Visualizing the strain field in semiflexible polymer networks: strain fluctuations and nonlinear rheology of F-actin gels

Strain-Controlled Critical Slowing Down in the Rheology of Disordered Networks

Networks and dense suspensions frequently reside near a boundary between soft (or fluidlike) and rigid (or solidlike) regimes. Transitions between these regimes can be driven by changes in structure, density, or applied stress or strain. In general, near the onset or loss of rigidity in these systems, dissipation-limiting heterogeneous nonaffine rearrangements dominate the macroscopic viscoelastic response, giving rise to diverging relaxation times and power-law rheology. Here, we describe a simple quantitative relationship between nonaffinity and the excess viscosity. We test this nonaffinity-viscosity relationship computationally and demonstrate its rheological consequences in simulations of strained filament networks and dense suspensions. We also predict critical signatures in the rheology of semiflexible and stiff biopolymer networks near the strain stiffening transition.

Nonlinear Microscale Mechanics of Actin Networks Governed by Coupling of Filament Crosslinking and Stabilization

Actin plays a vital role in maintaining the stability and rigidity of biological cells while allowing for cell motility and shape change. The semiflexible nature of actin filaments-along with the myriad actin-binding proteins (ABPs) that serve to crosslink, bundle, and stabilize filaments-are central to this multifunctionality. The effect of ABPs on the structural and mechanical properties of actin networks has been the topic of fervent investigation over the past few decades. Yet, the combined impact of filament stabilization, stiffening and crosslinking via ABPs on the mechanical response of actin networks has yet to be explored. Here, we perform optical tweezers microrheology measurements to characterize the nonlinear force response and relaxation dynamics of actin networks in the presence of varying concentrations of α-actinin, which transiently crosslinks actin filaments, and phalloidin, which stabilizes filamentous actin and increases its persistence length. We show that crosslinking and stabilization can act both synergistically and antagonistically to tune the network resistance to nonlinear straining. For example, phalloidin stabilization leads to enhanced elastic response and reduced dissipation at large strains and timescales, while the initial microscale force response is reduced compared to networks without phalloidin. Moreover, we find that stabilization switches this initial response from that of stress stiffening to softening despite the increased filament stiffness that phalloidin confers. Finally, we show that both crosslinking and stabilization are necessary to elicit these emergent features, while the effect of stabilization on networks without crosslinkers is much more subdued. We suggest that these intriguing mechanical properties arise from the competition and cooperation between filament connectivity, bundling, and rigidification, shedding light on how ABPs with distinct roles can act in concert to mediate diverse mechanical properties of the cytoskeleton and bio-inspired polymeric materials.

Molecular Characterization of Polymer Networks

Polymer networks are complex systems consisting of molecular components. Whereas the properties of the individual components are typically well understood by most chemists, translating that chemical insight into polymer networks themselves is limited by the statistical and poorly defined nature of network structures. As a result, it is challenging, if not currently impossible, to extrapolate from the molecular behavior of components to the full range of performance and properties of the entire polymer network. Polymer networks therefore present an unrealized, important, and interdisciplinary opportunity to exert molecular-level, chemical control on material macroscopic properties. A barrier to sophisticated molecular approaches to polymer networks is that the techniques for characterizing the molecular structure of networks are often unfamiliar to many scientists. Here, we present a critical overview of the current characterization techniques available to understand the relation between the molecular properties and the resulting performance and behavior of polymer networks, in the absence of added fillers. We highlight the methods available to characterize the chemistry and molecular-level properties of individual polymer strands and junctions, the gelation process by which strands form networks, the structure of the resulting network, and the dynamics and mechanics of the final material. The purpose is not to serve as a detailed manual for conducting these measurements but rather to unify the underlying principles, point out remaining challenges, and provide a concise overview by which chemists can plan characterization strategies that suit their research objectives. Because polymer networks cannot often be sufficiently characterized with a single method, strategic combinations of multiple techniques are typically required for their molecular characterization.

