Single cell mechanics: stress stiffening and kinematic hardening

Critical review of single-cell mechanotyping approaches for biomedical applications

Accurate mechanical measurements of cells has the potential to improve diagnostics, therapeutics and advance understanding of disease mechanisms, where high-resolution mechanical information can be measured by deforming individual cells. Here we evaluate recently developed techniques for measuring cell-scale stiffness properties; while many such techniques have been developed, much of the work examining single-cell stiffness is impacted by difficulties in standardization and comparability, giving rise to large variations in reported mechanical moduli. We highlight the role of underlying mechanical theories driving this variability, and note opportunities to develop novel mechanotyping devices and theoretical models that facilitate convenient and accurate mechanical characterisation. Moreover, many high-throughput approaches are confounded by factors including cell size, surface friction, natural population heterogeneity and convolution of elastic and viscous contributions to cell deformability. We nevertheless identify key approaches based on deformability cytometry as a promising direction for further development, where both high-throughput and accurate single-cell resolutions can be realized.

A comprehensive model of Drosophila epithelium reveals the role of embryo geometry and cell topology in mechanical responses

In order to understand morphogenesis, it is necessary to know the material properties or forces shaping the living tissue. In spite of this need, very few in vivo measurements are currently available. Here, using the early Drosophila embryo as a model, we describe a novel cantilever-based technique which allows for the simultaneous quantification of applied force and tissue displacement in a living embryo. By analyzing data from a series of experiments in which embryonic epithelium is subjected to developmentally relevant perturbations, we conclude that the response to applied force is adiabatic and is dominated by elastic forces and geometric constraints, or system size effects. Crucially, computational modeling of the experimental data indicated that the apical surface of the epithelium must be softer than the basal surface, a result which we confirmed experimentally. Further, we used the combination of experimental data and comprehensive computational model to estimate the elastic modulus of the apical surface and set a lower bound on the elastic modulus of the basal surface. More generally, our investigations revealed important general features that we believe should be more widely addressed when quantitatively modeling tissue mechanics in any system. Specifically, different compartments of the same cell can have very different mechanical properties; when they do, they can contribute differently to different mechanical stimuli and cannot be merely averaged together. Additionally, tissue geometry can play a substantial role in mechanical response, and cannot be neglected.

A New Elementary Method for Determining the Tip Radius and Young's Modulus in AFM Spherical Indentations

Atomic force microscopy (AFM) is a powerful tool for characterizing biological materials at the nanoscale utilizing the AFM nanoindentation method. When testing biological materials, spherical indenters are typically employed to reduce the possibility of damaging the sample. The accuracy of determining Young's modulus depends, among other factors, on the calibration of the indenter, i.e., the determination of the tip radius. This paper demonstrates that the tip radius can be approximately calculated using a single force-indentation curve on an unknown, soft sample without performing any additional experimental calibration process. The proposed method is based on plotting a tangent line on the force indentation curve at the maximum indentation depth. Subsequently, using equations that relate the applied force, maximum indentation depth, and the tip radius, the calculation of the tip radius becomes trivial. It is significant to note that the method requires only a single force-indentation curve and does not necessitate knowledge of the sample's Young's modulus. Consequently, the determination of both the sample's Young's modulus and the tip radius can be performed simultaneously. Thus, the experimental effort is significantly reduced. The method was tested on 80 force-indentation curves obtained on an agarose gel, and the results were accurate.

All-in-one rheometry and nonlinear rheology of multicellular aggregates

Tissues are generally subjected to external stresses, a potential stimulus for their differentiation or remodeling. While single-cell rheology has been extensively studied leading to controversial results about nonlinear response, mechanical tissue behavior under external stress is still poorly understood, in particular, the way individual cell properties translate at the tissue level. Herein, using magnetic cells we were able to form perfectly monitored cellular aggregates (magnetic molding) and to deform them under controlled applied stresses over a wide range of timescales and amplitudes (magnetic rheometer). We explore the rheology of these minimal tissue models using both standard assays (creep and oscillatory response) as well as an innovative broad spectrum solicitation coupled with inference analysis thus being able to determine in a single experiment the best rheological model. We find that multicellular aggregates exhibit a power-law response with nonlinearities leading to tissue stiffening at high stress. Moreover, we reveal the contribution of intracellular (actin network) and intercellular components (cell-cell adhesions) in this aggregate rheology.

