Deformation of cross-linked semiflexible polymer networks

Random fiber networks with inclusions: The mechanism of reinforcement

The mechanical behavior of athermal random fiber networks embedding particulate inclusions is studied in this work. Composites in which the filler size is comparable with the mean segment length of the network are considered. Inclusions are randomly distributed in the network at various volume fractions, and cases in which fibers are rigidly bonded to fillers and in which no such bonding is imposed are studied separately. In the presence of inclusions, the small strain modulus increases, while the ability of the network to strain stiffen decreases relative to the unfilled network case. The reinforcement induced by fillers is most pronounced in sparse networks of floppier filaments that deform in the bending-dominated mode in the unfilled state. As the unfilled network density or the bending stiffness of fibers increases, the effect of filling diminishes rapidly. Fillers lead to a transition from the soft, bending-dominated, to the stiffer, stretching-dominated, deformation mode of the network, a transition which is primarily responsible for the observed overall reinforcement. The confinement, i.e., the restriction on network kinematics imposed by fillers, causes this transition. These results provide a justification for the observed difference in reinforcement obtained in sparsely versus densely cross-linked networks at a given filling fraction and provide guidance for the further development of network-based materials.

Mechanical behavior of nonwoven non-crosslinked fibrous mats with adhesion and friction

We present a study of the mechanical behavior of planar fibrous mats stabilized by inter-fiber adhesion. Fibers of various degrees of tortuosity and of infinite and finite length are considered in separate models. Fibers are randomly distributed, are not cross-linked, and interact through adhesion and friction. The variation of structural parameters such as the mat thickness and the mean segment length between contacts along given fibers with the strength of adhesion is determined. These systems are largely dissipative in that most of the work performed during deformation is dissipated frictionally and only a small fraction is stored as strain energy. The response of the mats to tensile loading has three regimes: a short elastic regime in which no sliding at contacts is observed, a well-defined sliding regime characterized by strain hardening, and a rapid stiffening regime at larger strains. The third regime is due to the formation of stress paths after the fiber tortuosity is pulled out and is absent in mats of finite length fibers. Networks of finite length fibers lose stability during the second regime of deformation. The scaling of the yield stress, which characterizes the transition between the first and the second regimes, and of the second regime's strain hardening modulus, with system parameters such as the strength of adhesion and friction and the degree of fiber tortuosity are determined. The strength of mats of finite length fibers is also determined as a function of network parameters. These results are expected to become useful in the design of electrospun mats and other planar fibrous non-cross-linked networks.

Stress-stabilized subisostatic fiber networks in a ropelike limit

The mechanics of disordered fibrous networks such as those that make up the extracellular matrix are strongly dependent on the local connectivity or coordination number. For biopolymer networks this coordination number is typically between 3 and 4. Such networks are sub-isostatic and linearly unstable to deformation with only central force interactions, but exhibit a mechanical phase transition between floppy and rigid states under strain. The introduction of weak bending interactions stabilizes these networks and suppresses the critical signatures of this transition. We show that applying external stress can also stabilize subisostatic networks with only tensile central force interactions, i.e., a ropelike potential. Moreover, we find that the linear shear modulus shows a power-law scaling with the external normal stress, with a non-mean-field exponent. For networks with finite bending rigidity, we find that the critical stain shifts to lower values under prestress.

Cohesive and adhesive properties of crosslinked semiflexible biopolymer networks

Biomolecular semiflexible polymer networks with persistence lengths well above those of single polymeric chains serve important structural and adhesive roles in biology, biomaterials, food science and many other fields. While relationships between the structure and viscoelasticity of semiflexible polymer networks have been previously investigated, it remains challenging to systematically relate fibril and network properties to cohesive and adhesive properties that govern the function of these materials. To address this issue, here we utilize coarse-grained molecular dynamics simulations to thoroughly elucidate how the work of adhesion of a semiflexible polymer network to a surface depends on crosslink density and fibril persistence length. Two emergent characteristics of the network are its elasticity and its interfacial energy with the surface. Stiff networks that are either highly crosslinked or have high persistence length fibrils tend to have lower interfacial energy, and consequently, lower work of adhesion. For lightly crosslinked networks with flexible fibrils, considerable strain energy must be stored within the adhesive during detachment, which creates an additional penalty to detachment. Increasing persistence length while keeping crosslink density constant leads to porous, low density networks, leading to an optimal fibril persistence length at which maximum work of adhesion per mass density is attained for a given crosslink density. For any given fibril persistence length, increasing crosslink density has a slightly negative effect on network mass density and interfacial energy. A critical crosslink density is found, below which the networks have no significant load-bearing capacity. Lightly crosslinked networks above this threshold absorb more strain energy during desorption and consequently possess greater work of adhesion. The conflict between mass density and stiffness results in a non-monotonic trend between the ratio of work of adhesion to interfacial energy and persistence length. These findings provide physical insight into the adhesive mechanisms of biomaterials based on crosslinked semiflexible polymer networks, and reveal important design guidelines for bio-adhesives.

