High-resolution probing of cellular force transmission

Trapped tracer in a non-equilibrium bath: dynamics and energetics

We study the dynamics of a tracer that is elastically coupled to active particles being kept at two different temperatures, as a prototype of tracer dynamics in a non-equilibrium bath. Employing analytical techniques, we find the exact solution of the probability density function for the effective motion of the tracer. The analytical results are supported by numerical simulations. By combining the experimentally accessible quantities such as the response function and the power spectrum, we measure the non-equilibrium fluctuations in terms of the effective temperature. In addition, we compute the energy dissipation rate to find the precise effects of activity. Our study is relevant in understanding athermal fluctuations arising in cytoskeletal networks or inside a chromosome.

Activity-dependent glassy cell mechanics II: Nonthermal fluctuations under metabolic activity

The glassy cytoplasm, crowded with bio-macromolecules, is fluidized in living cells by mechanical energy derived from metabolism. Characterizing the living cytoplasm as a nonequilibrium system is crucial in elucidating the intricate mechanism that relates cell mechanics to metabolic activities. In this study, we conducted active and passive microrheology in eukaryotic cells, and quantified nonthermal fluctuations by examining the violation of the fluctuation-dissipation theorem. The power spectral density of active force generation was estimated following the Langevin theory extended to nonequilibrium systems. However, experiments performed while regulating cellular metabolic activity showed that the nonthermal displacement fluctuation, rather than the active nonthermal force, is linked to metabolism. We discuss that mechano-enzymes in living cells do not act as microscopic objects. Instead, they generate meso-scale collective fluctuations with displacements controlled by enzymatic activity. The activity induces structural relaxations in glassy cytoplasm. Even though the autocorrelation of nonthermal fluctuations is lost at long timescales due to the structural relaxations, the nonthermal displacement fluctuation remains regulated by metabolic reactions. Our results therefore demonstrate that nonthermal fluctuations serve as a valuable indicator of a cell's metabolic activities, regardless of the presence or absence of structural relaxations.

Simultaneous Nanorheometry and Nanothermometry Using Intracellular Diamond Quantum Sensors

The viscoelasticity of the cytoplasm plays a critical role in cell morphology, cell division, and intracellular transport. Viscoelasticity is also interconnected with other biophysical properties, such as temperature, which is known to influence cellular bioenergetics. Probing the connections between intracellular temperature and cytoplasmic viscoelasticity provides an exciting opportunity for the study of biological phenomena, such as metabolism and disease progression. The small length scales and transient nature of changes in these parameters combined with their complex interdependencies pose a challenge for biosensing tools, which are often limited to a single readout modality. Here, we present a dual-mode quantum sensor capable of performing simultaneous nanoscale thermometry and rheometry in dynamic cellular environments. We use nitrogen-vacancy centers in diamond nanocrystals as biocompatible sensors for in vitro measurements. We combine subdiffraction resolution single-particle tracking in a fluidic environment with optically detected magnetic resonance spectroscopy to perform simultaneous sensing of viscoelasticity and temperature. We use our sensor to demonstrate probing of the temperature-dependent viscoelasticity in complex media at the nanoscale. We then investigate the interplay between intracellular forces and the cytoplasmic rheology in live cells. Finally, we identify different rheological regimes and reveal evidence of active trafficking and details of the nanoscale viscoelasticity of the cytoplasm.

Centripetal nuclear shape fluctuations associate with chromatin condensation in early prophase

The nucleus plays a central role in several key cellular processes, including chromosome organisation, DNA replication and gene transcription. Recent work suggests an association between nuclear mechanics and cell-cycle progression, but many aspects of this connection remain unexplored. Here, by monitoring nuclear shape fluctuations at different cell cycle stages, we uncover increasing inward fluctuations in late G2 and in early prophase, which are initially transient, but develop into instabilities when approaching the nuclear-envelope breakdown. We demonstrate that such deformations correlate with chromatin condensation by perturbing both the chromatin and the cytoskeletal structures. We propose that the contrasting forces between an extensile stress and centripetal pulling from chromatin condensation could mechanically link chromosome condensation with nuclear-envelope breakdown, two main nuclear processes occurring during mitosis.

