Out-of-equilibrium microrheology inside living cells

Local changes in protein filament properties drive large-scale membrane transformations involved in endosome tethering and fusion

Large-scale cellular transformations are triggered by subtle physical and structural changes to individual biomacromolecular and membrane components. A prototypical example of such an event is the orchestrated fusion of membranes within an endosome that enables transport of cargo and processing of biochemical moieties. In this work, we demonstrate how protein filaments on the endosomal membrane surface can leverage a rigid-to-flexible transformation to elicit a large-scale change in membrane flexibility to enable membrane fusion. We develop a polymer field-theoretic model that captures molecular alignment arising from nematic interactions with varying surface density and fraction of flexible filaments, which are biologically controlled within the endosomal membrane. We then predict the collective elasticity of the filament brush in response to changes in the filament alignment, predicting a greater than 20-fold increase of the effective membrane elasticity over the bare membrane elasticity that is triggered by filament alignment. These results show that the endosome can modulate the filament properties to orchestrate membrane fluidization that facilitates vesicle fusion, providing an example of how active processes that modulate local molecular properties can result in large-scale transformations that are essential to cellular survival.

Characterizing intracellular mechanics via optical tweezers-based microrheology

Intracellular organization is a highly regulated homeostatic state maintained to ensure eukaryotic cells' correct and efficient functioning. Thanks to decades of research, vast knowledge of the proteins involved in intracellular transport and organization has been acquired. However, how these influence and potentially regulate the intracellular mechanical properties of the cell is largely unknown. There is a deep knowledge gap between the understanding of cortical mechanics, which is accessible by a series of experimental tools, and the intracellular situation that has been largely neglected due to the difficulty of performing intracellular mechanics measurements. Recently, tools required for such quantitative and localized analysis of intracellular mechanics have been introduced. Here, we review how these approaches and the resulting viscoelastic models lead the way to a full mechanical description of the cytoplasm, which is instrumental for a quantitative characterization of the intracellular life of cells.

Metabolically intact nuclei are fluidized by the activity of the chromatin remodeling motor BRG1

The structure and dynamics of the cell nucleus regulate nearly every facet of the cell. Changes in nuclear shape limit cell motility and gene expression. Although the nucleus is generally seen as the stiffest organelle in the cell, cells can nevertheless deform the nucleus to large strains by small mechanical stresses. Here, we show that the mechanical response of the cell nucleus exhibits active fluidization that is driven by the BRG 1 motor of the SWI/SNF/BAF chromatin-remodeling complex. Atomic force microscopy measurements show that the nucleus alters stiffness in response to the cell substrate stiffness, which is retained after the nucleus is isolated and that the work of nuclear compression is mostly dissipated rather than elastically stored. Inhibiting BRG 1 stiffens the nucleus and eliminates dissipation and nuclear remodeling both in isolated nuclei and in intact cells. These findings demonstrate a novel link between nuclear motor activity and global nuclear mechanics.

Confined run-and-tumble particles with non-Markovian tumbling statistics

Confined active particles constitute simple, yet realistic, examples of systems that converge into a nonequilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution g(t) of waiting times between tumbling events whose mean value is equal to τ. Unless g(t) is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviation and/or small-τ regime, where the SSD can be related to the the large-deviation function, s(x), via the scaling relation P_{st}(x)∼e^{-s(x)/τ}.

Studying fluctuating trajectories of optically confined passive tracers inside cells provides familiar active forces

In recent years, there has been a growing interest in studying the trajectories of microparticles inside living cells. Among other things, such studies are useful in understanding the spatio-temporal properties of a cell. In this work, we study the stochastic trajectories of a passive microparticle inside a cell using experiments and theory. Our theory is based on modeling the microparticle inside a cell as an active particle in a viscoelastic medium. The activity is included in our model from an additional stochastic term with non-zero persistence in the Langevin equation describing the dynamics of the microparticle. Using this model, we are able to predict the power spectral density (PSD) measured in the experiment and compute active forces. This caters to the situation where a tracer particle is optically confined and then yields a PSD for positional fluctuations. The low frequency part of the PSD yields information about the active forces that the particle feels. The fit to the model extracts such active force. Thus, we can conclude that trapping the particle does not affect the values of the forces extracted from the active fits if accounted for appropriately by proper theoretical models. In addition, the fit also provides system properties and optical tweezers trap stiffness.

