Nonlinearities of biopolymer gels increase the range of force transmission

Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions

We present a new model and extensive computations that explain the dramatic remodelling undergone by a fibrous collagen extracellular matrix (ECM), when subjected to contractile mechanical forces from embedded cells or cell clusters. This remodelling creates complex patterns, comprising multiple narrow localised bands of severe densification and fiber alignment, extending far into the ECM, often joining distant cells or cell clusters (such as tumours). Most previous models cannot capture this behaviour, as they assume stable mechanical fiber response with stress an increasing function of fiber stretch, and a restriction to small displacements. Our fully nonlinear network model distinguishes between two types of single-fiber nonlinearity: fibers that undergo stable (supercritical) buckling (as in previous work) versus fibers that suffer unstable (subcritical) buckling collapse. The model allows unrestricted, arbitrarily large displacements (geometric nonlinearity). Our assumptions on single-fiber instability are supported by recent simulations and experiments on buckling of individual beams with a hierarchical microstructure, such as collagen fibers. We use simple scenarios to illustrate, for the first time, two distinct compressive-instability mechanisms at work in our model: unstable buckling collapse of single fibers, and snap-through of multiple-fiber groups. The latter is possible even when single fibers are stable. Through simulations of large fiber networks, we show how these instabilities lead to spatially extended patterns of densification, fiber alignment and ECM remodelling induced by cell contraction. Our model is simple, but describes a very complex, multi-stable energy landscape, using sophisticated numerical optimisation methods that overcome the difficulties caused by instabilities in large systems. Our work opens up new ways of understanding the unique biomechanics of fibrous-network ECM, by fully accounting for nonlinearity and associated loss of stability in fiber networks. Our results provide new insights on tumour invasion and metastasis.

Mechanosensitivity of phase separation in an elastic gel

Liquid-liquid phase separation (LLPS) in binary or multi-component solutions is a well-studied subject in soft matter with extensive applications in biological systems. In recent years, several experimental studies focused on LLPS of solutes in hydrated gels, where the formation of coexisting domains induces elastic deformations within the gel. While the experimental studies report unique physical characteristics of these systems, such as sensitivity to mechanical forces and stabilization of multiple, periodic phase-separated domains, the theoretical understanding of such systems and the role of long-range interactions have not emphasized the nonlinear nature of the equilibrium binodal for strong segregation of the solute. In this paper, we formulate a generic, mean-field theory of a hydrated gel in the presence of an additional solute which changes the elastic properties of the gel. We derive equations for the equilibrium binodal of the phase separation of the solvent and solute and show that the deformations induced by the solute can result in effective long-range interactions between phase-separating solutes that can either enhance or, in the case of externally applied pressure, suppress phase separation of the solute relative to the case where there is no gel. This causes the coexisting concentrations at the binodal to depend on the system-wide average concentration, in contrast to the situation for phase separation in the absence of the gel.

Many-body interactions between contracting living cells

The organization of live cells into tissues and their subsequent biological function involves inter-cell mechanical interactions, which are mediated by their elastic environment. To model this interaction, we consider cells as spherical active force dipoles surrounded by an unbounded elastic matrix. Even though we assume that this elastic medium responds linearly, each cell's regulation of its mechanical activity leads to nonlinearities in the emergent interactions between cells. We study the many-body nature of these interactions by considering several geometries that include three or more cells. We show that for different regulatory behaviors of the cells' activity, the total elastic energy stored in the medium differs from the superposition of all two-body interactions between pairs of cells within the system. Specifically, we find that the many-body interaction energy between cells that regulate their position is smaller than the sum of interactions between all pairs of cells in the system, while for cells that do not regulate their position, the many-body interaction is larger than the superposition prediction. Thus, such higher-order interactions should be considered when studying the mechanics of multiple cells in proximity.

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.

