Uncoupling shear and uniaxial elastic moduli of semiflexible biopolymer networks: compression-softening and stretch-stiffening

Micro-tensile rheology of fibrous gels quantifies strain-dependent anisotropy

Semiflexible fiber gels such as collagen and fibrin have unique nonlinear mechanical properties that play an important role in tissue morphogenesis, wound healing, and cancer metastasis. Optical tweezers microrheology has greatly contributed to the understanding of the mechanics of fibrous gels at the microscale, including its heterogeneity and anisotropy. However, the explicit relationship between micromechanical properties and gel deformation has been largely overlooked. We introduce a unique gel-stretching apparatus and employ it to study the relationship between microscale strain and stiffening in fibrin and collagen gels, focusing on the development of anisotropy in the gel. We find that gels stretched by as much as 15 % stiffen significantly both in parallel and perpendicular to the stretching axis, and that the parallel axis is 2-3 times stiffer than the transverse axis. We also measure the stiffening and anisotropy along bands of aligned fibers created by aggregates of cancer cells, and find similar effects as in gels stretched with the tensile apparatus. Our results illustrate that the extracellular microenvironment is highly sensitive to deformation, with implications for tissue homeostasis and pathology. STATEMENT OF SIGNIFICANCE: The inherent fibrous architecture of the extracellular matrix (ECM) gives rise to unique strain-stiffening mechanics. The micromechanics of fibrous networks has been studied extensively, but the deformations involved in its stiffening at the microscale were not quantified. Here we introduce an apparatus that enables measuring the deformations in the gel as it is being stretched while simultaneously using optical tweezers to measure its microscale anisotropic stiffness. We reveal that fibrin and collagen both stiffen dramatically already at ∼10 % deformation, accompanied by the emergence of significant, yet moderate anisotropy. We measure similar stiffening and anisotropy in the matrix remodeled by the tensile apparatus to those found between cancer cell aggregates. Our results emphasize that small strains are enough to introduce substantial stiffening and anisotropy. These have been shown to result in directional cell migration and enhanced force propagation, and possibly control processes like morphogenesis and cancer metastasis.

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.

Multiscale elasticity mapping of biological samples in 3D at optical resolution

The mechanical properties of biological tissues have emerged as an integral determinant of tissue function in health and disease. Nonetheless, characterizing the elasticity of biological samples in 3D and at high resolution remains challenging. Here, we present a µElastography platform: a scalable elastography system that maps the elastic properties of tissues from cellular to organ scales. The platform leverages the use of a biocompatible, thermo-responsive hydrogel to deliver compressive stress to a biological sample and track its resulting deformation. By surrounding the specimen with a reference hydrogel of known Young's modulus, we are able to map the absolute values of elastic properties in biological samples. We validate the experimental and computational components of the platform using a hydrogel phantom and verify the system's ability to detect internal mechanical heterogeneities. We then apply the platform to map the elasticity of multicellular spheroids and the murine lymph node. With these applications, we demonstrate the platform's ability to map tissue elasticity at internal planes of interest, as well as capture mechanical heterogeneities neglected by most macroscale characterization techniques. The µElastography platform, designed to be implementable in any biology lab with access to 3D microscopy (e.g., confocal, multiphoton, or optical coherence microscopy), will provide the capability to characterize the mechanical properties of biological samples to labs across the large community of biological sciences by eliminating the need of specialized instruments such as atomic force microscopy. STATEMENT OF SIGNIFICANCE: Understanding the elasticity of biological tissues is of great importance, but characterizing these properties typically requires highly specialized equipment. Utilizing stimulus-responsive hydrogels, we present a scalable, hydrogel-based elastography method that uses readily available reagents and imaging modalities to generate resolved maps of internal elasticity within biomaterials and biological samples at optical resolution. This new approach is capable of detecting internal stiffness heterogeneities within the 3D bulk of samples and is highly scalable across both imaging modalities and biological length scales. Thus, it will have significant impact on the measurement capabilities of labs studying engineered biomaterials, mechanobiology, disease progression, and tissue engineering and development.

Collagen Networks under Indentation and Compression Behave Like Cellular Solids

Simple synthetic and natural hydrogels can be formulated to have elastic moduli that match biological tissues, leading to their widespread application as model systems for tissue engineering, medical device development, and drug delivery vehicles. However, two different hydrogels having the same elastic modulus but differing in microstructure or nanostructure can exhibit drastically different mechanical responses, including their poroelasticity, lubricity, and load bearing capabilities. Here, we investigate the mechanical response of collagen-1 networks to local and bulk compressive loads. We compare these results to the behavior of polyacrylamide, a fundamentally different class of hydrogel network consisting of flexible polymer chains. We find that the high bending rigidity of collagen fibers, which suppresses entropic bending fluctuations and osmotic pressure, facilitates the bulk compression of collagen networks under infinitesimal applied stress. These results are fundamentally different from the behavior of flexible polymer networks in which the entropic thermal fluctuations of the polymer chains result in an osmotic pressure that must first be overcome before bulk compression can occur. Furthermore, we observe minimal transverse strain during the axial loading of collagen networks, a behavior reminiscent of open-celled cellular solids. Inspired by these results, we applied mechanical models of cellular solids to predict the elastic moduli of the collagen networks and found agreement with the moduli values measured through contact indentation. Collectively, these results suggest that unlike flexible polymer networks that are often considered incompressible, collagen hydrogels behave like rigid porous solids that volumetrically compress and expel water rather than spreading laterally under applied normal loads.