Nonequilibrium steady state of trapped active particles

We consider an overdamped particle with a general physical mechanism that creates noisy active movement (e.g., a run-and-tumble particle or active Brownian particle, etc.), that is confined by an external potential. Focusing on the limit in which the correlation time τ of the active noise is small, we find the nonequilibrium steady-state distribution P_{st}(X) of the particle's position X. While typical fluctuations of X follow a Boltzmann distribution with an effective temperature that is not difficult to find, the tails of P_{st}(X) deviate from a Boltzmann behavior: In the limit τ→0, they scale as P_{st}(X)∼e^{-s(X)/τ}. We calculate the large-deviation function s(X) exactly for arbitrary trapping potential and active noise in dimension d=1, by relating it to the rate function that describes large deviations of the position of the same active particle in absence of an external potential at long times. We then extend our results to d>1 assuming rotational symmetry.

Nonequilibrium steady state for harmonically confined active particles

We study the full nonequilibrium steady-state distribution P_{st}(X) of the position X of a damped particle confined in a harmonic trapping potential and experiencing active noise whose correlation time τ_{c} is assumed to be very short. Typical fluctuations of X are governed by a Boltzmann distribution with an effective temperature that is found by approximating the noise as white Gaussian thermal noise. However, large deviations of X are described by a non-Boltzmann steady-state distribution. We find that, in the limit τ_{c}→0, they display the scaling behavior P_{st}(X)∼e^{-s(X)/τ_{c}}, where s(X) is the large-deviation function. We obtain an expression for s(X) for a general active noise and calculate it exactly for the particular case of telegraphic (dichotomous) noise.

Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self-propulsion in an orientation that stochastically rotates by 90^{∘} at a constant rate, clockwise or counterclockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness μ, and possibly diffuses. We find the exact time-dependent distribution P(x,y,t) of the particle's position, and in particular, the steady-state distribution P_{st}(x,y) that is reached in the long-time limit. We also find P(x,y,t) for a "free" particle, μ=0. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.

Hi-C embedded polymer model of Escherichia coli reveals the origin of heterogeneous subdiffusion in chromosomal loci

Underneath its apparently simple architecture, the circular chromosome of Escherichia coli is known for displaying complex dynamics in its cytoplasm, with past investigations hinting at inherently diverse mobilities of chromosomal loci across the genome. To decipher its origin, we simulate the dynamics of genome-wide spectrum of E. coli chromosomal loci, via integrating its experimentally derived Hi-C interaction matrix within a polymer-based model. Our analysis demonstrates that, while the dynamics of the chromosome is subdiffusive in a viscoelastic media, the diffusion constants are strongly dependent of chromosomal loci coordinates and diffusive exponents (α) are widely heterogenous with α ≈ 0.36-0.60. The loci-dependent heterogeneous dynamics and mean first-passage times of interloci encounter were found to be modulated via genetically distant interloci communications and is robust even in the presence of active, ATP-dependent noises. Control investigations reveal that the absence of Hi-C-derived interactions in the model would have abolished the traits of heterogeneous loci diffusion, underscoring the key role of loci-specific genetically distant interaction in modulating the underlying heterogeneity of the loci diffusion.

Tracer dynamics in crowded active-particle suspensions

We discuss the dynamics of active Brownian particles (ABPs) in crowded environments through the mean-squared displacement (MSD) of active and passive tracer particles in both active and passive host systems. Exact equations for the MSD are derived using a projection operator technique, extending to dense systems the known solution for a single ABP. The interaction of the tracer particle with the host particles gives rise to strong memory effects. Evaluating these approximately in the framework of a recently developed mode-coupling theory for active Brownian particles (ABP-MCT), we discuss the various dynamical regimes that emerge: While self-propelled motion gives rise to super-diffusive MSD, at high densities, this competes with an interaction-induced sub-diffusive regime. The predictions of the theory are shown to be in good agreement with results obtained from an event-driven Brownian dynamics (ED-BD) simulation scheme for the dynamics of two-dimensional active Brownian hard disks.

