Dynamics of Cell Ensembles on Adhesive Micropatterns: Bridging the Gap between Single Cell Spreading and Collective Cell Migration

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Modelling cell shape in 3D structured environments: A quantitative comparison with experiments

Cell shape plays a fundamental role in many biological processes, including adhesion, migration, division and development, but it is not clear which shape model best predicts three-dimensional cell shape in structured environments. Here, we compare different modelling approaches with experimental data. The shapes of single mesenchymal cells cultured in custom-made 3D scaffolds were compared by a Fourier method with surfaces that minimize area under the given adhesion and volume constraints. For the minimized surface model, we found marked differences to the experimentally observed cell shapes, which necessitated the use of more advanced shape models. We used different variants of the cellular Potts model, which effectively includes both surface and bulk contributions. The simulations revealed that the Hamiltonian with linear area energy outperformed the elastic area constraint in accurately modelling the 3D shapes of cells in structured environments. Explicit modelling the nucleus did not improve the accuracy of the simulated cell shapes. Overall, our work identifies effective methods for accurately modelling cellular shapes in complex environments.

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

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

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.

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.

Size-dependent response of cells in epithelial tissue modulated by contractile stress fibers

Although cells with distinct apical areas have been widely observed in epithelial tissues, how the size of cells affects their behavior during tissue deformation and morphogenesis as well as key physical factors modulating such influence remains elusive. Here, we showed that the elongation of cells within the monolayer under anisotropic biaxial stretching increases with their size because the strain released by local cell rearrangement (i.e., T1 transition) is more significant for small cells that possess higher contractility. On the other hand, by incorporating the nucleation, peeling, merging, and breakage dynamics of subcellular stress fibers into classical vertex formulation, we found that stress fibers with orientations predominantly aligned with the main stretching direction will be formed at tricellular junctions, in good agreement with recent experiments. The contractile forces generated by stress fibers help cells to resist imposed stretching, reduce the occurrence of T1 transitions, and, consequently, modulate their size-dependent elongation. Our findings demonstrate that epithelial cells could utilize their size and internal structure to regulate their physical and related biological behaviors. The theoretical framework proposed here can also be extended to investigate the roles of cell geometry and intracellular contraction in processes such as collective cell migration and embryo development.

Beyond the Genetic Code: A Tissue Code?

The genetic code determines how the precise amino acid sequence of proteins is specified by genomic information in cells. But what specifies the precise histologic organization of cells in plant and animal tissues is unclear. We now hypothesize that another code, the tissue code , exists at an even higher level of complexity which determines how tissue organization is dynamically maintained. Accordingly, we modeled spatial and temporal asymmetries of cell division and established that five simple mathematical laws ("the tissue code") convey a set of biological rules that maintain the specific organization and continuous self-renewal dynamics of cells in tissues. These laws might even help us understand wound healing, and how tissue disorganization leads to birth defects and tissue pathology like cancer.

Modelling of Tissue Invasion in Epithelial Monolayers

Mathematical and computational models are used to describe biomechanical processes in multicellular systems. Here, we develop a model to analyse how two types of epithelial cell layers interact during tissue invasion depending on their cellular properties, i.e., simulating cancer cells expanding into a region of normal cells. We model the tissue invasion process using the cellular Potts model and implement our two-dimensional computational simulations in the software package CompuCell3D. The model predicts that differences in mechanical properties of cells can lead to tissue invasion, even if the division rates and death rates of the two cell types are the same. We also show how the invasion speed varies depending on the cell division and death rates and the mechanical properties of the cells.

Simulating 3D Cell Shape with the Cellular Potts Model

Computer simulations have become a widely used method for the field of mechanobiology. An important question is whether one can predict the shape and forces of cells as a function of the extracellular environment. Different types of models have been described before to simulate cell and tissue shapes in structured environments. In this chapter, we give a brief overview of commonly used models and then describe the Cellular Potts Model, a lattice-based modelling framework, in more detail. We provide a hands-on guide on how to build a model that simulates the shape of a single cell on a micropattern in three dimensions in different open source software packages using the Cellular Potts framework. A simulation is set up with an initial configuration of generalized cells that change shape and position due to an energy function that incorporates cellular volume and surface area constraints as well as interaction energies between the generalized cells.

