Effective-medium approach for stiff polymer networks with flexible cross-links

Nonlinear Poisson effect in affine semiflexible polymer networks

Stretching an elastic material along one axis typically induces contraction along the transverse axes, a phenomenon known as the Poisson effect. From these strains, one can compute the specific volume, which generally either increases or, in the incompressible limit, remains constant as the material is stretched. However, in networks of semiflexible or stiff polymers, which are typically highly compressible yet stiffen significantly when stretched, one instead sees a significant reduction in specific volume under finite strains. This volume reduction is accompanied by increasing alignment of filaments along the strain axis and a nonlinear elastic response, with stiffening of the apparent Young's modulus. For semiflexible networks, in which entropic bending elasticity governs the linear elastic regime, the nonlinear Poisson effect is caused by the nonlinear force-extension relationship of the constituent filaments, which produces a highly asymmetric response of the constituent polymers to stretching and compression. The details of this relationship depend on the geometric and elastic properties of the underlying filaments, which can vary greatly in experimental systems. Here, we provide a comprehensive characterization of the nonlinear Poisson effect in an affine network model and explore the influence of filament properties on essential features of both microscopic and macroscopic response, including strain-driven alignment and volume reduction.

Myosin-independent stiffness sensing by fibroblasts is regulated by the viscoelasticity of flowing actin

The stiffness of the extracellular matrix induces differential tension within integrin-based adhesions, triggering differential mechanoresponses. However, it has been unclear if the stiffness-dependent differential tension is induced solely by myosin activity. Here, we report that in the absence of myosin contractility, 3T3 fibroblasts still transmit stiffness-dependent differential levels of traction. This myosin-independent differential traction is regulated by polymerizing actin assisted by actin nucleators Arp2/3 and formin where formin has a stronger contribution than Arp2/3 to both traction and actin flow. Intriguingly, despite only slight changes in F-actin flow speed observed in cells with the combined inhibition of Arp2/3 and myosin compared to cells with sole myosin inhibition, they show a 4-times reduction in traction than cells with myosin-only inhibition. Our analyses indicate that traditional models based on rigid F-actin are inadequate for capturing such dramatic force reduction with similar actin flow. Instead, incorporating the F-actin network's viscoelastic properties is crucial. Our new model including the F-actin viscoelasticity reveals that Arp2/3 and formin enhance stiffness sensitivity by mechanically reinforcing the F-actin network, thereby facilitating more effective transmission of flow-induced forces. This model is validated by cell stiffness measurement with atomic force microscopy and experimental observation of model-predicted stiffness-dependent actin flow fluctuation.

Topological mechanism in the nonlinear power-law relaxation of cell cortex

Different types of cells exhibit a universal power-law rheology, but the mechanism underneath is still unclear. Based on the exponential distribution of actin filament length, we treat the cell cortex as a collection of chains of crosslinkers with exponentially distributed binding energy, and show that the power-law exponent of its stress relaxation should scale with the chain length. Through this model, we are able to explain how the exponent can be regulated by the crosslinker number and imposed strain during cortex relaxation. Network statistics show that the average length of filament-crosslinker chains decreases with the crosslinker number, which endows a denser network with lower exponent. Due to gradual molecular alignment with the stretch direction, the number of effectively stretched crosslinkers in the network is found to increase with the imposed strain. This effective growth in network density diminishes the exponent under large strain. By incorporating the inclined angle of crosslinkers into the model without in-series structure, we show that the exponent cannot be altered by crosslinker rotation directly, refining our previous conjectures. This work may help to understand cellular mechanics from the molecular perspective.

Network dynamics of the nonlinear power-law relaxation of cell cortex

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.

Experimental Observation of Two Features Unexpected from the Classical Theories of Rubber Elasticity

Although the elastic modulus of a Gaussian chain network is thought to be successfully described by classical theories of rubber elasticity, such as the affine and phantom models, verification experiments are largely lacking owing to difficulties in precisely controlling of the network structure. We prepared well-defined model polymer networks experimentally, and measured the elastic modulus G for a broad range of polymer concentrations and connectivity probabilities, p. In our experiment, we observed two features that were distinct from those predicted by classical theories. First, we observed the critical behavior G∼|p-p_{c}|^{1.95} near the sol-gel transition. This scaling law is different from the prediction of classical theories, but can be explained by analogy between the electric conductivity of resistor networks and the elasticity of polymer networks. Here, p_{c} is the sol-gel transition point. Furthermore, we found that the experimental G-p relations in the region above C^{*} did not follow the affine or phantom theories. Instead, all the G/G_{0}-p curves fell onto a single master curve when G was normalized by the elastic modulus at p=1, G_{0}. We show that the effective medium approximation for Gaussian chain networks explains this master curve.