Mode-coupling theory for tagged-particle motion of active Brownian particles

We derive a mode-coupling theory (MCT) to describe the dynamics of a tracer particle that is embedded in a dense system of active Brownian particles (ABPs) in two spatial dimensions. The ABP undergo translational and rotational Brownian motion and are equipped with a fixed self-propulsion speed along their orientational vector that describes their active motility. The resulting equations of motion for the tagged-particle density-correlation functions describe the various cases of tracer dynamics close to the glass transition: that of a single active particle in a glass-forming passive host suspensions, that of a passive colloidal particle in a suspension of ABP, and that of active tracers in a bath of active particles. Numerical results are presented for these cases assuming hard-sphere interactions among the particles. The qualitative and quantitative accuracy of the theory is tested against event-driven Brownian dynamics (ED-BD) simulations of active and passive hard disks. Simulation and theory are found in quantitative agreement, provided one adjusts the overall density (as known from the passive description of glassy dynamics), and allows for a rescaling of self-propulsion velocities in the active host system. These adjustments account for the fact that ABP-MCT generally overestimates the tendency for kinetic arrest. We confirm in the simulations a peculiar feature of the transient and stationary dynamical density-correlation functions regarding their lack of symmetry under time reversal, demonstrating the nonequilibrium nature of the system and how it manifests itself in the theory.

Local detailed balance across scales: From diffusions to jump processes and beyond

Diffusive dynamics in presence of deep energy minima and weak nongradient forces can be coarse grained into a mesoscopic jump process over the various basins of attraction. Combining standard weak-noise results with a path integral expansion around equilibrium, we show that the emerging transition rates satisfy local detailed balance (LDB). Namely, the log ratio of the transition rates between nearby basins of attractions equals the free-energy variation appearing at equilibrium, supplemented by the work done by the nonconservative forces along the typical transition path. When the mesoscopic dynamics possesses a large-size deterministic limit, it can be further reduced to a jump process over macroscopic states satisfying LDB. The persistence of LDB under coarse graining of weakly nonequilibrium states is a generic consequence of the fact that only dissipative effects matter close to equilibrium.

Active carpets drive non-equilibrium diffusion and enhanced molecular fluxes

Biological activity is often highly concentrated on surfaces, across the scales from molecular motors and ciliary arrays to sessile and motile organisms. These 'active carpets' locally inject energy into their surrounding fluid. Whereas Fick's laws of diffusion are established near equilibrium, it is unclear how to solve non-equilibrium transport driven by such boundary-actuated fluctuations. Here, we derive the enhanced diffusivity of molecules or passive particles as a function of distance from an active carpet. Following Schnitzer's telegraph model, we then cast these results into generalised Fick's laws. Two archetypal problems are solved using these laws: First, considering sedimentation towards an active carpet, we find a self-cleaning effect where surface-driven fluctuations can repel particles. Second, considering diffusion from a source to an active sink, say nutrient capture by suspension feeders, we find a large molecular flux compared to thermal diffusion. Hence, our results could elucidate certain non-equilibrium properties of active coating materials and life at interfaces.

Optical Tweezers with Integrated Multiplane Microscopy (OpTIMuM): a new tool for 3D microrheology

We introduce a novel 3D microrheology system that combines for the first time Optical Tweezers with Integrated Multiplane Microscopy (OpTIMuM). The system allows the 3D tracking of an optically trapped bead, with ~ 20 nm accuracy along the optical axis. This is achieved without the need for a high precision z-stage, separate calibration sample, nor a priori knowledge of either the bead size or the optical properties of the suspending medium. Instead, we have developed a simple yet effective in situ spatial calibration method using image sharpness and exploiting the fact we image at multiple planes simultaneously. These features make OpTIMuM an ideal system for microrheology measurements, and we corroborate the effectiveness of this novel microrheology tool by measuring the viscosity of water in three dimensions, simultaneously.

Principles and Applications of Single Particle Tracking in Cell Research

It is a tough challenge for many decades to decipher the complex relationships between cell behaviors and cellular physical properties. Single particle tracking (SPT) with high spatial and temporal resolution has been applied extensively in cell research to understand physicochemical properties of cells and their bio-functions by tracking endogenous or exogenous probes. This review describes the fundamental principles of SPT as well as its applications in intracellular mechanics, membrane dynamics, organelles distribution, and processes of internalization and transport. Finally, challenges and future directions of SPT are also discussed.

Getting around the cell: physical transport in the intracellular world

Eukaryotic cells face the challenging task of transporting a variety of particles through the complex intracellular milieu in order to deliver, distribute, and mix the many components that support cell function. In this review, we explore the biological objectives and physical mechanisms of intracellular transport. Our focus is on cytoplasmic and intra-organelle transport at the whole-cell scale. We outline several key biological functions that depend on physically transporting components across the cell, including the delivery of secreted proteins, support of cell growth and repair, propagation of intracellular signals, establishment of organelle contacts, and spatial organization of metabolic gradients. We then review the three primary physical modes of transport in eukaryotic cells: diffusive motion, motor-driven transport, and advection by cytoplasmic flow. For each mechanism, we identify the main factors that determine speed and directionality. We also highlight the efficiency of each transport mode in fulfilling various key objectives of transport, such as particle mixing, directed delivery, and rapid target search. Taken together, the interplay of diffusion, molecular motors, and flows supports the intracellular transport needs that underlie a broad variety of biological phenomena.

