Collisions of deformable cells lead to collective migration

Differential Elasticity Affects Lineage Segregation of Embryonic Stem Cells

The question of what guides lineage segregation is central to development, where cellular differentiation leads to segregated cell populations destined for specialized functions. Here, using optical tweezers measurements of mouse embryonic stem cells, we reveal a mechanical mechanism based on differential elasticity in the second lineage segregation of the embryonic inner cell mass into epiblast (EPI) cells, which will develop into the fetus, and primitive endoderm (PrE), which will form extraembryonic structures such as the yolk sac. Remarkably, we find that these mechanical differences already occur during priming, not just after a cell has committed to differentiation. Specifically, we show that PrE-primed cells exhibit significantly higher elasticity than EPI-primed cells, characterized by lower power spectrum scaling exponents, higher Young's modulus, and lower loss tangent. Using a model of two cell types differing only in elasticity, we show that differential elasticity alone is sufficient to lead to segregation between cell types, suggesting that the mechanical attributes of the cells contribute to the segregation process. Importantly, we find that this process relies on cellular activity. Our findings present differential elasticity as a previously unknown mechanical contributor to lineage segregation during embryo morphogenesis.

Durotaxis and extracellular matrix degradation promote the clustering of cancer cells

Early stages of metastasis depend on the collective behavior of cancer cells and their interaction with the extracellular matrix (ECM). Cancer cell clusters are known to exhibit higher metastatic potential than single cells. To explore clustering dynamics, we developed a calibrated computational model describing how motile cancer cells biochemically and biomechanically interact with the ECM during the initial invasion phase, including ECM degradation and mechanical remodeling. The model reveals that cluster formation time, size, and shape are influenced by ECM degradation rates and cellular compliance to external stresses (durotaxis). The results align with experimental observations, demonstrating distinct cell trajectories and cluster morphologies shaped by biomechanical parameters. The simulations provide valuable insights into cancer invasion dynamics and may suggest potential therapeutic strategies targeting early-stage invasive cells.

Geometrical constraint change determines organized collective migration of follower cells

Spatial confinement plays a critical role in shaping collective cell migration, particularly in regulating interactions between leader and follower cells and among follower cells themselves. However, how changes in confinement geometry influence migration dynamics and cell-to-cell interactions remains poorly understood. This study leverages a novel microchannel design to systematically dissect the interplay between spatial confinement and collective cell behavior in endothelial-like cells (MILE SVEN 1). In a single-cell-wide T-shaped branching structure, rear cells selected alternate pathways, avoiding direct alignment with preceding cells. This highlights how spatial geometry mediates follower-follower interactions by encouraging dynamic rearrangements within the cell train. Ladder-like branching structures with consistent total pathway widths showed that dividing and reassembling cell trains had minimal impact on migration velocity, provided no compression or expansion occurred. Wide-narrow-wide patterns demonstrated distinct effects: stepwise transitions accelerated cells in narrow sections, increasing directional alignment driven by spatial restriction, followed by decreased alignment in wider regions. Gradual transitions maintained stable alignment and minimized disruptions, emphasizing the importance of smooth geometrical transitions in preserving robust collective behavior. These findings reveal how spatial confinement integrates follower-follower interactions and dynamic realignment. By linking geometric transitions to collective cell dynamics, our study advances the understanding of physical guidance mechanisms and offers a platform for investigating spatial influences on migrating cellular systems.

Collective mechanics of small migrating cell groups

Migration of adhesive cell groups is a fundamental part of wound healing, development and carcinogenesis. Intense research has been conducted on mechanisms of collective migration of adhesive groups of cells. Here we focus on mechanical and mechanistic lessons from small migrating cell groups. We review forces and locomotory dynamics of two- and three-cell clusters, rotation of cell doublets, self-organization of one-dimensional cell trains, nascent efforts to understand three-dimensional collective migration and border cell clusters in Drosophila embryo.

Derivation and simulation of a computational model of active cell populations: How overlap avoidance, deformability, cell-cell junctions and cytoskeletal forces affect alignment

Collective alignment of cell populations is a commonly observed phenomena in biology. An important example are aligning fibroblasts in healthy or scar tissue. In this work we derive and simulate a mechanistic agent-based model of the collective behaviour of actively moving and interacting cells, with a focus on understanding collective alignment. The derivation strategy is based on energy minimisation. The model ingredients are motivated by data on the behaviour of different populations of aligning fibroblasts and include: Self-propulsion, overlap avoidance, deformability, cell-cell junctions and cytoskeletal forces. We find that there is an optimal ratio of self-propulsion speed and overlap avoidance that maximises collective alignment. Further we find that deformability aids alignment, and that cell-cell junctions by themselves hinder alignment. However, if cytoskeletal forces are transmitted via cell-cell junctions we observe strong collective alignment over large spatial scales.

