Nonequilibrium fluctuations of a remodeling in vitro cytoskeleton

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}.

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

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

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.

Extending the analogy between intracellular motion in mammalian cells and glassy dynamics

How molecules, organelles, and foreign objects move within living cells has been studied in organisms ranging from bacteria to human cells. In mammalian cells, in particular, cellular vesicles move across the cell using motor proteins that carry the vesicle down the cytoskeleton to their destination. We have recently noted several similarities between the motion of such vesicles and that in disordered, "glassy", systems, but the generality of this observation remains unclear. Here we follow the motion of mitochondria, the organelles responsible for cell energy production, in mammalian cells over timescales from 50 ms to 70 s. Qualitative observations show that single mitochondria remain within a spatially limited region for extended periods of time, before moving longer distances relatively quickly. The displacement distribution is roughly Gaussian for shorter distances (≲0.05 μm) but exhibits exponentially decaying tails at longer distances (up to 0.40 μm). This behaviour is well-described by a model developed to describe the motion in glassy systems. These observations are extended to in total 3 different objects (mitochondria, lysosomes and nano-sized beads enclosed in vesicles), 3 different mammalian cell types (HEK 293, HeLa, and HT22), from 2 different organisms (human and mouse). Further evidence that supports glass-like characteristics of the motion is a difference between the time it takes to move a longer distance for the first time and subsequent times, as well as a weak ergodicity breaking of the motion. Overall, we demonstrate the ubiquity of glass-like motion in mammalian cells, providing a different perspective on intracellular motion.

F-actin architecture determines constraints on myosin thick filament motion

Active stresses are generated and transmitted throughout diverse F-actin architectures within the cell cytoskeleton, and drive essential behaviors of the cell, from cell division to migration. However, while the impact of F-actin architecture on the transmission of stress is well studied, the role of architecture on the ab initio generation of stresses remains less understood. Here, we assemble F-actin networks in vitro, whose architectures are varied from branched to bundled through F-actin nucleation via Arp2/3 and the formin mDia1. Within these architectures, we track the motions of embedded myosin thick filaments and connect them to the extent of F-actin network deformation. While mDia1-nucleated networks facilitate the accumulation of stress and drive contractility through enhanced actomyosin sliding, branched networks prevent stress accumulation through the inhibited processivity of thick filaments. The reduction in processivity is due to a decrease in translational and rotational motions constrained by the local density and geometry of F-actin.

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.

Emergence of local irreversibility in complex interacting systems

Living systems are fundamentally irreversible, breaking detailed balance and establishing an arrow of time. But how does the evident arrow of time for a whole system arise from the interactions among its multiple elements? We show that the local evidence for the arrow of time, which is the entropy production for thermodynamic systems, can be decomposed. First, it can be split into two components: an independent term reflecting the dynamics of individual elements and an interaction term driven by the dependencies among elements. Adapting tools from nonequilibrium physics, we further decompose the interaction term into contributions from pairs of elements, triplets, and higher-order terms. We illustrate our methods on models of cellular sensing and logical computations, as well as on patterns of neural activity in the retina as it responds to visual inputs. We find that neural activity can define the arrow of time even when the visual inputs do not, and that the dominant contribution to this breaking of detailed balance comes from interactions among pairs of neurons.

Molecular and Colloidal Transport in Bacterial Cellulose Hydrogels

Bacterial cellulose biofilms are complex networks of strong interwoven nanofibers that control transport and protect bacterial colonies in the film. The design of diverse applications of these bacterial cellulose films also relies on understanding and controlling transport through the fiber mesh, and transport simulations of the films are most accurate when guided by experimental characterization of the structures and the resultant diffusion inside. Diffusion through such films is a function of their key microstructural length scales, determining how molecules, as well as particles and microorganisms, permeate them. We use microscopy to study the unique bacterial cellulose film via its pore structure and quantify the mobility dynamics of various sizes of tracer particles and macromolecules. Mobility is hindered within the films, as confinement and local movement strongly depend on the void size relative to diffusing tracers. The biofilms have a naturally periodic structure of alternating dense and porous layers of nanofiber mesh, and we tune the magnitude of the spacing via fermentation conditions. Micron-sized particles can diffuse through the porous layers but cannot penetrate the dense layers. Tracer mobility in the porous layers is isotropic, indicating a largely random pore structure there. Molecular diffusion through the whole film is only slightly reduced by the structural tortuosity. Knowledge of transport variations within bacterial cellulose networks can be used to guide the design of symbiotic cultures in these structures and enhance their use in applications like biomedical implants, wound dressings, lab-grown meat, clothing textiles, and sensors.

