A mechanical model for the formation of vascular networks in vitro

Endothelial cells, when cultured on gelled basement membrane matrix exert forces of tension through which they deform the matrix and at the same time they aggregate into clusters. The cells eventually form a network of cord-like structures connecting cell aggregates. In this network, almost all of the matrix has been pulled underneath the cell cords and cell clusters. This phenomenon has been proposed as a possible model for the growth and development of planar vascular systems in vitro. Our hypothesis is that the matrix is reorganized and the cellular networks form as a result of traction forces exerted by the cells on the matrix and the latter's elasticity. We construct and analyze a mathematical model based on this hypothesis and examine conditions necessary for the formation of the pattern. We show cell migration is not necessary for pattern formation and that isotropic, strain-stimulated traction is sufficient to form the observed patterns.     

Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields

In vitro angiogenesis assays have shown that tubulogenesis of endothelial cells within biogels, like collagen or fibrin gels, only appears for a critical range of experimental parameter values. These experiments have enabled us to develop and validate a theoretical model in which mechanical interactions of endothelial cells with extracellular matrix influence both active cell migration--haptotaxis--and cellular traction forces. Depending on the number of cells, cell motility and biogel rheological properties, various 2D endothelial patterns can be generated, from non-connected stripe patterns to fully connected networks, which mimic the spatial organization of capillary structures. The model quantitatively and qualitatively reproduces the range of critical values of cell densities and fibrin concentrations for which these cell networks are experimentally observed. We illustrate how cell motility is associated to the self-enhancement of the local traction fields exerted within the biogel in order to produce a pre-patterning of this matrix and subsequent formation of tubular structures, above critical thresholds corresponding to bifurcation points of the mathematical model. The dynamics of this morphogenetic process is discussed in the light of videomicroscopy time lapse sequences of endothelial cells (EAhy926 line) in fibrin gels. Our modeling approach also explains how the progressive appearance and morphology of the cellular networks are modified by gradients of extracellular matrix thickness.     

Cell traction models for generating pattern and form in morphogenesis

During early development migratory mesenchymal cells navigate to distant sites where they aggregate to form a variety of embryonic organ rudiments. We present here a new model for mesenchymal cell morphogenesis based on the mechanical interaction between motile cells and their extracellular environment. The model is based on two properties of motile cells: (a) they are capable of generating large traction forces which can deform the extracellular matrix through which they move, and (b) the deformations they produce in their environment affect the direction of their movements. We derive field equations which describe the motion of cells in an elastic extracellular matrix and show that these equations can generate a variety of spatial patterns, such as the formations of skin organ primordia, especially feather germs, cartilage condensation patterns which presage bone formation in limb development, and melanocyte density patterns which form animal coat patterns.Support for this work was provided by NSF Grant # MCS-8110557 [GFO]     

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.     

Mapping three-dimensional stress and strain fields within a soft hydrogel using a fluorescence microscope

Three-dimensional cell culture is becoming mainstream as it is recognized that many animal cell types require the biophysical and biochemical cues within the extracellular matrices to perform truly physiologically realistic functions. However, tools for characterizing cellular mechanical environment are largely limited to cell culture plated on a two-dimensional substrate. We present a three-dimensional traction microscopy that is capable of mapping three-dimensional stress and strain within a soft and transparent extracellular matrix using a fluorescence microscope and a simple forward data analysis algorithm. We validated this technique by mapping the strain and stress field within the bulk of a thin polyacrylamide gel layer indented by a millimeter-size glass ball, together with a finite-element analysis. The experimentally measured stress and strain fields are in excellent agreements with results of the finite-element simulation. The unique contributions of the presented three-dimensional traction microscopy technique are: 1), the use of a fluorescence microscope in contrast with the confocal microscope that is required for the current three-dimensional traction microscopes in the literature; 2), the determination of the pressure field of an incompressible gel from strains; and 3), the simple forward-data-analysis algorithm. Future application of this technique for mapping animal cell traction in three-dimensional nonlinear biological gels is discussed.     

