Affine-nonaffine transition in networks of nematically ordered semiflexible polymers

Rigidity and fracture of biopolymer double networks

Tunable mechanics and fracture resistance are hallmarks of biological tissues whose properties arise from extracellular matrices comprised of double networks. To elucidate the origin of these desired properties, we study the shear modulus and fracture properties of a rigidly percolating double network model comprised of a primary network of stiff fibers and a secondary network of flexible fibers. We find that when the primary network density is just above its rigidity percolation threshold, the secondary network density can be tuned to facilitate stress relaxation via non-affine deformations and provide mechanical reinforcement. In contrast, when the primary network is far above its rigidity threshold, the double network is always stiff and brittle. These results highlight an important mechanism behind the tunability and resilience of biopolymer double networks: the secondary network can dramatically alter mechanical properties from compliant and ductile to stiff and brittle only when the primary network is marginally rigid.

Directed force propagation in semiflexible networks

We consider the propagation of tension along specific filaments of a semiflexible filament network in response to the application of a point force using a combination of numerical simulations and analytic theory. We find the distribution of force within the network is highly heterogeneous, with a small number of fibers supporting a significant fraction of the applied load over distances of multiple mesh sizes surrounding the point of force application. We suggest that these structures may be thought of as tensile force chains, whose structure we explore via simulation. We develop self-consistent calculations of the point-force response function and introduce a transfer matrix approach to explore the decay of tension (into bending) energy and the branching of tensile force chains in the network.

Anisotropic mechanical response of layered disordered fibrous materials

Mechanically bonded fabrics account for a significant portion of nonwoven products, and serve many niche areas of nonwoven manufacturing. Such fabrics are characterized by layers of disordered fibrous webs, but we lack an understanding of how such microstructures determine bulk material response. Here we numerically determine the linear shear response of needle-punched fabrics modeled as cross-linked sheets of two-dimensional (2D) Mikado networks. We systematically vary the intra-sheet fiber density, inter-sheet separation distance, and direction of shear, and quantify the macroscopic shear modulus alongside the degree of affinity and energy partition. For shear parallel to the sheets, the response is dominated by intrasheet fibers and follows known trends for 2D Mikado networks. By contrast, shears perpendicular to the sheets induce a softer response dominated by either intrasheet or intersheet fibers depending on a quadratic relation between sheet separation and fiber density. These basic trends are reproduced and elucidated by a simple scaling argument that we provide. We discuss the implications of our findings in the context of real nonwoven fabrics.

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.

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.

Elastic behavior and platelet retraction in low- and high-density fibrin gels

Fibrin is a biopolymer that gives thrombi the mechanical strength to withstand the forces imparted on them by blood flow. Importantly, fibrin is highly extensible, but strain hardens at low deformation rates. The density of fibrin in clots, especially arterial clots, is higher than that in gels made at plasma concentrations of fibrinogen (3-10 mg/mL), where most rheology studies have been conducted. Our objective in this study was to measure and characterize the elastic regimes of low (3-10 mg/mL) and high (30-100 mg/mL) density fibrin gels using shear and extensional rheology. Confocal microscopy of the gels shows that fiber density increases with fibrinogen concentration. At low strains, fibrin gels act as thermal networks independent of fibrinogen concentration. Within the low-strain regime, one can predict the mesh size of fibrin gels by the elastic modulus using semiflexible polymer theory. Significantly, this provides a link between gel mechanics and interstitial fluid flow. At moderate strains, we find that low-density fibrin gels act as nonaffine mechanical networks and transition to affine mechanical networks with increasing strains within the moderate regime, whereas high-density fibrin gels only act as affine mechanical networks. At high strains, the backbone of individual fibrin fibers stretches for all fibrin gels. Platelets can retract low-density gels by >80% of their initial volumes, but retraction is attenuated in high-density fibrin gels and with decreasing platelet density. Taken together, these results show that the nature of fibrin deformation is a strong function of fibrin fiber density, which has ramifications for the growth, embolization, and lysis of thrombi.

Mechanics of anisotropic spring networks

We construct and analyze a model for a disordered linear spring network with anisotropy. The modeling is motivated by, for example, granular systems, nematic elastomers, and ultimately cytoskeletal networks exhibiting some underlying anisotropy. The model consists of a triangular lattice with two different bond occupation probabilities, p(x) and p(y), for the linear springs. We develop an effective medium theory (EMT) to describe the network elasticity as a function of p(x) and p(y). We find that the onset of rigidity in the EMT agrees with Maxwell constraint counting. We also find beyond linear behavior in the shear and bulk modulus as a function of occupation probability in the rigid phase for small strains, which differs from the isotropic case. We compare our EMT with numerical simulations to find rather good agreement. Finally, we discuss the implications of extending the reach of effective medium theory as well as draw connections with prior work on both anisotropic and isotropic spring networks.

