The role of structure in the nonlinear mechanics of cross-linked semiflexible polymer networks

Review paper: The importance of consideration of collagen cross-links in computational models of collagen-based tissues

Collagen as the main protein in Extra Cellular Matrix (ECM) is the main load-bearing component of fibrous tissues. Nanostructure and architecture of collagen fibrils play an important role in mechanical behavior of these tissues. Extensive experimental and theoretical studies have so far been performed to capture these properties, but none of the current models realistically represent the complexity of network mechanics because still less is known about the collagen's inner structure and its effect on the mechanical properties of tissues. The goal of this review article is to emphasize the significance of cross-links in computational modeling of different collagen-based tissues, and to reveal the need for continuum models to consider cross-links properties to better reflect the mechanical behavior observed in experiments. In addition, this study outlines the limitations of current investigations and provides potential suggestions for the future work.

A High Coordination of Cross-Links Is Beneficial for the Strength of Cross-Linked Fibers

The influence of the coordination of (reversible) cross-links on the mechanical properties of aligned fiber bundles is investigated. Two polymeric systems containing cross-links of different coordination (two- and three-fold coordination) but having the same binding energy are investigated. In particular, the response to loading of these systems is compared. Mechanical parameters (strength, stiffness and work-to-fracture) are obtained by computational loading tests. The influence of coordination is studied for simple test systems with pre-defined topologies that maximize strength as well as for more realistic fiber bundles containing nine chains. The results show that a higher coordination of cross-links has a beneficial effect on the strength and the stiffness of the systems, while the work-to-fracture was found larger for the system having a smaller coordination of cross-links. It can be concluded that controlling the coordination of cross-links is a versatile tool to specifically tailor the mechanical properties of polymeric structures.

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.

Multi-scale strain-stiffening of semiflexible bundle networks

Bundles of polymer filaments are responsible for the rich and unique mechanical behaviors of many biomaterials, including cells and extracellular matrices. In fibrin biopolymers, whose nonlinear elastic properties are crucial for normal blood clotting, protofibrils self-assemble and bundle to form networks of semiflexible fibers. Here we show that the extraordinary strain-stiffening response of fibrin networks is a direct reflection of the hierarchical architecture of the fibrin fibers. We measure the rheology of networks of unbundled protofibrils and find excellent agreement with an affine model of extensible wormlike polymers. By direct comparison with these data, we show that physiological fibrin networks composed of thick fibers can be modeled as networks of tight protofibril bundles. We demonstrate that the tightness of coupling between protofibrils in the fibers can be tuned by the degree of enzymatic intermolecular crosslinking by the coagulation factor XIII. Furthermore, at high stress, the protofibrils contribute independently to the network elasticity, which may reflect a decoupling of the tight bundle structure. The hierarchical architecture of fibrin fibers can thus account for the nonlinearity and enormous elastic resilience characteristic of blood clots.

Mechanobiology of cell migration in the context of dynamic two-way cell-matrix interactions

Migration of cells is integral in various physiological processes in all facets of life. These range from embryonic development, morphogenesis, and wound healing, to disease pathology such as cancer metastasis. While cell migratory behavior has been traditionally studied using simple assays on culture dishes, in recent years it has been increasingly realized that the physical, mechanical, and chemical aspects of the matrix are key determinants of the migration mechanism. In this paper, we will describe the mechanobiological changes that accompany the dynamic cell-matrix interactions during cell migration. Furthermore, we will review what is to date known about how these changes feed back to the dynamics and biomechanical properties of the cell and the matrix. Elucidating the role of these intimate cell-matrix interactions will provide not only a better multi-scale understanding of cell motility in its physiological context, but also a more holistic perspective for designing approaches to regulate cell behavior.

The Elastic Behaviour of Sintered Metallic Fibre Networks: A Finite Element Study by Beam Theory

Background

The finite element method has complimented research in the field of network mechanics in the past years in numerous studies about various materials. Numerical predictions and the planning efficiency of experimental procedures are two of the motivational aspects for these numerical studies. The widespread availability of high performance computing facilities has been the enabler for the simulation of sufficiently large systems.

Objectives and Motivation

In the present study, finite element models were built for sintered, metallic fibre networks and validated by previously published experimental stiffness measurements. The validated models were the basis for predictions about so far unknown properties.

Materials and Methods

The finite element models were built by transferring previously published skeletons of fibre networks into finite element models. Beam theory was applied as simplification method.

Results and Conclusions

The obtained material stiffness isn’t a constant but rather a function of variables such as sample size and boundary conditions. Beam theory offers an efficient finite element method for the simulated fibre networks. The experimental results can be approximated by the simulated systems. Two worthwhile aspects for future work will be the influence of size and shape and the mechanical interaction with matrix materials.

The interplay of fibronectin functionalization and TGF-β1 presence on fibroblast proliferation, differentiation and migration in 3D matrices

TGF-β1 dependent fibroblast behaviour in a wound healing context is mimicked by topologically and mechanically defined collagen matrices with fibronectin functionalization.

Convolution theory for dynamic systems: a bottom-up approach to the viscoelasticity of polymeric networks

Biological materials such as the cytoskeleton are characterized by remarkable viscoelastic properties and therefore represent the subject of numerous micro- and macrorheological experimental studies. By generalizing the previously introduced dynamic convolution theory (DCT) to two dimensions, we devise a bottom-up approach for the viscoelastic properties of extended, crosslinked semiflexible polymer networks. Brownian dynamics (BD) simulations serve to determine the dynamic linear self- and cross-response properties of isolated semiflexible polymers to externally applied forces and torques; these response functions are used as input to the DCT. For a given network topology, the frequency-dependent response of the network subject to a given external force/torque distribution is calculated via the DCT allowing to resolve both micro- and macrorheological properties of the networks. A mapping on continuum viscoelastic theory yields the corresponding viscoelastic bulk moduli. Special attention is drawn to the flexibility of crosslinkers, which couple angular degrees of freedom at the network nodes and which are found to sensitively affect the resulting rheological properties of the polymeric meshwork.

Modeling of fibrin gels based on confocal microscopy and light-scattering data

Fibrin gels are biological networks that play a fundamental role in blood coagulation and other patho/physiological processes, such as thrombosis and cancer. Electron and confocal microscopies show a collection of fibers that are relatively monodisperse in diameter, not uniformly distributed, and connected at nodal points with a branching order of ∼3-4. Although in the confocal images the hydrated fibers appear to be quite straight (mass fractal dimension D(m) = 1), for the overall system 1<D(m)<2. Based on the confocal images, we developed a method to generate three-dimensional (3D) in silico gels made of cylindrical sticks of diameter d, density ρ, and average length <L>, joined at randomly distributed nodal points. The resulting 3D network strikingly resembles real fibrin gels and can be sketched as an assembly of densely packed fractal blobs, i.e., regions of size ξ, where the fiber concentration is higher than average. The blobs are placed at a distance ξ0 between their centers of mass so that they are overlapped by a factor η =ξ/ξ0 and have D(m) ∼1.2-1.6. The in silico gels' structure is quantitatively analyzed by its 3D spatial correlation function g(3D)(r) and corresponding power spectrum I(q) = FFT(3D[g3D(r)]), from which ρ, d, D(m), η, and ξ0 can be extracted. In particular, ξ0 provides an excellent estimate of the gel mesh size. The in silico gels' I(q) compares quite well with real gels' elastic light-scattering measurements. We then derived an analytical form factor for accurately fitting the scattering data, which allowed us to directly recover the gels' structural parameters.