Activity-dependent glassy cell mechanics Ⅰ: Mechanical properties measured with active microrheology

Active microrheology was conducted in living cells by applying an optical-trapping force to vigorously fluctuating tracer beads with feedback-tracking technology. The complex shear modulus G(ω)=G'(ω)-iG″(ω) was measured in HeLa cells in an epithelial-like confluent monolayer. We found that G(ω)∝(-iω)1/2 over a wide range of frequencies (1 Hz < ω/2π < 10 kHz). Actin disruption and cell-cycle progression from G1 to S and G2 phases only had a limited effect on G(ω) in living cells. On the other hand, G(ω) was found to be dependent on cell metabolism; ATP-depleted cells showed an increased elastic modulus G'(ω) at low frequencies, giving rise to a constant plateau such that G(ω)=G0+A(-iω)1/2. Both the plateau and the additional frequency dependency ∝(-iω)1/2 of ATP-depleted cells are consistent with a rheological response typical of colloidal jamming. On the other hand, the plateau G0 disappeared in ordinary metabolically active cells, implying that living cells fluidize their internal states such that they approach the critical jamming point.

Effect of shear flow on the hydrodynamic drag force of a spherical particle near a wall evaluated using optical tweezers and microfluidics

The hydrodynamic drag force on a spherical particle in shear flow near-wall is investigated using optical tweezers and microfluidics. Simple shear flow is applied using a microfluidic channel at different volumetric flow rates. The hydrodynamic drag force exerted on the particle is detected from the displacement of the trapped particle. The effect of the wall is obtained from the force balance of the trapping and hydrodynamic drag force employing the exact solution of the theoretical model using the lubrication theory for a sphere near the wall. Here, we report the experimentally obtained hydrodynamic drag force coefficient under the influence of shear flow. The drag correction factor increases with decreasing distance from the wall due to the effect of the wall surface. We found that the calculated hydrodynamic drag force coefficient is in quantitative comparison with the theoretical prediction for a shear flow past a sphere near-wall. This study provides a straightforward investigation of the effect of the shear flow on the hydrodynamic drag force coefficient on a particle near the wall. Furthermore, these pieces of information can be used in various applications, particularly in optimizing microfluidic designs for mixing and separations of particles or exploiting the formation of the concentration gradient of particles perpendicular to flow directions caused by the non-linear hydrodynamic interactions.

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.

Intracellular nonequilibrium fluctuating stresses indicate how nonlinear cellular mechanical properties adapt to microenvironmental rigidity

Living cells are known to be in thermodynamically nonequilibrium, which is largely brought about by intracellular molecular motors. The motors consume chemical energies to generate stresses and reorganize the cytoskeleton for the cell to move and divide. However, since there has been a lack of direct measurements characterizing intracellular stresses, questions remained unanswered on the intricacies of how cells use such stresses to regulate their internal mechanical integrity in different microenvironments. This report describes a new experimental approach by which we reveal an environmental rigidity-dependent intracellular stiffness that increases with intracellular stress - a revelation obtained, surprisingly, from a correlation between the fluctuations in cellular stiffness and that of intracellular stresses. More surprisingly, by varying two distinct parameters, environmental rigidity and motor protein activities, we observe that the stiffness-stress relationship follows the same curve. This finding provides some insight into the intricacies by suggesting that cells can regulate their responses to their mechanical microenvironment by adjusting their intracellular stress.

Probing force in living cells with optical tweezers: from single-molecule mechanics to cell mechanotransduction

The invention of optical tweezers more than three decades ago has opened new avenues in the study of the mechanical properties of biological molecules and cells. Quantitative force measurements still represent a challenging task in living cells due to the complexity of the cellular environment. Here, we review different methodologies to quantitatively measure the mechanical properties of living cells, the strength of adhesion/receptor bonds, and the active force produced during intracellular transport, cell adhesion, and migration. We discuss experimental strategies to attain proper calibration of optical tweezers and molecular resolution in living cells. Finally, we show recent studies on the transduction of mechanical stimuli into biomolecular and genetic signals that play a critical role in cell health and disease.

Distribution of active forces in the cell cortex

In this work, we study in detail the distribution of stochastic forces generated by the molecular motors activity, in the actin cortex of pre-muscular cells. By combining active and passive rheology experiments, performed on the same micro-bead bound to the actin network through membrane adhesive receptors, we measure the auto-correlation function C_ff(τ) of the average force pulling on the bead. As for any out-of-equilibrium system, the force distribution differs from the thermodynamical equilibrium one, especially at long time scale τ⪆ 1 s where the bead motion becomes partially directed. Thus the fluctuation-dissipation theorem does not apply and one can measure the distance from equilibrium through its violation. We investigate the influence of different parameters on the force distribution, focusing particularly on the role of ligand density: a detailed study shows how the amplitude of active forces increases when the bead is more tightly attached to the cortex. We introduce and study a model, which takes into account the number of bonds between the bead and the cytoskeleton, as well as the viscoelastic properties of the medium. This model faithfully accounts for the experimental observations. Also, it is shown that the amplitude of active forces increases with temperature. Finally, our data confirm that ATP depletion in the cell, or partial inhibition of the actomyosin activity, leads to a decrease of the amplitude of the force distribution. Altogether, we propose a consistent and quantitative description for the motion of a micrometric probe interacting with the actin network, and for the amplitude of the stochastic forces generated by molecular motors in the cortex surrounding this probe.