Intrinsic cell rheology drives junction maturation

A fundamental property of higher eukaryotes that underpins their evolutionary success is stable cell-cell cohesion. Yet, how intrinsic cell rheology and stiffness contributes to junction stabilization and maturation is poorly understood. We demonstrate that localized modulation of cell rheology governs the transition of a slack, undulated cell-cell contact (weak adhesion) to a mature, straight junction (optimal adhesion). Cell pairs confined on different geometries have heterogeneous elasticity maps and control their own intrinsic rheology co-ordinately. More compliant cell pairs grown on circles have slack contacts, while stiffer triangular cell pairs favour straight junctions with flanking contractile thin bundles. Counter-intuitively, straighter cell-cell contacts have reduced receptor density and less dynamic junctional actin, suggesting an unusual adaptive mechano-response to stabilize cell-cell adhesion. Our modelling informs that slack junctions arise from failure of circular cell pairs to increase their own intrinsic stiffness and resist the pressures from the neighbouring cell. The inability to form a straight junction can be reversed by increasing mechanical stress artificially on stiffer substrates. Our data inform on the minimal intrinsic rheology to generate a mature junction and provide a springboard towards understanding elements governing tissue-level mechanics.

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.

Micro-patterned cell populations as advanced pharmaceutical drugs with precise functional control

Human cells are both advanced pharmaceutical drugs and 'drug deliverers'. However, functional control prior to or after cell implantation remains challenging. Micro-patterning cells through geometrically defined adhesion sites allows controlling morphogenesis, polarity, cellular mechanics, proliferation, migration, differentiation, stemness, cell-cell interactions, collective cell behavior, and likely immuno-modulatory properties. Consequently, generating micro-patterned therapeutic cells is a promising idea that has not yet been realized and few if any steps have been undertaken in this direction. This review highlights potential therapeutic applications, summarizes comprehensively the many cell functions that have been successfully controlled through micro-patterning, details the established micro-pattern designs, introduces the available fabrication technologies to the non-specialized reader, and suggests a quality evaluation score. Such a broad review is not yet available but would facilitate the manufacturing of therapeutically patterned cell populations using micro-patterned cell-instructive biomaterials for improved functional control as drug delivery systems in the context of cells as pharmaceutical products.

Implementing computational modeling in tissue engineering: where disciplines meet

In recent years, the mathematical and computational sciences have developed novel methodologies and insights that can aid in designing advanced bioreactors, microfluidic set-ups or organ-on-chip devices, in optimizing culture conditions, or predicting long-term behavior of engineered tissues in vivo. In this review, we introduce the concept of computational models and how they can be integrated in an interdisciplinary workflow for Tissue Engineering and Regenerative Medicine (TERM). We specifically aim this review of general concepts and examples at experimental scientists with little or no computational modeling experience. We also describe the contribution of computational models in understanding TERM processes and in advancing the TERM field by providing novel insights.

Modelling the Collective Mechanical Regulation of the Structure and Morphology of Epithelial Cell Layers

The morphology and function of epithelial sheets play an important role in healthy tissue development and cancer progression. The maintenance of structure of closely packed epithelial layers requires the coordination of various mechanical forces due to intracellular activities and interactions with other cells and tissues. However, a general model for the combination of mechanical properties which determine the cell shape and the overall structure of epithelial layers remains elusive. Here, we propose a computational model, based on the Cellular Potts Model, to analyse the interplay between mechanical properties of cells and dynamical transitions in epithelial cell shapes and structures. We map out phase diagrams as functions of cellular properties and the orientation of cell division. Results show that monolayers of squamous, cuboidal, and columnar cells are formed when the axis of cell proliferation is perpendicular to the substrate or along the major axis of the cells. Monolayer-to-multilayer transition is promoted via cell extrusion, depending on the mechanical properties of cells and the orientation of cell division. The results and model predictions are discussed in the context of experimental observations.