Broken detailed balance and entropy production in the human brain

To perform biological functions, living systems must break detailed balance by consuming energy and producing entropy. At microscopic scales, broken detailed balance enables a suite of molecular and cellular functions, including computations, kinetic proofreading, sensing, adaptation, and transportation. But do macroscopic violations of detailed balance enable higher-order biological functions, such as cognition and movement? To answer this question, we adapt tools from nonequilibrium statistical mechanics to quantify broken detailed balance in complex living systems. Analyzing neural recordings from hundreds of human subjects, we find that the brain violates detailed balance at large scales and that these violations increase with physical and cognitive exertion. Generally, we provide a flexible framework for investigating broken detailed balance at large scales in complex systems.

Living systems break detailed balance at small scales, consuming energy and producing entropy in the environment to perform molecular and cellular functions. However, it remains unclear how broken detailed balance manifests at macroscopic scales and how such dynamics support higher-order biological functions. Here we present a framework to quantify broken detailed balance by measuring entropy production in macroscopic systems. We apply our method to the human brain, an organ whose immense metabolic consumption drives a diverse range of cognitive functions. Using whole-brain imaging data, we demonstrate that the brain nearly obeys detailed balance when at rest, but strongly breaks detailed balance when performing physically and cognitively demanding tasks. Using a dynamic Ising model, we show that these large-scale violations of detailed balance can emerge from fine-scale asymmetries in the interactions between elements, a known feature of neural systems. Together, these results suggest that violations of detailed balance are vital for cognition and provide a general tool for quantifying entropy production in macroscopic systems.

Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise

We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study the first-arrival properties of the system in the presence of a delta-function sink with strength κ, where κ=0 and κ=∞ correspond to fully nonreactive and fully reactive sinks, respectively. We explore the effect of two competitive mechanisms: the diffusive spread in the presence of two noises and the increase in probability density around the initial position due to stochastic resetting. We show that there exists an optimal resetting rate, which minimizes the mean first-arrival time (MFAT) to the sink for a given value of the sink strength. We also explore the effect of the strength of the Poissonian noise on MFAT, in addition to sink strength. Our formalism generalizes the diffusion-limited reaction under resetting in a nonequilibrium bath and provides an efficient search strategy for a reactant to find a target site, relevant in a range of biophysical processes.

Subtle changes in crosslinking drive diverse anomalous transport characteristics in actin-microtubule networks

Anomalous diffusion in crowded and complex environments is widely studied due to its importance in intracellular transport, fluid rheology and materials engineering. Specifically, diffusion through the cytoskeleton, a network comprised of semiflexible actin filaments and rigid microtubules that interact both sterically and via crosslinking, plays a principal role in viral infection, vesicle transport and targeted drug delivery. Here, we elucidate the impact of crosslinking on particle diffusion in composites of actin and microtubules with actin-actin, microtubule-microtubule and actin-microtubule crosslinking. We analyze a suite of transport metrics by coupling single-particle tracking and differential dynamic microscopy. Using these complementary techniques, we find that particles display non-Gaussian and non-ergodic subdiffusion that is markedly enhanced by cytoskeletal crosslinking, which we attribute to suppressed microtubule mobility. However, the extent to which transport deviates from normal Brownian diffusion depends strongly on the crosslinking motif - with actin-microtubule crosslinking inducing the most pronounced anomalous characteristics. Our results reveal that subtle changes to actin-microtubule interactions can have complex impacts on particle diffusion in cytoskeleton composites, and suggest that a combination of reduced filament mobility and more variance in actin mobilities leads to more strongly anomalous particle transport.