Range and strength of mechanical interactions of force dipoles in elastic fiber networks

Mechanical forces generated by myosin II molecular motors drive diverse cellular processes, most notably shape change, division and locomotion. These forces may be transmitted over long range through the cytoskeletal medium - a disordered, viscoelastic network of biopolymers. The resulting cell size scale force chains can in principle mediate mechanical interactions between distant actomyosin units, leading to self-organized structural order in the cell cytoskeleton. Inspired by such force transmission through elastic structures in the cytoskeleton, we consider a percolated fiber lattice network, where fibers are represented as linear elastic elements that can both bend and stretch, and the contractile activity of myosin motors is represented by force dipoles. Then, by using a variety of metrics, we show how two such contractile force dipoles interact with each other through their mutual mechanical deformations of the elastic fiber network. As a prelude to two-dipole interactions, we quantify how forces propagate through the network from a single anisotropic force dipole by analyzing clusters of nodes connected by highly strained bonds, as well as through the decay rate of strain energy with distance from a force dipole. We show that predominant fiber bending screens out force propagation, resulting in reduced and strongly network configuration-dependent dipole interactions. On the other hand, stretching-dominated networks support longer-ranged inter-dipole interactions that recapitulate the predictions of linear elasticity theory. By characterizing the differences between tensile and compressive force propagation in the fiber network, we show how inter-dipole interaction depends on the dipoles' mutual separation and orientation. The resulting elastic interaction energy may mediate a force between multiple distant dipoles, leading to their self-organization into ordered configurations. This provides a potential pathway for active mechanical force-driven structural order in elastic biopolymer networks.

Local response and emerging nonlinear elastic length scale in biopolymer matrices

Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale R* that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale R* can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.

Generic stress rectification in nonlinear elastic media

Stress propagation in nonlinear media is crucial in cell biology, where molecular motors exert anisotropic force dipoles on the fibrous cytoskeleton. While the force dipoles can be either contractile or expansile, a medium made of fibers which buckle under compression rectifies these stresses towards a biologically crucial contraction. A general understanding of this rectification phenomenon as a function of the medium's elasticity is however lacking. Here we use theoretical continuum elasticity to show that rectification is actually a very general effect in nonlinear materials subjected to anisotropic internal stresses. We analytically show that both bucklable and constitutively linear materials subjected to geometrical nonlinearities rectify small forces towards contraction, while granular-like materials rectify towards expansion. Using simulations, we moreover show that these results extend to larger forces. Beyond fiber networks, these results could shed light on the propagation of stresses in brittle or granular materials following a local plastic rearrangement.

Anomalous linear elasticity of disordered networks

Continuum elasticity is a powerful tool applicable in a broad range of physical systems and phenomena. Yet, understanding how and on what scales material disorder may lead to the breakdown of continuum elasticity is not fully understood. We show, based on recent theoretical developments and extensive numerical computations, that disordered elastic networks near a critical rigidity transition, such as strain-stiffened fibrous biopolymer networks that are abundant in living systems, reveal an anomalous long-range linear elastic response below a correlation length. This emergent anomalous elasticity, which is non-affine in nature, is shown to feature a qualitatively different multipole expansion structure compared to ordinary continuum elasticity, and a slower spatial decay of perturbations. The potential degree of universality of these results, their implications (e.g. for cell-cell communication through biological extracellular matrices) and open questions are briefly discussed.

Probing Local Force Propagation in Tensed Fibrous Gels

Fibrous hydrogels are a key component of soft animal tissues. They support cellular functions and facilitate efficient mechanical communication between cells. Due to their nonlinear mechanical properties, fibrous materials display non-trivial force propagation at the microscale, that is enhanced compared to that of linear-elastic materials. In the body, tissues are constantly subjected to external loads that tense or compress them, modifying their micro-mechanical properties into an anisotropic state. However, it is unknown how force propagation is modified by this isotropic-to-anisotropic transition. Here, force propagation in tensed fibrin hydrogels is directly measured. Local perturbations are induced by oscillating microspheres using optical tweezers. 1-point and 2-point microrheology are combined to simultaneously measure the shear modulus and force propagation. A mathematical framework to quantify anisotropic force propagation trends is suggested. Results show that force propagation becomes anisotropic in tensed gels, with, surprisingly, stronger response to perturbations perpendicular to the axis of tension. Importantly, external tension can also increase the range of force transmission. Possible implications and future directions for research are discussed. These results suggest a mechanism for favored directions of mechanical communication between cells in a tissue under external loads.

