Heterogeneity and nonaffinity of cell-induced matrix displacements

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.

Micromechanics of fibrous scaffolds and their stiffness sensing by cells

Mechanical properties of the tissue engineering scaffolds are known to play a crucial role in cell response. Therefore, an understanding of the cell-scaffold interactions is of high importance. Here, we have utilized discrete fiber network model to quantitatively study the micromechanics of fibrous scaffolds with different fiber arrangements and cross-linking densities. We observe that localized forces on the scaffold result in its anisotropic deformation even for isotropic fiber arrangements. We also see an exponential decay of the displacement field with distance from the location of applied force. This nature of the decay allows us to estimate the characteristic length for force transmission in fibrous scaffolds. Furthermore, we also looked at the stiffness sensing of fibrous scaffolds by individual cells and its dependence on the cellular sensing mechanism. For this, we considered two conditions- stress-controlled, and strain-controlled application of forces by a cell. With fixed strain, we find that the stiffness sensed by a cell is proportional to the scaffold's 'macroscopic' elastic modulus. However, under fixed stress application by the cell, the stiffness sensed by the cell also depends on the cell's own stiffness. In fact, the stiffness values for the same scaffold sensed by the stiff and soft cells can differ from each other by an order of magnitude. The insights from this work will help in designing tissue engineering scaffolds for applications where mechanical stimuli are a critical factor.

Electromechanical interactions between cell membrane and nuclear envelope: Beyond the standard Schwan's model of biological cells

We investigate little-appreciated features of the hierarchical core-shell (CS) models of the electrical, mechanical, and electromechanical interactions between the cell membrane (CM) and nuclear envelope (NE). We first consider a simple model of an individual cell based on a coupled resistor-capacitor (Schwan model (SM)) network and show that the CM, when exposed to ac electric fields, acts as a low pass filter while the NE acts as a wide and asymmetric bandpass filter. We provide a simplified calculation for characteristic time associated with the capacitive charging of the NE and parameterize its range of behavior. We furthermore observe several new features dealing with mechanical analogs of the SM based on elementary spring-damper combinations. The chief merit of these models is that they can predict creep compliance responses of an individual cell under static stress and their effective retardation time constants. Next, we use an alternative and a more accurate CS physical model solved by finite element simulations for which geometrical cell reshaping under electromechanical stress (electrodeformation (ED)) is included in a continuum approach with spatial resolution. We show that under an electric field excitation, the elongated nucleus scales differently compared to the electrodeformed cell.

Range and strength of mechanical interactions of force dipoles in elastic fiber networks

Mechanical forces generated by myosin II molecular motors drive diverse cellular processes, most notably shape change, division and locomotion. These forces may be transmitted over long range through the cytoskeletal medium - a disordered, viscoelastic network of biopolymers. The resulting cell size scale force chains can in principle mediate mechanical interactions between distant actomyosin units, leading to self-organized structural order in the cell cytoskeleton. Inspired by such force transmission through elastic structures in the cytoskeleton, we consider a percolated fiber lattice network, where fibers are represented as linear elastic elements that can both bend and stretch, and the contractile activity of myosin motors is represented by force dipoles. Then, by using a variety of metrics, we show how two such contractile force dipoles interact with each other through their mutual mechanical deformations of the elastic fiber network. As a prelude to two-dipole interactions, we quantify how forces propagate through the network from a single anisotropic force dipole by analyzing clusters of nodes connected by highly strained bonds, as well as through the decay rate of strain energy with distance from a force dipole. We show that predominant fiber bending screens out force propagation, resulting in reduced and strongly network configuration-dependent dipole interactions. On the other hand, stretching-dominated networks support longer-ranged inter-dipole interactions that recapitulate the predictions of linear elasticity theory. By characterizing the differences between tensile and compressive force propagation in the fiber network, we show how inter-dipole interaction depends on the dipoles' mutual separation and orientation. The resulting elastic interaction energy may mediate a force between multiple distant dipoles, leading to their self-organization into ordered configurations. This provides a potential pathway for active mechanical force-driven structural order in elastic biopolymer networks.

