Nonequilibrium mechanics of active cytoskeletal networks

Heat fluctuation of a harmonically trapped particle in an active bath

We study the heat fluctuation of an overdamped Brownian particle trapped in a harmonic potential and driven by active noise. Employing the phase-space path integral method we derive a general formula for the probability distribution of heat exchange in a generic model of an active bath. The work has been extended by considering two particular models of active noise and computing an exact analytical expression for distribution in Gaussian colored noise and a semianalytical result in the Poissonian bath. We corroborate the fluctuation theorem with our analytical findings by introducing the familiar concept of effective temperature and as a corollary the total entropy production is calculated.

Chain organization of human interphase chromosome determines the spatiotemporal dynamics of chromatin loci

We investigate spatiotemporal dynamics of human interphase chromosomes by employing a heteropolymer model that incorporates the information of human chromosomes inferred from Hi-C data. Despite considerable heterogeneities in the chromosome structures generated from our model, chromatins are organized into crumpled globules with space-filling (SF) statistics characterized by a single universal scaling exponent (ν = 1/3), and this exponent alone can offer a quantitative account of experimentally observed, many different features of chromosome dynamics. The local chromosome structures, whose scale corresponds to that of topologically associated domains (∼ 0.1 − 1 Mb), display dynamics with a fast relaxation time (≲ 1 − 10 sec); in contrast, the long-range spatial reorganization of the entire chromatin (≳O(102) Mb) occurs on a much slower time scale (≳ hour), providing the dynamic basis of cell-to-cell variability and glass-like behavior of chromosomes. Biological activities, modeled using stronger isotropic white noises added to active loci, accelerate the relaxation dynamics of chromatin domains associated with the low frequency modes and induce phase segregation between the active and inactive loci. Surprisingly, however, they do not significantly change the dynamics at local scales from those obtained under passive conditions. Our study underscores the role of chain organization of chromosome in determining the spatiotemporal dynamics of chromatin loci.

Chromosomes are giant chain molecules made of hundreds of megabase-long DNA intercalated with proteins. Structure and dynamics of interphase chromatin in space and time hold the key to understanding the cell type-dependent gene regulation. In this study, we establish that the crumpled and space-filling (SF) organization of chromatin fiber in the chromosome territory, characterized by a single scaling exponent, is sufficient to explain the complex spatiotemporal hierarchy in chromatin dynamics as well as the subdiffusive motion of the chromatin loci. While seemingly a daunting problem at a first glance, our study shows that relatively simple principles, rooted in polymer physics, can be used to grasp the essence of dynamical properties of the interphase chromatin.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

Entropy production rate is maximized in non-contractile actomyosin

The actin cytoskeleton is an active semi-flexible polymer network whose non-equilibrium properties coordinate both stable and contractile behaviors to maintain or change cell shape. While myosin motors drive the actin cytoskeleton out-of-equilibrium, the role of myosin-driven active stresses in the accumulation and dissipation of mechanical energy is unclear. To investigate this, we synthesize an actomyosin material in vitro whose active stress content can tune the network from stable to contractile. Each increment in activity determines a characteristic spectrum of actin filament fluctuations which is used to calculate the total mechanical work and the production of entropy in the material. We find that the balance of work and entropy does not increase monotonically and the entropy production rate is maximized in the non-contractile, stable state of actomyosin. Our study provides evidence that the origins of entropy production and activity-dependent dissipation relate to disorder in the molecular interactions between actin and myosin.

Nonequilibrium Energetics of Molecular Motor Kinesin

Nonequilibrium energetics of single molecule translational motor kinesin was investigated by measuring heat dissipation from the violation of the fluctuation-response relation of a probe attached to the motor using optical tweezers. The sum of the dissipation and work did not amount to the input free energy change, indicating large hidden dissipation exists. Possible sources of the hidden dissipation were explored by analyzing the Langevin dynamics of the probe, which incorporates the two-state Markov stepper as a kinesin model. We conclude that internal dissipation is dominant.

Active dumbbells: Dynamics and morphology in the coexisting region

With the help of molecular dynamics simulations we study an ensemble of active dumbbells in purely repulsive interaction. We derive the phase diagram in the density-activity plane and we characterise the various phases with liquid, hexatic and solid character. The analysis of the structural and dynamical properties, such as enstrophy, mean-square displacement, polarisation, and correlation functions, shows the continuous character of liquid and hexatic phases in the coexisting region when the activity is increased starting from the passive limit.