Nanocolloidal hydrogel mimics the structure and nonlinear mechanical properties of biological fibrous networks

Fibrous networks formed by biological polymers such as collagen or fibrin exhibit nonlinear mechanical behavior. They undergo strong stiffening in response to weak shear and elongational strains, but soften under compressional strain, in striking difference with the response to the deformation of flexible-strand networks formed by molecules. The nonlinear properties of fibrous networks are attributed to the mechanical asymmetry of the constituent filaments, for which a stretching modulus is significantly larger than the bending modulus. Studies of the nonlinear mechanical behavior are generally performed on hydrogels formed by biological polymers, which offers limited control over network architecture. Here, we report an engineered covalently cross-linked nanofibrillar hydrogel derived from cellulose nanocrystals and gelatin. The variation in hydrogel composition provided a broad-range change in its shear modulus. The hydrogel exhibited both shear-stiffening and compression-induced softening, in agreement with the predictions of the affine model. The threshold nonlinear stress and strain were universal for the hydrogels with different compositions, which suggested that nonlinear mechanical properties are general for networks formed by rigid filaments. The experimental results were in agreement with an affine model describing deformation of the network formed by rigid filaments. Our results lend insight into the structural features that govern the nonlinear biomechanics of fibrous networks and provide a platform for future studies of the biological impact of nonlinear mechanical properties.

Effective medium theory for mechanical phase transitions of fiber networks

Networks of stiff fibers govern the elasticity of biological structures such as the extracellular matrix of collagen. These networks are known to stiffen nonlinearly under shear or extensional strain. Recently, it has been shown that such stiffening is governed by a strain-controlled athermal but critical phase transition, from a floppy phase below the critical strain to a rigid phase above the critical strain. While this phase transition has been extensively studied numerically and experimentally, a complete analytical theory for this transition remains elusive. Here, we present an effective medium theory (EMT) for this mechanical phase transition of fiber networks. We extend a previous EMT appropriate for linear elasticity to incorporate nonlinear effects via an anharmonic Hamiltonian. The mean-field predictions of this theory, including the critical exponents, scaling relations and non-affine fluctuations qualitatively agree with previous experimental and numerical results.

The influence of static portal pressure on liver biophysical properties

The liver is a highly vascularized organ where fluid properties, including vascular pressure, vessel integrity and fluid viscosity, play a critical role in gross mechanical properties. To study the effects of portal pressure, liver confinement, fluid viscosity, and tissue crosslinking on liver stiffness, water diffusion, and vessel size, we applied multiparametric magnetic resonance imaging (mpMRI), including multifrequency magnetic resonance elastography (MRE) and apparent diffusion coefficient (ADC) measurements, to ex vivo livers from healthy male rats (13.6±1.6 weeks) at room temperature. Four scenarios including altered liver confinement, tissue crosslinking, and vascular fluid viscosity were investigated with mpMRI at different portal pressure levels (0-17.5 cmH2O). Our experiments demonstrated that, with increasing portal pressure, rat livers showed higher water content, water diffusivity, and increased vessel sizes quantified by the vessel tissue volume fraction (VTVF). These effects were most pronounced in native, unconfined livers (VTVF: 300±120%, p<0.05, ADC: 88±29%, p<0.01), while still significant under confinement (confined: VTVF: 53±32%, p<0.01, ADC: 28±19%, p<0.05; confined-fixed: VTVF: 52±20%, p<0.001, ADC: 11±2%, p<0.01; confined-viscous: VTVF: 210±110%, p<0.01, ADC: 26±9%, p<0.001). Softening with elevated portal pressure (-12±5, p<0.05) occurred regardless of confinement and fixation. However, the liver stiffened when exposed to a more viscous inflow fluid (11±4%, p<0.001). Taken together, our results elucidate the complex relationship between macroscopic-biophysical parameters of liver tissue measured by mpMRI and vascular-fluid properties. Influenced by portal pressure, vascular permeability, and matrix crosslinking, liver stiffness is sensitive to intrinsic poroelastic properties, which, alongside vascular architecture and water diffusivity, may aid in the differential diagnosis of liver disease. STATEMENT OF SIGNIFICANCE: Using highly controllable ex vivo rat liver phantoms, hepatic biophysical properties such as tissue-vascular structure, stiffness, and water diffusivity were investigated using multiparametric MRI including multifrequency magnetic resonance elastography (MRE) and diffusion-weighted imaging (DWI). Through elaborate tuning of the experimental conditions such as the static portal pressure, flow viscosity, amount and distribution of fluid content in the liver, we identified the contributions of the fluid component to the overall imaging-based biophysical properties of the liver. Our finding demonstrated the sensitivity of liver stiffness to the hepatic poroelastic properties, which may aid in the differential diagnosis of liver diseases.

Vaginal Fibroblast Behavior as a Function of Stiffness Changes in a Polyisocyanide Hydrogel for Prolapse Repair

There is an urgent need for improved outcomes in the treatment of pelvic organ prolapse (POP). Success of primary surgery relies on the load bearing capacity of plicated connective tissue underneath the vaginal wall, which is compromised due to an altered vaginal fibroblast function and collagen composition. There is an important factor in connective tissue repair that relates to changes in stiffness of the vaginal fibroblast microenvironment, which influences cell activity through cellular mechanosensing. The aim of this study is to investigate the effect of stiffness changes on vaginal fibroblast functions that relate to connective tissue healing in prolapse repair. The substrate stiffness was controlled by changing the polymer concentration in the fibrous and strongly biomimetic polyisocyanide (PIC) hydrogel. We analyzed stiffness during cell culture and assessed the consequential fibroblast proliferation, morphology, collagen deposition, and contraction. Our results show that increasing stiffness coincides with vaginal fibroblast alignment, promotes collagen deposition, and inhibits PIC gel contraction. These findings suggest that the matrix stiffness directly influences vaginal fibroblast functionality. Moreover, we observed a buildup in stiffness and collagen, with an enhanced fibroblast and collagen organization on the PIC-substrate, which indicate an enhanced structural integrity of the hydrogel-cell construct. An improved tissue structure during healing is relevant in the functional repair of POP. Therefore, this study encourages future research in the use of PIC gels as a supplement in prolapse surgery, whereby the hydrogel stiffness should be considered.

