Extracting the dynamic correlation length of actin networks from microrheology experiments

One- and two-particle microrheology of soft materials based on optical-flow image analysis

Particle-tracking microrheology probes the rheology of soft materials by accurately tracking an ensemble of embedded colloidal tracer particles. One-particle analysis, which focuses on the trajectory of individual tracers is ideal for homogeneous materials that do not interact with the particles. By contrast, the characterization of heterogeneous, micro-structured materials or those where particles interact directly with the medium requires a two-particle analysis that characterizes correlations between the trajectories of distinct particle pairs. Here, we propose an optical-flow image analysis as an alternative to the tracking-based algorithms to extract one and two-particle microrheology information from video microscopy images acquired using diverse imaging contrast modalities. This technique, termed optical-flow microrheology (OFM), represents a high-throughput, operator-free approach for the characterization of a broad range of soft materials, making microrheology accessible to a wider scientific community.

Micro-mechanical response and power-law exponents from the longitudinal fluctuations of F-actin solutions

We investigate the local fluctuations of filamentous actin (F-actin), with a focus on the skeletal thin filament, using single-particle optical trapping interferometry. This experimental technique allows us to detect the Brownian motion of a tracer bead immersed in a complex fluid with nanometric resolution at the microsecond time-scale. The mean square displacement, loss modulus, and velocity autocorrelation function (VAF) of the trapped microprobes in the fluid follow power-law behaviors, whose exponents can be determined in the short-time/high-frequency regime over several decades. We obtain 7/8 subdiffusive power-law exponents for polystyrene depleted microtracers at low optical trapping forces. Microrheologically, the elastic modulus of these suspensions is observed to be constant up to the limit of high frequencies, confirming that the origin of this subdiffusive exponent is the local longitudinal fluctuations of the polymers. Deviations from this value are measured and discussed in relation to the characteristic length scales of these F-actin networks and probes' properties, and also in connection with the different power-law exponents detected in the VAFs. Finally, we observed that the thin filament, composed of tropomyosin (Tm) and troponin (Tn) coupled to F-actin in the presence of Ca2+, shows exponent values less dispersed than that of F-actin alone, which we interpret as a micro-measurement of the filament stabilization.

Probing Local Force Propagation in Tensed Fibrous Gels

Fibrous hydrogels are a key component of soft animal tissues. They support cellular functions and facilitate efficient mechanical communication between cells. Due to their nonlinear mechanical properties, fibrous materials display non-trivial force propagation at the microscale, that is enhanced compared to that of linear-elastic materials. In the body, tissues are constantly subjected to external loads that tense or compress them, modifying their micro-mechanical properties into an anisotropic state. However, it is unknown how force propagation is modified by this isotropic-to-anisotropic transition. Here, force propagation in tensed fibrin hydrogels is directly measured. Local perturbations are induced by oscillating microspheres using optical tweezers. 1-point and 2-point microrheology are combined to simultaneously measure the shear modulus and force propagation. A mathematical framework to quantify anisotropic force propagation trends is suggested. Results show that force propagation becomes anisotropic in tensed gels, with, surprisingly, stronger response to perturbations perpendicular to the axis of tension. Importantly, external tension can also increase the range of force transmission. Possible implications and future directions for research are discussed. These results suggest a mechanism for favored directions of mechanical communication between cells in a tissue under external loads.

Motor-Driven Restructuring of Cytoskeleton Composites Leads to Tunable Time-Varying Elasticity

The composite cytoskeleton, comprising interacting networks of semiflexible actin and rigid microtubules, generates forces and restructures by using motor proteins such as myosins to enable key processes including cell motility and mitosis. Yet, how motor-driven activity alters the mechanics of cytoskeleton composites remains an open challenge. Here, we perform optical tweezers microrheology and confocal imaging of composites with varying actin-tubulin molar percentages (25-75, 50-50, and 75-25), driven by light-activated myosin II motors, to show that motor activity increases the elastic plateau modulus by over 2 orders of magnitude by active restructuring of both actin and microtubules that persists for hours after motor activation has ceased. Nonlinear microrheology measurements show that motor-driven restructuring increases the force response and stiffness and suppresses actin bending. The 50-50 composite exhibits the most dramatic mechanical response to motor activity due to the synergistic effects of added stiffness from the microtubules and sufficient motor substrate for pronounced activity.

