Computational model for cell migration in three-dimensional matrices

Although computational models for cell migration on two-dimensional (2D) substrata have described how various molecular and cellular properties and physiochemical processes are integrated to accomplish cell locomotion, the same issues, along with certain new ones, might contribute differently to a model for migration within three-dimensional (3D) matrices. To address this more complicated situation, we have developed a computational model for cell migration in 3D matrices using a force-based dynamics approach. This model determines an overall locomotion velocity vector, comprising speed and direction, for individual cells based on internally generated forces transmitted into external traction forces and considering a timescale during which multiple attachment and detachment events are integrated. Key parameters characterize cell and matrix properties, including cell/matrix adhesion and mechanical and steric properties of the matrix; critical underlying molecular properties are incorporated explicitly or implicitly. Model predictions agree well with experimental results for the limiting case of migration on 2D substrata as well as with recent experiments in 3D natural tissues and synthetic gels. Certain predicted features such as biphasic behavior of speed with density of matrix ligands for 3D migration are qualitatively similar to their 2D counterparts, but new effects generally absent in 2D systems, such as effects due to matrix sterics and mechanics, are now predicted to arise in many 3D situations. As one particular sample manifestation of these effects, the optimal levels of cell receptor expression and matrix ligand density yielding maximal migration are dependent on matrix mechanical compliance.     

Computational modelling and analysis of mechanical conditions on cell locomotion and cell–cell interaction

Between other parameters, cell migration is partially guided by the mechanical properties of its substrate. Although many experimental works have been developed to understand the effect of substrate mechanical properties on cell migration, accurate 3D cell locomotion models have not been presented yet. In this paper, we present a novel 3D model for cells migration. In the presented model, we assume that a cell follows two main processes: in the first process, it senses its interface with the substrate to determine the migration direction and in the second process, it exerts subsequent forces to move. In the presented model, cell traction forces are considered to depend on cell internal deformation during the sensing step. A random protrusion force is also considered which may change cell migration direction and/or speed. The presented model was applied for many cases of migration of the cells. The obtained results show high agreement with the available experimental and numerical data.     

A time-dependent phenomenological model for cell mechano-sensing

Adherent cells normally apply forces as a generic means of sensing and responding to the mechanical nature of their surrounding environment. How these forces vary as a function of the extracellular rigidity is critical to understanding the regulatory functions that drive important phenomena such as wound healing or muscle contraction. In recognition of this fact, experiments have been conducted to understand cell rigidity-sensing properties under known conditions of the extracellular environment, opening new possibilities for modeling this active behavior. In this work, we provide a physics-based constitutive model taking into account the main structural components of the cell to reproduce its most significant contractile properties such as the traction forces exerted as a function of time and the extracellular stiffness. This model shows how the interplay between the time-dependent response of the acto-myosin contractile system and the elastic response of the cell components determines the mechano-sensing behavior of single cells.     

Breast Cancer Cells Adapt Contractile Forces to Overcome Steric Hindrance

Cell migration through the extracellular matrix is governed by the interplay between cell-generated propulsion forces, adhesion forces, and resisting forces arising from the steric hindrance of the matrix. Steric hindrance in turn depends on matrix porosity, matrix deformability, cell size, and cell deformability. In this study, we investigate how cells respond to changes in steric hindrance that arise from altered cell mechanical properties. Specifically, we measure traction forces, cell morphology, and invasiveness of MDA-MB 231 breast cancer cells in three-dimensional collagen gels. To modulate cell mechanical properties, we either decrease nuclear deformability by twofold overexpression of the nuclear protein lamin A or we introduce into the cells stiff polystyrene beads with a diameter larger than the average matrix pore size. Despite this increase of steric hindrance, we find that cell invasion is only marginally inhibited, as measured by the fraction of motile cells and the mean invasion depth. To compensate for increased steric hindrance, cells employ two alternative strategies. Cells with higher nuclear stiffness increase their force polarity, whereas cells with large beads increase their net contractility. Under both conditions, the collagen matrix surrounding the cells stiffens dramatically and carries increased strain energy, suggesting that increased force polarity and increased net contractility are functionally equivalent strategies for overcoming an increased steric hindrance.     

Three-dimensional traction force microscopy: a new tool for quantifying cell-matrix interactions

The interactions between biochemical processes and mechanical signaling play important roles during various cellular processes such as wound healing, embryogenesis, metastasis, and cell migration. While traditional traction force measurements have provided quantitative information about cell matrix interactions in two dimensions, recent studies have shown significant differences in the behavior and morphology of cells when placed in three-dimensional environments. Hence new quantitative experimental techniques are needed to accurately determine cell traction forces in three dimensions. Recently, two approaches both based on laser scanning confocal microscopy have emerged to address this need. This study highlights the details, implementation and advantages of such a three-dimensional imaging methodology with the capability to compute cellular traction forces dynamically during cell migration and locomotion. An application of this newly developed three-dimensional traction force microscopy (3D TFM) technique to single cell migration studies of 3T3 fibroblasts is presented to show that this methodology offers a new quantitative vantage point to investigate the three-dimensional nature of cell-ECM interactions.     

The Role of Vinculin in the Regulation of the Mechanical Properties of Cells

Vinculin couples as a focal adhesion protein the extracellular matrix (ECM) through integrins to the actomyosin cytoskeleton. During the last years vinculin has become the focus of cell mechanical measurements and a key protein regulating the transmission of contractile forces. In earlier reports vinculin has been described as an inhibitor of cell migration on planar substrates, because knock-out of vinculin in F9 mouse embryonic carcinoma cells and mouse embryonic fibroblasts showed increased cell motility on 2D substrates. The role of vinculin in cell invasion through a 3D extracellular matrix is still fragmentarily investigated. This review presents vinculin in its role as a regulator of cellular mechanical functions. Contractile force generation is reduced when vinculin is absent, or enhanced when vinculin is present. Moreover, the generation of contractile forces is a prerequisite for cell invasion through a dense 3D ECM, where the pore-size is smaller than the diameter of the cell nucleus (<2 μm). Measurements of cell’s biophysical properties will be presented. In summary, vinculin’s leading role among focal adhesion proteins in regulating the mechanical properties of cells will be discussed.     

Acto-myosin based response to stiffness and rigidity sensing

Cells sense the rigidity of their environment and respond to it. Most studies have been focused on the role of adhesion complexes in rigidity sensing. In particular, it has been clearly shown that proteins of the adhesion complexes were stretch-sensitive and could thus trigger mechano-chemical signaling in response to applied forces. In order to understand how this local mechano-sensitivity could be coordinated at the cell scale, we have recently carried out single cell traction force measurements on springs of varying stiffness. We found that contractility at the cell scale (force, speed of contraction, mechanical power) was indeed adapted to external stiffness and reflected ATPase activity of non-muscle myosin II and acto-myosin response to load. Here we suggest a scenario of rigidity sensing where local adhesions sensitivity to force could be coordinated by adaptation of the acto-myosin dependent cortical tension at the global cell scale. Such a scenario could explain how spreading and migration are oriented by the rigidity of the cell environment.Key words: single cell, mechano-sensing, mechano-transduction, contractility, spreading, polarization     

Cell and tissue mechanics in cell migration

Migrating cells generate traction forces to counteract the movement-resisting forces arising from cell-internal stresses and matrix adhesions. In the case of collective migration in a cell colony, or in the case of 3-dimensional migration through connective tissue, movement-resisting forces arise also from external stresses. Although the deformation of a stiffer cell or matrix causes larger movement-resisting forces, at the same time a larger stiffness can also promote cell migration due to a feedback between forces, deformations, and deformation speed that is mediated by the acto-myosin contractile machinery of cells. This mechanical feedback is also important for stiffness sensing, durotaxis, plithotaxis, and collective migration in cell colonies.     

Mechanische Experimente an lebenden Zellen. Perspektiven für die Krebsforschung

The process of cell migration is definied by biochemical and biophysical properties of the invading cells. Accelarated investigations in the field of biomechanics are of high impact regarding the process of the migration into a 3‐D matrix of tumor cells. In contrast to use of several protein markers that determine tumor malignancy, a single cell mechanical parameter may delevop as a reliable, standard tool for tumor diagnostics. The cell mechanics of living cells shed light on tumor disease especially on the process of metastasis that involves the invasion of tumor cells through connective tissue.     