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.

Stiffening of Red Blood Cells Induced by Cytoskeleton Disorders: A Joint Theory-Experiment Study

The functions and elasticities of the cell are largely related to the structures of the cytoskeletons underlying the lipid bilayer. Among various cell types, the red blood cell (RBC) possesses a relatively simple cytoskeletal structure. Underneath the membrane, the RBC cytoskeleton takes the form of a two-dimensional triangular network, consisting of nodes of actins (and other proteins) and edges of spectrins. Recent experiments focusing on the malaria-infected RBCs (iRBCs) show that there is a correlation between the elongation of spectrins in the cytoskeletal network and the stiffening of the iRBCs. Here we rationalize the correlation between these two observations by combining the wormlike chain model for single spectrins and the effective medium theory for the network elasticity. We specifically focus on how the disorders in the cytoskeletal network affect its macroscopic elasticity. Analytical and numerical solutions from our model reveal that the stiffness of the membrane increases with increasing end-to-end distances of spectrins, but has a nonmonotonic dependence on the variance of the end-to-end distance distributions. These predictions are verified quantitatively by our atomic force microscopy and micropipette aspiration measurements of iRBCs. The model may, from a molecular level, provide guidelines for future identification of new treatment methods for RBC-related diseases, such as malaria infection.

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.

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

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

Molecular control of stress transmission in the microtubule cytoskeleton

In this article, we will summarize recent progress in understanding the mechanical origins of rigidity, strength, resiliency and stress transmission in the MT cytoskeleton using reconstituted networks formed from purified components. We focus on the role of network architecture, crosslinker compliance and dynamics, and molecular determinants of single filament elasticity, while highlighting open questions and future directions for this work.

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.

Direct visualization of flow-induced conformational transitions of single actin filaments in entangled solutions

While semi-flexible polymers and fibres are an important class of material due to their rich mechanical properties, it remains unclear how these properties relate to the microscopic conformation of the polymers. Actin filaments constitute an ideal model polymer system due to their micron-sized length and relatively high stiffness that allow imaging at the single filament level. Here we study the effect of entanglements on the conformational dynamics of actin filaments in shear flow. We directly measure the full three-dimensional conformation of single actin filaments, using confocal microscopy in combination with a counter-rotating cone-plate shear cell. We show that initially entangled filaments form disentangled orientationally ordered hairpins, confined in the flow-vorticity plane. In addition, shear flow causes stretching and shear alignment of the hairpin tails, while the filament length distribution remains unchanged. These observations explain the strain-softening and shear-thinning behaviour of entangled F-actin solutions, which aids the understanding of the flow behaviour of complex fluids containing semi-flexible polymers.

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.

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.

Microrheology of highly crosslinked microtubule networks is dominated by force-induced crosslinker unbinding

We determine the time- and force-dependent viscoelastic responses of reconstituted networks of microtubules that have been strongly crosslinked by biotin-streptavidin bonds. To measure the microscale viscoelasticity of such networks, we use a magnetic tweezers device to apply localized forces. At short time scales, the networks respond nonlinearly to applied force, with stiffening at small forces, followed by a reduction in the stiffening response at high forces, which we attribute to the force-induced unbinding of crosslinks. At long time scales, force-induced bond unbinding leads to local network rearrangement and significant bead creep. Interestingly, the network retains its elastic modulus even under conditions of significant plastic flow, suggesting that crosslinker breakage is balanced by the formation of new bonds. To better understand this effect, we developed a finite element model of such a stiff filament network with labile crosslinkers obeying force-dependent Bell model unbinding dynamics. The coexistence of dissipation, due to bond breakage, and the elastic recovery of the network is possible because each filament has many crosslinkers. Recovery can occur as long as a sufficient number of the original crosslinkers are preserved under the loading period. When these remaining original crosslinkers are broken, plastic flow results.

A new approach to model cross-linked actin networks: multi-scale continuum formulation and computational analysis

The mechanical properties of a cell are defined mainly by the cytoskeleton. One contributor within this three-dimensional structure is the actin cortex which is located underneath the lipid bilayer. It forms a nearly isotropic and densely cross-linked protein network. We present a continuum mechanical formulation for describing the mechanical properties of in vitro model systems based on their micro-structure, i.e. the behavior of a single filament and its spatial arrangement. The network is considered elastic, viscous effects being neglected. Filamentous actin is a biopolymer with a highly nonlinear force-stretch relationship. This can be well described by a worm-like chain model that includes extensibility of the filament, which we call the β-model. A comparison with experimental data shows good agreement with values for the physically interpretable parameters. To make these properties applicable to three dimensions we used a non-affine micro-sphere network, which accounts for filaments, equally distributed in space. The assembled model results in a strain-energy density which is a function of the deformation gradient, and it is validated with experimental data from rheological experiments of in vitro reconstituted actin networks. The Cauchy stress and elasticity tensors are obtained within the continuum mechanics framework and implemented into a finite element program to solve boundary-value problems.

