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

Burnecki, Krzysztof; Kepten, Eldad; Garini, Yuval; Sikora, Grzegorz; Weron, Aleksander

doi:10.1038/srep11306

Cited by 81 publications

(77 citation statements)

References 63 publications

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“…proposed to accurately estimate this exponent for single trajectories [7,8] in the presence of experimental limitations, such as optical diffraction and the finite length of the trajectory.…”

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confidence: 99%

“…In this scenario, determining whether a single-molecule trajectory (or a short segment of it) displays normal or anomalous behavior by its tMSD scaling exponent and associating the motion to a specific physical model are elements of paramount importance to gain insight about the biophysical mechanism underlying anomalous diffusion, thus providing a detailed picture of a variety of phenomena. Recent works in this direction have focused on classification schemes based on optimization procedures [8], power spectral density [25], or Bayesian approaches [26,27].…”

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confidence: 99%

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Single trajectory characterization via machine learning

et al. 2020

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In order to study transport in complex environments, it is extremely important to determine the physical mechanism underlying diffusion and precisely characterize its nature and parameters. Often, this task is strongly impacted by data consisting of trajectories with short length (either due to brief recordings or previous trajectory segmentation) and limited localization precision. In this paper, we propose a machine learning method based on a random forest architecture, which is able to associate single trajectories to the underlying diffusion mechanism with high accuracy. In addition, the algorithm is able to determine the anomalous exponent with a small error, thus inherently providing a classification of the motion as normal or anomalous (subor super-diffusion). The method provides highly accurate outputs even when working with very short trajectories and in the presence of experimental noise. We further demonstrate the application of transfer learning to experimental and simulated data not included in the training/test dataset. This allows for a full, high-accuracy characterization of experimental trajectories without the need of any prior information.In the last decades, the research in biophysics has conveyed large efforts toward the development of experimental techniques allowing the visualization of biological processes one molecule at a time [1-4]. These efforts have been mainly driven by the concept that ensemble-averaging hides important features that are relevant for cellular function. Somehow expectedly, experiments performed by means of these techniques have shown a large heterogeneity in the behavior of biological molecules, thus fully justifying the use of these raffinate tools.Besides, experiments performed using single particle tracking [3] have revealed that even chemically-identical molecules in biological media can display very different behaviors, as a consequence of the complex environment where diffusion takes place. By way of example, this heterogeneity is reflected in the broad distribution of dynamic parameters of distinct individual trajectories corresponding to the same molecular species, such as the diffusion coefficient, well above stochastic indetermination. Typically, the trajectories are analyzed by quantifying the (time-averaged) mean square displacement (tMSD) as a function of the time lag τ [5]:The calculation of this quantity-expected to scale linearly for a Brownian walker in a homogeneous environment-has provided a ubiquitous evidence of anomalous behaviors in biological systems, characterized by an asymptotic nonlinear scaling of the tMSD curve d t a 2. More experiments have shown that the anomalous exponent can vary from particle to particle ( figure 1(a)) as a consequence of molecular interactions and that these changes can be experienced by the same particle in space/time [6]. Several methods have been OPEN ACCESS RECEIVED

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Single trajectory characterization via machine learning

et al. 2020

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“…Pitfalls of MSD analysis, particularly with biological data(see Results and Discussion for examples), include the requirement for many data points (12), difficulties in selecting between competing models (13), difficulties with handling systems with two distinct subpopulations (12,14), and the relatively large errors in cases when trajectories are short. Several studies have tried to improve and expand upon MSD analysis, and also have proposed new methods of extracting models and parameters (12)(13)(14)(15)(16)(17)(18)(19).…”

Section: Mean Squared Displacementmentioning

confidence: 99%

A jump distance based parameter inference scheme for particulate trajectories in biological settings

