Fiber networks amplify active stress

Active biomaterials for mechanobiology

Active biomaterials offer novel approaches to study mechanotransduction in mammalian cells. These material systems probe cellular responses by dynamically modulating their resistance to endogenous forces or applying exogenous forces on cells in a temporally controlled manner. Stimuli-responsive molecules, polymers, and nanoparticles embedded inside cytocompatible biopolymer networks transduce external signals such as light, heat, chemicals, and magnetic fields into changes in matrix elasticity (few kPa to tens of kPa) or forces (few pN to several μN) at the cell-material interface. The implementation of active biomaterials in mechanobiology has generated scientific knowledge and therapeutic potential relevant to a variety of conditions including but not limited to cancer metastasis, fibrosis, and tissue regeneration. We discuss the repertoire of cellular responses that can be studied using these platforms including receptor signaling as well as downstream events namely, cytoskeletal organization, nuclear shuttling of mechanosensitive transcriptional regulators, cell migration, and differentiation. We highlight recent advances in active biomaterials and comment on their future impact.

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.

Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction

Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.

Rapid local compression in active gels is caused by nonlinear network response

The actin cytoskeleton in living cells generates forces in conjunction with myosin motor proteins to directly and indirectly drive essential cellular processes. The semiflexible filaments of the cytoskeleton can respond nonlinearly to the collective action of motors. We here investigate mechanics and force generation in a model actin cytoskeleton, reconstituted in vitro, by observing the response and fluctuations of embedded micron-scale probe particles. Myosin mini-filaments can be modeled as force dipoles and give rise to deformations in the surrounding network of cross-linked actin. Anomalously correlated probe fluctuations indicate the presence of rapid local compression or draining of the network that emerges in addition to the ordinary linear shear elastic (incompressible) response to force dipoles. The anomalous propagation of compression can be attributed to the nonlinear response of actin filaments to the microscopic forces, and is quantitatively consistent with motor-generated large-scale stiffening of the gels.

Compression stiffening of fibrous networks with stiff inclusions

Tissues commonly consist of cells embedded within a fibrous biopolymer network. Whereas cell-free reconstituted biopolymer networks typically soften under applied uniaxial compression, various tissues, including liver, brain, and fat, have been observed to instead stiffen when compressed. The mechanism for this compression-stiffening effect is not yet clear. Here, we demonstrate that when a material composed of stiff inclusions embedded in a fibrous network is compressed, heterogeneous rearrangement of the inclusions can induce tension within the interstitial network, leading to a macroscopic crossover from an initial bending-dominated softening regime to a stretching-dominated stiffening regime, which occurs before and independently of jamming of the inclusions. Using a coarse-grained particle-network model, we first establish a phase diagram for compression-driven, stretching-dominated stress propagation and jamming in uniaxially compressed two- and three-dimensional systems. Then, we demonstrate that a more detailed computational model of stiff inclusions in a subisostatic semiflexible fiber network exhibits quantitative agreement with the predictions of our coarse-grained model as well as qualitative agreement with experiments.

Introduction to Active Matter

Moumita Das, Michael Murrell and Christoph Schmidt introduce the Soft Matter collection on active matter.

Geometry underlies the mechanical stiffening and softening of an indented floating film

A basic paradigm underlying the Hookean mechanics of amorphous, isotropic solids is that small deformations are proportional to the magnitude of external forces. However, slender bodies may undergo large deformations even under minute forces, leading to nonlinear responses rooted in purely geometric effects. Here we study the indentation of a polymer film on a liquid bath. Our experiments and simulations support a recently-predicted stiffening response [D. Vella and B. Davidovitch, Phys. Rev. E, 2018, 98, 013003], and we show that the system softens at large slopes, in agreement with our theory that addresses small and large deflections. We show how stiffening and softening emanate from nontrivial yet generic features of the stress and displacement fields.

Three-dimensional pore network characterization of reconstructed extracellular matrix

The extracellular matrix (ECM) has a fiber network that provides physical scaffolds to cells and plays important roles by regulating cellular functions. Some previous works characterized the mechanical and geometrical properties of the ECM fiber network using reconstituted collagen-I. However, the characterization of the porous structure of reconstituted collagen-I has been limited to the pore diameter measurement, and pore network extraction has not been applied to reconstituted collagen-I despite the importance of pore interconnectivity. Here, we aim to show the importance of characterizing the pore network of reconstituted collagen-I by comparing the pore networks of structures that have different fiber alignments. We show that the fiber alignment significantly changes the pore throat area but not the pore diameter. Also, we demonstrate that larger pore throats are directed in the direction of the fiber alignment, which may help in understanding the enhanced cell migration when fibers are aligned.

Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling

Cell migration in fibrous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response, and cancer progression. During migration, individual cells can generate active pulling forces via actomyosin contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.

Traction Forces Control Cell-Edge Dynamics and Mediate Distance Sensitivity during Cell Polarization

Traction forces are generated by cellular actin-myosin system and transmitted to the environment through adhesions. They are believed to drive cell motion, shape changes, and extracellular matrix remodeling [1-3]. However, most of the traction force analysis has been performed on stationary cells, investigating forces at the level of individual focal adhesions or linking them to static cell parameters, such as area and edge curvature [4-10]. It is not well understood how traction forces are related to shape changes and motion, e.g., forces were reported to either increase or drop prior to cell retraction [11-15]. Here, we analyze the dynamics of traction forces during the protrusion-retraction cycle of polarizing fish epidermal keratocytes and find that forces fluctuate together with the cycle, increasing during protrusion and reaching maximum at the beginning of retraction. We relate force dynamics to the recently discovered phenomenological rule [16] that governs cell-edge behavior during keratocyte polarization: both traction forces and probability of switch from protrusion to retraction increase with the distance from the cell center. Diminishing forces with cell contractility inhibitor leads to decreased edge fluctuations and abnormal polarization, although externally applied force can induce protrusion-retraction switch. These results suggest that forces mediate distance sensitivity of the edge dynamics and organize cell-edge behavior, leading to spontaneous polarization. Actin flow rate did not exhibit the same distance dependence as traction stress, arguing against its role in organizing edge dynamics. Finally, using a simple model of actin-myosin network, we show that force-distance relationship might be an emergent feature of such networks.

Reversal of contractility as a signature of self-organization in cytoskeletal bundles

Bundles of cytoskeletal filaments and molecular motors generate motion in living cells, and have internal structures ranging from very organized to apparently disordered. The mechanisms powering the disordered structures are debated, and existing models predominantly predict that they are contractile. We reexamine this prediction through a theoretical treatment of the interplay between three well-characterized internal dynamical processes in cytoskeletal bundles: filament assembly and disassembly, the attachement-detachment dynamics of motors and that of crosslinking proteins. The resulting self-organization is easily understood in terms of motor and crosslink localization, and allows for an extensive control of the active bundle mechanics, including reversals of the filaments’ apparent velocities and the possibility of generating extension instead of contraction. This reversal mirrors some recent experimental observations, and provides a robust criterion to experimentally elucidate the underpinnings of both actomyosin activity and the dynamics of microtubule/motor assemblies in vitro as well as in diverse intracellular structures ranging from contractile bundles to the mitotic spindle.

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.

Dynamic motions of molecular motors in the actin cytoskeleton

During intracellular transport, cellular cargos, such as organelles, vesicles, and proteins, are transported within cells. Intracellular transport plays an important role in diverse cellular functions. Molecular motors walking on the cytoskeleton facilitate active intracellular transport, which is more efficient than diffusion-based passive transport. Active transport driven by kinesin and dynein walking on microtubules has been studied well during recent decades. However, mechanisms of active transport occurring in disorganized actin networks via myosin motors remain elusive. To provide physiologically relevant insights, we probed motions of myosin motors in actin networks under various conditions using our well-established computational model that rigorously accounts for the mechanical and dynamical behaviors of the actin cytoskeleton. We demonstrated that myosin motions can be confined due to three different reasons in the absence of F-actin turnover. We verified mechanisms of motor stalling using in vitro reconstituted actomyosin networks. We also found that with F-actin turnover, motors consistently move for a long time without significant confinement. Our study sheds light on the importance of F-actin turnover for effective active transport in the actin cytoskeleton.

Self-straining of actively crosslinked microtubule networks

Cytoskeletal networks are foundational examples of active matter and central to self-organized structures in the cell. In vivo, these networks are active and densely crosslinked. Relating their large-scale dynamics to the properties of their constituents remains an unsolved problem. Here, we study an in vitro active gel made from aligned microtubules and XCTK2 kinesin motors. Using photobleaching, we demonstrate that the gel's aligned microtubules, driven by motors, continually slide past each other at a speed independent of the local microtubule polarity and motor concentration. This phenomenon is also observed, and remains unexplained, in spindles. We derive a general framework for coarse graining microtubule gels crosslinked by molecular motors from microscopic considerations. Using microtubule-microtubule coupling through a force-velocity relationship for kinesin, this theory naturally explains the experimental results: motors generate an active strain rate in regions of changing polarity, which allows microtubules of opposite polarities to slide past each other without stressing the material.