Structure-function relations and rigidity percolation in the shear properties of articular cartilage

Among mammalian soft tissues, articular cartilage is particularly interesting because it can endure a lifetime of daily mechanical loading despite having minimal regenerative capacity. This remarkable resilience may be due to the depth-dependent mechanical properties, which have been shown to localize strain and energy dissipation. This paradigm proposes that these properties arise from the depth-dependent collagen fiber orientation. Nevertheless, this structure-function relationship has not yet been quantified. Here, we use confocal elastography, quantitative polarized light microscopy, and Fourier-transform infrared imaging to make same-sample measurements of the depth-dependent shear modulus, collagen fiber organization, and extracellular matrix concentration in neonatal bovine articular cartilage. We find weak correlations between the shear modulus |G(∗)| and both the collagen fiber orientation and polarization. We find a much stronger correlation between |G(∗)| and the concentration of collagen fibers. Interestingly, very small changes in collagen volume fraction vc lead to orders-of-magnitude changes in the modulus with |G(∗)| scaling as (vc - v0)(ξ). Such dependencies are observed in the rheology of other biopolymer networks whose structure exhibits rigidity percolation phase transitions. Along these lines, we propose that the collagen network in articular cartilage is near a percolation threshold that gives rise to these large mechanical variations and localization of strain at the tissue's surface.

Advances in the mechanical modeling of filamentous actin and its cross-linked networks on multiple scales

The protein actin is a part of the cytoskeleton and, therefore, responsible for the mechanical properties of the cells. Starting with the single molecule up to the final structure, actin creates a hierarchical structure of several levels exhibiting a remarkable behavior. The hierarchy spans several length scales and limitations in computational power; therefore, there is a call for different mechanical modeling approaches for the different scales. On the molecular level, we may consider each atom in molecular dynamics simulations. Actin forms filaments by combining the molecules into a double helix. In a model, we replace molecular subdomains using coarse-graining methods, allowing the investigation of larger systems of several atoms. These models on the nanoscale inform continuum mechanical models of large filaments, which are based on worm-like chain models for polymers. Assemblies of actin filaments are connected with cross-linker proteins. Models with discrete filaments, so-called Mikado models, allow us to investigate the dependence of the properties of networks on the parameters of the constituents. Microstructurally motivated continuum models of the networks provide insights into larger systems containing cross-linked actin networks. Modeling of such systems helps to gain insight into the processes on such small scales. On the other hand, they call for verification and hence trigger the improvement of established experiments and the development of new methods.

Predictive maps for stochastic nonaffine stiffening and damage in fibrous networks

The macroscopic responses of synthetic and natural filamentous networks are determined by a combination of microstructure and filament properties. Biofilament networks such as those of actin and fibrin have become vehicles for studying important concepts in mechanics such as rigidity percolation, linearity and nonlinearity, isotropy and anisotropy, affinity and nonaffinity, hardening and softening, bending and stretching transitions, etc. In this work, we consider generic fibrous network architectures to map out their mechanical responses over a wide range of filament properties. Using the finite element method, we perform two-dimensional simulations of discrete networks subjected to shear deformation. These simulations encompass stochastic effects arising from network topology (filament arrangement, orientation, and length distribution) and the thermally activated crosslink scission. We study the mechanics of these random networks up to a strain of 10%, including damage that is induced by crosslink scission. The response is nonlinear and the initial elastic modulus alone is not sufficient to give an understanding about the overall response. We show that the nonlinear elastic response of the network can be captured using a few parameters that depend on some well known length scales in network mechanics. For networks with filament density above the rigidity percolation threshold, by increasing filament density and bending stiffness, we observe a crossover from the bending dominated elastically compliant stiffening regime to a stretching dominated rigid nonstiffening regime. We show that in the bending dominated regime there are large deviations from the predictions of affine continuum theories. We also give a simple qualitative model for describing the contours of the incubation strain which marks the onset of damage in networks.

Modeling of fibrin gels based on confocal microscopy and light-scattering data

Fibrin gels are biological networks that play a fundamental role in blood coagulation and other patho/physiological processes, such as thrombosis and cancer. Electron and confocal microscopies show a collection of fibers that are relatively monodisperse in diameter, not uniformly distributed, and connected at nodal points with a branching order of ∼3-4. Although in the confocal images the hydrated fibers appear to be quite straight (mass fractal dimension D(m) = 1), for the overall system 1<D(m)<2. Based on the confocal images, we developed a method to generate three-dimensional (3D) in silico gels made of cylindrical sticks of diameter d, density ρ, and average length <L>, joined at randomly distributed nodal points. The resulting 3D network strikingly resembles real fibrin gels and can be sketched as an assembly of densely packed fractal blobs, i.e., regions of size ξ, where the fiber concentration is higher than average. The blobs are placed at a distance ξ0 between their centers of mass so that they are overlapped by a factor η =ξ/ξ0 and have D(m) ∼1.2-1.6. The in silico gels' structure is quantitatively analyzed by its 3D spatial correlation function g(3D)(r) and corresponding power spectrum I(q) = FFT(3D[g3D(r)]), from which ρ, d, D(m), η, and ξ0 can be extracted. In particular, ξ0 provides an excellent estimate of the gel mesh size. The in silico gels' I(q) compares quite well with real gels' elastic light-scattering measurements. We then derived an analytical form factor for accurately fitting the scattering data, which allowed us to directly recover the gels' structural parameters.

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.

Filament-length-controlled elasticity in 3D fiber networks

We present a model for disordered 3D fiber networks to study their linear and nonlinear elasticity. In contrast to previous 2D models, these 3D networks with binary crosslinks are underconstrained with respect to fiber stretching elasticity, suggesting that bending may dominate their response. We find that such networks exhibit a bending-dominated elastic regime controlled by fiber length, as well as a crossover to a stretch-dominated regime for long fibers. Finally, by extending the model to the nonlinear regime, we show that these networks become intrinsically nonlinear with a vanishing linear response regime in the limit of flexible or long filaments.

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.