Probing soft fibrous materials by indentation

Indentation is often used to measure the stiffness of soft materials whose main structural component is a network of filaments, such as the cellular cytoskeleton, connective tissue, gels, and the extracellular matrix. For elastic materials, the typical procedure requires fitting the experimental force-displacement curve with the Hertz model, which predicts that f=kδ1.5 and k is proportional to the reduced modulus of the indented material, E/(1-ν2). Here we show using explicit models of fiber networks that the Hertz model applies to indentation in network materials provided the indenter radius is larger than approximately 12lc, where lc is the mean segment length of the network. Using smaller indenters leads to a relation between force and indentation displacement of the form f=kδq, where q is observed to increase with decreasing indenter radius. Using the Hertz model to interpret results of indentations in network materials using small indenters leads to an inferred modulus smaller than the real modulus of the material. The origin of this departure from the classical Hertz model is investigated. A compacted, stiff network region develops under the indenter, effectively increasing the indenter size and modifying its shape. This modification is marginal when large indenters are used. However, when the indenter radius is small, the effect of the compacted layer is pronounced as it changes the indenter profile from spherical towards conical. This entails an increase of exponent q above the value of 1.5 corresponding to spherical indenters. STATEMENT OF SIGNIFICANCE: The article presents a study of indentation in network biomaterials and demonstrates a size effect which precludes the use of the Hertz model to infer the elastic constants of the material. The size effect occurs once the indenter radius is smaller than approximately 12 times the mean segment length of the network. This result provides guidelines for the selection of indentation conditions that guarantee the applicability of the Hertz model. At the same time, the finding may be used to infer the mean segment length of the network based on indentations with indenters of various sizes. Hence, the method can be used to evaluate this structural parameter which is not easily accessible in experiments.

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.

Random Fiber Network Loaded by a Point Force

This article presents the displacement field produced by a point force acting on an athermal random fiber network (the Green function for the network). The problem is defined within the limits of linear elasticity, and the field is obtained numerically for nonaffine networks characterized by various parameter sets. The classical Green function solution applies at distances from the point force larger than a threshold which is independent of the network parameters in the range studied. At smaller distances, the nonlocal nature of fiber interactions modifies the solution.

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.

Directional cues in the tumor microenvironment due to cell contraction against aligned collagen fibers

It is well established that collagen alignment in the breast tumor microenvironment provides biophysical cues to drive disease progression. Numerous mechanistic studies have demonstrated that tumor cell behavior is driven by the architecture and stiffness of the collagen matrix. However, the mechanical properties within a 3D collagen microenvironment, particularly at the scale of the cell, remain poorly defined. To investigate cell-scale mechanical cues with respect to local collagen architecture, we employed a combination of intravital imaging of the mammary tumor microenvironment and a 3D collagen gel system with both acellular pNIPAAm microspheres and MDA-MB-231 breast carcinoma cells. Within the in vivo tumor microenvironment, the displacement of collagen fiber was identified in response to tumor cells migrating through the stromal matrix. To further investigate cell-scale stiffness in aligned fiber architectures and the propagation of cell-induced fiber deformations, precise control of collagen architecture was coupled with innovative methodology to measure mechanical properties of the collagen fiber network. This method revealed up to a 35-fold difference in directional cell-scale stiffness resulting from contraction against aligned fibers. Furthermore, the local anisotropy of the matrix dramatically altered the rate at which contractility-induced fiber displacements decayed over distance. Together, our results reveal mechanical properties in aligned matrices that provide dramatically different cues to the cell in perpendicular directions. These findings are supported by the mechanosensing behavior of tumor cells and have important implications for cell-cell communication within the tissue microenvironment. STATEMENT OF SIGNIFICANCE: It is widely appreciated that the architecture of the extracellular matrix impacts cellular behavior in normal and disease states. Numerous studies have determined the fundamental role of collagen matrix architecture on cellular mechanosensing, but effectively quantifying anisotropic mechanical properties of the collagen matrix at the cell-scale remains challenging. Here, we developed innovative methodology to discover that collagen alignment results in a 35-fold difference in cell-scale stiffness and alters contractile force transmission through the fiber network. Furthermore, we identified bias in cell response along the axis of alignment, where local stiffness is highest. Overall, our results define cell-scale stiffness and fiber deformations due to collagen architecture that may instruct cell communication within a broad range of tissue microenvironments.

A computational framework for modeling cell-matrix interactions in soft biological tissues

Living soft tissues appear to promote the development and maintenance of a preferred mechanical state within a defined tolerance around a so-called set point. This phenomenon is often referred to as mechanical homeostasis. In contradiction to the prominent role of mechanical homeostasis in various (patho)physiological processes, its underlying micromechanical mechanisms acting on the level of individual cells and fibers remain poorly understood, especially how these mechanisms on the microscale lead to what we macroscopically call mechanical homeostasis. Here, we present a novel computational framework based on the finite element method that is constructed bottom up, that is, it models key mechanobiological mechanisms such as actin cytoskeleton contraction and molecular clutch behavior of individual cells interacting with a reconstructed three-dimensional extracellular fiber matrix. The framework reproduces many experimental observations regarding mechanical homeostasis on short time scales (hours), in which the deposition and degradation of extracellular matrix can largely be neglected. This model can serve as a systematic tool for future in silico studies of the origin of the numerous still unexplained experimental observations about mechanical homeostasis.

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

Through mechanical forces, biological cells remodel the surrounding collagen network, generating striking deformation patterns. Tethers-tracts of high densification and fibre alignment-form between cells, thinner bands emanate from cell clusters. While tethers facilitate cell migration and communication, how they form is unclear. Combining modelling, simulation and experiment, we show that tether formation is a densification phase transition of the extracellular matrix, caused by buckling instability of network fibres under cell-induced compression, featuring unexpected similarities with martensitic microstructures. Multiscale averaging yields a two-phase, bistable continuum energy landscape for fibrous collagen, with a densified/aligned second phase. Simulations predict strain discontinuities between the undensified and densified phase, which localizes within tethers as experimentally observed. In our experiments, active particles induce similar localized patterns as cells. This shows how cells exploit an instability to mechanically remodel the extracellular matrix simply by contracting, thereby facilitating mechanosensing, invasion and metastasis.