Menssen

Mani

2017

Preprint

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Modern biology is a treasure trove of data. With all this data, it is helpful to have analytical tools that are applicable regardless of context. One type of data that needs more quantitative analytical tools is particulate trajectories. This type of data appears in many different contexts and across scales in biology: from inferring statistics of a bacteria performing chemotaxis to the mobility of ms2 spots within nuclei. Presently, most analyses performed on data of this nature has been limited to mean square displacement (MSD) analyses. While simple, MSD analysis has several pitfalls, including difficulty in selecting between competing models, how to handle systems with multiple distinct sub-populations, and parameter extraction from limited time-series data sets. Here, we provide an alternative to MSD analysis using the jump distance distribution (JDD) [1,2]. The JDD resolves several issues: one can select between competing models of motion, have composite models that allow for multiple populations, and have improved error bounds on parameter estimates when data is limited. A major consequence is that you can perform analyses using a fraction of the data required to get similar results using MSD analyses, thereby giving access to a larger range of temporal dynamics when the underlying stochastic process is not stationary. In this paper, we construct and validate a derivation of the JDD for different transport models, explore the dependence on dimensionality of the process (1-3 dimensions), and implement a parameter estimation and model selection scheme. Finally, we discuss extensions of our scheme and its applications to biological data. Author summaryMean square displacement (MSD) analyses have been the standard for analyzing particulate trajectories, where its shortcomings have been overlooked in light of its simplicity. The Jump Distance Distribution (JDD) has been proposed by others in the past as a new way to analyze particulate trajectories, but has not been sufficiently analyzed in varying numbers of dimensions or given a robust analysis on performance and how it compares to MSD analysis. We present the forms of the JDD in 1, 2, and 3 dimensions for three different models for transport: pure diffusion, directed diffusion, and anomalous diffusion. We also discuss how to select between competing models, and verify our method with a rigorous analysis. Through this, we have a method that is superior to a MSD analysis. This method works across a wide range of parameters, which should make it broadly applicable to any system where the underlying motion is stochastic.

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“…For normal diffusion (i.e., pure Brownian motion), = 1, while for anomalous diffusion, the MSD can be subdiffusive ( < 1), or superdiffusive ( > 1), ( Fig. 1) (14,15).…”

Section: Introductionmentioning

confidence: 99%

“…For example, using only the first two time lags of the MSD curve to determine the diffusion coefficient in pure Brownian trajectories was shown to be preferable under most conditions because it was more robust to noise (18). To the best of our knowledge, identifying the optimal approach for the relevant parameters describing subdiffusive motion has not yet been investigated, despite the known inaccuracies of applying traditional methods to deficient datasets (14,15). Of particular interest is how to make use of short trajectories to accurately extract subdiffusive parameters, i.e.…”

Section: Introductionmentioning

confidence: 99%

Single particle diffusion characterization by deep learning

Granik

Nehme

Weiss

et al. 2019

Preprint

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Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion, but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools to distinguish between these processes are based on asymptotic behavior, which is inaccessible experimentally in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables, since different transport modes can result in the same diffusion power-law α, that is obtained from the commonly used mean-squared-displacement (MSD) in its various forms. The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the non-expert level.Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single particle trajectories according to diffusion type -Brownian motion, fractional Brownian motion (FBM) and Continuous Time Random Walk (CTRW). We further use the net to estimate the Hurst exponent for FBM, and the diffusion coefficient for Brownian motion, demonstrating its applicability on simulated and experimental data. The networks outperform time averaged MSD analysis on simulated trajectories while requiring as few as 25 time-steps. Furthermore, when tested on experimental data, both network and ensemble MSD analysis converge to similar values, with the network requiring half the trajectories required for ensemble MSD. Finally, we use the nets to extract diffusion parameters from multiple extremely short trajectories (10 steps).

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

Cited by 81 publications

References 63 publications

Single trajectory characterization via machine learning

Single trajectory characterization via machine learning

A jump distance based parameter inference scheme for particulate trajectories in biological settings

Single particle diffusion characterization by deep learning

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