Theory of applying shear strains from boundary walls: Linear response in glasses

We construct a linear response theory of applying shear deformations from boundary walls in the film geometry in Kubo's theoretical scheme. Our method is applicable to any solids and fluids. For glasses, we assume quasi-equilibrium around a fixed inherent state. Then, we obtain linear-response expressions for any variables including the stress and the particle displacements, even though the glass interior is elastically inhomogeneous. In particular, the shear modulus can be expressed in terms of the correlations between the interior stress and the forces from the walls. It can also be expressed in terms of the inter-particle correlations, as has been shown in the previous literature. Our stress relaxation function includes the effect of the boundary walls and can be used for inhomogeneous flow response. We show the presence of long-ranged, long-lived correlations among the fluctuations of the forces from the walls and the displacements of all the particles in the cell. We confirm these theoretical results numerically in a two-dimensional model glass. As an application, we describe emission and propagation of transverse sounds after boundary wall motions using these time-correlation functions. We also find resonant sound amplification when the frequency of an oscillatory shear approaches that of the first transverse sound mode.

Normal stress anisotropy and marginal stability in athermal elastic networks

Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component σzz on long (compressible) timescales, crossing over to the first normal stress difference, N1 = σxx - σzz at short (incompressible) times. While the behavior of N1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain.

Mimicking Active Biopolymer Networks with a Synthetic Hydrogel

Stiffening due to internal stress generation is of paramount importance in living systems and is the foundation for many biomechanical processes. For example, cells stiffen their surrounding matrix by pulling on collagen and fibrin fibers. At the subcellular level, molecular motors prompt fluidization and actively stiffen the cytoskeleton by sliding polar actin filaments in opposite directions. Here, we demonstrate that chemical cross-linking of a fibrous matrix of synthetic semiflexible polymers with thermoresponsive poly( N-isopropylacrylamide) (PNIPAM) produces internal stress by induction of a coil-to-globule transition upon crossing the lower critical solution temperature of PNIPAM, resulting in a macroscopic stiffening response that spans more than 3 orders of magnitude in modulus. The forces generated through collapsing PNIPAM are sufficient to drive a fluid material into a stiff gel within a few seconds. Moreover, rigidified networks dramatically stiffen in response to applied shear stress featuring power law rheology with exponents that match those of reconstituted collagen and actomyosin networks prestressed by molecular motors. This concept holds potential for the rational design of synthetic materials that are fluid at room temperature and rapidly rigidify at body temperature to form hydrogels mechanically and structurally akin to cells and tissues.

Inducing nematic ordering of cellulose nanofibers using osmotic dehydration

The formation of nematically-ordered cellulose nanofiber (CNF) suspensions with an order parameter f_max≈ 0.8 is studied by polarized optical microscopy, small-angle X-ray scattering (SAXS), and rheological measurements as a function of CNF concentration. The wide range of CNF concentrations, from 0.5 wt% to 4.9 wt%, is obtained using osmotic dehydration. The rheological measurements show a strong entangled network over all the concentration range whereas SAXS measurements indicate that at concentrations >1.05 wt% the CNF suspension crosses an isotropic-anisotropic transition that is accompanied by a dramatic increase of the optical birefringence. The resulting nanostructures are modelled as mass fractal structures that converge into co-existing nematically-ordered regions and network-like regions where the correlation distances decrease with concentration. The use of rapid, upscalable osmotic dehydration is an effective method to increase the concentration of CNF suspensions while partly circumventing the gel/glass formation. The facile formation of highly ordered fibers can result in materials with interesting macroscopic properties.

Fabrication and characterization of thiol-triacrylate polymer via Michael addition reaction for biomedical applications

Thiol-acrylate polymers have therapeutic potential as biocompatible scaffolds for bone tissue regeneration. Synthesis of a novel cyto-compatible and biodegradable polymer composed of trimethylolpropane ethoxylate triacrylate-trimethylolpropane tris (3-mercaptopropionate) (TMPeTA-TMPTMP) using a simple amine-catalyzed Michael addition reaction is reported in this study. This study explores the impact of molecular weight and crosslink density on the cyto-compatibility of human adipose derived mesenchymal stem cells. Eight groups were prepared with two different average molecular weights of trimethylolpropane ethoxylate triacrylate (TMPeTA 692 and 912) and four different concentrations of diethylamine (DEA) as catalyst. The materials were physically characterized by mechanical testing, wettability, mass loss, protein adsorption and surface topography. Cyto-compatibility of the polymeric substrates was evaluated by LIVE/DEAD staining^® and DNA content assay of cultured human adipose derived stem cells (hASCs) on the samples over over days. Surface topography studies revealed that TMPeTA (692) samples have island pattern features whereas TMPeTA (912) polymers showed pitted surfaces. Water contact angle results showed a significant difference between TMPeTA (692) and TMPeTA (912) monomers with the same DEA concentration. Decreased protein adsorption was observed on TMPeTA (912) -16% DEA compared to other groups. Fluorescent microscopy also showed distinct hASCs attachment behavior between TMPeTA (692) and TMPeTA (912), which is due to their different surface topography, protein adsorption and wettability. Our finding suggested that this thiol-acrylate based polymer is a versatile, cyto-compatible material for tissue engineering applications with tunable cell attachment property based on surface characteristics.

RIGID GRAPH COMPRESSION: MOTIF-BASED RIGIDITY ANALYSIS FOR DISORDERED FIBER NETWORKS

Using particle-scale models to accurately describe property enhancements and phase transitions in macroscopic behavior is a major engineering challenge in composite materials science. To address some of these challenges, we use the graph theoretic property of rigidity to model mechanical reinforcement in composites with stiff rod-like particles. We develop an efficient algorithmic approach called rigid graph compression (RGC) to describe the transition from floppy to rigid in disordered fiber networks ("rod-hinge systems"), which form the reinforcing phase in many composite systems. To establish RGC on a firm theoretical foundation, we adapt rigidity matroid theory to identify primitive topological network motifs that serve as rules for composing interacting rigid particles into larger rigid components. This approach is computationally efficient and stable, because RGC requires only topological information about rod interactions (encoded by a sparse unweighted network) rather than geometrical details such as rod locations or pairwise distances (as required in rigidity matroid theory). We conduct numerical experiments on simulated two-dimensional rod-hinge systems to demonstrate that RGC closely approximates the rigidity percolation threshold for such systems, through comparison with the pebble game algorithm (which is exact in two dimensions). Importantly, whereas the pebble game is derived from Laman's condition and is only valid in two dimensions, the RGC approach naturally extends to higher dimensions.