Measurements and characterization of the dynamics of tracer particles in an actin network

The underlying physics governing the diffusion of a tracer particle in a viscoelastic material is a topic of some dispute. The long-term memory in the mechanical response of such materials should induce diffusive motion with a memory kernel, such as fractional Brownian motion (fBM). This is the reason that microrheology is able to provide the shear modulus of polymer networks. Surprisingly, the diffusion of a tracer particle in a network of a purified protein, actin, was found to conform to the continuous time random walk type (CTRW). We set out to resolve this discrepancy by studying the tracer particle diffusion using two different tracer particle sizes, in actin networks of different mesh sizes. We find that the ratio of tracer particle size to the characteristic length scale of a bio-polymer network plays a crucial role in determining the type of diffusion it performs. We find that the diffusion of the tracer particles has features of fBm when the particle is large compared to the mesh size, of normal diffusion when the particle is much smaller than the mesh size, and of the CTRW in between these two limits. Based on our findings, we propose and verify numerically a new model for the motion of the tracer in all regimes. Our model suggests that diffusion in actin networks consists of fBm of the tracer particle coupled with caging events with power-law distributed escape times.

Cell nucleus as a microrheological probe to study the rheology of the cytoskeleton

Mechanical properties of the cell are important biomarkers for probing its architectural changes caused by cellular processes and/or pathologies. The development of microfluidic technologies has enabled measuring the cell's mechanical properties at high throughput so that mechanical phenotyping can be applied to large samples in reasonable timescales. These studies typically measure the stiffness of the cell as the only mechanical biomarker and do not disentangle the rheological contributions of different structural components of the cell, including the cell cortex, the interior cytoplasm and its immersed cytoskeletal structures, and the nucleus. Recent advancements in high-speed fluorescent imaging have enabled probing the deformations of the cell cortex while also tracking different intracellular components in rates applicable to microfluidic platforms. We present a, to our knowledge, novel method to decouple the mechanics of the cell cortex and the cytoplasm by analyzing the correlation between the cortical deformations that are induced by external microfluidic flows and the nucleus displacements, induced by those cortical deformations, i.e., we use the nucleus as a high-throughput microrheological probe to study the rheology of the cytoplasm, independent of the cell cortex mechanics. To demonstrate the applicability of this method, we consider a proof-of-concept model consisting of a rigid spherical nucleus centered in a spherical cell. We obtain analytical expressions for the time-dependent nucleus velocity as a function of the cell deformations when the interior cytoplasm is modeled as a viscous, viscoelastic, porous, and poroelastic material and demonstrate how the nucleus velocity can be used to characterize the linear rheology of the cytoplasm over a wide range of forces and timescales/frequencies.

Varying crosslinking motifs drive the mesoscale mechanics of actin-microtubule composites

The cytoskeleton precisely tunes its mechanics by altering interactions between semiflexible actin filaments, rigid microtubules, and crosslinking proteins. We use optical tweezers microrheology and confocal microscopy to characterize how varying crosslinking motifs impact the mesoscale mechanics and mobility of actin-microtubule composites. We show that, upon subtle changes in crosslinking patterns, composites can exhibit two distinct classes of force response - primarily elastic versus more viscous. For example, a composite in which actin and microtubules are crosslinked to each other but not to themselves is markedly more elastic than one in which both filaments are independently crosslinked. Notably, this distinction only emerges at mesoscopic scales in response to nonlinear forcing, whereas varying crosslinking motifs have little impact on the microscale mechanics and mobility. Our unexpected scale-dependent results not only inform the physics underlying key cytoskeleton processes and structures, but, more generally, provide valuable perspective to materials engineering endeavors focused on polymer composites.

