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.collagen networks | nonlinear elasticity | normal stress | tissue mechanics C ollagen type I is the most abundant protein in mammals where it serves as the primary component of many load-bearing tissues, including skin, ligaments, tendons, and bone. Networks of collagen type I fibers exhibit mechanical properties that are unmatched by manmade materials. A hallmark of collagen and collagenous tissues is a dramatic increase in stiffness when strained. Qualitatively, this property of strain stiffening is shared by many other biopolymers, including intracellular cytoskeletal networks of actin and intermediate filaments (1-5). On closer inspection, however, collagen stands out from the rest: it has been shown that collagenous tissues exhibit a regime in which the stress is approximately exponential in the applied strain (6). The origins of this nonlinearity are still not known (7,8), and existing models for biopolymer networks cannot account quantitatively for collagen. In particular, it is unknown whether the nonlinear mechanical response of collagen originates at the level of the individual fibers (1, 3, 9, 10) or arises from nonaffine network deformations as suggested by numerical simulations (11-17).Here, we present both experimental results on reconstituted collagen networks, as well as a model that quantitatively captures the observed nonlinear mechanics. Our model is a minimal one, of random networks of elastic fibers possessing only bending and stretching elasticity. This model can account for our striking experimental observation that the stiffness of collagen becomes independent of protein concentration in the nonlinear elastic regime, over a range of concentrations and applied shear stress. Our model highlights the importance of local network geometry in determining the strain threshold for the onset of nonlinear mechanics, which can account for the concentration independence of this threshold that is observed for collagen (8,17), in strong contrast...

Disordered fibrous networks are ubiquitous in nature as major structural components of living cells and tissues. The mechanical stability of networks generally depends on the degree of connectivity: only when the average number of connections between nodes exceeds the isostatic threshold are networks stable [1]. Upon increasing the connectivity through this point, such networks undergo a mechanical phase transition from a floppy to a rigid phase. However, even subisostatic networks become rigid when subjected to sufficiently large deformations. To study this strain-controlled transition, we perform a combination of computational modeling of fibre networks and experiments on networks of type I collagen fibers, which are crucial for the integrity of biological tissues. We show theoretically that the development of rigidity is characterized by a strain-controlled continuous phase transition with signatures of criticality. Our experiments demonstrate mechanical properties consistent with our model, including the predicted critical exponents. We show that the nonlinear mechanics of collagen networks can be quantitatively captured by the predictions of scaling theory for the straincontrolled critical behavior over a wide range of network concentrations and strains up to failure of the material.As shown by Maxwell, networks with only central-force interactions exhibit a transition from a floppy to rigid phase at the isostatic point, where the local coordination number, or connectivity z equals the threshold value of z = 2d in d dimensions [1]. At this point, the number of degrees of freedom is just balanced by the number of constraints, and the system is marginally stable to small deformations. The jamming transition [2][3][4][5][6] in granular materials and rigidity percolation [7][8][9][10] in disordered spring networks are examples of such a transition. An important feature of these systems is the order of the transition. Jamming exhibits signatures of both first-and second-order transitions, with discontinuous behaviour of the bulk modulus and continuous variation of the shear modulus [5,11,12]. For networks of springs or fibres, the transition from floppy to rigid is a continuous phase transition, in both bulk and shear moduli [2, 5, 7, 7, 8,14].Interestingly, the structural networks in biology almost always have connectivities below the central-force iso- static point. Networks such as the extracellular matrices of collagen that make up tissue are strictly sub-isostatic with respect to central forces: their typical connectivity is between 3 (local branching) and 4 (binary crosslinking), placing them below both 2D and 3D isostatic thresholds [17,18]. Such sub-isostatic networks can, nevertheless, become rigid as a result of other mechanical constraints, such as fibre bending [4, 7,19], or when subjected to external strain [21]. The threshold strain, at which the transition occurs, depends on the nature of applied deformation (shear or tensile) and on the average connectivity of the network, in particular, and othe...

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. Alvarado et al.Molecular motors robustly drive active gels to a critically connected state Page 2 of 21 behaviors has been proposed, since macroscopic contractions are known to occur above certain minimum values of crosslink or actin concentration 14,17,19,20 . We should expect remarkable critical behavior at the threshold of contraction. Recent theoretical models predict diverging correlation length-scales and a strong response to external fields 21-24 at the threshold of rigidity.In suspensions of self-propelled patches, critical slowing was predicted at the threshold of alignment 25 . Yet the threshold of contraction still remains poorly understood, and experimental evidence of criticality in active gels remains lacking. Alvarado et al. Molecular motors robustly drive active gels to a critically connected state Page 3 of 21Here, we experimentally study model cytoskeletal systems composed of actin filaments and myosin motors. We vary network connectivity over a broad range by adding controlled amounts of crosslink protein. We show that the motors can actively contract the networks into disjoint clusters that exhibit a power-law size distribution. This behavior is reminiscent of classical conductivity percolation 26 , for which a power-law size distribution of clusters occurs close to a critical point. However, in sharp contrast to this equilibrium phenomenon, we observe critical behavior over a wide range of initial network connectivities. To understand this robustness, we develop a general theoretical model of contractile gels that can quantitatively account for our observations. In this model, motors not only contract the network, but also reduce the connectivity of initially stable networks down to a marginal structure by promoting crosslink unbinding. Below this marginal connectivity, the network no longer supports stress and the system rapidly devolves to disjoint clusters which reflect the critical behavior of th...

We present a model for disordered 3D fiber networks to study their linear and nonlinear elasticity over a wide range of network densities and fiber lengths. In contrast to previous 2D models, these 3D networks with binary cross-links are under-constrained with respect to fiber stretching elasticity, suggesting that bending may dominate their response. We find that such networks exhibit a fiber length-controlled bending regime and a crossover to a stretch-dominated regime for lengths beyond a characteristic scale that depends on the fiber's elastic properties. 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 floppy or long filaments.