Two-tag correlations and nonequilibrium fluctuation-response relation in ageing single-file diffusion

Spatiotemporally correlated motions of interacting Brownian particles, confined in a narrow channel of infinite length, are studied in terms of statistical quantities involving two particles. A theoretical framework that allows analytical calculation of two-tag correlations is presented on the basis of the Dean-Kawasaki equation describing density fluctuations in colloidal systems. In the equilibrium case, the time-dependent Einstein relation holds between the two-tag displacement correlation and the response function corresponding to it, which is a manifestation of the fluctuation-dissipation theorem for the correlation of density fluctuations. While the standard procedure of closure approximation for nonlinear density fluctuations is known to be obstructed by inconsistency with the fluctuation-dissipation theorem, this difficulty is naturally avoided by switching from the standard Fourier representation of the density field to the label-based Fourier representation of the vacancy field. In the case of ageing dynamics started from equidistant lattice configuration, the time-dependent Einstein relation is violated, as the two-tag correlation depends on the waiting time for equilibration while the response function is not sensitive to it. Within linear approximation, however, there is a simple relation between the density (or vacancy) fluctuations and the corresponding response function, which is valid even if the system is out of equilibrium. This non-equilibrium fluctuation-response relation can be extended to the case of nonlinear fluctuations by means of closure approximation for the vacancy field.

Self-organized stress patterns drive state transitions in actin cortices

Biological functions rely on ordered structures and intricately controlled collective dynamics. This order in living systems is typically established and sustained by continuous dissipation of energy. The emergence of collective patterns of motion is unique to nonequilibrium systems and is a manifestation of dynamic steady states. Mechanical resilience of animal cells is largely controlled by the actomyosin cortex. The cortex provides stability but is, at the same time, highly adaptable due to rapid turnover of its components. Dynamic functions involve regulated transitions between different steady states of the cortex. We find that model actomyosin cortices, constructed to maintain turnover, self-organize into distinct nonequilibrium steady states when we vary cross-link density. The feedback between actin network structure and organization of stress-generating myosin motors defines the symmetries of the dynamic steady states. A marginally cross-linked state displays divergence-free long-range flow patterns. Higher cross-link density causes structural symmetry breaking, resulting in a stationary converging flow pattern. We track the flow patterns in the model actomyosin cortices using fluorescent single-walled carbon nanotubes as novel probes. The self-organization of stress patterns we have observed in a model system can have direct implications for biological functions.

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.

Universal glass-forming behavior of in vitro and living cytoplasm

Physiological processes in cells are performed efficiently without getting jammed although cytoplasm is highly crowded with various macromolecules. Elucidating the physical machinery is challenging because the interior of a cell is so complex and driven far from equilibrium by metabolic activities. Here, we studied the mechanics of in vitro and living cytoplasm using the particle-tracking and manipulation technique. The molecular crowding effect on cytoplasmic mechanics was selectively studied by preparing simple in vitro models of cytoplasm from which both the metabolism and cytoskeletons were removed. We obtained direct evidence of the cytoplasmic glass transition; a dramatic increase in viscosity upon crowding quantitatively conformed to the super-Arrhenius formula, which is typical for fragile colloidal suspensions close to jamming. Furthermore, the glass-forming behaviors were found to be universally conserved in all the cytoplasm samples that originated from different species and developmental stages; they showed the same tendency for diverging at the macromolecule concentrations relevant for living cells. Notably, such fragile behavior disappeared in metabolically active living cells whose viscosity showed a genuine Arrhenius increase as in typical strong glass formers. Being actively driven by metabolism, the living cytoplasm forms glass that is fundamentally different from that of its non-living counterpart.

