Role of catch bonds in actomyosin mechanics and cell mechanosensitivity

A mechanochemical model recapitulates distinct vertebrate gastrulation modes

During vertebrate gastrulation, an embryo transforms from a layer of epithelial cells into a multilayered gastrula. This process requires the coordinated movements of hundreds to tens of thousands of cells, depending on the organism. In the chick embryo, patterns of actomyosin cables spanning several cells drive coordinated tissue flows. Here, we derive a minimal theoretical framework that couples actomyosin activity to global tissue flows. Our model predicts the onset and development of gastrulation flows in normal and experimentally perturbed chick embryos, mimicking different gastrulation modes as an active stress instability. Varying initial conditions and a parameter associated with active cell ingression, our model recapitulates distinct vertebrate gastrulation morphologies, consistent with recently published experiments in the chick embryo. Altogether, our results show how changes in the patterning of critical cell behaviors associated with different force-generating mechanisms contribute to distinct vertebrate gastrulation modes via a self-organizing mechanochemical process.

Light-driven biological actuators to probe the rheology of 3D microtissues

The mechanical properties of biological tissues are key to their physical integrity and function. Although external loading or biochemical treatments allow the estimation of these properties globally, it remains difficult to assess how such external stimuli compare with cell-generated contractions. Here we engineer microtissues composed of optogenetically-modified fibroblasts encapsulated within collagen. Using light to control the activity of RhoA, a major regulator of cellular contractility, we induce local contractions within microtissues, while monitoring microtissue stress and strain. We investigate the regulation of these local contractions and their spatio-temporal distribution. We demonstrate the potential of our technique for quantifying tissue elasticity and strain propagation, before examining the possibility of using light to create and map local anisotropies in mechanically heterogeneous microtissues. Altogether, our results open an avenue to guide the formation of tissues while non-destructively charting their rheology in real time, using their own constituting cells as internal actuators.

Actomyosin contractility and buckling of microtubules in nucleation, growth and disassembling of focal adhesions

Building up and maintenance of cytoskeletal structure in living cells are force-dependent processes involving a dynamic chain of polymerization and depolymerization events, which are also at the basis of cells’ remodelling and locomotion. All these phenomena develop by establishing cell–matrix interfaces made of protein complexes, known as focal adhesions, which govern mechanosensing and mechanotransduction mechanisms mediated by stress transmission between cell interior and external environment. Within this framework, by starting from a work by Cao et al. (Biophys J 109:1807–1817, 2015), we here investigate the role played by actomyosin contractility of stress fibres in nucleation, growth and disassembling of focal adhesions. In particular, we propose a tensegrity model of an adherent cell incorporating nonlinear elasticity and unstable behaviours, which provides a new kinematical interpretation of cellular contractile forces and describes how stress fibres, microtubules and adhesion plaques interact mechanobiologically. The results confirm some experimental evidences and suggest how the actomyosin contraction level could be exploited by cells to actively control their adhesion, eventually triggering cytoskeleton reconfigurations and migration processes observed in both physiological conditions and diseases.

Epithelial cells fluidize upon adhesion but display mechanical homeostasis in the adherent state

Atomic force microscopy is used to study the viscoelastic properties of epithelial cells in three different states. Force relaxation data are acquired from cells in suspension, adhered but single cells, and polarized cells in a confluent monolayer using different indenter geometries comprising flat bars, pyramidal cones, and spheres. We found that the fluidity of cells increased substantially from the suspended to the adherent state. Along this line, the prestress of suspended cells generated by cortical contractility is also greater than that of cells adhering to a surface. Polarized cells that are part of a confluent monolayer form an apical cap that is soft and fluid enough to respond rapidly to mechanical challenges from wounding, changes in the extracellular matrix, osmotic stress, and external deformation. In contrast to adherent cells, cells in the suspended state show a pronounced dependence of fluidity on the external areal strain. With increasing areal strain, the suspended cells become softer and more fluid. We interpret the results in terms of cytoskeletal remodeling that softens cells in the adherent state to facilitate adhesion and spreading by relieving internal active stress. However, once the cells spread on the surface they maintain their mechanical phenotype displaying viscoelastic homeostasis.

