Ensemble-averaged Mean Square Displacement Research Articles

Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.

Tracking of fluorescently labeled chromosomal loci in live bacterial cells reveals a robust scaling of the mean square displacement (MSD) as τ(0.39). We propose that the observed motion arises from relaxation of the Rouse modes of the DNA polymer within the viscoelastic environment of the cytoplasm. The time-averaged and ensemble-averaged MSD of chromosomal loci exhibit ergodicity, and the velocity autocorrelation function is negative at short time lags. These observations are most consistent with fractional Langevin motion and rule out a continuous time random walk model as an explanation for anomalous motion in vivo.

Micro-heterogeneity of cellulosic fiber biopolymer prepared from corn hulls

Z-trim is a zero calorie cellulosic fiber biopolymer produced from corn hulls. The micro-structural heterogeneities of Z-trim biopolymer were investigated by monitoring the thermally driven displacements of well-dispersed micro-spheres via video fluorescence microscopy named multiple-particle tracking (MPT). By comparing the distribution of the time-dependent mean-square displacement (MSD) of polystyrene micro-spheres embedded in three concentrations of Z-trim, we found that the degree of heterogeneity of the suspensions increased dramatically within a narrow range of concentrations. The ensemble-averaged MSD of 0.5 g/100 g Z-trim biopolymer exhibited a power-law behavior that scaled log-linearly with time lag. This behavior was similar to that of homogeneous aqueous glycerol solution. But the MSD distribution for Z-trim was wider and more asymmetric than for glycerol. Increasing Z-trim concentration caused more deviation of micro-spheres colloidal dispersion from homogeneity as detected by the MPT.

Chromosomal Loci Move Subdiffusively Through a Viscoelastic Cytoplasm

Tracking of fluorescently labeled chromosomal loci in live bacterial cells reveals a robust scaling of the mean square displacement (MSD) as τ0.39. We use Brownian dynamics simulations to show that this anomalous behavior cannot be fully accounted for by the classic Rouse or reptation models for polymer dynamics. Instead, the motion seems to arise from the interaction of the Rouse modes of the DNA polymer with the viscoelastic environment of the cytoplasm. To demonstrate these physical effects, we present a general analytical derivation of the subdiffusive scaling for a monomer in a polymer within a viscoelastic medium. The time-averaged and ensemble-averaged MSD of chromosomal loci exhibit ergodicity, and the velocity autocorrelation function is negative at short time lags. These observations are most consistent with fractional Brownian motion and rule out a continuous time random walk model.

Mechanical changes of living oocytes at maturation investigated by multiple particle tracking

We have studied the mechanical properties of the cytoplasm of starfish oocytes before and after maturation using the multiple particle tracking technique. Fluorescent microbeads are microinjected in oocytes of Astropecten aranciacus starfishes. Mechanical properties are obtained analyzing the mean-square displacement (MSD) of the beads trajectories. The ensemble-averaged MSD provides information about diffusion coefficient D, and the cytoplasm viscosity η is estimated through the Stokes–Einstein equation. Results show that maturation is correlated with change of cytoplasm viscosity. The results can be ascribed to F-actin reorganization as also supported by fluorescence confocal microscopy.

Movements of HIV-virions in human cervical mucus.

Time-resolved confocal microscopy and fluorescence correlation spectroscopy were used to examine the movements of fluorescently labeled HIV-virions (approximately 100 nm) added to samples of human cervical mucus. Particle-tracking analysis indicates that the motion of most virions is decreased 200-fold compared to that in aqueous solution and is not driven by typical diffusion. Rather, the time-dependence of their ensemble-averaged mean-square displacements is proportional to tau(alpha) + v(2)tau(2), describing a combination of anomalous diffusion (alpha approximately 0.3) and flow-like behavior, with tau being the lag time. We attribute the flow-like behavior to slowly relaxing mucus matrix that follows mechanical perturbations such as stretching and twisting of the sample. Further analysis of the tracks and displacements of individual virions indicates differences in the local movements among the virions, including constrained motion and infrequent jumps, perhaps due to abrupt changes in matrix structure. Changes in the microenvironments due to slow structural changes may facilitate movement of the virions, allowing them to reach the epithelial layer.