Feedback-tracking microrheology in living cells

Living cells are composed of active materials, in which forces are generated by the energy derived from metabolism. Forces and structures self-organize to shape the cell and drive its dynamic functions. Understanding the out-of-equilibrium mechanics is challenging because constituent materials, the cytoskeleton and the cytosol, are extraordinarily heterogeneous, and their physical properties are strongly affected by the internally generated forces. We have analyzed dynamics inside two types of eukaryotic cells, fibroblasts and epithelial-like HeLa cells, with simultaneous active and passive microrheology using laser interferometry and optical trapping technology. We developed a method to track microscopic probes stably in cells in the presence of vigorous cytoplasmic fluctuations, by using smooth three-dimensional (3D) feedback of a piezo-actuated sample stage. To interpret the data, we present a theory that adapts the fluctuation-dissipation theorem (FDT) to out-of-equilibrium systems that are subjected to positional feedback, which introduces an additional nonequilibrium effect. We discuss the interplay between material properties and nonthermal force fluctuations in the living cells that we quantify through the violations of the FDT. In adherent fibroblasts, we observed a well-known polymer network viscoelastic response where the complex shear modulus scales as G* ∝ (-iω)3/4. In the more 3D confluent epithelial cells, we found glassy mechanics with G* ∝ (-iω)1/2 that we attribute to glassy dynamics in the cytosol. The glassy state in living cells shows characteristics that appear distinct from classical glasses and unique to nonequilibrium materials that are activated by molecular motors.

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.

Dissipation of contractile forces: the missing piece in cell mechanics

Mechanical forces are key regulators of cell and tissue physiology. The basic molecular mechanism of fiber contraction by the sliding of actin filament upon myosin leading to conformational change has been known for decades. The regulation of force generation at the level of the cell, however, is still far from elucidated. Indeed, the magnitude of cell traction forces on the underlying extracellular matrix in culture is almost impossible to predict or experimentally control. The considerable variability in measurements of cell-traction forces indicates that they may not be the optimal readout to properly characterize cell contractile state and that a significant part of the contractile energy is not transferred to cell anchorage but instead is involved in actin network dynamics. Here we discuss the experimental, numerical, and biological parameters that may be responsible for the variability in traction force production. We argue that limiting these sources of variability and investigating the dissipation of mechanical work that occurs with structural rearrangements and the disengagement of force transmission is key for further understanding of cell mechanics.

Introduction to Optical Tweezers

Thirty years after their invention by Arthur Ashkin and colleagues at Bell Labs in 1986 [1], optical tweezers (or traps) have become a versatile tool to address numerous biological problems. Put simply, an optical trap is a highly focused laser beam that is capable of holding and applying forces to micron-sized dielectric objects. However, their development over the last few decades has converted these tools from boutique instruments into highly versatile instruments of molecular biophysics. This introductory chapter intends to give a brief overview of the field, highlight some important scientific achievements, and demonstrate why optical traps have become a powerful tool in the biological sciences. We introduce a typical optical setup, describe the basic theoretical concepts of how trapping forces arise, and present the quantitative position and force measurement techniques that are most widely used today.

Analytical Limit Distributions from Random Power-Law Interactions

Nature is full of power-law interactions, e.g., gravity, electrostatics, and hydrodynamics. When sources of such fields are randomly distributed in space, the superposed interaction, which is what we observe, is naively expected to follow a Gauss or Lévy distribution. Here, we present an analytic expression for the actual distributions that converge to novel limits that are in between these already-known limit distributions, depending on physical parameters, such as the concentration of field sources and the size of the probe used to measure the interactions. By comparing with numerical simulations, the origin of non-Gauss and non-Lévy distributions are theoretically articulated.

Broken Detailed Balance of Filament Dynamics in Active Networks

Myosin motor proteins drive vigorous steady-state fluctuations in the actin cytoskeleton of cells. Endogenous embedded semiflexible filaments such as microtubules, or added filaments such as single-walled carbon nanotubes are used as novel tools to noninvasively track equilibrium and nonequilibrium fluctuations in such biopolymer networks. Here, we analytically calculate shape fluctuations of semiflexible probe filaments in a viscoelastic environment, driven out of equilibrium by motor activity. Transverse bending fluctuations of the probe filaments can be decomposed into dynamic normal modes. We find that these modes no longer evolve independently under nonequilibrium driving. This effective mode coupling results in nonzero circulatory currents in a conformational phase space, reflecting a violation of detailed balance. We present predictions for the characteristic frequencies associated with these currents and investigate how the temporal signatures of motor activity determine mode correlations, which we find to be consistent with recent experiments on microtubules embedded in cytoskeletal networks.

Nanomechanics of a fibroblast suspended using point-like anchors reveal cytoskeleton formation

Cells are suspended and stretched using two microbeads. The formation of inner cytoskeleton structures is reported using displacement, QPM phase change and fluorescent micrographs.

