Ensemble-averaged Mean Square Displacement Research Articles

Directional change during active diffusion of viral ribonucleoprotein particles through cytoplasm

A mesh of cytoskeletal fibers, consisting of microtubules, intermediate filaments, and fibrous actin, prevents the Brownian diffusion of particles with a diameter larger than 0.10 μm, such as vesicular stomatitis virus ribonucleoprotein (RNP) particles, in mammalian cells. Nevertheless, RNP particles do move in random directions but at a lower rate than Brownian diffusion, which is thermally driven. This nonthermal biological transport process is called “active diffusion” because it is driven by ATP. The ATP powers motor proteins such as myosin II. The motor proteins bend and cross-link actin fibers, causing the mesh to jiggle. Until recently, little was known about how RNP particles get through the mesh. It has been customary to analyze the tracks of particles like RNPs by computing the slope of the ensemble-averaged mean-squared displacement of the particles as a signature of mechanism. Although widely used, this approach “loses information” about the timing of the switches between physical mechanisms. It has been recently shown that machine learning composed of variational Bayesian analysis, Gaussian mixture models, and hidden Markov models can use “all the information” in a single track to reveal that that the positions of RNP particles are spatially clustered. Machine learning assigns a number, called a state, to each cluster. RNP particles remain in one state for 0.2–1.0 s before switching (hopping) to a different state. This earlier work is here extended to analyze the movements of a particle within a state and to determine particle directionality within and between states.

Landscapes of random diffusivity processes in harmonic potential

A huge number of experimental evidences support that Brownian yet non-Gaussian diffusion prevalently exists in biological and soft matter systems. To interpret the physical mechanism of the Brownian yet non-Gaussian diffusion, the superstatistical approach with random diffusivity and further general random diffusivity processes attract extensive attention. This paper focuses on the anomalous diffusion and nonergodic property of the random diffusivity processes confined in a harmonic potential. Based on the theoretical analyses in the framework of Langevin equation, we find the inherent disagreement between ensemble-averaged mean-squared displacement and time-averaged mean-squared displacement for any form of the random diffusivity D(t), which implies the ergodicity breaking of the confined dynamics. Further, the asymptotic distribution of the time-averaged mean-squared displacement is strictly derived and shows the explicit dependence on the time average of the diffusivity D(t), which means the randomness of the time-averaged mean-squared displacement is completely contributed by the random diffusivity D(t). These results hold true for a general diffusivity D(t), which has been verified through the simulations of three specific kinds of diffusivities D(t).

Inertia triggers nonergodicity of fractional Brownian motion.

How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

Ergodic property of Langevin systems with superstatistical, uncorrelated or correlated diffusivity

Brownian yet non-Gaussian diffusion has recently been observed in numerous biological and active matter system. The cause of the non-Gaussian distribution have been elaborately studied in the idea of a superstatistical dynamics or a diffusing diffusivity. Based on a random diffusivity model, we here focus on the ergodic property and the scatter of the amplitude of time-averaged mean-squared displacement (TAMSD). By investigating the random diffusivity model with three categories of diffusivities, including diffusivity being a random variable D, a time-dependent but uncorrelated diffusivity D(t), and a correlated stochastic process D(t), we find that ensemble-averaged TAMSDs are always normal while ensemble-averaged mean-squared displacement can be anomalous. Further, the scatter of dimensionless amplitude is completely determined by the time average of diffusivity D(t). Our results are valid for arbitrary diffusivity D(t).

Empirical anomaly measure for finite-variance processes

Anomalous diffusion phenomena are observed in many areas of interest. They manifest themselves in deviations from the laws of Brownian motion (BM), e.g. in the non-linear growth (mostly power-law) in time of the ensemble average mean squared displacement (MSD). When we analyze the real-life data in the context of anomalous diffusion, the primary problem is the proper identification of the type of the anomaly. In this paper, we introduce a new statistic, called empirical anomaly measure (EAM), that can be useful for this purpose. This statistic is the sum of the off-diagonal elements of the sample autocovariance matrix for the increments process. On the other hand, it can be represented as the convolution of the empirical autocovariance function with time lags. The idea of the EAM is intuitive. It measures dependence between the ensemble-averaged MSD of a given process from the ensemble-averaged MSD of the classical BM. Thus, it can be used to measure the distance between the anomalous diffusion process and normal diffusion. In this article, we prove the main probabilistic characteristics of the EAM statistic and construct the formal test for the recognition of the anomaly type. The advantage of the EAM is the fact that it can be applied to any data trajectories without the model specification. The only assumption is the stationarity of the increments process. The complementary summary of the paper constitutes of Monte Carlo simulations illustrating the effectiveness of the proposed test and properties of EAM for selected processes.