Variational methods and deep Ritz method for active elastic solids

Variational methods have been widely used in soft matter physics for both static and dynamic problems. These methods are mostly based on two variational principles: the variational principle of minimum free energy (MFEVP) and Onsager's variational principle (OVP). Our interests lie in the applications of these variational methods to active matter physics. In our former work [H. Wang, T. Qian and X. Xu, Soft Matter, 2021, 17, 3634-3653], we have explored the applications of OVP-based variational methods for the modeling of active matter dynamics. In the present work, we explore variational (or energy) methods that are based on MFEVP for static problems in active elastic solids. We show that MFEVP can be used not only to derive equilibrium equations, but also to develop approximate solution methods, such as the Ritz method, for active solid statics. Moreover, the power of the Ritz-type method can be further enhanced using deep learning methods if we use deep neural networks to construct the trial functions of the variational problems. We then apply these variational methods and the deep Ritz method to study the spontaneous bending and contraction of a thin active circular plate that is induced by internal asymmetric active contraction. The circular plate is found to be bent towards its contracting side. The study of such a simple toy system gives implications for understanding the morphogenesis of solid-like confluent cell monolayers. In addition, we introduce a so-called activogravity length to characterize the importance of gravitational forces relative to internal active contraction in driving the bending of the active plate. When the lateral plate dimension is larger than the activogravity length (about 100 micron), gravitational forces become important. Such gravitaxis behaviors at multicellular scales may play significant roles in the morphogenesis and in the up-down symmetry broken during tissue development.

Mean-field interactions between living cells in linear and nonlinear elastic matrices

Living cells respond to mechanical changes in the matrix surrounding them by applying contractile forces that are in turn transmitted to distant cells. We consider simple effective geometries for the spatial arrangement of cells, we calculate the mechanical work that each cell performs in order to deform the matrix, and study how that energy changes when a contracting cell is surrounded by other cells with similar properties and behavior. Cells regulating the displacements that they generate are attracted to each other in a manner that does not depend on the cell's rigidity. Whereas cells regulating the active stress that they apply repel each other. This repulsion depends on the cell's bulk modulus in spherical geometry, while in cylindrical geometries the interaction depends also on their shear modulus. In nonlinear, strain-stiffening matrices, for displacement regulation, in the presence of other cells, cell contraction is limited due to the divergence of the shear stress. For stress regulation, the interaction energy drops at the nonlinear stiffening stress. Our theoretical work provides insight into matrix-mediated interactions between contractile cells and on the role of their mechanical regulatory behavior.

Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells

The mechanical micro-environment of cells and tissues influences key aspects of cell structure and function, including cell motility. For proper tissue development, cells need to migrate, interact, and form contacts. Cells are known to exert contractile forces on underlying soft substrates and sense deformations in them. Here, we propose and analyze a minimal biophysical model for cell migration and long-range cell–cell interactions through mutual mechanical deformations of the substrate. We compute key metrics of cell motile behavior, such as the number of cell-cell contacts over a given time, the dispersion of cell trajectories, and the probability of permanent cell contact, and analyze how these depend on a cell motility parameter and substrate stiffness. Our results elucidate how cells may sense each other mechanically and generate coordinated movements and provide an extensible framework to further address both mechanical and short-range biophysical interactions.

Long-range mechanical signaling in biological systems

Cells can respond to signals generated by other cells that are remarkably far away. Studies from at least the 1920's showed that cells move toward each other when the distance between them is on the order of a millimeter, which is many times the cell diameter. Chemical signals generated by molecules diffusing from the cell surface would move too slowly and dissipate too fast to account for these effects, suggesting that they might be physical rather than biochemical. The non-linear elastic responses of sparsely connected networks of stiff or semiflexible filament such as those that form the extracellular matrix (ECM) and the cytoskeleton have unusual properties that suggest multiple mechanisms for long-range signaling in biological tissues. These include not only direct force transmission, but also highly non-uniform local deformations, and force-generated changes in fiber alignment and density. Defining how fibrous networks respond to cell-generated forces can help design new methods to characterize abnormal tissues and can guide development of improved biomimetic materials.