Force fluctuations in three-dimensional suspended fibroblasts

Cells are sensitive to mechanical cues from their environment and at the same time generate and transmit forces to their surroundings. To test quantitatively forces generated by cells not attached to a substrate, we used a dual optical trap to suspend 3T3 fibroblasts between two fibronectin-coated beads. In this simple geometry, we measured both the cells' elastic properties and the force fluctuations they generate with high bandwidth. Cell stiffness decreased substantially with both myosin inhibition by blebbistatin and serum-starvation, but not with microtubule depolymerization by nocodazole. We show that cortical forces generated by non-muscle myosin II deform the cell from its rounded shape in the frequency regime from 0.1 to 10 Hz. The amplitudes of these forces were strongly reduced by blebbistatin and serum starvation, but were unaffected by depolymerization of microtubules. Force fluctuations show a spectrum that is characteristic for an elastic network activated by random sustained stresses with abrupt transitions.

Nonthermal fluctuations of the mitotic spindle

We present direct measurements of fluctuations in the nucleus of yeast cells. While prior work has shown these fluctuations to be active and non-thermal in character, their origin and time dependence are not understood. We show that the nuclear fluctuations we observe are quantitatively consistent with uncorrelated, active force fluctuations driving a nuclear medium that is dominated by an uncondensed DNA solution, for which we perform rheological measurements on an in vitro model system under similar conditions to what is expected in the nucleus.

The mechanical properties of early Drosophila embryos measured by high-speed video microrheology

In early development, Drosophila melanogaster embryos form a syncytium, i.e., multiplying nuclei are not yet separated by cell membranes, but are interconnected by cytoskeletal polymer networks consisting of actin and microtubules. Between division cycles 9 and 13, nuclei and cytoskeleton form a two-dimensional cortical layer. To probe the mechanical properties and dynamics of this self-organizing pre-tissue, we measured shear moduli in the embryo by high-speed video microrheology. We recorded position fluctuations of injected micron-sized fluorescent beads with kHz sampling frequencies and characterized the viscoelasticity of the embryo in different locations. Thermal fluctuations dominated over nonequilibrium activity for frequencies between 0.3 and 1000 Hz. Between the nuclear layer and the yolk, the cytoplasm was homogeneous and viscously dominated, with a viscosity three orders of magnitude higher than that of water. Within the nuclear layer we found an increase of the elastic and viscous moduli consistent with an increased microtubule density. Drug-interference experiments showed that microtubules contribute to the measured viscoelasticity inside the embryo whereas actin only plays a minor role in the regions outside of the actin caps that are closely associated with the nuclei. Measurements at different stages of the nuclear division cycle showed little variation.

Universal axial fluctuations in optical tweezers

Optical tweezers (OTs) allow the measurement of fluctuations at the nanoscale, in particular fluctuations in the end-to-end distance in single molecules. Fluctuation spectra can yield valuable information, but they can easily be contaminated by instrumental effects. We identify axial fluctuations, i.e., fluctuations of the trapped beads in the direction of light propagation, as one of these instrumental effects. Remarkably, axial fluctuations occur on a characteristic timescale similar to that of conformational (folding) transitions, which may lead to misinterpretation of the experimental results. We show that a precise measurement of the effect of force on both axial and conformational fluctuations is crucial to disentangle them. Our results on axial fluctuations are captured by a simple and general formula valid for all OT setups and provide experimentalists with a general strategy to distinguish axial fluctuations from conformational transitions.

Bone cell mechanosensitivity, estrogen deficiency, and osteoporosis

Adaptation of bone to mechanical stresses normally produces a bone architecture that combines a proper resistance against failure with a minimal use of material. This adaptive process is governed by mechanosensitive osteocytes that transduce the mechanical signals into chemical responses, i.e. the osteocytes release signaling molecules, which orchestrate the recruitment and activity of bone forming osteoblasts and/or bone resorbing osteoclasts. Computer models have shown that the maintenance of a mechanically-efficient bone architecture depends on the intensity and spatial distribution of the mechanical stimulus as well as on the osteocyte response. Osteoporosis is a condition characterized by a reduced bone mass and a compromized resistance of bone against mechanical loads, which has led us to hypothesize that mechanotransduction by osteocytes is altered in osteoporosis. One of the major causal factors for osteoporosis is the loss of estrogen, the major hormonal regulator of bone metabolism. Loss of estrogen may increase osteocyte-mediated activation of bone remodeling, resulting in impaired bone mass and architecture. In this review we highlight current insights on how osteocytes perceive mechanical stimuli placed on whole bones. Particular emphasis is placed on the role of estrogen in signaling pathway activation by mechanical stimuli, and on computer simulation in combination with cell biology to unravel biological processes contributing to bone strength.