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.

Uncovering the dynamic precursors to motor-driven contraction of active gels

Cells and tissues have the remarkable ability to actively generate the forces required to change their shape. This active mechanical behavior is largely mediated by the actin cytoskeleton, a crosslinked network of actin filaments that is contracted by myosin motors. Experiments and active gel theories have established that the length scale over which gel contraction occurs is governed by a balance between molecular motor activity and crosslink density. By contrast, the dynamics that govern the contractile activity of the cytoskeleton remain poorly understood. Here we investigate the microscopic dynamics of reconstituted actin-myosin networks using simultaneous real-space video microscopy and Fourier-space dynamic light scattering. Light scattering reveals different regimes of microscopic dynamics as a function of sample age. We uncover two dynamical precursors that precede macroscopic gel contraction. One is characterized by a progressive acceleration of stress-induced rearrangements, while the other consists of sudden, heterogeneous rearrangements. Intriguingly, our findings suggest a qualitative analogy between self-driven rupture and collapse of active gels and the delayed rupture of passive gels observed in earlier studies of colloidal gels under external loads.

Mesoscopic non-equilibrium measures can reveal intrinsic features of the active driving

Biological assemblies such as chromosomes, membranes, and the cytoskeleton are driven out of equilibrium at the nanoscale by enzymatic activity and molecular motors. Similar non-equilibrium dynamics can be realized in synthetic systems, such as chemically fueled colloidal particles. Characterizing the stochastic non-equilibrium dynamics of such active soft assemblies still remains a challenge. Recently, new non-invasive approaches have been proposed to determine the non-equilibrium behavior, which are based on detecting broken detailed balance in the stochastic trajectories of several coordinates of the system. Inspired by the method of two-point microrheology, in which the equilibrium fluctuations of a pair of probe particles reveal the viscoelastic response of an equilibrium system, here, we investigate whether we can extend such an approach to non-equilibrium assemblies: can one extract information on the nature of the active driving in a system from the analysis of a two-point non-equilibrium measure? We address this question theoretically in the context of a class of elastic systems, driven out of equilibrium by a spatially heterogeneous stochastic internal driving. We consider several scenarios for the spatial features of the internal driving that may be relevant in biological and synthetic systems, and investigate how such features of the active noise may be reflected in the long-range scaling behavior of two-point non-equilibrium measures.

Absorbing-active transition in a multi-cellular system regulated by a dynamic force network

Collective cell migration in 3D extracellular matrix (ECM) is crucial to many physiological and pathological processes. Migrating cells can generate active pulling forces via actin filament contraction, which are transmitted to the ECM fibers and lead to a dynamically evolving force network in the system. Here, we elucidate the role of this force network in regulating collective cell behaviors using a minimal active-particle-on-network (APN) model, in which active particles can pull the fibers and hop between neighboring nodes of the network following local durotaxis. Our model reveals a dynamic transition as the particle number density approaches a critical value, from an "absorbing" state containing isolated stationary small particle clusters, to an "active" state containing a single large cluster undergoing constant dynamic reorganization. This reorganization is dominated by a subset of highly dynamic "radical" particles in the cluster, whose number also exhibits a transition at the same critical density. The transition is underlaid by the percolation of "influence spheres" due to the particle pulling forces. Our results suggest a robust mechanism based on ECM-mediated mechanical coupling for collective cell behaviors in 3D ECM.

Surface and Bulk Stresses Drive Morphological Changes in Fibrous Microtissues

Engineered fibrous tissues consisting of cells encapsulated within collagen gels are widely used three-dimensional in vitro models of morphogenesis and wound healing. Although cell-mediated matrix remodeling that occurs within these scaffolds has been extensively studied, less is known about the mesoscale physical principles governing the dynamics of tissue shape. Here, we show both experimentally and by using computer simulations how surface contraction through the development of surface stresses (analogous to surface tension in fluids) coordinates with bulk contraction to drive shape evolution in constrained three-dimensional microtissues. We used microelectromechanical systems technology to generate arrays of fibrous microtissues and robot-assisted microsurgery to perform local incisions and implantation. We introduce a technique based on phototoxic activation of a small molecule to selectively kill cells in a spatially controlled manner. The model simulations, which reproduced the experimentally observed shape changes after surgical and photochemical operations, indicate that fitting of only bulk and surface contractile moduli is sufficient for the prediction of the equilibrium shape of the microtissues. The computational and experimental methods we have developed provide a general framework to study and predict the morphogenic states of contractile fibrous tissues under external loading at multiple length scales.