Nonlinear Elastic and Inelastic Properties of Cells

Mechanical forces play an important role in various physiological processes, such as morphogenesis, cytokinesis, and migration. Thus, in order to illuminate mechanisms underlying these physiological processes, it is crucial to understand how cells deform and respond to external mechanical stimuli. During recent decades, the mechanical properties of cells have been studied extensively using diverse measurement techniques. A number of experimental studies have shown that cells are far from linear elastic materials. Cells exhibit a wide variety of nonlinear elastic and inelastic properties. Such complicated properties of cells are known to emerge from unique mechanical characteristics of cellular components. In this review, we introduce major cellular components that largely govern cell mechanical properties and provide brief explanations of several experimental techniques used for rheological measurements of cell mechanics. Then, we discuss the representative nonlinear elastic and inelastic properties of cells. Finally, continuum and discrete computational models of cell mechanics, which model both nonlinear elastic and inelastic properties of cells, will be described.

Loops versus lines and the compression stiffening of cells

Both animal and plant tissue exhibit a nonlinear rheological phenomenon known as compression stiffening, or an increase in moduli with increasing uniaxial compressive strain. Does such a phenomenon exist in single cells, which are the building blocks of tissues? One expects an individual cell to compression soften since the semiflexible biopolymer-based cytoskeletal network maintains the mechanical integrity of the cell and in vitro semiflexible biopolymer networks typically compression soften. To the contrary, we find that mouse embryonic fibroblasts (mEFs) compression stiffen under uniaxial compression via atomic force microscopy studies. To understand this finding, we uncover several potential mechanisms for compression stiffening. First, we study a single semiflexible polymer loop modeling the actomyosin cortex enclosing a viscous medium modeled as an incompressible fluid. Second, we study a two-dimensional semiflexible polymer/fiber network interspersed with area-conserving loops, which are a proxy for vesicles and fluid-based organelles. Third, we study two-dimensional fiber networks with angular-constraining crosslinks, i.e. semiflexible loops on the mesh scale. In the latter two cases, the loops act as geometric constraints on the fiber network to help stiffen it via increased angular interactions. We find that the single semiflexible polymer loop model agrees well with the experimental cell compression stiffening finding until approximately 35% compressive strain after which bulk fiber network effects may contribute. We also find for the fiber network with area-conserving loops model that the stress-strain curves are sensitive to the packing fraction and size distribution of the area-conserving loops, thereby creating a mechanical fingerprint across different cell types. Finally, we make comparisons between this model and experiments on fibrin networks interlaced with beads as well as discuss implications for single cell compression stiffening at the tissue scale.

The mechanical properties of microbial surfaces and biofilms

Microbes can modify their surface structure as an adaptive mechanism for survival and dissemination in the environment or inside the host. Altering their ability to respond to mechanical stimuli is part of this adaptive process. Since the 1990s, powerful micromanipulation tools have been developed that allow mechanical studies of microbial cell surfaces, exploring little known aspects of their dynamic behavior. This review concentrates on the study of mechanical and rheological properties of bacteria and fungi, focusing on their cell surface dynamics and biofilm formation.

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.

Programming the mechanics of cohesive fiber networks by compression

Fibrous networks are ideal functional materials since they provide mechanical rigidity at low weight. Here, we demonstrate that fibrous networks of the blood clotting protein fibrin undergo a strong and irreversible increase in their mechanical rigidity in response to uniaxial compression. This rigidification can be precisely controlled by the level of applied compressive strain, providing a means to program the network rigidity without having to change its composition. To identify the underlying mechanism we measure single fiber-fiber interactions using optical tweezers. We further develop a minimal computational model of cohesive fiber networks that shows that stiffening arises due to the formation of new bonds in the compressed state, which develop tensile stress when the network is re-expanded. The model predicts that the network stiffness after a compression cycle obeys a power-law dependence on tensile stress, which we confirm experimentally. This finding provides new insights into how biological tissues can adapt themselves independently of any cellular processes, offering new perspectives to inspire the design of reprogrammable materials.