The two faces of enhanced stroma: Stroma acts as a tumor promoter and a steric obstacle

In addition to genetic, morphological and biochemical alterations in cells, a key feature of the malignant progression of cancer is the stroma, including cancer cell motility as well as the emergence of metastases. Our current knowledge with regard to the biophysically driven experimental approaches of cancer progression indicates that mechanical aberrations are major contributors to the malignant progression of cancer. In particular, the mechanical probing of the stroma is of great interest. However, the impact of the tumor stroma on cellular motility, and hence the metastatic cascade leading to the malignant progression of cancer, is controversial as there are two different and opposing effects within the stroma. On the one hand, the stroma can promote and enhance the proliferation, survival and migration of cancer cells through mechanotransduction processes evoked by fiber alignment as a result of increased stroma rigidity. This enables all types of cancer to overcome restrictive biological capabilities. On the other hand, as a result of its structural constraints, the stroma acts as a steric obstacle for cancer cell motility in dense three-dimensional extracellular matrices, when the pore size is smaller than the cell's nucleus. The mechanical properties of the stroma, such as the tissue matrix stiffness and the entire architectural network of the stroma, are the major players in providing the optimal environment for cancer cell migration. Thus, biophysical methods determining the mechanical properties of the stroma, such as magnetic resonance elastography, are critical for the diagnosis and prediction of early cancer stages. Fibrogenesis and cancer are tightly connected, as there is an elevated risk of cancer on cystic fibrosis or, subsequently, cirrhosis. This also applies to the subsequent metastatic process.

Living Matter: Mesoscopic Active Materials

An introduction to the physical properties of living active matter at the mesoscopic scale (tens of nanometers to micrometers) and their unique features compared with "dead," nonactive matter is presented. This field of research is increasingly denoted as "biological physics" where physics includes chemical physics, soft matter physics, hydrodynamics, mechanics, and the related engineering sciences. The focus is on the emergent properties of these systems and their collective behavior, which results in active self-organization and how they relate to cellular-level biological function. These include locomotion (cell motility and migration) forces that give rise to cell division, the growth and form of cellular assemblies in development, the beating of heart cells, and the effects of mechanical perturbations such as shear flow (in the bloodstream) or adhesion to other cells or tissues. An introduction to the fundamental concepts and theory with selected experimental examples related to the authors' own research is presented, including red-blood-cell membrane fluctuations, motion of the nucleus within an egg cell, self-contracting acto-myosin gels, and structure and beating of heart cells (cardiomyocytes), including how they can be driven by an oscillating, mechanical probe.

Viscoelasticity of single cells-from subcellular to cellular level

This review deals with insights into complex cellular structures and processes obtained by measuring viscoelastic impedances of the cell envelope and the cytoplasm by colloidal bead microrheometry. I first introduce a mechanical cell model that allows us to understand their unique ability of mechanical self-stabilization by actin microtubule crosstalk. In the second part, I show how cell movements can be driven by pulsatile or propagating solitary actin gelatin waves (SAGW) that are generated on nascent adhesion domains by logistically controlled membrane recruitment of functional proteins by electrostatic-hydrophobic forces. The global polarization of cell migration is guided by actin-microtubule crosstalk that is mediated by the Ca⁺⁺ and strain-sensitive supramolecular scaffolding protein IQGAP. In the third part, I introduce the traction force microscopy as a tool to measure the forces between somatic cells and the tissue ´Here I show, how absolute values of viscoelastic impedances of the composite cell envelope can be obtained by deformation field mapping techniques. In the fourth part, it is shown how the dynamic mechanical properties of the active viscoplastic cytoplasmic space can be evaluated using colloidal beads as phantom endosomes. Separate measurements of velocity distributions of directed and random motions of phantom endosomes, yield local values of transport forces, viscosities and life times of directed motion along microtubules. The last part deals with biomimetic experiments allowing us to quantitatively evaluate the mechanical properties of passive and active actin networks on the basis of the percolation theory of gelation.

Substrate stiffness and mechanical stress due to intercellular cooperativity guides tissue structure

A key challenge in cell and tissue morphogenesis is to understand how a crucial balance between cell proliferation and apoptosis maintains an evolving tissue structure. These processes are mutually non-exclusive and require stiffness monitoring of the host substrate. Adhered cells actively mechanosense the tension in the extracellular matrix (ECM). They collectively alter self-organization and generate a host of tissue patterns. Using an in silico elastic fiber-network in two dimensions, we simulate cell-ECM composite structures and characterize features of the emerging tissue patterns during successive cell proliferation and apoptosis. Our data reveals that, in general, cell viability is a function of the cell-induced effective ECM stiffness supported by intercellular cooperativity. Translating this into a remodeling tissue, we find that average cell cycle duration in concert with the locally stressed regions of the ECM promote heterogeneous proliferation and apoptosis inducing finger-like protrusions along the tissue periphery - a feature normally observed during tumorigenesis. Further, we find that recovery of a scratch wound is delayed for cells harbored on a compliant or (and) in a highly collagen depleted ECM.