Glass-like characteristics of intracellular motion in human cells

The motion in the cytosol of microorganisms such as bacteria and yeast has been observed to undergo a dramatic slowing down upon cell energy depletion. These observations have been interpreted as the motion being "glassy," but whether this notion is useful also for active, motor-protein-driven transport in eukaryotic cells is less clear. Here, we use fluorescence microscopy of beads in human (HeLa) cells to probe the motion of membrane-surrounded structures that are carried along the cytoskeleton by motor proteins. Evaluating several hallmarks of glassy dynamics, we show that at short length scales, the motion is heterogeneous, is nonergodic, is well described by a model for the displacement distribution in glassy systems, and exhibits a decoupling of the exchange and persistence times. Overall, these results suggest that the short length scale behavior of objects that can be transported actively by motor proteins in human cells shares features with the motion in glassy systems.

Rapid local compression in active gels is caused by nonlinear network response

The actin cytoskeleton in living cells generates forces in conjunction with myosin motor proteins to directly and indirectly drive essential cellular processes. The semiflexible filaments of the cytoskeleton can respond nonlinearly to the collective action of motors. We here investigate mechanics and force generation in a model actin cytoskeleton, reconstituted in vitro, by observing the response and fluctuations of embedded micron-scale probe particles. Myosin mini-filaments can be modeled as force dipoles and give rise to deformations in the surrounding network of cross-linked actin. Anomalously correlated probe fluctuations indicate the presence of rapid local compression or draining of the network that emerges in addition to the ordinary linear shear elastic (incompressible) response to force dipoles. The anomalous propagation of compression can be attributed to the nonlinear response of actin filaments to the microscopic forces, and is quantitatively consistent with motor-generated large-scale stiffening of the gels.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

Kinetics of actin networks formation measured by time resolved particle-tracking microrheology

Actin is one of the most studied cytoskeleton proteins showing a very rich span of structures and functions. For example, adenosine triphosphate (ATP)-assisted polymerization of actin is used to push protrusions forward in a mechanism that enables cells to crawl on a substrate. In this process, the chemical energy released from the hydrolysis of ATP is what enables force generation. We study a minimal model system comprised of actin monomers in an excess of ATP concentration. In such a system polymerization proceeds in three stages: nucleation of actin filaments, elongation, and network formation. While the kinetics of filament growth was characterized previously, not much is known about the kinetics of network formation and the evolution of networks towards a steady-state structure. In particular, it is not clear how the non-equilibrium nature of this ATP-assisted polymerization manifests itself in the kinetics of self-assembly. Here, we use time-resolved microrheology to follow the kinetics of the three stages of self-assembly as a function of initial actin monomer concentration. Surprisingly, we find that at high enough initial monomer concentrations the effective elastic modulus of the forming actin networks overshoots and then relaxes with a -2/5 power law. We attribute the overshoot to the non-equilibrium nature of the polymerization and the relaxation to rearrangements of the network into a steady-state structure.

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.

Active Trap Model

Motivated by the dynamics of particles embedded in active gels, both in vitro and inside the cytoskeleton of living cells, we study an active generalization of the classical trap model. We demonstrate that activity leads to dramatic modifications in the diffusion compared to the thermal case: the mean square displacement becomes subdiffusive, spreading as a power law in time, when the trap depth distribution is a Gaussian and is slower than any power law when it is drawn from an exponential distribution. The results are derived for a simple, exactly solvable, case of harmonic traps. We then argue that the results are robust for more realistic trap shapes when the activity is strong.

Particle diffusion in extracellular hydrogels

Hyaluronic acid is an abundant polyelectrolyte in the human body that forms extracellular hydrogels in connective tissues. It is essential for regulating tissue biomechanics and cell-cell communication, yet hyaluronan overexpression is associated with pathological situations such as cancer and multiple sclerosis. Due to its enormous molecular weight (in the range of millions of Daltons), accumulation of hyaluronan hinders transport of macromolecules including nutrients and growth factors through tissues and also hampers drug delivery. However, the exact contribution of hyaluronan to tissue penetrability is poorly understood due to the complex structure and molecular composition of tissues. Here we reconstitute biomimetic hyaluronan gels and systematically investigate the effects of gel composition and crosslinking on the diffusion of microscopic tracer particles. We combine ensemble-averaged measurements via differential dynamic microscopy with single-particle tracking. We show that the particle diffusivity depends on the particle size relative to the network pore size and also on the stress relaxation dynamics of the network. We furthermore show that addition of collagen, the other major biopolymer in tissues, causes the emergence of caged particle dynamics. Our findings are useful for understanding macromolecular transport in tissues and for designing biomimetic extracellular matrix hydrogels for drug delivery and tissue regeneration.