Large Amplitude Oscillatory Shear Rheology of Living Fibroblasts: Path-Dependent Steady States

Mechanical properties of biological cells play a role in cell locomotion, embryonic tissue formation, and tumor migration among many other processes. Cells exhibit a complex nonlinear response to mechanical cues that is not understood. Cells may stiffen as well as soften, depending on the exact type of stimulus. Here we apply large-amplitude oscillatory shear to a monolayer of separated fibroblast cells suspended between two plates. Although we apply identical steady-state excitations, in response we observe different typical regimes that exhibit cell softening or cell stiffening to varying degrees. This degeneracy of the cell response can be linked to the initial paths that the instrument takes to go from cell rest to steady state. A model of cross-linked, force-bearing filaments submitted to steady-state excitation renders the different observed regimes with minor changes in parameters if the filaments are permitted to self-organize and form different spatially organized structures. We suggest that rather than a complex viscoelastic or plastic response, the different observed regimes reflect the emergence of different steady-state cytoskeletal conformations. A high sensitivity of the cytoskeletal rheology and structure to minor changes in parameters or initial conditions enables a cell to respond to mechanical requirements quickly and in various ways with only minor biochemical intervention. Probing path-dependent rheological changes constitutes a possibly very sensitive assessment of the cell cytoskeleton as a possible tool for medical diagnosis. Our observations show that the memory of subtle differences in earlier deformation paths must be taken into account when deciphering the cell mechanical response to large-amplitude deformations.

Fibrin Networks Support Recurring Mechanical Loads by Adapting their Structure across Multiple Scales

Tissues and cells sustain recurring mechanical loads that span a wide range of loading amplitudes and timescales as a consequence of exposure to blood flow, muscle activity, and external impact. Both tissues and cells derive their mechanical strength from fibrous protein scaffolds, which typically have a complex hierarchical structure. In this study, we focus on a prototypical hierarchical biomaterial, fibrin, which is one of the most resilient naturally occurring biopolymers and forms the structural scaffold of blood clots. We show how fibrous networks composed of fibrin utilize irreversible changes in their hierarchical structure at different scales to maintain reversible stress stiffening up to large strains. To trace the origin of this paradoxical resilience, we systematically tuned the microstructural parameters of fibrin and used a combination of optical tweezers and fluorescence microscopy to measure the interactions of single fibrin fibers for the first time, to our knowledge. We demonstrate that fibrin networks adapt to moderate strains by remodeling at the network scale through the spontaneous formation of new bonds between fibers, whereas they adapt to high strains by plastic remodeling of the fibers themselves. This multiscale adaptation mechanism endows fibrin gels with the remarkable ability to sustain recurring loads due to shear flows and wound stretching. Our findings therefore reveal a microscopic mechanism by which tissues and cells can balance elastic nonlinearity and plasticity, and thus can provide microstructural insights into cell-driven remodeling of tissues.

Using cell monolayer rheology to probe average single cell mechanical properties

The cell monolayer rheology technique consists of a commercial rotational rheometer that probes the mechanical properties of a monolayer of isolated cells. So far we have described properties of an entire monolayer. In this short communication, we show that we can deduce average single cell properties. Results are in very good agreement with earlier work on single cell mechanics. Our approach provides a mean of 105-106 adherent cells within a single experiment. This makes the results very reproducible. We extend our work on cell adhesion strength and deduce cell adhesion forces of fibroblast cells on fibronectin coated glass substrates.

Mechanical plasticity of cells

Under mechanical loading, most living cells show a viscoelastic deformation that follows a power law in time. After removal of the mechanical load, the cell shape recovers only incompletely to its original undeformed configuration. Here, we show that incomplete shape recovery is due to an additive plastic deformation that displays the same power-law dynamics as the fully reversible viscoelastic deformation response. Moreover, the plastic deformation is a constant fraction of the total cell deformation and originates from bond ruptures within the cytoskeleton. A simple extension of the prevailing viscoelastic power-law response theory with a plastic element correctly predicts the cell behaviour under cyclic loading. Our findings show that plastic energy dissipation during cell deformation is tightly linked to elastic cytoskeletal stresses, which suggests the existence of an adaptive mechanism that protects the cell against mechanical damage.