Multiple Particle Tracking Investigations of Acid Milk Gels Using Tracer Particles with Designed Surface Chemistries and Comparison with Diffusing Wave Spectroscopy Studies

Multiple particle tracking (MPT) has been used in an attempt to probe the heterogeneity of acid milk gels, made with and without added pectin, by following the distribution of the displacements of added tracer beads during and after gelation using the Van Hove distribution. Furthermore, the surface chemistry of the latex probe particles was modified in an attempt to control their location in the system and probe the microrheological properties of the protein network and aqueous-phase voids independently. In addition, the mean square displacement (MSD) of the casein micelles/casein aggregates themselves, obtained by diffusing wave spectroscopy (DWS), has been compared to the ensemble-averaged MSD calculated from the data obtained by tracking the movement of the added tracers, with and without a kappa-casein coating. For the kappa-casein-coated tracer particles, upon acidification and subsequent gel formation, the MSDs obtained by MPT superimpose remarkably well with the MSDs obtained by DWS, despite the fact that one is obtained by tracking the movement of the particle network elements themselves and the other is obtained from directly tracking added tracers. This result has important implications: (i) it demonstrates that, although the DWS measurement is intrinsically ensemble-averaged, it really gives insight into the dynamics of the colloidal gel network; (ii) it confirms that the kappa-casein-coated probes used in this MPT experiment are well incorporated into the gel network; and hence (iii) that at least in gelled systems kappa-casein-coated latex probes are interesting probes that reveal the dynamics of the casein network.

Subdiffusion in time-averaged, confined random walks

Certain techniques characterizing diffusive processes, such as single-particle tracking or molecular dynamics simulation, provide time averages rather than ensemble averages. Whereas the ensemble-averaged mean-squared displacement (MSD) of an unbounded continuous time random walk (CTRW) with a broad distribution of waiting times exhibits subdiffusion, the time-averaged MSD, delta2, does not. We demonstrate that, in contrast to the unbounded CTRW, in which delta2 is linear in the lag time Delta, the time-averaged MSD of the CTRW of a walker confined to a finite volume is sublinear in Delta, i.e., for long lag times delta2 approximately Delta1-alpha. The present results permit the application of CTRW to interpret time-averaged experimental quantities.

HIV-virions Appear To Be Trapped By Human Cervical Mucus

We apply time-resolved fluorescence confocal microscopy and fluorescence correlation spectroscopy to examine the movements of fluorescently-labeled HIV-1 virions (∼120 nm) embedded in crude human cervical mucus. Particle-tracking analysis indicates that the motion of most virions is decreased 200-fold compared to that in water and is not driven by typical diffusion. Rather, the time-dependence of their ensemble-averaged mean-square displacements is proportional to τα + v2τ2, describing a combination of anomalous diffusion at short time scales (α ∼ 0.3) and flow-like behavior at longer times, τ being the lag time. We attribute the flow-like behavior to slowly-relaxing mucus matrix that follows mechanical perturbations such as stretching and twisting of the sample. Further analysis of the tracks and displacements of individual virions indicates differences in the local movements among the virions, including constrained motion and infrequent jumps, perhaps due to abrupt changes in matrix structure. We surmise that these differences are related not only to possible variations in the local microenvironments experienced by each individual virion but, perhaps, also to variations in the surface structure of the virions themselves. Possible changes in the microenvironments due to slow structural changes may provide a means for some virions to move and reach the port of entry, the underlying cervical mucosa.

Micro-heterogeneity and micro-rheological properties of high-viscosity oat β-glucan solutions

Soluble fibre β-glucan is one of the key dietary materials in the healthy food products known for reducing serum cholesterol levels. The micro-structural heterogeneity and micro-rheology of high-viscosity oat β-glucan solutions were investigated by monitoring the thermally driven displacements of well-dispersed microspheres via video fluorescence microscopy. By comparing the distribution of the time-dependent mean-square displacement (MSD) and ensemble-averaged MSD of polystyrene microspheres imbedded in four concentrations of β-glucan solutions, we found that the solutions exhibited perfectly homogeneous behaviour at ⩽1%, but showed a certain degree of heterogeneity at 2%. Micro-rheology investigation revealed that β-glucan solutions displayed nearly perfect viscous behaviour at ⩽1%, but the property changed into viscoelastic at 2%. Both micro-structural heterogeneity and micro-rheological property shifts occurred over a small concentration range, between 1% and 2%, of β-glucan.