Normal Stresses, Contraction, and Stiffening in Sheared Elastic Networks

When elastic solids are sheared, a nonlinear effect named after Poynting gives rise to normal stresses or changes in volume. We provide a novel relation between the Poynting effect and the microscopic Grüneisen parameter, which quantifies how stretching shifts vibrational modes. By applying this relation to random spring networks, a minimal model for, e.g., biopolymer gels and solid foams, we find that networks contract or develop tension because they vibrate faster when stretched. The amplitude of the Poynting effect is sensitive to the network's linear elastic moduli, which can be tuned via its preparation protocol and connectivity. Finally, we show that the Poynting effect can be used to predict the finite strain scale where the material stiffens under shear.

Mechanical hysteresis in actin networks

Understanding the response of complex materials to external force is central to fields ranging from materials science to biology. Here, we describe a novel type of mechanical adaptation in cross-linked networks of F-actin, a ubiquitous protein found in eukaryotic cells. We show that shear stress changes the network's nonlinear mechanical response even long after that stress is removed. The duration, magnitude and direction of forcing history all change this mechanical response. While the mechanical hysteresis is long-lived, it can be simply erased by force application in the opposite direction. We further show that the observed mechanical adaptation is consistent with stress-dependent changes in the nematic order of the constituent filaments. Thus, this mechanical hysteresis arises from the changes in non-linear response that originates from stress-induced changes to filament orientation. This demonstrates that F-actin networks can exhibit analog read-write mechanical hysteretic properties, which can be used for adaptation to mechanical stimuli.

Topological Edge Floppy Modes in Disordered Fiber Networks

Disordered fiber networks are ubiquitous in a broad range of natural (e.g., cytoskeleton) and manmade (e.g., aerogels) materials. In this Letter, we discuss the emergence of topological floppy edge modes in two-dimensional fiber networks as a result of deformation or active driving. It is known that a network of straight fibers exhibits bulk floppy modes which only bend the fibers without stretching them. We find that, interestingly, with a perturbation in geometry, these bulk modes evolve into edge modes. We introduce a topological index for these edge modes and discuss their implications in biology.

Poisson's Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations

Connective tissue mechanics is highly nonlinear, exhibits a strong Poisson's effect, and is associated with significant collagen fiber re-arrangement. Although the general features of the stress-strain behavior have been discussed extensively, the Poisson's effect received less attention. In general, the relationship between the microscopic fiber network mechanics and the macroscopic experimental observations remains poorly defined. The objective of the present work is to provide additional insight into this relationship. To this end, results from models of random collagen networks are compared with experimental data on reconstructed collagen gels, mouse skin dermis, and the human amnion. Attention is devoted to the mechanism leading to the large Poisson's effect observed in experiments. The results indicate that the incremental Poisson's contraction is directly related to preferential collagen orientation. The experimentally observed downturn of the incremental Poisson's ratio at larger strains is associated with the confining effect of fibers transverse to the loading direction and contributing little to load bearing. The rate of collagen orientation increases at small strains, reaches a maximum, and decreases at larger strains. The peak in this curve is associated with the transition of the network deformation from bending dominated, at small strains, to axially dominated, at larger strains. The effect of fiber tortuosity on network mechanics is also discussed, and a comparison of biaxial and uniaxial loading responses is performed.

Hindrances to precise recovery of cellular forces in fibrous biopolymer networks

How cells move through the three-dimensional extracellular matrix (ECM) is of increasing interest in attempts to understand important biological processes such as cancer metastasis. Just as in motion on flat surfaces, it is expected that experimental measurements of cell-generated forces will provide valuable information for uncovering the mechanisms of cell migration. However, the recovery of forces in fibrous biopolymer networks may suffer from large errors. Here, within the framework of lattice-based models, we explore possible issues in force recovery by solving the inverse problem: how can one determine the forces cells exert to their surroundings from the deformation of the ECM? Our results indicate that irregular cell traction patterns, the uncertainty of local fiber stiffness, the non-affine nature of ECM deformations and inadequate knowledge of network topology will all prevent the precise force determination. At the end, we discuss possible ways of overcoming these difficulties.

Filament turnover tunes both force generation and dissipation to control long-range flows in a model actomyosin cortex

Actomyosin-based cortical flow is a fundamental engine for cellular morphogenesis. Cortical flows are generated by cross-linked networks of actin filaments and myosin motors, in which active stress produced by motor activity is opposed by passive resistance to network deformation. Continuous flow requires local remodeling through crosslink unbinding and and/or filament disassembly. But how local remodeling tunes stress production and dissipation, and how this in turn shapes long range flow, remains poorly understood. Here, we study a computational model for a cross-linked network with active motors based on minimal requirements for production and dissipation of contractile stress: Asymmetric filament compliance, spatial heterogeneity of motor activity, reversible cross-links and filament turnover. We characterize how the production and dissipation of network stress depend, individually, on cross-link dynamics and filament turnover, and how these dependencies combine to determine overall rates of cortical flow. Our analysis predicts that filament turnover is required to maintain active stress against external resistance and steady state flow in response to external stress. Steady state stress increases with filament lifetime up to a characteristic time τm, then decreases with lifetime above τm. Effective viscosity increases with filament lifetime up to a characteristic time τc, and then becomes independent of filament lifetime and sharply dependent on crosslink dynamics. These individual dependencies of active stress and effective viscosity define multiple regimes of steady state flow. In particular our model predicts that when filament lifetimes are shorter than both τc and τm, the dependencies of effective viscosity and steady state stress on filament turnover cancel one another, such that flow speed is insensitive to filament turnover, and shows a simple dependence on motor activity and crosslink dynamics. These results provide a framework for understanding how animal cells tune cortical flow through local control of network remodeling.