Probing shear-induced rearrangements in Fourier space. I. Dynamic light scattering

Understanding the microscopic origin of the rheological behavior of soft matter is a long-lasting endeavour. While early efforts concentrated mainly on the relationship between rheology and structure, current research focuses on the role of microscopic dynamics. We present in two companion papers a thorough discussion of how Fourier space-based methods may be coupled to rheology to shed light on the relationship between the microscopic dynamics and the mechanical response of soft systems. In this first companion paper, we report a theoretical, numerical and experimental investigation of dynamic light scattering coupled to rheology. While in ideal solids and simple viscous fluids the displacement field under a shear deformation is purely affine, additional non-affine displacements arise in many situations of great interest, for example in elastically heterogeneous materials or due to plastic rearrangements. We show how affine and non-affine displacements can be separately resolved by dynamic light scattering, and discuss in detail the effect of several non-idealities in typical experiments.

Synergistic Interactions Between DNA and Actin Trigger Emergent Viscoelastic Behavior

Composites of flexible and rigid polymers are ubiquitous in biology and industry alike, yet the physical principles determining their mechanical properties are far from understood. Here, we couple force spectroscopy with large-scale Brownian dynamics simulations to elucidate the unique viscoelastic properties of custom-engineered blends of entangled flexible DNA molecules and semiflexible actin filaments. We show that composites exhibit enhanced stress stiffening and prolonged mechanomemory compared to systems of actin or DNA alone, and that these nonlinear features display a surprising nonmonotonic dependence on the fraction of actin in the composite. Simulations reveal that these counterintuitive results arise from synergistic microscale interactions between the two biopolymers. Namely, DNA entropically drives actin filaments to form bundles that stiffen the network but reduce the entanglement density, while a uniform well-connected actin network is required to reinforce the DNA network against yielding and flow. The competition between bundling and connectivity triggers an unexpected stress response that leads equal mass DNA-actin composites to exhibit the most pronounced stress stiffening and the most long-lived entanglements.

Heterogeneity and nonaffinity of cell-induced matrix displacements

Cell contractile forces deform and reorganize the surrounding matrix, but the relationship between the forces and the resulting displacements is complicated by the fact that the fibrous structure brings about a complex set of mechanical properties. Many studies have quantified nonlinear and time-dependent properties at macroscopic scales, but it is unclear whether macroscopic properties apply to the scale of a cell, where the matrix is composed of a heterogeneous network of fibers. To address this question, we mimicked the contraction of a cell embedded within a fibrous collagen matrix and quantified the resulting displacements. The data revealed displacements that were heterogeneous and nonaffine. The heterogeneity was reproducible during cyclic loading, and it decreased with decreasing fiber length. Both the experiments and a fiber network model showed that the heterogeneous displacements decayed over distance at a rate no faster than the average displacement field, indicating no transition to homogeneous continuum behavior. Experiments with cells fully embedded in collagen matrices revealed the presence of heterogeneous displacements as well, exposing the dramatic heterogeneity in matrix reorganization that is induced by cells at different positions within the same fibrous matrix.

Crack Initiation in Viscoelastic Materials

In viscoelastic materials, individually short-lived bonds collectively result in a mechanical resistance which is long lived but finite as, ultimately, cracks appear. Here, we provide a microscopic mechanism by which a critical crack length emerges from the nonlinear local bond dynamics. Because of this emerging length scale, macroscopic viscoelastic materials fracture in a fundamentally different manner from microscopically small systems considered in previous models. We provide and numerically verify analytical equations for the dependence of the critical crack length on the bond kinetics and applied stress.