The role of the nucleus for cell mechanics: an elastic phase field approach

The nucleus of eukaryotic cells typically makes up around 30% of the cell volume and has significantly different mechanics, which can make it effectively up to ten times stiffer than the surrounding cytoplasm. Therefore it is an important element for cell mechanics, but a quantitative understanding of its mechanical role during whole cell dynamics is largely missing. Here we demonstrate that elastic phase fields can be used to describe dynamical cell processes in adhesive or confining environments in which the nucleus acts as a stiff inclusion in an elastic cytoplasm. We first introduce and verify our computational method and then study several prevalent cell-mechanical measurement methods. For cells on adhesive patterns, we find that nuclear stress is shielded by the adhesive pattern. For cell compression between two parallel plates, we obtain force-compression curves that allow us to extract an effective modulus for the cell-nucleus composite. For micropipette aspiration, the effect of the nucleus on the effective modulus is found to be much weaker, highlighting the complicated interplay between extracellular geometry and cell mechanics that is captured by our approach. We also show that our phase field approach can be used to investigate the effects of Kelvin-Voigt-type viscoelasticity and cortical tension.

Coordinated Motion of Epithelial Layers on Curved Surfaces

Coordinated cellular movements are key processes in tissue morphogenesis. Using a cell-based modeling approach we study the dynamics of epithelial layers lining surfaces with constant and varying curvature. We demonstrate that extrinsic curvature effects can explain the alignment of cell elongation with the principal directions of curvature. Together with specific self-propulsion mechanisms and cell-cell interactions this effect gets enhanced and can explain observed large-scale, persistent, and circumferential rotation on cylindrical surfaces. On toroidal surfaces the resulting curvature coupling is an interplay of intrinsic and extrinsic curvature effects. These findings unveil the role of curvature and postulate its importance for tissue morphogenesis.

Three-component contour dynamics model to simulate and analyze amoeboid cell motility in two dimensions

Amoeboid cell motility is relevant in a wide variety of biomedical processes such as wound healing, cancer metastasis, and embryonic morphogenesis. It is characterized by pronounced changes of the cell shape associated with expansions and retractions of the cell membrane, which result in a crawling kind of locomotion. Despite existing computational models of amoeboid motion, the inference of expansion and retraction components of individual cells, the corresponding classification of cells, and the a priori specification of the parameter regime to achieve a specific motility behavior remain challenging open problems. We propose a novel model of the spatio-temporal evolution of two-dimensional cell contours comprising three biophysiologically motivated components: a stochastic term accounting for membrane protrusions and two deterministic terms accounting for membrane retractions by regularizing the shape and area of the contour. Mathematically, these correspond to the intensity of a self-exciting Poisson point process, the area-preserving curve-shortening flow, and an area adjustment flow. The model is used to generate contour data for a variety of qualitatively different, e.g., polarized and non-polarized, cell tracks that visually resemble experimental data very closely. In application to experimental cell tracks, we inferred the protrusion component and examined its correlation to common biomarkers: the F-actin density close to the membrane and its local motion. Due to the low model complexity, parameter estimation is fast, straightforward, and offers a simple way to classify contour dynamics based on two locomotion types: the amoeboid and a so-called fan-shaped type. For both types, we use cell tracks segmented from fluorescence imaging data of the model organism Dictyostelium discoideum. An implementation of the model is provided within the open-source software package AmoePy, a Python-based toolbox for analyzing and simulating amoeboid cell motility.

A Vicsek-type model of confined cancer cells with variable clustering affinities

Clustering of cells is an essential component of many biological processes from tissue formation to cancer metastasis. We develop a minimal, Vicsek-based model of cellular interactions that robustly and accurately captures the variable propensity of different cells to form groups when confined. We calibrate and validate the model with experimental data on clustering affinities of four lines of tumor cells. We then show that cell clustering or separation tendencies are retained in environments with higher cell number densities and in cell mixtures. Finally, we calibrate our model with experimental measurements on the separation of cells treated with anti-clustering agents and find that treated cells maintain their distances in denser suspensions. We show that the model reconstructs several cell interaction mechanisms, which makes it suitable for exploring the dynamics of cell cluster formation as well as cell separation. Insight: We developed a model of cellular interactions that captures the clustering and separation of cells in an enclosure. Our model is particularly relevant for microfluidic systems with confined cells and we centered our work around one such emerging assay for the detection and research on clustering breast cancer cells. We calibrated our model using the existing experimental data and used it to explore the functionality of the assay under a broader set of conditions than originally considered. Future usages of our model can include purely theoretical and computational considerations, exploring experimental devices, and supporting research on small to medium-sized cell clusters.

Polarization and motility of one-dimensional multi-cellular trains

Collective cell migration, whereby cells adhere to form multi-cellular clusters that move as a single entity, play an important role in numerous biological processes, such as during development and cancer progression. Recent experimental work focused on migration of one-dimensional cellular clusters, confined to move along adhesive lanes, as a simple geometry in which to systematically study this complex system. One-dimensional migration also arises in the body when cells migrate along blood vessels, axonal projections, and narrow cavities between tissues. We explore here the modes of one-dimensional migration of cellular clusters ("trains") by implementing cell-cell interactions in a model of cell migration that contains a mechanism for spontaneous cell polarization. We go beyond simple phenomenological models of the cells as self-propelled particles by having the internal polarization of each cell depend on its interactions with the neighboring cells that directly affect the actin polymerization activity at the cell's leading edges. Both contact inhibition of locomotion and cryptic lamellipodia interactions between neighboring cells are introduced. We find that this model predicts multiple motility modes of the cell trains, which can have several different speeds for the same polarization pattern. Compared to experimental data, we find that Madin-Darby canine kidney cells are poised along the transition region where contact inhibition of locomotion and cryptic lamellipodia roughly balance each other, where collective migration speed is most sensitive to the values of the cell-cell interaction strength.

