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Recent Advances in Single-Particle Tracking: Experiment and Analysis

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (15 June 2021) | Viewed by 42538

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Special Issue Editors

Dr. Janusz Szwabiński

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Guest Editor

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Interests: complex systems; agent-based modeling; bio- and sociophysics; ergodicity testing; anomalous diffusion; ML in time series analysis

Prof. Dr. Aleksander Weron

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Guest Editor

Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Interests: stochastic processes; financial mathematics; anomalous diffusion; ergodicity testing; fractional dynamics; molecule imaging

Special Issue Information

Dear colleagues,

Recent advances in single-molecule microscopy and imaging technologies have made single-particle tracking (SPT) a popular method for the analysis of dynamic processes in a range of materials, including living cells. This is of particular importance, especially in the latter case, because it bridges the gap between biology, biochemistry, and biophysics and allows for an at least partial understanding of living cells on a microscopic basis.

Trajectories inferred from SPT data can be modeled according to different kinds of diffusion. The mean-square displacement (MSD) method is commonly used to discriminate the diffusion modes. However, due to finite localization precision, the stochastic nature of particle motion, and the commonly (too) short lengths of trajectories, it is a challenging task to extract a meaningful information from MSD data. Several alternative methods have already been introduced to overcome these problems, with the most recent ones rooted in the machine learning approach to classification. However, further progress in this field is required to provide robust methods that generalize well to experimental data.

Traditionally, SPT is performed in two dimensions because of the technical simplicity. However, life occurs in three dimensions, and restraining the analysis of trajectories to their 2D projections may also limit the accuracy of the obtained results. Thus, contributions addressing the issue of tracking and analysis of single particles in 3D are welcome.

The aim of this Special Issue is to gather the latest developments in single-particle tracking, with a focus on both the collection of data as well as classical statistics and machine learning approaches as applied to the characterization of trajectories.

Dr. Janusz Szwabiński
Prof. Aleksander Weron
Guest Editors

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Keywords

molecule imaging
single-particle tracking
anomalous diffusion
trajectory classification
machine learning
statistical analysis
time series classification

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Published Papers (13 papers)

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Research

16 pages, 2989 KiB

Open AccessArticle

Detecting Transient Trapping from a Single Trajectory: A Structural Approach

by Yann Lanoiselée, Jak Grimes, Zsombor Koszegi and Davide Calebiro

Entropy 2021, 23(8), 1044; https://doi.org/10.3390/e23081044 - 13 Aug 2021

Cited by 9 | Viewed by 2705

Abstract

In this article, we introduce a new method to detect transient trapping events within a single particle trajectory, thus allowing the explicit accounting of changes in the particle’s dynamics over time. Our method is based on new measures of a smoothed recurrence matrix. The newly introduced set of measures takes into account both the spatial and temporal structure of the trajectory. Therefore, it is adapted to study short-lived trapping domains that are not visited by multiple trajectories. Contrary to most existing methods, it does not rely on using a window, sliding along the trajectory, but rather investigates the trajectory as a whole. This method provides useful information to study intracellular and plasma membrane compartmentalisation. Additionally, this method is applied to single particle trajectory data of

β_{2}

-adrenergic receptors, revealing that receptor stimulation results in increased trapping of receptors in defined domains, without changing the diffusion of free receptors. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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15 pages, 1710 KiB

Open AccessArticle

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

by Nickolay Korabel, Daniel Han, Alessandro Taloni, Gianni Pagnini, Sergei Fedotov, Viki Allan and Thomas Andrew Waigh

Entropy 2021, 23(8), 958; https://doi.org/10.3390/e23080958 - 27 Jul 2021

Cited by 20 | Viewed by 2662

Abstract

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: “Recent Advances In Single-Particle Tracking: Experiment and Analysis” edited by Janusz Szwabiński and Aleksander Weron. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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16 pages, 1078 KiB