Physical basis of spindle self-organization

The cytoskeleton forms a variety of steady-state, subcellular structures that are maintained by continuous fluxes of molecules and energy. Understanding such self-organizing structures is not only crucial for cell biology but also poses a fundamental challenge for physics, since these systems are active materials that behave drastically differently from matter at or near equilibrium. Active liquid crystal theories have been developed to study the self-organization of cytoskeletal filaments in in vitro systems of purified components. However, it has been unclear how relevant these simplified approaches are for understanding biological structures, which can be composed of hundreds of distinct proteins. Here we show that a suitably constructed active liquid crystal theory produces remarkably accurate predictions of the behaviors of metaphase spindles-the cytoskeletal structure, composed largely of microtubules and associated proteins, that segregates chromosomes during cell division.

Probing the stochastic, motor-driven properties of the cytoplasm using force spectrum microscopy

Molecular motors in cells typically produce highly directed motion; however, the aggregate, incoherent effect of all active processes also creates randomly fluctuating forces, which drive diffusive-like, nonthermal motion. Here, we introduce force-spectrum-microscopy (FSM) to directly quantify random forces within the cytoplasm of cells and thereby probe stochastic motor activity. This technique combines measurements of the random motion of probe particles with independent micromechanical measurements of the cytoplasm to quantify the spectrum of force fluctuations. Using FSM, we show that force fluctuations substantially enhance intracellular movement of small and large components. The fluctuations are three times larger in malignant cells than in their benign counterparts. We further demonstrate that vimentin acts globally to anchor organelles against randomly fluctuating forces in the cytoplasm, with no effect on their magnitude. Thus, FSM has broad applications for understanding the cytoplasm and its intracellular processes in relation to cell physiology in healthy and diseased states.

Cells actively stiffen fibrin networks by generating contractile stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.

High-frequency affine mechanics and nonaffine relaxation in a model cytoskeleton

The cytoskeleton is a network of crosslinked, semiflexible filaments, and it has been suggested that it has properties of a glassy state. Here we employ optical-trap-based microrheology to apply forces to a model cytoskeleton and measure the high-bandwidth response at an anterior point. Simulating the highly nonlinear and anisotropic stress-strain propagation assuming affinity, we found that theoretical predictions for the quasistatic response of semiflexible polymers are only realized at high frequencies inaccessible to conventional rheometers. We give a theoretical basis for determining the frequency when both affinity and quasistaticity are valid, and we discuss with experimental evidence that the relaxations at lower frequencies can be characterized by the experimentally obtained nonaffinity parameter.

Studying single red blood cells under a tunable external force by combining passive microrheology with Raman spectroscopy

The dynamic micromechanical and structural properties of single human red blood cells are studied using a combination of dual trap optical tweezers and confocal Raman spectroscopy. Such a combination permits us to show a direct relationship between the rheological properties and chemical structure conformation. The frequency dependence of the complex stiffness of the cells was measured using both one and two probe response functions under identical experimental conditions. Both the microrheology and Raman measurements were performed at different stretching forces applied to the cell. A detailed analysis of the auto- and cross-correlated probe motions allows exploring the local and overall viscoelastic properties of the cells over a controlled range of the deformations. The observed growth of the cell viscoelasticity with stretching was associated with structural changes in the cell membrane monitored via the Raman spectroscopy.

Quantitative determination of optical trapping strength and viscoelastic moduli inside living cells

With the success of in vitro single-molecule force measurements obtained in recent years, the next step is to perform quantitative force measurements inside a living cell. Optical traps have proven excellent tools for manipulation, also in vivo, where they can be essentially non-invasive under correct wavelength and exposure conditions. It is a pre-requisite for in vivo quantitative force measurements that a precise and reliable force calibration of the tweezers is performed. There are well-established calibration protocols in purely viscous environments; however, as the cellular cytoplasm is viscoelastic, it would be incorrect to use a calibration procedure relying on a viscous environment. Here we demonstrate a method to perform a correct force calibration inside a living cell. This method (theoretically proposed in Fischer and Berg-Sørensen (2007 J. Opt. A: Pure Appl. Opt. 9 S239)) takes into account the viscoelastic properties of the cytoplasm and relies on a combination of active and passive recordings of the motion of the cytoplasmic object of interest. The calibration procedure allows us to extract absolute values for the viscoelastic moduli of the living cell cytoplasm as well as the force constant describing the optical trap, thus paving the way for quantitative force measurements inside the living cell. Here, we determine both the spring constant of the optical trap and the elastic contribution from the cytoplasm, influencing the motion of naturally occurring tracer particles. The viscoelastic moduli that we find are of the same order of magnitude as moduli found in other cell types by alternative methods.