Elastic compliance of fibrillar assemblies in type I collagen

Fibrillary assemblies of Type I collagen find important applications in tissue engineering and as matrices for biophysical studies. The mechanical and structural properties of these structures are governed by factors such as protein concentration, temperature, pH and ionic strength. This study reports on an impedance based analysis of the elastic compliance of fibrillary assemblies of Type I collagen using quartz crystal microbalance with dissipation (QCM-D) at a fundamental frequency of 5 MHz and overtones (n = 3,5,7,9,11). Here, In situ partial fibrillation of the adsorbing collagen followed by its fibrillary assemblies on hydrophilic gold coated quartz surface have been crosslinked using Gallic acid (GA), Chromium (III) gallate (Cr-GA), Catechin (Cat), Tetrakis(hydroxymethyl)phosphonium sulfate (THPS) and Oxazolidine (Ox). This approach allows direct comparison of how viscoelastic properties track the structural evolution of the fiber and network length scales. The collagen crosslinking shows significant positive impact on the protein's mechanical behaviour and on the type of crosslinking agents used. The elastic modulus increases as collagen <GA < THPS < Cr-GA < Cat < Ox. Atomic force microscopic studies on the adsorbed collagen after cross linking confirmed the presence of fibrous assemblies. The results indicate stabilization and reinforcement through strong physical entanglement between the molecules of collagen as well as chemical interaction between collagen matrix and fibrils during cross linking. The elastic compliance evaluated from ΔDissipation/Δfreq. from QCM-D showed that cross linking with GA, Cr-GA and Ox resulted in flexible fibrillary network while agents like THPS and Cat showed elastic moduli similar to that of pure collagen. Results suggest that optimal collagen-crosslinking agent ratio and degree of crosslinking of collagen can help tailor the mechanical properties for specific applications in design of bio-materials of these composites.

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.

Mechanical Properties of the Cytoskeleton and Cells

SUMMARYThe cytoskeleton is the major mechanical structure of the cell; it is a complex, dynamic biopolymer network comprising microtubules, actin, and intermediate filaments. Both the individual filaments and the entire network are not simple elastic solids but are instead highly nonlinear structures. Appreciating the mechanics of biopolymer networks is key to understanding the mechanics of cells. Here, we review the mechanical properties of cytoskeletal polymers and discuss the implications for the behavior of cells.

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.

Mechanical response of collagen networks to nonuniform microscale loads

As force is applied to fibrous proteins such as collagen or fibrin, the fibers respond by bending, stretching, or buckling, which together bring about a nonlinear relationship between force and displacement. The nonlinearity is typically understood in terms of strain stiffening in uniform extension or shear, but there remains a critical lack of data on how fibrous materials respond to other more complicated loadings. Here we study the mechanics of collagen networks in response to nonuniform loads applied on the local scale of the fibers. For this, we use particles made of an active hydrogel that undergoes a temperature-induced phase transition causing a large decrease in volume. We embed these particles in networks of fibrous collagen and use them as microactuators to apply controlled microscale loading. The resulting fiber displacements propagate over a long range with radial displacements u scaling as r^-n with n ≈ 1. By contrast, we find linear homogeneous materials have n ≈ 2, in agreement with classical linear elastic theory. Our experimental data supports the notion that the long range displacements result from buckling of fibers in compression and local straightening of fibers in tension, in agreement with previous studies. Surprisingly, global network anisotropy appears to have only a modest effect on the displacement propagation. These insights into the microscale mechanics demonstrate that the decay power n provides a useful metric to quantify the mechanics of fibrous materials. We therefore suggest it is a means to compare new theories with experimental data.

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.

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.

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.

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.

Multi-scale strain-stiffening of semiflexible bundle networks

Bundles of polymer filaments are responsible for the rich and unique mechanical behaviors of many biomaterials, including cells and extracellular matrices. In fibrin biopolymers, whose nonlinear elastic properties are crucial for normal blood clotting, protofibrils self-assemble and bundle to form networks of semiflexible fibers. Here we show that the extraordinary strain-stiffening response of fibrin networks is a direct reflection of the hierarchical architecture of the fibrin fibers. We measure the rheology of networks of unbundled protofibrils and find excellent agreement with an affine model of extensible wormlike polymers. By direct comparison with these data, we show that physiological fibrin networks composed of thick fibers can be modeled as networks of tight protofibril bundles. We demonstrate that the tightness of coupling between protofibrils in the fibers can be tuned by the degree of enzymatic intermolecular crosslinking by the coagulation factor XIII. Furthermore, at high stress, the protofibrils contribute independently to the network elasticity, which may reflect a decoupling of the tight bundle structure. The hierarchical architecture of fibrin fibers can thus account for the nonlinearity and enormous elastic resilience characteristic of blood clots.