Adiabatic-nonadiabatic transition in the diffusive Hamiltonian dynamics of a classical Holstein polaron

We study the Hamiltonian dynamics of a free particle injected onto a chain containing a periodic array of harmonic oscillators in thermal equilibrium. The particle interacts locally with each oscillator, with an interaction that is linear in the oscillator coordinate and independent of the particle's position when it is within a finite interaction range. At long times the particle exhibits diffusive motion, with an ensemble averaged mean-squared displacement that is linear in time. The diffusion constant at high temperatures follows a power law $D\ensuremath{\sim}{T}^{\phantom{\rule{0.1em}{0ex}}5∕2}$ for all parameter values studied. At low temperatures particle transport changes to a hopping process in which the particle is bound for considerable periods of time to a single oscillator before it is able to escape and explore the rest of the chain. A different power law, $D\ensuremath{\sim}{T}^{\phantom{\rule{0.1em}{0ex}}3∕4}$, emerges in this limit. A thermal distribution of particles exhibits thermally activated diffusion at low temperatures as a result of classically self-trapped polaronic states.

Anomalous diffusion probes microstructure dynamics of entangled F-actin networks.

We study the thermal motion of colloidal tracer particles in entangled actin filament (F-actin) networks, where the particle radius is comparable to the mesh size of the F-actin network. In this regime, the ensemble-averaged mean-squared displacement of the particles is proportional to tau(gamma), where 0<gamma<1 from 0.1<tau<100 s and depends only on the ratio of the probe radius to mesh size. By directly imaging hundreds of particles over 20 min, we determine this anomalous subdiffusion is due to the dynamics of infrequent and large jumps particles make between distinct pores in the network.

Effect of length, topology, and concentration on the microviscosity and microheterogeneity of DNA solutions.

The viscoelastic behavior of chromosomal DNA, which is heterogeneously distributed within the nucleus, may influence the diffusion of nuclear organelles and proteins. To identify some of the parameters that affect DNA viscoelasticity, we use the high-throughput method of multiple-particle nanotracking to measure the microviscosity and degree of heterogeneity of solutions of chromosomal DNA, linear DNA, and circular double-stranded DNA over a wide range of concentrations and lengths. The thermally excited displacements of multiple fluorescent microspheres imbedded in DNA solutions are monitored with 5nm spatial resolution and 30Hz temporal resolution, from which mean-squared displacement (MSD) and viscosity distributions are generated. For all probed DNA solutions but the most concentrated solution of the longest molecules, the ensemble-averaged MSD increases linearly with time at all probed time scales, a signature of viscous transport. The associated mean viscosity of the DNA solutions increases slowly with concentration for circular DNA and more rapidly for linear DNA, but more slowly than predicted by theory. The heterogeneity of the DNA solutions is assessed by computing the relative contributions of the 10%, 25%, and 50% highest values of MSD and viscosity to the ensemble-averaged MSD and viscosity. For both linear DNA and circular DNA, these contributions are much larger than observed in homogeneous liquids such as glycerol. The microheterogeneity of the linear DNA solutions increases with concentration more significantly for linear DNA than circular DNA. These in vitro results suggest that the topology, local concentration, and length of DNA influence the microrheology and microheterogeneity of the DNA within the nucleus.

Microheterogeneity and microrheology of wheat gliadin suspensions studied by multiple-particle tracking.

By monitoring the thermally driven displacements of imbedded polystyrene microspheres via video fluorescence microscopy, we quantified the microstructural and micromechanical heterogeneities of wheat gliadin suspensions. We found that the degree of heterogeneity of the suspensions, as measured by the width and skewness of the microspheres' mean squared displacement (MSD) distribution, increased dramatically over a narrow range of gliadin concentrations. The ensemble-averaged MSD of a 250 mg/mL gliadin suspension exhibited a power-law behavior scaling linearly with time, a behavior similar to that observed for a homogeneous aqueous glycerol solution. However, the MSD distribution was wider and more asymmetric than for glycerol. With increasing concentration of gliadin, the ensemble-averaged MSD rapidly displayed a plateau at small time scales, the MSD distribution became wider and more asymmetric, and the local viscoelastic moduli extracted from multiple-particle-tracking measurements showed an increasingly wide range.