Filament Rigidity Vies with Mesh Size in Determining Anomalous Diffusion in Cytoskeleton

The diffusion of microscopic particles through the cell, important to processes such as viral infection, gene delivery, and vesicle transport, is largely controlled by the complex cytoskeletal network, comprised of semiflexible actin filaments and rigid microtubules, that pervades the cytoplasm. By varying the relative concentrations of actin and microtubules, the cytoskeleton can display a host of different structural and dynamic properties that, in turn, impact the diffusion of particles through the composite network. Here, we couple single-particle tracking with differential dynamic microscopy to characterize the transport of microsphere tracers diffusing through composite in vitro networks with varying ratios of actin and microtubules. We analyze multiple complementary metrics for anomalous transport to show that particles exhibit anomalous subdiffusion in all networks, which our data suggest arises from caging by networks. Further, subdiffusive characteristics are markedly more pronounced in actin-rich networks, which exhibit similarly more prominent viscoelastic properties compared to microtubule-rich composites. While the smaller mesh size of actin-rich composites compared to microtubule-rich composites plays an important role in these results, the rigidity of the filaments comprising the network also influences the anomalous characteristics that we observe. Our results suggest that as microtubules in our composites are replaced with actin filaments, the decreasing filament rigidity competes with increasing network connectivity to drive anomalous transport.

Uncovering the dynamic precursors to motor-driven contraction of active gels

Cells and tissues have the remarkable ability to actively generate the forces required to change their shape. This active mechanical behavior is largely mediated by the actin cytoskeleton, a crosslinked network of actin filaments that is contracted by myosin motors. Experiments and active gel theories have established that the length scale over which gel contraction occurs is governed by a balance between molecular motor activity and crosslink density. By contrast, the dynamics that govern the contractile activity of the cytoskeleton remain poorly understood. Here we investigate the microscopic dynamics of reconstituted actin-myosin networks using simultaneous real-space video microscopy and Fourier-space dynamic light scattering. Light scattering reveals different regimes of microscopic dynamics as a function of sample age. We uncover two dynamical precursors that precede macroscopic gel contraction. One is characterized by a progressive acceleration of stress-induced rearrangements, while the other consists of sudden, heterogeneous rearrangements. Intriguingly, our findings suggest a qualitative analogy between self-driven rupture and collapse of active gels and the delayed rupture of passive gels observed in earlier studies of colloidal gels under external loads.

Heterogeneities Shape Passive Intracellular Transport

A living cell's interior is one of the most complex and intrinsically dynamic systems, providing an elaborate interplay between cytosolic crowding and ATP-driven motion that controls cellular functionality. Here, we investigated two distinct fundamental features of the merely passive, non-biomotor-shuttled material transport within the cytoplasm of Dictyostelium discoideum cells: the anomalous non-linear scaling of the mean-squared displacement of a 150-nm-diameter particle and non-Gaussian distribution of increments. Relying on single-particle tracking data of 320,000 data points, we performed a systematic analysis of four possible origins for non-Gaussian transport: 1) sample-based variability, 2) rarely occurring strong motion events, 3) ergodicity breaking/aging, and 4) spatiotemporal heterogeneities of the intracellular medium. After excluding the first three reasons, we investigated the remaining hypothesis of a heterogeneous cytoplasm as cause for non-Gaussian transport. A, to our knowledge, novel fit model with randomly distributed diffusivities implementing medium heterogeneities suits the experimental data. Strikingly, the non-Gaussian feature is independent of the cytoskeleton condition and lag time. This reveals that efficiency and consistency of passive intracellular transport and the related anomalous scaling of the mean-squared displacement are regulated by cytoskeleton components, whereas cytoplasmic heterogeneities are responsible for the generic, non-Gaussian distribution of increments.

