Elastic response of filamentous networks with compliant crosslinks

Stiffening mechanisms in stochastic athermal fiber networks

Stochastic athermal networks composed of fibers that deform axially and in bending strain stiffen much faster than thermal networks of axial elements, such as elastomers. Here we investigate the physical origin of stiffening in athermal network materials. To this end, we use models of stochastic networks subjected to uniaxial deformation and identify the emergence of two subnetworks, the stress path subnetwork (SPSN) and the bending support subnetwork (BSSN), which carry most of the axial and bending energies, respectively. The BSSN controls lateral contraction and modulates the organization of the SPSN during deformation. The SPSN is preferentially oriented in the loading direction, while the BSSN's preferential orientation is orthogonal to the SPSN. In nonaffine networks stiffening is exponential, while in close-to-affine networks it is quadratic. The difference is due to a much more modest lateral contraction in the approximately affine case and to a stiffer BSSN. Exponential stiffening emerges from the interplay of the axial and bending deformation modes at the scale of individual or small groups of fibers undergoing large deformations and being subjected to the constraint of rigid cross-links, and it is not necessarily a result of complex interactions involving many connected fibers. An apparent third regime of quadratic stiffening may be evidenced in nonaffinely deforming networks provided the nominal stress is observed. This occurs at large stretches, when the BSSN contribution of stiffening vanishes. However, this regime is not present if the Cauchy stress is used, in which case stiffening is exponential throughout the entire deformation. These results shed light on the physical nature of stiffening in a broad class of materials including connective tissue, the extracellular matrix, nonwovens, felt, and other athermal network materials.

Property Regulation of Conjugated Oligoelectrolytes with Polyisocyanide to Achieve Efficient Photodynamic Antibacterial Biomimetic Hydrogels

Fabricating antibacterial hydrogels with antimicrobial drugs and synthetic biocompatible biomimetic hydrogels is a promising strategy for practical medical applications. Here, we report a bicomponent hydrogel composed of a biomimetic polyisocyanopetide (PIC) hydrogel and a photodynamic antibacterial membrane-intercalating conjugated oligoelectrolyte (COE). The aggregation behavior and aggregate size of the COEs in water can be regulated using the PIC hydrogel, which could induce COEs with higher reactive oxygen species (ROS) production efficiency and increased association of COEs toward bacteria, therefore enhancing the antibacterial efficiency. This strategy provides a facile method for developing biomimetic hydrogels with high antibacterial capability.

Actin filament alignment causes mechanical hysteresis in cross-linked networks

Cells dynamically control their material properties through remodeling of the actin cytoskeleton, an assembly of cross-linked networks and bundles formed from the biopolymer actin. We recently found that cross-linked networks of actin filaments reconstituted in vitro can exhibit adaptive behavior and thus serve as a model system to understand the underlying mechanisms of mechanical adaptation of the cytoskeleton. In these networks, training, in the form of applied shear stress, can induce asymmetry in the nonlinear elasticity. Here, we explore control over this mechanical hysteresis by tuning the concentration and mechanical properties of cross-linking proteins in both experimental and simulated networks. We find that this effect depends on two conditions: the initial network must exhibit nonlinear strain stiffening, and filaments in the network must be able to reorient during training. Hysteresis depends strongly and non-monotonically on cross-linker concentration, with a peak at moderate concentrations. In contrast, at low concentrations, where the network does not strain stiffen, or at high concentrations, where filaments are less able to rearrange, there is little response to training. Additionally, we investigate the effect of changing cross-linker properties and find that longer or more flexible cross-linkers enhance hysteresis. Remarkably plotting hysteresis against alignment after training yields a single curve regardless of the physical properties or concentration of the cross-linkers.

Effect of crosslink torsional stiffness on elastic behavior of semiflexible polymer networks

Networks of semiflexible filaments are building blocks of different biological and structural materials such as cytoskeleton and extracellular matrix. The mechanical response of these systems when subjected to an applied strain at zero temperature is often investigated numerically using networks composed of filaments, which are either rigidly welded or pinned together at their crosslinks. In the latter, filaments during deformation are free to rotate about their crosslinks while the relative angles between filaments remain constant in the former. The behavior of crosslinks in actual semiflexible networks is different than these idealized models and there exists only partial constraint on torques at crosslinks. The present work develops a numerical model in which two intersecting filaments are connected to each other by torsional springs with arbitrary stiffness. We show that fiber networks composed of rigid and freely rotating crosslinks are the limiting case of the present model. Furthermore, we characterize the effects of stiffness of crosslinks on effective Young's modulus of semiflexible networks as a function of filament flexibility and crosslink density. The effective Young's modulus is determined as a function of the mechanical properties of crosslinks and is found to vanish for networks composed of very weak torsional springs. Independent of the stiffness of crosslinks, it is found that the effective Young's modulus is a function of fiber flexibility and crosslink density. In low density networks, filaments primarily bend and the effective Young's modulus is much lower than the affine estimate. With increasing filament bending stiffness and/or crosslink density, the mechanical behavior of the networks becomes more affine and the stretching of filaments depicts itself as the dominant mode of deformation. The torsional stiffness of the crosslinks significantly affects the effective Young's modulus of the semiflexible random fiber networks.

