Frequency-dependent stiffening of semiflexible networks: a dynamical nonaffine to affine transition

Revisiting the strain-induced softening behaviour in hydrogels

The strain-induced softening behaviour observed in the differential modulus K(T,γ) of hydrogels is typically attributed to the breakage of internal network structures, such as the cross-links that bind the polymer chains. In this study, however, we consider a stress-strain relationship derived from a coarse-grained model to demonstrate that rupture of the network is not necessary for rubber-like gels to exhibit such behaviour. In particular, we show that, in some cases, the decrease of K(T,γ) as a function of the strain γ can be associated with the energy-related contribution to the elastic modulus that has been experimentally observed, e.g., for tetra-PEG hydrogels. Our findings suggest that the softening behaviour can be also attributed to the effective interaction between polymer chains and their surrounding solvent molecules, rather than the breakage of structural elements. We compare our theoretical expressions with experimental data determined for several hydrogels to illustrate and validate our approach.

Structural Features and Nonlinear Rheology of Self-Assembled Networks of Cross-Linked Semiflexible Polymers

Disordered networks of semiflexible filaments are common support structures in biology. Familiar examples include fibrous matrices in blood clots, bacterial biofilms, and essential components of cells and tissues of plants, animals, and fungi. Despite the ubiquity of these networks in biomaterials, we have only a limited understanding of the relationship between their structural features and their highly strain-sensitive mechanical properties. In this work, we perform simulations of three-dimensional networks produced by the irreversible formation of cross-links between linker-decorated semiflexible filaments. We characterize the structure of networks formed by a simple diffusion-dependent assembly process and measure their associated steady-state rheological features at finite temperature over a range of applied prestrains that encompass the strain-stiffening transition. We quantify the dependence of network connectivity on cross-linker availability and detail the associated connectivity dependence of both linear elasticity and nonlinear strain-stiffening behavior, drawing comparisons with prior experimental measurements of the cross-linker concentration-dependent elasticity of actin gels.

Viscoelastic Scaling Regimes for Marginally Rigid Fractal Spring Networks

A family of marginally rigid (isostatic) spring networks with fractal structure up to a controllable length was devised, and the viscoelastic spectra G^{*}(ω) calculated. Two nontrivial scaling regimes were observed, (i) G^{'}≈G^{''}∝ω^{Δ} at low frequencies, consistent with Δ=1/2, and (ii) G^{'}∝G^{''}∝ω^{Δ^{'}} for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction Δ^{'}=(ln3-ln2)/(ln3+ln2). The crossover between these two regimes occurred at lower frequencies for larger fractals in a manner suggesting diffusivelike dispersion. Solid gels generated by introducing internal stresses exhibited similar behavior above a low-frequency cutoff, indicating the relevance of these findings to real-world applications.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

Anisotropic mechanical response of layered disordered fibrous materials

Mechanically bonded fabrics account for a significant portion of nonwoven products, and serve many niche areas of nonwoven manufacturing. Such fabrics are characterized by layers of disordered fibrous webs, but we lack an understanding of how such microstructures determine bulk material response. Here we numerically determine the linear shear response of needle-punched fabrics modeled as cross-linked sheets of two-dimensional (2D) Mikado networks. We systematically vary the intra-sheet fiber density, inter-sheet separation distance, and direction of shear, and quantify the macroscopic shear modulus alongside the degree of affinity and energy partition. For shear parallel to the sheets, the response is dominated by intrasheet fibers and follows known trends for 2D Mikado networks. By contrast, shears perpendicular to the sheets induce a softer response dominated by either intrasheet or intersheet fibers depending on a quadratic relation between sheet separation and fiber density. These basic trends are reproduced and elucidated by a simple scaling argument that we provide. We discuss the implications of our findings in the context of real nonwoven fabrics.

Nonaffinity and Fluid-Coupled Viscoelastic Plateau for Immersed Fiber Networks

We employ a matrix-based solver for the linear rheology of fluid-immersed disordered spring networks to reveal four distinct dynamic response regimes. One regime-completely absent in the known vacuum response-exhibits coupled fluid flow and network deformation, with both components responding nonaffinely. This regime contains an additional plateau (peak) in the frequency-dependent storage (loss) modulus-features that vanish without full hydrodynamic interactions. The mechanical response of immersed networks such as biopolymers and hydrogels is thus richer than previously established and offers additional modalities for design and control through fluid interactions.