Physically motivated models of polymer networks with dynamic cross-links: comparative study and future outlook

Polymer networks consisting of a mixture of chemical and physical cross-links are known to exhibit complex time-dependent behaviour due to the kinetics of bond association and dissociation. In this article, we highlight and compare two recent physically based constitutive models that describe the nonlinear viscoelastic behaviour of such transient networks. These two models are developed independently by two groups of researchers using different mathematical formulations. Here, we show that this difference can be attributed to different viewpoints: Lagrangian versus Eulerian. We establish the equivalence of the two models under the special situation where chains obey Gaussian statistics and steady-state bond dynamics. We provide experimental data demonstrating that both models can accurately predict the time-dependent uniaxial behaviour of a poly(vinylalcohol) dual cross-link hydrogel. We review the advantages and disadvantages of both approaches in applications and close by discussing a list of open challenges and questions regarding the mathematical modelling of soft, viscoelastic networks.

Mechanics of 3D Cell-Hydrogel Interactions: Experiments, Models, and Mechanisms

Hydrogels are highly water-swollen molecular networks that are ideal platforms to create tissue mimetics owing to their vast and tunable properties. As such, hydrogels are promising cell-delivery vehicles for applications in tissue engineering and have also emerged as an important base for ex vivo models to study healthy and pathophysiological events in a carefully controlled three-dimensional environment. Cells are readily encapsulated in hydrogels resulting in a plethora of biochemical and mechanical communication mechanisms, which recapitulates the natural cell and extracellular matrix interaction in tissues. These interactions are complex, with multiple events that are invariably coupled and spanning multiple length and time scales. To study and identify the underlying mechanisms involved, an integrated experimental and computational approach is ideally needed. This review discusses the state of our knowledge on cell-hydrogel interactions, with a focus on mechanics and transport, and in this context, highlights recent advancements in experiments, mathematical and computational modeling. The review begins with a background on the thermodynamics and physics fundamentals that govern hydrogel mechanics and transport. The review focuses on two main classes of hydrogels, described as semiflexible polymer networks that represent physically cross-linked fibrous hydrogels and flexible polymer networks representing the chemically cross-linked synthetic and natural hydrogels. In this review, we highlight five main cell-hydrogel interactions that involve key cellular functions related to communication, mechanosensing, migration, growth, and tissue deposition and elaboration. For each of these cellular functions, recent experiments and the most up to date modeling strategies are discussed and then followed by a summary of how to tune hydrogel properties to achieve a desired functional cellular outcome. We conclude with a summary linking these advancements and make the case for the need to integrate experiments and modeling to advance our fundamental understanding of cell-matrix interactions that will ultimately help identify new therapeutic approaches and enable successful tissue engineering.

Cell orientation under stretch: Stability of a linear viscoelastic model

The sensitivity of cells to alterations in the microenvironment and in particular to external mechanical stimuli is significant in many biological and physiological circumstances. In this regard, experimental assays demonstrated that, when a monolayer of cells cultured on an elastic substrate is subject to an external cyclic stretch with a sufficiently high frequency, a reorganization of actin stress fibres and focal adhesions happens in order to reach a stable equilibrium orientation, characterized by a precise angle between the cell major axis and the largest strain direction. To examine the frequency effect on the orientation dynamics, we propose a linear viscoelastic model that describes the coupled evolution of the cellular stress and the orientation angle. We find that cell orientation oscillates tending to an angle that is predicted by the minimization of a very general orthotropic elastic energy, as confirmed by a bifurcation analysis. Moreover, simulations show that the speed of convergence towards the predicted equilibrium orientation presents a changeover related to the viscous-elastic transition for viscoelastic materials. In particular, when the imposed oscillation period is lower than the characteristic turnover rate of the cytoskeleton and of adhesion molecules such as integrins, reorientation is significantly faster.

Onsager's variational principle in active soft matter

Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.

Detailed Balance Broken by Catch Bond Kinetics Enables Mechanical-Adaptation in Active Materials

Unlike nearly all engineered materials which contain bonds that weaken under load, biological materials contain "catch" bonds which are reinforced under load. Consequently, materials, such as the cell cytoskeleton, can adapt their mechanical properties in response to their state of internal, non-equilibrium (active) stress. However, how large-scale material properties vary with the distance from equilibrium is unknown, as are the relative roles of active stress and binding kinetics in establishing this distance. Through course-grained molecular dynamics simulations, the effect of breaking of detailed balance by catch bonds on the accumulation and dissipation of energy within a model of the actomyosin cytoskeleton is explored. It is found that the extent to which detailed balance is broken uniquely determines a large-scale fluid-solid transition with characteristic time-reversal symmetries. The transition depends critically on the strength of the catch bond, suggesting that active stress is necessary but insufficient to mount an adaptive mechanical response.