Quantifying the non-equilibrium activity of an active colloid

Active matter systems exhibit rich emergent behavior due to constant injection and dissipation of energy at the level of individual agents. Since these systems are far from equilibrium, their dynamics and energetics cannot be understood using the framework of equilibrium statistical mechanics. Recent developments in stochastic thermodynamics extend classical concepts of work, heat, and energy dissipation to fluctuating non-equilibrium systems. We use recent advances in experiment and theory to study the non-thermal dissipation of individual light-activated self-propelled colloidal particles. We focus on characterizing the transition from thermal to non-thermal fluctuations and show that energy dissipation rates on the order of ∼kBT s-1 are measurable from finite time series data.

Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems

We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single-particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent of the friction coefficient.

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.

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.

Active Mechanics Reveal Molecular-Scale Force Kinetics in Living Oocytes

Active diffusion of intracellular components is emerging as an important process in cell biology. This process is mediated by complex assemblies of molecular motors and cytoskeletal filaments that drive force generation in the cytoplasm and facilitate enhanced motion. The kinetics of molecular motors have been precisely characterized in vitro by single molecule approaches, but their in vivo behavior remains elusive. Here, we study the active diffusion of vesicles in mouse oocytes, where this process plays a key role in nuclear positioning during development, and combine an experimental and theoretical framework to extract molecular-scale force kinetics (force, power stroke, and velocity) of the in vivo active process. Assuming a single dominant process, we find that the nonequilibrium activity induces rapid kicks of duration τ ∼ 300 μs resulting in an average force of F ∼ 0.4 pN on vesicles in in vivo oocytes, remarkably similar to the kinetics of in vitro myosin-V. Our results reveal that measuring in vivo active fluctuations allows extraction of the molecular-scale activity in agreement with single-molecule studies and demonstrates a mesoscopic framework to access force kinetics.

A mesoscale study of creep in a microgel using the acoustic radiation force

We study the motion of a sphere of diameter 330 μm embedded in a Carbopol microgel under the effect of the acoustic radiation pressure exerted by a focused ultrasonic field. The sphere motion within the microgel is tracked using videomicroscopy and compared to conventional creep and recovery measurements performed with a rheometer. We find that under moderate ultrasonic intensities, the sphere creeps as a power law of time with an exponent α ≃ 0.2 that is significantly smaller than the one inferred from global creep measurements below the yield stress of the microgel (α ≃ 0.4). Moreover, the sphere relaxation motion after creep and the global recovery are respectively consistent with these two different exponents. By allowing a rheological characterization at the scale of the sphere with forces of the order of micronewtons, the present experiments pave the way for acoustic "mesorheology" which probes volumes and forces an intermediate between standard macroscopic rheology and classical microrheology. They also open new questions about the effects of the geometry of the deformation field and of the sphere size and surface properties on the creep behaviour of soft materials.

Anisotropic mechanics and dynamics of a living mammalian cytoplasm

During physiological processes, cells can undergo morphological changes that can result in a significant redistribution of the cytoskeleton causing anisotropic behavior. Evidence of anisotropy in cells under mechanical stimuli exists; however, the role of cytoskeletal restructuring resulting from changes in cell shape in mechanical anisotropy and its effects remain unclear. In the present study, we examine the role of cell morphology in inducing anisotropy in both intracellular mechanics and dynamics. We change the aspect ratio of cells by confining the cell width and measuring the mechanical properties of the cytoplasm using optical tweezers in both the longitudinal and transverse directions to quantify the degree of mechanical anisotropy. These active microrheology measurements are then combined with intracellular movement to calculate the intracellular force spectrum using force spectrum microscopy (FSM), from which the degree of anisotropy in dynamics and force can be quantified. We find that unrestricted cells with aspect ratio (AR) ∼1 are isotropic; however, when cells break symmetry, they exhibit significant anisotropy in cytoplasmic mechanics and dynamics.

