Strain-driven criticality underlies nonlinear mechanics of fibrous networks

Field Theory for Mechanical Criticality in Disordered Fiber Networks

Strain-controlled criticality governs the elasticity of jamming and fiber networks. While the upper critical dimension of jamming is believed to be d_{u}=2, non-mean-field exponents are observed in numerical studies of 2D and 3D fiber networks. The origins of this remains unclear. In this study we propose a minimal mean-field model for strain-controlled criticality of fiber networks. We then extend this to a phenomenological field theory, in which non-mean-field behavior emerges as a result of the disorder in the network structure. We predict that the upper critical dimension for such systems is d_{u}=4 using a Gaussian approximation. Moreover, we identify an order parameter for the phase transition, which has been lacking for fiber networks to date.

Effects of local incompressibility on the rheology of composite biopolymer networks

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. Theoretically and experimentally, we demonstrate 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.

Clots reveal anomalous elastic behavior of fiber networks

The adaptive mechanical properties of soft and fibrous biological materials are relevant to their functionality. The emergence of the macroscopic response of these materials to external stress and intrinsic cell traction from local deformations of their structural components is not well understood. Here, we investigate the nonlinear elastic behavior of blood clots by combining microscopy, rheology, and an elastic network model that incorporates the stretching, bending, and buckling of constituent fibrin fibers. By inhibiting fibrin cross-linking in blood clots, we observe an anomalous softening regime in the macroscopic shear response as well as a reduction in platelet-induced clot contractility. Our model explains these observations from two independent macroscopic measurements in a unified manner, through a single mechanical parameter, the bending stiffness of individual fibers. Supported by experimental evidence, our mechanics-based model provides a framework for predicting and comprehending the nonlinear elastic behavior of blood clots and other active biopolymer networks in general.

Mechanical criticality of fiber networks at a finite temperature

At zero temperature, spring networks with connectivity below Maxwell's isostatic threshold undergo a mechanical phase transition from a floppy state at small strains to a rigid state for applied shear strain above a critical strain threshold. Disordered networks in the floppy mechanical regime can be stabilized by entropic effects at finite temperature. We develop a scaling theory for this mechanical phase transition at finite temperature, yielding relationships between various scaling exponents. Using Monte Carlo simulations, we verify these scaling relations and identify anomalous entropic elasticity with sublinear T dependence in the linear elastic regime. While our results are consistent with prior studies of phase behavior near the isostatic point, the present work also makes predictions relevant to the broad class of disordered thermal semiflexible polymer networks for which the connectivity generally lies far below the isostatic threshold.

Effective medium theory for mechanical phase transitions of fiber networks

Networks of stiff fibers govern the elasticity of biological structures such as the extracellular matrix of collagen. These networks are known to stiffen nonlinearly under shear or extensional strain. Recently, it has been shown that such stiffening is governed by a strain-controlled athermal but critical phase transition, from a floppy phase below the critical strain to a rigid phase above the critical strain. While this phase transition has been extensively studied numerically and experimentally, a complete analytical theory for this transition remains elusive. Here, we present an effective medium theory (EMT) for this mechanical phase transition of fiber networks. We extend a previous EMT appropriate for linear elasticity to incorporate nonlinear effects via an anharmonic Hamiltonian. The mean-field predictions of this theory, including the critical exponents, scaling relations and non-affine fluctuations qualitatively agree with previous experimental and numerical results.

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.

Mechanics of fiber networks under a bulk strain

Biopolymer networks are common in biological systems from the cytoskeleton of individual cells to collagen in the extracellular matrix. The mechanics of these systems under applied strain can be explained in some cases by a phase transition from soft to rigid states. For collagen networks, it has been shown that this transition is critical in nature and it is predicted to exhibit diverging fluctuations near a critical strain that depends on the network's connectivity and structure. Whereas prior work focused mostly on shear deformation that is more accessible experimentally, here we study the mechanics of such networks under an applied bulk or isotropic extension. We confirm that the bulk modulus of subisostatic fiber networks exhibits similar critical behavior as a function of bulk strain. We find different nonmean-field exponents for bulk as opposed to shear. We also confirm a similar hyperscaling relation to what was previously found for shear.