Design, characterization and applications of nanocolloidal hydrogels

Nanocolloidal gels (NCGs) are an emerging class of soft matter, in which nanoparticles act as building blocks of the colloidal network. Chemical or physical crosslinking enables NCG synthesis and assembly from a broad range of nanoparticles, polymers, and low-molecular weight molecules. The synergistic properties of NCGs are governed by nanoparticle composition, dimensions and shape, the mechanism of nanoparticle bonding, and the NCG architecture, as well as the nature of molecular crosslinkers. Nanocolloidal gels find applications in soft robotics, bioengineering, optically active coatings and sensors, optoelectronic devices, and absorbents. This review summarizes currently scattered aspects of NCG formation, properties, characterization, and applications. We describe the diversity of NCG building blocks, discuss the mechanisms of NCG formation, review characterization techniques, outline NCG fabrication and processing methods, and highlight most common NCG applications. The review is concluded with the discussion of perspectives in the design and development of NCGs.

Generic stress rectification in nonlinear elastic media

Stress propagation in nonlinear media is crucial in cell biology, where molecular motors exert anisotropic force dipoles on the fibrous cytoskeleton. While the force dipoles can be either contractile or expansile, a medium made of fibers which buckle under compression rectifies these stresses towards a biologically crucial contraction. A general understanding of this rectification phenomenon as a function of the medium's elasticity is however lacking. Here we use theoretical continuum elasticity to show that rectification is actually a very general effect in nonlinear materials subjected to anisotropic internal stresses. We analytically show that both bucklable and constitutively linear materials subjected to geometrical nonlinearities rectify small forces towards contraction, while granular-like materials rectify towards expansion. Using simulations, we moreover show that these results extend to larger forces. Beyond fiber networks, these results could shed light on the propagation of stresses in brittle or granular materials following a local plastic rearrangement.

Probing Local Force Propagation in Tensed Fibrous Gels

Fibrous hydrogels are a key component of soft animal tissues. They support cellular functions and facilitate efficient mechanical communication between cells. Due to their nonlinear mechanical properties, fibrous materials display non-trivial force propagation at the microscale, that is enhanced compared to that of linear-elastic materials. In the body, tissues are constantly subjected to external loads that tense or compress them, modifying their micro-mechanical properties into an anisotropic state. However, it is unknown how force propagation is modified by this isotropic-to-anisotropic transition. Here, force propagation in tensed fibrin hydrogels is directly measured. Local perturbations are induced by oscillating microspheres using optical tweezers. 1-point and 2-point microrheology are combined to simultaneously measure the shear modulus and force propagation. A mathematical framework to quantify anisotropic force propagation trends is suggested. Results show that force propagation becomes anisotropic in tensed gels, with, surprisingly, stronger response to perturbations perpendicular to the axis of tension. Importantly, external tension can also increase the range of force transmission. Possible implications and future directions for research are discussed. These results suggest a mechanism for favored directions of mechanical communication between cells in a tissue under external loads.

Hyperelastic continuum models for isotropic athermal fibrous networks

Many biological materials contain fibrous protein networks as their main structural components. Understanding the mechanical properties of such networks is important for creating biomimicking materials for cell and tissue engineering, and for developing novel tools for detecting and diagnosing disease. In this work, we develop continuum models for isotropic, athermal fibrous networks by combining a single-fibre model that describes the axial response of individual fibres, with network models that assemble individual fibre properties into overall network behaviour. In particular, we consider four different network models, including the affine, three-chain, eight-chain, and micro-sphere models, which employ different assumptions about network structure and kinematics. We systematically investigate the ability of these models to describe the mechanical response of athermal collagen and fibrin networks by comparing model predictions with experimental data. We test how each model captures network behaviour under three different loading conditions: uniaxial tension, simple shear, and combined tension and shear. We find that the affine and three-chain models can accurately describe both the axial and shear behaviour, whereas the eight-chain and micro-sphere models fail to capture the shear response, leading to unphysical zero shear moduli at infinitesimal strains. Our study is the first to systematically investigate the applicability of popular network models for describing the macroscopic behaviour of athermal fibrous networks, offering insights for selecting efficient models that can be used for large-scale, finite-element simulations of athermal networks.

Mechanics of fiber networks under a bulk strain

Biopolymer networks are common in biological systems from the cytoskeleton of individual cells to collagen in the extracellular matrix. The mechanics of these systems under applied strain can be explained in some cases by a phase transition from soft to rigid states. For collagen networks, it has been shown that this transition is critical in nature and it is predicted to exhibit diverging fluctuations near a critical strain that depends on the network's connectivity and structure. Whereas prior work focused mostly on shear deformation that is more accessible experimentally, here we study the mechanics of such networks under an applied bulk or isotropic extension. We confirm that the bulk modulus of subisostatic fiber networks exhibits similar critical behavior as a function of bulk strain. We find different nonmean-field exponents for bulk as opposed to shear. We also confirm a similar hyperscaling relation to what was previously found for shear.

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.

