastasis.

Dual modes of motility at the leading edge of migrating epithelial cell sheets

Purse-string healing is driven by contraction of actin/myosin cables that span cells at wound edges, and it is the predominant mode of closing small round wounds in embryonic and some adult epithelia. Wounds can also heal by cell crawling, and my colleagues and I have shown previously that the presence of unconstrained, straight edges in sheets of epithelial cells is a sufficient signal to induce healing by crawling. Here, it is reported that the presence of highly concave edges, which are free or physically constrained by an inert material (agarose), is sufficient to induce formation of purse strings. It was determined that neither of the two types of healing required cell damage or other potential stimuli by using the particularly gentle procedure of introducing gaps by digesting agarose blocks imbedded in the cell sheets. Movement by crawling depends on signaling by the EGF receptor (EGFR); however, this was not required for purse-string contraction. A migrating epithelial cell sheet usually produces finger-like projections of crawling cells. The cells between fingers contain continuous actin cables, which were also determined to contain myosin IIA and exhibit additional characteristics of purse strings. When crawling was blocked by inhibition of EGFR signaling, the concave regions continued to move, suggesting that both mechanisms contribute to propel the sheets forward. Wounding epithelial cell sheets causes activation of the EGFR, which triggers movement by crawling. The EGFR was found to be activated only at straight and convex edges, which explains how both types of movement can coexist at leading epithelial edges.

Collective cell migration: leadership, invasion and segregation

A number of biological processes, such as embryo development, cancer metastasis or wound healing, rely on cells moving in concert. The mechanisms leading to the emergence of coordinated motion remain however largely unexplored. Although biomolecular signalling is known to be involved in most occurrences of collective migration, the role of physical and mechanical interactions has only been recently investigated. In this study, a versatile framework for cell motility is implemented in silico in order to study the minimal requirements for the coordination of a group of epithelial cells. We find that cell motility and cell-cell mechanical interactions are sufficient to generate a broad array of behaviours commonly observed in vitro and in vivo. Cell streaming, sheet migration and susceptibility to leader cells are examples of behaviours spontaneously emerging from these simple assumptions, which might explain why collective effects are so ubiquitous in nature. The size of the population and its confinement appear, in particular, to play an important role in the coordination process. In all cases, the complex response of the population can be predicted from the knowledge of the correlation length of the velocity field measured in the bulk of the epithelial layer. This analysis provides also new insights into cancer metastasis and cell sorting, suggesting, in particular, that collective invasion might result from an emerging coordination in a system where single cells are mechanically unable to invade.

Migration of adhesive glioma cells: front propagation and fingering

We investigate the migration of glioma cells as a front propagation phenomenon both theoretically (by using both discrete lattice modeling and a continuum approach) and experimentally. For small effective strength of cell-cell adhesion q, the front velocity does not depend on q. When q exceeds a critical threshold, a fingeringlike front propagation is observed due to cluster formation in the invasive zone. We show that the experiments correspond to the transient regime, before the regime of front propagation is established. We performed an additional experiment on cell migration. A detailed comparison with experimental observations showed that the theory correctly predicts the maximal migration distance but underestimates the migration of the main mass of cells.

Chemical and mechanical signaling in epithelial spreading

We propose a minimal mathematical model to explain long-range coordination of dynamics of multiple cells in epithelial spreading, which may be induced, under different conditions, by a chemical signal, or mechanically induced strain, or both. The model is based on chemo-mechanical interactions including a chemical effect of strain, chemically induced polarization and active traction, and interaction between polarized cells. The results, showing kinase concentration distribution and cell displacement, velocity, and stress fields, allow us to reproduce qualitatively available experimental data and distinguish between distinct dynamical patterns observed under conditions of injury or unconstraining.

Orientation and polarity in collectively migrating cell structures: statics and dynamics

Collective cell migration is often characterized by the spontaneous onset of multicellular protrusions (known as fingers) led by a single leader cell. Working with epithelial Madin-Darby canine kidney monolayers we show that cells within the fingers, as compared with the epithelium, are well oriented and polarized along the main finger direction, which suggests that these cells actively migrate. The cell orientation and polarity decrease continuously from the tip toward the epithelium over a penetration distance of typically two finger lengths. Furthermore, laser photoablation experiments at various locations along these fingers demonstrate that the cells in the fingers are submitted to a tensile stress whose value is larger close to the tip. From a dynamical point of view, cells entering a finger gradually polarize on timescales that depend upon their particular initial position. Selective laser nanosurgery of the leader lamellipodium shows not only that these structures need a leader to progress, but that this leader itself is the consequence of a prior self-organization of the cells forming the finger. These results highlight the complex interplay between the collective orientation within the fingers and the mechanical action of the leader.