Coupling of microtubule bundles isolates them from local disruptions to set the structural stability of the anaphase spindle

Chromosome segregation requires load-bearing interactions across kinetochore fibers and antiparallel microtubule bundles, which constitute the spindle midzone. Mechanical properties of kinetochore fibers have been characterized during metaphase, when the mitotic spindle achieves steady state. However, it has been difficult to probe the mechanics of the spindle midzone that elongates during anaphase. Here, we combine superresolution expansion and electron microscopies, lattice light-sheet imaging, and laser microsurgery to examine how midzone organization sets its mechanics. We find that individual midzone bundles extend out to multiple positions across chromosomes and form multiple apparent microtubule-based connections with each other. Across the spindle's short axis, these microtubule bundles exhibit restricted, submicrometer-amplitude motions, which are weakly correlated on <10s timescales. Severing individual midzone bundles near their center does not substantially affect positions of neighboring bundles, nor the overall structural stability of the midzone. In contrast, severing multiple midzone bundles or individual bundles at their chromosome-proximal ends significantly displaces neighboring microtubule bundles. Together, these data suggest a model wherein multiple midzone connections both reinforce its structure and mechanically isolate individual bundles from local perturbations. This feature sets the robust midzone architecture to accommodate disruptions, including those which result from lagging chromosomes, and achieve stereotypic outputs, such as proper chromosome separation.

Stiffening of under-constrained spring networks under isotropic strain

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e. networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the scaling behavior of the elastic network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus G scales linearly with isotropic stress T close to the rigidity transition. However, this seems to be at odds with recent numerical studies suggesting an exponent between G and T that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition a linear scaling, G ∼ T, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.

Force Transmission in Disordered Fibre Networks

Cells residing in living tissues apply forces to their immediate surroundings to promote the restructuration of the extracellular matrix fibres and to transmit mechanical signals to other cells. Here we use a minimalist model to study how these forces, applied locally by cell contraction, propagate through the fibrous network in the extracellular matrix. In particular, we characterize how the transmission of forces is influenced by the connectivity of the network and by the bending rigidity of the fibers. For highly connected fiber networks the stresses spread out isotropically around the cell over a distance that first increases with increasing contraction of the cell and then saturates at a characteristic length. For lower connectivity, however, the stress pattern is highly asymmetric and is characterised by force chains that can transmit stresses over very long distances. We hope that our analysis of force transmission in fibrous networks can provide a new avenue for future studies on how the mechanical feedback between the cell and the ECM is coupled with the microscopic environment around the cells.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Materials science and mechanosensitivity of living matter

Living systems are composed of molecules that are synthesized by cells that use energy sources within their surroundings to create fascinating materials that have mechanical properties optimized for their biological function. Their functionality is a ubiquitous aspect of our lives. We use wood to construct furniture, bacterial colonies to modify the texture of dairy products and other foods, intestines as violin strings, bladders in bagpipes, and so on. The mechanical properties of these biological materials differ from those of other simpler synthetic elastomers, glasses, and crystals. Reproducing their mechanical properties synthetically or from first principles is still often unattainable. The challenge is that biomaterials often exist far from equilibrium, either in a kinetically arrested state or in an energy consuming active state that is not yet possible to reproduce de novo. Also, the design principles that form biological materials often result in nonlinear responses of stress to strain, or force to displacement, and theoretical models to explain these nonlinear effects are in relatively early stages of development compared to the predictive models for rubberlike elastomers or metals. In this Review, we summarize some of the most common and striking mechanical features of biological materials and make comparisons among animal, plant, fungal, and bacterial systems. We also summarize some of the mechanisms by which living systems develop forces that shape biological matter and examine newly discovered mechanisms by which cells sense and respond to the forces they generate themselves, which are resisted by their environment, or that are exerted upon them by their environment. Within this framework, we discuss examples of how physical methods are being applied to cell biology and bioengineering.