Memory-based mediated interactions between rigid particulate inclusions in viscoelastic environments

Many practically relevant materials combine properties of viscous fluids and elastic solids to viscoelastic behavior. Our focus is on the induced dynamic behavior of damped finite-sized particulate inclusions in such substances. We explicitly describe history-dependent interactions that emerge between the embedded particles. These interactions are mediated by the viscoelastic surroundings. They result from the flows and distortions of the viscoelastic medium when induced by the rigid inclusions. Both viscoelastic environments of terminal fluidlike flow and of completely reversible damped elastic behavior are covered. For illustration and to highlight the role of the formalism in potential applications, we briefly address the relevant examples of dragging a rigid sphere through a viscoelastic environment together with subsequent relaxation dynamics, the switching dynamics of magnetic fillers in elastic gel matrices, and the swimming behavior of active microswimmers in viscoelastic solutions. The approach provides a basis for more quantitative and extended investigations of these and related systems in the future.

Reversible magnetomechanical collapse: virtual touching and detachment of rigid inclusions in a soft elastic matrix

Soft elastic composite materials containing particulate rigid inclusions in a soft elastic matrix are candidates for developing soft actuators or tunable damping devices. The possibility to reversibly drive the rigid inclusions within such a composite together to a close-to-touching state by an external stimulus would offer important benefits. Then, a significant tuning of the mechanical properties could be achieved due to the resulting mechanical hardening. For a long time, it has been argued whether a virtual touching of the embedded magnetic particles with subsequent detachment can actually be observed in real materials, and if so, whether the process is reversible. Here, we present experimental results that demonstrate this phenomenon in reality. Our system consists of two paramagnetic nickel particles embedded at finite initial distance in a soft elastic polymeric gel matrix. Magnetization in an external magnetic field tunes the magnetic attraction between the particles and drives the process. We quantify our experimental results by different theoretical tools, i.e., explicit analytical calculations in the framework of linear elasticity theory, a projection onto simplified dipole-spring models, as well as detailed finite-element simulations. From these different approaches, we conclude that in our case the cycle of virtual touching and detachment shows hysteretic behavior due to the mutual magnetization between the paramagnetic particles. Our results are important for the design and construction of reversibly tunable mechanical damping devices. Moreover, our projection on dipole-spring models allows the formal connection of our description to various related systems, e.g., magnetosome filaments in magnetotactic bacteria.

Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.

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.

Boltzmann-distribution-equivalent for Lévy noise and how it leads to thermodynamically consistent epicatalysis

Nonequilibrium systems commonly exhibit Lévy noise. This means that the distribution for the size of the Brownian fluctuations has a "fat" power-law tail. Large Brownian kicks are then more common as compared to the ordinary Gaussian distribution. We consider a two-state system, i.e., two wells and a barrier in between. The barrier is sufficiently high for a barrier crossing to be a rare event. When the noise is Lévy, we do not get a Boltzmann distribution between the two wells. Instead we get a situation where the distribution between the two wells also depends on the height of the barrier that is in between. Ordinarily, a catalyst, by lowering the barrier between two states, speeds up the relaxation to an equilibrium, but does not change the equilibrium distribution. In an environment with Lévy noise, on the other hand, we have the possibility of epicatalysis, i.e., a catalyst effectively altering the distribution between two states through the changing of the barrier height. After deriving formulas to quantitatively describe this effect, we discuss how this idea may apply in nuclear reactors and in the biochemistry of a living cell.

Self-assembly of collagen-based biomaterials: preparation, characterizations and biomedical applications

By combining regulatory parameters with characterization methods, researchers can selectively fabricate collagenous biomaterials with various functional responses for biomedical applications.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Nonequilibrium depletion interactions in active microrheology

Entropic depletion forces arise between mesoscopic bodies that are immersed in a suspension of macromolecules, such as colloid-polymer mixtures. Here we consider the case of a driven colloidal probe in the presence of another, passive colloidal particle, both solvated in an ideal bath of small spherical particles. We calculate the nonequilibrium forces mediated by the depletants on the two colloidal particles within a dynamical superposition approximation (DSA) scheme. In order to assess the quality of this approximation, and to obtain the colloidal microstructure around the driven probe, we corroborate our theoretical results with Brownian dynamics simulations.

Cytoplasmic Flow and Mixing Due to Deformation of Motile Cells

The cytoplasm of a living cell is a dynamic environment through which intracellular components must move and mix. In motile, rapidly deforming cells such as human neutrophils, bulk cytoplasmic flow couples cell deformation to the transport and dispersion of cytoplasmic particles. Using particle-tracking measurements in live neutrophil-like cells, we demonstrate that fluid flow associated with the cell deformation contributes to the motion of small acidic organelles, dominating over diffusion on timescales above a few seconds. We then use a general physical model of particle dispersion in a deforming fluid domain to show that transport of organelle-sized particles between the cell periphery and the bulk can be enhanced by dynamic deformation comparable to that observed in neutrophils. Our results implicate an important mechanism contributing to organelle transport in these motile cells: cytoplasmic flow driven by cell shape deformation.