Shear-wave elasticity measurements of three-dimensional cell cultures for mechanobiology

Studying mechanobiology in three-dimensional (3D) cell cultures better recapitulates cell behaviors in response to various types of mechanical stimuli in vivo. Stiffening of the extracellular matrix resulting from cell remodeling potentiates many pathological conditions, including advanced cancers. However, an effective tool for measuring the spatiotemporal changes in elastic properties of such 3D cell cultures without directly contacting the samples has not been reported previously. We describe an ultrasonic shear-wave-based platform for quantitatively evaluating the spatiotemporal dynamics of the elasticity of a matrix remodeled by cells cultured in 3D environments. We used this approach to measure the elasticity changes of 3D matrices grown with highly invasive lung cancer cells and cardiac myoblasts, and to delineate the principal mechanism underlying the stiffening of matrices remodeled by these cells. The described approach can be a useful tool in fields investigating and manipulating the mechanotransduction of cells in 3D contexts, and also has potential as a drug-screening platform.

Summary: Use of a non-direct-contact platform for measurement of the spatiotemporal dynamics of matrix elasticity when remodeled by cells cultured in three-dimensional contexts.

Soft viscoelastic properties of nuclear actin age oocytes due to gravitational creep

The actin cytoskeleton helps maintain structural organization within living cells. In large X. laevis oocytes, gravity becomes a dominant force and is countered by a nuclear actin network that prevents liquid-like nuclear bodies from immediate sedimentation and coalescence. However, nuclear actin's mechanical properties, and how they facilitate the stabilization of nuclear bodies, remain unknown. Using active microrheology, we find that nuclear actin forms a weak viscoelastic network, with a modulus of roughly 0.1 Pa. Embedded probe particles subjected to a constant force exhibit continuous displacement, due to viscoelastic creep. Gravitational forces also cause creep displacement of nuclear bodies, resulting in their asymmetric nuclear distribution. Thus, nuclear actin does not indefinitely support the emulsion of nuclear bodies, but only kinetically stabilizes them by slowing down gravitational creep to ~2 months. This is similar to the viability time of large oocytes, suggesting gravitational creep ages oocytes, with fatal consequences on long timescales.

Inelastic mechanics: A unifying principle in biomechanics

Many soft materials are classified as viscoelastic. They behave mechanically neither quite fluid-like nor quite solid-like - rather a bit of both. Biomaterials are often said to fall into this class. Here, we argue that this misses a crucial aspect, and that biomechanics is essentially damage mechanics, at heart. When deforming an animal cell or tissue, one can hardly avoid inducing the unfolding of protein domains, the unbinding of cytoskeletal crosslinkers, the breaking of weak sacrificial bonds, and the disruption of transient adhesions. We classify these activated structural changes as inelastic. They are often to a large degree reversible and are therefore not plastic in the proper sense, but they dissipate substantial amounts of elastic energy by structural damping. We review recent experiments involving biological materials on all scales, from single biopolymers over cells to model tissues, to illustrate the unifying power of this paradigm. A deliberately minimalistic yet phenomenologically very rich mathematical modeling framework for inelastic biomechanics is proposed. It transcends the conventional viscoelastic paradigm and suggests itself as a promising candidate for a unified description and interpretation of a wide range of experimental data. This article is part of a Special Issue entitled: Mechanobiology.