Morphogens enable interacting supracellular phases that generate organ architecture

During vertebrate organogenesis, increases in morphological complexity are tightly coupled to morphogen expression. In this work, we studied how morphogens influence self-organizing processes at the collective or "supra"-cellular scale in avian skin. We made physical measurements across length scales, which revealed morphogen-enabled material property differences that were amplified at supracellular scales in comparison to cellular scales. At the supracellular scale, we found that fibroblast growth factor (FGF) promoted "solidification" of tissues, whereas bone morphogenetic protein (BMP) promoted fluidity and enhanced mechanical activity. Together, these effects created basement membrane-less compartments within mesenchymal tissue that were mechanically primed to drive avian skin tissue budding. Understanding this multiscale process requires the ability to distinguish between proximal effects of morphogens that occur at the cellular scale and their functional effects, which emerge at the supracellular scale.

Motility induced phase separation of deformable cells

Using a multi-phase field model, we examine how particle deformability, which is a proxy for cell stiffness, affects motility induced phase separation (MIPS). We show that purely repulsive deformable, i.e., squishy, cells phase separate more effectively than their rigid counterparts. This can be understood as due to the fact that deformability increases the effective duration of collisions. In addition, the dense regions become increasingly disordered as deformability increases. Our results contextualize the applicability of MIPS to biological systems and have implications for how cells in biological systems may self-organize.

Effect of Porous Substrate Topographies on Cell Dynamics: A Computational Study

Controlling cell-substrate interactions via the microstructural characteristics of biomaterials offers an advantageous path for modulating cell dynamics, mechanosensing, and migration, as well as for designing immune-modulating implants, all without the drawbacks of chemical-based triggers. Specifically, recent in vivo studies have suggested that a porous implant's microscale curvature landscape can significantly impact cell behavior and ultimately the immune response. To investigate such cell-substrate interactions, we utilized a 3D computational model incorporating the minimum necessary physics of cell migration and cell-substrate interactions needed to replicate known in vitro behaviors. This model specifically incorporates the effect of membrane tension, which was found to be necessary to replicate in vitro cell behavior on curved surfaces. Our simulated substrates represent two classes of porous materials recently used in implant studies, which have markedly different microscale curvature distributions and pore geometries. We found distinct differences between the overall migration behaviors, shapes, and actin polymerization dynamics of cells interacting with the two substrates. These differences were correlated to the shape energy of the cells as they interacted with the porous substrates, in effect interpreting substrate topography as an energetic landscape interrogated by cells. Our results demonstrate that microscale curvature directly influences cell shape and migration and, therefore, is likely to influence cell behavior. This supports further investigation of the relationship between the surface topography of implanted materials and the characteristic immune response, a complete understanding of which would broadly advance principles of biomaterial design.

Coarse-graining the vertex model and its response to shear

Tissue dynamics and collective cell motion are crucial biological processes. Their biological machinery is mostly known, and simulation models such as the active vertex model exist and yield reasonable agreement with experimental observations such as tissue fluidization or fingering. However, a good and well-founded continuum description for tissues remains to be developed. In this work, we derive a macroscopic description for a two-dimensional cell monolayer by coarse-graining the vertex model through the Poisson bracket approach. We obtain equations for cell density, velocity, and the cellular shape tensor. We then study the homogeneous steady states, their stability (which coincides with thermodynamic stability), and especially their behavior under an externally applied shear. Our results contribute to elucidate the interplay between flow and cellular shape. The obtained macroscopic equations present a good starting point for adding cell motion, morphogenetic, and other biologically relevant processes.

Spontaneous motion of a passive fluid droplet in an active microchannel

We numerically study the dynamics of a passive fluid droplet confined within a microchannel whose walls are covered with a thin layer of active gel. The latter represents a fluid of extensile material modelling, for example, a suspension of cytoskeletal filaments and molecular motors. Our results show that the layer is capable of producing a spontaneous flow triggering a rectilinear motion of the passive droplet. For a hybrid design (a single wall covered by the active layer), at the steady state the droplet attains an elliptical shape, resulting from an asymmetric saw-toothed structure of the velocity field. In contrast, if the active gel covers both walls, the velocity field exhibits a fully symmetric pattern considerably mitigating morphological deformations. We further show that the structure of the spontaneous flow in the microchannel can be controlled by the anchoring conditions of the active gel at the wall. These findings are also confirmed by selected 3D simulations. Our results may stimulate further research addressed to design novel microfludic devices whose functioning relies on the collective properties of active gels.

Migration and division in cell monolayers on substrates with topological defects

Collective movement and organization of cell monolayers are important for wound healing and tissue development. Recent experiments highlighted the importance of liquid crystal order within these layers, suggesting that +1 topological defects have a role in organizing tissue morphogenesis. We study fibroblast organization, motion, and proliferation on a substrate with micron-sized ridges that induce +1 and -1 topological defects using simulation and experiment. We model cells as self-propelled deformable ellipses that interact via a Gay-Berne potential. Unlike earlier work on other cell types, we see that density variation near defects is not explained by collective migration. We propose instead that fibroblasts have different division rates depending on their area and aspect ratio. This model captures key features of our previous experiments: the alignment quality worsens at high cell density and, at the center of the +1 defects, cells can adopt either highly anisotropic or primarily isotropic morphologies. Experiments performed with different ridge heights confirm a prediction of this model: Suppressing migration across ridges promotes higher cell density at the +1 defect. Our work enables a mechanism for tissue patterning using topological defects without relying on cell migration.

