On the mechanochemical theory of biological pattern formation with application to vasculogenesis

We first describe the Murray-Oster mechanical theory of pattern formation, the biological basis of which is experimentally well documented. The model quantifies the interaction of cells and the extracellular matrix via the cell-generated forces. The model framework is described in quantitative detail. Vascular endothelial cells, when cultured on gelled basement membrane matrix, rapidly aggregate into clusters while deforming the matrix into a network of cord-like structures tessellating the planar culture. We apply the mechanical theory of pattern formation to this culture system and show that neither strain-biased anisotropic cell traction nor cell migration are necessary for pattern formation: isotropic, strain-stimulated cell traction is sufficient to form the observed patterns. Predictions from the model were confirmed experimentally.     

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]     

A Novel Cell Traction Force Microscopy to Study Multi-Cellular System

Traction forces exerted by adherent cells on their microenvironment can mediate many critical cellular functions. Accurate quantification of these forces is essential for mechanistic understanding of mechanotransduction. However, most existing methods of quantifying cellular forces are limited to single cells in isolation, whereas most physiological processes are inherently multi-cellular in nature where cell-cell and cell-microenvironment interactions determine the emergent properties of cell clusters. In the present study, a robust finite-element-method-based cell traction force microscopy technique is developed to estimate the traction forces produced by multiple isolated cells as well as cell clusters on soft substrates. The method accounts for the finite thickness of the substrate. Hence, cell cluster size can be larger than substrate thickness. The method allows computing the traction field from the substrate displacements within the cells'' and clusters'' boundaries. The displacement data outside these boundaries are not necessary. The utility of the method is demonstrated by computing the traction generated by multiple monkey kidney fibroblasts (MKF) and human colon cancerous (HCT-8) cells in close proximity, as well as by large clusters. It is found that cells act as individual contractile groups within clusters for generating traction. There may be multiple of such groups in the cluster, or the entire cluster may behave a single group. Individual cells do not form dipoles, but serve as a conduit of force (transmission lines) over long distances in the cluster. The cell-cell force can be either tensile or compressive depending on the cell-microenvironment interactions.     

How Nucleus Mechanics and ECM Microstructure Influence the Invasion of Single Cells and Multicellular Aggregates

In order to move in a three-dimensional extracellular matrix, the nucleus of a cell must squeeze through the narrow spacing among the fibers and, by adhering to them, the cell needs to exert sufficiently strong traction forces. If the nucleus is too stiff, the spacing too narrow, or traction forces too weak, the cell is not able to penetrate the network. In this article, we formulate a mathematical model based on an energetic approach, for cells entering cylindrical channels composed of extracellular matrix fibers. Treating the nucleus as an elastic body covered by an elastic membrane, the energetic balance leads to the definition of a necessary criterion for cells to pass through the regular network of fibers, depending on the traction forces exerted by the cells (or possibly passive stresses), the stretchability of the nuclear membrane, the stiffness of the nucleus, and the ratio of the pore size within the extracellular matrix with respect to the nucleus diameter. The results obtained highlight the importance of the interplay between mechanical properties of the cell and microscopic geometric characteristics of the extracellular matrix and give an estimate for a critical value of the pore size that represents the physical limit of migration and can be used in tumor growth models to predict their invasive potential in thick regions of ECM.     

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.     

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.     

ECM concentration and cell-mediated traction forces play a role in vascular network assembly in 3D bioprinted tissue

Tissue vascularization is critical to enable oxygen and nutrient supply. Therefore, establishing expedient vasculature is necessary for the survival of tissue after transplantation. The use of biomechanical forces, such as cell-induced traction forces, may be a promising method to encourage growth of the vascular network. Three-dimensional (3D) bioprinting, which offers unprecedented versatility through precise control over spatial distribution and structure of tissue constructs, can be used to generate capillary-like structures in vitro that would mimic microvessels. This study aimed to develop an in vitro, 3D bioprinted tissue model to study the effect of cellular forces on the spatial organization of vascular structures and tissue maturation. The developed in vitro model consists of a 3D bioprinted polycaprolactone (PCL) frame with a gelatin spacer hydrogel layer and a gelatin–fibrin–hyaluronic acid hydrogel layer containing normal human dermal fibroblasts and human umbilical vein endothelial cells printed as vessel lines on top. The formation of vessel-like networks and vessel lumens in the 3D bioprinted in vitro model was assessed at different fibrinogen concentrations with and without inhibitors of cell-mediated traction forces. Constructs containing 5 mg/ml fibrinogen had longer vessels compared to the other concentrations of fibrinogen used. Also, for all concentrations of fibrinogen used, most of the vessel-like structures grew parallel to the direction the PCL frame-mediated tensile forces, with very few branching structures observed. Treatment of the 3D bioprinted constructs with traction inhibitors resulted in a significant reduction in length of vessel-like networks. The 3D bioprinted constructs also had better lumen formation, increased collagen deposition, more elaborate actin networks, and well-aligned matrix fibers due to the increased cell-mediated traction forces present compared to the non-anchored, floating control constructs. This study showed that cell traction forces from the actomyosin complex are critical for vascular network assembly in 3D bioprinted tissue. Strategies involving the use of cell-mediated traction forces may be promising for the development of bioprinting approaches for fabrication of vascularized tissue constructs.     