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.

Entangled F-actin displays a unique crossover to microscale nonlinearity dominated by entanglement segment dynamics

We drive optically trapped microspheres through entangled F-actin at constant speeds and distances well beyond the linear regime, and measure the microscale force response of the entangled filaments during and following strain. Our results reveal a unique crossover to appreciable nonlinearity at a strain rate of [small gamma, Greek, dot above]c ≈ 3 s(-1) which corresponds remarkably well with the theoretical rate of relaxation of entanglement length deformations 1/τent. Above [small gamma, Greek, dot above]c, we observe stress stiffening which occurs over very short time scales comparable to the predicted timescale over which mesh size deformations relax. Stress softening then takes over, yielding to an effectively viscous regime over a timescale comparable to the entanglement length relaxation time, τent. The viscous regime displays shear thinning but with a less pronounced viscosity scaling with strain rate compared to flexible polymers. The relaxation of induced force on filaments following strain shows that the relative relaxation proceeds more quickly for increasing strain rates; and for rates greater than [small gamma, Greek, dot above]c, the relaxation displays a complex power-law dependence on time. Our collective results reveal that molecular-level nonlinear viscoelasticity is driven by non-classical dynamics of individual entanglement segments that are unique to semiflexible polymers.

Elasticity of 3D networks with rigid filaments and compliant crosslinks

Disordered filamentous networks with compliant crosslinks exhibit a low linear elastic shear modulus at small strains, but stiffen dramatically at high strains. Experiments have shown that the elastic modulus can increase by up to three orders of magnitude while the networks withstand relatively large stresses without rupturing. Here, we perform an analytical and numerical study on model networks in three dimensions. Our model consists of a collection of randomly oriented rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Due to zero probability of filament intersection in three dimensions, our model networks are by construction prestressed in terms of initial tension in the crosslinks. We demonstrate how the linear elastic modulus can be related to the prestress in these networks. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. We further show that there is no qualitative change in the stiffening behavior of filamentous networks even if the filaments are bending-compliant. Consequently, unlike suggested in prior work, the model system studied here cannot provide an explanation for the experimentally observed linear scaling of the modulus with the stress in filamentous networks.

Stiffest elastic networks

The rigidity of a network of elastic beams is closely related to its microstructure. We show both numerically and theoretically that there is a class of isotropic networks, which are stiffer than any other isotropic network of same density. The elastic moduli of these stiffest elastic networks are explicitly given. They constitute upper-bounds, which compete or improve the well-known Hashin–Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions. In particular, we present an optimal and isotropic three-dimensional structure which, to our knowledge, is the first one to be presented as such.

Wrinkling of random and regular semiflexible polymer networks

We investigate wrinkling of two-dimensional random and triangular semiflexible polymer networks under shear. Both types of semiflexible networks exhibit wrinkling above a small critical shear angle, which scales with an exponent of the bending modulus between 1.9 and 2.0. Random networks exhibit hysteresis at the wrinkling threshold. Wrinkling lowers the total elastic energy by up to 20% and strongly affects the elastic properties of all semiflexible networks such as the crossover between bending and stretching dominated behavior. In random networks, we also find evidence for metastable wrinkled configurations. While the disordered microstructure of random networks affects the scaling behavior of wrinkle amplitudes, it has little effect on wrinkle wavelength. Therefore, wrinkles represent a robust, microstructure-independent assay of shear strain or elastic properties.