Focused ultrasound excites cortical neurons via mechanosensitive calcium accumulation and ion channel amplification

Ultrasonic neuromodulation has the unique potential to provide non-invasive control of neural activity in deep brain regions with high spatial precision and without chemical or genetic modification. However, the biomolecular and cellular mechanisms by which focused ultrasound excites mammalian neurons have remained unclear, posing significant challenges for the use of this technology in research and potential clinical applications. Here, we show that focused ultrasound excites primary murine cortical neurons in culture through a primarily mechanical mechanism mediated by specific calcium-selective mechanosensitive ion channels. The activation of these channels results in a gradual build-up of calcium, which is amplified by calcium- and voltage-gated channels, generating a burst firing response. Cavitation, temperature changes, large-scale deformation, and synaptic transmission are not required for this excitation to occur. Pharmacological and genetic inhibition of specific ion channels leads to reduced responses to ultrasound, while over-expressing these channels results in stronger ultrasonic stimulation. These findings provide a mechanistic explanation for the effect of ultrasound on neurons to facilitate the further development of ultrasonic neuromodulation and sonogenetics as tools for neuroscience research.

Spatial collagen stiffening promotes collective breast cancer cell invasion by reinforcing extracellular matrix alignment

The tumor micro-environment often contains stiff and irregular-bundled collagen fibers that are used by tumor cells to disseminate. It is still unclear how and to what extent, extracellular matrix (ECM) stiffness versus ECM bundle size and alignment dictate cancer cell invasion. Here, we have uncoupled Collagen-I bundling from stiffness by introducing inter-collagen crosslinks, combined with temperature induced aggregation of collagen bundling. Using organotypic models from mouse invasive ductal and invasive lobular breast cancers, we show that increased collagen bundling in 3D induces a generic increase in breast cancer invasion that is independent of migration mode. However, systemic collagen stiffening using advanced glycation end product (AGE) crosslinking prevents collective invasion, while leaving single cell invasion unaffected. Collective invasion into collagen matrices by ductal breast cancer cells depends on Lysyl oxidase-like 3 (Loxl3), a factor produced by tumor cells that reinforces local collagen stiffness. Finally, we present clinical evidence that collectively invading cancer cells at the invasive front of ductal breast carcinoma upregulate LOXL3. By uncoupling the mechanical, chemical, and structural cues that control invasion of breast cancer in three dimensions, our data reveal that spatial control over stiffness and bundling underlie collective dissemination of ductal-type breast cancers.

Shear-induced phase transition and critical exponents in three-dimensional fiber networks

When subject to applied strain, fiber networks exhibit nonlinear elastic stiffening. Recent theory and experiments have shown that this phenomenon is controlled by an underlying mechanical phase transition that is critical in nature. Growing simulation evidence points to non-mean-field behavior for this transition and a hyperscaling relation has been proposed to relate the corresponding critical exponents. Here, we report simulations on two distinct network structures in three dimensions. By performing a finite-size scaling analysis, we test hyperscaling and identify various critical exponents. From the apparent validity of hyperscaling, as well as the non-mean-field exponents we observe, our results suggest that the upper critical dimension for the strain-controlled phase transition is above three, in contrast to the jamming transition that represents another athermal, mechanical phase transition.

How Structural Features of a Spring-Based Model of Fibrous Collagen Tissue Govern the Overall Young's Modulus

While much study has been dedicated to investigating biopolymers' stress-strain response at low strain levels, little research has been done to investigate the linear region of biopolymers' stress-strain response and how the microstructure affects it. We propose a mathematical model of fibrous networks which reproduces qualitative features of collagen gel's stress-strain response and provides insight into the key features which impact the Young's Modulus of similar fibrous tissues. This model analyzes the relationship of the Young's Modulus of the lattice to internodal fiber length, number of connection points or nodes per unit area, and average number of connections to each node. Our results show that fiber length, nodal density, and level of connectivity each uniquely impact the Young's Modulus of the lattice. Furthermore, our model indicates that the Young's Modulus of a lattice can be estimated using the effective resistance of the network, a graph theory technique that measures distances across a network. Our model thus provides insight into how the organization of fibers in a biopolymer impact its linear Young&#39;s Modulus.

Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.

Strain Stiffening and Negative Normal Force of Agarose Hydrogels

Inspired by the specific strain stiffening and negative normal force phenomena in several biological networks, herein, we show strain stiffening and negative normal force in agarose hydrogels. We use both pre-strain and strain amplitude sweep protocols in dynamic rheological measurements where the gel slip was suppressed by the in situ gelation in the cross-hatched parallel plate rheometer geometry. Within the stiffening region, we show the scaling relation for the differential modulus K ∝ σ1, where σ is stress. The strain at the onset of stiffening is almost constant throughout the concentration range. The gels show negative apparent normal stress difference when sheared as a result of the gel contraction. The pore size of the hydrogel is large enough to allow water to move with respect to the network to balance the pressure difference caused by the hoop stress. The rheological analysis together with scanning electron microscopy suggests that the agarose gels can be described using subisostatic athermal network models where the connectivity dictates the stiffening behavior. Therefore, the simple agarose gels appear to capture several of the viscoelastic properties, which were previously thought to be characteristic to biological protein macromolecules.