Chemoenvironmental modulators of fluidity in the suspended biological cell

Biological cells can be characterized as "soft matter" with mechanical characteristics potentially modulated by external cues such as pharmaceutical dosage or fever temperature. Further, quantifying the effects of chemical and physical stimuli on a cell's mechanical response informs models of living cells as complex materials. Here, we investigate the mechanical behavior of single biological cells in terms of fluidity, or mechanical hysteresivity normalized to the extremes of an elastic solid or a viscous liquid. This parameter, which complements stiffness when describing whole-cell viscoelastic response, can be determined for a suspended cell within subsecond times. Questions remain, however, about the origin of fluidity as a conserved parameter across timescales, the physical interpretation of its magnitude, and its potential use for high-throughput sorting and separation of interesting cells by mechanical means. Therefore, we exposed suspended CH27 lymphoma cells to various chemoenvironmental conditions--temperature, pharmacological agents, pH, and osmolarity--and measured cell fluidity with a non-contact technique to extend familiarity with suspended-cell mechanics in the context of both soft-matter physics and mechanical flow cytometry development. The actin-cytoskeleton-disassembling drug latrunculin exacted a large effect on mechanical behavior, amenable to dose-dependence analysis of coupled changes in fluidity and stiffness. Fluidity was minimally affected by pH changes from 6.5 to 8.5, but strongly modulated by osmotic challenge to the cell, where the range spanned halfway from solid to liquid behavior. Together, these results support the interpretation of fluidity as a reciprocal friction within the actin cytoskeleton, with implications both for cytoskeletal models and for expectations when separating interesting cell subpopulations by mechanical means in the suspended state.

Quantification of surface tension and internal pressure generated by single mitotic cells

During mitosis, adherent cells round up, by increasing the tension of the contractile actomyosin cortex while increasing the internal hydrostatic pressure. In the simple scenario of a liquid cell interior, the surface tension is related to the local curvature and the hydrostatic pressure difference by Laplace's law. However, verification of this scenario for cells requires accurate measurements of cell shape. Here, we use wedged micro-cantilevers to uniaxially confine single cells and determine confinement forces while concurrently determining cell shape using confocal microscopy. We fit experimentally measured confined cell shapes to shapes obeying Laplace's law with uniform surface tension and find quantitative agreement. Geometrical parameters derived from fitting the cell shape, and the measured force were used to calculate hydrostatic pressure excess and surface tension of cells. We find that HeLa cells increase their internal hydrostatic pressure excess and surface tension from ≈ 40 Pa and 0.2 mNm(-1) during interphase to ≈ 400 Pa and 1.6 mNm(-1) during metaphase. The method introduced provides a means to determine internal pressure excess and surface tension of rounded cells accurately and with minimal cellular perturbation, and should be applicable to characterize the mechanical properties of various cellular systems.

Mechanical fluidity of fully suspended biological cells

Mechanical characteristics of single biological cells are used to identify and possibly leverage interesting differences among cells or cell populations. Fluidity-hysteresivity normalized to the extremes of an elastic solid or a viscous liquid-can be extracted from, and compared among, multiple rheological measurements of cells: creep compliance versus time, complex modulus versus frequency, and phase lag versus frequency. With multiple strategies available for acquisition of this nondimensional property, fluidity may serve as a useful and robust parameter for distinguishing cell populations, and for understanding the physical origins of deformability in soft matter. Here, for three disparate eukaryotic cell types deformed in the suspended state via optical stretching, we examine the dependence of fluidity on chemical and environmental influences at a timescale of ∼1 s. We find that fluidity estimates are consistent in the time and frequency domains under a structural damping (power-law or fractional-derivative) model, but not under an equivalent-complexity, lumped-component (spring-dashpot) model; the latter predicts spurious time constants. Although fluidity is suppressed by chemical cross-linking, we find that ATP depletion in the cell does not measurably alter the parameter, and we thus conclude that active ATP-driven events are not a crucial enabler of fluidity during linear viscoelastic deformation of a suspended cell. Finally, by using the capacity of optical stretching to produce near-instantaneous increases in cell temperature, we establish that fluidity increases with temperature-now measured in a fully suspended, sortable cell without the complicating factor of cell-substratum adhesion.