Large Amplitude Oscillatory Shear Rheology of Living Fibroblasts: Path-Dependent Steady States

Mechanical properties of biological cells play a role in cell locomotion, embryonic tissue formation, and tumor migration among many other processes. Cells exhibit a complex nonlinear response to mechanical cues that is not understood. Cells may stiffen as well as soften, depending on the exact type of stimulus. Here we apply large-amplitude oscillatory shear to a monolayer of separated fibroblast cells suspended between two plates. Although we apply identical steady-state excitations, in response we observe different typical regimes that exhibit cell softening or cell stiffening to varying degrees. This degeneracy of the cell response can be linked to the initial paths that the instrument takes to go from cell rest to steady state. A model of cross-linked, force-bearing filaments submitted to steady-state excitation renders the different observed regimes with minor changes in parameters if the filaments are permitted to self-organize and form different spatially organized structures. We suggest that rather than a complex viscoelastic or plastic response, the different observed regimes reflect the emergence of different steady-state cytoskeletal conformations. A high sensitivity of the cytoskeletal rheology and structure to minor changes in parameters or initial conditions enables a cell to respond to mechanical requirements quickly and in various ways with only minor biochemical intervention. Probing path-dependent rheological changes constitutes a possibly very sensitive assessment of the cell cytoskeleton as a possible tool for medical diagnosis. Our observations show that the memory of subtle differences in earlier deformation paths must be taken into account when deciphering the cell mechanical response to large-amplitude deformations.

Nonlinear Actin Deformations Lead to Network Stiffening, Yielding, and Nonuniform Stress Propagation

We use optical tweezers microrheology and fluorescence microscopy to apply nonlinear microscale strains to entangled and cross-linked actin networks, and measure the resulting stress and actin filament deformations. We couple nonlinear stress response and relaxation to the velocities and displacements of individual fluorescent-labeled actin segments, at varying times throughout the strain and varying distances from the strain path, to determine the underlying molecular dynamics that give rise to the debated nonlinear response and stress propagation of cross-linked and entangled actin networks at the microscale. We show that initial stress stiffening arises from acceleration of strained filaments due to molecular extension along the strain, while softening and yielding is coupled to filament deceleration, halting, and recoil. We also demonstrate a surprising nonmonotonic dependence of filament deformation on cross-linker concentration. Namely, networks with no cross-links or substantial cross-links both exhibit fast initial filament velocities and reduced molecular recoil while intermediate cross-linker concentrations display reduced velocities and increased recoil. We show that these collective results are due to a balance of network elasticity and force-induced cross-linker unbinding and rebinding. We further show that cross-links dominate entanglement dynamics when the length between cross-linkers becomes smaller than the length between entanglements. In accord with recent simulations, we demonstrate that post-strain stress can be long-lived in cross-linked networks by distributing stress to a small fraction of highly strained connected filaments that span the network and sustain the load, thereby allowing the rest of the network to recoil and relax.

Force percolation of contractile active gels

Living systems provide a paradigmatic example of active soft matter. Cells and tissues comprise viscoelastic materials that exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, an active gel of semiflexible filaments, crosslinks, and molecular motors inside cells. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (∼nN and greater) on cellular, tissue, and organismal length scales. Here we review experimental and theoretical studies on contractile active gels composed of actin filaments and myosin motors. Unlike other active soft matter systems, which tend to form ordered patterns, actin-myosin systems exhibit a generic tendency to contract. Experimental studies of reconstituted actin-myosin model systems have long suggested that a mechanical interplay between motor activity and the network's connectivity governs this contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, extended to include effects of motor activity on the network connectivity. Based on concepts from percolation theory, we propose a state diagram that unites a large body of experimental observations. This framework provides valuable insights into the mechanisms that drive cellular shape changes and also provides design principles for synthetic active materials.

Nonlinear mechanics of hybrid polymer networks that mimic the complex mechanical environment of cells

The mechanical properties of cells and the extracellular environment they reside in are governed by a complex interplay of biopolymers. These biopolymers, which possess a wide range of stiffnesses, self-assemble into fibrous composite networks such as the cytoskeleton and extracellular matrix. They interact with each other both physically and chemically to create a highly responsive and adaptive mechanical environment that stiffens when stressed or strained. Here we show that hybrid networks of a synthetic mimic of biological networks and either stiff, flexible and semi-flexible components, even very low concentrations of these added components, strongly affect the network stiffness and/or its strain-responsive character. The stiffness (persistence length) of the second network, its concentration and the interaction between the components are all parameters that can be used to tune the mechanics of the hybrids. The equivalence of these hybrids with biological composites is striking.