Nonequilibrium fluctuations of mechanically stretched single red blood cells detected by optical tweezers

We study the thermal and out-of-equilibrium mechanical dynamics of single, living human red blood cells (RBCs) by combining two-probe passive and active microrheology techniques. Both experiments were performed quasisimultaneously on the same cell using two identical polystyrene probes, biochemically attached to the cell membrane. We obtained compelling evidence of nonequilibrium fluctuations in the RBCs under physiological condition and without the influence of any external chemicals. The spectral distributions of metabolically driven forces and viscoelastic response were evaluated in the relaxed and stretched states, intended to simulate the varying natural environment of the cells during blood circulation. We found that the internally generated forces are more pronounced in the stretched state, suggesting a stress-dependent RBC activity.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.

Cell visco-elasticity measured with AFM and optical trapping at sub-micrometer deformations

The measurement of the elastic properties of cells is widely used as an indicator for cellular changes during differentiation, upon drug treatment, or resulting from the interaction with the supporting matrix. Elasticity is routinely quantified by indenting the cell with a probe of an AFM while applying nano-Newton forces. Because the resulting deformations are in the micrometer range, the measurements will be affected by the finite thickness of the cell, viscous effects and even cell damage induced by the experiment itself. Here, we have analyzed the response of single 3T3 fibroblasts that were indented with a micrometer-sized bead attached to an AFM cantilever at forces from 30–600 pN, resulting in indentations ranging from 0.2 to 1.2 micrometer. To investigate the cellular response at lower forces up to 10 pN, we developed an optical trap to indent the cell in vertical direction, normal to the plane of the coverslip. Deformations of up to two hundred nanometers achieved at forces of up to 30 pN showed a reversible, thus truly elastic response that was independent on the rate of deformation. We found that at such small deformations, the elastic modulus of 100 Pa is largely determined by the presence of the actin cortex. At higher indentations, viscous effects led to an increase of the apparent elastic modulus. This viscous contribution that followed a weak power law, increased at larger cell indentations. Both AFM and optical trapping indentation experiments give consistent results for the cell elasticity. Optical trapping has the benefit of a lower force noise, which allows a more accurate determination of the absolute indentation. The combination of both techniques allows the investigation of single cells at small and large indentations and enables the separation of their viscous and elastic components.

Mechanical control of nanomaterials and nanosystems

In situations of power outage or shortage, such as periods just following a seismic disaster, the only reliable power source available is the most fundamental of forces i.e., manual mechanical stimuli. Although there are many macroscopic mechanical tools, mechanical control of nanomaterials and nanosystems has not been an easy subject to develop even by using advanced nanotechnological concepts. However, this challenge has now become a hot topic and many new ideas and strategies have been proposed recently. This report summarizes recent research examples of mechanical control of nanomaterials and nanosystems. Creation of macroscopic mechanical outputs by efficient accumulation of molecular-level phenomena is first briefly introduced. We will then introduce the main subject: control of molecular systems by macroscopic mechanical stimuli. The research described is categorized according to the respective areas of mechanical control of molecular structure, molecular orientation, molecular interaction including cleavage and healing, and biological and micron-level phenomena. Finally, we will introduce two more advanced approaches, namely, mechanical strategies for microdevice fabrication and mechanical control of molecular machines. As mechanical forces are much more reliable and widely applicable than other stimuli, we believe that development of mechanically responsive nanomaterials and nanosystems will make a significant contribution to fundamental improvements in our lifestyles and help to maintain and stabilize our society.

Active rheology of phospholipid vesicles

Optical tweezers are used to manipulate the shape of artificial dioleoyl-phosphatidylcholine (DOPC) phospholipid vesicles of around 30 μm diameter. Using a time-shared trapping system, a complex of traps drives oscillations of the vesicle equator, with a sinusoidal time dependence and over a range of spatial and temporal frequencies. The mechanical response of the vesicle membrane as a function of the frequency and wavelength of the driving oscillation is monitored. A simple model of the vesicles as spherical elastic membranes immersed in a newtonian fluid, driven by a harmonic trapping potential, describes the experimental data. The bending modulus of the membrane is recovered. The method has potential for future investigation of nonthermally driven systems, where comparison of active and passive rheology can help to distinguish nonthermal forces from equilibrium fluctuations.