Focus on time: dynamic imaging reveals stretch-dependent cell relaxation and nuclear deformation

Among the stimuli to which cells are exposed in vivo, it has been shown that tensile deformations induce specific cellular responses in musculoskeletal, cardiovascular, and stromal tissues. However, the early response of cells to sustained substrate-based stretch has remained elusive because of the short timescale at which it occurs. To measure the tensile mechanical properties of adherent cells immediately after the application of substrate deformations, we have developed a dynamic traction force microscopy method that enables subsecond temporal resolution imaging of transient subcellular events. The system employs a novel, to our knowledge, tracking approach with minimal computational overhead to compensate substrate-based, stretch-induced motion/drift of stretched single cells in real time, allowing capture of biophysical phenomena on multiple channels by fluorescent multichannel imaging on a single camera, thus avoiding the need for beam splitting with the associated loss of light. Using this tool, we have characterized the transient subcellular forces and nuclear deformations of single cells immediately after the application of equibiaxial strain. Our experiments reveal significant differences in the cell relaxation dynamics and in the intracellular propagation of force to the nuclear compartment in cells stretched at different strain rates and exposes the need for time control for the correct interpretation of dynamic cell mechanics experiments.

Crosslinking activity of non-muscle myosin II is not sufficient for embryonic cytokinesis in C. elegans

Cytokinesis in animal cells requires the assembly and constriction of a contractile actomyosin ring. Non-muscle myosin II is essential for cytokinesis, but the role of its motor activity remains unclear. Here, we examine cytokinesis in C. elegans embryos expressing non-muscle myosin motor mutants generated by genome editing. Two non-muscle motor-dead myosins capable of binding F-actin do not support cytokinesis in the one-cell embryo, and two partially motor-impaired myosins delay cytokinesis and render rings more sensitive to reduced myosin levels. Further analysis of myosin mutants suggests that it is myosin motor activity, and not the ability of myosin to crosslink F-actin, that drives the alignment and compaction of F-actin bundles during contractile ring assembly, and that myosin motor activity sets the pace of contractile ring constriction. We conclude that myosin motor activity is required at all stages of cytokinesis. Finally, characterization of the corresponding motor mutations in C. elegans major muscle myosin shows that motor activity is required for muscle contraction but is dispensable for F-actin organization in adult muscles.

This article has an associated ‘The people behind the papers’ interview.

Highlighted Article: The motor activity of non-muscle myosin II is essential for cytokinesis and contributes to all stages of the process in C. elegans embryos.

Computational modeling of the large deformation and flow of viscoelastic polymers

Deformation of soft polymeric materials often involves complex nonlinear or transient mechanical behaviors. This is due to the dynamic behaviors of polymer chains at the molecular level within the polymer network. In this paper, we present a computational formulation to describe the transient behavior (e.g., viscoelasticity) of soft polymer networks with dynamic bonds undergoing large to extreme deformation. This formulation is based on an Eulerian description of kinematics and a theoretical framework that directly connects the molecular-level kinetics of dynamic bonds to the macroscopic mechanical behavior of the material. An extended finite element method is used to discretize the field variables and the governing equations in an axisymmetric domain. In addition to validating the framework, this model is used to study how the chain dynamics affect the macroscopic response of material as they undergo a combination of flow and elasticity. The problems of cavitation rheology and polymer indentation under extreme deformation are investigated in this context.

Thermodynamic Modeling of the Statistics of Cell Spreading on Ligand-Coated Elastic Substrates

Biological spread cells exist in a perpetually fluctuating state and therefore cannot be described in terms of a unique deterministic system. For modeling approaches to provide novel insight and uncover new mechanisms that drive cell behavior, a framework is required that progresses from traditional deterministic methods (whereby simulation of an experiment predicts a single outcome). In this study, we implement a new, to our knowledge, modeling approach for the analysis of cell spreading on ligand-coated substrates, extending the framework for nonequilibrium thermodynamics of cells developed by Shishvan et al. to include active focal adhesion assembly. We demonstrate that the model correctly predicts the coupled influence of surface collagen density and substrate stiffness on cell spreading, as reported experimentally by Engler et al. Low surface collagen densities are shown to result in a high probability that cells will be restricted to low spread areas. Furthermore, elastic free energy induced by substrate deformation lowers the probability of observing a highly spread cell, and, consequentially, lower cell tractions affect the assembly of focal adhesions. Experimentally measurable observables such as cell spread area and aspect ratio can be directly postprocessed from the computed homeostatic ensemble of (several million) spread states. This allows for the prediction of mean and SDs of such experimental observables. This class of cell mechanics modeling presents a significant advance on conventional deterministic approaches.