The role of temperature in the rigidity-controlled fracture of elastic networks

We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the microscopic damage spreads over a larger area and a more ductile fracture process is observed. These effects are modulated by network rigidity and can therefore be tuned via the network connectivity and the rupture threshold of the springs. Knowledge of the interplay between temperature and rigidity improves our understanding of the fracture of elastic network materials, such as (biological) polymer networks, and can help to refine design principles for tough soft materials.

Mechanical properties of subisostatic random networks composed of nonlinear fibers

Fibrous protein networks provide structural integrity to different biological materials such as soft tissues. These networks display an unusual exponential strain-stiffening behavior when subjected to mechanical loads. This nonlinear strain-stiffening behavior has so far been explained in terms of the network microstructure and the flexibility of constituting fibers. Here, we conduct a comprehensive computational study to characterize the importance of material properties of individual fibers in the overall nonlinear mechanical response of random fiber networks. To this end, we consider three nonlinear material models, ranging from an almost linear form to a highly nonlinear one, for the fibers of subisostatic disordered networks. We characterize the amount of strain-stiffening as a function of bending rigidity of the fibers, the amount of nonlinearity of the fibers, and the connectivity of random networks. We find that networks composed of highly nonlinear fibers exhibit much more strain-stiffening than networks made up of linear fibers. Furthermore, the local strain distribution becomes more homogenous as the amount of nonlinearity in the material models increases. Increasing the network connectivity signifies the importance of the nonlinear material response of individual fibers in the overall mechanical behavior of networks. The constitutive behavior of fibers plays an important role in defining the failure response of networks particularly in the damage initiation and evolution. These important findings for how the mechanical response of individual fibers affects the overall mechanical properties of random networks could find applications in designing new biomimetic materials and could help scientists better understand the mechanical properties of biological materials.

Finite size effects in critical fiber networks

Fibrous networks such as collagen are common in physiological systems. One important function of these networks is to provide mechanical stability for cells and tissues. At physiological levels of connectivity, such networks would be mechanically unstable with only central-force interactions. While networks can be stabilized by bending interactions, it has also been shown that they exhibit a critical transition from floppy to rigid as a function of applied strain. Beyond a certain strain threshold, it is predicted that underconstrained networks with only central-force interactions exhibit a discontinuity in the shear modulus. We study the finite-size scaling behavior of this transition and identify both the mechanical discontinuity and critical exponents in the thermodynamic limit. We find both non-mean-field behavior and evidence for a hyperscaling relation for the critical exponents, for which the network stiffness is analogous to the heat capacity for thermal phase transitions. Further evidence for this is also found in the self-averaging properties of fiber networks.

Nonlinear Poisson Effect Governed by a Mechanical Critical Transition

Under extensional strain, fiber networks can exhibit an anomalously large and nonlinear Poisson effect accompanied by a dramatic transverse contraction and volume reduction for applied strains as small as a few percent. We demonstrate that this phenomenon is controlled by a collective mechanical phase transition that occurs at a critical uniaxial strain that depends on network connectivity. This transition is punctuated by an anomalous peak in the apparent Poisson's ratio and other critical signatures such as diverging nonaffine strain fluctuations.

Force chains in cell-cell mechanical communication

Force chains (FCs) are a key determinant of the micromechanical properties and behaviour of heterogeneous materials, such as granular systems. However, less is known about FCs in fibrous materials, such as the networks composing the extracellular matrix (ECM) of biological systems. Using a finite-element computational model, we simulated the contraction of a single cell and two nearby cells embedded in two-dimensional fibrous elastic networks and analysed the tensile FCs that developed in the ECM. The role of ECM nonlinear elasticity on FC formation was evaluated by considering linear and nonlinear, i.e. exhibiting 'buckling' and/or 'strain-stiffening', stress-strain curves. The effect of the degree of cell contraction and network coordination value was assessed. We found that nonlinear elasticity of the ECM fibres influenced the structure of the FCs, facilitating the transition towards more distinct chains that were less branched and more radially oriented than the chains formed in linear elastic networks. When two neighbouring cells contract, a larger number of FCs bridged between the cells in nonlinear networks, and these chains had a larger effective rigidity than the chains that did not reach a neighbouring cell. These results suggest that FCs function as a route for mechanical communication between distant cells and highlight the contribution of ECM fibre nonlinear elasticity to the formation of FCs.