Continuum elastic models for force transmission in biopolymer gels

We review continuum elastic models for the transmission of both external forces and internal active cellular forces in biopolymer gels, and relate them to recent experiments. Rather than being exhaustive, we focus on continuum elastic models for small affine deformations and intend to provide a systematic continuum method and some analytical perspectives on the study of force transmission in biopolymer gels. We start from a very brief review of the nonlinear mechanics of individual biopolymers and a summary of constitutive models for the nonlinear elasticity of biopolymer gels. We next show that the simple 3-chain model can give predictions that fit well the shear experiments of some biopolymer gels, including the effects of strain-stiffening and negative normal stress. We then review continuum models for the transmission of internal active forces that are induced by a spherically contracting cell embedded in a three-dimensional biopolymer gel. Various scaling regimes for the decay of cell-induced displacements are identified for linear isotropic and anisotropic materials, and for biopolymer gels with nonlinear compressive-softening and strain-stiffening elasticity, respectively. After that, we present (using an energetic approach) the generic and unified continuum theory proposed in [D. Ben-Yaakov et al., Soft Matter, 2015, 11, 1412] about how the transmission of forces in the biogel matrix can mediate long-range interactions between cells with mechanical homeostasis. We show the predictions of the theory in a special hexagonal multicellular array, and relate them to recent experiments. Finally, we conclude this paper with comments on the limitations and outlook of continuum modeling, and highlight the need for complementary theoretical approaches, such as discrete network simulations, to force transmission in biopolymer gels and phenomenological active gel theories for multicellular systems.

A Change in ECM Composition Affects Sensory Organ Mechanics and Function

Proprioception requires the transduction of muscle-generated deformations into sensory neuronal impulses. In proprioceptive organs, the mechanical coupling between the sensory neuron and the muscle is mediated by a connective structure composed of accessory cells and an extracellular matrix (ECM). Here, we use the fly chordotonal organ (ChO) to investigate how the mechanical properties of the connective element affect mechanosensing. We show that the loss of Pericardin, a major constituent of the ChO ECM, alters the mechanical properties of the ChO resulting in short-wavelength buckling of the accessory cells upon muscle contraction and low compressive strain within the organ. We explain these results using a simplified theoretical model of an elastic beam interacting with an elastic network under a compressive force. We further demonstrate that the transition from compression to bending interferes with the ability of the accessory cells to propagate muscle-generated deformations correctly to the neuron and hence with proper sensing.

Long-range mechanical coupling of cells in 3D fibrin gels

When seeded in fibrous gels, pairs of cells or cell aggregates can induce bands of deformed gel, extending to surprisingly long distances in the intercellular medium. The formation of bands has been previously shown and studied in collagen systems. In this study, we strive to further our understanding of this fundamental mechanical mechanism in fibrin, a key element in wound healing and angiogenesis processes. We embedded fibroblast cells in 3D fibrin gels, and monitored band formation by real-time confocal microscopy. Quantitative dynamic analysis of band formation revealed a gradual increase in fiber density and alignment between pairs of cells. Such intercellular bands extended into a large-scale network of mechanically connected cells, in which the connected cells exhibited a more spread morphology than the isolated cells. Moreover, computational modeling demonstrated that the direction of cell-induced force triggering band formation can be applied in a wide range of angles relative to a neighboring cell. Our findings indicate that long-range mechanical coupling between cells is an important mechanism in regulating multicellular processes in reconstituted fibrin gels. As such, it should motivate exploration of this mechanism in studies in vivo, in wound healing or angiogenesis, in which fibrin is contracted by fibroblast cells.