Small-world rheology: an introduction to probe-based active microrheology

We introduce active, probe-based microrheological techniques for measuring the flow and deformation of complex fluids. These techniques are ideal for mechanical characterization either when little sample is available, or when samples show significant spatial heterogeneity. We review recent results, paying particular attention to comparing and contrasting rheological parameters obtained from micro- and macro-rheological techniques.

An AFM-based stiffness clamp for dynamic control of rigidity

Atomic force microscopy (AFM) has become a powerful tool for measuring material properties in biology and imposing mechanical boundary conditions on samples from single molecules to cells and tissues. Constant force or constant height can be maintained in an AFM experiment through feedback control of cantilever deflection, known respectively as a ‘force clamp’ or ‘position clamp’. However, stiffness, the third variable in the Hookean relation F = kx that describes AFM cantilever deflection, has not been dynamically controllable in the same way. Here we present and demonstrate a ‘stiffness clamp’ that can vary the apparent stiffness of an AFM cantilever. This method, employable on any AFM system by modifying feedback control of the cantilever, allows rapid and reversible tuning of the stiffness exposed to the sample in a way that can decouple the role of stiffness from force and deformation. We demonstrated the AFM stiffness clamp on two different samples: a contracting fibroblast cell and an expanding polyacrylamide hydrogel. We found that the fibroblast, a cell type that secretes and organizes the extracellular matrix, exhibited a rapid, sub-second change in traction rate (dF/dt) and contraction velocity (dx/dt) in response to step changes in stiffness between 1–100 nN/µm. This response was independent of the absolute contractile force and cell height, demonstrating that cells can react directly to changes in stiffness alone. In contrast, the hydrogel used in our experiment maintained a constant expansion velocity (dx/dt) over this range of stiffness, while the traction rate (dF/dt) changed with stiffness, showing that passive materials can also behave differently in different stiffness environments. The AFM stiffness clamp presented here, which is applicable to mechanical measurements on both biological and non-biological samples, may be used to investigate cellular mechanotransduction under a wide range of controlled mechanical boundary conditions.

Moving into the cell: single-molecule studies of molecular motors in complex environments

Much has been learned in the past decades about molecular force generation. Single-molecule techniques, such as atomic force microscopy, single-molecule fluorescence microscopy and optical tweezers, have been key in resolving the mechanisms behind the power strokes, 'processive' steps and forces of cytoskeletal motors. However, it remains unclear how single force generators are integrated into composite mechanical machines in cells to generate complex functions such as mitosis, locomotion, intracellular transport or mechanical sensory transduction. Using dynamic single-molecule techniques to track, manipulate and probe cytoskeletal motor proteins will be crucial in providing new insights.

Real-time single-cell response to stiffness

Living cells adapt to the stiffness of their environment. However, cell response to stiffness is mainly thought to be initiated by the deformation of adhesion complexes under applied force. In order to determine whether cell response was triggered by stiffness or force, we have developed a unique method allowing us to tune, in real time, the effective stiffness experienced by a single living cell in a uniaxial traction geometry. In these conditions, the rate of traction force buildup dF/dt was adapted to stiffness in less than 0.1 s. This integrated fast response was unambiguously triggered by stiffness, and not by force. It suggests that early cell response could be mechanical in nature. In fact, local force-dependent signaling through adhesion complexes could be triggered and coordinated by the instantaneous cell-scale adaptation of dF/dt to stiffness. Remarkably, the effective stiffness method presented here can be implemented on any mechanical setup. Thus, beyond single-cell mechanosensing, this method should be useful to determine the role of rigidity in many fundamental phenomena such as morphogenesis and development.

Nonlocal fluctuation correlations in active gels

Many active materials and biological systems are driven far from equilibrium by embedded agents that spontaneously generate forces and distort the surrounding material. Probing and characterizing these athermal fluctuations are essential to understand the properties and behaviors of such systems. Here we present a mathematical procedure to estimate the local action of force-generating agents from the observed fluctuating displacement fields. The active agents are modeled as oriented force dipoles or isotropic compression foci, and the matrix on which they act is assumed to be either a compressible elastic continuum or a coupled network-solvent system. Correlations at a single point and between points separated by an arbitrary distance are obtained, giving a total of three independent fluctuation modes that can be tested with microrheology experiments. Since oriented dipoles and isotropic compression foci give different contributions to these fluctuation modes, ratiometric analysis allows us characterize the force generators. We also predict and experimentally find a high-frequency ballistic regime, arising from individual force-generating events in the form of the slow buildup of stress followed by rapid but finite decay. Finally, we provide a quantitative statistical model to estimate the mean filament tension from these athermal fluctuations, which leads to stiffening of active networks.