Cells exploit a phase transition to mechanically remodel the fibrous extracellular matrix

Unexpected softening of a fibrous matrix by contracting inclusions

Cells respond to the stiffness of their surrounding environment, but quantifying the stiffness of a fibrous matrix at the scale of a cell is complicated, due to the effects of nonlinearity and complex force transmission pathways resulting from randomness in fiber density and connections. While it is known that forces produced by individual contractile cells can stiffen the matrix, it remains unclear how simultaneous contraction of multiple cells in a fibrous matrix alters the stiffness at the scale of a cell. Here, we used computational modeling and experiments to quantify the stiffness of a random fibrous matrix embedded with multiple contracting inclusions, which mimicked the contractile forces of a cell. The results showed that when the matrix was free to contract as a result of the forces produced by the inclusions, the matrix softened rather than stiffened, which was surprising given that the contracting inclusions applied tensile forces to the matrix. Using the computational model, we identified that the underlying cause of the softening was that the majority of the fibers were under a local state of axial compression, causing buckling. We verified that this buckling-induced matrix softening was sufficient for cells to sense and respond by altering their morphology and force generation. Our findings reveal that the localized forces induced by cells do not always stiffen the matrix; rather, softening can occur in instances wherein the matrix can contract in response to the cell-generated forces. This study opens up new possibilities to investigate whether cell-induced softening contributes to maintenance of homeostatic conditions or progression of disease. STATEMENT OF SIGNIFICANCE: Mechanical interactions between cells and the surrounding matrix strongly influence cellular functions. Cell-induced forces can alter matrix properties, and much prior literature in this area focused on the influence of individual contracting cells. Cells in tissues are rarely solitary; rather, they are interspersed with neighboring cells throughout the matrix. As a result, the mechanics are complicated, leaving it unclear how the multiple contracting cells affect matrix stiffness. Here, we show that multiple contracting inclusions within a fibrous matrix can cause softening that in turn affects cell sensing and response. Our findings provide new directions to determine impacts of cell-induced softening on maintenance of tissue or progression of disease.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

Micropatterning the organization of multicellular structures in 3D biological hydrogels; insights into collective cellular mechanical interactions

Control over the organization of cells at the microscale level within supporting biomaterials can push forward the construction of complex tissue architectures for tissue engineering applications and enable fundamental studies of how tissue structure relates to its function. While cells patterning on 2D substrates is a relatively established and available procedure, micropatterning cells in biomimetic 3D hydrogels has been more challenging, especially with micro-scale resolution, and currently relies on sophisticated tools and protocols. We present a robust and accessible 'peel-off' method to micropattern large arrays of individual cells or cell-clusters of precise sizes in biological 3D hydrogels, such as fibrin and collagen gels, with control over cell-cell separation distance and neighboring cells position. We further demonstrate partial control over cell position in thez-dimension by stacking two layers in varying distances between the layers. To demonstrate the potential of the micropatterning gel platform, we study the matrix-mediated mechanical interaction between array of cells that are accurately separated in defined distances. A collective process of intense cell-generated densified bands emerging in the gel between near neighbors was identified, along which cells preferentially migrate, a process relevant to tissue morphogenesis. The presented 3D gel micropatterning method can be used to reveal fundamental morphogenetic processes, and to reconstruct any tissue geometry with micrometer resolution in 3D biomimetic gel environments, leveraging the engineering of tissues in complex architectures.

Modeling collagen fibril self-assembly from extracellular medium in embryonic tendon

Collagen is a key structural component of multicellular organisms and is arranged in a highly organized manner. In structural tissues such as tendons, collagen forms bundles of parallel fibers between cells, which appear within a 24-h window between embryonic day 13.5 (E13.5) and E14.5 during mouse embryonic development. Current models assume that the organized structure of collagen requires direct cellular control, whereby cells actively lay down collagen fibrils from cell surfaces. However, such models appear incompatible with the time and length scales of fibril formation. We propose a phase-transition model to account for the rapid development of ordered fibrils in embryonic tendon, reducing reliance on active cellular processes. We develop phase-field crystal simulations of collagen fibrillogenesis in domains derived from electron micrographs of inter-cellular spaces in embryonic tendon and compare results qualitatively and quantitatively to observed patterns of fibril formation. To test the prediction of this phase-transition model that free protomeric collagen should exist in the inter-cellular spaces before the formation of observable fibrils, we use laser-capture microdissection, coupled with mass spectrometry, which demonstrates steadily increasing free collagen in inter-cellular spaces up to E13.5, followed by a rapid reduction of free collagen that coincides with the appearance of less-soluble collagen fibrils. The model and measurements together provide evidence for extracellular self-assembly of collagen fibrils in embryonic mouse tendon, supporting an additional mechanism for rapid collagen fibril formation during embryonic development.

