Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking (original) (raw)
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Physical Chemistry Chemical Physics, 2011
Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.
Analysis of single particle trajectories: from normal to anomalous diffusion
2009
With modern experimental tools it is possible to track the motion of single nanoparticles in real time, even in complex environments such as biological cells. The quest is then to reliably evaluate the time series of individual trajectories. While this is straightforward for particles performing normal Brownian motion, interesting subtleties occur in the case of anomalously diffusing particles: it is no longer granted that the long time average equals the ensemble average. We here discuss for two different models of anomalous diffusion the detailed behaviour of time averaged mean squared displacement and related quantities, and present possible criteria to analyse single particle trajectories. An important finding is that although the time average may suggest normal diffusion the actual process may in fact be subdiffusive.
The role of ergodicity in anomalous stochastic processes: analysis of single-particle trajectories
Physica Scripta, 2012
Single-particle experiments produce time series x(t) of individual particle trajectories, frequently revealing anomalous diffusion behaviour. Typically, individual x(t) are evaluated in terms of time-averaged quantities instead of ensemble averages. Here we discuss the behaviour of the time-averaged mean squared displacement of different stochastic processes giving rise to anomalous diffusion. In particular, we pay attention to the ergodic properties of these processes, i.e. the (non)equivalence of time and ensemble averages.
PloS one, 2015
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 estima...
Improved analysis of anomalous diffusion data in single particle tracking experiments
arXiv (Cornell University), 2012
The Mean Square Displacement is a central tool in the analysis of Single Particle Tracking experiments, shedding light on various biophysical phenomena. However, as we show, it suffers from two systematic errors when analysing tracks of anomalous diffusing particles. The first is significant at short time differences and is induced by measurement errors. The second arises from the natural heterogeneity in biophysical systems. We show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution. As a consequence we manage to characterise ensembles of heterogeneous particles even at very short and noisy measurements where regular time averaged mean square displacement analysis fails. This procedure has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity. Notation Anomalous diffusion constant-D α Apparent anomalous exponent due to heterogeneity-α S Apparent anomalous exponent due to noise-α N Apparent location-x (t) Apparent TAMSD-δ 2 ∆ Control experiment noise MSD-Ñ c Dynamic functional-ϕ Error in anomalous exponent due to heterogeneity-d α Mean Logarithmic Square Displacement (MLSD)-λ (∆) Mean of anomalous exponent distribution-µ α Noise and heterogeneity corrected MSD-ν C(∆) Noise Corrected MSD (NC-MSD)-ν (∆) Noise MSD-Ñ Relative error caused by noise-N Standard deviation of anomalous exponent distribution-σ α Standard deviation of noise-ρ Time averaged MSD-δ 2 ∆ Time difference-∆ True location-x (t) True particle anomalous exponent-α
Improved estimation of anomalous diffusion exponents in single-particle tracking experiments
Physical Review E, 2013
The mean square displacement is a central tool in the analysis of single-particle tracking experiments, shedding light on various biophysical phenomena. Frequently, parameters are extracted by performing time averages on single-particle trajectories followed by ensemble averaging. This procedure, however, suffers from two systematic errors when applied to particles that perform anomalous diffusion. The first is significant at short-time lags and is induced by measurement errors. The second arises from the natural heterogeneity in biophysical systems. We show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution. As a consequence, we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time-averaged mean square displacement analysis fails. We apply this method to both simulations and in vivo measurements of telomere diffusion in 3T3 mouse embryonic fibroblast cells. The motion of telomeres is found to be subdiffusive with an average exponent constant in time. Individual telomere exponents are normally distributed around the average exponent. The proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity.
Physical Chemistry Chemical Physics, 2014
Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is x 2 (t) ≃ K (t)t with K (t) ≃ t α−1 for 0 < α < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used.
Scientific Reports, 2015
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.
Anomalous diffusion: A dynamic perspective
Physica A: Statistical Mechanics and its Applications, 1990
This paper investigates whether spontaneous, stationary velocity fluctuations can lead to deviations from the regular Fickian diffusion. A kinematic analysis reveals that anomalous diffusion, both fast and slow, arises from long-tailed velocity auto-correlation functions (VACF). This infinite span of interdependence of the random velocity leads to the breakdown of the central limit theorem for particle displacements. A generalized Langevin equation, which features a retarded friction, has been used to describe the particle dynamics in the long-time limit. The analysis reveals that simple power-law decay models for the friction kernel are adequate to yield the pathological VACFs which imply anomalous diffusion. The fluctuation dissipation theorem is invoked to infer that a fractional noise gives rise to anomalous diffusion. Such a Langevin equation represents a mean-field description of disorder effects and the friction kernel then becomes a constitutive property of the medium.
Improved analysis of experimental data from anomalous diffusion measurements
arXiv (Cornell University), 2012
The Mean Square Displacement is a central tool in the analysis of Single Particle Tracking experiments, shedding light on various biophysical phenomena. Frequently, parameters are extracted by performing time-averages on single particle trajectories followed by ensemble averaging. This procedure however, suffers from two systematic errors when applied to particles that perform anomalous diffusion. The first is significant at short time lags and is induced by measurement errors. The second arises from the natural heterogeneity in biophysical systems. We show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution. As a consequence we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time averaged mean square displacement analysis fails. We apply this method to both simulations and in vivo measurements of telomere diffusion in 3T3 mouse embryonic fibroblast cells. The motion of telomeres is found to be subdiffusive with an average exponent constant in time. Individual telomere exponents are normally distributed around the average exponent. The proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity.