Nonlinear Poisson Effect Governed by a Mechanical Critical Transition.

Under extensional strain, fiber networks can exhibit an anomalously large and nonlinear Poisson effect accompanied by a dramatic transverse contraction and volume reduction for applied strains as small as a few percent. We demonstrate that this phenomenon is controlled by a collective mechanical phase transition that occurs at a critical uniaxial strain that depends on network connectivity. This transition is punctuated by an anomalous peak in the apparent Poisson's ratio and other critical signatures such as diverging nonaffine strain fluctuations.

Origin of Slow Stress Relaxation in the Cytoskeleton.

Dynamically cross-linked semiflexible biopolymers such as the actin cytoskeleton govern the mechanical behavior of living cells. Semiflexible biopolymers nonlinearly stiffen in response to mechanical loads, whereas the cross-linker dynamics allow for stress relaxation over time. Here we show, through rheology and theoretical modeling, that the combined nonlinearity in time and stress leads to an unexpectedly slow stress relaxation, similar to the dynamics of disordered systems close to the glass transition. Our work suggests that transient cross-linking combined with internal stress can explain prior reports of soft glassy rheology of cells, in which the shear modulus increases weakly with frequency.

Normal stress anisotropy and marginal stability in athermal elastic networks.

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

Reply to the 'Comment on "A symmetrical method to obtain shear moduli from microrheology"' by M. Tassieri, Soft Matter, 2018, 14, DOI.

The Comment on our paper introducing "a symmetric method to obtain shear moduli from microrheology" proposes an interpolation method to generate oversampled data from an original time series that are then used to approximate shear moduli at frequencies "beyond the Nyquist frequency." The author states that this can be done without the use of "preconceived fitting functions," implying that the results are unique and reliable. We disagree with these assertions. While it is possible to generate reasonable looking transforms at frequencies above the Nyquist limit by interpolation, any results obtained above the Nyquist limit will be questionable at best. Moreover, while the cubic spline interpolation the author uses may be standard, it constitutes a particular "preconceived" fit and produces oversampled data that are not unique.

A symmetrical method to obtain shear moduli from microrheology.

Passive microrheology typically deduces shear elastic loss and storage moduli from displacement time series or mean-squared displacements (MSD) of thermally fluctuating probe particles in equilibrium materials. Common data analysis methods use either Kramers-Kronig (KK) transformation or functional fitting to calculate frequency-dependent loss and storage moduli. We propose a new analysis method for passive microrheology that avoids the limitations of both of these approaches. In this method, we determine both real and imaginary components of the complex, frequency-dependent response function χ(ω) = χ'(ω) + iχ''(ω) as direct integral transforms of the MSD of thermal particle motion. This procedure significantly improves the high-frequency fidelity of χ(ω) relative to the use of KK transformation, which has been shown to lead to artifacts in χ'(ω). We test our method on both model and experimental data. Experiments were performed on solutions of worm-like micelles and dilute collagen solutions. While the present method agrees well with established KK-based methods at low frequencies, we demonstrate significant improvement at high frequencies using our symmetric analysis method, up to almost the fundamental Nyquist limit.

Broken Detailed Balance of Filament Dynamics in Active Networks.

Myosin motor proteins drive vigorous steady-state fluctuations in the actin cytoskeleton of cells. Endogenous embedded semiflexible filaments such as microtubules, or added filaments such as single-walled carbon nanotubes are used as novel tools to noninvasively track equilibrium and nonequilibrium fluctuations in such biopolymer networks. Here, we analytically calculate shape fluctuations of semiflexible probe filaments in a viscoelastic environment, driven out of equilibrium by motor activity. Transverse bending fluctuations of the probe filaments can be decomposed into dynamic normal modes. We find that these modes no longer evolve independently under nonequilibrium driving. This effective mode coupling results in nonzero circulatory currents in a conformational phase space, reflecting a violation of detailed balance. We present predictions for the characteristic frequencies associated with these currents and investigate how the temporal signatures of motor activity determine mode correlations, which we find to be consistent with recent experiments on microtubules embedded in cytoskeletal networks.

Porosity Governs Normal Stresses in Polymer Gels.

When sheared, most elastic solids including metals, rubbers, and polymer gels dilate perpendicularly to the shear plane. This behavior, known as the Poynting effect, is characterized by a positive normal stress. Surprisingly, fibrous biopolymer gels exhibit a negative normal stress under shear. Here we show that this anomalous behavior originates from the open-network structure of biopolymer gels. Using fibrin networks with a controllable pore size as a model system, we show that the normal-stress response to an applied shear is positive at short times, but decreases to negative values with a characteristic time scale set by pore size. Using a two-fluid model, we develop a quantitative theory that unifies the opposite behaviors encountered in synthetic and biopolymer gels.

Critical behaviour in the nonlinear elastic response of hydrogels.

In this paper we study the elastic response of synthetic hydrogels to an applied shear stress. The hydrogels studied here have previously been shown to mimic the behaviour of biopolymer networks when they are sufficiently far above the gel point. We show that near the gel point they exhibit an elastic response that is consistent with the predicted critical behaviour of networks near or below the isostatic point of marginal stability. This point separates rigid and floppy states, distinguished by the presence or absence of finite linear elastic moduli. Recent theoretical work has also focused on the response of such networks to finite or large deformations, both near and below the isostatic point. Despite this interest, experimental evidence for the existence of criticality in such networks has been lacking. Using computer simulations, we identify critical signatures in the mechanical response of sub-isostatic networks as a function of applied shear stress. We also present experimental evidence consistent with these predictions. Furthermore, our results show the existence of two distinct critical regimes, one of which arises from the nonlinear stretch response of semi-flexible polymers.