How do fire ants control the rheology of their aggregations? A statistical mechanics approach

Active networks are omnipresent in nature, from the molecular to the macro-scale. In this study, we explore the mechanical behaviour of fire ant aggregations, closely knit swarms that display impressive dynamics culminating with the aggregations' capacity to self-heal and adapt to the environment. Although the combined elasticity and rheology of the ant aggregation can be characterized by phenomenological mechanical models (e.g. linear Maxwell or Kelvin-Voigt model), it is not clear how the behaviour of individual ants affects the aggregations' emerging responses. Here, we explore an alternative way to think about these materials, describing them as a collection of individuals connected via elastic chains that associate and dissociate over time. Using our knowledge of these connections-e.g. their elasticity and attachment/dissociation rates-we construct a statistical description of connection stretch and derive an evolution equation for the corresponding stretch distribution. This time-evolving stretch distribution is then used to determine important macroscopic measures, e.g. stress, energy storage and energy dissipation, in the network. In this context, we show how the physical characteristics and activities of individual ants can explain the elasticity, flow and shear thinning of the aggregation. In particular, we find that experimental results are matched if the detachment rate between two individuals increases with tension in the connection.

The Chain Distribution Tensor: Linking Nonlinear Rheology and Chain Anisotropy in Transient Polymers

Transient polymer networks are ubiquitous in natural and engineered materials and contain cross-links that can reversibly break and re-form. The dynamic nature of these bonds allows for interesting mechanical behavior, some of which include nonlinear rheological phenomena such as shear thickening and shear thinning. Specifically, physically cross-linked networks with reversible bonds are typically observed to have viscosities that depend nonlinearly on shear rate and can be characterized by three flow regimes. In slow shear, they behave like Newtonian fluids with a constant viscosity. With further increase in shear rate, the viscosity increases nonlinearly to subsequently reach a maximum value at the critical shear rate. At this point, network fracture occurs followed by a reduction in viscosity (shear-thinning) with a further increase in shear rate. The underlying mechanism of shear thickening in this process is still unclear with debates between a conversion of intra-chain to inter-chain cross-linking and nonlinear chain stretch under high tension. In this paper, we provide a new framework to describe the nonlinear rheology of transient polymer networks with the so-called chain distribution tensor using recent advances from the transient network theory. This tensor contains quantitatively and statistical information of the chain alignment and possible anisotropy that affect network behavior and mechanics. We investigate shear thickening as a primary result of non-Gaussian chain behavior and derive a relationship for the nonlinear viscosity in terms of the non-dimensional Weissenberg number. We further address the criterion for network fracture at the critical shear rate by introducing a critical chain force when bond dissociation is suddenly accelerated. Finally, we discuss the role of cross-linker density on viscosity using a "sticky" reptation mechanism in the context of previous studies on metallo-supramolecular networks with reversible cross-linkers.

Theoretical Analysis of Stress Distribution and Cell Polarization Surrounding a Model Wound

A growing amount of experimental evidence shows that the local elastic field acting on cells governs their spatial organization and polarity in a tissue. Interestingly, experiments on wound healing reveal a universal formation of thick actomyosin bundles around the margins of epithelial gaps. Although the forces involved in this process have been measured, the mechanisms governing cellular alignment and contractile ring formation are still not fully understood. To theoretically investigate this process, we have carried out a self-consistent calculation of the elastic field that is actively generated around a circular gap in a contractile cell monolayer that is adhered to an elastic substrate, taking into account the responsiveness of actomyosin activity to the locally generated stress. We model actomyosin contractility by a radial distribution of point force dipoles that may alter in magnitude and orientation in response to the local elastic stress. In addition, the model takes into account the forces exerted by leader cells on the margins of the cell monolayer. Our model suggests that the presence of a hole in the center of a contractile cell monolayer creates a mechanical tendency for actomyosin forces to polarize tangentially around the hole margin. In addition, it predicts that this tendency optimizes with substrate rigidity, thickness, and strength of cell adhesion to the substrate. Our calculations support the view that the universal formation of a peripheral contractile ring is a consequence of actomyosin contractility in the bulk and its inherent responsiveness to the local stress.