Collective forces of tumor spheroids in three-dimensional biopolymer networks

We describe a method for quantifying the contractile forces that tumor spheroids collectively exert on highly nonlinear three-dimensional collagen networks. While three-dimensional traction force microscopy for single cells in a nonlinear matrix is computationally complex due to the variable cell shape, here we exploit the spherical symmetry of tumor spheroids to derive a scale-invariant relationship between spheroid contractility and the surrounding matrix deformations. This relationship allows us to directly translate the magnitude of matrix deformations to the total contractility of arbitrarily sized spheroids. We show that our method is accurate up to strains of 50% and remains valid even for irregularly shaped tissue samples when considering only the deformations in the far field. Finally, we demonstrate that collective forces of tumor spheroids reflect the contractility of individual cells for up to 1 hr after seeding, while collective forces on longer timescales are guided by mechanical feedback from the extracellular matrix.

Elastic Anisotropy Governs the Range of Cell-Induced Displacements

The unique nonlinear mechanics of the fibrous extracellular matrix (ECM) facilitates long-range cell-cell mechanical communications that would be impossible for linear elastic substrates. Past research has described the contribution of two separated effects on the range of force transmission, including ECM elastic nonlinearity and fiber alignment. However, the relation between these different effects is unclear, and how they combine to dictate force transmission range is still elusive. Here, we combine discrete fiber simulations with continuum modeling to study the decay of displacements induced by a contractile cell in fibrous networks. We demonstrate that fiber nonlinearity and fiber reorientation both contribute to the strain-induced elastic anisotropy of the cell's local environment. This elastic anisotropy is a "lumped" parameter that governs the slow decay of displacements, and it depends on the magnitude of applied strain, either an external tension or an internal contraction, as a model of the cell. Furthermore, we show that accounting for artificially prescribed elastic anisotropy dictates the decay of displacements induced by a contracting cell. Our findings unify previous single effects into a mechanical theory that explains force transmission in fibrous networks. This work may provide insights into biological processes that involve communication of distant cells mediated by the ECM, such as those occurring in morphogenesis, wound healing, angiogenesis, and cancer metastasis. It may also provide design parameters for biomaterials to control force transmission between cells as a way to guide morphogenesis in tissue engineering.

Mechanical Interaction between Cells Facilitates Molecular Transport

In vivo, eukaryotic cells are embedded in a matrix environment, where they grow and develop. Generally, this extracellular matrix (ECM) is an anisotropic fibrous structure, through which macromolecules and biochemical signaling molecules at the nanometer scale diffuse. The ECM is continuously remodeled by cells, via mechanical interactions, which lead to a potential link between biomechanical and biochemical cell-cell interactions. Here, it is studied how cell-induced forces applied on the ECM impact the biochemical transport of molecules between distant cells. It is experimentally observed that cells remodel the ECM by increasing fiber alignment and density of the matrix between them over time. Using random walk simulations on a 3D lattice, elongated fixed obstacles are implemented that mimic the fibrous ECM structure. Both diffusion of a tracer molecule and the mean first-passage time a molecule secreted from one cell takes to reach another cell are measured. The model predicts that cell-induced remodeling can lead to a dramatic speedup in the transport of molecules between cells. Fiber alignment and densification cause reduction of the transport dimensionality from a 3D to a much more rapid 1D process. Thus, a novel mechanism of mechano-biochemical feedback in the regulation of long-range cell-cell communication is suggested.

Modulus of Fibrous Collagen at the Length Scale of a Cell

The extracellular matrix provides macroscale structural support to tissues as well as microscale mechanical cues, like stiffness, to the resident cells. As those cues modulate gene expression, proliferation, differentiation, and motility, quantifying the stiffness that cells sense is crucial to understanding cell behavior. Whereas the macroscopic modulus of a collagen network can be measured in uniform extension or shear, quantifying the local stiffness sensed by a cell remains a challenge due to the inhomogeneous and nonlinear nature of the fiber network at the scale of the cell. To address this challenge, we designed an experimental method to measure the modulus of a network of collagen fibers at this scale. We used spherical particles of an active hydrogel (poly N-isopropylacrylamide) that contract when heated, thereby applying local forces to the collagen matrix and mimicking the contractile forces of a cell. After measuring the particles' bulk modulus and contraction in networks of collagen fibers, we applied a nonlinear model for fibrous materials to compute the modulus of the local region surrounding each particle. We found the modulus at this length scale to be highly heterogeneous, with modulus varying by a factor of 3. In addition, at different values of applied strain, we observed both strain stiffening and strain softening, indicating nonlinearity of the collagen network. Thus, this experimental method quantifies local mechanical properties in a fibrous network at the scale of a cell, while also accounting for inherent nonlinearity.