Molecular-scale substrate anisotropy, crowding and division drive collective behaviours in cell monolayers

The ability of cells to reorganize in response to external stimuli is important in areas ranging from morphogenesis to tissue engineering. While nematic order is common in biological tissues, it typically only extends to small regions of cells interacting via steric repulsion. On isotropic substrates, elongated cells can co-align due to steric effects, forming ordered but randomly oriented finite-size domains. However, we have discovered that flat substrates with nematic order can induce global nematic alignment of dense, spindle-like cells, thereby influencing cell organization and collective motion and driving alignment on the scale of the entire tissue. Remarkably, single cells are not sensitive to the substrate's anisotropy. Rather, the emergence of global nematic order is a collective phenomenon that requires both steric effects and molecular-scale anisotropy of the substrate. To quantify the rich set of behaviours afforded by this system, we analyse velocity, positional and orientational correlations for several thousand cells over days. The establishment of global order is facilitated by enhanced cell division along the substrate's nematic axis, and associated extensile stresses that restructure the cells' actomyosin networks. Our work provides a new understanding of the dynamics of cellular remodelling and organization among weakly interacting cells.

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.

Mathematical model of mechano-sensing and mechanically induced collective motility of cells on planar elastic substrates

Cells mechanically interact with their environment to sense, for example, topography, elasticity and mechanical cues from other cells. Mechano-sensing has profound effects on cellular behaviour, including motility. The current study aims to develop a mathematical model of cellular mechano-sensing on planar elastic substrates and demonstrate the model's predictive capabilities for the motility of individual cells in a colony. In the model, a cell is assumed to transmit an adhesion force, derived from a dynamic focal adhesion integrin density, that locally deforms a substrate, and to sense substrate deformation originating from neighbouring cells. The substrate deformation from multiple cells is expressed as total strain energy density with a spatially varying gradient. The magnitude and direction of the gradient at the cell location define the cell motion. Cell-substrate friction, partial motion randomness, and cell death and division are included. The substrate deformation by a single cell and the motility of two cells are presented for several substrate elasticities and thicknesses. The collective motility of 25 cells on a uniform substrate mimicking the closure of a circular wound of 200 µm is predicted for deterministic and random motion. Cell motility on substrates with varying elasticity and thickness is explored for four cells and 15 cells, the latter again mimicking wound closure. Wound closure by 45 cells is used to demonstrate the simulation of cell death and division during migration. The mathematical model can adequately simulate the mechanically induced collective cell motility on planar elastic substrates. The model is suitable for extension to other cell and substrates shapes and the inclusion of chemotactic cues, offering the potential to complement in vitro and in vivo studies.

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.

The crucial role of adhesion in the transmigration of active droplets through interstitial orifices

Active fluid droplets are a class of soft materials exhibiting autonomous motion sustained by an energy supply. Such systems have been shown to capture motility regimes typical of biological cells and are ideal candidates as building-block for the fabrication of soft biomimetic materials of interest in pharmacology, tissue engineering and lab on chip devices. While their behavior is well established in unconstrained environments, much less is known about their dynamics under strong confinement. Here, we numerically study the physics of a droplet of active polar fluid migrating within a microchannel hosting a constriction with adhesive properties, and report evidence of a striking variety of dynamic regimes and morphological features, whose properties crucially depend upon droplet speed and elasticity, degree of confinement within the constriction and adhesiveness to the pore. Our results suggest that non-uniform adhesion forces are instrumental in enabling the crossing through narrow orifices, in contrast to larger gaps where a careful balance between speed and elasticity is sufficient to guarantee the transition. These observations may be useful for improving the design of artificial micro-swimmers, of interest in material science and pharmaceutics, and potentially for cell sorting in microfluidic devices.

MorphoSim: an efficient and scalable phase-field framework for accurately simulating multicellular morphologies

The phase field model can accurately simulate the evolution of microstructures with complex morphologies, and it has been widely used for cell modeling in the last two decades. However, compared to other cellular models such as the coarse-grained model and the vertex model, its high computational cost caused by three-dimensional spatial discretization hampered its application and scalability, especially for multicellular organisms. Recently, we built a phase field model coupled with in vivo imaging data to accurately reconstruct the embryonic morphogenesis of Caenorhabditis elegans from 1- to 8-cell stages. In this work, we propose an improved phase field model by using the stabilized numerical scheme and modified volume constriction. Then we present a scalable phase-field framework, MorphoSim, which is 100 times more efficient than the previous one and can simulate over 100 mechanically interacting cells. Finally, we demonstrate how MorphoSim can be successfully applied to reproduce the assembly, self-repairing, and dissociation of a synthetic artificial multicellular system - the synNotch system.

Active Forces in Confluent Cell Monolayers

We use a computational phase-field model together with analytical analysis to study how intercellular active forces can mediate individual cell morphology and collective motion in a confluent cell monolayer. We explore the regime where intercellular forces dominate the tissue dynamics, and polar forces are negligible. Contractile intercellular interactions lead to cell elongation, nematic ordering, and active turbulence characterized by motile topological defects. Extensile interactions result in frustration, and perpendicular cell orientations become more prevalent. Furthermore, we show that contractile behavior can change to extensile behavior if anisotropic fluctuations in cell shape are considered.