Polymer physics of the cytoskeleton

The cytoskeleton is generally visualized by light or electron microscopy as a meshwork of protein filaments that spans the space between the nuclear envelope and the plasma membrane. In most cell types, this meshwork is formed by a three dimensional composite network of actin filaments, microtubules (MT), and intermediate filaments (IF) together with the host of proteins that bind to the sides or ends of these linear polymers. Cytoskeletal binding proteins regulate filament length, crosslink filaments to each other, and apply forces to the filaments. One approach to modeling the mechanical properties of the cytoskeleton and of cell in general is to consider the elements of the cytoskeleton as polymers, using experimental methods and theoretical models developed for traditional polymers but modified for the much larger, stiffer, and fragile biopolymers comprising the cytoskeleton. The presence of motor proteins that move actin filaments and microtubules also creates a new class of active materials that are out of thermodynamic equilibrium, and unconstrained by limitations of the fluctuation-dissipation theorem. These active materials create rich opportunities for experimental design and theoretical developments. The degree to which the mechanics of live cells can usefully be modeled as highly complex polymer networks is by no means certain, and this article will discuss recent progress in quantitatively measuring cytoskeletal polymer systems and relating them to the properties of the cell.

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.

Cytoskeleton fluidization versus resolidification: prestress effect

The differential elastic modulus of an active actomyosin network is computed as a function of applied stress, taking into account both thermal and motor contributions to filament compliance in the low-frequency domain. It is shown that, due to a dual nature of motor activity, increasing motor concentration may either stiffen the network due to stronger prestress or soften it due to motor agitation, in accordance with experimental data. Prestress anisotropy, which may be induced by redistribution of motors triggered by external force, causes anisotropy of the elastic moduli. This helps to explain the contradictory phenomena of cell fluidization and resolidification in response to transient stretch observed in recent experiments.

Elasticity in ionically cross-linked neurofilament networks

Neurofilaments are found in abundance in the cytoskeleton of neurons, where they act as an intracellular framework protecting the neuron from external stresses. To elucidate the nature of the mechanical properties that provide this protection, we measure the linear and nonlinear viscoelastic properties of networks of neurofilaments. These networks are soft solids that exhibit dramatic strain stiffening above critical strains of 30-70%. Surprisingly, divalent ions such as Mg(2+), Ca(2+), and Zn(2+) act as effective cross-linkers for neurofilament networks, controlling their solidlike elastic response. This behavior is comparable to that of actin-binding proteins in reconstituted filamentous actin. We show that the elasticity of neurofilament networks is entropic in origin and is consistent with a model for cross-linked semiflexible networks, which we use to quantify the cross-linking by divalent ions.

Actin filament length tunes elasticity of flexibly cross-linked actin networks

Networks of the cytoskeletal biopolymer actin cross-linked by the compliant protein filamin form soft gels that stiffen dramatically under shear stress. We demonstrate that the elasticity of these networks shows a strong dependence on the mean length of the actin polymers, unlike networks with small, rigid cross-links. This behavior is in agreement with a model of rigid filaments connected by multiple flexible linkers.

Divalent cations crosslink vimentin intermediate filament tail domains to regulate network mechanics

Intermediate filament networks in the cytoplasm and nucleus are critical for the mechanical integrity of metazoan cells. However, the mechanism of crosslinking in these networks and the origins of their mechanical properties are not understood. Here, we study the elastic behavior of in vitro networks of the intermediate filament protein vimentin. Rheological experiments reveal that vimentin networks stiffen with increasing concentrations of Ca(2+) and Mg(2+), showing that divalent cations act as crosslinkers. We quantitatively describe the elastic response of vimentin networks over five decades of applied stress using a theory that treats the divalent cations as crosslinkers: at low stress, the behavior is entropic in origin, and increasing stress pulls out thermal fluctuations from single filaments, giving rise to a nonlinear response; at high stress, enthalpic stretching of individual filaments significantly modifies the nonlinearity. We investigate the elastic properties of networks formed by a series of protein variants with stepwise tail truncations and find that the last 11 amino acids of the C-terminal tail domain mediate crosslinking by divalent ions. We determined the single-filament persistence length, l(P) approximately 0.5 mum, and Young's modulus, Y approximately 9 MPa; both are consistent with literature values. Our results provide insight into a crosslinking mechanism for vimentin networks and suggest that divalent ions may help regulate the cytoskeletal structure and mechanical properties of cells.

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.