Nonequilibrium Scaling Behavior in Driven Soft Biological Assemblies

Measuring and quantifying nonequilibrium dynamics in active biological systems is a major challenge because of their intrinsic stochastic nature and the limited number of variables accessible in any real experiment. We investigate what nonequilibrium information can be extracted from noninvasive measurements using a stochastic model of soft elastic networks with a heterogeneous distribution of activities, representing enzymatic force generation. In particular, we use this model to study how the nonequilibrium activity, detected by tracking two probes in the network, scales as a function of the distance between the probes. We quantify the nonequilibrium dynamics through the cycling frequencies, a simple measure of circulating currents in the phase space of the probes. We find that these cycling frequencies exhibit power-law scaling behavior with the distance between probes. In addition, we show that this scaling behavior governs the entropy production rate that can be recovered from the two traced probes. Our results provide insight into how internal enzymatic driving generates nonequilibrium dynamics on different scales in soft biological assemblies.

Eukaryotic Cell Dynamics from Crawlers to Swimmers

Movement requires force transmission to the environment, and motile cells are robustly, though not elegantly, designed nanomachines that often can cope with a variety of environmental conditions by altering the mode of force transmission used. As with humans, the available modes range from momentary attachment to a substrate when crawling, to shape deformations when swimming, and at the cellular level this involves sensing the mechanical properties of the environment and altering the mode appropriately. While many types of cells can adapt their mode of movement to their microenvironment (ME), our understanding of how they detect, transduce and process information from the ME to determine the optimal mode is still rudimentary. The shape and integrity of a cell is determined by its cytoskeleton (CSK), and thus the shape changes that may be required to move involve controlled remodeling of the CSK. Motion in vivo is often in response to extracellular signals, which requires the ability to detect such signals and transduce them into the shape changes and force generation needed for movement. Thus the nanomachine is complex, and while much is known about individual components involved in movement, an integrated understanding of motility in even simple cells such as bacteria is not at hand. In this review we discuss recent advances in our understanding of cell motility and some of the problems remaining to be solved.

The homeostatic ensemble for cells

Cells are quintessential examples of out-of-equilibrium systems, but they maintain a homeostatic state over a timescale of hours to days. As a consequence, the statistics of all observables is remarkably consistent. Here, we develop a statistical mechanics framework for living cells by including the homeostatic constraint that exists over the interphase period of the cell cycle. The consequence is the introduction of the concept of a homeostatic ensemble and an associated homeostatic temperature, along with a formalism for the (dynamic) homeostatic equilibrium that intervenes to allow living cells to evade thermodynamic decay. As a first application, the framework is shown to accurately predict the observed effect of the mechanical environment on the in vitro response of smooth muscle cells. This includes predictions that both the mean values and diversity/variability in the measured values of observables such as cell area, shape and tractions decrease with decreasing stiffness of the environment. Thus, we argue that the observed variabilities are inherent to the entropic nature of the homeostatic equilibrium of cells and not a result of in vitro experimental errors.

Active Viscoelasticity of Sarcomeres

The perturbation response of muscle is important for the versatile, stable and agile control capabilities of animals. Muscle resists being stretched by developing forces in the passive tissues and in the active crossbridges. This review focuses on the active perturbation response of the sarcomere. The active response exhibits typical stress relaxation, and thus approximated by a Maxwell material that has a spring and dashpot arranged in series. The ratio of damping to stiffness in this approximation defines the relaxation timescale for dissipating stresses that are developed in the crossbridges due to external perturbations. Current understanding of sarcomeres suggests that stiffness varies nearly linearly with neural excitation, but not much is known about damping. But if both stiffness and damping have the same functional (linear or not) dependence on neural excitation, then the stress relaxation timescale cannot be varied depending on the demands of the task. This implies an unavoidable and biologically unrealistic trade-off between how freely the crossbridges can yield and dissipate stresses when stretched (injury avoidance in agile motions) vs. how long they can maintain perturbation-induced stresses and behave like a solid material (stiffness maintenance for stability). We hypothesize that muscle circumvents this trade-off by varying damping in a nonlinear manner with neural excitation, unlike stiffness that varies linearly. Testing this hypothesis requires new experimental and mathematical characterization of muscle mechanics, and also identifies new design goals for robotic actuators.

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.

Inferring energy dissipation from violation of the fluctuation-dissipation theorem

The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation.