Dynamic filopodial forces induce accumulation, damage, and plastic remodeling of 3D extracellular matrices

The mechanical properties of the extracellular matrix (ECM)–a complex, 3D, fibrillar scaffold of cells in physiological environments–modulate cell behavior and can drive tissue morphogenesis, regeneration, and disease progression. For simplicity, it is often convenient to assume these properties to be time-invariant. In living systems, however, cells dynamically remodel the ECM and create time-dependent local microenvironments. Here, we show how cell-generated contractile forces produce substantial irreversible changes to the density and architecture of physiologically relevant ECMs–collagen I and fibrin–in a matter of minutes. We measure the 3D deformation profiles of the ECM surrounding cancer and endothelial cells during stages when force generation is active or inactive. We further correlate these ECM measurements to both discrete fiber simulations that incorporate fiber crosslink unbinding kinetics and continuum-scale simulations that account for viscoplastic and damage features. Our findings further confirm that plasticity, as a mechanical law to capture remodeling in these networks, is fundamentally tied to material damage via force-driven unbinding of fiber crosslinks. These results characterize in a multiscale manner the dynamic nature of the mechanical environment of physiologically mimicking cell-in-gel systems.

Many cells in the body are surrounded by a 3D extracellular matrix of interconnected protein fibers. The density and architecture of this protein fiber network can play important roles in controlling cell behavior. Deregulated biophysical properties of the extracellular environment are observed in diseases such as cancer. We demonstrate, through an integrated computational and experimental study, that cell-generated dynamic local forces rapidly and mechanically remodel the matrix, creating a non-homogeneous, densified region around the cell. This substantially increases extracellular matrix protein concentration in the vicinity of cells and alters matrix mechanical properties over time, creating a new microenvironment. Cells are known to respond to both biochemical and biomechanical properties of their surroundings. Our findings show that for mechanically active cells that exert dynamic forces onto the extracellular matrix, the physical properties of the surrounding environment that they sense are dynamic, and these dynamic properties should be taken into consideration in studies involving cell-matrix interactions, such as 3D traction force microscopy experiments in physiologically relevant environments.

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.

Capturing 3D large-strain Euler-bending filament dynamics in fibrous media simulations; sample case of compression collapse in dendritic actin network

Cytoskeletal networks to transmission towers are comprised of slender elements. Slender filaments bend and buckle more easily than stretch. Therefore a deforming network is expected to exhaust all possible bending-based modes before engaging filament stretch. While the large-strain bending critically determines fibrous-media response, simulations use small-strain and jointed approximations. At low resolution, these approximations inflate bending resistance and delay buckling onset. The proposed string-of-continuous-beams (SOCB) approach captures 3D nonlinear Euler bending of filaments with high fidelity at low cost. Bending geometry (i.e. angles and its differentials) is solved as primary variables, to fit a 5th order polynomial of the contour angle. Displacement, solved simultaneously as length conservation, is predicted with C3 and C6 smoothness between and within segments, using only 2 nodes. In the chosen analysis frame, in-plane and out-plane moments can be decoupled for arbitrarily-curved segments. Complex crosslink force-transfers can be specified. Simulations show that when a daughter branch is appended, the buckling resistance of a filament changes from linear to nonlinear before reversible collapse. An actin outcrop with 8 generations of mother-daughter branching produced the linear, nonlinear, and collapse regimes observed in compression experiments. 'Collapse' was a redistribution of outcrop forces following the buckling of few strands.

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.

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.

Cytoskeletal stiffening in synthetic hydrogels

Although common in biology, controlled stiffening of hydrogels in vitro is difficult to achieve; the required stimuli are commonly large and/or the stiffening amplitudes small. Here, we describe the hierarchical mechanics of ultra-responsive hybrid hydrogels composed of two synthetic networks, one semi-flexible and stress-responsive, the other flexible and thermoresponsive. Heating collapses the flexible network, which generates internal stress that causes the hybrid gel to stiffen up to 50 times its original modulus; an effect that is instantaneous and fully reversible. The average generated forces amount to ~1 pN per network fibre, which are similar to values found for stiffening resulting from myosin molecular motors in actin. The excellent control, reversible nature and large response gives access to many biological and bio-like applications, including tissue engineering with truly dynamic mechanics and life-like matter.

