We describe a technique for the quantitative measurement of cell-generated forces in highly nonlinear three-dimensional biopolymer networks that mimic the physiological situation of living cells. We computed forces of MDA-MB-231 breast carcinoma cells from the measured network deformations around the cells using a finite-element approach based on a constitutive equation that captures the complex mechanical properties of diverse biopolymers such as collagen gels, fibrin gels and Matrigel. Our measurements show that breast carcinoma cells cultured in collagen gels generated nearly constant forces regardless of the collagen concentration and matrix stiffness. Furthermore, time-lapse force measurements showed that these cells migrated in a gliding motion with alternating phases of high and low contractility, elongation, migratory speed and persistence.

Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress-strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.collagen networks | nonlinear elasticity | normal stress | tissue mechanics C ollagen type I is the most abundant protein in mammals where it serves as the primary component of many load-bearing tissues, including skin, ligaments, tendons, and bone. Networks of collagen type I fibers exhibit mechanical properties that are unmatched by manmade materials. A hallmark of collagen and collagenous tissues is a dramatic increase in stiffness when strained. Qualitatively, this property of strain stiffening is shared by many other biopolymers, including intracellular cytoskeletal networks of actin and intermediate filaments (1-5). On closer inspection, however, collagen stands out from the rest: it has been shown that collagenous tissues exhibit a regime in which the stress is approximately exponential in the applied strain (6). The origins of this nonlinearity are still not known (7,8), and existing models for biopolymer networks cannot account quantitatively for collagen. In particular, it is unknown whether the nonlinear mechanical response of collagen originates at the level of the individual fibers (1, 3, 9, 10) or arises from nonaffine network deformations as suggested by numerical simulations (11-17).Here, we present both experimental results on reconstituted collagen networks, as well as a model that quantitatively captures the observed nonlinear mechanics. Our model is a minimal one, of random networks of elastic fibers possessing only bending and stretching elasticity. This model can account for our striking experimental observation that the stiffness of collagen becomes independent of protein concentration in the nonlinear elastic regime, over a range of concentrations and applied shear stress. Our model highlights the importance of local network geometry in determining the strain threshold for the onset of nonlinear mechanics, which can account for the concentration independence of this threshold that is observed for collagen (8,17), in strong contrast...

Cell invasion through a dense three-dimensional (3D) matrix is believed to depend on the ability of cells to generate traction forces. To quantify the role of cell tractions during invasion in 3D, we present a technique to measure the elastic strain energy stored in the matrix due to traction-induced deformations. The matrix deformations around a cell were measured by tracking the 3D positions of fluorescent beads tightly embedded in the matrix. The bead positions served as nodes for a finite element tessellation. From the strain in each element and the known matrix elasticity, we computed the local strain energy in the matrix surrounding the cell. We applied the technique to measure the strain energy of highly invasive MDA-MB-231 breast carcinoma and A-125 lung carcinoma cells in collagen gels. The results were compared to the strain energy generated by non-invasive MCF-7 breast and A-549 lung carcinoma cells. In all cases, cells locally contracted the matrix. Invasive breast and lung carcinoma cells showed a significantly higher contractility compared to non-invasive cells. Higher contractility, however, was not universally associated with higher invasiveness. For instance, non-invasive A-431 vulva carcinoma cells were the most contractile cells among all cell lines tested. As a universal feature, however, we found that invasive cells assumed an elongated spindle-like morphology as opposed to a more spherical shape of non-invasive cells. Accordingly, the distribution of strain energy density around invasive cells followed patterns of increased complexity and anisotropy. These results suggest that not so much the magnitude of traction generation but their directionality is important for cancer cell invasion.

We show that the nonlinear mechanical response of networks formed from un-cross-linked fibrin or collagen type I continually changes in response to repeated large-strain loading. We demonstrate that this dynamic evolution of the mechanical response arises from a shift of a characteristic nonlinear stress-strain relationship to higher strains. Therefore, the imposed loading does not weaken the underlying matrices but instead delays the occurrence of the strain stiffening. Using confocal microscopy, we present direct evidence that this behavior results from persistent lengthening of individual fibers caused by an interplay between fiber stretching and fiber buckling when the networks are repeatedly strained. Moreover, we show that covalent cross-linking of fibrin or collagen inhibits the shift of the nonlinear material response, suggesting that the molecular origin of individual fiber lengthening may be slip of monomers within the fibers. Thus, a fibrous architecture in combination with constituents that exhibit internal plasticity creates a material whose mechanical response adapts to external loading conditions. This design principle may be useful to engineer novel materials with this capability.ECM | nonlinear rheology | factor XIII | blood clot N etworks of stiff biopolymer fibers are a major component of the structural architecture of multicellular organisms; their unique material properties provide rigidity and protect structural integrity. These networks are particularly important in the extracellular matrix (ECM) where they provide mechanical support to living cells and form many of the load-carrying structures in the body. One important example is fibrin, which forms the underlying scaffold of blood clots and the provisional matrix (1). Another important example is collagen type I, the major structural constituent of all connective tissue, tendons, ligaments, and bone (2). Because the in vivo structure of these fiber networks is so complex, investigations of in vitro networks of both proteins have been used to explore their structure and unique mechanical properties, and to elucidate their underlying design principles. Interestingly, fibrin and collagen exhibit many similar features: Both proteins self-assemble into thick, hierarchically ordered, rather stiff fibers through electrostatic and hydrophobic interactions (3, 4); these fibers associate into sparse, 3D networks that possess unusual mechanical properties not seen in synthetic polymers. In both cases, these networks display highly nonlinear mechanics and stiffen significantly as the strain increases (5-8). In addition, they are both also viscoelastic: They partially store elastic energy and partially relax internal stress through dissipative processes (9-12). All of these properties are delicately influenced by the structure of the networks, by the molecular interactions between monomers, and by the addition of covalent cross-links (7,13,14). This creates a delicate interplay between the viscoelastic and nonlinear mechanical properties of these ne...