The Effects of the Interplay between Motor and Brownian Forces on the Rheology of Active Gels

Active gels perform key mechanical roles inside the cell, such as cell division, motion, and force sensing. The unique mechanical properties required to perform such functions arise from the interactions between molecular motors and semiflexible polymeric filaments. Molecular motors can convert the energy released in the hydrolysis of ATP into forces of up to piconewton magnitudes. Moreover, the polymeric filaments that form active gels are flexible enough to respond to Brownian forces but also stiff enough to support the large tensions induced by the motor-generated forces. Brownian forces are expected to have a significant effect especially at motor activities at which stable noncontractile in vitro active gels are prepared for rheological measurements. Here, a microscopic mean-field theory of active gels originally formulated in the limit of motor-dominated dynamics is extended to include Brownian forces. In the model presented here, Brownian forces are included accurately, at real room temperature, even in systems with high motor activity. It is shown that a subtle interplay, or competition, between motor-generated forces and Brownian forces has an important impact on the mass transport and rheological properties of active gels. The model predictions show that at low frequencies the dynamic modulus of active gels is determined mostly by motor protein dynamics. However, Brownian forces significantly increase the breadth of the relaxation spectrum and can affect the shape of the dynamic modulus over a wide frequency range even for ratios of motor to Brownian forces of more than a hundred. Since the ratio between motor and Brownian forces is sensitive to ATP concentration, the results presented here shed some light on how the transient mechanical response of active gels changes with varying ATP concentration.

Rapid, topology-based particle tracking for high-resolution measurements of large complex 3D motion fields

Spatiotemporal tracking of tracer particles or objects of interest can reveal localized behaviors in biological and physical systems. However, existing tracking algorithms are most effective for relatively low numbers of particles that undergo displacements smaller than their typical interparticle separation distance. Here, we demonstrate a single particle tracking algorithm to reconstruct large complex motion fields with large particle numbers, orders of magnitude larger than previously tractably resolvable, thus opening the door for attaining very high Nyquist spatial frequency motion recovery in the images. Our key innovations are feature vectors that encode nearest neighbor positions, a rigorous outlier removal scheme, and an iterative deformation warping scheme. We test this technique for its accuracy and computational efficacy using synthetically and experimentally generated 3D particle images, including non-affine deformation fields in soft materials, complex fluid flows, and cell-generated deformations. We augment this algorithm with additional particle information (e.g., color, size, or shape) to further enhance tracking accuracy for high gradient and large displacement fields. These applications demonstrate that this versatile technique can rapidly track unprecedented numbers of particles to resolve large and complex motion fields in 2D and 3D images, particularly when spatial correlations exist.

Spatial Cytoskeleton Organization Supports Targeted Intracellular Transport

The efficiency of intracellular cargo transport from specific sources to target locations is strongly dependent upon molecular motor-assisted motion along the cytoskeleton. Radial transport along microtubules and lateral transport along the filaments of the actin cortex underneath the cell membrane are characteristic for cells with a centrosome. The interplay between the specific cytoskeleton organization and the motor performance results in a spatially inhomogeneous intermittent search strategy. To analyze the efficiency of such intracellular search strategies, we formulate a random velocity model with intermittent arrest states. We evaluate efficiency in terms of mean first passage times for three different, frequently encountered intracellular transport tasks: 1) the narrow escape problem, which emerges during cargo transport to a synapse or other specific region of the cell membrane; 2) the reaction problem, which considers the binding time of two particles within the cell; and 3) the reaction-escape problem, which arises when cargo must be released at a synapse only after pairing with another particle. Our results indicate that cells are able to realize efficient search strategies for various intracellular transport tasks economically through a spatial cytoskeleton organization that involves only a narrow actin cortex rather than a cell body filled with randomly oriented actin filaments.

Sensing Magnetic Fields with Magnetosensitive Ion Channels

[-15]Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k B T in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.

Topology Counts: Force Distributions in Circular Spring Networks

Filamentous polymer networks govern the mechanical properties of many biological materials. Force distributions within these networks are typically highly inhomogeneous, and, although the importance of force distributions for structural properties is well recognized, they are far from being understood quantitatively. Using a combination of probabilistic and graph-theoretical techniques, we derive force distributions in a model system consisting of ensembles of random linear spring networks on a circle. We show that characteristic quantities, such as the mean and variance of the force supported by individual springs, can be derived explicitly in terms of only two parameters: (i) average connectivity and (ii) number of nodes. Our analysis shows that a classical mean-field approach fails to capture these characteristic quantities correctly. In contrast, we demonstrate that network topology is a crucial determinant of force distributions in an elastic spring network. Our results for 1D linear spring networks readily generalize to arbitrary dimensions.

Dynamics of a bilayer membrane coupled to a two-dimensional cytoskeleton: Scale transfers of membrane deformations

We theoretically investigate the dynamics of a floating lipid bilayer membrane coupled with a two-dimensional cytoskeleton network, taking into account explicitly the intermonolayer friction, the discrete lattice structure of the cytoskeleton, and its prestress. The lattice structure breaks lateral continuous translational symmetry and couples Fourier modes with different wave vectors. It is shown that within a short time interval a long-wavelength deformation excites a collection of modes with wavelengths shorter than the lattice spacing. These modes relax slowly with a common renormalized rate originating from the long-wavelength mode. As a result, and because of the prestress, the slowest relaxation is governed by the intermonolayer friction. Conversely, and most interestingly, forces applied at the scale of the cytoskeleton for a sufficiently long time can cooperatively excite large-scale modes.