Although common in biology, controlled stiffening of hydrogels in vitro is difficult to achieve. Here the authors show how a biomimetic hybrid hydrogel can be stiffened instantaneously and reversibly up to 50 times.

Cell force-mediated matrix reorganization underlies multicellular network assembly

Vasculogenesis is the de novo formation of a vascular network from individual endothelial progenitor cells occurring during embryonic development, organogenesis, and adult neovascularization. Vasculogenesis can be mimicked and studied in vitro using network formation assays, in which endothelial cells (ECs) spontaneously form capillary-like structures when seeded in the appropriate microenvironment. While the biochemical regulators of network formation have been well studied using these assays, the role of mechanical and topographical properties of the extracellular matrix (ECM) is less understood. Here, we utilized both natural and synthetic fibrous materials to better understand how physical attributes of the ECM influence the assembly of EC networks. Our results reveal that active cell-mediated matrix recruitment through actomyosin force generation occurs concurrently with network formation on Matrigel, a reconstituted basement membrane matrix regularly used to promote EC networks, and on synthetic matrices composed of electrospun dextran methacrylate (DexMA) fibers. Furthermore, modulating physical attributes of DexMA matrices that impair matrix recruitment consequently inhibited the formation of cellular networks. These results suggest an iterative process in which dynamic cell-induced changes to the physical microenvironment reciprocally modulate cell behavior to guide the formation and stabilization of multicellular networks.

Chronology of motor-mediated microtubule streaming

We introduce a filament-based simulation model for coarse-grained, effective motor-mediated interaction between microtubule pairs to study the time-scales that compose cytoplasmic streaming. We characterise microtubule dynamics in two-dimensional systems by chronologically arranging five distinct processes of varying duration that make up streaming, from microtubule pairs to collective dynamics. The structures found were polarity sorted due to the propulsion of antialigned microtubules. This also gave rise to the formation of large polar-aligned domains, and streaming at the domain boundaries. Correlation functions, mean squared displacements, and velocity distributions reveal a cascade of processes ultimately leading to microtubule streaming and advection, spanning multiple microtubule lengths. The characteristic times for the processes extend over three orders of magnitude from fast single-microtubule processes to slow collective processes. Our approach can be used to directly test the importance of molecular components, such as motors and crosslinking proteins between microtubules, on the collective dynamics at cellular scale.

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.

From cytoskeletal assemblies to living materials

Many subcellular structures contain large numbers of cytoskeletal filaments. Such assemblies underlie much of cell division, motility, signaling, metabolism, and growth. Thus, understanding cell biology requires understanding the properties of networks of cytoskeletal filaments. While there are well established disciplines in biology dedicated to studying isolated proteins - their structure (Structural Biology) and behaviors (Biochemistry) - it is much less clear how to investigate, or even just describe, the structure and behaviors of collections of cytoskeletal filaments. One approach is to use methodologies from Mechanics and Soft Condensed Matter Physics, which have been phenomenally successful in the domains where they have been traditionally applied. From this perspective, collections of cytoskeletal filaments are viewed as materials, albeit very complex, 'active' materials, composed of molecules which use chemical energy to perform mechanical work. A major challenge is to relate these material level properties to the behaviors of the molecular constituents. Here we discuss this materials perspective and review recent work bridging molecular and network scale properties of the cytoskeleton, focusing on the organization of microtubules by dynein as an illustrative example.

Heterogeneity and nonaffinity of cell-induced matrix displacements

Cell contractile forces deform and reorganize the surrounding matrix, but the relationship between the forces and the resulting displacements is complicated by the fact that the fibrous structure brings about a complex set of mechanical properties. Many studies have quantified nonlinear and time-dependent properties at macroscopic scales, but it is unclear whether macroscopic properties apply to the scale of a cell, where the matrix is composed of a heterogeneous network of fibers. To address this question, we mimicked the contraction of a cell embedded within a fibrous collagen matrix and quantified the resulting displacements. The data revealed displacements that were heterogeneous and nonaffine. The heterogeneity was reproducible during cyclic loading, and it decreased with decreasing fiber length. Both the experiments and a fiber network model showed that the heterogeneous displacements decayed over distance at a rate no faster than the average displacement field, indicating no transition to homogeneous continuum behavior. Experiments with cells fully embedded in collagen matrices revealed the presence of heterogeneous displacements as well, exposing the dramatic heterogeneity in matrix reorganization that is induced by cells at different positions within the same fibrous matrix.