Criticality and mechanical enhancement in composite fiber networks

Many biological materials consist of sparse networks of disordered fibers, embedded in a soft elastic matrix. The interplay between rigid and soft elements in such composite networks leads to mechanical properties that can go far beyond the sum of those of the constituents. Here we present lattice-based simulations to unravel the microscopic origins of this mechanical synergy. We show that the competition between fiber stretching and bending and elastic deformations of the matrix gives rise to distinct mechanical regimes, with phase transitions between them that are characterized by critical behavior and diverging strain fluctuations and with different mechanisms leading to mechanical enhancement.

Mechanical Cell-Cell Communication in Fibrous Networks: The Importance of Network Geometry

Cells contracting in extracellular matrix (ECM) can transmit stress over long distances, communicating their position and orientation to cells many tens of micrometres away. Such phenomena are not observed when cells are seeded on substrates with linear elastic properties, such as polyacrylamide (PA) gel. The ability for fibrous substrates to support far reaching stress and strain fields has implications for many physiological processes, while the mechanical properties of ECM are central to several pathological processes, including tumour invasion and fibrosis. Theoretical models have investigated the properties of ECM in a variety of network geometries. However, the effects of network architecture on mechanical cell-cell communication have received little attention. This work investigates the effects of geometry on network mechanics, and thus the ability for cells to communicate mechanically through different networks. Cell-derived displacement fields are quantified for various network geometries while controlling for network topology, cross-link density and micromechanical properties. We find that the heterogeneity of response, fibre alignment, and substrate displacement fields are sensitive to network choice. Further, we show that certain geometries support mechanical communication over longer distances than others. As such, we predict that the choice of network geometry is important in fundamental modelling of cell-cell interactions in fibrous substrates, as well as in experimental settings, where mechanical signalling at the cellular scale plays an important role. This work thus informs the construction of theoretical models for substrate mechanics and experimental explorations of mechanical cell-cell communication.

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality [Sharma et al., Nature Phys. 12, 584 (2016)1745-247310.1038/nphys3628]. The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative characterization of critical behavior in subisostatic networks. Using finite-size scaling we demonstrate the divergence of strain fluctuations in the network at well-defined critical strain. We show that the characteristic strain corresponding to the onset of strain stiffening is distinct from but related to this critical strain in a way that depends on critical exponents. We confirm this prediction experimentally for collagen networks. Moreover, we find that the apparent critical exponents are largely independent of the spatial dimensionality. With subisostaticity as the only required condition, strain-driven criticality is expected to be a general feature of biologically relevant fibrous networks.

Inferring spatial variations of microstructural properties from macroscopic mechanical response

Disease alters tissue microstructure, which in turn affects the macroscopic mechanical properties of tissue. In elasticity imaging, the macroscopic response is measured and is used to infer the spatial distribution of the elastic constitutive parameters. When an empirical constitutive model is used, these parameters cannot be linked to the microstructure. However, when the constitutive model is derived from a microstructural representation of the material, it allows for the possibility of inferring the local averages of the spatial distribution of the microstructural parameters. This idea forms the basis of this study. In particular, we first derive a constitutive model by homogenizing the mechanical response of a network of elastic, tortuous fibers. Thereafter, we use this model in an inverse problem to determine the spatial distribution of the microstructural parameters. We solve the inverse problem as a constrained minimization problem and develop efficient methods for solving it. We apply these methods to displacement fields obtained by deforming gelatin-agar co-gels and determine the spatial distribution of agar concentration and fiber tortuosity, thereby demonstrating that it is possible to image local averages of microstructural parameters from macroscopic measurements of deformation.

Automatic control: the vertebral column of dogfish sharks behaves as a continuously variable transmission with smoothly shifting functions

During swimming in dogfish sharks, Squalus acanthias, both the intervertebral joints and the vertebral centra undergo significant strain. To investigate this system, unique among vertebrates, we cyclically bent isolated segments of 10 vertebrae and nine joints. For the first time in the biomechanics of fish vertebral columns, we simultaneously characterized non-linear elasticity and viscosity throughout the bending oscillation, extending recently proposed techniques for large-amplitude oscillatory shear (LAOS) characterization to large-amplitude oscillatory bending (LAOB). The vertebral column segments behave as non-linear viscoelastic springs. Elastic properties dominate for all frequencies and curvatures tested, increasing as either variable increases. Non-linearities within a bending cycle are most in evidence at the highest frequency, 2.0 Hz, and curvature, 5 m−1. Viscous bending properties are greatest at low frequencies and high curvatures, with non-linear effects occurring at all frequencies and curvatures. The range of mechanical behaviors includes that of springs and brakes, with smooth transitions between them that allow for continuously variable power transmission by the vertebral column to assist in the mechanics of undulatory propulsion.

Nonlinear elasticity of semiflexible filament networks

We develop a continuum theory for equilibrium elasticity of a network of crosslinked semiflexible filaments, spanning the full range between flexible entropy-driven chains to stiff athermal rods. We choose the 3-chain constitutive model of network elasticity over several plausible candidates, and derive analytical expressions for the elastic energy at arbitrary strain, with the corresponding stress-strain relationship. The theory fits well to a wide range of experimental data on simple shear in different filament networks, quantitatively matching the differential shear modulus variation with stress, with only two adjustable parameters (which represent the filament stiffness and the pre-tension in the network, respectively). The general theory accurately describes the crossover between the positive and negative Poynting effect (normal stress on imposed shear) on increasing the stiffness of filaments forming the network. We discuss the network stability (the point of marginal rigidity) and the phenomenon of tensegrity, showing that filament pre-tension on crosslinking into the network determines the magnitude of linear modulus G0.