Continuum model of collective cell migration in wound healing and colony expansion

Collective cell migration plays an important role during wound healing and embryo development. Although the exact mechanisms that coordinate such migration are still unknown, experimental studies of moving cell layers have shown that the primary interactions governing the motion of the layer are the force of lamellipodia, the adhesion of cells to the substrate, and the adhesion of cells to each other. Here, we derive a two-dimensional continuum mechanical model of cell-layer migration that is based on a novel assumption of elastic deformation of the layer and incorporates basic mechanical interactions of cells as well as cell proliferation and apoptosis. The evolution equations are solved numerically using a level set method. The model successfully reproduces data from two types of experiments: 1), the contraction of an enterocyte cell layer during wound healing; and 2), the expansion of a radially symmetric colony of MDCK cells, both in the edge migration velocity and in cell-layer density. In accord with experimental observations, and in contrast to reaction-diffusion models, this model predicts a partial wound closure if lamellipod formation is inhibited at the wound edge and gives implications of the effect of spatially restricted proliferation.

Collective behavior of brain tumor cells: the role of hypoxia

We consider emergent collective behavior of a multicellular biological system. Specifically, we investigate the role of hypoxia (lack of oxygen) in migration of brain tumor cells. We performed two series of cell migration experiments. In the first set of experiments, cell migration away from a tumor spheroid was investigated. The second set of experiments was performed in a typical wound-healing geometry: Cells were placed on a substrate, a scratch was made, and cell migration into the gap was investigated. Experiments show a surprising result: Cells under normal and hypoxic conditions have migrated the same distance in the "spheroid" experiment, while in the "scratch" experiment cells under normal conditions migrated much faster than under hypoxic conditions. To explain this paradox, we formulate a discrete stochastic model for cell dynamics. The theoretical model explains our experimental observations and suggests that hypoxia decreases both the motility of cells and the strength of cell-cell adhesion. The theoretical predictions were further verified in independent experiments.

Velocity fields in a collectively migrating epithelium

We report quantitative measurements of the velocity field of collectively migrating cells in a motile epithelium. The migration is triggered by presenting free surface to an initially confluent monolayer by using a microstencil technique that does not damage the cells. To avoid the technical difficulties inherent in the tracking of single cells, the field is mapped using the technique of particle image velocimetry. The main relevant parameters, such as the velocity module, the order parameter, and the velocity correlation function, are then extracted from this cartography. These quantities are dynamically measured on two types of cells (collectively migrating Madin-Darby canine kidney (MDCK) cells and fibroblastlike normal rat kidney (NRK) cells), first as they approach confluence, and then when the geometrical constraints are released. In particular, for MDCK cells filling up the patterns, we observe a sharp decrease in the average velocity after the point of confluence, whereas the densification of the monolayer is much more regular. After the peeling off of the stencil, a velocity correlation length of approximately 200 microm is measured for MDCK cells versus only approximately 40 microm for the more independent NRK cells. Our conclusions are supported by parallel single-cell tracking experiments. By using the biorthogonal decomposition of the velocity field, we conclude that the velocity field of MDCK cells is very coherent in contrast with the NRK cells. The displacements in the fingers arising from the border of MDCK epithelia are very oriented along their main direction. They influence the velocity field in the epithelium over a distance of approximately 200 microm.

Physical model of the dynamic instability in an expanding cell culture

Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands in a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.

Traction forces during collective cell motion

Collective motion of cell cultures is a process of great interest, as it occurs during morphogenesis, wound healing, and tumor metastasis. During these processes cell cultures move due to the traction forces induced by the individual cells on the surrounding matrix. A recent study [Trepat, et al. (2009). Nat. Phys. 5, 426-430] measured for the first time the traction forces driving collective cell migration and found that they arise throughout the cell culture. The leading 5-10 rows of cell do play a major role in directing the motion of the rest of the culture by having a distinct outwards traction. Fluctuations in the traction forces are an order of magnitude larger than the resultant directional traction at the culture edge and, furthermore, have an exponential distribution. Such exponential distributions are observed for the sizes of adhesion domains within cells, the traction forces produced by single cells, and even in nonbiological nonequilibrium systems, such as sheared granular materials. We discuss these observations and their implications for our understanding of cellular flows within a continuous culture.