Multiscale architecture: Mechanics of composite cytoskeletal networks

Different types of biological cells respond differently to mechanical stresses, and these responses are mainly governed by the cytoskeleton. The main components of this biopolymer network are actin filaments, microtubules, and intermediate filaments, whose mechanical and dynamic properties are highly distinct, thus opening up a large mechanical parameter space. Aside from experiments on whole, living cells, "bottom-up" approaches, utilizing purified, reconstituted protein systems, tremendously help to shed light on the complex mechanics of cytoskeletal networks. Such experiments are relevant in at least three aspects: (i) from a fundamental point of view, cytoskeletal networks provide a perfect model system for polymer physics; (ii) in materials science and "synthetic cell" approaches, one goal is to fully understand properties of cellular materials and reconstitute them in synthetic systems; (iii) many diseases are associated with cell mechanics, so a thorough understanding of the underlying phenomena may help solving pressing biomedical questions. In this review, we discuss the work on networks consisting of one, two, or all three types of filaments, entangled or cross-linked, and consider active elements such as molecular motors and dynamically growing filaments. Interestingly, tuning the interactions among the different filament types results in emergent network properties. We discuss current experimental challenges, such as the comparability of different studies, and recent methodological advances concerning the quantification of attractive forces between filaments and their influence on network mechanics.

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.

Stiffening of under-constrained spring networks under isotropic strain

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e. networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the scaling behavior of the elastic network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus G scales linearly with isotropic stress T close to the rigidity transition. However, this seems to be at odds with recent numerical studies suggesting an exponent between G and T that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition a linear scaling, G ∼ T, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.

Rheology of fibrous gels under compression

A number of biological tissues and synthetic gels consist of a fibrous network infused with liquid. There have been a few experimental studies of the rheological properties of such gels under applied compressive strain. Their results suggest that a plot of rheological moduli as a function of applied compressive strain has a long plateau flanked by a steeply increasing curve for large compressive strains and a slowly decreasing curve for small strains. In this paper we explain these trends in rheological properties using a chemo-elastic model characterized by a double-well strain energy function for the underlying fibrous network. The wells correspond to rarefied and densified phases of the fibrous network at low and high strains, respectively. These phases can co-exist across a movable transition front in the gel in order to accommodate overall applied compression. We find that the rheological properties of fibrous gels share similarities with a Kelvin-Voigt visco-elastic solid. The storage modulus has its origins in the elasticity of the fibrous network, while the loss modulus is determined by the dissipation caused by liquid flow through pores. The rheological properties can depend on the number of phase transition fronts present in a compressed sample. Our analysis may explain the dependence of storage and loss moduli of fibrin gels on the loading history. We also point to the need for combining rheological measurements on gels with a microstructural analysis that could shed light on various dissipation mechanisms.

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.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Materials science and mechanosensitivity of living matter

Living systems are composed of molecules that are synthesized by cells that use energy sources within their surroundings to create fascinating materials that have mechanical properties optimized for their biological function. Their functionality is a ubiquitous aspect of our lives. We use wood to construct furniture, bacterial colonies to modify the texture of dairy products and other foods, intestines as violin strings, bladders in bagpipes, and so on. The mechanical properties of these biological materials differ from those of other simpler synthetic elastomers, glasses, and crystals. Reproducing their mechanical properties synthetically or from first principles is still often unattainable. The challenge is that biomaterials often exist far from equilibrium, either in a kinetically arrested state or in an energy consuming active state that is not yet possible to reproduce de novo. Also, the design principles that form biological materials often result in nonlinear responses of stress to strain, or force to displacement, and theoretical models to explain these nonlinear effects are in relatively early stages of development compared to the predictive models for rubberlike elastomers or metals. In this Review, we summarize some of the most common and striking mechanical features of biological materials and make comparisons among animal, plant, fungal, and bacterial systems. We also summarize some of the mechanisms by which living systems develop forces that shape biological matter and examine newly discovered mechanisms by which cells sense and respond to the forces they generate themselves, which are resisted by their environment, or that are exerted upon them by their environment. Within this framework, we discuss examples of how physical methods are being applied to cell biology and bioengineering.

Cell-matrix reciprocity in 3D culture models with nonlinear elasticity

Three-dimensional (3D) matrix models using hydrogels are powerful tools to understand and predict cell behavior. The interactions between the cell and its matrix, however is highly complex: the matrix has a profound effect on basic cell functions but simultaneously, cells are able to actively manipulate the matrix properties. This (mechano)reciprocity between cells and the extracellular matrix (ECM) is central in regulating tissue functions and it is fundamentally important to broadly consider the biomechanical properties of the in vivo ECM when designing in vitro matrix models. This manuscript discusses two commonly used biopolymer networks, i.e. collagen and fibrin gels, and one synthetic polymer network, polyisocyanide gel (PIC), which all possess the characteristic nonlinear mechanics in the biological stress regime. We start from the structure of the materials, then address the uses, advantages, and limitations of each material, to provide a guideline for tissue engineers and biophysicists in utilizing current materials and also designing new materials for 3D cell culture purposes.

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.

Shear-induced phase transition and critical exponents in three-dimensional fiber networks

When subject to applied strain, fiber networks exhibit nonlinear elastic stiffening. Recent theory and experiments have shown that this phenomenon is controlled by an underlying mechanical phase transition that is critical in nature. Growing simulation evidence points to non-mean-field behavior for this transition and a hyperscaling relation has been proposed to relate the corresponding critical exponents. Here, we report simulations on two distinct network structures in three dimensions. By performing a finite-size scaling analysis, we test hyperscaling and identify various critical exponents. From the apparent validity of hyperscaling, as well as the non-mean-field exponents we observe, our results suggest that the upper critical dimension for the strain-controlled phase transition is above three, in contrast to the jamming transition that represents another athermal, mechanical phase transition.

