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Fibrous networks such as collagen are common in biological systems. Recent theoretical and experimental efforts have shed light on the mechanics of single component networks. Most real biopolymer networks, however, are composites made of elements with different rigidity. For instance, the extracellular matrix in mammalian tissues consists of stiff collagen fibers in a background matrix of flexible polymers such as hyaluronic acid (HA). The interplay between different biopolymer components in such composite networks remains unclear. In this work, we use 2D coarse-grained models to study the nonlinear strain-stiffening behavior of composites. We introduce a local volume constraint to model the incompressibility of HA. We also perform rheology experiments on composites of collagen with HA. We demonstrate both theoretically and experimentally that the linear shear modulus of composite networks can be increased by approximately an order of magnitude above the corresponding moduli of the pure components. Our model shows that this synergistic effect can be understood in terms of the local incompressibility of HA, which acts to suppress density fluctuations of the collagen matrix with which it is entangled.

Self-organisation is the spontaneous emergence of spatio-temporal structures and patterns from the interaction of smaller individual units. Examples are found across many scales in very different systems and scientific disciplines, from physics, materials science and robotics to biology, geophysics and astronomy. Recent research has highlighted how self-organisation can be both mediated and controlled by confinement. Confinement occurs through interactions with boundaries, and can function as either a catalyst or inhibitor of self-organisation. It can then become a means to actively steer the emergence or suppression of collective phenomena in space and time. Here, to provide a common framework for future research, we examine the role of confinement in self-organisation and identify overarching scientific challenges across disciplines that need to be addressed to harness its full scientific and technological potential. This framework will not only accelerate the generation of a common deeper understanding of self-organisation but also trigger the development of innovative strategies to steer it through confinement, with impact, e.g., on the design of smarter materials, tissue engineering for biomedicine and crowd management.

Living tissue is able to withstand large stresses in everyday life, yet it also actively adapts to dynamic loads. This remarkable mechanical behaviour emerges from the interplay between living cells and their non-living extracellular environment. Here we review recent insights into the biophysical mechanisms involved in the reciprocal interplay between cells and the extracellular matrix and how this interplay determines tissue mechanics, with a focus on connective tissues. We first describe the roles of the main macromolecular components of the extracellular matrix in regards to tissue mechanics. We then proceed to highlight the main routes via which cells sense and respond to their biochemical and mechanical extracellular environment. Next we introduce the three main routes via which cells can modify their extracellular environment: exertion of contractile forces, secretion and deposition of matrix components, and matrix degradation. Finally we discuss how recent insights in the mechanobiology of cell-matrix interactions are furthering our understanding of the pathophysiology of connective tissue diseases and cancer, and facilitating the design of novel strategies for tissue engineering., Comment: 24 pages, 4 figures, review article

Collagen forms the structural scaffold of connective tissues in all mammals. Tissues are remarkably resistant against mechanical deformations because collagen molecules hierarchically self-assemble in fibrous networks that stiffen with increasing strain. Nevertheless, collagen networks do fracture when tissues are overloaded or subject to pathological conditions such as aneurysms. Prior studies of the role of collagen in tissue fracture have mainly focused on tendons, which contain highly aligned bundles of collagen. By contrast, little is known about fracture of the orientationally more disordered collagen networks present in many other tissues such as skin and cartilage. Here, we combine shear rheology of reconstituted collagen networks with computer simulations to investigate the primary determinants of fracture in disordered collagen networks. We show that the fracture strain is controlled by the coordination number of the network junctions, with less connected networks fracturing at larger strains. The hierarchical structure of collagen fine-tunes the fracture strain by providing structural plasticity at the network and fiber level. Our findings imply that structural disorder provides a protective mechanism against network fracture that can optimize the strength of biological tissues.

Soft living tissues like cartilage can be considered as biphasic materials comprised of a fibrous complex biopolymer network and a viscous background liquid. Here, we show by a combination of experiment and theoretical analysis that both the hydraulic permeability and the elastic properties of (bio)polymer networks can be determined with simple ramp compression experiments in a commercial rheometer. In our approximate closed-form solution of the poroelastic equations of motion, we find the normal force response during compression as a combination of network stress and fluid pressure. Choosing fibrin as a biopolymer model system with controllable pore size, measurements of the full time-dependent normal force during compression are found to be in excellent agreement with the theoretical calculations. The inferred elastic response of large-pore ($\mathrm{\mu m}$) fibrin networks depends on the strain rate, suggesting a strong interplay between network elasticity and fluid flow. Phenomenologically extending the calculated normal force into the regime of nonlinear elasticity, we find strain-stiffening of small-pore (sub-$\mathrm{\mu m}$) fibrin networks to occur at an onset average tangential stress at the gel-plate interface that depends on the polymer concentration in a power-law fashion. The inferred permeability of small-pore fibrin networks scales approximately inverse squared with the fibrin concentration, implying with a microscopic cubic lattice model that the thickness of the fibrin fibers decreases with protein concentration. Our theoretical model provides a new method to obtain the hydraulic permeability and the elastic properties of biopolymer networks and hydrogels with simple compression experiments, and paves the way to study the relation between fluid flow and elasticity in biopolymer networks during dynamical compression.