Single-molecule tracking measurement of PDMS layer during curing process

The curing process of poly(dimethylsiloxane) (PDMS) was microscopically investigated by the single-molecule tracking method based on the diffusion motion of fluorescent dye molecules adding to the PDMS layer stored at a temperature of 308K. The PDMS layer was completely cured at 900 min after adding a curing agent. We compared the time- and ensemble-averaged mean square displacements (MSDs) of the tracked molecules at 90, 510, and 900 min after adding the curing agent into the PDMS layer. The discrepancies were observed between the time- and ensemble-averaged MSDs, indicating weak ergodicity breaking. The spatially averaged diffusion coefficient exhibited a two-step decrease: first step was rapid decrease suggesting the extent of crosslinking, and second step was slow one suggesting the increase of crosslink density. The single-molecule trajectory scale analysis revealed the heterogeneous distribution of the diffusion coefficient. By calculating the heat map from the slope of moment scaling spectrum (MSS) of each single-molecule trajectory, cluster structures were recognized. The spatial correlation of the slope of MSS decreased with the time elapsed. These results suggested the existence of the heterogeneous structure in the PDMS layer during the curing process.

Dynamics of polydisperse hard-spheres under strong confinement

We use molecular simulation to probe the connection between local structure and the unusual re-entrant dynamics observed for polydisperse hard-sphere liquids confined in thin slit pores. The local structure is characterised by calculating 2-D bond-orientational order parameters associated with square and hexatic order for particles in the layer adjacent to the confining walls. When the wall separation is commensurate with the average particle size, the particles primarily exhibit local hexatic order, whereas local square order increases in prevalence for incommensurate geometries. The relaxation time extracted from the ensemble-averaged mean-square displacement increases exponentially with the static correlation length associated with hexatic local order in strongly confined commensurate geometries, in agreement with theoretical predictions for dynamical slowing. Square order, by contrast, is not associated with a growing length scale for either commensurate or incommensurate geometries, indicating that it is strongly geometrically frustrated. Our results suggest that the influence of bond-orientational order on dynamical slowing may be altered by changing the extent of confinement.

Gelation mechanism and network structure of mixed kappa carrageenan/lambda carrageenan gels studied by macroscopic and microscopic observation methods

Dynamic rheology and particle tracking experiments were carried out to investigate the gelation mechanism and network structure of mixed κ-carrageenan (KC) and λ-carrageenan (LC) gels on the macroscopic and microscopic levels. Mixtures of KC and LC (total carrageenan concentration = 1.5%, K+ concentration = 15 mM) were prepared at different mixing ratios. The storage modulus (G′) was highest for pure KC and decreased monotonously with decreasing KC content. The mean square displacement (MSD) of probe particles was used to characterize the network on cooling and after one day of storage. On cooling, a single step decrease in the ensemble-averaged MSD was observed, in agreement with the rheological measurements. Time-averaged MSD values of individual particles showed a large distribution after one day in storage, especially for gels with low content of KC. This heterogeneity reflects microstructural heterogeneity, suggesting phase separation into KC-rich and LC-rich domains. Individual particle MSD values showed a bimodal distribution. The MSD could be described as a power-law, MSD~τα, with τ the lag time. The exponent α also showed a bimodal distribution. These findings suggest that two groups of probe particles with distinctly different mobilities are present in the mixed gels with particles of higher and lower mobilities present in the LC-rich and KC-rich phases. Based on the rheological results, as well as the particle tracking experiments, we propose that mixtures of KC and LC separate into a continuous phase and a non-continuous filler phase, with the continuous phase being made up of the more concentrated carrageenan. When the concentrations of the two carrageenans are roughly equal, we suggest that a bicontinuous phase-separated network is formed.

Subdiffusion of loci and cytoplasmic particles are different in compressed Escherichia coli cells

The complex physical nature of the bacterial intracellular environment remains largely unknown, and has relevance for key biochemical and biological processes of the cell. Although recent work has addressed the role of non-equilibrium sources of activity and crowding, the consequences of mechanical perturbations are relatively less explored. Here we use a microfabricated valve system to track both fluorescently labeled chromosomal loci and cytoplasmic particles in Escherichia coli cells shortly after applying a compressive force, observing the response on time scales that are too sudden to allow for biochemical response from the cell. Cytoplasmic diffusion slows markedly on compression but the exponent governing the growth of the ensemble-averaged mean-squared displacement of cytoplasmic particles is unaffected. In contrast, the corresponding exponent for DNA loci changes significantly. These results suggest that DNA elasticity and nucleoid organization play a more important role in loci subdiffusion than cytoplasmic viscoelasticity under such short time scales.