Open AccessArticle

Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts

by Konstantin Speckner and Matthias Weiss

Entropy 2021, 23(7), 892; https://doi.org/10.3390/e23070892 - 13 Jul 2021

Cited by 19 | Viewed by 3212

Abstract

Single-particle tracking (SPT) has become a powerful tool to quantify transport phenomena in complex media with unprecedented detail. Based on the reconstruction of individual trajectories, a wealth of informative measures become available for each particle, allowing for a detailed comparison with theoretical predictions. While SPT has been used frequently to explore diffusive transport in artificial fluids and inside living cells, intermediate systems, i.e., biochemically active cell extracts, have been studied only sparsely. Extracts derived from the eggs of the clawfrog Xenopus laevis, for example, are known for their ability to support and mimic vital processes of cells, emphasizing the need to explore also the transport phenomena of nano-sized particles in such extracts. Here, we have performed extensive SPT on beads with 20 nm radius in native and chemically treated Xenopus extracts. By analyzing a variety of distinct measures, we show that these beads feature an anti-persistent subdiffusion that is consistent with fractional Brownian motion. Chemical treatments did not grossly alter this finding, suggesting that the high degree of macromolecular crowding in Xenopus extracts equips the fluid with a viscoelastic modulus, hence enforcing particles to perform random walks with a significant anti-persistent memory kernel. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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21 pages, 4623 KiB

Open AccessArticle

Detection of Anomalous Diffusion with Deep Residual Networks

by Miłosz Gajowczyk and Janusz Szwabiński

Entropy 2021, 23(6), 649; https://doi.org/10.3390/e23060649 - 22 May 2021

Cited by 10 | Viewed by 3009

Abstract

Identification of the diffusion type of molecules in living cells is crucial to deduct their driving forces and hence to get insight into the characteristics of the cells. In this paper, deep residual networks have been used to classify the trajectories of molecules. We started from the well known ResNet architecture, developed for image classification, and carried out a series of numerical experiments to adapt it to detection of diffusion modes. We managed to find a model that has a better accuracy than the initial network, but contains only a small fraction of its parameters. The reduced size significantly shortened the training time of the model. Moreover, the resulting network has less tendency to overfitting and generalizes better to unseen data. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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10 pages, 2520 KiB

Open AccessEditor’s ChoiceArticle

Information-Efficient, Off-Center Sampling Results in Improved Precision in 3D Single-Particle Tracking Microscopy

by Chen Zhang and Kevin Welsher

Entropy 2021, 23(5), 498; https://doi.org/10.3390/e23050498 - 22 Apr 2021

Cited by 10 | Viewed by 3082

Abstract

In this work, we present a 3D single-particle tracking system that can apply tailored sampling patterns to selectively extract photons that yield the most information for particle localization. We demonstrate that off-center sampling at locations predicted by Fisher information utilizes photons most efficiently. When performing localization in a single dimension, optimized off-center sampling patterns gave doubled precision compared to uniform sampling. A ~20% increase in precision compared to uniform sampling can be achieved when a similar off-center pattern is used in 3D localization. Here, we systematically investigated the photon efficiency of different emission patterns in a diffraction-limited system and achieved higher precision than uniform sampling. The ability to maximize information from the limited number of photons demonstrated here is critical for particle tracking applications in biological samples, where photons may be limited. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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11 pages, 5778 KiB

Open AccessArticle

Study of Wound Healing Dynamics by Single Pseudo-Particle Tracking in Phase Contrast Images Acquired in Time-Lapse

by Riccardo Scheda, Silvia Vitali, Enrico Giampieri, Gianni Pagnini and Isabella Zironi

Entropy 2021, 23(3), 284; https://doi.org/10.3390/e23030284 - 26 Feb 2021

Viewed by 2391

Abstract

Cellular contacts modify the way cells migrate in a cohesive group with respect to a free single cell. The resulting motion is persistent and correlated, with cells’ velocities self-aligning in time. The presence of a dense agglomerate of cells makes the application of single particle tracking techniques to define cells dynamics difficult, especially in the case of phase contrast images. Here, we propose an original pipeline for the analysis of phase contrast images of the wound healing scratch assay acquired in time-lapse, with the aim of extracting single particle trajectories describing the dynamics of the wound closure. In such an approach, the membrane of the cells at the border of the wound is taken as a unicum, i.e., the wound edge, and the dynamics is described by the stochastic motion of an ensemble of points on such a membrane, i.e., pseudo-particles. For each single frame, the pipeline of analysis includes: first, a texture classification for separating the background from the cells and for identifying the wound edge; second, the computation of the coordinates of the ensemble of pseudo-particles, chosen to be uniformly distributed along the length of the wound edge. We show the results of this method applied to a glioma cell line (T98G) performing a wound healing scratch assay without external stimuli. We discuss the efficiency of the method to assess cell motility and possible applications to other experimental layouts, such as single cell motion. The pipeline is developed in the Python language and is available upon request. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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32 pages, 1324 KiB

Open AccessArticle

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

by M. Hidalgo-Soria, E. Barkai and S. Burov

Entropy 2021, 23(2), 231; https://doi.org/10.3390/e23020231 - 17 Feb 2021

Cited by 19 | Viewed by 2800

Abstract

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants,

D_{+} > D_{-}

, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, [...] Read more.