Dynamic Mechanisms of Cell Rigidity Sensing: Insights from a Computational Model of Actomyosin Networks

Cells modulate themselves in response to the surrounding environment like substrate elasticity, exhibiting structural reorganization driven by the contractility of cytoskeleton. The cytoskeleton is the scaffolding structure of eukaryotic cells, playing a central role in many mechanical and biological functions. It is composed of a network of actins, actin cross-linking proteins (ACPs), and molecular motors. The motors generate contractile forces by sliding couples of actin filaments in a polar fashion, and the contractile response of the cytoskeleton network is known to be modulated also by external stimuli, such as substrate stiffness. This implies an important role of actomyosin contractility in the cell mechano-sensing. However, how cells sense matrix stiffness via the contractility remains an open question. Here, we present a 3-D Brownian dynamics computational model of a cross-linked actin network including the dynamics of molecular motors and ACPs. The mechano-sensing properties of this active network are investigated by evaluating contraction and stress in response to different substrate stiffness. Results demonstrate two mechanisms that act to limit internal stress: (i) In stiff substrates, motors walk until they exert their maximum force, leading to a plateau stress that is independent of substrate stiffness, whereas (ii) in soft substrates, motors walk until they become blocked by other motors or ACPs, leading to submaximal stress levels. Therefore, this study provides new insights into the role of molecular motors in the contraction and rigidity sensing of cells.     

Galectin-3 secretion and tyrosine phosphorylation is dependent on the calpain small subunit, Calpain 4

Cell adhesion and migration are important events that occur during embryonic development, immune surveillance, wound healing and in tumor metastasis. It is a multi-step process that involves both mechanical and biochemical signaling that results in cell protrusion, adhesion, contraction and retraction. Each of these events generates mechanical forces into the environment measured as traction forces. We have previously found that the calpain small subunit, Calpain 4, is required for normal traction forces, and that this mechanism is independent of the catalytic activities of the holoenzymes that are formed between Calpain 4 and each of the proteolytic heavy chains of Calpain 1 and 2. To define a potential mechanism for the Calpain 4 regulation of traction force, we have evaluated the levels of tyrosine phosphorylation, a hallmark of force dependent signaling within focal adhesions. Using 2D gel electrophoresis we compared tyrosine phosphorylation profiles of Calpain 4 deficient mouse embryonic fibroblasts (MEFs) to the levels in wildtype MEFs and MEF’s deficient in the large catalytic subunits, Capn1 and Capn2. Of particular interest, was the identification of Galectin-3, a galactose binding protein known to interact with integrins. Galectin-3 has previously been shown to regulate cell adhesion and migration in both normal and tumor cells; however its full mechanism remains elusive. We have found that Calpain 4 is essential for the tyrosine phosphorylation of galectin-3, and its ultimate secretion from the cell, and speculate that its secretion interferes with the production of traction forces.     

The role of the cytoskeleton in cellular force generation in 2D and 3D environments

To adhere and migrate, cells generate forces through the cytoskeleton that are transmitted to the surrounding matrix. While cellular force generation has been studied on 2D substrates, less is known about cytoskeletal-mediated traction forces of cells embedded in more in vivo-like 3D matrices. Recent studies have revealed important differences between the cytoskeletal structure, adhesion, and migration of cells in 2D and 3D. Because the cytoskeleton mediates force, we sought to directly compare the role of the cytoskeleton in modulating cell force in 2D and 3D. MDA-MB-231 cells were treated with agents that perturbed actin, microtubules, or myosin, and analyzed for changes in cytoskeletal organization and force generation in both 2D and 3D. To quantify traction stresses in 2D, traction force microscopy was used; in 3D, force was assessed based on single cell-mediated collagen fibril reorganization imaged using confocal reflectance microscopy. Interestingly, even though previous studies have observed differences in cell behaviors like migration in 2D and 3D, our data indicate that forces generated on 2D substrates correlate with forces within 3D matrices. Disruption of actin, myosin or microtubules in either 2D or 3D microenvironments disrupts cell-generated force. These data suggest that despite differences in cytoskeletal organization in 2D and 3D, actin, microtubules and myosin contribute to contractility and matrix reorganization similarly in both microenvironments.     

Changes in surface topologies of chondrocytes subjected to mechanical forces: an AFM analysis

The cartilage is composed of chondrocytes embedded in a matrix of collagen fibrils interspersed within a network of proteoglycans and is constantly exposed to biomechanical forces during normal joint movement. Characterization of the surface morphology, cytoskeletal structure, adherance and elastic properties of these mechanosensitive cells are crucial in understanding the effects of mechanical forces around a cell and how a cell responds to changes in its physical environment. In this work, we employed the atomic force microscope (AFM) to image cultured chondrocytes before and after subjecting them to mechanical forces in the presence or absence of interleukin-1β to mimic inflammatory conditions. Nanoscale imaging and quantitative measurements from AFM data revealed that there are distinct changes in cell-surface topology and cytoskeleton arrangement in the cells following treatment with mechanical forces, IL-1β or both. Our findings for the first time demonstrate that cultured chondrocytes are amenable to high-resolution AFM imaging and dynamic tensile forces may help overcome the effect of inflammatory factors on chondrocyte response.     