Hyaluronic acid is an abundant polyelectrolyte in the human body that forms extracellular hydrogels in connective tissues. It is essential for regulating tissue biomechanics and cell-cell communication, yet hyaluronan overexpression is associated with pathological situations such as cancer and multiple sclerosis. Due to its enormous molecular weight (in the range of millions of Daltons), accumulation of hyaluronan hinders transport of macromolecules including nutrients and growth factors through tissues and also hampers drug delivery. However, the exact contribution of hyaluronan to tissue penetrability is poorly understood due to the complex structure and molecular composition of tissues. Here we reconstitute biomimetic hyaluronan gels and systematically investigate the effects of gel composition and crosslinking on the diffusion of microscopic tracer particles. We combine ensemble-averaged measurements via differential dynamic microscopy with single-particle tracking. We show that the particle diffusivity depends on the particle size relative to the network pore size and also on the stress relaxation dynamics of the network. We furthermore show that addition of collagen, the other major biopolymer in tissues, causes the emergence of caged particle dynamics, which is not present in the one-component systems. Our findings are useful for understanding macromolecular transport in tissues and for designing biomimetic extracellular matrix hydrogels for drug delivery and tissue regeneration.

Intermediate filaments are cytoskeletal proteins that are key regulators of cell mechanics, a role which is intrinsically tied to their hierarchical structure and their unique ability to accommodate large axial strains. However, how the single-filament response to applied strains translates to networks remains unclear, particularly with regards to the crosslinking role played by the filaments' disordered "tail" domains. Here we test the role of these noncovalent crosslinks in the nonlinear rheology of reconstituted networks of the intermediate filament protein vimentin, probing their stress- and rate- dependent mechanics. Similarly to previous studies we observe elastic stress-stiffening but unlike previous work we identify a characteristic yield stress $\sigma^*$, above which the networks exhibit rate-dependent softening of the network, referred to as $\textit{inelastic fluidization}$. By investigating networks formed from tail-truncated vimentin, in which noncovalent crosslinking is suppressed, and glutaraldehyde-treated vimentin, in which crosslinking is made permanent, we show that rate-dependent inelastic fluidization is a direct consequence of vimentin's transient crosslinking. Surprisingly, although the tail-tail crosslinks are individually weak, the effective timescale for stress relaxation of the network exceeds 1000s at $\sigma^*$. Vimentin networks can therefore maintain their integrity over a large range of strains (up to ~1000\%) and loading rates (10$^{-3}$ to 10$^{3}s^{-1}$). Our results provide insight into how the hierarchical structure of vimentin networks contributes to the cell's ability to be deformable yet strong.

The cells and tissues that make up our body juggle contradictory mechanical demands. It is crucial for their survival to be able to withstand large mechanical loads, but it is equally crucial for them to produce forces and actively change shape during biological processes such as tissue growth and repair. The mechanics of cell and tissues is determined by scaffolds of protein polymers known as the cytoskeleton and the extracellular matrix, respectively. Experiments on model systems reconstituted from purified components combined with polymer physics concepts have already successfully uncovered some of the mechanisms that underlie the paradoxical mechanics of living matter. Initial work focussed on explaining universal features such as the nonlinear elasticity of cells and tissues in terms of polymer network models. However, living matter exhibits many advanced mechanical functionalities that are not captured by these coarse-grained theories. In this Review, we focus on recent experimental and theoretical insights revealing how their porous structure, structural hierarchy, transient crosslinking, and mechanochemical activity confer resilience combined with the ability to adapt and self-heal. These physical insights improve our understanding of cell and tissue biology and also provide a source of inspiration for synthetic life-like materials.

Dynamically crosslinked 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 crosslinker 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 crosslinking combined with internal stress can explain prior reports of soft glassy rheology of cells, in which the shear modulus increases weakly with frequency., Comment: 6 pages, 4 figures

Living tissues show an extraordinary adaptiveness to strain, which is crucial for their proper biological functioning. The physical origin of this mechanical behaviour has been widely investigated using reconstituted networks of collagen fibres, the principal load-bearing component of tissues. However, collagen fibres in tissues are embedded in a soft hydrated polysaccharide matrix which generates substantial internal stresses whose effect on tissue mechanics is unknown. Here, by combining mechanical measurements and computer simulations, we show that networks composed of collagen fibres and a hyaluronan matrix exhibit synergistic mechanics characterized by an enhanced stiffness and delayed strain-stiffening. We demonstrate that the polysaccharide matrix has a dual effect on the composite response involving both internal stress and elastic reinforcement. Our findings elucidate how tissues can tune their strain-sensitivity over a wide range and provide a novel design principle for synthetic materials with programmable mechanical properties.