Deviation from fluctuation-dissipation relation for driven superdiffusion: Polymer stretching as an example

We discuss a deviation of the fluctuation-dissipation relation (FDR) in a driven superdiffusive system as exemplified by polymer stretching. The superdiffusion is found by monitoring momentum transfer to a tracer, which is a conjugate observable with the position. Molecular-dynamics simulation demonstrates that the FDR deviates during the nonequilibrium transient process. We then propose nonequilibrium mode analysis for superdiffusion, which is a counterpart to that for driven subdiffusion. The mode analysis yields results that are in qualitative agreement with the simulation results, suggesting that the fluctuations of the stiffness in the system from initial equilibrium to stretching account for the FDR deviation.

A disassembly-driven mechanism explains F-actin-mediated chromosome transport in starfish oocytes

While contraction of sarcomeric actomyosin assemblies is well understood, this is not the case for disordered networks of actin filaments (F-actin) driving diverse essential processes in animal cells. For example, at the onset of meiosis in starfish oocytes a contractile F-actin network forms in the nuclear region transporting embedded chromosomes to the assembling microtubule spindle. Here, we addressed the mechanism driving contraction of this 3D disordered F-actin network by comparing quantitative observations to computational models. We analyzed 3D chromosome trajectories and imaged filament dynamics to monitor network behavior under various physical and chemical perturbations. We found no evidence of myosin activity driving network contractility. Instead, our observations are well explained by models based on a disassembly-driven contractile mechanism. We reconstitute this disassembly-based contractile system in silico revealing a simple architecture that robustly drives chromosome transport to prevent aneuploidy in the large oocyte, a prerequisite for normal embryonic development.

Physical constraints in cell fate specification. A case in point: Microgravity and phenotypes differentiation

Data obtained by studying mammalian cells in absence of gravity strongly support the notion that cell fate specification cannot be understood according to the current molecular model. A paradigmatic case in point is provided by studying cell populations growing in absence of gravity. When the physical constraint (gravity) is 'experimentally removed', cells spontaneously allocate into two morphologically different phenotypes. Such phenomenon is likely enacted by the intrinsic stochasticity, which, in turn, is successively 'canalized' by a specific gene regulatory network. Both phenotypes are thermodynamically and functionally 'compatibles' with the new, modified environment. However, when the two cell subsets are reseeded into the 1g gravity field the two phenotypes collapse into one. Gravity constraints the system in adopting only one phenotype, not by selecting a pre-existing configuration, but more precisely shaping it de-novo through the modification of the cytoskeleton three-dimensional structure. Overall, those findings highlight how macro-scale features are irreducible to lower-scale explanations. The identification of macroscale control parameters - as those depending on the field (gravity, electromagnetic fields) or emerging from the cooperativity among the field's components (tissue stiffness, cell-to-cell connectivity) - are mandatory for assessing boundary conditions for models at lower scales, thus providing a concrete instantiation of top-down effects.

Multiple sources of slow activity fluctuations in a bacterial chemosensory network

Cellular networks are intrinsically subject to stochastic fluctuations, but analysis of the resulting noise remained largely limited to gene expression. The pathway controlling chemotaxis of Escherichia coli provides one example where posttranslational signaling noise has been deduced from cellular behavior. This noise was proposed to result from stochasticity in chemoreceptor methylation, and it is believed to enhance environment exploration by bacteria. Here we combined single-cell FRET measurements with analysis based on the fluctuation-dissipation theorem (FDT) to characterize origins of activity fluctuations within the chemotaxis pathway. We observed surprisingly large methylation-independent thermal fluctuations of receptor activity, which contribute to noise comparably to the energy-consuming methylation dynamics. Interactions between clustered receptors involved in amplification of chemotactic signals are also necessary to produce the observed large activity fluctuations. Our work thus shows that the high response sensitivity of this cellular pathway also increases its susceptibility to noise, from thermal and out-of-equilibrium processes.

Mechanics and Buckling of Biopolymeric Shells and Cell Nuclei

We study a Brownian dynamics simulation model of a biopolymeric shell deformed by axial forces exerted at opposing poles. The model exhibits two distinct, linear force-extension regimes, with the response to small tensions governed by linear elasticity and the response to large tensions governed by an effective spring constant that scales with radius as R^-0.25. When extended beyond the initial linear elastic regime, the shell undergoes a hysteretic, temperature-dependent buckling transition. We experimentally observe this buckling transition by stretching and imaging the lamina of isolated cell nuclei. Furthermore, the interior contents of the shell can alter mechanical response and buckling, which we show by simulating a model for the nucleus that quantitatively agrees with our micromanipulation experiments stretching individual nuclei.

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.