Crowding-Activity Coupling Effect on Conformational Change of a Semi-Flexible Polymer

The behavior of a polymer in a passive crowded medium or in a very dilute active bath has been well studied, while a polymer immersed in an environment featured by both crowding and activity remains an open problem. In this paper, a systematic Langevin simulation is performed to investigate the conformational change of a semi-flexible chain in a concentrated solution packed with spherical active crowders. A very novel shrinkage-to-swelling transition is observed for a polymer with small rigidity. The underlying phase diagram is constructed in the parameter space of active force and crowder size. Moreover, the variation of the polymer gyration radius demonstrates a non-monotonic dependence on the dynamical persistence length of the active particle. Lastly, the activity-crowding coupling effect in different crowder size baths is clarified. In the case of small crowders, activity strengthens the crowding-induced shrinkage to the chain. As crowder size increases, activity turns out to be a contrasting factor to crowding, resulting in a competitive shrinkage and swelling. In the large size situation, the swelling effect arising from activity eventually becomes dominant. The present study provides a deeper understanding of the unusual behavior of a semi-flexible polymer in an active and crowded medium, associated with the nontrivial activity-crowding coupling and the cooperative crowder size effect.

Signatures of motor susceptibility to forces in the dynamics of a tracer particle in an active gel

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

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.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α<2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

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.

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.

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?

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.

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.

Dynamics in steady state in vitro acto-myosin networks

It is well known that many biochemical processes in the cell such as gene regulation, growth signals and activation of ion channels, rely on mechanical stimuli. However, the mechanism by which mechanical signals propagate through cells is not as well understood. In this review we focus on stress propagation in a minimal model for cell elasticity, actomyosin networks, which are comprised of a sub-family of cytoskeleton proteins. After giving an overview of th actomyosin network components, structure and evolution we review stress propagation in these materials as measured through the correlated motion of tracer beads. We also discuss the possibility to extract structural features of these networks from the same experiments. We show that stress transmission through these networks has two pathways, a quickly dissipative one through the bulk, and a long ranged weakly dissipative one through the pre-stressed actin network.

The effect of active fluctuations on the dynamics of particles, motors and DNA-hairpins

Inspired by recent experiments on the dynamics of particles and polymers in artificial cytoskeletons and in cells, we introduce a modified Langevin equation for a particle in an environment that is a viscoelastic medium and that is brought out of equilibrium by the action of active fluctuations caused by molecular motors. We show that within such a model, the motion of a free particle crosses over from superdiffusive to subdiffusive as observed for tracer particles in an in vitro cytoskeleton or in a cell. We investigate the dynamics of a particle confined by a harmonic potential as a simple model for the motion of the tethered head of kinesin-1. We find that the probability that the head is close to its binding site on the microtubule can be enhanced by a factor of two due to active forces. Finally, we study the dynamics of a particle in a double well potential as a model for the dynamics of DNA-hairpins. We show that the active forces effectively lower the potential barrier between the two minima and study the impact of this phenomenon on the zipping/unzipping rate.

Cytoplasmic RNA-Protein Particles Exhibit Non-Gaussian Subdiffusive Behavior

The cellular cytoplasm is a complex, heterogeneous environment (both spatially and temporally) that exhibits viscoelastic behavior. To further develop our quantitative insight into cellular transport, we analyze data sets of mRNA molecules fluorescently labeled with MS2-GFP tracked in real time in live Escherichia coli and Saccharomyces cerevisiae cells. As shown previously, these RNA-protein particles exhibit subdiffusive behavior that is viscoelastic in its origin. Examining the ensemble of particle displacements reveals a Laplace distribution at all observed timescales rather than the Gaussian distribution predicted by the central limit theorem. This ensemble non-Gaussian behavior is caused by a combination of an exponential distribution in the time-averaged diffusivities and non-Gaussian behavior of individual trajectories. We show that the non-Gaussian behavior is a consequence of significant heterogeneity between trajectories and dynamic heterogeneity along single trajectories. Informed by theory and simulation, our work provides an in-depth analysis of the complex diffusive behavior of RNA-protein particles in live cells.