Interacting active surfaces: A model for three-dimensional cell aggregates

We introduce a modelling and simulation framework for cell aggregates in three dimensions based on interacting active surfaces. Cell mechanics is captured by a physical description of the acto-myosin cortex that includes cortical flows, viscous forces, active tensions, and bending moments. Cells interact with each other via short-range forces capturing the effect of adhesion molecules. We discretise the model equations using a finite element method, and provide a parallel implementation in C++. We discuss examples of application of this framework to small and medium-sized aggregates: we consider the shape and dynamics of a cell doublet, a planar cell sheet, and a growing cell aggregate. This framework opens the door to the systematic exploration of the cell to tissue-scale mechanics of cell aggregates, which plays a key role in the morphogenesis of embryos and organoids.

Computational approaches for simulating luminogenesis

Lumens, liquid-filled cavities surrounded by polarized tissue cells, are elementary units involved in the morphogenesis of organs. Theoretical modeling and computations, which can integrate various factors involved in biophysics of morphogenesis of cell assembly and lumens, may play significant roles to elucidate the mechanisms in formation of such complex tissue with lumens. However, up to present, it has not been documented well what computational approaches or frameworks can be applied for this purpose and how we can choose the appropriate approach for each problem. In this review, we report some typical lumen morphologies and basic mechanisms for the development of lumens, focusing on three keywords - mechanics, hydraulics and geometry - while outlining pros and cons of the current main computational strategies. We also describe brief guidance of readouts, i.e., what we should measure in experiments to make the comparison with the model's assumptions and predictions.

Spontaneous transitions between amoeboid and keratocyte-like modes of migration

The motility of adherent eukaryotic cells is driven by the dynamics of the actin cytoskeleton. Despite the common force-generating actin machinery, different cell types often show diverse modes of locomotion that differ in their shape dynamics, speed, and persistence of motion. Recently, experiments in Dictyostelium discoideum have revealed that different motility modes can be induced in this model organism, depending on genetic modifications, developmental conditions, and synthetic changes of intracellular signaling. Here, we report experimental evidence that in a mutated D. discoideum cell line with increased Ras activity, switches between two distinct migratory modes, the amoeboid and fan-shaped type of locomotion, can even spontaneously occur within the same cell. We observed and characterized repeated and reversible switchings between the two modes of locomotion, suggesting that they are distinct behavioral traits that coexist within the same cell. We adapted an established phenomenological motility model that combines a reaction-diffusion system for the intracellular dynamics with a dynamic phase field to account for our experimental findings.

Numerical Study on Dynamics of Blood Cell Migration and Deformation in Atherosclerotic Vessels

A phase field model is used to study the effect of atherosclerotic plaque on hemodynamics. The migration of cells in blood flows is described by a set of multiple phase field equations, which incorporate elastic energies and the interacting effects of cells. Several simulations are carried out to reveal the influences of initial velocities of blood cells, cellular elasticity and block rates of hemodynamic vessels. The results show that the cell deformation increases with the growth of the initial active velocity and block rate but with the decrease of the cellular elasticity. The atherosclerotic plaque not only affects the deformation and migration of cells but also can promote the variation in hemodynamic properties. The atherosclerotic plaque causes a burst in cell velocity, and the greater the block rate and cellular elasticity, the more dramatic the variation of instantaneous velocity. The present work demonstrates that the phase field method could be extended to reveal formation atherosclerosis at the microscopic level from the perspective of hemodynamics.

The solid tumor microenvironment-Breaking the barrier for T cells: How the solid tumor microenvironment influences T cells: How the solid tumor microenvironment influences T cells

The tumor microenvironment (TME) plays a pivotal role in the behavior and development of solid tumors as well as shaping the immune response against them. As the tumor cells proliferate, the space they occupy and their physical interactions with the surrounding tissue increases. The growing tumor tissue becomes a complex dynamic structure, containing connective tissue, vascular structures, and extracellular matrix (ECM) that facilitates stimulation, oxygenation, and nutrition, necessary for its fast growth. Mechanical cues such as stiffness, solid stress, interstitial fluid pressure (IFP), matrix density, and microarchitecture influence cellular functions and ultimately tumor progression and metastasis. In this fight, our body is equipped with T cells as its spearhead against tumors. However, the altered biochemical and mechanical environment of the tumor niche affects T cell efficacy and leads to their exhaustion. Understanding the mechanobiological properties of the TME and their effects on T cells is key for developing novel adoptive tumor immunotherapies.

Impact of contact inhibition on collective cell migration and proliferation

Contact inhibition limits migration and proliferation of cells in cell colonies. We consider a multiphase field model to investigate the growth dynamics of a cell colony, composed of proliferating cells. The model takes into account the mechanism of contact inhibition of proliferation by local mechanical interactions. We compare nonmigrating and migrating cells, in order to provide a quantitative characterization of the dynamics and analyze the velocity of the colony boundary for both cases. Additionally, we measure single cell velocities, number of neighbor distributions, as well as the influence of stress and age on positions of the cells and with respect to each other.