Transient response of nonlinear polymer networks: A kinetic theory

Dynamic networks are found in a majority of natural materials, but also in engineering materials, such as entangled polymers and physically cross-linked gels. Owing to their transient bond dynamics, these networks display a rich class of behaviors, from elasticity, rheology, self-healing, or growth. Although classical theories in rheology and mechanics have enabled us to characterize these materials, there is still a gap in our understanding on how individuals (i.e., the mechanics of each building blocks and its connection with others) affect the emerging response of the network. In this work, we introduce an alternative way to think about these networks from a statistical point of view. More specifically, a network is seen as a collection of individual polymer chains connected by weak bonds that can associate and dissociate over time. From the knowledge of these individual chains (elasticity, transient attachment, and detachment events), we construct a statistical description of the population and derive an evolution equation of their distribution based on applied deformation and their local interactions. We specifically concentrate on nonlinear elastic response that follows from the strain stiffening response of individual chains of finite size. Upon appropriate averaging operations and using a mean field approximation, we show that the distribution can be replaced by a so-called chain distribution tensor that is used to determine important macroscopic measures such as stress, energy storage and dissipation in the network. Prediction of the kinetic theory are then explored against known experimental measurement of polymer responses under uniaxial loading. It is found that even under the simplest assumptions of force-independent chain kinetics, the model is able to reproduce complex time-dependent behaviors of rubber and self-healing supramolecular polymers.

Molecular force spectroscopy of kinetochore-microtubule attachment in silico: Mechanical signatures of an unusual catch bond and collective effects

Measurement of the lifetime of attachments formed by a single microtubule (MT) with a single kinetochore (kt) in vitro under force-clamp conditions had earlier revealed a catch-bond-like behavior. In the past, the physical origin of this apparently counterintuitive phenomenon was traced to the nature of the force dependence of the (de)polymerization kinetics of the MTs. Here, first the same model MT-kt attachment is subjected to external tension that increases linearly with time until rupture occurs. In our force-ramp experiments in silico, the model displays the well known "mechanical signatures" of a catch bond probed by molecular force spectroscopy. Exploiting this evidence, we have further strengthened the analogy between MT-kt attachments and common ligand-receptor bonds in spite of the crucial differences in their underlying physical mechanisms. We then extend the formalism to model the stochastic kinetics of an attachment formed by a bundle of multiple parallel microtubules with a single kt considering the effect of rebinding under force-clamp and force-ramp conditions. From numerical studies of the model we predict the trends of variation of the mean lifetime and mean rupture force with the increasing number of MTs in the bundle. Both the mean lifetime and the mean rupture force display nontrivial nonlinear dependence on the maximum number of MTs that can attach simultaneously to the same kt.

Heterogeneity is key to hydrogel-based cartilage tissue regeneration

Degradable hydrogels have been developed to provide initial mechanical support to encapsulated cells while facilitating the growth of neo-tissues. When cells are encapsulated within degradable hydrogels, the process of neo-tissue growth is complicated by the coupled phenomena of transport of large extracellular matrix macromolecules and the rate of hydrogel degradation. If hydrogel degradation is too slow, neo-tissue growth is hindered, whereas if it is too fast, complete loss of mechanical integrity can occur. Therefore, there is a need for effective modelling techniques to predict hydrogel designs based on the growth parameters of the neo-tissue. In this article, hydrolytically degradable hydrogels are investigated due to their promise in tissue engineering. A key output of the model focuses on the ability of the construct to maintain overall structural integrity as the construct transitions from a pure hydrogel to engineered neo-tissue. We show that heterogeneity in cross-link density and cell distribution is the key to this successful transition and ultimately to achieve tissue growth. Specifically, we find that optimally large regions of weak cross-linking around cells in the hydrogel and well-connected and dense cell clusters create the optimum conditions needed for neo-tissue growth while maintaining structural integrity. Experimental observations using cartilage cells encapsulated in a hydrolytically degradable hydrogel are compared with model predictions to show the potential of the proposed model.

Structural Modeling of Mechanosensitivity in Non-Muscle Cells: Multiscale Approach to Understand Cell Sensing

The contraction and spreading of nonmuscle cells are important phenomena in a number of cellular processes such as differentiation, morphogenesis, and tissue growth. Recent experimental work has shown that the topology and the mechanical properties of the underlying substrate play a significant role in directing the cell's response. In this work, we introduce a multiscale model to understand the sensing, activation, and contraction of the actin cytoskeleton of nonmuscle cells based on the idea that acto-myosin cross-bridges display a catch-bond response. After investigating the respective roles of bond catchiness and acto-myosin assembly on the mechano-sensitivity of stress fibers, we present full simulations of cells laying on arrays of micropillars. Model predictions show good qualitative agreements with experimental observation, suggesting that acto-myosin catch bonds are a major mechano-sensing element in nonmuscle cells.