High ligand density drives extensive spreading and motility on soft GelMA gels

In comparison to synthetic hydrogels where ligand density and stiffness can be independently tuned, cell responses are expected to deviate on native biopolymer networks where ligand density and stiffness are coupled. Here we probe the tensional homeostasis of fibroblasts on methacrylated gelatin (GelMA) gels, which are widely used in tissue engineering applications. On 5%-15% GelMA gels which are very soft (10-100's of Pa's in stiffness), fibroblasts were found to spread extensively and assemble prominent stress fibers and focal adhesions. Probing of contractile mechanics using trypsin-induced detachment revealed adhesive drag, but not contractility, was sensitive to GelMA concentration. Contractility-altering drugs blebbistatin and nocodazole, which exhibited opposite effects on focal adhesion size, both led to reduction in adhesive drag and cell rounding. However, cell motility was impacted only in nocodazole-treated cells. Collectively, our experiments suggest that on soft GelMA gels, contractility-independent adhesion clustering mediated by high ligand density can drive cell spreading and motility.

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.

A tough act to follow: collagen hydrogel modifications to improve mechanical and growth factor loading capabilities

Collagen hydrogels are among ​the most well-studied platforms for drug delivery and in situ tissue engineering, thanks to their low cost, low immunogenicity, versatility, biocompatibility, and similarity to the natural extracellular matrix (ECM). Despite collagen being largely responsible for the tensile properties of native connective tissues, collagen hydrogels have relatively low mechanical properties in the absence of covalent cross-linking. This is particularly problematic when attempting to regenerate stiffer and stronger native tissues such as bone. Furthermore, in contrast to hydrogels based on ECM proteins such as fibronectin, collagen hydrogels do not have any growth factor (GF)-specific binding sites and often cannot sequester physiological (small) amounts of the protein. GF binding and in situ presentation are properties that can aid significantly in the tissue regeneration process by dictating cell fate without causing adverse effects such as malignant tumorigenic tissue growth. To alleviate these issues, researchers have developed several strategies to increase the mechanical properties of collagen hydrogels using physical or chemical modifications. This can expand the applicability of collagen hydrogels to tissues subject to a continuous load. GF delivery has also been explored, mathematically and experimentally, through the development of direct loading, chemical cross-linking, electrostatic interaction, and other carrier systems. This comprehensive article explores the ways in which these parameters, mechanical properties and GF delivery, have been optimized in collagen hydrogel systems ​and examines their in vitro or in vivo biological effect. This article can, therefore, be a useful tool to streamline future studies in the field, by pointing researchers into the appropriate direction according to their collagen hydrogel design requirements.

Modeling the Mechanobiology of Cancer Cell Migration Using 3D Biomimetic Hydrogels

Understanding how cancer cells migrate, and how this migration is affected by the mechanical and chemical composition of the extracellular matrix (ECM) is critical to investigate and possibly interfere with the metastatic process, which is responsible for most cancer-related deaths. In this article we review the state of the art about the use of hydrogel-based three-dimensional (3D) scaffolds as artificial platforms to model the mechanobiology of cancer cell migration. We start by briefly reviewing the concept and composition of the extracellular matrix (ECM) and the materials commonly used to recreate the cancerous ECM. Then we summarize the most relevant knowledge about the mechanobiology of cancer cell migration that has been obtained using 3D hydrogel scaffolds, and relate those discoveries to what has been observed in the clinical management of solid tumors. Finally, we review some recent methodological developments, specifically the use of novel bioprinting techniques and microfluidics to create realistic hydrogel-based models of the cancer ECM, and some of their applications in the context of the study of cancer cell migration.

Long-range mechanical signaling in biological systems

Cells can respond to signals generated by other cells that are remarkably far away. Studies from at least the 1920's showed that cells move toward each other when the distance between them is on the order of a millimeter, which is many times the cell diameter. Chemical signals generated by molecules diffusing from the cell surface would move too slowly and dissipate too fast to account for these effects, suggesting that they might be physical rather than biochemical. The non-linear elastic responses of sparsely connected networks of stiff or semiflexible filament such as those that form the extracellular matrix (ECM) and the cytoskeleton have unusual properties that suggest multiple mechanisms for long-range signaling in biological tissues. These include not only direct force transmission, but also highly non-uniform local deformations, and force-generated changes in fiber alignment and density. Defining how fibrous networks respond to cell-generated forces can help design new methods to characterize abnormal tissues and can guide development of improved biomimetic materials.

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.

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.

The role of temperature in the rigidity-controlled fracture of elastic networks

We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the microscopic damage spreads over a larger area and a more ductile fracture process is observed. These effects are modulated by network rigidity and can therefore be tuned via the network connectivity and the rupture threshold of the springs. Knowledge of the interplay between temperature and rigidity improves our understanding of the fracture of elastic network materials, such as (biological) polymer networks, and can help to refine design principles for tough soft materials.

Rapid local compression in active gels is caused by nonlinear network response

The actin cytoskeleton in living cells generates forces in conjunction with myosin motor proteins to directly and indirectly drive essential cellular processes. The semiflexible filaments of the cytoskeleton can respond nonlinearly to the collective action of motors. We here investigate mechanics and force generation in a model actin cytoskeleton, reconstituted in vitro, by observing the response and fluctuations of embedded micron-scale probe particles. Myosin mini-filaments can be modeled as force dipoles and give rise to deformations in the surrounding network of cross-linked actin. Anomalously correlated probe fluctuations indicate the presence of rapid local compression or draining of the network that emerges in addition to the ordinary linear shear elastic (incompressible) response to force dipoles. The anomalous propagation of compression can be attributed to the nonlinear response of actin filaments to the microscopic forces, and is quantitatively consistent with motor-generated large-scale stiffening of the gels.