Measurements and characterization of the dynamics of tracer particles in an actin network

The underlying physics governing the diffusion of a tracer particle in a viscoelastic material is a topic of some dispute. The long-term memory in the mechanical response of such materials should induce diffusive motion with a memory kernel, such as fractional Brownian motion (fBM). This is the reason that microrheology is able to provide the shear modulus of polymer networks. Surprisingly, the diffusion of a tracer particle in a network of a purified protein, actin, was found to conform to the continuous time random walk type (CTRW). We set out to resolve this discrepancy by studying the tracer particle diffusion using two different tracer particle sizes, in actin networks of different mesh sizes. We find that the ratio of tracer particle size to the characteristic length scale of a bio-polymer network plays a crucial role in determining the type of diffusion it performs. We find that the diffusion of the tracer particles has features of fBm when the particle is large compared to the mesh size, of normal diffusion when the particle is much smaller than the mesh size, and of the CTRW in between these two limits. Based on our findings, we propose and verify numerically a new model for the motion of the tracer in all regimes. Our model suggests that diffusion in actin networks consists of fBm of the tracer particle coupled with caging events with power-law distributed escape times.

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.

Surface Response of a Polymer Network: Semi-infinite Network

We study theoretically the surface response of a semi-infinite viscoelastic polymer network using the two-fluid model. We focus on the overdamped limit and on the effect of the network's intrinsic length scales. We calculate the decay rate of slow surface fluctuations, and the surface displacement in response to a localized force. Deviations from the large-scale continuum response are found at length scales much larger than the network's mesh size. We discuss implications for surface scattering and microrheology. We provide closed-form expressions that can be used for surface microrheology: the extraction of viscoelastic moduli and intrinsic length scales from the motions of tracer particles lying on the surface without doping the bulk material.

Single-Particle Diffusion Characterization by Deep Learning

Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle-tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools for distinguishing between these processes are based on asymptotic behavior, which is experimentally inaccessible in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables because different transport modes can result in the same diffusion power-law α, which is typically obtained from the mean-square displacements (MSDs). The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the nonexpert level. Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single-particle trajectories by diffusion type: Brownian motion, fractional Brownian motion and continuous time random walk. Further, we demonstrate the applicability of our network architecture for estimating the Hurst exponent for fractional Brownian motion and the diffusion coefficient for Brownian motion on both simulated and experimental data. These networks achieve greater accuracy than time-averaged MSD analysis on simulated trajectories while only requiring as few as 25 steps. When tested on experimental data, both net and ensemble MSD analysis converge to similar values; however, the net needs only half the number of trajectories required for ensemble MSD to achieve the same confidence interval. Finally, we extract diffusion parameters from multiple extremely short trajectories (10 steps) using our approach.

Flow Arrest in the Plasma Membrane

The arrangement of receptors in the plasma membrane strongly affects the ability of a cell to sense its environment both in terms of sensitivity and in terms of spatial resolution. The spatial and temporal arrangement of the receptors is affected in turn by the mechanical properties and the structure of the cell membrane. Here, we focus on characterizing the flow of the membrane in response to the motion of a protein embedded in it. We do so by measuring the correlated diffusion of extracellularly tagged transmembrane neurotrophin receptors TrkB and p75 on transfected neuronal cells. In accord with previous reports, we find that the motion of single receptors exhibits transient confinement to submicron domains. We confirm predictions based on hydrodynamics of fluid membranes, finding long-range correlations in the motion of the receptors in the plasma membrane. However, we discover that these correlations do not persist for long ranges, as predicted, but decay exponentially, with a typical decay length on the scale of the average confining domain size.

On the calculation of the potential of mean force between atomistic nanoparticles

We study the potential of mean force (PMF) between atomistic silica and gold nanoparticles in the vacuum by using molecular dynamics simulations. Such an investigation is devised in order to fully characterize the effective interactions between atomistic nanoparticles, a crucial step to describe the PMF in high-density coarse-grained polymer nanocomposites. In our study, we first investigate the behavior of silica nanoparticles, considering cases corresponding to different particle sizes and assessing results against an analytic theory developed by Hamaker for a system of Lennard-Jones interacting particles (H.C. Hamaker, Physica A 4, 1058 (1937)). Once validated the procedure, we calculate effective interactions between gold nanoparticles, which are considered both bare and coated with polyethylene chains, in order to investigate the effects of the grafting density [Formula: see text] on the PMF. Upon performing atomistic molecular dynamics simulations, it turns out that silica nanoparticles experience similar interactions regardless of the particle size, the most remarkable difference being a peak in the PMF due to surface interactions, clearly apparent for the larger size. As for bare gold nanoparticles, they are slightly interacting, the strength of the effective force increasing for the coated cases. The profile of the resulting PMF resembles a Lennard-Jones potential for intermediate [Formula: see text], becoming progressively more repulsive for high [Formula: see text] and low interparticle separations.

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.

Fast Diffusion of Long Guest Rods in a Lamellar Phase of Short Host Particles

We investigate the dynamic behavior of long guest rodlike particles immersed in liquid crystalline phases formed by shorter host rods, tracking both guest and host particles by fluorescence microscopy. Counterintuitively, we evidence that long rods diffuse faster than short rods forming the one-dimensional ordered smectic-A phase. This results from the larger and noncommensurate size of the guest particles as compared to the wavelength of the energy landscape set by the lamellar stack of liquid slabs. The long guest particles are also shown to be still mobile in the crystalline smectic-B phase, as they generate their own voids in the adjacent layers.