Elastic response of filamentous networks with compliant crosslinks

Experiments have shown that elasticity of disordered filamentous networks with compliant crosslinks is very different from networks with rigid crosslinks. Here, we model and analyze filamentous networks as a collection of randomly oriented rigid filaments connected to each other by flexible crosslinks that are modeled as wormlike chains. For relatively large extensions we allow for enthalpic stretching of crosslink backbones. We show that for sufficiently high crosslink density, the network linear elastic response is affine on the scale of the filaments' length. The nonlinear regime can become highly nonaffine and is characterized by a divergence of the elastic modulus at finite strain. In contrast to the prior predictions, we do not find an asymptotic regime in which the differential elastic modulus scales linearly with the stress, although an approximate linear dependence can be seen in a transition from entropic to enthalpic regimes. We discuss our results in light of recent experiments.

Local mechanical response in semiflexible polymer networks subjected to an axisymmetric prestress

Analytical and numerical calculations are presented for the mechanical response of fiber networks in a state of axisymmetric prestress, in the limit where geometric nonlinearities such as fiber rotation are negligible. This allows us to focus on the anisotropy deriving purely from the nonlinear force-extension curves of individual fibers. The number of independent elastic coefficients for isotropic, axisymmetric, and fully anisotropic networks are enumerated before deriving expressions for the response to a locally applied force that can be tested against, e.g., microrheology experiments. Localized forces can generate anisotropy away from the point of application, so numerical integration of nonlinear continuum equations is employed to determine the stress field, and induced mechanical anisotropy, at points located directly behind and in front of a force monopole. Results are presented for the wormlike chain model in normalized forms, allowing them to be easily mapped to a range of systems. Finally, the relevance of these findings to naturally occurring systems and directions for future investigation are discussed.

Determining the structure-mechanics relationships of dense microtubule networks with confocal microscopy and magnetic tweezers-based microrheology

The microtubule (MT) cytoskeleton is essential in maintaining the shape, strength, and organization of cells. Its spatiotemporal organization is fundamental for numerous dynamic biological processes, and mechanical stress within the MT cytoskeleton provides an important signaling mechanism in mitosis and neural development. This raises important questions about the relationships between structure and mechanics in complex MT structures. In vitro, reconstituted cytoskeletal networks provide a minimal model of cell mechanics while also providing a testing ground for the fundamental polymer physics of stiff polymer gels. Here, we describe our development and implementation of a broad tool kit to study structure-mechanics relationships in reconstituted MT networks, including protocols for the assembly of entangled and cross-linked MT networks, fluorescence imaging, microstructure characterization, construction and calibration of magnetic tweezers devices, and mechanical data collection and analysis. In particular, we present the design and assembly of three neodymium iron boron (NdFeB)-based magnetic tweezers devices optimized for use with MT networks: (1) high-force magnetic tweezers devices that enable the application of nano-Newton forces and possible meso- to macroscale materials characterization; (2) ring-shaped NdFeB-based magnetic tweezers devices that enable oscillatory microrheology measurements; and (3) portable magnetic tweezers devices that enable direct visualization of microscale deformation in soft materials under applied force.

Model selection for athermal cross-linked fiber networks

Athermal random fiber networks are usually modeled by representing each fiber as a truss, a Euler-Bernoulli or a Timoshenko beam, and, in the case of cross-linked networks, each cross-link as a pinned, rotating, or welded joint. In this work we study the effect of these various modeling options on the dependence of the overall network stiffness on system parameters. We conclude that Timoshenko beams can be used for the entire range of density and beam stiffness parameters, while the Euler-Bernoulli model can be used only at relatively low network densities. In the high density-high bending stiffness range, strain energy is stored predominantly in the axial and shear deformation modes, while in the other extreme range of parameters, the energy is stored in the bending mode. The effect of the model size on the network stiffness is also discussed.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Non-affine deformations in polymer hydrogels

