Deformation of cross-linked semiflexible polymer networks

Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior

The microstructure of tissues and tissue equivalents (TEs) plays a critical role in determining the mechanical properties thereof. One of the key challenges in constitutive modeling of TEs is incorporating the kinematics at both the macroscopic and the microscopic scale. Models of fibrous microstructure commonly assume fibrils to move homogeneously, that is affine with the macroscopic deformation. While intuitive for situations of fibril-matrix load transfer, the relevance of the affine assumption is less clear when primary load transfer is from fibril to fibril. The microstructure of TEs is a hydrated network of collagen fibrils, making its microstructural kinematics an open question. Numerical simulation of uniaxial extensile behavior in planar TE networks was performed with fibril kinematics dictated by the network model and by the affine model. The average fibril orientation evolved similarly with strain for both models. The individual fibril kinematics, however, were markedly different. There was no correlation between fibril strain and orientation in the network model, and fibril strains were contained by extensive reorientation. As a result, the macroscopic stress given by the network model was roughly threefold lower than the affine model. Also, the network model showed a toe region, where fibril reorientation precluded the development of significant fibril strain. We conclude that network fibril kinematics are not governed by affine principles, an important consideration in the understanding of tissue and TE mechanics, especially when load bearing is primarily by an interconnected fibril network.

Mechanobiology in the third dimension

Cells are mechanically coupled to their extracellular environments, which play critical roles in both communicating the state of the mechanical environment to the cell as well as in mediating cellular response to a variety of stimuli. Along with the molecular composition and mechanical properties of the extracellular matrix (ECM), recent work has demonstrated the importance of dimensionality in cell-ECM associations for controlling the sensitive communication between cells and the ECM. Matrix forces are generally transmitted to cells differently when the cells are on two-dimensional (2D) vs. within three-dimensional (3D) matrices, and cells in 3D environments may experience mechanical signaling that is unique vis-à-vis cells in 2D environments, such as the recently described 3D-matrix adhesion assemblies. This review examines how the dimensionality of the extracellular environment can affect in vitro cell mechanobiology, focusing on collagen and fibrin systems.

A master relation defines the nonlinear viscoelasticity of single fibroblasts

Cell mechanical functions such as locomotion, contraction, and division are controlled by the cytoskeleton, a dynamic biopolymer network whose mechanical properties remain poorly understood. We perform single-cell uniaxial stretching experiments on 3T3 fibroblasts. By superimposing small amplitude oscillations on a mechanically prestressed cell, we find a transition from linear viscoelastic behavior to power law stress stiffening. Data from different cells over several stress decades can be uniquely scaled to obtain a master relation between the viscoelastic moduli and the average force. Remarkably, this relation holds independently of deformation history, adhesion biochemistry, and intensity of active contraction. In particular, it is irrelevant whether force is actively generated by the cell or externally imposed by stretching. We propose that the master relation reflects the mechanical behavior of the force-bearing actin cytoskeleton, in agreement with stress stiffening known from semiflexible filament networks.

Prestressed F-actin networks cross-linked by hinged filamins replicate mechanical properties of cells

We show that actin filaments, shortened to physiological lengths by gelsolin and cross-linked with recombinant human filamins (FLNs), exhibit dynamic elastic properties similar to those reported for live cells. To achieve elasticity values of comparable magnitude to those of cells, the in vitro network must be subjected to external prestress, which directly controls network elasticity. A molecular requirement for the strain-related behavior at physiological conditions is a flexible hinge found in FLNa and some FLNb molecules. Basic physical properties of the in vitro filamin-F-actin network replicate the essential mechanical properties of living cells. This physical behavior could accommodate passive deformation and internal organelle trafficking at low strains yet resist externally or internally generated high shear forces.

Stiff polymers, foams, and fiber networks

We study the elasticity of fibrous materials composed of generalized stiff polymers. It is shown that, in contrast to cellular foam-like structures, affine strain fields are generically unstable. Instead, a subtle interplay between the architecture of the network and the elastic properties of its building blocks leads to intriguing mechanical properties with intermediate asymptotic scaling regimes. We present exhaustive numerical studies based on a finite element method complemented by scaling arguments.