Modeling and simulation of cell migration on the basis of force equilibrium

To study cell behavior, we developed a cell model to simulate cell movements and the interacting forces among cells and between cells and obstacles. The developed model simulates several cells simultaneously and examines correlations among characteristic parameters between cells and substrates during migration. We modified Odde's model to develop fundamental model, applied Gillespie's stochastic algorithm to design time during in the migration simulation, and employed Keren's membrane theory to analyze the equilibrium at the leading edges. Thus, the proposed model can analyze stresses due to substrate, the intracellular body, and the external interaction between cells and obstacles. Simulation results indicate that cell-cell interaction depends on the equilibrium between the forces at the leading edge of the membrane, namely the cell-substrate interaction force, cell-cell interaction forces, and the cell membrane force. These results also indicate that the migration direction is dependent on the resultant forces. The membrane force and substrate force directions are "low correlation," and the polymerization rate exhibits "little correlative" with the migration direction. We propose a modified cell migration model for simulating allocation and interaction among multiple cells. This model helps indicate the weightings of characteristic parameters that affect the cell migration direction and velocity.

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

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

High-Throughput Methods in the Discovery and Study of Biomaterials and Materiobiology

The complex interaction of cells with biomaterials (i.e., materiobiology) plays an increasingly pivotal role in the development of novel implants, biomedical devices, and tissue engineering scaffolds to treat diseases, aid in the restoration of bodily functions, construct healthy tissues, or regenerate diseased ones. However, the conventional approaches are incapable of screening the huge amount of potential material parameter combinations to identify the optimal cell responses and involve a combination of serendipity and many series of trial-and-error experiments. For advanced tissue engineering and regenerative medicine, highly efficient and complex bioanalysis platforms are expected to explore the complex interaction of cells with biomaterials using combinatorial approaches that offer desired complex microenvironments during healing, development, and homeostasis. In this review, we first introduce materiobiology and its high-throughput screening (HTS). Then we present an in-depth of the recent progress of 2D/3D HTS platforms (i.e., gradient and microarray) in the principle, preparation, screening for materiobiology, and combination with other advanced technologies. The Compendium for Biomaterial Transcriptomics and high content imaging, computational simulations, and their translation toward commercial and clinical uses are highlighted. In the final section, current challenges and future perspectives are discussed. High-throughput experimentation within the field of materiobiology enables the elucidation of the relationships between biomaterial properties and biological behavior and thereby serves as a potential tool for accelerating the development of high-performance biomaterials.

Intermediate adhesion maximizes migration velocity of multicellular clusters

Collections of cells exhibit coherent migration during morphogenesis, cancer metastasis, and wound healing. In many cases, bigger clusters split, smaller subclusters collide and reassemble, and gaps continually emerge. The connections between cell-level adhesion and cluster-level dynamics, as well as the resulting consequences for cluster properties such as migration velocity, remain poorly understood. Here we investigate collective migration of one- and two-dimensional cell clusters that collectively track chemical gradients using a mechanism based on contact inhibition of locomotion. We develop both a minimal description based on the lattice gas model of statistical physics and a more realistic framework based on the cellular Potts model which captures cell shape changes and cluster rearrangement. In both cases, we find that cells have an optimal adhesion strength that maximizes cluster migration speed. The optimum negotiates a tradeoff between maintaining cell-cell contact and maintaining configurational freedom, and we identify maximal variability in the cluster aspect ratio as a revealing signature. Our results suggest a collective benefit for intermediate cell-cell adhesion.

The Research Advance of Cell Bridges in vitro

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

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.

Cell-substrate mechanics guide collective cell migration through intercellular adhesion: a dynamic finite element cellular model

During the process of tissue formation and regeneration, cells migrate collectively while remaining connected through intercellular adhesions. However, the roles of cell-substrate and cell-cell mechanical interactions in regulating collective cell migration are still unclear. In this study, we employ a newly developed finite element cellular model to study collective cell migration by exploring the effects of mechanical feedback between cell and substrate and mechanical signal transmission between adjacent cells. Our viscoelastic model of cells consists many triangular elements and is of high resolution. Cadherin adhesion between cells is modeled explicitly as linear springs at subcellular level. In addition, we incorporate a mechano-chemical feedback loop between cell-substrate mechanics and Rac-mediated cell protrusion. Our model can reproduce a number of experimentally observed patterns of collective cell migration during wound healing, including cell migration persistence, separation distance between cell pairs and migration direction. Moreover, we demonstrate that cell protrusion determined by the cell-substrate mechanics plays an important role in guiding persistent and oriented collective cell migration. Furthermore, this guidance cue can be maintained and transmitted to submarginal cells of long distance through intercellular adhesions. Our study illustrates that our finite element cellular model can be employed to study broad problems of complex tissue in dynamic changes at subcellular level.