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.

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.

Relevance of interfacial viscoelasticity in stability and conformation of biomolecular organizates at air/fluid interface

Soft materials are complex macromolecular systems often exhibiting perplexing non-Newtonian viscoelastic properties, especially when the macromolecules are entangled, crowded or cross-linked. These materials are ubiquitous in the biology, food and pharma industry and have several applications in biotechnology and in the field of biosensors. Based on the length scales, topologies, flexibility and concentration, the systems behave both as liquids (viscous) and solids (elastic). Particularly, for proteins and protein-lipid systems, viscoelasticity is an important parameter because it often relates directly to stability and thermodynamic interactions of the pure biological components as well as their mixtures. Despite the large body of work that is available in solution macro-rheometry, there are still a number of issues that need to be addressed in dealing with proteins at air/fluid interfaces and with protein-polymer or protein-lipid interfaces that often exhibit very low interfacial viscosity values. Considering the important applications that they have in biopharmaceutical, biotechnological and nutraceutical industries, there is a need for developing methods that meet the following three specific issues: small volume, large dynamic range of shear rates and interfacial properties of different biomolecules. Further, the techniques that are developed should include Newtonian, shear thinning and yielding properties, which are representative of the different solution behaviors typically encountered. The review presented here is a comprehensive account of the rheological properties of different biomolecules at air/fluid and solid/fluid interfaces. It addresses the usefulness of 'viscoelasticity' of the systems at the interfaces analyzed at the molecular level that can be correlated with the microscopic material properties and touches upon some recent techniques in microrheology that are being used to measure the unusually low viscosity values sensitively.

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.

Actin kinetics shapes cortical network structure and mechanics

The actin cortex of animal cells is the main determinant of cellular mechanics. The continuous turnover of cortical actin filaments enables cells to quickly respond to stimuli. Recent work has shown that most of the cortical actin is generated by only two actin nucleators, the Arp2/3 complex and the formin Diaph1. However, our understanding of their interplay, their kinetics, and the length distribution of the filaments that they nucleate within living cells is poor. Such knowledge is necessary for a thorough comprehension of cellular processes and cell mechanics from basic polymer physics principles. We determined cortical assembly rates in living cells by using single-molecule fluorescence imaging in combination with stochastic simulations. We find that formin-nucleated filaments are, on average, 10 times longer than Arp2/3-nucleated filaments. Although formin-generated filaments represent less than 10% of all actin filaments, mechanical measurements indicate that they are important determinants of cortical elasticity. Tuning the activity of actin nucleators to alter filament length distribution may thus be a mechanism allowing cells to adjust their macroscopic mechanical properties to their physiological needs.

Tracking of colloids close to contact

The precise tracking of micron sized colloidal particles - held in the vicinity of each other using optical tweezers - is an elegant way to gain information about the particle-particle pair interaction potential. The accuracy of the method, however, relies strongly on the tracking precision. Particularly the elimination of systematic errors in the position detection due to overlapping particle diffraction patterns remains a great challenge. Here we propose a template based particle finding algorithm that circumvents these problems by tracking only a fraction of the particle image that is insignificantly affected by nearby colloids. Under realistic experimental conditions we show that our algorithm significantly reduces systematic errors compared to standard tracking methods. Moreover our approach should in principle be applicable to almost arbitrary shaped particles as the template can be adapted to any geometry.

Response of a polymer network to the motion of a rigid sphere

In view of recent microrheology experiments we re-examine the problem of a rigid sphere oscillating inside a dilute polymer network. The network and its solvent are treated using the two-fluid model. We show that the dynamics of the medium can be decomposed into two independent incompressible flows. The first, dominant at large distances and obeying the Stokes equation, corresponds to the collective flow of the two components as a whole. The other, governing the dynamics over an intermediate range of distances and following the Brinkman equation, describes the flow of the network and solvent relative to one another. The crossover between these two regions occurs at a dynamic length scale which is much larger than the network's mesh size. The analysis focuses on the spatial structure of the medium's response and the role played by the dynamic crossover length. We examine different boundary conditions at the sphere surface. The large-distance collective flow is shown to be independent of boundary conditions and network compressibility, establishing the robustness of two-point microrheology at large separations. The boundary conditions that fit the experimental results for inert spheres in entangled F-actin networks are those of a free network, which does not interact directly with the sphere. Closed-form expressions and scaling relations are derived, allowing for the extraction of material parameters from a combination of one- and two-point microrheology. We discuss a basic deficiency of the two-fluid model and a way to bypass it when analyzing microrheological data.