Viscoelasticity of reversibly crosslinked networks of semiflexible polymers

We present a theoretical framework for the linear and nonlinear viscoelastic properties of reversibly crosslinked networks of semiflexible polymers. In contrast to affine models where network strain couples to the polymer end-to-end distance, in our model strain rather serves to locally distort the network structure. This induces bending modes in the polymer filaments, the properties of which are slaved to the surrounding network structure. Specifically, we investigate the frequency-dependent linear rheology, in particular in combination with crosslink binding-unbinding processes. We also develop schematic extensions to describe the nonlinear response during creep measurements as well as during constant strain-rate ramps.

Elasticity of fibrous networks under uniaxial prestress

We present theoretical and experimental studies of the elastic response of fibrous networks subjected to uniaxial strain. Uniaxial compression or extension is applied to extracellular networks of fibrin and collagen using a shear rheometer with free water in/outflow. Both uniaxial stress and the network shear modulus are measured. Prior work [van Oosten, et al., Sci. Rep., 2015, 6, 19270] has shown softening/stiffening of these networks under compression/extension, together with a nonlinear response to shear, but the origin of such behaviour remains poorly understood. Here, we study how uniaxial strain influences the nonlinear mechanics of fibrous networks. Using a computational network model with bendable and stretchable fibres, we show that the softening/stiffening behaviour can be understood for fixed lateral boundaries in 2D and 3D networks with comparable average connectivities to the experimental extracellular networks. Moreover, we show that the onset of stiffening depends strongly on the imposed uniaxial strain. Our study highlights the importance of both uniaxial strain and boundary conditions in determining the mechanical response of hydrogels.

Flexibility and rigidity of cross-linked Straight Fibrils under axial motion constraints

The Straight Fibrils are stiff rod-like filaments and play a significant role in cellular processes as structural stability and intracellular transport. Introducing a 3D mechanical model for the motion of braced cylindrical fibrils under axial motion constraint; we provide some mechanism and a graph theoretical model for fibril structures and give the characterization of the flexibility and the rigidity of this bar-and-joint spatial framework. The connectedness and the circuit of the bracing graph characterize the flexibility of these structures. In this paper, we focus on the kinematical properties of hierarchical levels of fibrils and evaluate the number of the bracing elements for the rigidity and its computational complexity. The presented model is a good characterization of the frameworks of bio-fibrils such as microtubules, cellulose, which inspired this work.

Importance of non-affine viscoelastic response in disordered fibre networks

Disordered fibre networks are ubiquitous in nature and have a wide range of industrial applications as novel biomaterials. Predicting their viscoelastic response is straightforward for affine deformations that are uniform over all length scales, but when affinity fails, as has been observed experimentally, modelling becomes challenging. Here we present a numerical methodology, related to an existing framework for amorphous packings, to predict the steady-state viscoelastic spectra and degree of affinity for disordered fibre networks driven at arbitrary frequencies. Applying this method to a peptide gel model reveals a monotonic increase of the shear modulus as the soft, non-affine normal modes are successively suppressed as the driving frequency increases. In addition to being dominated by fibril bending, these low frequency network modes are also shown to be delocalised. The presented methodology provides insights into the importance of non-affinity in the viscoelastic response of peptide gels, and is easily extendible to all types of fibre networks.

Elasticity of randomly diluted honeycomb and diamond lattices with bending forces

We use numerical simulations and an effective-medium theory to study the rigidity percolation transition of the honeycomb and diamond lattices when weak bond-bending forces are included. We use a rotationally invariant bond-bending potential, which, in contrast to the Keating potential, does not involve any stretching. As a result, the bulk modulus does not depend on the bending stiffness κ. We obtain scaling functions for the behavior of some elastic moduli in the limits of small ΔP = 1-P, and small δP = P-Pc, where P is an occupation probability of each bond, and Pc is the critical probability at which rigidity percolation occurs. We find good quantitative agreement between effective-medium theory and simulations for both lattices for P close to one.

Nonaffine behavior of three-dimensional semiflexible polymer networks

Three-dimensional semiflexible polymer networks are the structural building blocks of various biological and structural materials. Previous studies have primarily used two-dimensional models for understanding the behavior of these networks. In this paper, we develop a three-dimensional nonaffinity measure capable of providing direct comparison with continuum level homogenized quantities, i.e., strain field. The proposed nonaffinity measure is capable of capturing possible anisotropic microstructures of the filamentous networks. This strain-based nonaffinity measure is used to probe the mechanical behavior at different length scales and investigate the effects of network mechanical and microstructural properties. Specifically, it is found that although all nonaffinity measure components have a power-law variation with the probing length scale, the degree of nonaffinity decreases with increasing the length scale of observation. Furthermore, the amount of nonaffinity is a function of network fiber density, bending stiffness of the constituent filaments, and the network architecture. Finally, it is found that the two power-law scaling regimes previously reported for two-dimensional systems do not appear in three-dimensional networks. Also, unlike two-dimensional models, the exponent of the power-law relation depends weakly on the density of the three-dimensional networks.