Biochemical and mechanical extracellular matrix properties dictate mammary epithelial cell motility and assembly

Biochemical and mechanical cues of the extracellular matrix have been shown to play important roles in cell-matrix and cell-cell interactions. We have experimentally tested the combined influence of these cues to better understand cell motility, force generation, cell-cell interaction, and assembly in an in vitro breast cancer model. MCF-10A non-tumorigenic mammary epithelial cells were observed on surfaces with varying fibronectin ligand concentration and polyacrylamide gel rigidity. Our data show that cell velocity is biphasic in both matrix rigidity and adhesiveness. The maximum cell migration velocity occurs only at specific combination of substrate stiffness and ligand density. We found cell-cell interactions reduce migration velocity. However, the traction forces cells exert onto the substrate increase linearly with both cues, with cells in pairs exerting higher maximum tractions observed over single cells. A relationship between force and motility shows a maximum in single cell velocity not observed in cell pairs. Cell-cell adhesion becomes strongly favored on softer gels with elasticity ≤ 1250 Pascals (Pa), implying the existence of a compliance threshold that promotes cell-cell over cell-matrix adhesion. Finally on gels with stiffness similar to pre-malignant breast tissue, 400 Pa, cells undergo multicellular assembly and division into 3D spherical aggregates on a 2D surface.     

Unified multiscale theory of cellular mechanical adaptations to substrate stiffness

Rigidity of the extracellular matrix markedly regulates many cellular processes. However, how cells detect and respond to matrix rigidity remains incompletely understood. Here, we propose a unified two-dimensional multiscale framework accounting for the chemomechanical feedback to explore the interrelated cellular mechanosensing, polarization, and migration, which constitute the dynamic cascade in cellular response to matrix stiffness but are often modeled separately in previous theories. By combining integrin dynamics and intracellular force transduction, we show that substrate stiffness can act as a switch to activate or deactivate cell polarization. Our theory quantitatively reproduces rich stiffness-dependent cellular dynamics, including spreading, polarity selection, migration pattern, durotaxis, and even negative durotaxis, reported in a wide spectrum of cell types, and reconciles some inconsistent experimental observations. We find that a specific bipolarized mode can determine the optimal substrate stiffness, which enables the fastest cell migration rather than the largest traction forces that cells apply on the substrate. We identify that such a mechanical adaptation stems from the force balance across the whole cell. These findings could yield universal insights into various stiffness-mediated cellular processes within the context of tissue morphogenesis, wound healing, and cancer invasion.     

Modelling Biological Gel Contraction by Cells: Mechanocellular Formulation and Cell Traction Force Quantification

Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This modelis then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cell death and lumen formation in spheroids of MCF‐7 cells

3D (three‐dimensional) cell culture permits a more integrated analysis of the relationship between cells, inserting them into a structure more closely resembling the cellular microenvironment in vivo. The development of in vitro parameters to approximate in vivo 3D cellular environments makes a less reductionist interpretation of cell biology possible. For breast cells, in vitro 3D culture has proven to be an important tool for the analysis of luminal morphogenesis. A greater understanding of this process is necessary because alterations in the lumen arrangement are associated with carcinogenesis. Following lumen formation in 3D cell culture using laser scanning confocal microscopy, we observed alterations in the arrangement of cytoskeletal components (F‐actin and microtubules) and increasing levels of cell death associated with lumen formation. The formation of a polarized monolayer facing the lumen was characterized through 3D reconstructions and the use of TEM (transmission electron microscopy), and this process was found to occur through the gradual clearing of cells from the medullary region of the spheroids. This process was associated with different types of cell death, such as apoptosis, autophagy and entosis. The present study showed that changes in the extracellular matrix associated with long periods of time in 3D cell culture lead to the formation of a lumen in MCF‐7 cell spheroids and that features of differentiation such as lumen and budding formation occur after long periods in 3D culture, even in the absence of exogenous extracellular compounds.     