On the importance of protein diffusion in biological systems: The example of the Bicoid morphogen gradient

Morphogens are proteins that form concentration gradients in embryos and developing tissues, where they act as postal codes, providing cells with positional information and allowing them to behave accordingly. Bicoid was the first discovered morphogen, and remains one of the most studied. It regulates segmentation in flies, forming a striking exponential gradient along the anterior-posterior axis of early Drosophila embryos, and activating the transcription of multiple target genes in a concentration-dependent manner. In this review, the work done by us and by others to characterize the mobility of Bicoid in D. melanogaster embryos is presented. The central role played by the diffusion of Bicoid in both the establishment of the gradient and the activation of target genes is discussed, and placed in the context of the need for these processes to be all at once rapid, precise and robust. The Bicoid system, and morphogen gradients in general, remain amongst the most amazing examples of the coexistence, often observed in living systems, of small-scale disorder and large-scale spatial order. This article is part of a Special Issue entitled: Biophysics in Canada, edited by Lewis Kay, John Baenziger, Albert Berghuis and Peter Tieleman.

Unraveling intermittent features in single-particle trajectories by a local convex hull method

We propose a model-free method to detect change points between distinct phases in a single random trajectory of an intermittent stochastic process. The local convex hull (LCH) is constructed for each trajectory point, while its geometric properties (e.g., the diameter or the volume) are used as discriminators between phases. The efficiency of the LCH method is validated for six models of intermittent motion, including Brownian motion with different diffusivities or drifts, fractional Brownian motion with different Hurst exponents, and surface-mediated diffusion. We discuss potential applications of the method for detection of active and passive phases in the intracellular transport, temporal trapping or binding of diffusing molecules, alternating bulk and surface diffusion, run and tumble (or search) phases in the motion of bacteria and foraging animals, and instantaneous firing rates in neurons.

Impact of a mechanical shear stress on intracellular trafficking

Intracellular trafficking mainly takes place along the microtubules, and its efficiency depends on the local architecture and organization of the cytoskeletal network. In this work, the cytoplasm of stem cells is subjected to mechanical vortexing at a frequency of up to 1 Hz, by using magnetic chains of endosomes embedded in the cell body, in order to locally perturb the network structure. The consequences are evaluated on the directionality and processivity of the spontaneous motion of endosomes. When the same chains are used both to shear the cell medium and to probe the intracellular traffic, a substantial decrease in transport efficiency is detected after applying the mechanical shear. Interestingly, when using different objects to apply the shear and to probe the spontaneous motion, no alteration of the transport efficiency can be detected. We conclude that shaking the vesicles mainly causes their unbinding from the cytoskeletal tracks, but has little influence on the integrity of the network itself. This is corroborated by active microrheology measurements, performed with chains actuated by a magnetic field, and showing that the mechanical compliance of the cytoplasm is similar before and after slow vortexing.

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Diffusion in systems crowded by active force-dipole molecules

Experimental studies of systems containing active proteins that undergo conformational changes driven by catalytic chemical reactions have shown that the diffusion coefficients of passive tracer particles and active molecules are larger than the corresponding values when chemical activity is absent. Various mechanisms have been proposed for such behavior, including, among others, force dipole interactions of molecular motors moving on filaments and collective hydrodynamic effects arising from active proteins. Simulations of a multi-component system containing active dumbbell molecules that cycle between open and closed states, a passive tracer particle and solvent molecules are carried out. Consistent with experiments, it is shown that the diffusion coefficients of both passive particles and the dumbbells themselves are enhanced when the dumbbells are active. The dependence of the diffusion enhancement on the volume fraction of dumbbells is determined, and the effects of crowding by active dumbbell molecules are shown to differ from those due to inactive molecules.