Viscoelastic and elastomeric active matter: Linear instability and nonlinear dynamics

We consider a continuum model of active viscoelastic matter, whereby an active nematic liquid crystal is coupled to a minimal model of polymer dynamics with a viscoelastic relaxation time τ(C). To explore the resulting interplay between active and polymeric dynamics, we first generalize a linear stability analysis (from earlier studies without polymer) to derive criteria for the onset of spontaneous heterogeneous flows (strain rate) and/or deformations (strain). We find two modes of instability. The first is a viscous mode, associated with strain rate perturbations. It dominates for relatively small values of τ(C) and is a simple generalization of the instability known previously without polymer. The second is an elastomeric mode, associated with strain perturbations, which dominates at large τ(C) and persists even as τ(C)→∞. We explore the dynamical states to which these instabilities lead by means of direct numerical simulations. These reveal oscillatory shear-banded states in one dimension and activity-driven turbulence in two dimensions even in the elastomeric limit τ(C)→∞. Adding polymer can also have calming effects, increasing the net throughput of spontaneous flow along a channel in a type of drag reduction. The effect of including strong antagonistic coupling between the nematic and polymer is examined numerically, revealing a rich array of spontaneously flowing states.

Response of biopolymer networks governed by the physical properties of cross-linking molecules

In this study, we examine how the physical properties of cross-linking molecules affect the bulk response of bio-filament networks, an outstanding question in the study of biological gels and the cytoskeleton. We show that the stress-strain relationship of such networks typically undergoes linear increase - strain hardening - stress serration - total fracture transitions due to the interplay between the bending and stretching of individual filaments and the deformation and breakage of cross-linkers. Interestingly, the apparent network modulus is found to scale with the linear and rotational stiffness of the crosslinks to a power exponent of 0.78 and 0.13, respectively. In addition, the network fracture energy will reach its minimum at intermediate rotational compliance values, reflecting the fact that most of the strain energy will be stored in the distorted filaments with rigid cross-linkers while the imposed deformation will be "evenly" distributed among significantly more crosslinking molecules with high rotational compliance.

Nonlinear Mechanics of Athermal Branched Biopolymer Networks

Naturally occurring biopolymers such as collagen and actin form branched fibrous networks. The average connectivity in branched networks is generally below the isostatic threshold at which central force interactions marginally stabilize the network. In the submarginal regime, for connectivity below this threshold, such networks are unstable toward small deformations unless stabilized by additional interactions such as bending. Here we perform a numerical study on the elastic behavior of such networks. We show that the nonlinear mechanics of branched networks is qualitatively similar to that of filamentous networks with freely hinged cross-links. In agreement with a recent theoretical study,1 we find that branched networks also exhibit nonlinear mechanics consistent with athermal critical phenomena controlled by strain. We obtain the critical exponents capturing the nonlinear elastic behavior near the critical point by performing scaling analysis of the stiffening curves. We find that the exponents evolve with the connectivity in the network. We show that the nonlinear mechanics of disordered networks, independent of the detailed microstructure, can be characterized by a strain-driven second-order phase transition, and that the primary quantitative differences among different architectures are in the critical exponents describing the transition.

Softening in Random Networks of Non-Identical Beams

Random fiber networks are assemblies of elastic elements connected in random configurations. They are used as models for a broad range of fibrous materials including biopolymer gels and synthetic nonwovens. Although the mechanics of networks made from the same type of fibers has been studied extensively, the behavior of composite systems of fibers with different properties has received less attention. In this work we numerically and theoretically study random networks of beams and springs of different mechanical properties. We observe that the overall network stiffness decreases on average as the variability of fiber stiffness increases, at constant mean fiber stiffness. Numerical results and analytical arguments show that for small variabilities in fiber stiffness the amount of network softening scales linearly with the variance of the fiber stiffness distribution. This result holds for any beam structure and is expected to apply to a broad range of materials including cellular solids.

Virtual design of electrospun-like gelatin scaffolds: the effect of three-dimensional fibre orientation on elasticity behaviour

Remarkable mechanical performance of biological tissues is explained by a hierarchical fibrous structure. Designing materials that have similar properties is challenging because of the need to assess complex deformation mechanisms. In order to shed more light on architectural possibilities of biopolymer fibrous networks, we propose a numerical study that relates the fibre arrangement to the elastic modulus of a gelatin scaffold obtained using electrospinning. The adopted approach is based on the virtual designing of scaffolds using all possible combinations of Euler angles that define fibre orientations including preferable alignment. The generated networks are converted into a finite element model and the predicted elastic behaviour is examined. Predictions show that the fibre alignment achieved experimentally in biopolymer fibrous networks is for most of the fibres exhibiting an orthotropic behaviour. Some particular combinations of Euler angles allow transverse isotropic architectures while only limited cases are isotropic. A large sensitivity of Young's moduli to Euler angles is achieved describing multiple scenarios of independent anisotropic behaviours. An anisotropy ratio of the elastic behaviour is suggested based on a suitable combination of elastic moduli. Such a ratio exhibits a wide variation depending on individual and coupled effects of Euler angles. The finite element model predicts 2D, 3D and 4D maps representing all possible configurations of fibre alignment and their consequences on elastic behaviour. The predicted fibre orientation representing the observed anisotropic behaviour of electrospun gelatin networks demonstrates unbalanced contributions of in-plane and out-of plane fibres for a large range of processing conditions.

Elastic regimes of subisostatic athermal fiber networks

Athermal models of disordered fibrous networks are highly useful for studying the mechanics of elastic networks composed of stiff biopolymers. The underlying network architecture is a key aspect that can affect the elastic properties of these systems, which include rich linear and nonlinear elasticity. Existing computational approaches have focused on both lattice-based and off-lattice networks obtained from the random placement of rods. It is not obvious, a priori, whether the two architectures have fundamentally similar or different mechanics. If they are different, it is not clear which of these represents a better model for biological networks. Here, we show that both approaches are essentially equivalent for the same network connectivity, provided the networks are subisostatic with respect to central force interactions. Moreover, for a given subisostatic connectivity, we even find that lattice-based networks in both two and three dimensions exhibit nearly identical nonlinear elastic response. We provide a description of the linear mechanics for both architectures in terms of a scaling function. We also show that the nonlinear regime is dominated by fiber bending and that stiffening originates from the stabilization of subisostatic networks by stress. We propose a generalized relation for this regime in terms of the self-generated normal stresses that develop under deformation. Different network architectures have different susceptibilities to the normal stress but essentially exhibit the same nonlinear mechanics. Such a stiffening mechanism has been shown to successfully capture the nonlinear mechanics of collagen networks.

