Time-averaged Mean Square Displacement Research Articles

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement.

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Mean-squared-displacement statistical test for fractional Brownian motion.

Anomalous diffusion in crowded fluids, e.g., in cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous diffusion of a single particle can be classified is the time-averaged mean square displacement (TAMSD). A classical mechanism leading to the anomalous diffusion is the fractional Brownian motion (FBM). A validation of such process for single-particle tracking data is of great interest for experimentalists. In this paper we propose a rigorous statistical test for FBM based on TAMSD. To this end we analyze the distribution of the TAMSD statistic, which is given by the generalized chi-squared distribution. Next, we study the power of the test by means of Monte Carlo simulations. We show that the test is very sensitive for changes of the Hurst parameter. Moreover, it can easily distinguish between two models of subdiffusion: FBM and continuous-time random walk.

Distributional behavior of diffusion coefficients obtained by single trajectories in annealed transit time model

Local diffusion coefficients in disordered systems such as spin glass systems and living cells are highly heterogeneous and may change over time. Such a time-dependent and spatially heterogeneous environment results in irreproducibility of single-particle-tracking measurements. Irreproducibility of time-averaged observables has been theoretically studied in the context of weak ergodicity breaking in stochastic processes. Here, we provide rigorous descriptions of equilibrium and non-equilibrium diffusion processes for the annealed transit time model, which is a heterogeneous diffusion model in living cells. We give analytical solutions for the mean square displacement (MSD) and the relative standard deviation of the time-averaged MSD for equilibrium and non-equilibrium situations. We find that the time-averaged MSD grows linearly with time and that the time-averaged diffusion coefficients are intrinsically random (irreproducible) even in the long-time measurements in non-equilibrium situations. Furthermore, the distribution of the time-averaged diffusion coefficients converges to a universal distribution in the sense that it does not depend on initial conditions. Our findings pave the way for a theoretical understanding of distributional behavior of the time-averaged diffusion coefficients in disordered systems.

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.

Distributional behaviors of time-averaged observables in the Langevin equation with fluctuating diffusivity: Normal diffusion but anomalous fluctuations.

We consider the Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously over time, in order to study fluctuations of time-averaged observables in temporally heterogeneous diffusion processes. We find that the time-averaged mean-square displacement (TMSD) can be represented by the occupation time of a state in the asymptotic limit of the measurement time and hence occupation time statistics is a powerful tool for calculating the TMSD in the model. We show that the TMSD increases linearly with time (normal diffusion) but the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges, i.e., intrinsic nonequilibrium processes. Thus, we find that temporally heterogeneous environments provide anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficients obtained by single-particle-tracking trajectories in experiments.

Revealing nonergodic dynamics in living cells from a single particle trajectory.

We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.

Superdiffusive Motion of Membrane-Targeting Domains

Membrane-targeting domains play crucial roles in the association of signaling molecules to the plasma membrane. For most peripheral proteins, the protein-to-membrane interaction is transient; but after proteins dissociate from the membrane they are observed to rebind following a brief excursion in the bulk solution. These membrane hops can have broad implications on protein dynamics. We study the diffusion of membrane-targeting domains using single-molecule tracking in supported lipid bilayers. The ensemble-averaged mean square displacement (MSD) exhibits superdiffusive behavior. However, the time-averaged MSD of individual trajectories is found to be linear with respect to lag time, as in Brownian diffusion. These observations are explained in terms of bulk excursions that introduce jumps with a heavy-tail distribution, rapidly increasing the area explored. Furthermore, this type of motion is characterized by a pronounced scattering in the time-averaged MSD of individual trajectories.

Superdiffusive motion of membrane-targeting C2 domains

Membrane-targeting domains play crucial roles in the recruitment of signalling molecules to the plasma membrane. For most peripheral proteins, the protein-to-membrane interaction is transient. After proteins dissociate from the membrane they have been observed to rebind following brief excursions in the bulk solution. Such membrane hops can have broad implications for the efficiency of reactions on membranes. We study the diffusion of membrane-targeting C2 domains using single-molecule tracking in supported lipid bilayers. The ensemble-averaged mean square displacement (MSD) exhibits superdiffusive behaviour. However, traditional time-averaged MSD analysis of individual trajectories remains linear and does not reveal superdiffusion. Our observations are explained in terms of bulk excursions that introduce jumps with a heavy-tail distribution. These hopping events allow proteins to explore large areas in a short time. The experimental results are shown to be consistent with analytical models of bulk-mediated diffusion and numerical simulations.

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.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach.

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments.

Single particle tracking is an essential tool in the study of complex systems and biophysics and it is commonly analyzed by the time-averaged mean square displacement (MSD) of the diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors and biases, preventing the comparison and assessment of experimental studies. Here, we attempt to extract practical guidelines for the estimation of anomalous time averaged MSDs through the simulation of multiple scenarios with fractional Brownian motion as a representative of a large class of fractional ergodic processes. We extract the precision and accuracy of the fitted MSD for various anomalous exponents and measurement errors with respect to measurement length and maximum time lags. Based on the calculated precision maps, we present guidelines to improve accuracy in single particle studies. Importantly, we find that in some experimental conditions, the time averaged MSD should not be used as an estimator.

Anomalous diffusion in a quenched-trap model on fractal lattices

Models with mixed origins of anomalous subdiffusion have been considered important for understanding transport in biological systems. Here one such mixed model, the quenched-trap model (QTM) on fractal lattices, is investigated. It is shown that both ensemble- and time-averaged mean-square displacements (MSDs) show subdiffusion with different scaling exponents, i.e., this system shows weak ergodicity breaking. Moreover, time-averaged MSD exhibits aging and converges to a random variable following the modified Mittag-Leffler distribution. It is also shown that the QTM on a fractal lattice cannot be reduced to the continuous-time random walks if the spectral dimension of the fractal lattice is less than 2.

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.

Confined subdiffusion in three dimensions

Three-dimensional (3D) Fick's diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model (CTRW) to investigate the time-averaged mean square displacement (MSD) of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag—Leffler function. Our new expression is in agreement with previous formulas in two limiting cases: in short lag time and in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density (PSD) further indicate the subdiffsive behavior of an individual trajectory.

Phase Diagram in Stored-Energy-Driven Lévy Flight

Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven L\'evy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.

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.

Transient aging in fractional Brownian and Langevin-equation motion

Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t=0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on t(a) is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.

Optimal and suboptimal quadratic forms for noncentered Gaussian processes

Individual random trajectories of stochastic processes are often analyzed by using quadratic forms such as time averaged (TA) mean square displacement (MSD) or velocity auto-correlation function (VACF). The appropriate quadratic form is expected to have a narrow probability distribution in order to reduce statistical uncertainty of a single measurement. We consider the problem of finding the optimal quadratic form that minimizes a chosen cumulant moment (e.g., the variance) of the probability distribution, under the constraint of fixed mean value. For discrete noncentered Gaussian processes, we construct the optimal quadratic form by using the spectral representation of cumulant moments. Moreover, we obtain a simple explicit formula for the smallest achievable cumulant moment that may serve as a quality benchmark for other quadratic forms. We illustrate the optimality issues by comparing the optimal variance with the variances of the TA MSD and TA VACF of fractional Brownian motion superimposed with a constant drift and independent Gaussian noise.