Autonomous Actuation of Zero Modes in Mechanical Networks Far from Equilibrium

A zero mode, or floppy mode, is a nontrivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this Letter, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by nonequilibrium activity. We find that a correlated active bath actuates an infinitesimal zero mode while simultaneously suppressing fluctuations in higher modes compared to thermal fluctuations, which we experimentally mimic by high frequency shaking of a physical network. Furthermore, self-propulsive dynamics spontaneously mobilize finite mechanisms as exemplified by a self-propelled topological soliton. Nonequilibrium activity thus enables autonomous actuation of coordinated mechanisms engineered through network topology.

Realizations of highly heterogeneous collagen networks via stochastic reconstruction for micromechanical analysis of tumor cell invasion

Three-dimensional (3D) collective cell migration in a collagen-based extracellular matrix (ECM) is among one of the most significant topics in developmental biology, cancer progression, tissue regeneration, and immune response. Recent studies have suggested that collagen-fiber mediated force transmission in cellularized ECM plays an important role in stress homeostasis and regulation of collective cellular behaviors. Motivated by the recent in vitro observation that oriented collagen can significantly enhance the penetration of migrating breast cancer cells into dense Matrigel which mimics the intravasation process in vivo [Han et al. Proc. Natl. Acad. Sci. USA 113, 11208 (2016)PNASA60027-842410.1073/pnas.1610347113], we devise a procedure for generating realizations of highly heterogeneous 3D collagen networks with prescribed microstructural statistics via stochastic optimization. Specifically, a collagen network is represented via the graph (node-bond) model and the microstructural statistics considered include the cross-link (node) density, valence distribution, fiber (bond) length distribution, as well as fiber orientation distribution. An optimization problem is formulated in which the objective function is defined as the squared difference between a set of target microstructural statistics and the corresponding statistics for the simulated network. Simulated annealing is employed to solve the optimization problem by evolving an initial network via random perturbations to generate realizations of homogeneous networks with randomly oriented fibers, homogeneous networks with aligned fibers, heterogeneous networks with a continuous variation of fiber orientation along a prescribed direction, as well as a binary system containing a collagen region with aligned fibers and a dense Matrigel region with randomly oriented fibers. The generation and propagation of active forces in the simulated networks due to polarized contraction of an embedded ellipsoidal cell and a small group of cells are analyzed by considering a nonlinear fiber model incorporating strain hardening upon large stretching and buckling upon compression. Our analysis shows that oriented fibers can significantly enhance long-range force transmission in the network. Moreover, in the oriented-collagen-Matrigel system, the forces generated by a polarized cell in collagen can penetrate deeply into the Matrigel region. The stressed Matrigel fibers could provide contact guidance for the migrating cell cells, and thus enhance their penetration into Matrigel. This suggests a possible mechanism for the observed enhanced intravasation by oriented collagen.

Nonequilibrium phase diagrams for actomyosin networks

Living cells dynamically modulate the local morphologies of their actin networks to perform biological functions, including force transduction, intracellular transport, and cell division. A major challenge is to understand how diverse structures of the actin cytoskeleton are assembled from a limited set of molecular building blocks. Here we study the spontaneous self-assembly of a minimal model of cytoskeletal materials, consisting of semiflexible actin filaments, crosslinkers, and molecular motors. Using coarse-grained simulations, we demonstrate that by changing concentrations and kinetics of crosslinkers and motors, as well as filament lengths, we can generate three distinct structural phases of actomyosin assemblies: bundled, polarity-sorted, and contracted. We introduce new metrics to distinguish these structural phases and demonstrate their functional roles. We find that the binding kinetics of motors and crosslinkers can be tuned to optimize contractile force generation, motor transport, and mechanical response. By quantitatively characterizing the relationships between the modes of cytoskeletal self-assembly, the resulting structures, and their functional consequences, our work suggests new principles for the design of active materials.

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.

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.

Balancing forces in migration

The integrated molecular interactions of proteins can create active biological networks whose material properties and actions can impact a variety of physiological processes. Chief among these is the ability to generate and respond to physical forces. The cytoskeleton plays a key role in this behavior, characterized by active self-reorganization to control a cell's shape and mediate its physical interactions. This review discusses our current understanding of how the material properties of the cytoskeleton and its physical interactions with the extracellular environment impact cell migration.

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.