Subdiffusion and ergodicity breaking in heterogeneous environments subject to Lévy noise

The anomalous diffusion and ergodicity breaking of Brownian particles in heterogeneous environments (space-dependent diffusion coefficient D(x)=D0|x|α) subject to Lévy noise are investigated, by both the analytically theory and the numerically simulating schemes. Within the tailored parameter, the competition of the power-law exponent α and the Lévy index μ ensures the long tail of probability density function with the form P(x)∼1|x|μ∕p+1 with p=22−α, falling rapidly when μ∕p>2. Meanwhile, we observe the subdiffusive motion that is characterized by the scale of ensemble averaged mean-square displacement (MSD), 〈x2(t)〉∝tκ with κ=2pμ<1. Furthermore, we also observe the weak ergodicity breaking which means the inequivalence between individual time averaged MSD and the corresponding ensemble averaged MSD. We quantify the weak ergodicity breaking behavior in terms of the dimensionless parameters EB and EB. For 1≤μ<2 (except for μ=2), the ergodicity breaking enhances as the α increases, but weakens as μ or D0. Overall, the EB follows the from, EB∝(Δ∕T)(12−2pμ), in the large time. Additionally, the anomalous diffusion above might be particularly relevant for ultracold atoms diffusion in heterogeneous media.

Langevin equation with fluctuating diffusivity: A two-state model.

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.

Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions

We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction ϕ of the crowders and the crowder–crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational—but not rotational—diffusivity of the particles D is a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law with the area fraction ϕ occupied by the crowders and the critical value Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.

Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity.

The mean-square displacement (MSD) is widely utilized to study the dynamical properties of stochastic processes. The time-averaged MSD (TAMSD) provides some information on the dynamics which cannot be extracted from the ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of the TAMSD can be utilized to study the long-time relaxation behavior. In this work, we consider a class of Langevin equations which are multiplicatively coupled to time-dependent and fluctuating diffusivities. Various interesting dynamics models such as entangled polymers and supercooled liquids can be interpreted as the Langevin equations with time-dependent and fluctuating diffusivities. We derive a general formula for the RSD of the TAMSD for the Langevin equation with the time-dependent and fluctuating diffusivity. We show that the RSD can be expressed in terms of the correlation function of the diffusivity. The RSD exhibits the crossover at the long time region. The crossover time is related to a weighted average relaxation time for the diffusivity. Thus the crossover time gives some information on the relaxation time of fluctuating diffusivity which cannot be extracted from the ensemble-averaged MSD. We discuss the universality and possible applications of the formula via some simple examples.

Transition from distributional to ergodic behavior in an inhomogeneous diffusion process: Method revealing an unknown surface diffusivity.

Diffusion of molecules in cells plays an important role in providing a biological reaction on the surface by finding a target on the membrane surface. The water retardation (slow diffusion) near the target assists the searching molecules to recognize the target. Here, we consider effects of the surface diffusivity on the effective diffusivity, where diffusion on the surface is slower than that in bulk. We show that the ensemble-averaged mean-square displacements increase linearly with time when the desorption rate from the surface is finite, which is valid even when the diffusion on the surface is anomalous (subdiffusion). Moreover, this slow diffusion on the surface affects the fluctuations of the time-averaged mean-square displacements (TAMSDs). We find that fluctuations of the TAMSDs remain large when the measurement time is smaller than a characteristic time, and decays according to an increase of the measurement time for a relatively large measurement time. Therefore, we find a transition from nonergodic (distributional) to ergodic diffusivity in a target search process. Moreover, this fluctuation analysis provides a method to estimate an unknown surface diffusivity.

Time-average based on scaling law in anomalous diffusions

To solve the obscureness in measurement brought about from the weak ergodicity breaking appeared in anomalous diffusions, we have suggested the time-averaged mean squared displacement (MSD) [Formula: see text] with an integral interval depending linearly on the lag time τ. For the continuous time random walk describing a subdiffusive behavior, we have found that [Formula: see text] like that of the ensemble-averaged MSD, which makes it be possible to measure the proper exponent values through time-average in experiments like a single molecule tracking. Also, we have found that it has originated from the scaling nature of the MSD at an aging time in anomalous diffusion and confirmed them through numerical results of the other microscopic non-Markovian model showing subdiffusions and superdiffusions with the origin of memory enhancement.

Localisation and universal fluctuations in ultraslow diffusion processes

We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) . Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD.

Weak ergodicity breaking and aging of chaotic transport in Hamiltonian systems.