Eigenstrain as a mechanical set-point of cells

Cell contraction regulates how cells sense their mechanical environment. We sought to identify the set-point of cell contraction, also referred to as tensional homeostasis. In this work, bovine aortic endothelial cells (BAECs), cultured on substrates with different stiffness, were characterized using traction force microscopy (TFM). Numerical models were developed to provide insights into the mechanics of cell–substrate interactions. Cell contraction was modeled as eigenstrain which could induce isometric cell contraction without external forces. The predicted traction stresses matched well with TFM measurements. Furthermore, our numerical model provided cell stress and displacement maps for inspecting the fundamental regulating mechanism of cell mechanosensing. We showed that cell spread area, traction force on a substrate, as well as the average stress of a cell were increased in response to a stiffer substrate. However, the cell average strain, which is cell type-specific, was kept at the same level regardless of the substrate stiffness. This indicated that the cell average strain is the tensional homeostasis that each type of cell tries to maintain. Furthermore, cell contraction in terms of eigenstrain was found to be the same for both BAECs and fibroblast cells in different mechanical environments. This implied a potential mechanical set-point across different cell types. Our results suggest that additional measurements of contractility might be useful for monitoring cell mechanosensing as well as dynamic remodeling of the extracellular matrix (ECM). This work could help to advance the understanding of the cell-ECM relationship, leading to better regenerative strategies.     

Cell Traction Forces Direct Fibronectin Matrix Assembly

Interactions between cells and the surrounding matrix are critical to the development and engineering of tissues. We have investigated the role of cell-derived traction forces in the assembly of extracellular matrix using what we believe is a novel assay that allows for simultaneous measurement of traction forces and fibronectin fibril growth at discrete cell-matrix attachment sites. NIH3T3 cells were plated onto arrays of deformable cantilever posts for 2-24 h. Data indicate that developing fibril orientation is guided by the direction of the traction force applied to that fibril. In addition, cells initially establish a spatial distribution of traction forces that is largest at the cell edge and decreases toward the cell center. This distribution progressively shifts from a predominantly peripheral pattern to a more uniform pattern as compressive strain at the cell perimeter decreases with time. The impact of these changes on fibrillogenesis was tested by treating cells with blebbistatin or calyculin A to tonically block or augment, respectively, myosin II activity. Both treatments blocked the inward translation of traction forces, the dissipation of compressive strain, and fibronectin fibrillogenesis over time. These data indicate that dynamic spatial and temporal changes in traction force and local strain may contribute to successful matrix assembly.     

The effects of coating culture dishes with collagen on fibroblast cell shape and swirling pattern formation

Motile human-skin fibroblasts form macroscopic swirling patterns when grown to confluence on a culture dish. In this paper, we investigate the effect of coating the culture-dish surface with collagen on the resulting pattern, using human-skin fibroblast NB1RGB cells as the model system. The presence of the collagen coating is expected to enhance the adherence of the fibroblasts to the dish surface, and thereby also enhance the traction that the fibroblasts have as they move. We find that, contrary to our initial expectation, the coating does not significantly affect the motility of the fibroblasts. Their eventual number density at confluence is also unaffected. However, the coherence length of cell orientation in the swirling pattern is diminished. We also find that the fibroblasts cultured in collagen-coated dishes are rounder in shape and shorter in perimeter, compared with those cultured in uncoated polystyrene or glass culture dishes. We hypothesise that the rounder cell-shape which weakens the cell–cell nematic contact interaction is responsible for the change in coherence length. A simple mathematical model of the migrating fibroblasts is constructed, which demonstrates that constant motility with weaker nematic interaction strength does indeed lead to the shortening of the coherence length.Electronic supplementary materialThe online version of this article (10.1007/s10867-020-09556-3) contains supplementary material, which is available to authorized users.     