Active contractions in single suspended epithelial cells

Investigations of active contractions in tissue cells to date have been focused on cells that exert forces via adhesion sites to substrates or to other cells. In this study we show that also suspended epithelial cells exhibit contractility, revealing that contractions can occur independently of focal adhesions. We employ the Optical Stretcher to measure adhesion-independent mechanical properties of an epithelial cell line transfected with a heat-sensitive cation channel. During stretching the heat transferred to the ion channel causes a pronounced Ca(2+) influx through the plasma membrane that can be blocked by adequate drugs. This way the contractile forces in suspended cells are shown to be partially triggered by Ca(2+) signaling. A phenomenological mathematical model is presented, incorporating a term accounting for the active stress exerted by the cell, which is both necessary and sufficient to describe the observed increase in strain when the Ca(2+) influx is blocked. The median and the shape of the strain distributions depend on the activity of the cells. Hence, it is unlikely that they can be described by a simple Gaussian or log normal distribution, but depend on specific cellular properties such as active contractions. Our results underline the importance of considering activity when measuring cellular mechanical properties even in the absence of measurable contractions. Thus, the presented method to quantify active contractions of suspended cells offers new perspectives for a better understanding of cellular force generation with possible implications for medical diagnosis and therapy.

Microactuator device for integrated measurement of epithelium mechanics

Mechanical forces are among important factors that drive cellular function and organization. We present a microfabricated device with on-chip actuation for mechanical testing of single cells. An integrated immersible electrostatic actuator system is demonstrated that applies calibrated forces to cells. We conduct stretching experiments by directly applying forces to epithelial cells adhered to device surfaces functionalized with collagen. We measure mechanical properties including stiffness, hysteresis and visco-elasticity of adherent cells.

Minimal model for the inelastic mechanics of biopolymer networks and cells

We explore the mechanism behind the ambiguous nonlinear mechanical response of biopolymer networks and cells. Our theoretical analysis is based on the inelastic glassy wormlike chain model (iGWLC), which accounts for simultaneous softening and stiffening in terms of two antagonistic mechanisms: viscoelastic stress stiffening caused by polymer stretching, and inelastic fluidization caused by bond breaking. On this basis, we derive a set of simple schematic constitutive equations that faithfully reproduce the rich inelastic phenomenology of biopolymer networks and cells.

Development of a Compact 1-D Micromanipulator with Flexure Manufactured Using Rapid Prototyping

Presented is the design and initial experimental results of a compact 1-D micromanipulator. The presented manipulator is a piezoelectric actuator based complaint mechanism with a compact translational flexure manufactured using rapid prototyping. Rapid prototyping allows complicated designs with low manufacturing costs. Analytical and Finite Element (FE) models for designing the flexure are presented. The simulation results are compared with experiments conducted on a prototype. Nonlinear stiffness is measured and evaluated. The importance of pre-loading force is investigated. Trajectory tracking tests at different frequencies are performed on the manipulator. The maximum and root mean square errors are analyzed.

Low intensity ultrasound perturbs cytoskeleton dynamics

Therapeutic ultrasound is widely employed in clinical applications but its mechanism of action remains unclear. Here we report prompt fluidization of a cell and dramatic acceleration of its remodeling dynamics when exposed to low intensity ultrasound. These physical changes are caused by very small strains (10^-5) at ultrasonic frequencies (10⁶ Hz), but are closely analogous to those caused by relatively large strains (10^-1) at physiological frequencies (10⁰ Hz). Moreover, these changes are reminiscent of rejuvenation and aging phenomena that are well-established in certain soft inert materials. As such, we suggest cytoskeletal fluidization together with resulting acceleration of cytoskeletal remodeling events as a mechanism contributing to the salutary effects of low intensity therapeutic ultrasound.

The influence of disorder on deformations in semiflexible networks

We present a model that assesses the different elastic responses of a semiflexible network, which either (i) is constrained to deform in an affine way or (ii) is permitted to thermally fluctuate and deviate from affine response. The thermal, non-affine response of the network is achieved using a Metropolis Monte Carlo algorithm with dynamic step size. We find that non-affine deformations soften the network dramatically at low strains and make the eventual nonlinear strain stiffening far more pronounced. We show that the effect of these non-affine deformations are very sensitive to the degree variation in the lengths of filaments connecting cross-links. Where there is high variation, non-affine deformations allow internal stresses to relax, giving rise to a smaller range of tensile forces in filaments and a dramatic reduction of network stiffness. This highlights that non-affine deformations are crucial in small strain response of stiff polymer networks.