Rheological Properties and Growth Factors Content of Platelet-Rich Plasma: Relevance in Veterinary Biomedical Treatments

Platelet-rich plasma (PRP) is a nontransfusional hemocomponent, considered as a powerful concentrate of growth factors (GFs) therapeutically used to stimulate tissue regeneration. The use of autologous PRP, as the patient’s own biological material, for therapeutic purposes represents a safe and effective alternative to conventional treatments in both human and veterinary medicine. The aim of this study was the characterization of canine PRP from rheological and biological points of view. Thus, a characterization of the viscoelastic properties of the PRP systems was performed in order to clarify the influence of different calcium concentrations, in the presence of autologous thrombin-rich solution, on the PRP gels’ mechanical properties, from which the applicability of these systems in biomedical treatments is strongly dependent. Then, an evaluation of the content of GFs in PRP, activated or not with thrombin, and stored at different temperatures (37 °C and −20 °C) was performed over time, outlining, for the first time, the importance of the effect of physiological temperature (37 °C) on the production of GFs. A clinical case study conducted in a dog with a complete rupture of the common calcaneal tendon (Achilles tendon) confirmed the relevance of this hemocomponent in the daily veterinary clinical activity and the potential translational value for human health.

Tissue Rheology as a Possible Complementary Procedure to Advance Histological Diagnosis of Colon Cancer

In recent years, rheological measurements of cells and tissues at physiological and pathological stages have become an essential method to determine how forces and changes in mechanical properties contribute to disease development and progression, but there is no standardization of this procedure so far. In this study, we evaluate the potential of nanoscale atomic force microscopy (AFM) and macroscopic shear rheometry to assess the mechanical properties of healthy and cancerous human colon tissues. The direct comparison of tissue mechanical behavior under uniaxial and shear deformation shows that cancerous tissues not only are stiffer compared to healthy tissue but also respond differently when shear and compressive stresses are applied. These results suggest that rheological parameters can be useful measures of colon cancer mechanopathology. Additionally, we extend the list of biological materials exhibiting compressional stiffening and shear weakening effects to human colon tumors. These mechanical responses might be promising mechanomarkers and become part of the new procedures in colon cancer diagnosis. Enrichment of histopathological grading with rheological assessment of tissue mechanical properties will potentially allow more accurate colon cancer diagnosis and improve prognosis.

A novel method to make viscoelastic polyacrylamide gels for cell culture and traction force microscopy

Polyacrylamide hydrogels are commonly used in cell biology, notably to cultivate cells on soft surfaces. Polyacrylamide gels are purely elastic and well adapted to cell culture as they are inert and can be conjugated with adhesion proteins. Here, we report a method to make viscoelastic polyacrylamide gels with mechanical properties more closely resembling biological tissues and suitable for cell culture in vitro. We demonstrate that these gels can be used for traction force microscopy experiments. We also show that multiple cell types respond to the viscoelasticity of their substrate and that viscous dissipation has an influence on cell spreading, contractility, and motility. This new material provides new opportunities for investigating how normal or malignant cells sense and respond to viscous dissipation within the extra-cellular matrix.

What Caging Force Cells Feel in 3D Hydrogels: A Rheological Perspective

It has been established that the mechanical properties of hydrogels control the fate of (stem) cells. However, despite its importance, a one-to-one correspondence between gels' stiffness and cell behavior is still missing from literature. In this work, the viscoelastic properties of poly(ethylene-glycol) (PEG)-based hydrogels are investigated by means of rheological measurements performed at different length scales. The outcomes of this work reveal that PEG-based hydrogels show significant stiffening when subjected to a compressional deformation, implying that conventional bulk rheology measurements may overestimate the stiffness of hydrogels by up to an order of magnitude. It is hypothesized that this apparent stiffening is caused by an induced "tensional state" of the gel network, due to the application of a compressional normal force during sample loading. Moreover, it is shown that the actual stiffness of the hydrogels is instead accurately determined by means of both passive-video-particle-tracking (PVPT) microrheology and nanoindentation measurements, which are inherently performed at the cell's length scale and in absence of any externally applied force in the case of PVPT. These results underpin a methodology for measuring hydrogels' linear viscoelastic properties that are representative of the mechanical constraints perceived by cells in 3D hydrogel cultures.

Effects of extracellular matrix viscoelasticity on cellular behaviour

Substantial research over the past two decades has established that extracellular matrix (ECM) elasticity, or stiffness, affects fundamental cellular processes, including spreading, growth, proliferation, migration, differentiation and organoid formation. Linearly elastic polyacrylamide hydrogels and polydimethylsiloxane (PDMS) elastomers coated with ECM proteins are widely used to assess the role of stiffness, and results from such experiments are often assumed to reproduce the effect of the mechanical environment experienced by cells in vivo. However, tissues and ECMs are not linearly elastic materials-they exhibit far more complex mechanical behaviours, including viscoelasticity (a time-dependent response to loading or deformation), as well as mechanical plasticity and nonlinear elasticity. Here we review the complex mechanical behaviours of tissues and ECMs, discuss the effect of ECM viscoelasticity on cells, and describe the potential use of viscoelastic biomaterials in regenerative medicine. Recent work has revealed that matrix viscoelasticity regulates these same fundamental cell processes, and can promote behaviours that are not observed with elastic hydrogels in both two- and three-dimensional culture microenvironments. These findings have provided insights into cell-matrix interactions and how these interactions differentially modulate mechano-sensitive molecular pathways in cells. Moreover, these results suggest design guidelines for the next generation of biomaterials, with the goal of matching tissue and ECM mechanics for in vitro tissue models and applications in regenerative medicine.