Momentum diffusion is a widespread phenomenon in generic Hamiltonian systems. We show for the prototypical standard map that this implies weak ergodicity breaking for the superdiffusive transport in coordinate direction with an averaging-dependent quadratic and cubic increase of the mean-squared displacement (MSD), respectively. This is explained via integrated Brownian motion, for which we derive aging time dependent expressions for the ensemble-averaged MSD, the distribution of time-averaged MSDs, and the ergodicity breaking parameter. Generalizations to other systems showing momentum diffusion are pointed out.

Random time averaged diffusivities for Lévy walks

We investigate a L\'evy-Walk alternating between velocities $\pm v_0$ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is $< x^2 > \propto t^2$, the latter to enhanced diffusion with $< x^2 > \propto t^\nu$, $1<\nu<2$. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.

Novel algorithm and MATLAB-based program for automated power law analysis of single particle, time-dependent mean-square displacement

In many physical and biophysical studies, single-particle tracking is utilized to reveal interactions, diffusion coefficients, active modes of driving motion, dynamic local structure, micromechanics, and microrheology. The basic analysis applied to those data is to determine the time-dependent mean-square displacement (MSD) of particle trajectories and perform time- and ensemble-averaging of similar motions. The motion of particles typically exhibits time-dependent power-law scaling, and only trajectories with qualitatively and quantitatively comparable MSD should be ensembled. Ensemble averaging trajectories that arise from different mechanisms, e.g., actively driven and diffusive, is incorrect and can result inaccurate correlations between structure, mechanics, and activity. We have developed an algorithm to automatically and accurately determine power-law scaling of experimentally measured single-particle MSD. Trajectories can then categorized and grouped according to user defined cutoffs of time, amplitudes, scaling exponent values, or combinations. Power-law fits are then provided for each trajectory alongside categorized groups of trajectories, histograms of power laws, and the ensemble-averaged MSD of each group. The codes are designed to be easily incorporated into existing user codes. We expect that this algorithm and program will be invaluable to anyone performing single-particle tracking, be it in physical or biophysical systems. Program summaryProgram title: LINSA (acronym: linear segment analysis)Catalogue identifier: AEMD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEMD_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 25 892No. of bytes in distributed program, including test data, etc.: 5 572 780Distribution format: tar.gzProgramming language: MATLAB (MathWorks Inc.) version 7.11 (2010b) or higher, program should also be backwards compatible. Symbolic Math Toolboxes (5.5) is required. The Curve Fitting Toolbox (3.0) is recommended.Computer: Tested on Windows only, yet should work on any computer running MATLAB. In Windows 7, should be used as administrator, if the user is not the administrator the program may not be able to save outputs and temporary outputs to all locations.Operating system: Any supporting MATLAB (MathWorks Inc.) v7.11 / 2010b or higher.Supplementary material: Sample output files (approx. 30 MBytes) are available.Classification: 12External routines: Several MATLAB subfunctions (m-files), freely available on the web, were used as part of and included in, this code: count, NaN suite, parseArgs, roundsd, subaxis, wcov, wmean, and the executable pdfTK.exe.Nature of problem: In many physical and biophysical areas employing single-particle tracking, having the time-dependent power-laws governing the time-averaged meansquare displacement (MSD) of a single particle is crucial. Those power laws determine the mode-of-motion and hint at the underlying mechanisms driving motion. Accurate determination of the power laws that describe each trajectory will allow categorization into groups for further analysis of single trajectories or ensemble analysis, e.g. ensemble and time-averaged MSD.Solution method: The algorithm in the provided program automatically analyzes and fits time-dependent power laws to single particle trajectories, then group particles according to user defined cutoffs. It accepts time-dependent trajectories of several particles, each trajectory is run through the program, its time-averaged MSD is calculated, and power laws are determined in regions where the MSD is linear on a log–log scale. Our algorithm searches for high-curvature points in experimental data, here time-dependent MSD. Those serve as anchor points for determining the ranges of the power-law fits. Power-law scaling is then accurately determined and error estimations of the parameters and quality of fit are provided. After all single trajectory time-averaged MSDs are fit, we obtain cutoffs from the user to categorize and segment the power laws into groups; cutoff are either in exponents of the power laws, time of appearance of the fits, or both together. The trajectories are sorted according to the cutoffs and the time- and ensemble-averaged MSD of each group is provided, with histograms of the distributions of the exponents in each group. The program then allows the user to generate new trajectory files with trajectories segmented according to the determined groups, for any further required analysis.Additional comments: README file giving the names and a brief description of all the files that make-up the package and clear instructions on the installation and execution of the program is included in the distribution package.Running time: On an i5 Windows 7 machine with 4 GB RAM the automated parts of the run (excluding data loading and user input) take less than 45 minutes to analyze and save all stages for an 844 trajectory file, including optional PDF save. Trajectory length did not affect run time (tested up to 3600 frames/trajectory), which was on average 3.2±0.4 seconds per trajectory.

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.