A phenomenological cohesive model for the macroscopic simulation of cell–matrix adhesions

Cell adhesion is crucial for cells to not only physically interact with each other but also sense their microenvironment and respond accordingly. In fact, adherent cells can generate physical forces that are transmitted to the surrounding matrix, regulating the formation of cell–matrix adhesions. The main purpose of this work is to develop a computational model to simulate the dynamics of cell–matrix adhesions through a cohesive formulation within the framework of the finite element method and based on the principles of continuum damage mechanics. This model enables the simulation of the mechanical adhesion between cell and extracellular matrix (ECM) as regulated by local multidirectional forces and thus predicts the onset and growth of the adhesion. In addition, this numerical approach allows the simulation of the cell as a whole, as it models the complete mechanical interaction between cell and ECM. As a result, we can investigate and quantify how different mechanical conditions in the cell (e.g., contractile forces, actin cytoskeletal properties) or in the ECM (e.g., stiffness, external forces) can regulate the dynamics of cell–matrix adhesions.     

A hybrid computational model for collective cell durotaxis

Collective cell migration is regulated by a complex set of mechanical interactions and cellular mechanisms. Collective migration emerges from mechanisms occurring at single cell level, involving processes like contraction, polymerization and depolymerization, of cell–cell interactions and of cell–substrate adhesion. Here, we present a computational framework which simulates the dynamics of this emergent behavior conditioned by substrates with stiffness gradients. The computational model reproduces the cell’s ability to move toward the stiffer part of the substrate, process known as durotaxis. It combines the continuous formulation of truss elements and a particle-based approach to simulate the dynamics of cell–matrix adhesions and cell–cell interactions. Using this hybrid approach, researchers can quickly create a quantitative model to understand the regulatory role of different mechanical conditions on the dynamics of collective cell migration. Our model shows that durotaxis occurs due to the ability of cells to deform the substrate more in the part of lower stiffness than in the stiffer part. This effect explains why cell collective movement is more effective than single cell movement in stiffness gradient conditions. In addition, we numerically evaluate how gradient stiffness properties, cell monolayer size and force transmission between cells and extracellular matrix are crucial in regulating durotaxis.     

3D Traction Stresses Activate Protease-Dependent Invasion of Cancer Cells

Cell invasion and migration that occurs, for example, in cancer metastasis is rooted in the ability of cells to navigate through varying levels of physical constraint exerted by the extracellular matrix. Cancer cells can invade matrices in either a protease-independent or a protease-dependent manner. An emerging critical component that influences the mode of cell invasion is the traction stresses generated by the cells in response to the physicostructural properties of the extracellular matrix. In this study, we have developed a reference-free quantitative assay for measuring three-dimensional (3D) traction stresses generated by cells during the initial stages of invasion into matrices exerting varying levels of mechanical resistance. Our results show that as cells encounter higher mechanical resistance, a larger fraction of them shift to protease-mediated invasion, and this process begins at lower values of cell invasion depth. On the other hand, the compressive stress generated by the cells at the onset of protease-mediated invasion is found to be independent of matrix stiffness, suggesting that 3D traction stress is a key factor in triggering protease-mediated cancer cell invasion. At low 3D compressive traction stresses, cells utilize bleb formation to indent the matrix in a protease independent manner. However, at higher stress values, cells utilize invadopodia-like structures to mediate protease-dependent invasion into the 3D matrix. The critical value of compressive traction stress at the transition from a protease-independent to a protease-dependent mode of invasion is found to be ∼165 Pa.     