Most theories of soft matter elasticity assume that the local strain in a sample after deformation is identical everywhere and equal to the macroscopic strain, or equivalently that the deformation is affine. We discuss the elasticity of hydrogels of crosslinked polymers with special attention to affine and non-affine theories of elasticity. Experimental procedures to measure non-affine deformations are also described. Entropic theories, which account for gel elasticity based on stretching out individual polymer chains, predict affine deformations. In contrast, simulations of network deformation that result in bending of the stiff constituent filaments generally predict non-affine behavior. Results from experiments show significant non-affine deformation in hydrogels even when they are formed by flexible polymers for which bending would appear to be negligible compared to stretching. However, this finding is not necessarily an experimental proof of the non-affine model for elasticity. We emphasize the insights gained from experiments using confocal rheoscope and show that, in addition to filament bending, sample micro-inhomogeneity can be a significant alternative source of non-affine deformation.

Mechanics of soft composites of rods in elastic gels

We report detailed theoretical investigations of the micromechanics and bulk elastic properties of composites consisting of randomly distributed stiff fibers embedded in an elastic matrix in two and three dimensions. Recent experiments [V. Pelletier, N. Gal, P. Fournier, and M. L. Kilfoil, Phys. Rev. Lett. 102, 188303 (2009)] have suggested that the inclusion of stiff microtubules in a softer, nearly incompressible biopolymer matrix can lead to emergent compressibility. This can be understood in terms of the enhancement of the compressibility of the composite relative to its shear compliance as a result of the addition of stiff rodlike inclusions. We show that the Poisson's ratio ν of such a composite evolves with increasing rod density toward a particular value, or fixed point, independent of the material properties of the matrix, as long as it has a finite initial compressibility. This fixed point is ν = 1/4 in three dimensions and ν = 1/3 in two dimensions. Our results suggest an important role for stiff filaments such as microtubules and stress fibers in cell mechanics. At the same time, our work has a wider elasticity context, with potential applications to composite elastic media with a wide separation of scales in stiffness of its constituents such as carbon nanotube-polymer composites, which have been shown to have highly tunable mechanics.

On the role of the filament length distribution in the mechanics of semiflexible networks

This paper explores the effects of filament length polydispersity on the mechanical properties of semiflexible crosslinked polymer networks. Extending previous studies on monodisperse networks, we compute numerically the response of crosslinked networks of elastic filaments of bimodal and exponential length distributions. These polydisperse networks are subject to the same affine to nonaffine (A/NA) transition observed previously for monodisperse networks, wherein the decreases in either crosslink density or bending stiffness lead to a shift from affine, stretching-dominated deformations to nonaffine, bending-dominated deformations. We find that the onset of this transition is generally more sensitive to changes in the density of longer filaments than shorter filaments, meaning that longer filaments have greater mechanical efficiency. Moreover, in polydisperse networks, mixtures of long and short filaments interact cooperatively to generally produce a nonaffine mechanical response closer to the affine prediction than comparable monodisperse networks of either long or short filaments. Accordingly, the mechanical affinity of polydisperse networks is dependent on the filament length composition. Overall, length polydispersity has the effect of sharpening and shifting the A/NA transition to lower network densities. We discuss the implications of these results on experimental observation of the A/NA transition, and on the design of advanced materials.

Size-dependent rheology of type-I collagen networks

We investigate the system size-dependent rheological response of branched type I collagen gels. When subjected to a shear strain, the highly interconnected mesh dynamically reorients, resulting in overall stiffening of the network. When a continuous shear strain is applied to a collagen network, we observe that the local apparent modulus, in the strain-stiffening regime, is strongly dependent on the gel thickness. In addition, we demonstrate that the overall network failure is determined by the ratio of the gel thickness to the mesh size. These findings have broad implications for cell-matrix interactions, the interpretation of rheological tissue data, and the engineering of biomimetic scaffolds.