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.

Elastic interactions between anisotropically contracting circular cells

We study interactions between biological cells that apply anisotropic active mechanical forces on an elastic substrate. We model the cells as thin disks that along their perimeters apply radial, but angle-dependent forces on the substrate. We obtain analytical expressions for the elastic energy stored in the substrate as a function of the distance between the cells, the Fourier modes of applied forces, and their phase angles. We show how the relative phases of the forces applied by the cells can switch the interaction between attractive and repulsive, and relate our results to those for linear force dipoles. For long enough distances, the interaction energy decays in magnitude as a power law of the cell-cell distance with an integer exponent that generally increases with the Fourier modes of the applied forces.

Fiber plucking by molecular motors yields large emergent contractility in stiff biopolymer networks

The mechanical properties of the cell depend crucially on the tension of its cytoskeleton, a biopolymer network that is put under stress by active motor proteins. While the fibrous nature of the network is known to strongly affect the transmission of these forces to the cellular scale, our understanding of this process remains incomplete. Here we investigate the transmission of forces through the network at the individual filament level, and show that active forces can be geometrically amplified as a transverse motor-generated force "plucks" the fiber and induces a nonlinear tension. In stiff and densely connected networks, this tension results in large network-wide tensile stresses that far exceed the expectation drawn from a linear elastic theory. This amplification mechanism competes with a recently characterized network-level amplification due to fiber buckling, suggesting that that fiber networks provide several distinct pathways for living systems to amplify their molecular forces.

Stress-dependent amplification of active forces in nonlinear elastic media

The production of mechanical stresses in living organisms largely relies on localized, force-generating active units embedded in filamentous matrices. Numerical simulations of discrete fiber networks with fixed boundaries have shown that buckling in the matrix dramatically amplifies the resulting active stresses. Here we extend this result to a continuum elastic medium prone to buckling subjected to an arbitrary external stress, and derive analytical expressions for the active, nonlinear constitutive relations characterizing the full active medium. Inserting these relations into popular "active gel" descriptions of living tissues and the cytoskeleton will enable investigations into nonlinear regimes previously inaccessible due to the phenomenological nature of these theories.

Nonlinear Elasticity of the ECM Fibers Facilitates Efficient Intercellular Communication

Biological cells embedded in fibrous matrices have been observed to form intercellular bands of dense and aligned fibers through which they mechanically interact over long distances. Such matrix-mediated cellular interactions have been shown to regulate various biological processes. This study aimed to explore the effects of elastic nonlinearity of the fibers contained in the extracellular matrix (ECM) on the transmission of mechanical loads between contracting cells. Based on our biological experiments, we developed a finite-element model of two contracting cells embedded within a fibrous network. The individual fibers were modeled as showing linear elasticity, compression microbuckling, tension stiffening, or both of the latter two. Fiber compression buckling resulted in smaller loads in the ECM, which were primarily directed toward the neighboring cell. These loads decreased with increasing cell-to-cell distance; when cells were >9 cell diameters apart, no such intercellular interaction was observed. Tension stiffening further contributed to directing the loads toward the neighboring cell, though to a smaller extent. The contraction of two neighboring cells resulted in mutual attraction forces, which were considerably increased by tension stiffening and decayed with increasing cell-to-cell distances. Nonlinear elasticity contributed also to the onset of force polarity on the cell boundaries, manifested by larger contractile forces pointing toward the neighboring cell. The density and alignment of the fibers within the intercellular band were greater when fibers buckled under compression, with tension stiffening further contributing to this structural remodeling. Although previous studies have established the role of the ECM nonlinear mechanical behavior in increasing the range of force transmission, our model demonstrates the contribution of nonlinear elasticity of biological gels to directional and efficient mechanical signal transfer between distant cells, and rehighlights the importance of using fibrous gels in experimental settings for facilitating intercellular communication. VIDEO ABSTRACT.