Anomalous discontinuity at the percolation critical point of active gels.

We develop a percolation model motivated by recent experimental studies of gels with active network remodeling by molecular motors. This remodeling was found to lead to a critical state reminiscent of random percolation (RP), but with a cluster distribution inconsistent with RP. Our model not only can account for these experiments, but also exhibits an unusual type of mixed phase transition: We find that the transition is characterized by signatures of criticality, but with a discontinuity in the order parameter.

Nonthermal fluctuations of the mitotic spindle.

We present direct measurements of fluctuations in the nucleus of yeast cells. While prior work has shown these fluctuations to be active and non-thermal in character, their origin and time dependence are not understood. We show that the nuclear fluctuations we observe are quantitatively consistent with uncorrelated, active force fluctuations driving a nuclear medium that is dominated by an uncondensed DNA solution, for which we perform rheological measurements on an in vitro model system under similar conditions to what is expected in the nucleus.

Fluctuation-stabilized marginal networks and anomalous entropic elasticity.

We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have a vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work, we show that thermal networks exhibit a nonzero shear modulus G well below the isostatic point and that this modulus exhibits an anomalous, sublinear dependence on temperature T. At the isostatic point, G increases as the square root of T, while we find G∝Tα below the isostatic point, where α≃0.8. We show that this anomalous T dependence is entropic in origin.

Actively stressed marginal networks.

We study the effects of motor-generated stresses in disordered three-dimensional fiber networks using a combination of a mean-field theory, scaling analysis, and a computational model. We find that motor activity controls the elasticity in an anomalous fashion close to the point of marginal stability by coupling to critical network fluctuations. We also show that motor stresses can stabilize initially floppy networks, extending the range of critical behavior to a broad regime of network connectivities below the marginal point. Away from this regime, or at high stress, motors give rise to a linear increase in stiffness with stress. Finally, we demonstrate that our results are captured by a simple, constitutive scaling relation highlighting the important role of nonaffine strain fluctuations as a susceptibility to motor stress.

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.

Nonlinear elasticity in biological gels.

The mechanical properties of soft biological tissues are essential to their physiological function and cannot easily be duplicated by synthetic materials. Unlike simple polymer gels, many biological materials--including blood vessels, mesentery tissue, lung parenchyma, cornea and blood clots--stiffen as they are strained, thereby preventing large deformations that could threaten tissue integrity. The molecular structures and design principles responsible for this nonlinear elasticity are unknown. Here we report a molecular theory that accounts for strain-stiffening in a range of molecularly distinct gels formed from cytoskeletal and extracellular proteins and that reveals universal stress-strain relations at low to intermediate strains. The input to this theory is the force-extension curve for individual semi-flexible filaments and the assumptions that biological networks composed of these filaments are homogeneous, isotropic, and that they strain uniformly. This theory shows that systems of filamentous proteins arranged in an open crosslinked mesh invariably stiffen at low strains without requiring a specific architecture or multiple elements with different intrinsic stiffness.

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.

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.

The mechanics and fluctuation spectrum of active gels.

Recent experiments on molecular motor driven in vitro F-Actin networks have found anomalously large strain fluctuations at low frequency. In addition, the shear modulus of these active networks becomes as much as one hundred times larger than that of the same system in equilibrium. We develop a two-fluid model of a low-density semiflexible network driven by molecular motors to explore these effects and show that, relying on only simple assumptions regarding the motor activity in the system we can quantitatively understand both the low-frequency fluctuation enhancement and the nonequilibrium stiffening of the network. These results have implications for the interpretation of microrheology in such active networks including the cytoskeleton of living cells. In addition, they may form the basis for theoretical studies of biomimetic nonequilibrium gels whose mechanical properties are tunable through the control of their nonequilibrium steady-state.

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

Recent experiments have demonstrated that the nonlinear elasticity of in vitro networks of the biopolymer actin is dramatically altered in the presence of a flexible cross-linker such as the abundant cytoskeletal protein filamin. The basic principles of such networks remain poorly understood. Here we describe an effective-medium theory of flexibly cross-linked stiff polymer networks. We argue that the response of the cross-links can be fully attributed to entropic stiffening, while softening due to domain unfolding can be ignored. The network is modeled as a collection of randomly oriented rods connected by flexible cross-links to an elastic continuum. This effective medium is treated in a linear elastic limit as well as in a more general framework, in which the medium self-consistently represents the nonlinear network behavior. This model predicts that the nonlinear elastic response sets in at strains proportional to cross-linker length and inversely proportional to filament length. Furthermore, we find that the differential modulus scales linearly with the stress in the stiffening regime. These results are in excellent agreement with bulk rheology data.

Nonlinear elasticity of stiff biopolymers connected by flexible linkers.

Networks of the biopolymer actin, cross-linked by the compliant protein filamin, form soft gels. They can, however, withstand large shear stresses due to their pronounced nonlinear elastic behavior. The nonlinear elasticity can be controlled by varying the number of cross-links per actin filament. We propose and test a model of rigid filaments decorated by multiple flexible linkers that is in quantitative agreement with experiment. This allows us to estimate loads on individual cross-links, which we find to be less than 10 pN.