Treating inertia in passive microbead rheology

Nonequilibrium dynamical structure factor of a dilute suspension of active particles in a viscoelastic fluid

In this work we investigate the dynamics of the number-density fluctuations of a dilute suspension of active particles in a linear viscoelastic fluid. We propose a model for the frequency-dependent diffusion coefficient of the active particles which captures the effect of rotational diffusion on the persistence of their self-propelled motion and the viscoelasticity of the medium. Using fluctuating hydrodynamics, the linearized equations for the active suspension are derived, from which we calculate its dynamic structure factor and the corresponding intermediate scattering function. For a Maxwell-type rheological model, we find an intricate dependence of these functions on the parameters that characterize the viscoelasticity of the solvent and the activity of the particles, which can significantly deviate from those of an inert suspension of passive particles and of an active suspension in a Newtonian solvent. In particular, in some regions of the parameter space we uncover the emergence of oscillations in the intermediate scattering function at certain wave numbers which represent the hallmark of the nonequilibrium particle activity in the dynamical structure of the suspension and also encode the viscoelastic properties of the medium.

From generalized Langevin stochastic dynamics to anomalous diffusion

Scaling methods are fundamental in all branches of physics. In stochastic process, we usually try to describe the long time behavior of a given time correlation function. In this work we investigate a scaling method for anomalous diffusion in systems with memory that produces good results for long and intermediate times. We will initially present a generalization of the diffusion exponent. Then, we present an asymptotic method to obtain an analytical expression for the diffusion coefficient by introducing a time scale factor λ(t). We found an exact expression for the function λ(t), which allows us to describe the diffusive process. For large times, λ(t) becomes a universal parameter determined by the diffusion exponent. In turn, the analytical results are then compared to the numerical results, with a good matching. Then, we'll show the practical effects of scaling. An important first result is that λ(t) quickly converges to a constant. Another very important point was the classification of new forms of diffusion due to the generalized exponent. In previous works, we verified the existence of ergodic ballistic diffusion with diffusion exponent α=2^{-}. Here, we verify the existence of the nonergodic ballistic diffusion type with the obtainment of the diffusion coefficient α=2. Finally, we show that the scaling works. This method is general and can be applied to various types of stochastic problems.

Passive and Active Microrheology for Biomedical Systems

Microrheology encompasses a range of methods to measure the mechanical properties of soft materials. By characterizing the motion of embedded microscopic particles, microrheology extends the probing length scale and frequency range of conventional bulk rheology. Microrheology can be characterized into either passive or active methods based on the driving force exerted on probe particles. Tracer particles are driven by thermal energy in passive methods, applying minimal deformation to the assessed medium. In active techniques, particles are manipulated by an external force, most commonly produced through optical and magnetic fields. Small-scale rheology holds significant advantages over conventional bulk rheology, such as eliminating the need for large sample sizes, the ability to probe fragile materials non-destructively, and a wider probing frequency range. More importantly, some microrheological techniques can obtain spatiotemporal information of local microenvironments and accurately describe the heterogeneity of structurally complex fluids. Recently, there has been significant growth in using these minimally invasive techniques to investigate a wide range of biomedical systems both in vitro and in vivo. Here, we review the latest applications and advancements of microrheology in mammalian cells, tissues, and biofluids and discuss the current challenges and potential future advances on the horizon.

Active Brownian motion with memory delay induced by a viscoelastic medium

By now active Brownian motion is a well-established model to describe the motion of mesoscopic self-propelled particles in a Newtonian fluid. On the basis of the generalized Langevin equation, we present an analytic framework for active Brownian motion with memory delay assuming time-dependent friction kernels for both translational and orientational degrees of freedom to account for the time-delayed response of a viscoelastic medium. Analytical results are obtained for the orientational correlation function, mean displacement, and mean-square displacement which we evaluate in particular for a Maxwell fluid characterized by a kernel which decays exponentially in time. Further, we identify a memory-induced delay between the effective self-propulsion force and the particle orientation which we quantify in terms of a special dynamical correlation function. In principle, our predictions can be verified for an active colloidal particle in various viscoelastic environments such as a polymer solution.

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.

Negative friction memory induces persistent motion

We investigate the mean-square displacement (MSD) for random motion governed by the generalized Langevin equation for memory functions that contain two different time scales: In the first model, the memory kernel consists of a delta peak and a single-exponential and in the second model of the sum of two exponentials. In particular, we investigate the scenario where the long-time exponential kernel contribution is negative. The competition between positive and negative friction memory contributions produces an enhanced transient persistent regime in the MSD, which is relevant for biological motility and active matter systems.