Cohesive and adhesive properties of crosslinked semiflexible biopolymer networks

Biomolecular semiflexible polymer networks with persistence lengths well above those of single polymeric chains serve important structural and adhesive roles in biology, biomaterials, food science and many other fields. While relationships between the structure and viscoelasticity of semiflexible polymer networks have been previously investigated, it remains challenging to systematically relate fibril and network properties to cohesive and adhesive properties that govern the function of these materials. To address this issue, here we utilize coarse-grained molecular dynamics simulations to thoroughly elucidate how the work of adhesion of a semiflexible polymer network to a surface depends on crosslink density and fibril persistence length. Two emergent characteristics of the network are its elasticity and its interfacial energy with the surface. Stiff networks that are either highly crosslinked or have high persistence length fibrils tend to have lower interfacial energy, and consequently, lower work of adhesion. For lightly crosslinked networks with flexible fibrils, considerable strain energy must be stored within the adhesive during detachment, which creates an additional penalty to detachment. Increasing persistence length while keeping crosslink density constant leads to porous, low density networks, leading to an optimal fibril persistence length at which maximum work of adhesion per mass density is attained for a given crosslink density. For any given fibril persistence length, increasing crosslink density has a slightly negative effect on network mass density and interfacial energy. A critical crosslink density is found, below which the networks have no significant load-bearing capacity. Lightly crosslinked networks above this threshold absorb more strain energy during desorption and consequently possess greater work of adhesion. The conflict between mass density and stiffness results in a non-monotonic trend between the ratio of work of adhesion to interfacial energy and persistence length. These findings provide physical insight into the adhesive mechanisms of biomaterials based on crosslinked semiflexible polymer networks, and reveal important design guidelines for bio-adhesives.

Linear rheology of reversibly cross-linked biopolymer networks

We suggest a simple model for reversible cross-links, binding, and unbinding to/from a network of semiflexible polymers. The resulting frequency dependent response of the network to an applied shear is calculated via Brownian dynamics simulations. It is shown to be rather complex with the time scale of the linkers competing with the excitations of the network. If the lifetime of the linkers is the longest time scale, as is indeed the case in most biological networks, then a distinct low frequency peak of the loss modulus develops. The storage modulus shows a corresponding decay from its plateau value, which for irreversible cross-linkers extends all the way to the static limit. This additional relaxation mechanism can be controlled by the relative weight of reversible and irreversible linkers.

Microstructure of Sheared Entangled Solutions of Semiflexible Polymers

We study the influence of finite shear deformations on the microstructure and rheology of solutions of entangled semiflexible polymers theoretically and by numerical simulations and experiments with filamentous actin. Based on the tube model of semiflexible polymers, we predict that large finite shear deformations strongly affect the average tube width and curvature, thereby exciting considerable restoring stresses. In contrast, the associated shear alignment is moderate, with little impact on the average tube parameters, and thus expected to be long-lived and detectable after cessation of shear. Similarly, topologically preserved hairpin configurations are predicted to leave a long-lived fingerprint in the shape of the distributions of tube widths and curvatures. Our numerical and experimental data support the theory.

Importance of non-affine viscoelastic response in disordered fibre networks

Disordered fibre networks are ubiquitous in nature and have a wide range of industrial applications as novel biomaterials. Predicting their viscoelastic response is straightforward for affine deformations that are uniform over all length scales, but when affinity fails, as has been observed experimentally, modelling becomes challenging. Here we present a numerical methodology, related to an existing framework for amorphous packings, to predict the steady-state viscoelastic spectra and degree of affinity for disordered fibre networks driven at arbitrary frequencies. Applying this method to a peptide gel model reveals a monotonic increase of the shear modulus as the soft, non-affine normal modes are successively suppressed as the driving frequency increases. In addition to being dominated by fibril bending, these low frequency network modes are also shown to be delocalised. The presented methodology provides insights into the importance of non-affinity in the viscoelastic response of peptide gels, and is easily extendible to all types of fibre networks.

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.

High-frequency affine mechanics and nonaffine relaxation in a model cytoskeleton

The cytoskeleton is a network of crosslinked, semiflexible filaments, and it has been suggested that it has properties of a glassy state. Here we employ optical-trap-based microrheology to apply forces to a model cytoskeleton and measure the high-bandwidth response at an anterior point. Simulating the highly nonlinear and anisotropic stress-strain propagation assuming affinity, we found that theoretical predictions for the quasistatic response of semiflexible polymers are only realized at high frequencies inaccessible to conventional rheometers. We give a theoretical basis for determining the frequency when both affinity and quasistaticity are valid, and we discuss with experimental evidence that the relaxations at lower frequencies can be characterized by the experimentally obtained nonaffinity parameter.

Normal stresses in elastic networks

When loaded in simple shear deformation, polymeric materials may develop so-called normal stresses: stresses perpendicular to the direction of the applied shear. These normal stresses are intrinsically nonlinear: basic symmetry considerations dictate they may only enter at O(γ(2)), with γ the dimensionless shear strain. There is no fundamental restriction on their sign, and normal stresses may be positive (pushing outward) or negative (pulling inward). Most materials tend to dilate in the normal direction, but a wide variety of biopolymer networks including fibrin and actin gels have been reported to present anomalously large, negative normal stresses-a feature which has been ascribed to the intrinsic elastic nonlinearity of semiflexible fibers. In this work, we present analytical results on a model nonlinear network, which we expand to the required nonlinear order to show that due to geometric, rather than elastic, nonlinearities (negative) normal stresses generically arise in filamentous networks-even in networks composed of linear, Hookean springs. We investigate analytically and numerically how the subsequent addition of elastic nonlinearities, nonaffine deformations, and filament persistence through cross-linkers augment this basic behavior.