Transient networks comprised of polymers connected by short-lived bonds are a common design theme for both biological and synthetic materials. Transient bonds can provide mechanical rigidity, while still allowing for visco-elastic flows on timescales longer than the bond lifetime. In many biological polymer networks such as the actin cytoskeleton, the short-lived bonds are formed by accessory proteins that diffuse away after unbinding. By contrast, bonds in synthetic networks, such as the pendant groups of telechelic polymers, can only rebind in the same location. Using a recently developed theoretical model of the fracturing of visco-elastic materials, we here investigate the effect of linker mobility on the bond dynamics of a network under stress. We find that although mean field properties such as the average bond linker lifetime are barely affected by bond mobility, networks cross linked by mobile bonds fracture more readily due to 'leaking' of linkers from crack areas to less stressed regions within the network. We propose a theoretical model to describe the redistribution of mobile linkers, which we validate by simulations. Our work offers insight into a potential trade-off that cells face, between fracture strength versus the modularity and tight dynamic control offered by mobile linkers., Comment: 8 pages, 4 figures

Cells and tissues exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, which includes an active gel of actin filaments, crosslinks, and myosin molecular motors. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (order nN and greater) on cellular, tissue, and organismal length scales (order 10 ${\mu}$m and greater). Experimental studies of reconstituted active-network model systems have long suggested that a mechanical interplay between the network's connectivity and motor activity dominates contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, where the network connectivity is influenced by motor activity. Here we review experimental and theoretical studies of model contractile active gels and propose a state diagram that unites a large body of experimental and theoretical observations based on concepts from percolation theory.

Fibrous networks are ideal functional materials since they provide mechanical rigidity at low weight. Such structures are omnipresent in natural biomaterials from cells to tissues, as well as in man-made materials from polymeric composites to paper and textiles. Here, we demonstrate that fibrous networks of the blood clotting protein fibrin undergo a strong and irreversible increase in their mechanical rigidity in response to compression. This rigidification can be precisely predetermined from the level of applied compressive strain, providing a means to program the network rigidity without having to change its composition. To identify the mechanism underlying this programmable rigidification, we measure single fiber-fiber interactions using optical tweezers. We further develop a minimal computational model of adhesive fiber networks that shows that load-induced bond formation can explain the adaptation of the fibrin networks to compressive loading. The model predicts that the network stiffness after compressive programming obeys a universal power-law dependence on the prestress built in by new bond formation, which we confirm experimentally. The generality of this functional stiffening mechanism together with our ability to quantitatively predict it provides a new powerful approach to program the stiffness of fibrous materials and dynamically adapt them to different loading conditions.

While semi-flexible polymers and fibers are an important class of material due to their rich mechanical properties, it remains unclear how these properties relate to the microscopic conformation of the polymers. Actin filaments constitute an ideal model polymer system due to their micron-sized length and relatively high stiffness that allow imaging at the single filament level. Here we study the effect of entanglements on the conformational dynamics of actin filaments in shear flow. We directly measure the full three-dimensional conformation of single actin filaments, using confocal microscopy in combination with a counter-rotating cone-plate shear cell. We show that initially entangled filaments form disentangled orientationally ordered hairpins, confined in the flow-vorticity plane. In addition, shear flow causes stretching and shear alignment of the hairpin tails, while the filament length distribution remains unchanged. These observations explain the strain-softening and shear-thinning behavior of entangled F-actin solutions, which aids the understanding of the flow behavior of complex fluids containing semi-flexible polymers., Comment: supplementary material; see Journal reference

Living systems often exhibit internal driving: active, molecular processes drive nonequilibrium phenomena such as metabolism or migration. Active gels constitute a fascinating class of internally driven matter, where molecular motors exert localized stresses inside polymer networks. There is evidence that network crosslinking is required to allow motors to induce macroscopic contraction. Yet a quantitative understanding of how network connectivity enables contraction is lacking. Here we show experimentally that myosin motors contract crosslinked actin polymer networks to clusters with a scale-free size distribution. This critical behavior occurs over an unexpectedly broad range of crosslink concentrations. To understand this robustness, we develop a quantitative model of contractile networks that takes into account network restructuring: motors reduce connectivity by forcing crosslinks to unbind. Paradoxically, to coordinate global contractions, motor activity should be low. Otherwise, motors drive initially well-connected networks to a critical state where ruptures form across the entire network., Comment: Main text: 21 pages, 5 figures. Supplementary Information: 13 pages, 8 figures