Statistical field theory for nonlinear elasticity of polymer networks with excluded volume interactions

Polymer networks formed by cross linking flexible polymer chains are ubiquitous in many natural and synthetic soft-matter systems. Current micromechanics models generally do not account for excluded volume interactions except, for instance, through imposing a phenomenological incompressibility constraint at the continuum scale. This work aims to examine the role of excluded volume interactions on the mechanical response. The approach is based on the framework of the self-consistent statistical field theory of polymers, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the end-to-end vectors of the chains by this deformation. In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in three dimensions using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (i) the linearized Poisson's ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (ii) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real polymer networks, driven by the interplay between the entropy and the excluded volume interactions; and (iii) the emergence of a deformation-sensitive localization instability at large excluded volumes.

Does the Extracellular Matrix Support Cell-Cell Communication by Elastic Wave Packets?

The extracellular matrix (ECM) is a fibrous network supporting biological cells and provides them a medium for interaction. Cells modify the ECM by applying traction forces, and these forces can propagate to long ranges and establish a mechanism of mechanical communication between neighboring cells. Previous studies have mainly focused on analysis of static force transmission across the ECM. In this study, we explore the plausibility of dynamic mechanical interaction, expressed as vibrations or abrupt fluctuations, giving rise to elastic waves propagating along ECM fibers. We use a numerical mass-spring model to simulate the longitudinal and transversal waves propagating along a single ECM fiber and across a 2D random fiber network. The elastic waves are induced by an active contracting cell (signaler) and received by a passive neighboring cell (receiver). We show that dynamic wave propagation may amplify the signal at the receiver end and support up to an order of magnitude stronger mechanical cues and longer-ranged communication relative to static transmission. Also, we report an optimal impulse duration corresponding to the most effective transmission, as well as extreme fast impulses, in which the waves are encaged around the active cell and do not reach the neighboring cell, possibly due to the Anderson localization effect. Finally, we also demonstrate that extracellular fluid viscosity reduces, but still allows, dynamic propagation along embedded ECM fibers. Our results motivate future biological experiments in mechanobiology to investigate, on the one hand, the mechanosensitivity of cells to dynamic forces traveling and guided by the ECM and, on the other hand, the impact of ECM architecture and remodeling on dynamic force transmission and its spectral filtering, dispersion, and decay.

Disorder to order transition in cell-ECM systems mediated by cell-cell collective interactions

Cells in functional tissues execute various collective activities to achieve diverse ordered processes including wound healing, organogenesis, and tumor formation. How a group of individually operating cells initiate such complex collective processes is still not clear. Here, we report that cells in 3D extracellular matrix (ECM) initiate collective behavior by forming cell-ECM network when the cells are within a critical distance from each other. We employed compaction of free-floating (FF) 3D collagen gels with embedded fibroblasts as a model system to study collective behavior and found a sharp transition in the amount of compaction as a function of cell-cell distance, reminiscent of phase transition in materials. Within the critical distance, cells remodel the ECM irreversibly, and form dense collagen bridges between each other resulting in the formation of a network. Beyond the critical distance, cells exhibit Brownian dynamics and only deform the matrix reversibly in a transient manner with no memory of history, thus maintaining the disorder. Network formation seems to be a necessary and sufficient condition to trigger collective behavior and a disorder-to order transition. STATEMENT OF SIGNIFICANCE: Macroscopic compaction of in vitro collagen gels is mediated by collective mechanical interaction of cells. Previous studies on cell-induced ECM compaction suggest the existence of a critical cell density and phase transition associated with this phenomenon. Cell-mediated mechanical remodeling and global compaction of ECM has mostly been studied at steady state. Our study reveals a link between a transition in cell dynamics and material microstructure as cells collectively compact collagen gels. It underscores the significance of temporal evolution of these cell-ECM systems in understanding the mechanism of such collective action and provides insights on the process from a mechanistic viewpoint. These insights can be valuable in understanding dynamic pathological processes such as, cancer progression and wound healing, as well as engineering biomaterials and regenerative tissue mimics.

Effect of hyaluronic acid on microscale deformations of collagen gels

As fibrous collagen is the most abundant protein in mammalian tissues, gels of collagen fibers have been extensively used as an extracellular matrix scaffold to study how cells sense and respond to cues from their microenvironment. Other components of native tissues, such as glycosaminoglycans like hyaluronic acid, can affect cell behavior in part by changing the mechanical properties of the collagen gel. Prior studies have quantified the effects of hyaluronic acid on the mechanical properties of collagen gels in experiments of uniform shear or compression at the macroscale. However, there remains a lack of experimental studies of how hyaluronic acid changes the mechanical properties of collagen gels at the scale of a cell. Here, we studied how addition of hyaluronic acid to gels of collagen fibers affects the local field of displacements in response to contractile loads applied on length scales similar to those of a contracting cell. Using spherical poly(N-isopropylacrylamide) particles, which contract when heated, we induced displacement in gels of collagen and collagen with hyaluronic acid. Displacement fields were quantified using a combination of confocal microscopy and digital image correlation. Results showed that hyaluronic acid suppressed the distance over which displacements propagated, suggesting that it caused the network to become more linear. Additionally, hyaluronic acid had no statistical effect on heterogeneity of the displacement fields, but it did make the gels more elastic by substantially reducing the magnitude of permanent deformations. Lastly, we examined the effect of hyaluronic acid on fiber remodeling due to localized forces and found that hyaluronic acid partially - but not fully - inhibited remodeling. This result is consistent with prior studies suggesting that fiber remodeling is associated with a phase transition resulting from an instability caused by nonlinearity of the collagen gel.