Deformable active nematic particles and emerging edge currents in circular confinements

We consider a microscopic field theoretical approach for interacting active nematic particles. With only steric interactions the self-propulsion strength in such systems can lead to different collective behaviour, e.g. synchronized self-spinning and collective translation. The different behaviour results from the delicate interplay between internal nematic structure, particle shape deformation and particle-particle interaction. For intermediate active strength an asymmetric particle shape emerges and leads to chirality and self-spinning crystals. For larger active strength the shape is symmetric and translational collective motion emerges. Within circular confinements, depending on the packing fraction, the self-spinning regime either stabilizes positional and orientational order or can lead to edge currents and global rotation which destroys the synchronized self-spinning crystalline structure.

Cell behaviors within a confined adhesive area fabricated using novel micropatterning methods

In the field of cell and tissue engineering, there is an increasing demand for techniques to spatially control the adhesion of cells to substrates of desired sizes and shapes. Here, we describe two novel methods for fabricating a substrate for adhesion of cells to a defined area. In the first method, the surface of the coverslip or plastic dish was coated with Lipidure, a non-adhesive coating material, and air plasma was applied through a mask with holes, to confer adhesiveness to the surface. In the second method, after the surface of the coverslip was coated with gold by sputtering and then with Lipidure; the Lipidure coat was locally removed using a novel scanning laser ablation method. These methods efficiently confined cells within the adhesive area and enabled us to follow individual cells for a longer duration, compared to the currently available commercial substrates. By following single cells within the confined area, we were able to observe several new aspects of cell behavior in terms of cell division, cell–cell collisions, and cell collision with the boundary between adhesive and non-adhesive areas.

Computable early Caenorhabditis elegans embryo with a phase field model

Morphogenesis is a precise and robust dynamic process during metazoan embryogenesis, consisting of both cell proliferation and cell migration. Despite the fact that much is known about specific regulations at molecular level, how cell proliferation and migration together drive the morphogenesis at cellular and organismic levels is not well understood. Using Caenorhabditis elegans as the model animal, we present a phase field model to compute early embryonic morphogenesis within a confined eggshell. With physical information about cell division obtained from three-dimensional time-lapse cellular imaging experiments, the model can precisely reproduce the early morphogenesis process as seen in vivo, including time evolution of location and morphology of each cell. Furthermore, the model can be used to reveal key cell-cell attractions critical to the development of C. elegans embryo. Our work demonstrates how genetic programming and physical forces collaborate to drive morphogenesis and provides a predictive model to decipher the underlying mechanism.

Embryonic development is a precise process involving cell division, cell-cell interaction, and cell migration. During the process, how each cell reaches its supposed location and be in contact with the right neighbors, and what roles genetic factors and physical forces play are important and fascinating questions. Using the worm Caenorhabditis elegans as a model system, we build a phase field model to simulate early morphogenesis. With a few physical inputs, the model can precisely reproduce the early morphological development of the worm. Such an accurate simulator can not only teach us how physical forces work together with genetic factors to shape up the complex process of development, but also make predictions, such as key cell-cell attractions critical in the process.

Multiphase field models for collective cell migration

Confluent cell monolayers and epithelia tissues show remarkable patterns and correlations in structural arrangements and actively driven collective flows. We simulate these properties using multiphase field models. The models are based on cell deformations and cell-cell interactions and we investigate the influence of microscopic details to incorporate active forces on emerging phenomena. We compare four different approaches, one in which the activity is determined by a random orientation, one where the activity is related to the deformation of the cells, and two models with subcellular details to resolve the mechanochemical interactions underlying cell migration. The models are compared with respect to generic features, such as coordination number distribution, cell shape variability, emerging nematic properties, as well as vorticity correlations and flow patterns in large confluent monolayers and confinements. All results are compared with experimental data for a large variety of cell cultures. The appearing qualitative differences of the models show the importance of microscopic details and provide a route towards predictive simulations of patterns and correlations in cell colonies.

A mesoscale mechanical model of cellular interactions

Computational models of cell mechanics allow the precise interrogation of cell shape change. These morphological changes are required for cells to survive in diverse tissue environments. Here, we present a mesoscale mechanical model of cell-substrate interactions using the level set method based on experimentally measured parameters. By implementing a viscoelastic mechanical equivalent circuit, we accurately model whole-cell deformations that are important for a variety of cellular processes. To effectively model shape changes as a cell interacts with a substrate, we have included receptor-mediated adhesion, which is governed by catch-slip bond behavior. The effect of adhesion was explored by subjecting cells to a variety of different substrates including flat, curved, and deformable surfaces. Finally, we increased the accuracy of our simulations by including a deformable nucleus in our cells. This model sets the foundation for further exploration into computational analyses of multicellular interactions.