Monitoring matrix remodeling in the cellular microenvironment using microrheology for complex cellular systems

Multiple particle tracking (MPT) microrheology was employed for monitoring the development of extracellular matrix (ECM) mechanical properties in the direct microenvironment of living cells. A customized setup enabled us to overcome current limitations: (i) Continuous measurements were enabled using a cell culture chamber, with this, matrix remodeling by fibroblasts in the heterogeneous environment of macroporous scaffolds was monitored continuously. (ii) Employing tracer laden porous scaffolds for seeding human mesenchymal stem cells (hMSCs), we followed conventional differentiation protocols. Thus, we were, for the first time able to study the massive alterations in ECM elasticity during hMSC differentiation. (iii) MPT measurements in 2D cell cultures were enabled using a long distance objective. Exemplarily, local mechanical properties of the ECM in human umbilical vein endothelial cell (HUVEC) cultures, that naturally form 2D layers, were investigated scaffold-free. Using our advanced setup, we measured local, apparent elastic moduli G_0,app in a range between 0.08 and 60 Pa. For fibroblasts grown in collagen-based scaffolds, a continuous decrease of local matrix elasticity resulted during the first 10 hours after seeding. The osteogenic differentiation of hMSC cells cultivated in similar scaffolds, led to an increase of G_0,app by 100 %, whereas after adipogenic differentiation it was reduced by 80 %. The local elasticity of ECM that was newly secreted by HUVECs increased significantly upon addition of protease inhibitor and in high glucose conditions even a twofold increase in G_0,app was observed. The combination of these advanced methods opens up new avenues for a broad range of investigations regarding cell-matrix interactions and the propagation of ECM mechanical properties in complex biological systems. STATEMENT OF SIGNIFICANCE: Cells sense the elasticity of their environment on a micrometer length scale. For studying the local elasticity of extracellular matrix (ECM) in the direct environment of living cells, we employed an advanced multipleparticle tracking microrheology setup. MPT is based on monitoring the Brownian motion oftracer particles, which is restricted by the surrounding network. Network elasticity can thusbe quantified. Overcoming current limitations, we realized continuous investigations of ECM elasticityduring fibroblast growth. Furthermore, MPT measurements of stem cell ECM showed ECMstiffening during osteogenic differentiation and softening during adipogenic differentiation.Finally, we characterized small amounts of delicate ECM newly secreted in scaffold-freecultures of endothelial cells, that naturally form 2D layers.

Microstructure, local viscoelasticity and cell culture suitability of 3D hybrid HA/collagen scaffolds

As mechanical properties of cell culture substrates matter, methods for mechanical characterization of scaffolds on a relevant length scale are required. We used multiple particle tracking microrheology to close the gap between elasticity determined from bulk measurements and elastic properties sensed by cells. Structure and elasticity of macroporous, three-dimensional cryogel scaffolds from mixtures of hyaluronic acid (HA) and collagen (Coll) were characterized. Both one-component gels formed homogeneous networks, whereas hybrid gels were heterogeneous in terms of elasticity. Most strikingly, local elastic moduli were significantly lower than bulk moduli presumably due to non-equilibrium chain conformations between crosslinks. This was more pronounced in Coll and hybrid gels than in pure HA gels. Local elastic moduli were similar for all gels, irrespective of their different swelling ratio and bulk moduli. Fibroblast cell culture proved the biocompatibility of all investigated compositions. Coll containing gels enabled cell migration, adhesion and proliferation inside the gels.

Microrheology, advances in methods and insights

Microrheology is an emerging technique that probes mechanical response of soft material at micro-scale. Generally, microrheology technique can be divided into active and passive versions. During last two decades, extensive efforts have been paid to improve both the experiment techniques and data analysis methods, especially about how to link consequential particle positions into trajectories. We review the recent advances in microrheology, including improvements in labeling, imaging, data acquiring, data processing and data interpretation. Some of the recent insights in soft matter and living systems gained by using this technique are given. Before these, we also give a very brief description of the basic principles of both active and passive microrheology techniques, and some details about optical particle tracking and DWS.

Nanorheology of Entangled Polymer Melts

We use molecular simulations to probe the local viscoelasticity of an entangled polymer melt by tracking the motion of embedded nonsticky nanoparticles (NPs). As in conventional microrheology, the generalized Stokes-Einstein relation is employed to extract an effective stress relaxation function G_{GSE}(t) from the mean square displacement of NPs. G_{GSE}(t) for different NP diameters d are compared with the stress relaxation function G(t) of a pure polymer melt. The deviation of G_{GSE}(t) from G(t) reflects the incomplete coupling between NPs and the dynamic modes of the melt. For linear polymers, a plateau in G_{GSE}(t) emerges as d exceeds the entanglement mesh size a and approaches the entanglement plateau in G(t) for a pure melt with increasing d. For ring polymers, as d increases towards the spanning size R of ring polymers, G_{GSE}(t) approaches G(t) of the ring melt with no entanglement plateau.