Cell-cell mechanical communication through compliant substrates

The role of matrix mechanics on cell behavior is under intense investigation. Cells exert contractile forces on their matrix and the matrix elasticity can alter these forces and cell migratory behavior. However, little is known about the contribution of matrix mechanics and cell-generated forces to stable cell-cell contact and tissue formation. Using matrices of varying stiffness and measurements of endothelial cell migration and traction stresses, we find that cells can detect and respond to substrate strains created by the traction stresses of a neighboring cell, and that this response is dependent on matrix stiffness. Specifically, pairs of endothelial cells display hindered migration on gels with elasticity below 5500 Pa in comparison to individual cells, suggesting these cells sense each other through the matrix. We believe that these results show for the first time that matrix mechanics can foster tissue formation by altering the relative motion between cells, promoting the formation of cell-cell contacts. Moreover, our data indicate that cells have the ability to communicate mechanically through their matrix. These findings are critical for the understanding of cell-cell adhesion during tissue formation and disease progression, and for the design of biomaterials intended to support both cell-matrix and cell-cell adhesion.     

A predictive model of cell traction forces based on cell geometry

Recent work has indicated that the shape and size of a cell can influence how a cell spreads, develops focal adhesions, and exerts forces on the substrate. However, it is unclear how cell shape regulates these events. Here we present a computational model that uses cell shape to predict the magnitude and direction of forces generated by cells. The predicted results are compared to experimentally measured traction forces, and show that the model can predict traction force direction, relative magnitude, and force distribution within the cell using only cell shape as an input. Analysis of the model shows that the magnitude and direction of the traction force at a given point is proportional to the first moment of area about that point in the cell, suggesting that contractile forces within the cell act on the entire cytoskeletal network as a single cohesive unit. Through this model, we demonstrate that intrinsic properties of cell shape can facilitate changes in traction force patterns, independently of heterogeneous mechanical properties or signaling events within the cell.     

Investigating metalloproteinases MMP-2 and MMP-9 mechanosensitivity to feedback loops involved in the regulation of in vitro angiogenesis by endogenous mechanical stresses

Angiogenesis is a complex morphogenetic process regulated by growth factors, but also by the force balance between endothelial cells (EC) traction stresses and extracellular matrix (ECM) viscoelastic resistance. Studies conducted with in vitro angiogenesis assays demonstrated that decreasing ECM stiffness triggers an angiogenic switch that promotes organization of EC into tubular cords or pseudo-capillaries. Thus, mechano-sensitivity of EC with regard to proteases secretion, and notably matrix metalloproteinases (MMPs), should likely play a pivotal role in this switching mechanism. While most studies analysing strain regulation of MMPs used cell cultured on stretched membranes, this work focuses on MMP expression during self-assembly of EC into capillary-like structures within fibrin gels, i.e. on conditions that mimics more closely the in vivo cellular mechanical microenvironment. The activity of MMP-2 and MMP-9, two MMPs that have a pivotal role in capillaries formation, has been monitored in pace with the progressive elongation of EAhy926 cells that takes place during the emergence of cellular cords. We found an increase of the zymogen proMMP-2 that correlates with the initial stages of EC cords formation. However, MMP-2 was not detected. ProMMP-9 secretion decreased, with levels of MMP-9 kept at a rather low value. In order to analyse more precisely the observed differences of EAhy926 response on fibrin and plastic substrates, we proposed a theoretical model of the mechano-regulation of proMMP-2 activation in the presence of type 2 tissue inhibitor of MMPs (TIMP-2). Using association/dissociation rates experimentally reported for this enzymatic network, the model adequately describes the synergism of proMMP-2 and TIMP-2 strain activation during pseudo-capillary morphogenesis. All together, these results provide a first step toward a systems biology approach of angiogenesis mechano-regulation by cell-generated extracellular stresses and strains.     