Contact inhibition of locomotion generates collective cell migration without chemoattractants in an open domain

Neural crest cells are embryonic stem cells that migrate throughout embryos and, at different target locations, give rise to the formation of a variety of tissues and organs. The directional migration of the neural crest cells is experimentally described using a process referred to as contact inhibition of locomotion, by which cells redirect their movement upon the cell-cell contacts. However, it is unclear how the migration alignment is affected by the motility properties of the cells. Here, we theoretically model the migration alignment as a function of the motility dynamics and interaction of the cells in an open domain with a channel geometry. The results indicate that by increasing the influx rate of the cells into the domain a transition takes place from random movement to an organized collective migration, where the migration alignment is maximized and the migration time is minimized. This phase transition demonstrates that the cells can migrate efficiently over long distances without any external chemoattractant information about the direction of migration just based on local interactions with each other. The analysis of the dependence of this transition on the characteristic properties of cellular motility shows that the cell density determines the coordination of collective migration whether the migration domain is open or closed. In the open domain, this density is determined by a feedback mechanism between the flux and order parameter, which characterises the alignment of collective migration. The model also demonstrates that the coattraction mechanism proposed earlier is not necessary for collective migration and a constant flux of cells moving into the channel is sufficient to produce directed movement over arbitrary long distances.

Field-theoretic model for chemotaxis in run and tumble particles

In this paper, we develop a field-theoretic description for run and tumble chemotaxis, based on a density-functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its in-built multiparticle interactions, soft-core repulsion, and elasticity, is ideal for describing continuum collective phases with particle resolution, but on diffusive timescales. We show that our model exhibits particle aggregation in an externally imposed constant attractant field, as is observed for phototactic or thermotactic agents. We also show that this model captures particle aggregation through self-chemotaxis, an important mechanism that aids quorum-dependent cellular interactions.

More than just a barrier: using physical models to couple membrane shape to cell function

The correct execution of many cellular processes, such as division and motility, requires the cell to adopt a specific shape. Physically, these shapes are determined by the interplay of the plasma membrane and internal cellular driving factors. While the plasma membrane defines the boundary of the cell, processes inside the cell can result in the generation of forces that deform the membrane. These processes include protein binding, the assembly of protein superstructures, and the growth and contraction of cytoskeletal networks. Due to the complexity of the cell, relating observed membrane deformations back to internal processes is a challenging problem. Here, we review cell shape changes in endocytosis, cell adhesion, cell migration and cell division and discuss how by modeling membrane deformations we can investigate the inner working principles of the cell.

Learning the dynamics of cell-cell interactions in confined cell migration

The migratory dynamics of cells in physiological processes, ranging from wound healing to cancer metastasis, rely on contact-mediated cell-cell interactions. These interactions play a key role in shaping the stochastic trajectories of migrating cells. While data-driven physical formalisms for the stochastic migration dynamics of single cells have been developed, such a framework for the behavioral dynamics of interacting cells still remains elusive. Here, we monitor stochastic cell trajectories in a minimal experimental cell collider: a dumbbell-shaped micropattern on which pairs of cells perform repeated cellular collisions. We observe different characteristic behaviors, including cells reversing, following, and sliding past each other upon collision. Capitalizing on this large experimental dataset of coupled cell trajectories, we infer an interacting stochastic equation of motion that accurately predicts the observed interaction behaviors. Our approach reveals that interacting noncancerous MCF10A cells can be described by repulsion and friction interactions. In contrast, cancerous MDA-MB-231 cells exhibit attraction and antifriction interactions, promoting the predominant relative sliding behavior observed for these cells. Based on these experimentally inferred interactions, we show how this framework may generalize to provide a unifying theoretical description of the diverse cellular interaction behaviors of distinct cell types.

Excitable actin dynamics and amoeboid cell migration

Amoeboid cell migration is characterized by frequent changes of the direction of motion and resembles a persistent random walk on long time scales. Although it is well known that cell migration is typically driven by the actin cytoskeleton, the cause of this migratory behavior remains poorly understood. We analyze the spontaneous dynamics of actin assembly due to nucleation promoting factors, where actin filaments lead to an inactivation of these factors. We show that this system exhibits excitable dynamics and can spontaneously generate waves, which we analyze in detail. By using a phase-field approach, we show that these waves can generate cellular random walks. We explore how the characteristics of these persistent random walks depend on the parameters governing the actin-nucleator dynamics. In particular, we find that the effective diffusion constant and the persistence time depend strongly on the speed of filament assembly and the rate of nucleator inactivation. Our findings point to a deterministic origin of the random walk behavior and suggest that cells could adapt their migration pattern by modifying the pool of available actin.

Dynamic Self-Organization of Idealized Migrating Cells by Contact Communication

This Letter investigates what forms of cellular dynamic self-organization are caused through intercellular contact communication based on a theoretical model in which migrating cells perform contact following and contact inhibition and attraction of locomotion. Tuning those strengths causes varieties of dynamic patterns. This further includes a novel form of collective migration, snakelike dynamic assembly. Scrutinizing this pattern reveals that cells in this state can accurately respond to an external directional cue but have no spontaneous global polar order.

Tools for computational analysis of moving boundary problems in cellular mechanobiology

A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non-expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology.

Bridging from single to collective cell migration: A review of models and links to experiments

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.

Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction

Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.

Plasticity of cancer cell invasion: Patterns and mechanisms

Cancer cell migration and invasion are integral components of metastatic disease, which is the major cause of death in cancer patients. Cancer cells can disseminate and migrate via several alternative mechanisms including amoeboid cell migration, mesenchymal cell migration, and collective cell migration. These diverse movement strategies display certain specific and distinct hallmarks in cell-cell junctions, actin cytoskeleton, matrix adhesion, and protease activity. During tumor progression, cells pass through complex microenvironments and adapt their migration strategies by reversible mesenchymal-amoeboid and individual-collective transitions. This plasticity in motility patterns enables cancer cells disseminate further and thus limit the efficiency of anti-metastasis therapies. In this review, we discuss the modes and mechanisms of cancer cell migration and focus on the plasticity of tumor cell movement as well as potential emerging therapeutic options for reducing cancer cell invasion.