Mechanisms of mechanical signaling in development and disease

The responses of cells to chemical signals are relatively well characterized and understood. Cells also respond to mechanical signals in the form of externally applied force and forces generated by cell-matrix and cell-cell contacts. Many features of cell function that are generally considered to be under the control of chemical stimuli, such as motility, proliferation, differentiation and survival, can also be altered by changes in the stiffness of the substrate to which the cells are adhered, even when their chemical environment remains unchanged. Many examples from clinical and whole animal studies have shown that changes in tissue stiffness are related to specific disease characteristics and that efforts to restore normal tissue mechanics have the potential to reverse or prevent cell dysfunction and disease. How cells detect stiffness is largely unknown, but the cellular structures that measure stiffness and the general principles by which they work are beginning to be revealed. This Commentary highlights selected recent reports of mechanical signaling during disease development, discusses open questions regarding the physical mechanisms by which cells sense stiffness, and examines the relationship between studies in vitro on flat substrates and the more complex three-dimensional setting in vivo.

Frequency-dependent stiffening of semiflexible networks: a dynamical nonaffine to affine transition

By combining the force-extension relation of single semiflexible polymers with a Langevin equation to capture the dissipative dynamics of chains moving through a viscous medium we study the dynamical response of cross-linked biopolymer materials. We find that at low frequencies the network deformations are highly nonaffine, and show a low plateau in the modulus. At higher frequencies, this nonaffinity decreases while the elastic modulus increases. With increasing frequency, more and more nonaffine network relaxation modes are suppressed, resulting in a stiffening. This effect is fundamentally different from the high-frequency stiffening due to the single-filament relaxation modes [F. Gittes and F. C. MacKintosh, Phys. Rev. E 58, R1241 (1998)], not only in terms of its mechanism but also in its resultant scaling: G'(ω) ∼ ω(α) with α > 3/4. This may determine nonlinear material properties at low, physiologically relevant frequencies.

Poisson's ratio in composite elastic media with rigid rods

We calculate both the micromechanical response and bulk elastic constants of composites of rods embedded in elastic media. We find two fixed points for Poisson's ratio with respect to rod density: there is an unstable fixed point for Poisson's ratio =1/2 (an incompressible system) and a stable fixed point for Poisson's ratio =1/4 (a compressible system). We also derive approximate expressions for the elastic constants for arbitrary rod density, which agree with exact results for both low and high density. These results may help to explain recent experiments [Phys. Rev. Lett. 102, 188303 (2009)10.1103/PhysRevLett.102.188303] that reported compressibility for composites of microtubules in filamentous actin networks.

Fluctuations and scaling in creep deformation

The spatial fluctuations of deformation are studied in the creep in Andrade's power law and the logarithmic phases, using paper samples. Measurements by the digital image correlation technique show that the relative strength of the strain rate fluctuations increases with time, in both creep regimes. In the Andrade creep phase characterized by a power-law decay of the strain rate ϵt∼t(-θ), with θ≈0.7, the fluctuations obey Δϵt∼t(-γ), with γ≈0.5. The local deformation follows a data collapse appropriate for a phase transition. Similar behavior is found in a crystal plasticity model, with a jamming or yielding transition.

Nonlocal fluctuation correlations in active gels

Many active materials and biological systems are driven far from equilibrium by embedded agents that spontaneously generate forces and distort the surrounding material. Probing and characterizing these athermal fluctuations are essential to understand the properties and behaviors of such systems. Here we present a mathematical procedure to estimate the local action of force-generating agents from the observed fluctuating displacement fields. The active agents are modeled as oriented force dipoles or isotropic compression foci, and the matrix on which they act is assumed to be either a compressible elastic continuum or a coupled network-solvent system. Correlations at a single point and between points separated by an arbitrary distance are obtained, giving a total of three independent fluctuation modes that can be tested with microrheology experiments. Since oriented dipoles and isotropic compression foci give different contributions to these fluctuation modes, ratiometric analysis allows us characterize the force generators. We also predict and experimentally find a high-frequency ballistic regime, arising from individual force-generating events in the form of the slow buildup of stress followed by rapid but finite decay. Finally, we provide a quantitative statistical model to estimate the mean filament tension from these athermal fluctuations, which leads to stiffening of active networks.