Further characterization of human fetal osteoblastic hFOB 1.19 and hFOB/ER alpha cells: bone formation in vivo and karyotype analysis using multicolor fluorescent in situ hybridization

We have previously generated an immortalized human fetal osteoblastic cell line (hFOB) using stably transfected temperature sensitive SV40 T-antigen (Harris et al. [1995a] J. Bone. Miner. Res. 10:178-1860). To characterize these cells for phenotypic/genotypic attributes desired for a good cell model system, we performed karyotype analysis by multicolor fluorescent in situ hybridization (M-FISH), their ability to form bone in vivo without developing cell transformation, and finally their ability to form extracellular matrix formation in vitro. The karyotype analysis of hFOB cells revealed structural or numeric anomalies involving 1-2 chromosomes. In contrast, the human osteosarcoma MG63 cells displayed multiple, and often complex, numeric, and structural abnormalities. Subcutaneous injection of hFOB cells in the presence of Matrigel into nude mice resulted in bone formation after 2-3 weeks. Electron microscopic analysis of the extracellular matrix deposited by hFOB cells in culture revealed a parallel array of lightly banded fibrils typical of the fibrillar collagens such as type I and III. These results demonstrate that the hFOB cell line has minimal chromosome abnormalities, exhibit the matrix synthetic properties of differentiated osteoblasts, and are immortalized but non-transformed cell line. These hFOB cells thus appear to be an excellent model system for the study of osteoblast biology in vitro.     

Emergence of complex patterns induced by dynamic systems implemented in dynamic structures (DS)2

We present the evolution of the simple system of Meinhardt implemented in both static or dynamic two-dimensional structures of almost-squared cells. In a static structure of 8 x 4=32 to 128 x 128=16384 cells, the pattern observed is periodic. An algorithm allows us to divide the cells following the greater size, and to define a dynamic structure. The implementation of the same Meinhardt system in this dynamic structure varying from 32 to 16 384 cells and a context of the same genotypic complexity for the model provides aperiodic patterns, with a higher phenotypic complexity than those observed in static structures, while the number of computations is comparable. We define that emergence occurs each time the ratio of phenotypic/genotypic complexities increases.     

Toward single cell traction microscopy within 3D collagen matrices

Mechanical interaction between the cell and its extracellular matrix (ECM) regulates cellular behaviors, including proliferation, differentiation, adhesion, and migration. Cells require the three-dimensional (3D) architectural support of the ECM to perform physiologically realistic functions. However, current understanding of cell–ECM and cell–cell mechanical interactions is largely derived from 2D cell traction force microscopy, in which cells are cultured on a flat substrate. 3D cell traction microscopy is emerging for mapping traction fields of single animal cells embedded in either synthetic or natively derived fibrous gels. We discuss here the development of 3D cell traction microscopy, its current limitations, and perspectives on the future of this technology. Emphasis is placed on strategies for applying 3D cell traction microscopy to individual tumor cell migration within collagen gels.     

A Novel Method for Localizing Reporter Fluorescent Beads Near the Cell Culture Surface for Traction Force Microscopy

PA gels have long been used as a platform to study cell traction forces due to ease of fabrication and the ability to tune their elastic properties. When the substrate is coated with an extracellular matrix protein, cells adhere to the gel and apply forces, causing the gel to deform. The deformation depends on the cell traction and the elastic properties of the gel. If the deformation field of the surface is known, surface traction can be calculated using elasticity theory. Gel deformation is commonly measured by embedding fluorescent marker beads uniformly into the gel. The probes displace as the gel deforms. The probes near the surface of the gel are tracked. The displacements reported by these probes are considered as surface displacements. Their depths from the surface are ignored. This assumption introduces error in traction force evaluations. For precise measurement of cell forces, it is critical for the location of the beads to be known. We have developed a technique that utilizes simple chemistry to confine fluorescent marker beads, 0.1 and 1 µm in diameter, in PA gels, within 1.6 μm of the surface. We coat a coverslip with poly-D-lysine (PDL) and fluorescent beads. PA gel solution is then sandwiched between the coverslip and an adherent surface. The fluorescent beads transfer to the gel solution during curing. After polymerization, the PA gel contains fluorescent beads on a plane close to the gel surface.