Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.
Multicellular collective migration is a ubiquitous strategy of cells to translocate spatially in diverse tissue environments to accomplish a wide variety of biological phenomena, viz. embryonic development, wound healing, and tumor progression. Diverse cellular functions and behaviors, for instance, cell protrusions, active contractions, cell-cell adhesion, biochemical signaling, remodeling of tissue micro-environment, etc., play their own role concomitantly to have a single concerted consequence of multicellular migration. Thus unveiling the driving principles, both biochemical and biophysica…
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 mec…
The importance of collective cellular migration during embryogenesis and tissue repair asks for a sound understanding of underlying principles and mechanisms. Here, we address recent in vitro experiments on cell monolayers which show that the advancement of the leading edge relies on cell proliferation and protrusive activity at the tissue margin. Within a simple viscoelastic mechanical model amenable to detailed analysis, we identify a key parameter responsible for tissue expansion, and we determine the dependence of the monolayer velocity as a function of measurable rheological parameters. O…
Cell migration is essential for regulating many biological processes in physiological or pathological conditions, including embryonic development and cancer invasion. In vitro and in silico studies suggest that collective cell migration is associated with some biomechanical particularities, such as restructuring of extracellular matrix, stress and force distribution profiles, and reorganization of cytoskeleton. Therefore, the phenomenon could be understood by an in-depth study of cells' behavior determinants, including but not limited to mechanical cues from the environment and from fellow tra…
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 paper, 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 …
Modeling cell interactions such as co-attraction and contact-inhibition of locomotion is essential for understanding collective cell migration. Here, we propose a novel deep reinforcement learning model for collective neural crest cell migration. We apply the deep deterministic policy gradient algorithm in association with a particle dynamics simulation environment to train agents to determine the migration path. Because of the different migration mechanisms of leader and follower neural crest cells, we train two types of agents (leaders and followers) to learn the collective cell migration be…
Tumor cells invade individually or in groups, mediated by mechanical interactions between cells and their surrounding matrix. These multicellular dynamics are reminiscent of leader-follower coordination and epithelial-mesenchymal transitions (EMT) in tissue development, which may occur via dysregulation of associated molecular or physical mechanisms. However, it remains challenging to elucidate such phenotypic heterogeneity and plasticity without precision measurements of single cell behavior. The convergence of technological developments in live cell imaging, biophysical measurements, and 3D …
Cells collectively migrate as a cluster in three-dimensional (3D) tissues, such as in embryogenesis and cancer invasion. Here, numerical simulations using a 3D vertex model show that polarized interfacial tension, expressing cell adhesion and cortex contractility, induces the cluster migration in the 3D space. The mechanism is that polarized interfacial tension induced a directional flow of cell-cell interfaces from the front to rear within the whole cluster, producing a driving force, i.e., cells move forward as a cluster by simply expanding and contracting cell-cell boundaries.
We have recorded the swarming-like collective migration of a large number of keratocytes (tissue cells obtained from the scales of goldfish) using long-term videomicroscopy. By increasing the overall density of the migrating cells, we have been able to demonstrate experimentally a kinetic phase transition from a disordered into an ordered state. Near the critical density a complex picture emerges with interacting clusters of cells moving in groups. Motivated by these experiments we have constructed a flocking model that exhibits a continuous transition to the ordered phase, while assuming only…
Metastasis is a process of cell migration that can be collective and guided by chemical cues. Viewing metastasis in this way, as a physical phenomenon, allows one to draw upon insights from other studies of collective sensing and migration in cell biology. Here we review recent progress in the study of cell sensing and migration as collective phenomena, including in the context of metastatic cells. We describe simple physical models that yield the limits to the precision of cell sensing, and we review experimental evidence that cells operate near these limits. Models of collective migration ar…
Movement of cell clusters along extracellular matrices (ECM) during tissue development, wound healing, and early stage of cancer invasion involve various inter-connected migration modes such as: (1) cell movement within clusters, (2) cluster extension (wetting) and compression (de-wetting), and (3) directional cluster movement. It has become increasingly evident that dilational and volumetric viscoelasticity of cell clusters and their surrounding substrate significantly influence these migration modes through physical parameters such as: cell and matrix surface tensions, interfacial tension be…
The spontaneous formation of vortices is a hallmark of collective cellular activity. Here, we study the onset and persistence of coherent angular motion (CAMo) as a function of the number of cells $N$ confined in circular micropatterns. We find that the persistence of CAMo increases with $N$ but exhibits a pronounced discontinuity accompanied by a geometric rearrangement of cells to a configuration containing a central cell. Computer simulations based on a generalized Potts model reproduce the emergence of vortex states and show in agreement with experiment that their stability depends on the …
Correlated velocity patterns and associated large length-scale transmission of traction forces have been observed in collective live cell migration as a response to a "wound". We argue that a simple physical model of a force-driven heterogeneous elastic membrane sliding over a viscous substrate can qualitatively explain a few experimentally observed facts: (i) the growth of velocity ordering which spreads from the wound boundary to the interior, (ii) the exponential tails of the traction force distributions, and (iii) the swirling pattern of velocities in the interior of the tissue.