Physical Modeling of Dynamic Coupling between Chromosomal Loci

The motion of chromosomal DNA is essential to many biological processes, including segregation, transcriptional regulation, recombination, and packaging. Physical understanding of these processes would be dramatically enhanced through predictive, quantitative modeling of chromosome dynamics of multiple loci. Using a polymer dynamics framework, we develop a prediction for the correlation in the velocities of two loci on a single chromosome or otherwise connected by chromatin. These predictions reveal that the signature of correlated motion between two loci can be identified by varying the lag time between locus position measurements. In general, this theory predicts that as the lag time interval increases, the dual-loci dynamic behavior transitions from being completely uncorrelated to behaving as an effective single locus. This transition corresponds to the timescale of the stress communication between loci through the intervening segment. This relatively simple framework makes quantitative predictions based on a single timescale fit parameter that can be directly compared to the in vivo motion of fluorescently labeled chromosome loci. Furthermore, this theoretical framework enables the detection of dynamically coupled chromosome regions from the signature of their correlated motion.

Determination of linear viscoelastic properties of an entangled polymer melt by probe rheology simulations

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)PLEEE81539-375510.1103/PhysRevE.86.051501], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length d_{T}, in addition to those length scales already relevant: monomer bead size d, probe size R, polymer radius of gyration R_{g}, simulation box size L, shear wave penetration length Δ, and wave period Λ. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when d≳Λ; medium inertia is important and IGSER is required when R≳Λ; and the probe should not experience hydrodynamic interaction with its periodic images, L≳Δ. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as M_{w}^{3}. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness L_{S} of the shell around the particle that contains the added mass, L_{S}>d. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

Active one-particle microrheology of an unentangled polymer melt studied by molecular dynamics simulation

We present molecular dynamics simulations for active one-particle microrheology of an unentangled polymer melt. The tracer particle is forced to oscillate by an oscillating harmonic potential, which models an experiment using optical tweezers. The amplitude and phase shift of this oscillation are related to the complex shear modulus of the polymer melt. In the linear response regime at low frequencies, the active microrheology gives the same result as the passive microrheology, where the thermal motion of a tracer particle is related to the complex modulus. We expand the analysis to include full hydrodynamic effects instead of stationary Stokes friction only, and show that different approaches suggested in the literature lead to completely different results, and that none of them improves on the description using the stationary Stokes friction.

Trajectories of probe spheres in generalized linear viscoelastic complex fluids

We have developed a fast simulation that generates a random walk of an isolated probe sphere in a generalized linear viscoelastic complex fluid over a highly extended dynamic range. We introduce a coupled harmonically bound Brownian particle (c-HBBP) model, in which the relaxation modes of the viscoelastic medium are treated as harmonic wells. These wells are coupled to the probe sphere and perform Brownian motion in bound harmonic potentials corresponding to the next-longer relaxation mode, according to the relaxation spectrum of the viscoelastic material. We implement this c-HBBP model by generating variable temporal step sizes that have a uniform distribution in logarithmic time. We create and analyze trajectories for several different viscoelastic complex fluids: a polymer system at its gel point, a dense emulsion system, a blend of two monodisperse polystyrene polymers for which the relaxation spectrum has been measured, and a model anisotropic soft system that shows dense emulsion-like and gel-point behaviors along two orthogonal directions. Except for unusual viscoelastic materials, such as the polymer system at its gel point, the generated trajectories are neither self-similar nor self-affine. The resulting mean square displacements predicted by the c-HBBP model are consistent with the single-particle generalized Stokes-Einstein relation of linear passive microrheology.

Passive one-particle microrheology of an unentangled polymer melt studied by molecular dynamics simulation

We present a molecular dynamics simulation study of the possibility of performing a microrheological analysis of a polymer melt by following the Brownian motion of a dispersed nanoparticle. We study the influence of the size of the nanoparticle, taken to be comparable to the radius of gyration of the chains, and of the strength of the interaction between the nanoparticle and the repeat units of the polymer chains. The influence of the presence of the nanoparticle on the melt mechanical behavior is analyzed, and the importance of effects of different levels of hydrodynamic analysis on the frequency-dependent dynamic shear modulus derived from the particle motion is worked out.