Coupling traction force patterns and actomyosin wave dynamics reveals mechanics of cell motion

Motile cells can use and switch between different modes of migration. Here, we use traction force microscopy and fluorescent labeling of actin and myosin to quantify and correlate traction force patterns and cytoskeletal distributions in Dictyostelium discoideum cells that move and switch between keratocyte‐like fan‐shaped, oscillatory, and amoeboid modes. We find that the wave dynamics of the cytoskeletal components critically determine the traction force pattern, cell morphology, and migration mode. Furthermore, we find that fan‐shaped cells can exhibit two different propulsion mechanisms, each with a distinct traction force pattern. Finally, the traction force patterns can be recapitulated using a computational model, which uses the experimentally determined spatiotemporal distributions of actin and myosin forces and a viscous cytoskeletal network. Our results suggest that cell motion can be generated by friction between the flow of this network and the substrate.     

Measurement of mechanical tractions exerted by cells in three-dimensional matrices

Quantitative measurements of cell-generated forces have heretofore required that cells be cultured on two-dimensional substrates. We describe a technique to quantitatively measure three-dimensional traction forces exerted by cells fully encapsulated in well-defined elastic hydrogel matrices. Using this approach we measured traction forces for several cell types in various contexts and revealed patterns of force generation attributable to morphologically distinct regions of cells as they extend into the surrounding matrix.     

A novel functional assessment of the differentiation of micropatterned muscle cells

Duchenne muscular dystrophy (DMD) is a lethal disease characterized by rapid, progressive atrophy of muscle tissues. Timely screening of therapeutic interventions is necessary for the development of effective treatment approaches for DMD. We have developed an in vitro model using a combination of micropatterning of C2C12 skeletal muscle cells and cell traction force microscopy (CTFM). In this model, C2C12 cells were micropatterned on a highly elongated adhesive island such that the cells assumed a shape typical of a myotube. During differentiation, these cells gradually fused together and began expressing dystrophin, a structural protein of myotubes, meanwhile, their contractile forces, represented by cell traction forces, continually increased until the myotubes reached maturation. In addition, the high-degree alignment of cells favored myotube differentiation and dystrophin expression. Since the fundamental structural unit of muscle tissue is myofiber, which is responsible for muscle contraction, such a technology that can directly quantify the contractile forces of the myotube, a precursor of myofiber, may constitute a fast and efficient screening approach for DMD therapies.     

A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

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.     

Measuring traction forces of motile dendritic cells on micropost arrays

Dendritic cells (DCs) migrate from sites of inflammation to secondary lymphoid organs where they initiate the adaptive immune response. Although motility is essential to DC function, the mechanisms by which they migrate are not fully understood. We incorporated micropost array detectors into a microfluidic gradient generator to develop what we consider to be a novel method for probing low magnitude traction forces during directional migration. We found migration of primary murine DCs is driven by short-lived traction stresses at the leading edge or filopodia. The traction forces generated by DCs are smaller in magnitude than found in neutrophils, and of similar magnitude during chemotaxis and chemokinesis, at 18 ± 1.4 and 16 ± 1.3 nN/cell, respectively. The characteristic duration of local DC traction forces was 3 min. The maximum principal stress in the cell occurred in the plane perpendicular to the axis of motion, forward of the centroid. We illustrate that the spatiotemporal pattern of traction stresses can be used to predict the direction of future DC motion. Overall, DCs show a mode of migration distinct from both mesenchymal cells and neutrophils, characterized by rapid turnover of traction forces in leading filopodia.     

Separation of propulsive and adhesive traction stresses in locomoting keratocytes

Strong, actomyosin-dependent, pinching tractions in steadily locomoting (gliding) fish keratocytes revealed by traction imaging present a paradox, since only forces perpendicular to the direction of locomotion are apparent, leaving the actual propulsive forces unresolved. When keratocytes become transiently "stuck" by their trailing edge and adopt a fibroblast-like morphology, the tractions opposing locomotion are concentrated into the tail, leaving the active pinching and propulsive tractions clearly visible under the cell body. Stuck keratocytes can develop approximately 1 mdyn (10,000 pN) total propulsive thrust, originating in the wings of the cell. The leading lamella develops no detectable propulsive traction, even when the cell pulls on its transient tail anchorage. The separation of propulsive and adhesive tractions in the stuck phenotype leads to a mechanically consistent hypothesis that resolves the traction paradox for gliding keratocytes: the propulsive tractions driving locomotion are normally canceled by adhesive tractions resisting locomotion, leaving only the pinching tractions as a resultant. The resolution of the traction pattern into its components specifies conditions to be met for models of cytoskeletal force production, such as the dynamic network contraction model (Svitkina, T.M., A.B. Verkhovsky, K.M. McQuade, and G.G. Borisy. 1997. J. Cell Biol. 139:397-415). The traction pattern associated with cells undergoing sharp turns differs markedly from the normal pinching traction pattern, and can be accounted for by postulating an asymmetry in contractile activity of the opposed lateral wings of the cell.     