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.

Broken detailed balance and non-equilibrium dynamics in living systems: a review

Living systems operate far from thermodynamic equilibrium. Enzymatic activity can induce broken detailed balance at the molecular scale. This molecular scale breaking of detailed balance is crucial to achieve biological functions such as high-fidelity transcription and translation, sensing, adaptation, biochemical patterning, and force generation. While biological systems such as motor enzymes violate detailed balance at the molecular scale, it remains unclear how non-equilibrium dynamics manifests at the mesoscale in systems that are driven through the collective activity of many motors. Indeed, in several cellular systems the presence of non-equilibrium dynamics is not always evident at large scales. For example, in the cytoskeleton or in chromosomes one can observe stationary stochastic processes that appear at first glance thermally driven. This raises the question how non-equilibrium fluctuations can be discerned from thermal noise. We discuss approaches that have recently been developed to address this question, including methods based on measuring the extent to which the system violates the fluctuation-dissipation theorem. We also review applications of this approach to reconstituted cytoskeletal networks, the cytoplasm of living cells, and cell membranes. Furthermore, we discuss a more recent approach to detect actively driven dynamics, which is based on inferring broken detailed balance. This constitutes a non-invasive method that uses time-lapse microscopy data, and can be applied to a broad range of systems in cells and tissue. We discuss the ideas underlying this method and its application to several examples including flagella, primary cilia, and cytoskeletal networks. Finally, we briefly discuss recent developments in stochastic thermodynamics and non-equilibrium statistical mechanics, which offer new perspectives to understand the physics of living systems.

Cell contraction induces long-ranged stress stiffening in the extracellular matrix

Animal cells in tissues are supported by biopolymer matrices, which typically exhibit highly nonlinear mechanical properties. While the linear elasticity of the matrix can significantly impact cell mechanics and functionality, it remains largely unknown how cells, in turn, affect the nonlinear mechanics of their surrounding matrix. Here, we show that living contractile cells are able to generate a massive stiffness gradient in three distinct 3D extracellular matrix model systems: collagen, fibrin, and Matrigel. We decipher this remarkable behavior by introducing nonlinear stress inference microscopy (NSIM), a technique to infer stress fields in a 3D matrix from nonlinear microrheology measurements with optical tweezers. Using NSIM and simulations, we reveal large long-ranged cell-generated stresses capable of buckling filaments in the matrix. These stresses give rise to the large spatial extent of the observed cell-induced matrix stiffness gradient, which can provide a mechanism for mechanical communication between cells.

Displacement Propagation in Fibrous Networks Due to Local Contraction

The extracellular matrix provides macroscale structure to tissues and microscale guidance for cell contraction, adhesion, and migration. The matrix is composed of a network of fibers, which each deform by stretching, bending, and buckling. Whereas the mechanics has been well characterized in uniform shear and extension, the response to more general loading conditions remains less clear, because the associated displacement fields cannot be predicted a priori. Studies simulating contraction, such as due to a cell, have observed displacements that propagate over a long range, suggesting mechanisms such as reorientation of fibers toward directions of tensile force and nonlinearity due to buckling of fibers under compression. It remains unclear which of these two mechanisms produces the long-range displacements and how properties like fiber bending stiffness and fiber length affect the displacement field. Here, we simulate contraction of an inclusion within a fibrous network and fit the resulting radial displacements to ur ∼ r−n where the power n quantifies the decay of displacements over distance, and a value of n less than that predicted by classical linear elasticity indicates displacements that propagate over a long range. We observed displacements to propagate over a longer range for greater contraction of the inclusion, for networks having longer fibers, and for networks with lower fiber bending stiffness. Contraction of the inclusion also caused fibers to reorient into the radial direction, but, surprisingly, the reorientation was minimally affected by bending stiffness. We conclude that both reorientation and nonlinearity are responsible for the long-range displacements.