Microrheology of colloidal systems

Microrheology was proposed almost twenty years ago as a technique to obtain rheological properties in soft matter from the microscopic motion of colloidal tracers used as probes, either freely diffusing in the host medium, or subjected to external forces. The former case is known as passive microrheology, and is based on generalizations of the Stokes-Einstein relation between the friction experienced by the probe and the host-fluid viscosity. The latter is termed active microrheology, and extends the measurement of the friction coefficient to the nonlinear-response regime of strongly driven probes. In this review article, we discuss theoretical models available in the literature for both passive and active microrheology, focusing on the case of single-probe motion in model colloidal host media. A brief overview of the theory of passive microrheology is given, starting from the work of Mason and Weitz. Further developments include refined models of the host suspension beyond that of a Newtonian-fluid continuum, and the investigation of probe-size effects. Active microrheology is described starting from microscopic equations of motion for the whole system including both the host-fluid particles and the tracer; the many-body Smoluchowski equation for the case of colloidal suspensions. At low fluid densities, this can be simplified to a two-particle equation that allows the calculation of the friction coefficient with the input of the density distribution around the tracer, as shown by Brady and coworkers. The results need to be upscaled to agree with simulations at moderate density, in both the case of pulling the tracer with a constant force or dragging it at a constant velocity. The full many-particle equation has been tackled by Fuchs and coworkers, using a mode-coupling approximation and the scheme of integration through transients, valid at high densities. A localization transition is predicted for a probe embedded in a glass-forming host suspension. The nonlinear probe-friction coefficient is calculated from the tracer's position correlation function. Computer simulations show qualitative agreement with the theory, but also some unexpected features, such as superdiffusive motion of the probe related to the breaking of nearest-neighbor cages. We conclude with some perspectives and future directions of theoretical models of microrheology.

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: subdiffusion of heavy tracers

We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.

Determination of viscoelastic properties by analysis of probe-particle motion in molecular simulations

We present a technique for the determination of viscoelastic properties of a medium by tracking the motion of an embedded probe particle by using molecular dynamics simulations. The approach involves the analysis of the simulated particle motion by continuum theory; it is shown to work in both passive and active modes. We demonstrate that, for passive rheology, an analysis based on the generalized Stokes-Einstein relationship is not adequate to obtain the values of the viscoelastic moduli over the frequency range studied. For both passive and active modes, it is necessary to account for the medium and particle inertia when analyzing the particle motion. For a polymer melt system consisting of short chains, the values calculated from the proposed approach are in good quantitative agreement with previous literature results that were obtained using completely different simulation approaches. The proposed particle rheology simulation technique is general and could provide insight into the characterization of the mechanical properties in biological systems, such as cellular environments and polymeric systems, such as thin films and nanocomposites that exhibit spatial variation in properties over the nanoscale.

Competing effects of particle and medium inertia on particle diffusion in viscoelastic materials, and their ramifications for passive microrheology

We analyze the appropriate form for the generalized Stokes-Einstein relation (GSER) for viscoelastic solids and fluids when bead inertia and medium inertia are taken into account, which we call the inertial GSER. It was previously shown for Maxwell fluids that the Basset (or Boussinesq) force arising from medium inertia can act purely dissipatively at high frequencies, where elasticity of the medium is dominant. In order to elucidate the cause of this counterintuitive result, we consider Brownian motion in a purely elastic solid where ordinary Stokes-type dissipation is not possible. The fluctuation-dissipation theorem requires the presence of a dissipative mechanism for the particle to experience fluctuating Brownian forces in a purely elastic solid. We show that the mechanism for such dissipation arises from the radiation of elastic waves toward the system boundaries. The frictional force associated with this mechanism is the Basset force, and it exists only when medium inertia is taken into consideration in the analysis of such a system. We consider first a one-dimensional harmonic lattice where all terms in the generalized Langevin equation--i.e., the elastic term, the memory kernel, and Brownian forces-can be found analytically from projection-operator methods. We show that the dissipation is purely from radiation of elastic waves. A similar analysis is made on a particle in a continuum, three-dimensional purely elastic solid, where the memory kernel is determined from continuum mechanics. Again, dissipation arises only from radiation of elastic shear waves toward infinite boundaries when medium inertia is taken into account. If the medium is a viscoelastic solid, Stokes-type dissipation is possible in addition to radiational dissipation so that the wave decays at the penetration depth. Inertial motion of the bead couples with the elasticity of the viscoelastic material, resulting in a possible resonant oscillation of the mean-square displacement (MSD) of the bead. On the other hand, medium inertia (the Basset force) tends to attenuate the oscillations by the dissipation mechanism described above. Thus competition between bead inertia and medium inertia determines whether or not the MSD oscillates. We find that, if the medium density is larger than 4/7 of the bead density, the Basset damping will suppress oscillations in the MSD; this criterion is sufficient but not necessary to present oscillations.