Extracellular Matrix Geometry and Initial Adhesive Position Determine Stress Fiber Network Organization during Cell Spreading

Three-dimensional matrices often contain highly structured adhesive tracks that require cells to turn corners and bridge non-adhesive areas. Here, we investigate these complex processes using micropatterned cell adhesive frames. Spreading kinetics on these matrices depend strongly on initial adhesive position and are predicted by a cellular Potts model (CPM), which reflects a balance between adhesion and intracellular tension. As cells spread, new stress fibers (SFs) assemble periodically and parallel to the leading edge, with spatial intervals of ∼2.5 μm, temporal intervals of ∼15 min, and characteristic lifetimes of ∼50 min. By incorporating these rules into the CPM, we can successfully predict SF network architecture. Moreover, we observe broadly similar behavior when we culture cells on arrays of discrete collagen fibers. Our findings show that ECM geometry and initial cell position strongly determine cell spreading and that cells encode a memory of their spreading history through SF network organization.

Multiscale modelling of motility wave propagation in cell migration

The collective motion of cell monolayers within a tissue is a fundamental biological process that occurs during tissue formation, wound healing, cancerous invasion, and viral infection. Experiments have shown that at the onset of migration, the motility is self-generated as a polarisation wave starting from the leading edge of the monolayer and progressively propagates into the bulk. However, it is unclear how the propagation of this motility wave is influenced by cellular properties. Here, we investigate this question using a computational model based on the Potts model coupled to the dynamics of intracellular polarisation. The model captures the propagation of the polarisation wave and suggests that the cells cortex can regulate the migration modes: strongly contractile cells may depolarise the monolayer, whereas less contractile cells can form swirling movement. Cortical contractility is further found to limit the cells motility, which (i) decelerates the wave speed and the leading edge progression, and (ii) destabilises the leading edge. Together, our model describes how different mechanical properties of cells can contribute to the regulation of collective cell migration.

A Lock Free Approach To Parallelize The Cellular Potts Model: Application To Ductal Carcinoma In Situ

In the field of computational biology, in order to simulate multiscale biological systems, the Cellular Potts Model (CPM) has been used, which determines the actions that simulated cells can perform by determining a hamiltonian of energy that takes into account the influence that neighboring cells exert, under a wide range of parameters. There are some proposals in the literature that parallelize the CPM; in all cases, either lock-based techniques or other techniques that require large amounts of information to be disseminated among parallel tasks are used to preserve data coherence. In both cases, computational performance is limited. This work proposes an alternative approach for the parallelization of the model that uses transactional memory to maintain the coherence of the information. A Java implementation has been applied to the simulation of the ductal adenocarcinoma of breast in situ (DCIS). Times and speedups of the simulated execution of the model on the cluster of our university are analyzed. The results show a good speedup.

Disentangling the behavioural variability of confined cell migration

Cell-to-cell variability is inherent to numerous biological processes, including cell migration. Quantifying and characterizing the variability of migrating cells is challenging, as it requires monitoring many cells for long time windows under identical conditions. Here, we observe the migration of single human breast cancer cells (MDA-MB-231) in confining two-state micropatterns. To describe the stochastic dynamics of this confined migration, we employ a dynamical systems approach. We identify statistics to measure the behavioural variance of the migration, which significantly exceeds that predicted by a population-averaged stochastic model. This additional variance can be explained by the combination of an ‘ageing’ process and population heterogeneity. To quantify population heterogeneity, we decompose the cells into subpopulations of slow and fast cells, revealing the presence of distinct classes of dynamical systems describing the migration, ranging from bistable to limit cycle behaviour. Our findings highlight the breadth of migration behaviours present in cell populations.