Contact guidance mediated three-dimensional cell migration is regulated by Rho/ROCK-dependent matrix reorganization

Cells generate mechanical force to organize the extracellular matrix (ECM) and drive important developmental and reparative processes. Likewise, tumor cells invading into three-dimensional (3D) matrices remodel the ECM microenvironment. Importantly, we previously reported a distinct radial reorganization of the collagen matrix surrounding tumors that facilitates local invasion. Here we describe a mechanism by which cells utilize contractility events to reorganize the ECM to provide contact guidance that facilitates 3D migration. Using novel assays to differentially organize the collagen matrix we show that alignment of collagen perpendicular to the tumor-explant boundary promotes local invasion of both human and mouse mammary epithelial cells. In contrast, organizing the collagen matrix to mimic the ECM organization associated with noninvading regions of tumors suppresses 3D migration/invasion. Moreover, we demonstrate that matrix reorganization is contractility-dependent and that the Rho/Rho kinase pathway is necessary for collagen alignment to provide contact guidance. Yet, if matrices are prealigned, inhibiting neither Rho nor Rho kinase inhibits 3D migration, which supports our conclusion that Rho-mediated matrix alignment is an early step in the invasion process, preceding and subsequently facilitating 3D migration.     

Cell traction force and measurement methods

Cell traction forces (CTFs) are crucial to many biological processes such as inflammation, wound healing, angiogenesis, and metastasis. CTFs are generated by actomyosin interactions and actin polymerization and regulated by intracellular proteins such as alpha-smooth muscle actin (α-SMA) and soluble factors such as transforming growth factor-β (TGF-β). Once transmitted to the extracellular matrix (ECM) through stress fibers via focal adhesions, which are assemblies of ECM proteins, transmembrane receptors, and cytoplasmic structural and signaling proteins (e.g., integrins), CTFs direct many cellular functions, including cell migration, ECM organization, and mechanical signal generation. Various methods have been developed over the years to measure CTFs of both populations of cells and of single cells. At present, cell traction force microscopy (CTFM) is among the most efficient and reliable method for determining CTF field of an entire cell spreading on a two-dimensional (2D) substrate surface. There are currently three CTFM methods, each of which is unique in both how displacement field is extracted from images and how CTFs are subsequently estimated. A detailed review and comparison of these methods are presented. Future research should improve CTFM methods such that they can automatically track dynamic CTFs, thereby providing new insights into cell motility in response to altered biological conditions. In addition, research effort should be devoted to developing novel experimental and theoretical methods for determining CTFs in three-dimensional (3D) matrix, which better reflects physiological conditions than 2D substrate used in current CTFM methods.     

Crossing barriers: The new dimension of 2D cell migration assays

In our body cells move in three dimensions, embedded in an extracellular matrix that varies in composition, density and stiffness, and this movement is fundamental to life. Next to 3D cell migration assays, representing these physiological circumstances, still we need 2D migrations assays to perform detailed studies on the contribution of matrix‐components and (extra)cellular proteins to cell movements. Next to the debate on differences between 3D and 2D migration, there also are many new perspectives on the use and development of novel or modified 2D cell migration assays. Of special significance is the introduction of so‐called barrier migration assays, methods that avoid cell and matrix damage, as complementation or replacement of scratch/wound healing assays. Here, we discuss the possibilities and limitations of different 2D barrier migration assays. J. Cell. Physiol. 226: 288–290, 2010. © 2010 Wiley‐Liss, Inc.     

3D culture model and computer-assisted videomicroscopy to analyze migratory behavior of noninvasive and invasive bronchial epithelial cells

To date, most of the studies in the field of cell migration have been applied to two-dimensional (2D) models. To mimic the three-dimensional (3D) conditions similar to those observed in vivo during tumor invasion, we developed a 3D model of cell migration in which cells were embedded in a collagen I matrix placed in a double-compartment chamber. Using time-lapse videomicroscopy and interactive cell tracking in a four-dimensional data set, we determined the cell trajectories and their migration kinetics. We compared the 2D and 3D migratory behavior of a noninvasive cell line (16HBE) with the migratory behavior of an invasive cell line (BZR). Our results show that the 3D migration kinetics of the noninvasive cell line were lower than the migration kinetics of the invasive cell line. In contrast, in 2D models, no significant difference was observed between the two cell lines. To validate our 3D model, we further investigated the effect of epidermal growth factor (EGF), a promoter of tumor cell motility and invasion on the noninvasive cell line (16HBE). EGF increased significantly the migration kinetics of the noninvasive cell line. Our results show that the 3D model of cell migration allowed us to differentiate the migratory behavior of invasive and noninvasive cells and that such a model can help in the development of molecular targeted therapy as it approaches the in vivo conditions. tumor invasion; metastasis; image analysis; kinetic migration; epidermal growth factor