Compressive elasticity of polydisperse biopolymer gels

Dynamic remodeling of fiber networks with stiff inclusions under compressive loading

The ability of tissues to sustain and withstand mechanical stress is critical to tissue development and healthy tissue maintenance. The mechanical properties of tissues are typically considered to be dominated by the fibrous extracellular matrix (ECM) component of tissues. Fiber network mechanics can capture certain mechanical features of tissues, such as shear strain stiffening, but is insufficient in describing the compressive response of certain tissues and blood clots that are rich in extracellular matrix. To understand the mechanical response of tissues, we employ a contemporary mechanical model, a fibrous network of fibrin embedded with inert bead inclusions that preserve the volume-conserving constraints of cells in tissues. Combining bulk mechanical rheology and a custom imaging device, we show that the presence of inclusions alters the local dynamic remodeling of the networks undergoing uniaxial compressive strains and demonstrate non-affine correlated motion within a fiber-bead network, predicted to stretch fibers in the network and lead to the ability of the network to stiffen under compression, a key feature of real tissues. These findings have important implications for understanding how local structural properties of cells and ECM fibers impact the bulk mechanical response of real tissues. STATEMENT OF SIGNIFICANCE: To understand why real tissue stiffens under compression, we study a model tissue system which also stiffens: a fibrin network embedded with stiff beads. We design a device that images compression of both fiber and fiber-bead networks. Distinct from previous imaging studies, this setup can dynamically capture network deformation on scales larger than single fibers. From the videos, we see fluid flow and network remodeling are both critical to compression behavior. The fiber-bead network has faster fluid flow, reduced network recovery, and correlated motion during network relaxation. We hypothesize that the beads hinder network relaxation of stretched fibers, thus retaining the applied stress and exhibiting stiffening. Our findings reveal important details for modeling tissue mechanics towards optimizing healthcare.

Unique Role of Vimentin Networks in Compression Stiffening of Cells and Protection of Nuclei from Compressive Stress

In this work, we investigate whether stiffening in compression is a feature of single cells and whether the intracellular polymer networks that comprise the cytoskeleton (all of which stiffen with increasing shear strain) stiffen or soften when subjected to compressive strains. We find that individual cells, such as fibroblasts, stiffen at physiologically relevant compressive strains, but genetic ablation of vimentin diminishes this effect. Further, we show that unlike networks of purified F-actin or microtubules, which soften in compression, vimentin intermediate filament networks stiffen in both compression and extension, and we present a theoretical model to explain this response based on the flexibility of vimentin filaments and their surface charge, which resists volume changes of the network under compression. These results provide a new framework by which to understand the mechanical responses of cells and point to a central role of intermediate filaments in response to compression.

Astral hydrogels mimic tissue mechanics by aster-aster interpenetration

Many soft tissues are compression-stiffening and extension-softening in response to axial strains, but common hydrogels are either inert (for ideal chains) or tissue-opposite (for semiflexible polymers). Herein, we report a class of astral hydrogels that are structurally distinct from tissues but mechanically tissue-like. Specifically, hierarchical self-assembly of amphiphilic gemini molecules produces radial asters with a common core and divergently growing, semiflexible ribbons; adjacent asters moderately interpenetrate each other via interlacement of their peripheral ribbons to form a gel network. Resembling tissues, the astral gels stiffen in compression and soften in extension with all the experimental data across different gel compositions collapsing onto a single master curve. We put forward a minimal model to reproduce the master curve quantitatively, underlying the determinant role of aster-aster interpenetration. Compression significantly expands the interpenetration region, during which the number of effective crosslinks is increased and the network strengthened, while extension does the opposite. Looking forward, we expect this unique mechanism of interpenetration to provide a fresh perspective for designing and constructing mechanically tissue-like materials.

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.

Long-range mechanical coupling of cells in 3D fibrin gels

When seeded in fibrous gels, pairs of cells or cell aggregates can induce bands of deformed gel, extending to surprisingly long distances in the intercellular medium. The formation of bands has been previously shown and studied in collagen systems. In this study, we strive to further our understanding of this fundamental mechanical mechanism in fibrin, a key element in wound healing and angiogenesis processes. We embedded fibroblast cells in 3D fibrin gels, and monitored band formation by real-time confocal microscopy. Quantitative dynamic analysis of band formation revealed a gradual increase in fiber density and alignment between pairs of cells. Such intercellular bands extended into a large-scale network of mechanically connected cells, in which the connected cells exhibited a more spread morphology than the isolated cells. Moreover, computational modeling demonstrated that the direction of cell-induced force triggering band formation can be applied in a wide range of angles relative to a neighboring cell. Our findings indicate that long-range mechanical coupling between cells is an important mechanism in regulating multicellular processes in reconstituted fibrin gels. As such, it should motivate exploration of this mechanism in studies in vivo, in wound healing or angiogenesis, in which fibrin is contracted by fibroblast cells.