Hydrodynamic effects on the motility of crawling eukaryotic cells

Eukaryotic cell motility is crucial during development, wound healing, the immune response, and cancer metastasis. Some eukaryotic cells can swim, but cells more commonly adhere to and crawl along the extracellular matrix. We study the relationship between hydrodynamics and adhesion that describe whether a cell is swimming, crawling, or combining these motions. Our simple model of a cell, based on the three-sphere swimmer, is capable of both swimming and crawling. As cell-matrix adhesion strength increases, the influence of hydrodynamics on migration diminishes. Cells with significant adhesion can crawl with speeds much larger than their nonadherent, swimming counterparts. We predict that, while most eukaryotic cells are in the strong-adhesion limit, increasing environment viscosity or decreasing cell-matrix adhesion could lead to significant hydrodynamic effects even in crawling cells. Signatures of hydrodynamic effects include a dependence of cell speed on the presence of a nearby substrate or interactions between noncontacting cells. These signatures will be suppressed at large adhesion strengths, but even strongly adherent cells will generate relevant fluid flows that will advect nearby passive particles and swimmers.

From energy to cellular forces in the Cellular Potts Model: An algorithmic approach

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = −∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments.

Cells exert forces on their surroundings and on one another. In simulations of cell shape using the Cellular Potts Model (CPM), the dynamics of deforming cell shapes is traditionally represented by an energy-minimization method. We use this CPM energy, the Hamiltonian, to derive and visualize the corresponding forces exerted by the cells. We use the fact that force is the negative gradient of energy to assign forces to the CPM cell edges, and then extend the results to approximate interior forces by interpolation. We show that this method works for single as well as multiple interacting model cells, both static and motile. Finally, we show favorable comparison between predicted forces and real forces measured experimentally.

Bridging the gap between single-cell migration and collective dynamics

Motivated by the wealth of experimental data recently available, we present a cellular-automaton-based modeling framework focussing on high-level cell functions and their concerted effect on cellular migration patterns. Specifically, we formulate a coarse-grained description of cell polarity through self-regulated actin organization and its response to mechanical cues. Furthermore, we address the impact of cell adhesion on collective migration in cell cohorts. The model faithfully reproduces typical cell shapes and movements down to the level of single cells, yet allows for the efficient simulation of confluent tissues. In confined circular geometries, we find that specific properties of individual cells (polarizability; contractility) influence the emerging collective motion of small cell cohorts. Finally, we study the properties of expanding cellular monolayers (front morphology; stress and velocity distributions) at the level of extended tissues.

Continuum Models of Collective Cell Migration

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

Computational Modeling of Collective Cell Migration: Mechanical and Biochemical Aspects

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

Cellular Response to Surface Morphology: Electrospinning and Computational Modeling

Surface properties of biomaterials, such as chemistry and morphology, have a major role in modulating cellular behavior and therefore impact on the development of high-performance devices for biomedical applications, such as scaffolds for tissue engineering and systems for drug delivery. Opportunely-designed micro- and nanostructures provides a unique way of controlling cell-biomaterial interaction. This mini-review discusses the current research on the use of electrospinning (extrusion of polymer nanofibers upon the application of an electric field) as effective technique to fabricate patterns of micro- and nano-scale resolution, and the corresponding biological studies. The focus is on the effect of morphological cues, including fiber alignment, porosity and surface roughness of electrospun mats, to direct cell migration and to influence cell adhesion, differentiation and proliferation. Experimental studies are combined with computational models that predict and correlate the surface composition of a biomaterial with the response of cells in contact with it. The use of predictive models can facilitate the rational design of new bio-interfaces.

Cooperation of dual modes of cell motility promotes epithelial stress relaxation to accelerate wound healing

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 show that a balance of protrusive motility and actomyosin contractility is optimized for accelerating the rate of wound repair, which is robust to variations in cell and substrate mechanical properties. This balance underlies rapid collective cell motion during wound healing, resulting from a tradeoff between tension mediated collective cell guidance and active stress relaxation in the tissue.