Forces and torques on rigid inclusions in an elastic environment: Resulting matrix-mediated interactions, displacements, and rotations

Embedding rigid inclusions into elastic matrix materials is a procedure of high practical relevance, for instance, for the fabrication of elastic composite materials. We theoretically analyze the following situation. Rigid spherical inclusions are enclosed by a homogeneous elastic medium under stick boundary conditions. Forces and torques are directly imposed from outside onto the inclusions or are externally induced between them. The inclusions respond to these forces and torques by translations and rotations against the surrounding elastic matrix. This leads to elastic matrix deformations, and in turn results in mutual long-ranged matrix-mediated interactions between the inclusions. Adapting a well-known approach from low-Reynolds-number hydrodynamics, we explicitly calculate the displacements and rotations of the inclusions from the externally imposed or induced forces and torques. Analytical expressions are presented as a function of the inclusion configuration in terms of displaceability and rotateability matrices. The role of the elastic environment is implicitly included in these relations. That is, the resulting expressions allow a calculation of the induced displacements and rotations directly from the inclusion configuration, without having to explicitly determine the deformations of the elastic environment. In contrast to the hydrodynamic case, compressibility of the surrounding medium is readily taken into account. We present the complete derivation based on the underlying equations of linear elasticity theory. In the future, the method will, for example, be helpful to characterize the behavior of externally tunable elastic composite materials, to accelerate numerical approaches, as well as to improve the quantitative interpretation of microrheological results.

Anomalous diffusion in viscoelastic media with active force dipoles

With the use of the "two-fluid model," we discuss anomalous diffusion induced by active force dipoles in viscoelastic media. Active force dipoles, such as proteins and bacteria, generate nonthermal fluctuating flows that lead to a substantial increment of the diffusion. Using the partial Green's function of the two-fluid model, we first obtain passive (thermal) two-point correlation functions such as the displacement cross-correlation function between the two-point particles separated by a finite distance. We then calculate active (nonthermal) one-point and two-point correlation functions due to active force dipoles. The time correlation of a force dipole is assumed to decay exponentially with a characteristic time scale. We show that the active component of the displacement cross-correlation function exhibits various crossovers from super-diffusive to subdiffusive behaviors depending on the characteristic time scales and the particle separation. Our theoretical results are intimately related to the microrheology technique to detect fluctuations in nonequilibrium environment.

Forces on Rigid Inclusions in Elastic Media and Resulting Matrix-Mediated Interactions

To describe many-particle systems suspended in incompressible low-Reynolds-number fluids, effective hydrodynamic interactions can be introduced. Here, we consider particles embedded in elastic media. The effective elastic interactions between spherical particles are calculated analytically, inspired by the approach in the fluid case. Our experiments on interacting magnetic particles confirm the theory. In view of the huge success of the method in hydrodynamics, we similarly expect many future applications in the elastic case, e.g., for elastic composite materials.

Discontinuous thinning in active microrheology of soft complex matter

Employing theory and numerical simulations, we demonstrate discontinuous force thinning due to the driven motion of an external probe in a host medium. We consider two cases: an ideal structureless medium (modeling ultrasoft materials such as polymer melts) and a dilute bath of interacting repulsive particles. When the driving of the probe exceeds a critical force, the microviscosity of the medium drops abruptly by about an order of magnitude. This phenomenon occurs for strong attractive interactions between a large probe and a sufficiently dense host medium.

How Far from Equilibrium Is Active Matter?

Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relation between injection and dissipation of energy at the microscopic scale. We consider such a system of interacting particles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics still satisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that, for short persistent times, the entropy production rate vanishes. This endows such systems with an effective fluctuation-dissipation theorem akin to that of thermal equilibrium systems. Last, we show how interacting particle systems with viscous drags and correlated noises can be seen as in equilibrium with a viscoelastic bath but driven out of equilibrium by nonconservative forces, hence providing energetic insight into the departure of active systems from equilibrium.

Advances in the microrheology of complex fluids

New developments in the microrheology of complex fluids are considered. Firstly the requirements for a simple modern particle tracking microrheology experiment are introduced, the error analysis methods associated with it and the mathematical techniques required to calculate the linear viscoelasticity. Progress in microrheology instrumentation is then described with respect to detectors, light sources, colloidal probes, magnetic tweezers, optical tweezers, diffusing wave spectroscopy, optical coherence tomography, fluorescence correlation spectroscopy, elastic- and quasi-elastic scattering techniques, 3D tracking, single molecule methods, modern microscopy methods and microfluidics. New theoretical techniques are also reviewed such as Bayesian analysis, oversampling, inversion techniques, alternative statistical tools for tracks (angular correlations, first passage probabilities, the kurtosis, motor protein step segmentation etc), issues in micro/macro rheological agreement and two particle methodologies. Applications where microrheology has begun to make some impact are also considered including semi-flexible polymers, gels, microorganism biofilms, intracellular methods, high frequency viscoelasticity, comb polymers, active motile fluids, blood clots, colloids, granular materials, polymers, liquid crystals and foods. Two large emergent areas of microrheology, non-linear microrheology and surface microrheology are also discussed.