Random fiber networks with inclusions: The mechanism of reinforcement

The mechanical behavior of athermal random fiber networks embedding particulate inclusions is studied in this work. Composites in which the filler size is comparable with the mean segment length of the network are considered. Inclusions are randomly distributed in the network at various volume fractions, and cases in which fibers are rigidly bonded to fillers and in which no such bonding is imposed are studied separately. In the presence of inclusions, the small strain modulus increases, while the ability of the network to strain stiffen decreases relative to the unfilled network case. The reinforcement induced by fillers is most pronounced in sparse networks of floppier filaments that deform in the bending-dominated mode in the unfilled state. As the unfilled network density or the bending stiffness of fibers increases, the effect of filling diminishes rapidly. Fillers lead to a transition from the soft, bending-dominated, to the stiffer, stretching-dominated, deformation mode of the network, a transition which is primarily responsible for the observed overall reinforcement. The confinement, i.e., the restriction on network kinematics imposed by fillers, causes this transition. These results provide a justification for the observed difference in reinforcement obtained in sparsely versus densely cross-linked networks at a given filling fraction and provide guidance for the further development of network-based materials.

Structural Redundancy in Supracellular Actomyosin Networks Enables Robust Tissue Folding

Tissue morphogenesis is strikingly robust. Yet, how tissues are sculpted under challenging conditions is unknown. Here, we combined network analysis, experimental perturbations, and computational modeling to determine how network connectivity between hundreds of contractile cells on the ventral side of the Drosophila embryo ensures robust tissue folding. We identified two network properties that mechanically promote robustness. First, redundant supracellular cytoskeletal network paths ensure global connectivity, even with network degradation. By forming many more connections than are required, morphogenesis is not disrupted by local network damage, analogous to the way redundancy guarantees the large-scale function of vasculature and transportation networks. Second, directional stiffening of edges oriented orthogonal to the folding axis promotes furrow formation at lower contractility levels. Structural redundancy and directional network stiffening ensure robust tissue folding with proper orientation.

Stress-stabilized subisostatic fiber networks in a ropelike limit

The mechanics of disordered fibrous networks such as those that make up the extracellular matrix are strongly dependent on the local connectivity or coordination number. For biopolymer networks this coordination number is typically between 3 and 4. Such networks are sub-isostatic and linearly unstable to deformation with only central force interactions, but exhibit a mechanical phase transition between floppy and rigid states under strain. The introduction of weak bending interactions stabilizes these networks and suppresses the critical signatures of this transition. We show that applying external stress can also stabilize subisostatic networks with only tensile central force interactions, i.e., a ropelike potential. Moreover, we find that the linear shear modulus shows a power-law scaling with the external normal stress, with a non-mean-field exponent. For networks with finite bending rigidity, we find that the critical stain shifts to lower values under prestress.

Scaling Theory for Mechanical Critical Behavior in Fiber Networks

As a function of connectivity, spring networks exhibit a critical transition between floppy and rigid phases at an isostatic threshold. For connectivity below this threshold, fiber networks were recently shown theoretically to exhibit a rigidity transition with corresponding critical signatures as a function of strain. Experimental collagen networks were also shown to be consistent with these predictions. We develop a scaling theory for this strain-controlled transition. Using a real-space renormalization approach, we determine relations between the critical exponents governing the transition, which we verify for the strain-controlled transition using numerical simulations of both triangular lattice-based and packing-derived fiber networks.