Force localization modes in dynamic epithelial colonies

Collective cell behaviors, including tissue remodeling, morphogenesis, and cancer metastasis, rely on dynamics among cells, their neighbors, and the extracellular matrix. The lack of quantitative models precludes understanding of how cell-cell and cell-matrix interactions regulate tissue-scale force transmission to guide morphogenic processes. We integrate biophysical measurements on model epithelial tissues and computational modeling to explore how cell-level dynamics alter mechanical stress organization at multicellular scales. We show that traction stress distribution in epithelial colonies can vary widely for identical geometries. For colonies with peripheral localization of traction stresses, we recapitulate previously described mechanical behavior of cohesive tissues with a continuum model. By contrast, highly motile cells within colonies produce traction stresses that fluctuate in space and time. To predict the traction force dynamics, we introduce an active adherent vertex model (AAVM) for epithelial monolayers. AAVM predicts that increased cellular motility and reduced intercellular mechanical coupling localize traction stresses in the colony interior, in agreement with our experimental data. Furthermore, the model captures a wide spectrum of localized stress production modes that arise from individual cell activities including cell division, rotation, and polarized migration. This approach provides a robust quantitative framework to study how cell-scale dynamics influence force transmission in epithelial tissues.

Epithelial vertex models with active biochemical regulation of contractility can explain organized collective cell motility

Collective motions of groups of cells are observed in many biological settings such as embryo development, tissue formation, and cancer metastasis. To effectively model collective cell movement, it is important to incorporate cell specific features such as cell size, cell shape, and cell mechanics, as well as active behavior of cells such as protrusion and force generation, contractile forces, and active biochemical signaling mechanisms that regulate cell behavior. In this paper, we develop a comprehensive model of collective cell migration in confluent epithelia based on the vertex modeling approach. We develop a method to compute cell-cell viscous friction based on the vertex model and incorporate RhoGTPase regulation of cortical myosin contraction. Global features of collective cell migration are examined by computing the spatial velocity correlation function. As active cell force parameters are varied, we found rich dynamical behavior. Furthermore, we find that cells exhibit nonlinear phenomena such as contractile waves and vortex formation. Together our work highlights the importance of active behavior of cells in generating collective cell movement. The vertex modeling approach is an efficient and versatile approach to rigorously examine cell motion in the epithelium.

A DISCRETE MATHEMATICAL MODEL FOR SINGLE AND COLLECTIVE MOVEMENT IN AMOEBOID CELLS

In this paper, we develop a new discrete mathematical model for individual and collective cell motility. We introduce a mechanical model for the movement of a cell on a two-dimensional rigid surface to describe and investigate the cell–cell and cell–substrate interactions. The cell cytoskeleton is modeled as a series of springs and dashpots connected in parallel. The cell–substrate attachments and the cell protrusions are also included. In particular, this model is used to describe the directed movement of endothelial cells on a Matrigel plate. We compare the results from our model with experimental data. We show that cell density and substrate rigidity play an important role in network formation.

Multicompartment cell-based modeling of confined migration: regulation by cell intrinsic and extrinsic factors

Though cell and nuclear deformability are expected to influence efficiency of confined migration, their individual and collective influence on migration efficiency remains incompletely understood. In addition to cell intrinsic properties, the relevance of cell extrinsic factors on confined migration, if any, has not been adequately explored. Here we address these questions using a statistical mechanics-based stochastic modeling approach where cell/nuclear dimensions and their deformability are explicitly taken into consideration. In addition to demonstrating the importance of cell softness in sustaining confined migration, our results suggest that dynamic tuning of cell and nuclear properties at different stages of migration is essential for maximizing migration efficiency. Our simulations also implicate confinement shape and confinement history as two important cell extrinsic regulators of cell invasiveness. Together, our findings illustrate the strength of a multicompartment model in dissecting the contributions of multiple factors that collectively influence confined cell migration.

Collective cell migration without proliferation: density determines cell velocity and wave velocity

Collective cell migration contributes to embryogenesis, wound healing and tumour metastasis. Cell monolayer migration experiments help in understanding what determines the movement of cells far from the leading edge. Inhibiting cell proliferation limits cell density increase and prevents jamming; we observe long-duration migration and quantify space-time characteristics of the velocity profile over large length scales and time scales. Velocity waves propagate backwards and their frequency depends only on cell density at the moving front. Both cell average velocity and wave velocity increase linearly with the cell effective radius regardless of the distance to the front. Inhibiting lamellipodia decreases cell velocity while waves either disappear or have a lower frequency. Our model combines conservation laws, monolayer mechanical properties and a phenomenological coupling between strain and polarity: advancing cells pull on their followers, which then become polarized. With reasonable values of parameters, this model agrees with several of our experimental observations. Together, our experiments and model disantangle the respective contributions of active velocity and of proliferation in monolayer migration, explain how cells maintain their polarity far from the moving front, and highlight the importance of strain-polarity coupling and density in long-range information propagation.