Architecture shapes contractility in actomyosin networks

Myosin-driven contraction of the actin cytoskeleton is at the base of cell and tissue morphogenesis. At the molecular level, myosin motors drive contraction by sliding actin filaments past one another using energy produced by ATP hydrolysis. How this microscopic sliding activity gives rise to cell-scale contractions has been an active research question first in muscle cells, and over the last few decades in non-muscle cells. While many early investigations focused on myosin motor activity, increasingly, the nanoscale architecture of the actin network emerges as a key regulator of contractility. Here we review theoretical and in vitro reconstitution studies that have uncovered some of the key mechanisms by which actin network organization controls contractile tension generation. We then discuss recent findings indicating that similar principles apply in cells.

Traction Force Microscopy for Noninvasive Imaging of Cell Forces

The forces exerted by cells on their surroundings play an integral role in both physiological processes and disease progression. Traction force microscopy is a noninvasive technique that enables the in vitro imaging and quantification of cell forces. Utilizing expertise from a variety of disciplines, recent developments in traction force microscopy are enhancing the study of cell forces in physiologically relevant model systems, and hold promise for further advancing knowledge in mechanobiology. In this chapter, we discuss the methods, capabilities, and limitations of modern approaches for traction force microscopy, and highlight ongoing efforts and challenges underlying future innovations.

Polarity sorting drives remodeling of actin-myosin networks

Cytoskeletal networks of actin filaments and myosin motors drive many dynamic cell processes. A key characteristic of these networks is their contractility. Despite intense experimental and theoretical efforts, it is not clear what mechanism favors network contraction over expansion. Recent work points to a dominant role for the nonlinear mechanical response of actin filaments, which can withstand stretching but buckle upon compression. Here we present an alternative mechanism. We study how interactions between actin and myosin-2 at the single filament level translate into contraction at the network scale by performing time-lapse imaging on reconstituted quasi-2D-networks mimicking the cell cortex. We observe myosin end-dwelling after it runs processively along actin filaments. This leads to transport and clustering of actin filament ends and the formation of transiently stable bipolar structures. Further we show that myosin-driven polarity sorting produces polar actin asters, which act as contractile nodes that drive contraction in crosslinked networks. Computer simulations comparing the roles of the end-dwelling mechanism and a buckling-dependent mechanism show that the relative contribution of end-dwelling contraction increases as the network mesh-size decreases.

Enhanced Dynamics of Confined Cytoskeletal Filaments Driven by Asymmetric Motors

Cytoskeletal filaments and molecular motors facilitate the micron-scale force generation necessary for the distribution of organelles and the restructuring of the cytoskeleton within eukaryotic cells. Although the mesoscopic structure and the dynamics of such filaments have been studied in vitro and in vivo, their connection with filament polarity-dependent motor-mediated force generation is not well understood. Using 2D Brownian dynamics simulations, we study a dense, confined mixture of rigid microtubules (MTs) and active springs that have arms that cross-link neighboring MT pairs and move unidirectionally on the attached MT. We simulate depletion interactions between MTs using an attractive potential. We show that dimeric motors, with a motile arm on only one of the two MTs, produce large polarity-sorted MT clusters, whereas tetrameric motors, with motile arms on both microtubules, produce bundles. Furthermore, dimeric motors induce, on average, higher velocities between antialigned MTs than tetrameric motors. Our results, where MTs move faster near the confining wall, are consistent with experimental observations in Drosophila oocytes where enhanced microtubule activity is found close to the confining plasma membrane.

Filament rigidity and connectivity tune the deformation modes of active biopolymer networks

Molecular motors embedded within collections of actin and microtubule filaments underlie the dynamics of cytoskeletal assemblies. Understanding the physics of such motor-filament materials is critical to developing a physical model of the cytoskeleton and designing biomimetic active materials. Here, we demonstrate through experiments and simulations that the rigidity and connectivity of filaments in active biopolymer networks regulates the anisotropy and the length scale of the underlying deformations, yielding materials with variable contractility. We find that semiflexible filaments can be compressed and bent by motor stresses, yielding materials that undergo predominantly biaxial deformations. By contrast, rigid filament bundles slide without bending under motor stress, yielding materials that undergo predominantly uniaxial deformations. Networks dominated by biaxial deformations are robustly contractile over a wide range of connectivities, while networks dominated by uniaxial deformations can be tuned from extensile to contractile through cross-linking. These results identify physical parameters that control the forces generated within motor-filament arrays and provide insight into the self-organization and mechanics of cytoskeletal assemblies.