Fitting of random tessellation models to keratin filament networks

The role of specific structural patterns in keratin filament networks for regulating biophysical properties of epithelial cells is poorly understood. This is at least partially due to a lack of methods for the analysis of filament network morphology. We have previously developed a statistical approach to the analysis of keratin filament networks imaged by scanning electron microscopy. The segmentation of images in this study resulted in graph structures, i.e. tessellations, whose structural characteristics are now further investigated by iteratively fitting geometrical statistical models. An optimal model as well as corresponding optimal parameters are detected from a given set of possible random tessellation models, i.e. Poisson-Line tessellations (PLT), Poisson-Voronoi tessellations (PVT) and Poisson-Delaunay tessellations (PDT). Using this method, we investigated the remodeling of keratin filament networks in pancreatic cancer cells in response to transforming growth factor alpha (TGFalpha), which is involved in pancreatic cancer progression. The results indicate that the fitting of random tessellation models represents a suitable method for the description of complex filament networks.

Nonaffine correlations in random elastic media

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements from a uniform reference state grow linearly with position , and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements as the difference between and affine displacements, and we investigate the nonaffinity correlation function and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants (none of which are infinite), by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that scales as where the amplitude is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short- and long-range correlations in random elastic moduli.

Mechanical response of semiflexible networks to localized perturbations

Previous research on semiflexible polymers including cytoskeletal networks in cells has suggested the existence of distinct regimes of elastic response, in which the strain field is either uniform (affine) or nonuniform (nonaffine) under external stress. Associated with these regimes, it has been further suggested that a mesoscopic length scale emerges, which characterizes the scale for the crossover from nonaffine to affine deformations. Here, we extend these studies by probing the response to localized forces and force dipoles. We show that the previously identified nonaffinity length [D. A. Head, Phys. Rev. E 68, 061907 (2003)] controls the mesoscopic response to point forces and the crossover to continuum elastic behavior at large distances.

Quantitative analysis of keratin filament networks in scanning electron microscopy images of cancer cells

The keratin filament network is an important part of the cytoskeleton. It is involved in the regulation of shape and viscoelasticity of epithelial cells. The morphology of keratin networks depends on post-translational modifications of keratin monomers. In-vitro studies indicated that network characteristics, such as filament crosslink density, determines the biophysical properties of the filament network. This report presents a quantitative method for the morphological analysis of keratin filament networks. Visualization of filaments was based on prefixation extraction of epithelial cells and scanning electron microscopy (SEM). SEM images were processed by a skeletonization algorithm to obtain a graph structure that represents individual filaments as well as their connections. This method was applied to investigate the effects of transforming growth factor alpha (TGFalpha) on the morphology of keratin networks in pancreatic cancer cells. TGFalpha contributes to pancreatic cancer progression and activates signalling pathways phosphorylating keratin monomers. Using this new method, a significant alteration to the keratin network morphology could be detected in response to TGFalpha.

Alternative explanation of stiffening in cross-linked semiflexible networks

Strain stiffening of filamentous protein networks is explored by means of a finite strain analysis of a two-dimensional network model of cross-linked semiflexible filaments. The results show that stiffening is caused by nonaffine network rearrangements that govern a transition from a bending-dominated response at small strains to a stretching-dominated response at large strains. Filament undulations, which are key in the existing explanation of stiffening, merely postpone the transition.

Discrete modeling of the mechanics of entangled materials

We employ a discrete computational model to study the entanglement transition of non-cross-linked semiflexible fibers during isostatic compressions. We determine, as a function of the fiber aspect ratio, packing densities and caging numbers, i.e., the density and number of contacts per fiber at the entanglement transition. The caging number is found to be 8 for short fibers and to drop down to 4 for longer fibers. Compressions beyond the entanglement transition allow us to determine, for these networks that deform primarily by bending, the scaling exponents of the pressure and of the bulk modulus (=3), as well as of the number of contacts per fiber (=1).

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.

Generic theory of active polar gels: a paradigm for cytoskeletal dynamics

We develop a general theory for active viscoelastic materials made of polar filaments. This theory is motivated by the dynamics of the cytoskeleton. The continuous consumption of a fuel leads to a non equilibrium state characterized by the generation of flows and stresses. Our theory can be applied to experiments in which cytoskeletal patterns are set in motion by active processes such as those which are at work in cells.