Topography on a subcellular scale modulates cellular adhesions and actin stress fiber dynamics in tumor associated fibroblasts

Cells can sense and adapt to mechanical properties of their environment. The local geometry of the extracellular matrix, such as its topography, has been shown to modulate cell morphology, migration, and proliferation. Here we investigate the effect of micro/nanotopography on the morphology and cytoskeletal dynamics of human pancreatic tumor-associated fibroblast cells (TAFs). We use arrays of parallel nanoridges with variable spacings on a subcellular scale to investigate the response of TAFs to the topography of their environment. We find that cell shape and stress fiber organization both align along the direction of the nanoridges. Our analysis reveals a strong bimodal relationship between the degree of alignment and the spacing of the nanoridges. Furthermore, focal adhesions align along ridges and form preferentially on top of the ridges. Tracking actin stress fiber movement reveals enhanced dynamics of stress fibers on topographically patterned surfaces. We find that components of the actin cytoskeleton move preferentially along the ridges with a significantly higher velocity along the ridges than on a flat surface. Our results suggest that a complex interplay between the actin cytoskeleton and focal adhesions coordinates the cellular response to micro/nanotopography.

Mechanical Properties of the Cytoskeleton and Cells

SUMMARYThe cytoskeleton is the major mechanical structure of the cell; it is a complex, dynamic biopolymer network comprising microtubules, actin, and intermediate filaments. Both the individual filaments and the entire network are not simple elastic solids but are instead highly nonlinear structures. Appreciating the mechanics of biopolymer networks is key to understanding the mechanics of cells. Here, we review the mechanical properties of cytoskeletal polymers and discuss the implications for the behavior of cells.

Scale dependence of the mechanics of active gels with increasing motor concentration

Actin is a protein that plays an essential role in maintaining the mechanical integrity of cells. In response to strong external stresses, it can assemble into large bundles, but it grows into a fine branched network to induce cell motion. In some cases, the self-organization of actin fibers and networks involves the action of bipolar filaments of the molecular motor myosin. Such self-organization processes mediated by large myosin bipolar filaments have been studied extensively in vitro. Here we create active gels, composed of single actin filaments and small myosin bipolar filaments. The active steady state in these gels persists long enough to enable the characterization of their mechanical properties using one and two point microrheology. We study the effect of myosin concentration on the mechanical properties of this model system for active matter, for two different motor assembly sizes. In contrast to previous studies of networks with large motor assemblies, we find that the fluctuations of tracer particles embedded in the network decrease in amplitude as motor concentration increases. Nonetheless, we show that myosin motors stiffen the actin networks, in accordance with bulk rheology measurements of networks containing larger motor assemblies. This implies that such stiffening is of universal nature and may be relevant to a wider range of cytoskeleton-based structures.

The effect of antibiotics on protein diffusion in the Escherichia coli cytoplasmic membrane

Accumulating evidence suggests that molecular motors contribute to the apparent diffusion of molecules in cells. However, current literature lacks evidence for an active process that drives diffusive-like motion in the bacterial membrane. One possible mechanism is cell wall synthesis, which involves the movement of protein complexes in the cell membrane circumferentially around the cell envelope and may generate currents in the lipid bilayer that advectively transport other transmembrane proteins. We test this hypothesis in Escherichia coli using drug treatments that slow cell wall synthesis and measure their effect on the diffusion of the transmembrane protein mannitol permease using fluorescence recovery after photobleaching. We found no clear decrease in diffusion in response to vancomycin and no decrease in response to mecillinam treatment. These results suggest that cell wall synthesis is not an active contributor to mobility in the cytoplasmic membrane.

A theory that predicts behaviors of disordered cytoskeletal networks

Morphogenesis in animal tissues is largely driven by actomyosin networks, through tensions generated by an active contractile process. Although the network components and their properties are known, and networks can be reconstituted in vitro, the requirements for contractility are still poorly understood. Here, we describe a theory that predicts whether an isotropic network will contract, expand, or conserve its dimensions. This analytical theory correctly predicts the behavior of simulated networks, consisting of filaments with varying combinations of connectors, and reveals conditions under which networks of rigid filaments are either contractile or expansile. Our results suggest that pulsatility is an intrinsic behavior of contractile networks if the filaments are not stable but turn over. The theory offers a unifying framework to think about mechanisms of contractions or expansion. It provides the foundation for studying a broad range of processes involving cytoskeletal networks and a basis for designing synthetic networks.