Reversible elastic phase field approach and application to cell monolayers

Motion and generation of forces by single cells and cell collectives are essential elements of many biological processes, including development, wound healing and cancer cell migration. Quantitative wound healing assays have demonstrated that cell monolayers can be both dynamic and elastic at the same time. However, it is very challenging to model this combination with conventional approaches. Here we introduce an elastic phase field approach that allows us to predict the dynamics of elastic sheets under the action of active stresses and localized forces, e.g. from leader cells. Our method ensures elastic reversibility after release of forces. We demonstrate its potential by studying several paradigmatic situations and geometries relevant for single cells and cell monolayers, including elastic bars, contractile discs and expanding monolayers with leader cells.

Aster swarming by symmetry breaking of cortical dynein transport and coupling kinesins

Microtubule (MT) radial arrays or asters establish the internal topology of a cell by interacting with organelles and molecular motors. We proceed to understand the general pattern forming potential of aster-motor systems using a computational model of multiple MT asters interacting with motors in cellular confinement. In this model dynein motors are attached to the cell cortex and plus-ended motors resembling kinesin-5 diffuse in the cell interior. The introduction of 'noise' in the form of MT length fluctuations spontaneously results in the emergence of coordinated, achiral vortex-like rotation of asters. The coherence and persistence of rotation require a threshold density of both cortical dyneins and coupling kinesins, while the onset is diffusion-limited with relation to the cortical dynein mobility. The coordinated rotational motion emerges due to the resolution of a 'tug-of-war' of multiple cortical dynein motors bound to MTs of the same aster by 'noise' in the form of MT dynamic instability. This transient symmetry breaking is amplified by local coupling by kinesin-5 complexes. The lack of widespread aster rotation across cell types suggests that biophysical mechanisms that suppress such intrinsic dynamics may have evolved. This model is analogous to more general models of locally coupled self-propelled particles (SPP) that spontaneously undergo collective transport in the presence of 'noise' that have been invoked to explain swarming in birds and fish. However, the aster-motor system is distinct from SPP models with regard to the particle density and 'noise' dependence, providing a set of experimentally testable predictions for a novel sub-cellular pattern forming system.

Active inter-cellular forces in collective cell motility

The collective behaviour of confluent cell sheets is strongly influenced both by polar forces, arising through cytoskeletal propulsion, and by active inter-cellular forces, which are mediated by interactions across cell-cell junctions. We use a phase-field model to explore the interplay between these two contributions and compare the dynamics of a cell sheet when the polarity of the cells aligns to (i) their main axis of elongation, (ii) their velocity and (iii) when the polarity direction executes a persistent random walk. In all three cases, we observe a sharp transition from a jammed state (where cell rearrangements are strongly suppressed) to a liquid state (where the cells can move freely relative to each other) when either the polar or the inter-cellular forces are increased. In addition, for case (ii) only, we observe an additional dynamical state, flocking (solid or liquid), where the majority of the cells move in the same direction. The flocking state is seen for strong polar forces, but is destroyed as the strength of the inter-cellular activity is increased.

Rules of collective migration: from the wildebeest to the neural crest

Collective migration, the movement of groups in which individuals affect the behaviour of one another, occurs at practically every scale, from bacteria up to whole species' populations. Universal principles of collective movement can be applied at all levels. In this review, we will describe the rules governing collective motility, with a specific focus on the neural crest, an embryonic stem cell population that undergoes extensive collective migration during development. We will discuss how the underlying principles of individual cell behaviour, and those that emerge from a supracellular scale, can explain collective migration. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling

Cell migration in fibrous 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 actomyosin 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. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.

A theoretical model of collective cell polarization and alignment

Collective cell polarization and alignment play important roles in tissue morphogenesis, wound healing and cancer metastasis. How cells sense the direction and position in these processes, however, has not been fully understood. Here we construct a theoretical model based on describing cell layer as a nemato-elastic medium, by which the cell polarization, cell alignment and cell active contraction are explicitly expressed as functions of components of the nematic order parameter. To determine the order parameter we derive two sets of governing equations, one for the force equilibrium of the system, and the other for the minimization of the system's free energy including the energy of cell polarization and alignment. By solving these coupled governing equations, we can predict the effects of substrate stiffness, geometries of cell layers, external forces and myosin activity on the direction- and position-dependent cell aspect ratio and cell orientation. Moreover, the axisymmetric problem with cells on a ring-like pattern is solved analytically, and the analytical solution for cell aspect ratio are governed by parameter groups which include the stiffness of the cell and the substrate, the strength of myosin activity and the external forces. Our predictions of the cell aspect ratio and orientation are generally comparable to experimental observations. These results show that the pattern of cell polarization is determined by the anisotropic degree of active contractile stress, and suggest a stress-driven polarization mechanism that enables cells to sense their spatial positions to develop direction- and position-dependent behavior. This, in turn, sheds light on the ways to control pattern formation in tissue engineering for potential biomedical applications.