Diffusing colloidal probes of cell surfaces

Measurements and analyses are reported to quantify dynamic and equilibrium interactions between colloidal particles and live cell surfaces using dark field video microscopy. Two-dimensional trajectories of micron-sized polyethylene glycol (PEG)-coated silica colloids relative to adherent epithelial breast cancer cell perimeters are determined allowing measurement of position dependent diffusivities and interaction potentials. PEG was chosen as the material system of interest to assess non-specific interactions with cell surfaces and establishes a basis for investigation of specific interactions in future studies. Analysis of measured potential energies on cell surfaces reveals the spatial dependence in cell topography. With the measured cell topography and models for particle-cell surface hydrodynamic interactions, excellent agreement is obtained between theoretical and measured colloidal transport on cell surfaces. Quantitative analyses of association lifetimes showed that PEG coatings act to stabilize colloids above the cell surface through net repulsive, steric interactions. Our results demonstrate a self-consistent analysis of diffusing colloidal probe interactions due to conservative and non-conservative forces to characterize biophysical cell surface properties.

Buckling of paramagnetic chains in soft gels

We study the magneto-elastic coupling behavior of paramagnetic chains in soft polymer gels exposed to external magnetic fields. To this end, a laser scanning confocal microscope is used to observe the morphology of the paramagnetic chains together with the deformation field of the surrounding gel network. The paramagnetic chains in soft polymer gels show rich morphological shape changes under oblique magnetic fields, in particular a pronounced buckling deformation. The details of the resulting morphological shapes depend on the length of the chain, the strength of the external magnetic field, and the modulus of the gel. Based on the observation that the magnetic chains are strongly coupled to the surrounding polymer network, a simplified model is developed to describe their buckling behavior. A coarse-grained molecular dynamics simulation model featuring an increased matrix stiffness on the surfaces of the particles leads to morphologies in agreement with the experimentally observed buckling effects.

Modeling magnetosensitive ion channels in the viscoelastic environment of living cells

We propose and study a model of hypothetical magnetosensitive ionic channels which are long thought to be a possible candidate to explain the influence of weak magnetic fields on living organisms ranging from magnetotactic bacteria to fishes, birds, rats, bats, and other mammals including humans. The core of the model is provided by a short chain of magnetosomes serving as a sensor, which is coupled by elastic linkers to the gating elements of ion channels forming a small cluster in the cell membrane. The magnetic sensor is fixed by one end on cytoskeleton elements attached to the membrane and is exposed to viscoelastic cytosol. Its free end can reorient stochastically and subdiffusively in viscoelastic cytosol responding to external magnetic field changes and can open the gates of coupled ion channels. The sensor dynamics is generally bistable due to bistability of the gates which can be in two states with probabilities which depend on the sensor orientation. For realistic parameters, it is shown that this model channel can operate in the magnetic field of Earth for a small number (five to seven) of single-domain magnetosomes constituting the sensor rod, each of which has a typical size found in magnetotactic bacteria and other organisms or even just one sufficiently large nanoparticle of a characteristic size also found in nature. It is shown that, due to the viscoelasticity of the medium, the bistable gating dynamics generally exhibits power law and stretched exponential distributions of the residence times of the channels in their open and closed states. This provides a generic physical mechanism for the explanation of the origin of such anomalous kinetics for other ionic channels whose sensors move in a viscoelastic environment provided by either cytosol or biological membrane, in a quite general context, beyond the fascinating hypothesis of magnetosensitive ionic channels we explore.

Modeling the dynamics of a tracer particle in an elastic active gel

The internal dynamics of active gels both in artificial (in vitro) model systems and inside the cytoskeleton of living cells has been extensively studied with experiments of recent years. These dynamics are probed using tracer particles embedded in the network of biopolymers together with molecular motors, and distinct nonthermal behavior is observed. We present a theoretical model of the dynamics of a trapped active particle, which allows us to quantify the deviations from equilibrium behavior, using both analytic and numerical calculations. We map the different regimes of dynamics in this system and highlight the different manifestations of activity: breakdown of the virial theorem and equipartition, different elasticity-dependent "effective temperatures," and distinct non-Gaussian distributions. Our results shed light on puzzling observations in active gel experiments and provide physical interpretation of existing observations, as well as predictions for future studies.