Force percolation of contractile active gels

Living systems provide a paradigmatic example of active soft matter. Cells and tissues comprise viscoelastic materials that exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, an active gel of semiflexible filaments, crosslinks, and molecular motors inside cells. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (∼nN and greater) on cellular, tissue, and organismal length scales. Here we review experimental and theoretical studies on contractile active gels composed of actin filaments and myosin motors. Unlike other active soft matter systems, which tend to form ordered patterns, actin-myosin systems exhibit a generic tendency to contract. Experimental studies of reconstituted actin-myosin model systems have long suggested that a mechanical interplay between motor activity and the network's connectivity governs this contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, extended to include effects of motor activity on the network connectivity. Based on concepts from percolation theory, we propose a state diagram that unites a large body of experimental observations. This framework provides valuable insights into the mechanisms that drive cellular shape changes and also provides design principles for synthetic active materials.

Nonequilibrium dynamics of probe filaments in actin-myosin networks

Active dynamic processes of cells are largely driven by the cytoskeleton, a complex and adaptable semiflexible polymer network, motorized by mechanoenzymes. Small dimensions, confined geometries, and hierarchical structures make it challenging to probe dynamics and mechanical response of such networks. Embedded semiflexible probe polymers can serve as nonperturbing multiscale probes to detect force distributions in active polymer networks. We show here that motor-induced forces transmitted to the probe polymers are reflected in nonequilibrium bending dynamics, which we analyze in terms of spatial eigenmodes of an elastic beam under steady-state conditions. We demonstrate how these active forces induce correlations among the mode amplitudes, which furthermore break time-reversal symmetry. This leads to a breaking of detailed balance in this mode space. We derive analytical predictions for the magnitude of resulting probability currents in mode space in the white-noise limit of motor activity. We relate the structure of these currents to the spatial profile of motor-induced forces along the probe polymers and provide a general relation for observable currents on two-dimensional hyperplanes.

Physical limits to biomechanical sensing in disordered fibre networks

Cells actively probe and respond to the stiffness of their surroundings. Since mechanosensory cells in connective tissue are surrounded by a disordered network of biopolymers, their in vivo mechanical environment can be extremely heterogeneous. Here we investigate how this heterogeneity impacts mechanosensing by modelling the cell as an idealized local stiffness sensor inside a disordered fibre network. For all types of networks we study, including experimentally-imaged collagen and fibrin architectures, we find that measurements applied at different points yield a strikingly broad range of local stiffnesses, spanning roughly two decades. We verify via simulations and scaling arguments that this broad range of local stiffnesses is a generic property of disordered fibre networks. Finally, we show that to obtain optimal, reliable estimates of global tissue stiffness, a cell must adjust its size, shape, and position to integrate multiple stiffness measurements over extended regions of space.

Cells in the connective tissue are surrounded by a heterogeneous network of biopolymers. Here, the authors investigate how such heterogeneity affects cellular mechanosensing by simulating the deformation response of experimental and modelled biopolymer networks to locally applied forces.

Mode Selection in Compressible Active Flow Networks

Coherent, large-scale dynamics in many nonequilibrium physical, biological, or information transport networks are driven by small-scale local energy input. Here, we introduce and explore an analytically tractable nonlinear model for compressible active flow networks. In contrast to thermally driven systems, we find that active friction selects discrete states with a limited number of oscillation modes activated at distinct fixed amplitudes. Using perturbation theory, we systematically predict the stationary states of noisy networks and find good agreement with a Bayesian state estimation based on a hidden Markov model applied to simulated time series data. Our results suggest that the macroscopic response of active network structures, from actomyosin force networks to cytoplasmic flows, can be dominated by a significantly reduced number of modes, in contrast to energy equipartition in thermal equilibrium. The model is also well suited to study topological sound modes and spectral band gaps in active matter.

Vascular Mechanobiology: Towards Control of In Situ Regeneration

The paradigm of regenerative medicine has recently shifted from in vitro to in situ tissue engineering: implanting a cell-free, biodegradable, off-the-shelf available scaffold and inducing the development of functional tissue by utilizing the regenerative potential of the body itself. This approach offers a prospect of not only alleviating the clinical demand for autologous vessels but also circumventing the current challenges with synthetic grafts. In order to move towards a hypothesis-driven engineering approach, we review three crucial aspects that need to be taken into account when regenerating vessels: (1) the structure-function relation for attaining mechanical homeostasis of vascular tissues, (2) the environmental cues governing cell function, and (3) the available experimental platforms to test instructive scaffolds for in situ tissue engineering. The understanding of cellular responses to environmental cues leads to the development of computational models to predict tissue formation and maturation, which are validated using experimental platforms recapitulating the (patho)physiological micro-environment. With the current advances, a progressive shift is anticipated towards a rational and effective approach of building instructive scaffolds for in situ vascular tissue regeneration.