The role of the cytoskeleton in volume regulation and beading transitions in PC12 neurites

We present investigations on volume regulation and beading shape transitions in PC12 neurites, conducted using a flow-chamber technique. By disrupting the cell cytoskeleton with specific drugs, we investigate the role of its individual components in the volume regulation response. We find that microtubule disruption increases both swelling rate and maximum volume attained, but does not affect the ability of the neurite to recover its initial volume. In addition, investigation of axonal beading-also known as pearling instability-provides additional clues on the mechanical state of the neurite. We conclude that volume recovery is driven by passive diffusion of osmolites, and propose that the initial swelling phase is mechanically slowed down by microtubules. Our experiments provide a framework to investigate the role of cytoskeletal mechanics in volume homeostasis.

Remodeling of integrated contractile tissues and its dependence on strain-rate amplitude

Here we investigate the origin of relaxation times governing the mechanical response of an integrated contractile tissue to imposed cyclic changes of length. When strain-rate amplitude is held constant as frequency is varied, fast events are accounted for by actomyosin cross-bridge cycling, but slow events reveal relaxation processes associated with ongoing cytoskeletal length adaptation. Although both relaxation regimes are innately nonlinear, these regimes are unified and their positions along the frequency axis are set by the imposed strain-rate amplitude.

Cytoskeletal forces produced by extremely low-frequency electric fields acting on extracellular glycoproteins

The physical mechanism by which cells transduce an applied electric field is not well understood. This article establishes for the first time a direct, quantitative model that links the field to cytoskeletal forces. In a previous article, applied electric fields of physiological strength were shown to produce significant mechanical torques at the cellular level. In this article, the corresponding forces exerted on the cytoskeleton are computed and found to be comparable in magnitude to mechanical forces known to produce physiological effects. In addition to the electrical force, the viscous drag force exerted by the surrounding medium and the restoring force exerted by the neighboring structures are considered in the analysis. For an applied electric field of 10 V/m, the force transmitted to the CD44 receptor of a hyaluronan chain in cartilage is about 1 pN at 10 Hz and 7 pN at 1 Hz. For an applied electric field of 100 V/m, the force transmitted to the cytoskeleton at one focus of the glycocalyx is about 0.5 pN at 10 Hz and 1.3 pN at 1 Hz. Mechanical forces of similar magnitude have been observed to produce physiological effects. Hence, this electromechanical transduction process is a plausible mechanism for the production of physiological effects by such electric fields.

Cell mechanics: dissecting the physical responses of cells to force

It is now widely appreciated that normal tissue morphology and function rely upon cells' ability to sense and generate forces appropriate to their correct tissue context. Although the effects of forces on cells have been studied for decades, our understanding of how those forces propagate through and act on different cell substructures remains at an early stage. The past decade has seen a resurgence of interest, with a variety of different micromechanical methods in current use that probe cells' dynamic deformation in response to a time-varying force. The ability of researchers to carefully measure the mechanical properties of cells subjected to a variety of pharmacological and genetic interventions, however, currently outstrips our ability to quantitatively interpret the data in many cases. Despite these challenges, the stage is now set for the development of detailed models for cell deformability, motility, and mechanosensing that are rooted at the molecular level.

The compaction of gels by cells: a case of collective mechanical activity

To understand mechanotransduction, purely mechanical phenomena resulting from the crosstalk between contractile cells and their elastic surroundings must be distinguished from adaptive responses to mechanical cues. Here, we revisit the compaction of freely suspended collagen gels by embedded cells, where a small volume fraction of cells (osteoblasts and fibroblasts) compacts the surrounding matrix by two orders of magnitude. Combining micropatterning with time-lapse strain mapping, we find gel compaction to be crucially determined by mechanical aspects of the surrounding matrix. First, it is a boundary effect: the compaction propagates from the edges of the matrix into the bulk. Second, the stress imposed by the cells irreversibly compacts the matrix and renders it anisotropic as a consequence of its nonlinear mechanics and the boundary conditions. Third, cell polarization and alignment follow in time and seem to be a consequence of gel compaction, at odds with current mechanosensing conceptions. Finally, our observation of a threshold cell density shows gel compaction to be a cooperative effect, revealing a mechanical interaction between cells through the matrix. The intricate interplay between cell contractility and surrounding matrix mechanics provides an important organizing principle with implications for many physiological processes such as tissue development.