Unravelling the origins of anomalous diffusion: from molecules to migrating storks (original) (raw)
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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.
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
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.
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 : PCCP, 2014
Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ...
Objective comparison of methods to decode anomalous diffusion
Nature Communications, 2021
Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.
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.
A Method for Identifying Diffusive Trajectories with Stochastic Models
Journal of Statistical Physics, 2014
Single particle tracking is a tool that is being increasingly used to study diffusive or dispersive processes in many branches of natural science. Often the ability to collect these trajectories experimentally or produce them numerically outpaces the ability to understand them theoretically. On the other hand many stochastic models have been developed and continue to be developed capable of capturing complex diffusive behavior such as heavy tails, long-range correlations, nonstationarity, and combinations of these things. We describe a computational method for connecting particle trajectory data with stochastic models of diffusion. Several tests are performed to demonstrate the efficacy of the method, and the method is applied to polymer diffusion, RNA diffusion in E. coli, and RAFOS dispersion in the Gulf of Mexico.