Pore Shape Defines Paths of Metastatic Cell Migration

Invasion of dense tissues by cancer cells involves the interplay between the penetration resistance offered by interstitial pores and the deformability of cells. Metastatic cancer cells find optimal paths of minimal resistance through an adaptive path-finding process, which leads to successful dissemination. The physical limit of nuclear deformation is related to the minimal cross section of pores that can be successfully penetrated. However, this single biophysical parameter does not fully describe the architectural complexity of tissues featuring pores of variable area and shape. Here, employing laser nanolithography, we fabricate pore microenvironment models with well-controlled pore shapes, through which human breast cells (MCF10A) and their metastatic offspring (MCF10CA1a.cl1) could pervade. In these experimental settings, we demonstrate that the actual pore shape, and not only the cross section, is a major and independent determinant of cancer penetration efficiency. In complex architectures containing pores demanding large deformations from invading cells, tall and narrow rectangular openings facilitate cancer migration. In addition, we highlight the characteristic traits of the explorative behavior enabling metastatic cells to identify and select such pore shapes in a complex multishape pore environment, pinpointing paths of least resistance to invasion.

Role of contact inhibition of locomotion and junctional mechanics in epithelial collective responses to injury

Epithelial tissues form physically integrated barriers against the external environment protecting organs from infection and invasion. Within each tissue, epithelial cells respond to different challenges that can potentially compromise tissue integrity. In particular, cells collectively respond to injuries by reorganizing their cell-cell junctions and migrating directionally towards the sites of damage. Notwithstanding, the mechanisms that drive collective responses in epithelial aggregates remain poorly understood. In this work, we develop a minimal mechanistic model that is able to capture the essential features of epithelial collective responses to injuries. We show that a model that integrates the mechanics of cells at the cell-cell and cell-substrate interfaces as well as contact inhibition of locomotion (CIL) correctly predicts two key properties of epithelial response to injury as: (1) local relaxation of the tissue and (2) collective reorganization involving the extension of cryptic lamellipodia that extend, on average, up to 3 cell diameters from the site of injury and morphometric changes in the basal regions. Our model also suggests that active responses (like the actomyosin purse string and softening of cell-cell junctions) are needed to drive morphometric changes in the apical region. Therefore, our results highlight the importance of the crosstalk between junctional biomechanics, cell substrate adhesion, and CIL, as well as active responses, in guiding the collective rearrangements that are required to preserve the epithelial barrier in response to injury.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Dynamic cellular finite-element method for modelling large-scale cell migration and proliferation under the control of mechanical and biochemical cues: a study of re-epithelialization

Computational modelling of cells can reveal insight into the mechanisms of the important processes of tissue development. However, current cell models have limitations and are challenged to model detailed changes in cellular shapes and physical mechanics when thousands of migrating and interacting cells need to be modelled. Here we describe a novel dynamic cellular finite-element model (DyCelFEM), which accounts for changes in cellular shapes and mechanics. It also models the full range of cell motion, from movements of individual cells to collective cell migrations. The transmission of mechanical forces regulated by intercellular adhesions and their ruptures are also accounted for. Intra-cellular protein signalling networks controlling cell behaviours are embedded in individual cells. We employ DyCelFEM to examine specific effects of biochemical and mechanical cues in regulating cell migration and proliferation, and in controlling tissue patterning using a simplified re-epithelialization model of wound tissue. Our results suggest that biochemical cues are better at guiding cell migration with improved directionality and persistence, while mechanical cues are better at coordinating collective cell migration. Overall, DyCelFEM can be used to study developmental processes when a large population of migrating cells under mechanical and biochemical controls experience complex changes in cell shapes and mechanics.