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.     

Tissue engineering science: Consequences of cell traction force

Blood and tissue cells mechanically interact with soft tissues and tissue-equivalent reconstituted collagen gels in a variety of situations relevant to biomedicine and biotechnology. A key phenomenon in these interactions is the exertion of traction force by cells on local collagen fibers which typically constitute the solid network of these tissues and gels and impart gross mechanical integrity. Two important consequences of cells exerting traction on such collagen networks are first, when the cells co-ordinate their traction, resulting in cell migration, and second, when their traction is sufficient to deform the network. Such cell-collagen network interactions are coupled in a number of ways. Network deformation, for example, can result in net alignment of collagen fibers, eliciting contact guidance, wherein cells move with bidirectional bias along an axis of fiber alignment, potentially leading to a nonuniform cell distribution. This may govern cell accumulation in wounds and be exploited to control cell infiltration of bioartificial tissues and organs. Another consequence of cell traction is the resultant stress and strain in the network which modulate cell protein and DNA synthesis and differentiation. We summarize, here, relevant mathematical theories which we have used to describe the inherent coupling of cell dynamics and tissue mechanics in cell-populated collagen gels via traction. The development of appropriate models based on these theories, in an effort to understand how events in wound healing govern the rate and extent of wound contraction, and to measure cell traction forces in vitro, are described. Relevant observations and speculation from cell biology and medicine that motivate or serve to critique the assumptions made in the theories and models are also summarized.Abbreviations ECM Extracellular Matrix - FPCL Fibroblast-Populated Collagen Lattice - FPCM Fibroblast-Populated Collagen Microsphere     

Simulations of the contractile cycle in cell migration using a bio-chemical-mechanical model

Cell migration relies on traction forces in order to propel a cell. Several computational models have been developed that help explain the trajectory that cells take during migration, but little attention has been placed on traction forces during this process. Here, we investigated the spatiotemporal dynamics of cell migration by using a bio-chemical-mechanical contractility model that incorporates the first steps of cell migration on an array of posts. In the model, formation of a new adhesion causes a reactivation of stress fibre assembly within a cell. The model was able to predict the spatial distribution of traction forces observed with previous experiments. Moreover, the model found that the strain energy exerted by the traction forces of a migrating cell underwent a cyclic relationship that rose with the formation of a new adhesion and fell with the release of an adhesion at its rear.     

Traction force microscopy of migrating normal and H-ras transformed 3T3 fibroblasts

Mechanical interactions between cell and substrate are involved in vital cellular functions from migration to signal transduction. A newly developed technique, traction force microscopy, makes it possible to visualize the dynamic characteristics of mechanical forces exerted by fibroblasts, including the magnitude, direction, and shear. In the present study such analysis is applied to migrating normal and transformed 3T3 cells. For normal cells, the lamellipodium provides almost all the forces for forward locomotion. A zone of high shear separates the lamellipodium from the cell body, suggesting that they are mechanically distinct entities. Timing and distribution of tractions at the leading edge bear no apparent relationship to local protrusive activities. However, changes in the pattern of traction forces often precede changes in the direction of migration. These observations suggest a frontal towing mechanism for cell migration, where dynamic traction forces at the leading edge actively pull the cell body forward. For H-ras transformed cells, pockets of weak, transient traction scatter among small pseudopods and appear to act against one another. The shear pattern suggests multiple disorganized mechanical domains. The weak, poorly coordinated traction forces, coupled with weak cell-substrate adhesions, are likely responsible for the abnormal motile behavior of H-ras transformed cells.     

Contractile forces in tumor cell migration

Cancer is a deadly disease primarily because of the ability of tumor cells to spread from the primary tumor, to invade into the connective tissue, and to form metastases at distant sites. In contrast to cell migration on a planar surface where large cell tractions and contractile forces are not essential, tractions and forces are thought to be crucial for overcoming the resistance and steric hindrance of a dense three-dimensional connective tissue matrix. In this review, we describe recently developed biophysical tools, including 2-D and 3-D traction microscopy to measure contractile forces of cells. We discuss evidence indicating that tumor cell invasiveness is associated with increased contractile force generation.