M. Sheinman

Stress controls the mechanics of collagen networks

Significance We report nonlinear rheology experiments on collagen type I networks, which demonstrate a surprising concentration independence of the network stiffness in the nonlinear elastic regime. We develop a model that can account for this, as well as the classical observations of an approximate exponential stress–strain relationship in collagenous tissues, for which a microscopic model has been lacking. Our model also demonstrates the importance of normal stresses in controlling the nonlinear mechanics of fiber networks. Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress–strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.

Molecular motors robustly drive active gels to a critically connected state

A study of an actomyosin active gel now demonstrates the importance of the crosslinking density of actin polymers in enabling myosin motors to internally drive contraction and rupture the network into clusters. These results could help us to better understand the role of the cytoskeleton in cell division and tissue morphogenesis.

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.

Strain-controlled criticality governs the nonlinear mechanics of fibre networks

Fibre networks become rigid at a critical connectivity, but biopolymers giving structure to cells aren’t always well connected. Modelling and experiments on collagen networks show that their rigidity constitutes strain-controlled critical behaviour.

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.

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.

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.

Inherently unstable networks collapse to a critical point.

M. Sheinman, A. Sharma, J. Alvarado, G. H. Koenderink, F. C. MacKintosh Department of Physics and Astronomy, VU University, Amsterdam, The Netherlands Max Planck Institute for Molecular Genetics, 14195 Berlin, Germany FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands Department of Mechanical Engineering, Hatsopoulos Microfluids Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States (Dated: February 12, 2014)

Dynamical behavior of disordered spring networks

We study the dynamical rheology of spring networks with a percolation model constructed by bond dilution in a two-dimensional triangular lattice. Hydrodynamic interactions are implemented by a Stokesian viscous coupling between the network nodes and a uniformly deforming liquid. Our simulations show that in a critical connectivity regime, these systems display weak power law rheology in which the complex shear modulus scales with frequency as G* ∼ (iω)Δ where Δ = 0.41, in discord with a mean field prediction of Δ = 1/2. The weak power law rheology in the critical regime can be understood from a simple scaling relation between the macroscopic rheology and the nonaffine strain fluctuations, which diverge with vanishing frequency for isostatic networks. We expand on a dynamic effective medium theory, showing that it quantitatively describes the rheology of a diluted triangular lattice far from isostaticity; although the EMT correctly predicts the scaling form for the rheology of near-isostatic networks, there remains a quantitative disparity due to the mean-field nature of the EMT. Surprisingly, by connecting this critical scaling of the rheology with that of the strain fluctuations, we find that the dynamical behavior of disordered spring networks is fully determined by the critical exponents that govern the behavior of elastic network in the absence of viscous interactions.

Sheinman, Sharma, and MacKintosh Reply.

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.This is consistent with Eq. (1), while satisfying Eq. (3) for τ < 2. In general, with no information about τ being larger or smaller than 2, one should analyze the numerical data for both cases. We do this in Fig. 1, e.g., by plotting sns=M vs s=M for the case τ < 2. We find good collapse and near constancy of sns=M for τ 1⁄4 1.82 and over a wide range of s=M up to ∼0.1. By contrast, attempting the same collapse for τ 1⁄4 2, where both our ansatz and that of the Comment [1] are equivalent, we do not find the expected near constancy of sns=M. Thus, while it may not be possible to entirely rule out τ 1⁄4 2with significant logarithmic corrections, our results appear to be more consistent with τ 1⁄4 1.82. In the inset, however, we have plotted the distribution log-linear, in a way closely analogous to the Comment [1]. Here, we do not find evidence of a logarithmic dependence. Our data are, in fact, consistent with a weak exponent 0.18, as indicated by the thick line. We thank the authors of the Comment [1] for their interest and the useful discussion of subtleties in interpreting the numerical data. But, we fundamentally disagree with their approach that tacitly assumes τ ≥ 2.