Elastic Anisotropy Governs the Range of Cell-Induced Displacements

The unique nonlinear mechanics of the fibrous extracellular matrix (ECM) facilitates long-range cell-cell mechanical communications that would be impossible for linear elastic substrates. Past research has described the contribution of two separated effects on the range of force transmission, including ECM elastic nonlinearity and fiber alignment. However, the relation between these different effects is unclear, and how they combine to dictate force transmission range is still elusive. Here, we combine discrete fiber simulations with continuum modeling to study the decay of displacements induced by a contractile cell in fibrous networks. We demonstrate that fiber nonlinearity and fiber reorientation both contribute to the strain-induced elastic anisotropy of the cell's local environment. This elastic anisotropy is a "lumped" parameter that governs the slow decay of displacements, and it depends on the magnitude of applied strain, either an external tension or an internal contraction, as a model of the cell. Furthermore, we show that accounting for artificially prescribed elastic anisotropy dictates the decay of displacements induced by a contracting cell. Our findings unify previous single effects into a mechanical theory that explains force transmission in fibrous networks. This work may provide insights into biological processes that involve communication of distant cells mediated by the ECM, such as those occurring in morphogenesis, wound healing, angiogenesis, and cancer metastasis. It may also provide design parameters for biomaterials to control force transmission between cells as a way to guide morphogenesis in tissue engineering.

Elastic interactions between anisotropically contracting circular cells

We study interactions between biological cells that apply anisotropic active mechanical forces on an elastic substrate. We model the cells as thin disks that along their perimeters apply radial, but angle-dependent forces on the substrate. We obtain analytical expressions for the elastic energy stored in the substrate as a function of the distance between the cells, the Fourier modes of applied forces, and their phase angles. We show how the relative phases of the forces applied by the cells can switch the interaction between attractive and repulsive, and relate our results to those for linear force dipoles. For long enough distances, the interaction energy decays in magnitude as a power law of the cell-cell distance with an integer exponent that generally increases with the Fourier modes of the applied forces.

Strong triaxial coupling and anomalous Poisson effect in collagen networks

While cells within tissues generate and sense 3D states of strain, the current understanding of the mechanics of fibrous extracellular matrices (ECMs) stems mainly from uniaxial, biaxial, and shear tests. Here, we demonstrate that the multiaxial deformations of fiber networks in 3D cannot be inferred solely based on these tests. The interdependence of the three principal strains gives rise to anomalous ratios of biaxial to uniaxial stiffness between 8 and 9 and apparent Poisson's ratios larger than 1. These observations are explained using a microstructural network model and a coarse-grained constitutive framework that predicts the network Poisson effect and stress-strain responses in uniaxial, biaxial, and triaxial modes of deformation as a function of the microstructural properties of the network, including fiber mechanics and pore size of the network. Using this theoretical approach, we found that accounting for the Poisson effect leads to a 100-fold increase in the perceived elastic stiffness of thin collagen samples in extension tests, reconciling the seemingly disparate measurements of the stiffness of collagen networks using different methods. We applied our framework to study the formation of fiber tracts induced by cellular forces. In vitro experiments with low-density networks showed that the anomalous Poisson effect facilitates higher densification of fibrous tracts, associated with the invasion of cancerous acinar cells. The approach developed here can be used to model the evolving mechanics of ECM during cancer invasion and fibrosis.

Nonlinear Elasticity of the ECM Fibers Facilitates Efficient Intercellular Communication

Biological cells embedded in fibrous matrices have been observed to form intercellular bands of dense and aligned fibers through which they mechanically interact over long distances. Such matrix-mediated cellular interactions have been shown to regulate various biological processes. This study aimed to explore the effects of elastic nonlinearity of the fibers contained in the extracellular matrix (ECM) on the transmission of mechanical loads between contracting cells. Based on our biological experiments, we developed a finite-element model of two contracting cells embedded within a fibrous network. The individual fibers were modeled as showing linear elasticity, compression microbuckling, tension stiffening, or both of the latter two. Fiber compression buckling resulted in smaller loads in the ECM, which were primarily directed toward the neighboring cell. These loads decreased with increasing cell-to-cell distance; when cells were >9 cell diameters apart, no such intercellular interaction was observed. Tension stiffening further contributed to directing the loads toward the neighboring cell, though to a smaller extent. The contraction of two neighboring cells resulted in mutual attraction forces, which were considerably increased by tension stiffening and decayed with increasing cell-to-cell distances. Nonlinear elasticity contributed also to the onset of force polarity on the cell boundaries, manifested by larger contractile forces pointing toward the neighboring cell. The density and alignment of the fibers within the intercellular band were greater when fibers buckled under compression, with tension stiffening further contributing to this structural remodeling. Although previous studies have established the role of the ECM nonlinear mechanical behavior in increasing the range of force transmission, our model demonstrates the contribution of nonlinear elasticity of biological gels to directional and efficient mechanical signal transfer between distant cells, and rehighlights the importance of using fibrous gels in experimental settings for facilitating intercellular communication. VIDEO ABSTRACT.

Programming the mechanics of cohesive fiber networks by compression

Fibrous networks are ideal functional materials since they provide mechanical rigidity at low weight. Here, we demonstrate that fibrous networks of the blood clotting protein fibrin undergo a strong and irreversible increase in their mechanical rigidity in response to uniaxial compression. This rigidification can be precisely controlled by the level of applied compressive strain, providing a means to program the network rigidity without having to change its composition. To identify the underlying mechanism we measure single fiber-fiber interactions using optical tweezers. We further develop a minimal computational model of cohesive fiber networks that shows that stiffening arises due to the formation of new bonds in the compressed state, which develop tensile stress when the network is re-expanded. The model predicts that the network stiffness after a compression cycle obeys a power-law dependence on tensile stress, which we confirm experimentally. This finding provides new insights into how biological tissues can adapt themselves independently of any cellular processes, offering new perspectives to inspire the design of reprogrammable materials.