Dense monolayers of living cells display intriguing relaxation dynamics, reminiscent of soft and glassy materials close to the jamming transition, and migrate collectively when space is available, as in wound healing or in cancer invasion. Here we show that collective cell migration occurs in bursts that are similar to those recorded in the propagation of cracks, fluid fronts in porous media and ferromagnetic domain walls. In analogy with these systems, the distribution of activity bursts displays scaling laws that are universal in different cell types and for cells moving on different substra…
Collective behavior in cellular populations is coordinated by biochemical signaling networks within individual cells. Connecting the dynamics of these intracellular networks to the population phenomena they control poses a considerable challenge because of network complexity and our limited knowledge of kinetic parameters. However, from physical systems we know that behavioral changes in the individual constituents of a collectively-behaving system occur in a limited number of well-defined classes, and these can be described using simple models. Here we apply such an approach to the emergence …
Substrate-based cell motility is essential for fundamental biological processes, such as tissue growth, wound healing and immune response. Even if a comprehensive understanding of this motility mode remains elusive, progress has been achieved in its modeling using a whole cell physical model. The model takes into account the main mechanisms of cell motility - actin polymerization, substrate mediated adhesion and actin-myosin dynamics and combines it with steric cell-cell and hydrodynamic interactions. The model predicts the onset of collective cell migration, which emerges spontaneously as a r…
The migration behaviors of cancer cells are known to be heterogeneous. However, the interplay between the adhesion interactions, dynamical shape changes and fluid flows in regulating cell migration heterogeneity and plasticity during cancer metastasis is still elusive. To further quantitative understanding of cell motility and morphology, we develop a theory using stochastic quantization method that describes the role of biophysical cues in regulating diverse cell motility. We show that the cumulative effect of time dependent adhesion interactions that determine the structural rearrangements a…
Cell monolayers provide an interesting example of active matter, exhibiting a phase transition from a flowing to jammed state as they age. Here we report experiments and numerical simulations illustrating how a jammed cellular layer rapidly reverts to a flowing state after a wound. Quantitative comparison between experiments and simulations shows that cells change their self-propulsion and alignement strength so that the system crosses a phase transition line, which we characterize by finite-size scaling in an active particle model. This wound-induced unjamming transition is found to occur gen…
The multicellular organization of diverse systems, such as embryos, intestines and tumours, relies on the coordinated migration of cells in 3D curved environments. In these settings, cells establish supracellular patterns of motion, including collective rotation and invasion. While such collective modes are increasingly well understood in 2D flat systems, the consequences of geometrical and topological constraints on collective cell migration in 3D curved tissues are largely unknown. Here, we study 3D collective migration in mammary cell spheroids, which represent a common and conceptually sim…
Collective motion of cells is critical to some of the most vital tasks including wound healing, development, and immune response [Friedl and Gilmour 2009; Tokarski et al. 2012; Lee et al. 2012; Beltman et al. 2009], and is common to many pathological processes including cancer cell invasion and teratogenesis [Khalil and Friedl 2010]. The extensive understanding of movement by single cells [Rørth 2011; Insall and Machesky 2011; Houk et al. 2012] is insufficient to predict the behavior of cellular groups [Theveneau et al. 2013; Trepat, X. and Fredberg 2011], and identifying underlying rules of c…
The mechanisms underlying collective migration, or the coordinated movement of a population of cells, are not well understood despite its ubiquitous nature. As a means to investigate collective migration, we consider a wound healing scenario in which a population of cells fills in the empty space left from a scratch wound. Here we present a simplified mathematical model that uses reaction-diffusion equations to model collective migration during wound healing with an emphasis on cell movement and its response to both cell signaling and cell-cell adhesion. We use the model to investigate the eff…
Migration of cells can be characterized by two, prototypical types of motion: individual and collective migration. We propose a statistical-inference approach designed to detect the presence of cell-cell interactions that give rise to collective behaviors in cell-motility experiments. Such inference method has been first successfully tested on synthetic motional data, and then applied to two experiments. In the first experiment, cell migrate in a wound-healing model: when applied to this experiment, the inference method predicts the existence of cell-cell interactions, correctly mirroring the …
Cell migration is often accompanied by collisions with other cells, which can lead to cessation of movement, repolarization, and migration away from the contact site - a process termed contact inhibition of locomotion (CIL). During CIL, the coupling between actomyosin contractilityand cell-substrate adhesions is modified. However, mathematical models describing stochastic cell migration and collision outcomes as a result of the coupling remain elusive. Here, we extend our previously developed stochastic model of single cell migration to include CIL. Our simulation results explain, in terms of …
Collective cell migration in cohesive units is vital for tissue morphogenesis, wound repair, and immune response. While the fundamental driving forces for collective cell motion stem from contractile and protrusive activities of individual cells, it remains unknown how their balance is optimized to maintain tissue cohesiveness and the fluidity for motion. Here we present a cell-based computational model for collective cell migration during wound healing that incorporates mechanochemical coupling of cell motion and adhesion kinetics with stochastic transformation of active motility forces. We s…
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 f…
The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecula…
A simple model is proposed that describes the various morphodynamic principles of migrating cells from polar to amoeboidal motions. The model equation is derived using competing internal cellular compass variables and symmetries of the system. Fixed points for the $N = 2$ system are closely investigated to clarify how the competition among polaritors explains the observed morphodynamics. Response behaviors of cell--to--signal stimuli are also investigated. This model will be useful for classifying high-dimensional cell motions and investigating collective cellular behaviors.