Theory of frequency response of mechanically driven cardiomyocytes

We theoretically predict and compare with experiments, transitions from spontaneous beating to dynamical entrainment of cardiomyocytes induced by an oscillating, external mechanical probe. In accord with recent experiments, we predict the dynamical behavior as a function of the probe amplitude and frequency. The theory is based on a phenomenological model for a non-linear oscillator, motivated by acto-myosin contractility. The generic behavior is independent of the detailed, molecular origins of the dynamics and, consistent with experiment, we find three regimes: spontaneous beating with the natural frequency of the cell, entrained beating with the frequency of the probe, and a “bursting” regime where the two frequencies alternate in time. We quantitatively predict the properties of the “bursting” regime as a function of the amplitude and frequency of the probe. Furthermore, we examine the pacing process in the presence of weak noise and explain how this might relate to cardiomyocyte physiology.

Mechanical response of collagen networks to nonuniform microscale loads

As force is applied to fibrous proteins such as collagen or fibrin, the fibers respond by bending, stretching, or buckling, which together bring about a nonlinear relationship between force and displacement. The nonlinearity is typically understood in terms of strain stiffening in uniform extension or shear, but there remains a critical lack of data on how fibrous materials respond to other more complicated loadings. Here we study the mechanics of collagen networks in response to nonuniform loads applied on the local scale of the fibers. For this, we use particles made of an active hydrogel that undergoes a temperature-induced phase transition causing a large decrease in volume. We embed these particles in networks of fibrous collagen and use them as microactuators to apply controlled microscale loading. The resulting fiber displacements propagate over a long range with radial displacements u scaling as r^-n with n ≈ 1. By contrast, we find linear homogeneous materials have n ≈ 2, in agreement with classical linear elastic theory. Our experimental data supports the notion that the long range displacements result from buckling of fibers in compression and local straightening of fibers in tension, in agreement with previous studies. Surprisingly, global network anisotropy appears to have only a modest effect on the displacement propagation. These insights into the microscale mechanics demonstrate that the decay power n provides a useful metric to quantify the mechanics of fibrous materials. We therefore suggest it is a means to compare new theories with experimental data.

Talking over the extracellular matrix: How do cells communicate mechanically?

Communication between cells enables them to coordinate their activity and is crucial for the differentiation, development, and function of tissues and multicellular organisms. Cell-cell communication is discussed almost exclusively as having a chemical or electrical origin. Only recently, a new mode of cell communication was elucidated: mechanical communication through the extracellular matrix (ECM). Cells can communicate mechanically by responding either to mechanical deformations generated by their neighbors or to a change in the mechanical properties of the ECM induced by a neighboring cell. This newly resolved mode of communication possesses unique features that complement the cellular ability to receive and share information, and to consequently act in a cooperative way with surrounding cells. Herein, we review several examples of mechanical communication, discuss their unique properties, and comment on the major challenges facing the field.

Compressive elasticity of polydisperse biopolymer gels

We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels.

Mechanical Cell-Cell Communication in Fibrous Networks: The Importance of Network Geometry

Cells contracting in extracellular matrix (ECM) can transmit stress over long distances, communicating their position and orientation to cells many tens of micrometres away. Such phenomena are not observed when cells are seeded on substrates with linear elastic properties, such as polyacrylamide (PA) gel. The ability for fibrous substrates to support far reaching stress and strain fields has implications for many physiological processes, while the mechanical properties of ECM are central to several pathological processes, including tumour invasion and fibrosis. Theoretical models have investigated the properties of ECM in a variety of network geometries. However, the effects of network architecture on mechanical cell-cell communication have received little attention. This work investigates the effects of geometry on network mechanics, and thus the ability for cells to communicate mechanically through different networks. Cell-derived displacement fields are quantified for various network geometries while controlling for network topology, cross-link density and micromechanical properties. We find that the heterogeneity of response, fibre alignment, and substrate displacement fields are sensitive to network choice. Further, we show that certain geometries support mechanical communication over longer distances than others. As such, we predict that the choice of network geometry is important in fundamental modelling of cell-cell interactions in fibrous substrates, as well as in experimental settings, where mechanical signalling at the cellular scale plays an important role. This work thus informs the construction of theoretical models for substrate mechanics and experimental explorations of mechanical cell-cell communication.