Cell migration in microengineered tumor environments

Recent advances in microengineered cell migration platforms are discussed critically with a focus on how cell migration is influenced by engineered tumor microenvironments, the medical relevance being to understand how tumor microenvironments may promote or suppress the progression of cancer. We first introduce key findings in cancer cell migration under the influence of the physical environment, which is systematically controlled by microengineering technology, followed by multi-cues of physico-chemical factors, which represent the complexity of the tumor environment. Recognizing that cancer cells constantly communicate not only with each other but also with tumor-associated cells such as vascular, fibroblast, and immune cells, and also with non-cellular components, it follows that cell motility in tumor microenvironments, especially metastasis via the invasion of cancer cells into the extracellular matrix and other tissues, is closely related to the malignancy of cancer-related mortality. Medical relevance of forefront research realized in microfabricated devices, such as single cell sorting based on the analysis of cell migration behavior, may assist personalized theragnostics based on the cell migration phenotype. Furthermore, we urge development of theory and numerical understanding of single or collective cell migration in microengineered platforms to gain new insights in cancer metastasis and in therapeutic strategies.

Development of a Virtual Cell Model to Predict Cell Response to Substrate Topography

Cells can sense and respond to changes in the topographical, chemical, and mechanical information in their environment. Engineered substrates are increasingly being developed that exploit these physical attributes to direct cell responses (most notably mesenchymal stem cells) and therefore control cell behavior toward desired applications. However, there are very few methods available for robust and accurate modeling that can predict cell behavior prior to experimental evaluations, and this typically means that many cell test iterations are needed to identify best material features. Here, we developed a unifying computational framework to create a multicomponent cell model, called the "virtual cell model" that has the capability to predict changes in whole cell and cell nucleus characteristics (in terms of shape, direction, and even chromatin conformation) on a range of cell substrates. Modeling data were correlated with cell culture experimental outcomes in order to confirm the applicability of the virtual cell model and demonstrating the ability to reflect the qualitative behavior of mesenchymal stem cells. This may provide a reliable, efficient, and fast high-throughput approach for the development of optimized substrates for a broad range of cellular applications including stem cell differentiation.

Deciphering the Complex Signaling Systems That Regulate Intestinal Epithelial Cell Death Processes and Shedding

Intestinal epithelial cells play a fundamental role in maintaining homeostasis. Shedding of intestinal cells in a controlled manner is critical to maintenance of barrier function. Barrier function is maintained during this shedding process by a redistribution of tight junctional proteins to facilitate closure of the gap left by the shedding cell. However, despite the obvious importance of epithelial cell shedding to gut health, a central question is how the extrusion of epithelial cells is achieved, enabling barrier integrity to be maintained in the healthy gut and restored during inflammation remains largely unanswered. Recent studies have provided evidence that excessive epithelial cell shedding and loss of epithelial barrier integrity is triggered by exposure to lipopolysaccharide or tumor necrosis factor alpha. Subsequent studies have provided evidence of the involvement of specific cellular components and signaling mechanisms as well as the functionality of microbiota that can be either detrimental or beneficial for intestinal barrier integrity. This review will focus on the evidence and decipher how the signaling systems through which the mucosal immune system and microbiota can regulate epithelial cell shedding and how these mechanisms interact to preserve the viability of the epithelium.

Cell Contractility Facilitates Alignment of Cells and Tissues to Static Uniaxial Stretch

During animal development and homeostasis, the structure of tissues, including muscles, blood vessels, and connective tissues, adapts to mechanical strains in the extracellular matrix (ECM). These strains originate from the differential growth of tissues or forces due to muscle contraction or gravity. Here we show using a computational model that by amplifying local strain cues, active cell contractility can facilitate and accelerate the reorientation of single cells to static strains. At the collective cell level, the model simulations show that active cell contractility can facilitate the formation of strings along the orientation of stretch. The computational model is based on a hybrid cellular Potts and finite-element simulation framework describing a mechanical cell-substrate feedback, where: 1) cells apply forces on the ECM, such that 2) local strains are generated in the ECM and 3) cells preferentially extend protrusions along the strain orientation. In accordance with experimental observations, simulated cells align and form stringlike structures parallel to static uniaxial stretch. Our model simulations predict that the magnitude of the uniaxial stretch and the strength of the contractile forces regulate a gradual transition between stringlike patterns and vascular networklike patterns. Our simulations also suggest that at high population densities, less cell cohesion promotes string formation.