Cell migration in fibreous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response and cancer progression. During migration, individual cells can generate active pulling forces via actin filament contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Her…
Contact inhibition plays a crucial role in the motility of cells, the process of wound healing, and the formation of tumors. By mimicking the mechanical motion of calls crawling on a substrate using a pseudopod, we constructed a minimal model for migrating cells which gives rise to contact inhibition of locomotion (CIL) naturally. The model cell consists of two disks, one in the front (a pseudopod) and the other one in the back (cell body), connected by a finitely extensible spring. Despite the simplicity of the model, the cells' collective behavior is highly nontrivial, depending on the shape…
Many cell types display random motility on two-dimensional substrates, but crawl persistently in a single direction when confined in a microchannel or on an adhesive micropattern. Does this imply that the motility mechanism of confined cells is fundamentally different from that of unconfined cells? We argue that both free- and confined- cell migration may be described by a generic model of cells as "velocity aligning" active Brownian particles previously proposed to solve a completely separate problem in collective cell migration. Our model can be mapped to a diffusive escape over a barrier an…
Drosophila melanogaster hemocytes are highly motile cells that are crucial for successful embryogenesis and have important roles in the organism's immunological response. Hemocyte motion was measured using selective plane illumination microscopy. Every hemocyte cell in one half of an embryo was tracked during embryogenesis and analysed using a deep learning neural network. The anomalous transport of the cells was well described by fractional Brownian motion that was heterogeneous in both time and space. Hemocyte motion became less persistent over time. LanB1 and SCAR mutants disrupted the coll…
Although understanding the collective migration of cells, such as that seen in epithelial sheets, is essential for understanding diseases such as metastatic cancer, this motion is not yet as well characterized as individual cell migration. Here we adapt quantitative metrics used to characterize the flow and deformation of soft matter to contrast different types of motion within a migrating sheet of cells. Using a Finite-Time Lyapunov Exponent (FTLE) analysis, we find that - in spite of large fluctuations - the flow field of an epithelial cell sheet is not chaotic. Stretching of a sheet of cell…
Mathematical models are vital interpretive and predictive tools used to assist in the understanding of cell migration. There are typically two approaches to modelling cell migration: either micro-scale, discrete or macro-scale, continuum. The discrete approach, using agent-based models (ABMs), is typically stochastic and accounts for properties at the cell-scale. Conversely, the continuum approach, in which cell density is often modelled as a system of deterministic partial differential equations (PDEs), provides a global description of the migration at the population level. Deterministic mode…
Collective migration dominates many phenomena, from cell movement in living systems to abiotic self-propelling particles. Focusing on the early stages of tumor evolution, we enunciate the principles involved in cell dynamics and highlight their implications in understanding similar behavior in seemingly unrelated soft glassy materials and possibly chemokine-induced migration of CD8$^{+}$ T cells. We performed simulations of tumor invasion using a minimal three dimensional model, accounting for cell elasticity and adhesive cell-cell interactions as well as cell birth and death to establish that…
Physical Confinement and Cell Proximity Increase Cell Migration Rates and Invasiveness: A Mathematical Model of Cancer Cell Invasion through Flexible Channels
Collective cell migration in 3D extracellular matrix (ECM) is crucial to many physiological and pathological processes. Migrating cells can generate active pulling forces via actin filament contraction, which are transmitted to the ECM fibers and lead to a dynamically evolving force network in the system. Here, we elucidate the role of such force network in regulating collective cell behaviors using a minimal active-particle-on-network (APN) model, in which active particles can pull the fibers and hop between neighboring nodes of the network following local durotaxis. Our model reveals a dynam…