Multi-step control over self-assembled hydrogels of peptide-derived building blocks and a polymeric cross-linker

We present a detailed study of self-assembled hydrogels of bundled and cross-linked networks consisting of positively charged amyloid-like nanofibers and a triblock copolymer with negatively charged end blocks as a cross-linker. In a first step small oligopeptides self-assemble into macrocycles which are held together by reversible disulfide bonds. Interactions between the peptides cause the macrocycles to assemble into nanofibers, which form a reversible hydrogel. The physical properties of the hydrogel are tuned using various methods such as control over the fibre length, addition of a cross-linking copolymer, and addition of salt. We establish a relationship between the bulk mechanical properties, the properties of the individual fibers and the hydrogel morphology using characterization techniques operating at different length scales such as rheology, atomic force microscopy (AFM) and cryo transmission electron microscopy (Cryo-TEM). This allows for a precise control of the elastic behaviour of these networks.

Active Entanglement-Tracking Microrheology Directly Couples Macromolecular Deformations to Nonlinear Microscale Force Response of Entangled Actin

We track the deformation of discrete entangled actin segments while simultaneously measuring the resistive force the deformed filaments exert in response to an optically driven microsphere. We precisely map the network deformation field to show that local microscale stresses can induce filament deformations that propagate beyond mesoscopic length scales (60 μm, >3 persistence lengths l_p). We show that the filament persistence length controls the critical length scale at which distinct entanglement deformations become driven by collective network mechanics. Mesoscale propagation beyond l_p is coupled with nonlinear local stresses arising from steric entanglements mimicking cross-links.

Totally asymmetric simple exclusion process simulations of molecular motor transport on random networks with asymmetric exit rates

Using the totally asymmetric simple-exclusion-process and mean-field transport theory, we investigate the transport in closed random networks with simple crossing topology-two incoming, two outgoing segments, as a model for molecular motor motion along biopolymer networks. Inspired by in vitro observation of molecular motor motion, we model the motor behavior at the intersections by introducing different exit rates for the two outgoing segments. Our simulations of this simple network reveal surprisingly rich behavior of the transport current with respect to the global density and exit rate ratio. For asymmetric exit rates, we find a broad current plateau at intermediate motor densities resulting from the competition of two subnetwork populations. This current plateau leads to stabilization of transport properties within such networks.

Stability and anomalous entropic elasticity of subisostatic random-bond networks

We study the elasticity of thermalized spring networks under an applied bulk strain. The networks considered are subisostatic random-bond networks that, in the athermal limit, are known to have vanishing bulk and linear shear moduli at zero bulk strain. Above a bulk strain threshold, however, these networks become rigid, although surprisingly the shear modulus remains zero until a second, higher, strain threshold. We find that thermal fluctuations stabilize all networks below the rigidity transition, resulting in systems with both finite bulk and shear moduli. Our results show a T(0.66) temperature dependence of the moduli in the region below the bulk strain threshold, resulting in networks with anomalously high rigidity as compared to ordinary entropic elasticity. Furthermore, we find a second regime of anomalous temperature scaling for the shear modulus at its zero-temperature rigidity point, where it scales as T(0.5), behavior that is absent for the bulk modulus since its athermal rigidity transition is discontinuous.

Stress controls the mechanics of collagen networks

Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress-strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.

Filamin A Mediates Wound Closure by Promoting Elastic Deformation and Maintenance of Tension in the Collagen Matrix

Cell-mediated remodeling and wound closure are critical for efficient wound healing, but the contribution of actin-binding proteins to contraction of the extracellular matrix is not defined. We examined the role of filamin A (FLNa), an actin filament cross-linking protein, in wound contraction and maintenance of matrix tension. Conditional deletion of FLNa in fibroblasts in mice was associated with ~4 day delay of full-thickness skin wound contraction compared with wild-type (WT) mice. We modeled the healing wound matrix using cultured fibroblasts plated on grid-supported collagen gels that create lateral boundaries, which are analogues to wound margins. In contrast to WT cells, FLNa knockdown (KD) cells could not completely maintain tension when matrix compaction was resisted by boundaries, which manifested as relaxed matrix tension. Similarly, WT cells on cross-linked collagen, which requires higher levels of sustained tension, exhibited approximately fivefold larger deformation fields and approximately twofold greater fiber alignment compared with FLNa KD cells. Maintenance of boundary-resisted tension markedly influenced the elongation of cell extensions: in WT cells, the number (~50%) and length (~300%) of cell extensions were greater than FLNa KD cells. We conclude that FLNa is required for wound contraction, in part by enabling elastic deformation and maintenance of tension in the matrix.

Phonons and elasticity in critically coordinated lattices

Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site z. When z  <  zc  ≈  2d, where d is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on the properties of frames with z at or near zc, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, N0  -NS  =  dN  -NB relating the number of sites, N, and number of bonds, NB, to the number, N0, of modes of zero energy and the number, NS, of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which z  =  zc and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.