Collagen fibril assembly: New approaches to unanswered questions

Collagen fibrils are essential for metazoan life. They are the largest, most abundant, and most versatile protein polymers in animals, where they occur in the extracellular matrix to form the structural basis of tissues and organs. Collagen fibrils were first observed at the turn of the 20th century. During the last 40 years, the genes that encode the family of collagens have been identified, the structure of the collagen triple helix has been solved, the many enzymes involved in the post-translational modifications of collagens have been identified, mutations in the genes encoding collagen and collagen-associated proteins have been linked to heritable disorders, and changes in collagen levels have been associated with a wide range of diseases, including cancer. Yet despite extensive research, a full understanding of how cells assemble collagen fibrils remains elusive. Here, we review current models of collagen fibril self-assembly, and how cells might exert control over the self-assembly process to define the number, length and organisation of fibrils in tissues.

Effect of matrix heterogeneity on cell mechanosensing

Cells sense mechanical signals within the extracellular matrix, the most familiar being stiffness, but matrix stiffness cannot be simply described by a single value. Randomness in matrix structure causes stiffness at the scale of a cell to vary by more than an order of magnitude. Additionally, the extracellular matrix contains ducts, blood vessels, and, in cancer or fibrosis, regions with abnormally high stiffness. These different features could alter the stiffness sensed by a cell, but it is unclear whether the change in stiffness is large enough to overcome the noise caused by heterogeneity due to the random fibrous structure. Here we used a combination of experiments and modeling to determine the extent to which matrix heterogeneity disrupts the potential for cell sensing of a locally stiff feature in the matrix. Results showed that, at the scale of a single cell, spatial heterogeneity in local stiffness was larger than the increase in stiffness due to a stiff feature. The heterogeneity was reduced only for large length scales compared to the fiber length. Experiments verified this conclusion, showing spheroids of cells, which were large compared to the average fiber length, spreading preferentially toward stiff inclusions. Hence, the propagation of mechanical cues through the matrix depends on length scale, with single cells being able to sense only the stiffness of the nearby fibers and multicellular structures, such as tumors, also sensing the stiffness of distant matrix features.

Percolation of collagen stress in a random network model of the alveolar wall

Fibrotic diseases are characterized by progressive and often irreversible scarring of connective tissue in various organs, leading to substantial changes in tissue mechanics largely as a result of alterations in collagen structure. This is particularly important in the lung because its bulk modulus is so critical to the volume changes that take place during breathing. Nevertheless, it remains unclear how fibrotic abnormalities in the mechanical properties of pulmonary connective tissue can be linked to the stiffening of its individual collagen fibers. To address this question, we developed a network model of randomly oriented collagen and elastin fibers to represent pulmonary alveolar wall tissue. We show that the stress–strain behavior of this model arises via the interactions of collagen and elastin fiber networks and is critically dependent on the relative fiber stiffnesses of the individual collagen and elastin fibers themselves. We also show that the progression from linear to nonlinear stress–strain behavior of the model is associated with the percolation of stress across the collagen fiber network, but that the location of the percolation threshold is influenced by the waviness of collagen fibers.

Nonlinear statistical mechanics drives intrinsic electrostriction and volumetric torque in polymer networks

Statistical mechanics is an important tool for understanding polymer electroelasticity because the elasticity of polymers is primarily due to entropy. However, a common approach for the statistical mechanics of polymer chains, the Gaussian chain approximation, misses key physics. By considering the nonlinearities of the problem, we show a strong coupling between the deformation of a polymer chain and its dielectric response, that is, its net dipole. When chains with this coupling are cross linked in an elastomer network and an electric field is applied, the field breaks the symmetry of the elastomer's elastic properties and, combined with electrostatic torque and incompressibility, leads to intrinsic electrostriction. Conversely, deformation can break the symmetry of the dielectric response, leading to volumetric torque and asymmetric actuation. Both phenomena have important implications for designing high-efficiency soft actuators and soft electroactive materials, and the presence of mechanisms for volumetric torque, in particular, can be used to develop higher degree of freedom actuators and to achieve bioinspired locomotion.