Continuum elastic models for force transmission in biopolymer gels

We review continuum elastic models for the transmission of both external forces and internal active cellular forces in biopolymer gels, and relate them to recent experiments. Rather than being exhaustive, we focus on continuum elastic models for small affine deformations and intend to provide a systematic continuum method and some analytical perspectives on the study of force transmission in biopolymer gels. We start from a very brief review of the nonlinear mechanics of individual biopolymers and a summary of constitutive models for the nonlinear elasticity of biopolymer gels. We next show that the simple 3-chain model can give predictions that fit well the shear experiments of some biopolymer gels, including the effects of strain-stiffening and negative normal stress. We then review continuum models for the transmission of internal active forces that are induced by a spherically contracting cell embedded in a three-dimensional biopolymer gel. Various scaling regimes for the decay of cell-induced displacements are identified for linear isotropic and anisotropic materials, and for biopolymer gels with nonlinear compressive-softening and strain-stiffening elasticity, respectively. After that, we present (using an energetic approach) the generic and unified continuum theory proposed in [D. Ben-Yaakov et al., Soft Matter, 2015, 11, 1412] about how the transmission of forces in the biogel matrix can mediate long-range interactions between cells with mechanical homeostasis. We show the predictions of the theory in a special hexagonal multicellular array, and relate them to recent experiments. Finally, we conclude this paper with comments on the limitations and outlook of continuum modeling, and highlight the need for complementary theoretical approaches, such as discrete network simulations, to force transmission in biopolymer gels and phenomenological active gel theories for multicellular systems.

Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction

Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.

Non-affinity in multi-material mechanical metamaterials

Non-affine deformations enable mechanical metamaterials to achieve their unusual properties while imposing implications for their structural integrity. The presence of multiple phases with different mechanical properties results in additional non-affinity of the deformations, a phenomenon that has never been studied before in the area of extremal mechanical metamaterials. Here, we studied the degree of non-affinity, [Formula: see text], resulting from the random substitution of a fraction of the struts,[Formula: see text], that make up a lattice structure and are printed using a soft material (elastic modulus = [Formula: see text]) by those printed using a hard material ([Formula: see text]). Depending on the unit cell angle (i.e., [Formula: see text] = 60°, 90°, or 120°), the lattice structures exhibited negative, near-zero, or positive values of the Poisson's ratio, respectively. We found that the auxetic structures exhibit the highest levels of non-affinity, followed by the structures with positive and near-zero values of the Poisson's ratio. We also observed an increase in [Formula: see text] with [Formula: see text] and [Formula: see text] until [Formula: see text] =104 and [Formula: see text]= 75%-90% after which [Formula: see text] saturated. The dependency of [Formula: see text] upon [Formula: see text] was therefore found to be highly asymmetric. The positive and negative values of the Poisson's ratio were strongly correlated with [Formula: see text]. Interestingly, achieving extremely high or extremely low values of the Poisson's ratio required highly affine deformations. In conclusion, our results clearly show the importance of considering non-affinity when trying to achieve a specific set of mechanical properties and underscore the structural integrity implications in multi-material mechanical metamaterials.

Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling

Cell migration in fibrous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response, and cancer progression. During migration, individual cells can generate active pulling forces via actomyosin contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.

Modulus of Fibrous Collagen at the Length Scale of a Cell

The extracellular matrix provides macroscale structural support to tissues as well as microscale mechanical cues, like stiffness, to the resident cells. As those cues modulate gene expression, proliferation, differentiation, and motility, quantifying the stiffness that cells sense is crucial to understanding cell behavior. Whereas the macroscopic modulus of a collagen network can be measured in uniform extension or shear, quantifying the local stiffness sensed by a cell remains a challenge due to the inhomogeneous and nonlinear nature of the fiber network at the scale of the cell. To address this challenge, we designed an experimental method to measure the modulus of a network of collagen fibers at this scale. We used spherical particles of an active hydrogel (poly N-isopropylacrylamide) that contract when heated, thereby applying local forces to the collagen matrix and mimicking the contractile forces of a cell. After measuring the particles' bulk modulus and contraction in networks of collagen fibers, we applied a nonlinear model for fibrous materials to compute the modulus of the local region surrounding each particle. We found the modulus at this length scale to be highly heterogeneous, with modulus varying by a factor of 3. In addition, at different values of applied strain, we observed both strain stiffening and strain softening, indicating nonlinearity of the collagen network. Thus, this experimental method quantifies local mechanical properties in a fibrous network at the scale of a cell, while also accounting for inherent nonlinearity.