Semiflexible chain networks formed via self-assembly of beta-hairpin molecules

We report experimental results from a de novo designed oligopeptide that intermolecularly self-assembles into rigid hydrogel networks after an intramolecular folding event. Microscopy and neutron scattering reveal a fibril local structure that is approximately 3 nm in diameter and over several hundred nanometers in length. Oscillatory rheology suggests that the peptidic network viscoelastic behavior follows that theoretically predicted for heavily cross-linked, semiflexible polymer networks.

Scaling of F-actin network rheology to probe single filament elasticity and dynamics

The linear and nonlinear viscoelastic response of networks of cross-linked and bundled cytoskeletal filaments demonstrates remarkable scaling with both frequency and applied prestress, which helps elucidate the origins of the viscoelasticity. The frequency dependence of the shear modulus reflects the underlying single-filament relaxation dynamics for 0.1-10 rad/sec. Moreover, the nonlinear strain stiffening of such networks exhibits a universal form as a function of prestress; this is quantitatively explained by the full force-extension relation of single semiflexible filaments.

Viscoelasticity of single wall carbon nanotube suspensions

We investigate the viscoelastic properties of an associating rigid rod network: aqueous suspensions of surfactant stabilized single wall carbon nanotubes (SWNTs). The SWNT suspensions exhibit a rigidity percolation transition with an onset of solidlike elasticity at a volume fraction of 0.0026; the percolation exponent is 2.3+/-0.1. At large strain, the solidlike samples show volume fraction dependent yielding. We develop a simple model to understand these rheological responses and show that the shear dependent stresses can be scaled onto a single master curve to obtain an internanotube interaction energy per bond approximately 40k(B)T. Our experimental observations suggest SWNTs in suspension form interconnected networks with bonds that freely rotate and resist stretching. Suspension elasticity originates from bonds between SWNTs rather than from the stiffness or stretching of individual SWNTs.

Anomalous diffusion in living yeast cells

The viscoelastic properties of the cytoplasm of living yeast cells were investigated by studying the motion of lipid granules naturally occurring in the cytoplasm. A large frequency range of observation was obtained by a combination of video-based and laser-based tracking methods. At time scales from 10(-4) to 10(2) s, the granules typically perform subdiffusive motion with characteristics different from previous measurements in living cells. This subdiffusive behavior is thought to be due to the presence of polymer networks and membranous structures in the cytoplasm. Consistent with this hypothesis, we observe that the motion becomes less subdiffusive upon actin disruption.

Elastic behavior of cross-linked and bundled actin networks

Networks of cross-linked and bundled actin filaments are ubiquitous in the cellular cytoskeleton, but their elasticity remains poorly understood. We show that these networks exhibit exceptional elastic behavior that reflects the mechanical properties of individual filaments. There are two distinct regimes of elasticity, one reflecting bending of single filaments and a second reflecting stretching of entropic fluctuations of filament length. The mechanical stiffness can vary by several decades with small changes in cross-link concentration, and can increase markedly upon application of external stress. We parameterize the full range of behavior in a state diagram and elucidate its origin with a robust model.

Transverse fluctuations of grafted polymers

We study the statistical mechanics of grafted polymers of arbitrary stiffness in a two-dimensional embedding space with Monte Carlo simulations. The probability distribution function of the free end is found to be highly anisotropic and non-Gaussian for typical semiflexible polymers. The reduced distribution in the transverse direction, a Gaussian in the stiff and flexible limits, shows a double-peak structure at intermediate stiffnesses. We also explore the response to a transverse force applied at the polymer free end. We identify F-Actin as an ideal benchmark for the effects discussed.

Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks

Semiflexible polymers such as filamentous actin (F-actin) play a vital role in the mechanical behavior of cells, yet the basic properties of cross-linked F-actin networks remain poorly understood. To address this issue, we have performed numerical studies of the linear response of homogeneous and isotropic two-dimensional networks subject to an applied strain at zero temperature. The elastic moduli are found to vanish for network densities at a rigidity percolation threshold. For higher densities, two regimes are observed: one in which the deformation is predominately affine and the filaments stretch and compress; and a second in which bending modes dominate. We identify a dimensionless scalar quantity, being a combination of the material length scales, that specifies to which regime a given network belongs. A scaling argument is presented that approximately agrees with this crossover variable. By a direct geometric measure, we also confirm that the degree of affinity under strain correlates with the distinct elastic regimes. We discuss the implications of our findings and suggest possible directions for future investigations.