Chase-and-run dynamics in cell motility and the molecular rupture of interacting active elastic dimers

Cell migration in morphogenesis and cancer metastasis typically involves interplay between different cell types. We construct and study a minimal, one-dimensional model composed of two different motile cells with each cell represented as an active elastic dimer. The interaction between the two cells via cadherins is modeled as a spring that can rupture beyond a threshold force as it undergoes dynamic loading from the interacting motile cells. We obtain a phase diagram consisting of chase-and-run dynamics and clumping dynamics as a function of the stiffness of the interaction spring and the threshold force and, therefore, posit that active rupture, or rupture via active forces, is a mechanosensitive means to regulate dynamics between cells. Since the parameters in the model differentiate between N- and E-cadherins, we make predictions for the interactions between a placodelike cell and a neural crestlike cell in a microchannel as well as discuss how our results inform chase-and-run dynamics found in a group of placode cells interacting with a group of neural crest cells. In particular, an argument was made in the latter case that the feedback between cadherins and cell-substrate interaction via integrins was necessary to obtain the chase-and-run behavior. Based on our two-cell results, we argue that this feedback accentuates, but is not necessary for, the chase-and-run behavior.

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.

Parallel Compression Is a Fast Low-Cost Assay for the High-Throughput Screening of Mechanosensory Cytoskeletal Proteins in Cells

Cellular mechanosensing is critical for many biological processes, including cell differentiation, proliferation, migration, and tissue morphogenesis. The actin cytoskeletal proteins play important roles in cellular mechanosensing. Many techniques have been used to investigate the mechanosensory behaviors of these proteins. However, a fast, low-cost assay for the quantitative characterization of these proteins is still lacking. Here, we demonstrate that compression assay using agarose overlay is suitable for the high throughput screening of mechanosensory proteins in live cells while requiring minimal experimental setup. We used several well-studied myosin II mutants to assess the compression assay. On the basis of elasticity theories, we simulated the mechanosensory accumulation of myosin II's and quantitatively reproduced the experimentally observed protein dynamics. Combining the compression assay with confocal microscopy, we monitored the polarization of myosin II oligomers at the subcellular level. The polarization was dependent on the ratio of the two principal strains of the cellular deformations. Finally, we demonstrated that this technique could be used on the investigation of other mechanosensory proteins.

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.

Fluctuation spectra and force generation in nonequilibrium systems

Many biological systems are appropriately viewed as passive inclusions immersed in an active bath: from proteins on active membranes to microscopic swimmers confined by boundaries. The nonequilibrium forces exerted by the active bath on the inclusions or boundaries often regulate function, and such forces may also be exploited in artificial active materials. Nonetheless, the general phenomenology of these active forces remains elusive. We show that the fluctuation spectrum of the active medium, the partitioning of energy as a function of wavenumber, controls the phenomenology of force generation. We find that, for a narrow, unimodal spectrum, the force exerted by a nonequilibrium system on two embedded walls depends on the width and the position of the peak in the fluctuation spectrum, and oscillates between repulsion and attraction as a function of wall separation. We examine two apparently disparate examples: the Maritime Casimir effect and recent simulations of active Brownian particles. A key implication of our work is that important nonequilibrium interactions are encoded within the fluctuation spectrum. In this sense, the noise becomes the signal.

Mechanosensing drives acuity of αβ T-cell recognition

T lymphocytes use surface [Formula: see text] T-cell receptors (TCRs) to recognize peptides bound to MHC molecules (pMHCs) on antigen-presenting cells (APCs). How the exquisite specificity of high-avidity T cells is achieved is unknown but essential, given the paucity of foreign pMHC ligands relative to the ubiquitous self-pMHC array on an APC. Using optical traps, we determine physicochemical triggering thresholds based on load and force direction. Strikingly, chemical thresholds in the absence of external load require orders of magnitude higher pMHC numbers than observed physiologically. In contrast, force applied in the shear direction ([Formula: see text]10 pN per TCR molecule) triggers T-cell Ca²⁺ flux with as few as two pMHC molecules at the interacting surface interface with rapid positional relaxation associated with similarly directed motor-dependent transport via [Formula: see text]8-nm steps, behaviors inconsistent with serial engagement during initial TCR triggering. These synergistic directional forces generated during cell motility are essential for adaptive T-cell immunity against infectious pathogens and cancers.

Integrated Analysis of Intracellular Dynamics of MenaINV Cancer Cells in a 3D Matrix

The intracellular environment is composed of a filamentous network that exhibits dynamic turnover of cytoskeletal components and internal force generation from molecular motors. Particle tracking microrheology enables a means to probe the internal mechanics and dynamics. Here, we develop an analytical model to capture the basic features of the active intracellular mechanical environment, including both thermal and motor-driven effects, and show consistency with a diverse range of experimental microrheology data. We further perform microrheology experiments, integrated with Brownian dynamics simulations of the active cytoskeleton, on metastatic breast cancer cells embedded in a three-dimensional collagen matrix with and without the presence of epidermal growth factor to probe the intracellular mechanical response in a physiologically mimicking scenario. Our results demonstrate that EGF stimulation can alter intracellular stiffness and power output from molecular motor-driven fluctuations in cells overexpressing an invasive isoform of the actin-associated protein Mena.