Functional hyperemia drives fluid exchange in the paravascular space

The brain lacks a conventional lymphatic system to remove metabolic waste. It has been proposed that directional fluid movement through the arteriolar paravascular space (PVS) promotes metabolite clearance. We performed simulations to examine if arteriolar pulsations and dilations can drive directional CSF flow in the PVS and found that arteriolar wall movements do not drive directional CSF flow. We propose an alternative method of metabolite clearance from the PVS, namely fluid exchange between the PVS and the subarachnoid space (SAS). In simulations with compliant brain tissue, arteriolar pulsations did not drive appreciable fluid exchange between the PVS and the SAS. However, when the arteriole dilated, as seen during functional hyperemia, there was a marked exchange of fluid. Simulations suggest that functional hyperemia may serve to increase metabolite clearance from the PVS. We measured blood vessels and brain tissue displacement simultaneously in awake, head-fixed mice using two-photon microscopy. These measurements showed that brain deforms in response to pressure changes in PVS, consistent with our simulations. Our results show that the deformability of the brain tissue needs to be accounted for when studying fluid flow and metabolite transport.

Compression stiffening of fibrous networks with stiff inclusions

Tissues commonly consist of cells embedded within a fibrous biopolymer network. Whereas cell-free reconstituted biopolymer networks typically soften under applied uniaxial compression, various tissues, including liver, brain, and fat, have been observed to instead stiffen when compressed. The mechanism for this compression-stiffening effect is not yet clear. Here, we demonstrate that when a material composed of stiff inclusions embedded in a fibrous network is compressed, heterogeneous rearrangement of the inclusions can induce tension within the interstitial network, leading to a macroscopic crossover from an initial bending-dominated softening regime to a stretching-dominated stiffening regime, which occurs before and independently of jamming of the inclusions. Using a coarse-grained particle-network model, we first establish a phase diagram for compression-driven, stretching-dominated stress propagation and jamming in uniaxially compressed two- and three-dimensional systems. Then, we demonstrate that a more detailed computational model of stiff inclusions in a subisostatic semiflexible fiber network exhibits quantitative agreement with the predictions of our coarse-grained model as well as qualitative agreement with experiments.

Loops versus lines and the compression stiffening of cells

Both animal and plant tissue exhibit a nonlinear rheological phenomenon known as compression stiffening, or an increase in moduli with increasing uniaxial compressive strain. Does such a phenomenon exist in single cells, which are the building blocks of tissues? One expects an individual cell to compression soften since the semiflexible biopolymer-based cytoskeletal network maintains the mechanical integrity of the cell and in vitro semiflexible biopolymer networks typically compression soften. To the contrary, we find that mouse embryonic fibroblasts (mEFs) compression stiffen under uniaxial compression via atomic force microscopy studies. To understand this finding, we uncover several potential mechanisms for compression stiffening. First, we study a single semiflexible polymer loop modeling the actomyosin cortex enclosing a viscous medium modeled as an incompressible fluid. Second, we study a two-dimensional semiflexible polymer/fiber network interspersed with area-conserving loops, which are a proxy for vesicles and fluid-based organelles. Third, we study two-dimensional fiber networks with angular-constraining crosslinks, i.e. semiflexible loops on the mesh scale. In the latter two cases, the loops act as geometric constraints on the fiber network to help stiffen it via increased angular interactions. We find that the single semiflexible polymer loop model agrees well with the experimental cell compression stiffening finding until approximately 35% compressive strain after which bulk fiber network effects may contribute. We also find for the fiber network with area-conserving loops model that the stress-strain curves are sensitive to the packing fraction and size distribution of the area-conserving loops, thereby creating a mechanical fingerprint across different cell types. Finally, we make comparisons between this model and experiments on fibrin networks interlaced with beads as well as discuss implications for single cell compression stiffening at the tissue scale.

On the mechanical response of the actomyosin cortex during cell indentations

A mechanical model is presented to analyze the mechanics and dynamics of the cell cortex during indentation. We investigate the impact of active contraction on the cross-linked actin network for different probe sizes and indentation rates. The essential molecular mechanisms of filament stretching, cross-linking and motor activity, are represented by an active and viscous mechanical continuum. The filaments behave as worm-like chains linked either by passive rigid linkers or by myosin motors. In the first example, the effects of probe size and loading rate are evaluated using the model for an idealized rounded cell shape in which properties are based on the results of parallel-plate rheometry available in the literature. Extreme cases of probe size and indentation rate are taken into account. Afterward, AFM experiments were done by engaging smooth muscle cells with both sharp and spherical probes. By inverse analysis with finite element software, our simulations mimicking the experimental conditions show the model is capable of fitting the AFM data. The results provide spatiotemporal dependence on the size and rate of the mechanical stimuli. The model captures the general features of the cell response. It characterizes the actomyosin cortex as an active solid at short timescales and as a fluid at longer timescales by showing (1) higher levels of contraction in the zones of high curvature; (2) larger indentation forces as the probe size increases; and (3) increase in the apparent modulus with the indentation depth but no dependence on the rate of the mechanical stimuli. The methodology presented in this work can be used to address and predict microstructural dependence on the force generation of living cells, which can contribute to understanding the broad spectrum of results in cell experiments.