Non-equilibrium cytoquake dynamics in cytoskeletal remodeling and stabilization

The cytoskeleton (CSK) is a tensed fiber framework that supports, shapes and stabilizes the cell. The CSK is in a constant state of remodeling, moreover, which is an active non-equilibrium thermodynamic process. We report here that cytoskeletal remodeling involves reconfigurations that are not only sudden but also are transmitted to great distances within the cell in a fashion reminiscent of quakes in the Earth's crust. Remarkably, these events in the cell conform both qualitatively and quantitatively to empirical laws typical of earthquakes, including hierarchical fault structures, cumulative energy distributions following the Gutenberg-Richter law, and rate of after-shocks following Omori's law. While it is well-established that remodeling and stabilization of the cytoskeleton are non-equilibrium process, these new unanticipated observations establish that these processes are also remarkably non-local and strongly cooperative.

Actomyosin dynamics drive local membrane component organization in an in vitro active composite layer

The surface of a living cell provides a platform for receptor signaling, protein sorting, transport, and endocytosis, whose regulation requires the local control of membrane organization. Previous work has revealed a role for dynamic actomyosin in membrane protein and lipid organization, suggesting that the cell surface behaves as an active composite composed of a fluid bilayer and a thin film of active actomyosin. We reconstitute an analogous system in vitro that consists of a fluid lipid bilayer coupled via membrane-associated actin-binding proteins to dynamic actin filaments and myosin motors. Upon complete consumption of ATP, this system settles into distinct phases of actin organization, namely bundled filaments, linked apolar asters, and a lattice of polar asters. These depend on actin concentration, filament length, and actin/myosin ratio. During formation of the polar aster phase, advection of the self-organizing actomyosin network drives transient clustering of actin-associated membrane components. Regeneration of ATP supports a constitutively remodeling actomyosin state, which in turn drives active fluctuations of coupled membrane components, resembling those observed at the cell surface. In a multicomponent membrane bilayer, this remodeling actomyosin layer contributes to changes in the extent and dynamics of phase-segregating domains. These results show how local membrane composition can be driven by active processes arising from actomyosin, highlighting the fundamental basis of the active composite model of the cell surface, and indicate its relevance to the study of membrane organization.

Non-periodic oscillatory deformation of an actomyosin microdroplet encapsulated within a lipid interface

Active force generation in living organisms, which is mainly involved in actin cytoskeleton and myosin molecular motors, plays a crucial role in various biological processes. Although the contractile properties of actomyosin have been extensively investigated, their dynamic contribution to a deformable membrane remains unclear because of the cellular complexities and the difficulties associated with in vitro reconstitution. Here, by overcoming these experimental difficulties, we demonstrate the dynamic deformation of a reconstituted lipid interface coupled with self-organized structure of contractile actomyosin. Therein, the lipid interface repeatedly oscillates without any remarkable periods. The oscillatory deformation of the interface is caused by the aster-like three-dimensional hierarchical structure of actomyosin inside the droplet, which is revealed that the oscillation occurs stochastically as a Poisson process.

Thermal breakage of a semiflexible polymer: breakage profile and rate

Understanding fluctuation-induced breakages in polymers has important implications for basic and applied sciences. Here I present for the first time an analytical treatment of the thermal breakage problem of a semi-flexible polymer model that is asymptotically exact in the low temperature and high friction limits. Specifically, I provide analytical expressions for the breakage propensity and rate, and discuss the generalities of the results and their relevance to biopolymers. This work is fundamental to our understanding of the kinetics of living polymerisation.

Inelastic mechanics: A unifying principle in biomechanics

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

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.

Mechanics and dynamics of reconstituted cytoskeletal systems

The intracellular cytoskeleton is an active dynamic network of filaments and associated binding proteins that control key cellular properties, such as cell shape and mechanics. Due to the inherent complexity of the cell, reconstituted model systems have been successfully employed to gain an understanding of the fundamental physics governing cytoskeletal processes. Here, we review recent advances and key aspects of these reconstituted systems. We focus on the importance of assembly kinetics and dynamic arrest in determining network mechanics, and highlight novel emergent behavior occurring through interactions between cytoskeletal components in more complex networks incorporating multiple biopolymers and molecular motors.

The role of filament length, finite-extensibility and motor force dispersity in stress relaxation and buckling mechanisms in non-sarcomeric active gels

After relaxing some assumptions we apply a single-chain mean-field mathematical model recently introduced [RSC Adv. (2014)] to describe the role of molecular motors in the mechanical properties of active gels. The model allows physics that are not available in models postulated on coarser levels of description. Moreover it proposes a level of description that allows the prediction of observables at time scales too difficult to achieve in multi-chain simulations for realistic filament lengths and densities. We model the semiflexible filaments that compose the active gel as bead-spring chains; molecular motors are accounted for by using a mean-field approach, in which filaments undergo transitions of one motor attachment state depending on the state of the probe filament. The level of description includes the end-to-end distance and attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman-Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces. We consider bead-spring chains with multiple beads, explore the effect of finite-extensibility of the strands and incorporate into the model motor force distributions that have been measured experimentally. The model can no longer be solved analytically but is amenable to numerical simulation. This version of the model allows a more quantitative description of buckling dynamics [Lenz et. al. PRL, 2012, 108, 238107] and the dynamic modulus of active gels. The effect of finite extensibility of the filament strands on the dynamic modulus was also found to be in agreement with the microrheology experiments of Mizuno et. al., [Science, 2007, 315, 370-373].

Energetics of active fluctuations in living cells

The nonequilibrium activity taking place in a living cell can be monitored with a tracer embedded in the medium. While microrheology experiments based on optical manipulation of such probes have become increasingly standard, we put forward a number of experiments with alternative protocols that, we claim, will provide insight into the energetics of active fluctuations. These are based on either performing thermodynamiclike cycles in control-parameter space or determining response to external perturbations of the confining trap beyond simple translation. We illustrate our proposals on an active itinerant Brownian oscillator modeling the dynamics of a probe embedded in a living medium.

Active transport of vesicles in neurons is modulated by mechanical tension

Effective intracellular transport of proteins and organelles is critical in cells, and is especially important for ensuring proper neuron functionality. In neurons, most proteins are synthesized in the cell body and must be transported through thin structures over long distances where normal diffusion is insufficient. Neurons transport subcellular cargo along axons and neurites through a stochastic interplay of active and passive transport. Mechanical tension is critical in maintaining proper function in neurons, but its role in transport is not well understood. To this end, we investigate the active and passive transport of vesicles in Aplysia neurons while changing neurite tension via applied strain, and quantify the resulting dynamics. We found that tension in neurons modulates active transport of vesicles by increasing the probability of active motion, effective diffusivity, and induces a retrograde bias. We show that mechanical tension modulates active transport processes in neurons and that external forces can couple to internal (subcellular) forces and change the overall transport dynamics.

Impact of photosensitizers activation on intracellular trafficking and viscosity

The intracellular microenvironment is essential for the efficiency of photo-induced therapies, as short-lived reactive oxygen species generated must diffuse through their intracellular surrounding medium to reach their cellular target. Here, by combining measurements of local cytoplasmic dissipation and active trafficking, we found that photosensitizers activation induced small changes in surrounding viscosity but a massive decrease in diffusion. These effects are the signature of a return to thermodynamic equilibrium of the system after photo-activation and correlated with depolymerization of the microtubule network, as shown in a reconstituted system. These mechanical measurements were performed with two intracellular photosensitizing chlorins having similar quantum yield of singlet oxygen production but different intracellular localizations (cytoplasmic for mTHPC, endosomal for TPCS2a). These two agents demonstrated different intracellular impact.

Revealing hidden dynamics within living soft matter

In the study of living soft matter, we often seek to understand the mechanisms underlying the motion of a single molecule, an organelle, or some other tracer. The experimentally observed signature of the tracer is masked by its thermal fluctuations, inherent drift of the system, and instrument noise. In addition, the timing or length scales of the events of interest are often unknown. In the current issue of ACS Nano, Chen et al. present a general method for extracting the underlying dynamics from time series. Here, we provide an easily accessible introduction to the method, put it into perspective with the field, and exemplify how it can be used to answer important out-standing questions within soft matter and living systems.