A minimal-length approach unifies rigidity in underconstrained materials

We present an approach to understand geometric-incompatibility-induced rigidity in underconstrained materials, including subisostatic 2D spring networks and 2D and 3D vertex models for dense biological tissues. We show that in all these models a geometric criterion, represented by a minimal length [Formula: see text], determines the onset of prestresses and rigidity. This allows us to predict not only the correct scalings for the elastic material properties, but also the precise magnitudes for bulk modulus and shear modulus discontinuities at the rigidity transition as well as the magnitude of the Poynting effect. We also predict from first principles that the ratio of the excess shear modulus to the shear stress should be inversely proportional to the critical strain with a prefactor of 3. We propose that this factor of 3 is a general hallmark of geometrically induced rigidity in underconstrained materials and could be used to distinguish this effect from nonlinear mechanics of single components in experiments. Finally, our results may lay important foundations for ways to estimate [Formula: see text] from measurements of local geometric structure and thus help develop methods to characterize large-scale mechanical properties from imaging data.

Normal stress anisotropy and marginal stability in athermal elastic networks

Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component σzz on long (compressible) timescales, crossing over to the first normal stress difference, N1 = σxx - σzz at short (incompressible) times. While the behavior of N1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain.

The Role of Network Architecture in Collagen Mechanics

Collagen forms fibrous networks that reinforce tissues and provide an extracellular matrix for cells. These networks exhibit remarkable strain-stiffening properties that tailor the mechanical functions of tissues and regulate cell behavior. Recent models explain this nonlinear behavior as an intrinsic feature of disordered networks of stiff fibers. Here, we experimentally validate this theoretical framework by measuring the elastic properties of collagen networks over a wide range of self-assembly conditions. We show that the model allows us to quantitatively relate both the linear and nonlinear elastic behavior of collagen networks to their underlying architecture. Specifically, we identify the local coordination number (or connectivity) 〈z〉 as a key architectural parameter that governs the elastic response of collagen. The network elastic response reveals that 〈z〉 decreases from 3.5 to 3 as the polymerization temperature is raised from 26 to 37°C while being weakly dependent on concentration. We furthermore infer a Young's modulus of 1.1 MPa for the collagen fibrils from the linear modulus. Scanning electron microscopy confirms that 〈z〉 is between three and four but is unable to detect the subtle changes in 〈z〉 with polymerization conditions that rheology is sensitive to. Finally, we show that, consistent with the model, the initial stress-stiffening response of collagen networks is controlled by the negative normal stress that builds up under shear. Our work provides a predictive framework to facilitate future studies of the regulatory effect of extracellular matrix molecules on collagen mechanics. Moreover, our findings can aid mechanobiological studies of wound healing, fibrosis, and cancer metastasis, which require collagen matrices with tunable mechanical properties.

Poisson's Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations

Connective tissue mechanics is highly nonlinear, exhibits a strong Poisson's effect, and is associated with significant collagen fiber re-arrangement. Although the general features of the stress-strain behavior have been discussed extensively, the Poisson's effect received less attention. In general, the relationship between the microscopic fiber network mechanics and the macroscopic experimental observations remains poorly defined. The objective of the present work is to provide additional insight into this relationship. To this end, results from models of random collagen networks are compared with experimental data on reconstructed collagen gels, mouse skin dermis, and the human amnion. Attention is devoted to the mechanism leading to the large Poisson's effect observed in experiments. The results indicate that the incremental Poisson's contraction is directly related to preferential collagen orientation. The experimentally observed downturn of the incremental Poisson's ratio at larger strains is associated with the confining effect of fibers transverse to the loading direction and contributing little to load bearing. The rate of collagen orientation increases at small strains, reaches a maximum, and decreases at larger strains. The peak in this curve is associated with the transition of the network deformation from bending dominated, at small strains, to axially dominated, at larger strains. The effect of fiber tortuosity on network mechanics is also discussed, and a comparison of biaxial and uniaxial loading responses is performed.

Geometry and the onset of rigidity in a disordered network

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture, the number of nodes and the number of bonds between them, this rigidity transition must have a geometric origin. Naive, degree-of-freedom-based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks and demonstrate that the onset of rigidity, at a finite simple shear strain γ^{★}, coincides with the appearance of a single state of self-stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self-stress but imparts a finite shear modulus to the spring network. Beyond the critical shear, the network acquires a highly anisotropic elastic modulus, resisting further deformation most strongly in the direction of the rigidifying shear. We confirm previously reported critical scaling of the corresponding differential shear modulus. In the subcritical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to the nascent state of self-stress.