Engineered Biomaterial Platforms to Study Fibrosis

Many pathologic conditions lead to the development of tissue scarring and fibrosis, which are characterized by the accumulation of abnormal extracellular matrix (ECM) and changes in tissue mechanical properties. Cells within fibrotic tissues are exposed to dynamic microenvironments that may promote or prolong fibrosis, which makes it difficult to treat. Biomaterials have proved indispensable to better understand how cells sense their extracellular environment and are now being employed to study fibrosis in many tissues. As mechanical testing of tissues becomes more routine and biomaterial tools become more advanced, the impact of biophysical factors in fibrosis are beginning to be understood. Herein, fibrosis from a materials perspective is reviewed, including the role and mechanical properties of ECM components, the spatiotemporal mechanical changes that occur during fibrosis, current biomaterial systems to study fibrosis, and emerging biomaterial systems and tools that can further the understanding of fibrosis initiation and progression. This review concludes by highlighting considerations in promoting wide-spread use of biomaterials for fibrosis investigations and by suggesting future in vivo studies that it is hoped will inspire the development of even more advanced biomaterial systems.

Hierarchically structured hydrogels utilizing multifunctional assembling peptides for 3D cell culture

Approaches for the creation of soft materials, particularly hydrogels, with hierarchical structure are of interest in a variety of applications owing to their unique properties. In the context of tissue mimics, hydrogels with multiscale structures more accurately capture the complexities of tissues within the body (e.g., fibrous collagen-rich microenvironments). However, cytocompatible fabrication of such materials with hierarchical structures and independent control of mechanical and biochemical properties remains challenging and is needed for probing and directing cell-microenvironment interactions for three-dimensional (3D) cell encapsulation and culture applications. To address this, we have designed innovative multifunctional assembling peptides: these unique peptides contain a core block that mimics the structure of collagen for achieving relevant melting temperatures; 'sticky' ends to promote assembly of long fibrils; and a biocompatible reactive handle that is orthogonal to assembly to allow the formation of desired multiscale structures and their subsequent rapid, light-triggered integration within covalently crosslinked synthetic hydrogels. Nano- to micro-fibrils were observed to form in physiologically-relevant aqueous solutions, where both underlying peptide chemical structure and assembly conditions were observed to impact the resulting fibril sizes. These assembled structures were 'locked' into place and integrated as linkers within cell-degradable, bioactive hydrogels formed with photoinitiated thiol-ene 'click' chemistry. Hydrogel compositions were identified for achieving robust mechanical properties like those of soft tissues while also retaining higher ordered structures after photopolymerization. The utility of these innovative materials for 3D cell culture was demonstrated with human mesenchymal stem cells, where cell morphologies reminiscent of responses to assembled native collagen were observed now with a fully synthetic material. Using a bottom-up approach, a new materials platform has been established that combines the advantageous properties of covalent and assembling chemistries for the creation of synthetic hydrogels with controllable nanostructure, mechanical properties, and biochemical content.

Liver Bioreactor Design Issues of Fluid Flow and Zonation, Fibrosis, and Mechanics: A Computational Perspective

Tissue engineering, with the goal of repairing or replacing damaged tissue and organs, has continued to make dramatic science-based advances since its origins in the late 1980’s and early 1990’s. Such advances are always multi-disciplinary in nature, from basic biology and chemistry through physics and mathematics to various engineering and computer fields. This review will focus its attention on two topics critical for tissue engineering liver development: (a) fluid flow, zonation, and drug screening, and (b) biomechanics, tissue stiffness, and fibrosis, all within the context of 3D structures. First, a general overview of various bioreactor designs developed to investigate fluid transport and tissue biomechanics is given. This includes a mention of computational fluid dynamic methods used to optimize and validate these designs. Thereafter, the perspective provided by computer simulations of flow, reactive transport, and biomechanics responses at the scale of the liver lobule and liver tissue is outlined, in addition to how bioreactor-measured properties can be utilized in these models. Here, the fundamental issues of tortuosity and upscaling are highlighted, as well as the role of disease and fibrosis in these issues. Some idealized simulations of the effects of fibrosis on lobule drug transport and mechanics responses are provided to further illustrate these concepts. This review concludes with an outline of some practical applications of tissue engineering advances and how efficient computational upscaling techniques, such as dual continuum modeling, might be used to quantify the transition of bioreactor results to the full liver scale.

Poroelasticity of (bio)polymer networks during compression: theory and experiment

Soft living tissues like cartilage can be considered as biphasic materials comprising a fibrous complex biopolymer network and a viscous background liquid. Here, we show by a combination of experiment and theoretical analysis that both the hydraulic permeability and the elastic properties of (bio)polymer networks can be determined with simple ramp compression experiments in a commercial rheometer. In our approximate closed-form solution of the poroelastic equations of motion, we find the normal force response during compression as a combination of network stress and fluid pressure. Choosing fibrin as a biopolymer model system with controllable pore size, measurements of the full time-dependent normal force during compression are found to be in excellent agreement with the theoretical calculations. The inferred elastic response of large-pore (μm) fibrin networks depends on the strain rate, suggesting a strong interplay between network elasticity and fluid flow. Phenomenologically extending the calculated normal force into the regime of nonlinear elasticity, we find strain-stiffening of small-pore (sub-μm) fibrin networks to occur at an onset average tangential stress at the gel-plate interface that depends on the polymer concentration in a power-law fashion. The inferred permeability of small-pore fibrin networks scales approximately inverse squared with the fibrin concentration, implying with a microscopic cubic lattice model that the number of protofibrils per fibrin fiber cross-section decreases with protein concentration. Our theoretical model provides a new method to obtain the hydraulic permeability and the elastic properties of biopolymer networks and hydrogels with simple compression experiments, and paves the way to study the relation between fluid flow and elasticity in biopolymer networks during dynamical compression.