Dynamics of Fractal Cluster Gels with Embedded Active Colloids

We find that embedded active colloids increase the ensemble-averaged mean squared displacement of particles in otherwise passively fluctuating fractal cluster gels. The enhancement in dynamics occurs by a mechanism in which the active colloids contribute to the average dynamics both directly through their own active motion and indirectly through their excitation of neighboring passive colloids in the fractal network. Fractal cluster gels are synthesized by addition of magnesium chloride to an initially stable suspension of 1.0  μm polystyrene colloids in which a dilute concentration of platinum coated Janus colloids has been dispersed. The Janus colloids are thereby incorporated into the fractal network. We measure the ensemble-averaged mean squared displacement of all colloids in the gel before and after the addition of hydrogen peroxide, a fuel that drives diffusiophoretic motion of the Janus particles. The gel mean squared displacement increases by up to a factor of 3 for an active to passive particle ratio of 1∶20 and inputted active energy-defined based on the hydrogen peroxide's effect on colloid swim speed and run length-that is up to 9.5 times thermal energy, on a per particle basis. We model the enhancement in gel particle dynamics as the sum of a direct contribution from the displacement of the Janus particles themselves and an indirect contribution from the strain field that the active colloids induce in the surrounding passive particles.

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.

Nonequilibrium steady states in Langevin thermal systems

Equilibrium is characterized by its fundamental properties, such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relation mutually exclude each other, and no equivalence relation is possible between any two of the three properties. This implies that a steady state of Langevin systems with velocity-dependent forces may maintain some equilibrium properties but not all of them. Our results are illustrated with a few example systems.

High-frequency microrheology reveals cytoskeleton dynamics in living cells

Living cells are viscoelastic materials, with the elastic response dominating at long timescales (≳1 ms)1. At shorter timescales, the dynamics of individual cytoskeleton filaments are expected to emerge, but active microrheology measurements on cells accessing this regime are scarce2. Here, we develop high-frequency microrheology (HF-MR) to probe the viscoelastic response of living cells from 1Hz to 100 kHz. We report the viscoelasticity of different cell types and upon cytoskeletal drug treatments. At previously inaccessible short timescales, cells exhibit rich viscoelastic responses that depend on the state of the cytoskeleton. Benign and malignant cancer cells revealed remarkably different scaling laws at high frequency, providing a univocal mechanical fingerprint. Microrheology over a wide dynamic range up to the frequency of action of the molecular components provides a mechanistic understanding of cell mechanics.

Spatial distributions of pericellular stiffness in natural extracellular matrices are dependent on cell-mediated proteolysis and contractility

Bulk tissue stiffness has been correlated with regulation of cellular processes and conversely cells have been shown to remodel their pericellular tissue according to a complex feedback mechanism critical to development, homeostasis, and disease. However, bulk rheological methods mask the dynamics within a heterogeneous fibrous extracellular matrix (ECM) in the region proximal to a cell (pericellular region). Here, we use optical tweezers active microrheology (AMR) to probe the distribution of the complex material response function (α=α'+α″, in units of µm/nN) within a type I collagen ECM, a biomaterial commonly used in tissue engineering. We discovered cells both elastically and plastically deformed the pericellular material. α' is wildly heterogeneous, with 1/α' values spanning three orders of magnitude around a single cell. This was observed in gels having a cell-free 1/α' of approximately 0.5nN/µm. We also found that inhibition of cell contractility instantaneously softens the pericellular space and reduces stiffness heterogeneity, suggesting the system was strain hardened and not only plastically remodeled. The remaining regions of high stiffness suggest cellular remodeling of the surrounding matrix. To test this hypothesis, cells were incubated within the type I collagen gel for 24-h in a media containing a broad-spectrum matrix metalloproteinase (MMP) inhibitor. While pericellular material maintained stiffness asymmetry, stiffness magnitudes were reduced. Dual inhibition demonstrates that the combination of MMP activity and contractility is necessary to establish the pericellular stiffness landscape. This heterogeneity in stiffness suggests the distribution of pericellular stiffness, and not bulk stiffness alone, must be considered in the study of cell-ECM interactions and design of complex biomaterial scaffolds.

STATEMENT OF SIGNIFICANCE

Collagen is a fibrous extracellular matrix (ECM) protein widely used to study cell-ECM interactions. Stiffness of ECM has been shown to instruct cells, which can in turn modify their ECM, as has been shown in the study of cancer and regenerative medicine. Here we measure the stiffness of the collagen microenvironment surrounding cells and quantitatively measure the dependence of pericellular stiffness on MMP activity and cytoskeletal contractility. Competent cell-mediated stiffening results in a wildly heterogeneous micromechanical topography, with values spanning orders of magnitude around a single cell. We speculate studies must consider this notable heterogeneity generated by cells when testing theories regarding the role of ECM mechanics in health and disease.