Crosslinking of fibrous hydrogels

In contrast to most synthetic hydrogels, biological gels are made of fibrous networks. This architecture gives rise to unique properties, like low concentration, high porosity gels with a high mechanical responsiveness as a result of strain-stiffening. Here, we used a synthetic polymer model system, based on polyisocyanides, that we crosslinked selectively inside the bundles. This approach allows us to lock in the fibrous network present at the crosslinking conditions. At minimum crosslink densities, we are able to freeze in the architecture, as well as the associated mechanical properties. Rheology and X-ray scattering experiments show that we able to accurately tailor network mechanics, not by changing the gel composition or architecture, but rather by tuning its (thermal) history. Selective crosslinking is a crucial step in making biomimetic networks with a controlled architecture.

Unlike synthetic hydrogels, biological gels are made of fibrous networks which give rise to unique properties, such as high porosity and mechanical responsiveness. Here the authors use polyisocyanide-based gels and selectively crosslink inside the bundles to lock the fibrous network and thus control the architecture and the mechanics.

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.

Topology determines force distributions in one-dimensional random spring networks

Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber-reinforced materials are also common in technology. An important characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore, it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of N nodes and have an average degree of connectivity z but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters (N,z). Despite the universal properties of such (N,z) ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks.

Topology Counts: Force Distributions in Circular Spring Networks

Filamentous polymer networks govern the mechanical properties of many biological materials. Force distributions within these networks are typically highly inhomogeneous, and, although the importance of force distributions for structural properties is well recognized, they are far from being understood quantitatively. Using a combination of probabilistic and graph-theoretical techniques, we derive force distributions in a model system consisting of ensembles of random linear spring networks on a circle. We show that characteristic quantities, such as the mean and variance of the force supported by individual springs, can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes. Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in an elastic spring network. Our results for 1D linear spring networks readily generalize to arbitrary dimensions.

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.

Sulfo-SMCC Prevents Annealing of Taxol-Stabilized Microtubules In Vitro

Microtubule structure and functions have been widely studied in vitro and in cells. Research has shown that cysteines on tubulin play a crucial role in the polymerization of microtubules. Here, we show that blocking sulfhydryl groups of cysteines in taxol-stabilized polymerized microtubules with a commonly used chemical crosslinker prevents temporal end-to-end annealing of microtubules in vitro. This can dramatically affect the length distribution of the microtubules. The crosslinker sulfosuccinimidyl 4-(N-maleimidomethyl)cyclohexane-1-carboxylate, sulfo-SMCC, consists of a maleimide and an N-hydroxysuccinimide ester group to bind to sulfhydryl groups and primary amines, respectively. Interestingly, addition of a maleimide dye alone does not show the same interference with annealing in stabilized microtubules. This study shows that the sulfhydryl groups of cysteines of tubulin that are vital for the polymerization are also important for the subsequent annealing of microtubules.

Two fundamental mechanisms govern the stiffening of cross-linked networks

Biopolymer networks, such as those constituting the cytoskeleton of a cell or biological tissue, exhibit a nonlinear strain-stiffening behavior when subjected to large deformations. Interestingly, rheological experiments on various in vitro biopolymer networks have shown similar strain-stiffening trends regardless of the differences in their microstructure or constituents, suggesting a universal stiffening mechanism. In this article, we use computer simulations of a random network comprised of cross-linked biopolymer-like fibers to substantiate the notion that this universality lies in the existence of two fundamental stiffening mechanisms. After showing that the large strain response is accompanied by the development of a stress path, i.e., a percolating path of axially stressed fibers and cross-links, we demonstrate that the strain stiffening can be caused by two distinctly different mechanisms: 1) the pulling out of stress-path undulations; and 2) reorientation of the stress path. The former mechanism is bending-dominated and can be recognized by a power-law dependence with exponent 3/2 of the shear modulus on stress, whereas the latter mechanism is stretching-dominated and characterized by a power-law exponent 1/2. We demonstrate how material properties of the constituents, as well as the network microstructure, can affect the transition between the two stiffening mechanisms and, as such, control the dominant power-law scaling behavior.

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.

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.