Elasticity of a filamentous kagome lattice

The diluted kagome lattice, in which bonds are randomly removed with probability 1-p, consists of straight lines that intersect at points with a maximum coordination number of 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective-medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus μ and bending modulus κ, are used to study the elasticity of this lattice as functions of p and κ. At p=1, elastic response is purely affine, and the macroscopic elastic modulus G is independent of κ. When κ=0, the lattice undergoes a first-order rigidity-percolation transition at p=1. When κ>0, G decreases continuously as p decreases below one, reaching zero at a continuous rigidity-percolation transition at p=p(b)≈0.605 that is the same for all nonzero values of κ. The effective-medium theories predict scaling forms for G, which exhibit crossover from bending-dominated response at small κ/μ to stretching-dominated response at large κ/μ near both p=1 and p(b), that match simulations with no adjustable parameters near p=1. The affine response as p→1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

Two-photon time-lapse microscopy of BODIPY-cholesterol reveals anomalous sterol diffusion in chinese hamster ovary cells

Background

Cholesterol is an important membrane component, but our knowledge about its transport in cells is sparse. Previous imaging studies using dehydroergosterol (DHE), an intrinsically fluorescent sterol from yeast, have established that vesicular and non-vesicular transport modes contribute to sterol trafficking from the plasma membrane. Significant photobleaching, however, limits the possibilities for in-depth analysis of sterol dynamics using DHE. Co-trafficking studies with DHE and the recently introduced fluorescent cholesterol analog BODIPY-cholesterol (BChol) suggested that the latter probe has utility for prolonged live-cell imaging of sterol transport.

Results

We found that BChol is very photostable under two-photon (2P)-excitation allowing the acquisition of several hundred frames without significant photobleaching. Therefore, long-term tracking and diffusion measurements are possible. Two-photon temporal image correlation spectroscopy (2P-TICS) provided evidence for spatially heterogeneous diffusion constants of BChol varying over two orders of magnitude from the cell interior towards the plasma membrane, where D ~ 1.3 μm²/s. Number and brightness (N&B) analysis together with stochastic simulations suggest that transient partitioning of BChol into convoluted membranes slows local sterol diffusion. We observed sterol endocytosis as well as fusion and fission of sterol-containing endocytic vesicles. The mobility of endocytic vesicles, as studied by particle tracking, is well described by a model for anomalous subdiffusion on short time scales with an anomalous exponent α ~ 0.63 and an anomalous diffusion constant of D_α = 1.95 x 10^-3 μm²/s^α. On a longer time scale (t > ~5 s), a transition to superdiffusion consistent with slow directed transport with an average velocity of v ~ 6 x 10^-3 μm/s was observed. We present an analytical model that bridges the two regimes and fit this model to vesicle trajectories from control cells and cells with disrupted microtubule or actin filaments. Both treatments reduced the anomalous diffusion constant and the velocity by ~40-50%.

Conclusions

The mobility of sterol-containing vesicles on the short time scale could reflect dynamic rearrangements of the cytoskeleton, while directed transport of sterol vesicles occurs likely along both, microtubules and actin filaments. Spatially varying anomalous diffusion could contribute to fine-tuning and local regulation of intracellular sterol transport.

Non-affine deformations in polymer hydrogels

Most theories of soft matter elasticity assume that the local strain in a sample after deformation is identical everywhere and equal to the macroscopic strain, or equivalently that the deformation is affine. We discuss the elasticity of hydrogels of crosslinked polymers with special attention to affine and non-affine theories of elasticity. Experimental procedures to measure non-affine deformations are also described. Entropic theories, which account for gel elasticity based on stretching out individual polymer chains, predict affine deformations. In contrast, simulations of network deformation that result in bending of the stiff constituent filaments generally predict non-affine behavior. Results from experiments show significant non-affine deformation in hydrogels even when they are formed by flexible polymers for which bending would appear to be negligible compared to stretching. However, this finding is not necessarily an experimental proof of the non-affine model for elasticity. We emphasize the insights gained from experiments using confocal rheoscope and show that, in addition to filament bending, sample micro-inhomogeneity can be a significant alternative source of non-affine deformation.

Gelation of semiflexible polyelectrolytes by multivalent counterions

Filamentous polyelectrolytes in aqueous solution aggregate into bundles by interactions with multivalent counterions. These effects are well documented by experiment and theory. Theories also predict a gel phase in isotropic rodlike polyelectrolyte solutions caused by multivalent counterion concentrations much lower than those required for filament bundling. We report here the gelation of Pf1 virus, a model semiflexible polyelectrolyte, by the counterions Mg(2+), Mn(2+) and spermine(4+). Gelation can occur at 0.04% Pf1 volume fraction, which is far below the isotropic-nematic transition of 0.7% for Pf1 in monovalent salt. Unlike strongly crosslinked gels of semiflexible polymers, which stiffen at large strains, Pf1 gels reversibly soften at high strain. The onset strain for softening depends on the strength of interaction between counterions and the polyelectrolyte. Simulations show that the elasticity of counterion crosslinked gels is consistent with a model of semiflexible filaments held by weak crosslinks that reversibly rupture at a critical force.