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants,

D_{+} > D_{-}

, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient

D_{-} ⟶ 0

is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements

P (x, t)

for

x ⟶ 0

. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state

D_{+}

to a

δ

-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in

P (x, t)

. We demonstrate how super-statistical framework is a zeroth order short time expansion of

P (x, t)

, in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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13 pages, 2702 KiB

Open AccessArticle

Extracting Work Optimally with Imprecise Measurements

by Luis Dinis and Juan Manuel Rodríguez Parrondo

Entropy 2021, 23(1), 8; https://doi.org/10.3390/e23010008 - 23 Dec 2020

Cited by 7 | Viewed by 2197

Abstract

Measurement and feedback allows for an external agent to extract work from a system in contact with a single thermal bath. The maximum amount of work that can be extracted in a single measurement and the corresponding feedback loop is given by the information that is acquired via the measurement, a result that manifests the close relation between information theory and stochastic thermodynamics. In this paper, we show how to reversibly confine a Brownian particle in an optical tweezer potential and then extract the corresponding increase of the free energy as work. By repeatedly tracking the position of the particle and modifying the potential accordingly, we can extract work optimally, even with a high degree of inaccuracy in the measurements. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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25 pages, 4430 KiB

Open AccessArticle

Impact of Feature Choice on Machine Learning Classification of Fractional Anomalous Diffusion

by Hanna Loch-Olszewska and Janusz Szwabiński

Entropy 2020, 22(12), 1436; https://doi.org/10.3390/e22121436 - 19 Dec 2020

Cited by 23 | Viewed by 4420

Abstract

The growing interest in machine learning methods has raised the need for a careful study of their application to the experimental single-particle tracking data. In this paper, we present the differences in the classification of the fractional anomalous diffusion trajectories that arise from the selection of the features used in random forest and gradient boosting algorithms. Comparing two recently used sets of human-engineered attributes with a new one, which was tailor-made for the problem, we show the importance of a thoughtful choice of the features and parameters. We also analyse the influence of alterations of synthetic training data set on the classification results. The trained classifiers are tested on real trajectories of G proteins and their receptors on a plasma membrane. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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17 pages, 1928 KiB

Open AccessArticle

Testing of Multifractional Brownian Motion

by Michał Balcerek and Krzysztof Burnecki

Entropy 2020, 22(12), 1403; https://doi.org/10.3390/e22121403 - 12 Dec 2020

Cited by 11 | Viewed by 6044

Abstract

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index

0 < H < 1

. In nature one often observes changes in the dynamics of a system over [...] Read more.

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index

0 < H < 1

. In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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16 pages, 2315 KiB

Open AccessArticle

Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks

by Dawid Szarek, Grzegorz Sikora, Michał Balcerek, Ireneusz Jabłoński and Agnieszka Wyłomańska

Entropy 2020, 22(11), 1322; https://doi.org/10.3390/e22111322 - 20 Nov 2020

Cited by 5 | Viewed by 3473

Abstract

Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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16 pages, 858 KiB

Open AccessArticle

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

by Aleksander Stanislavsky and Aleksander Weron

Entropy 2020, 22(11), 1317; https://doi.org/10.3390/e22111317 - 18 Nov 2020

Cited by 4 | Viewed by 2470

Abstract

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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11 pages, 357 KiB

Open AccessArticle

Diauxic Growth at the Mesoscopic Scale

by Mirosław Lachowicz and Mateusz Dȩbowski

Entropy 2020, 22(11), 1280; https://doi.org/10.3390/e22111280 - 12 Nov 2020

Viewed by 2392

Abstract

In the present paper, we study a diauxic growth that can be generated by a class of model at the mesoscopic scale. Although the diauxic growth can be related to the macroscopic scale, similarly to the logistic scale, one may ask whether models on mesoscopic or microscopic scales may lead to such a behavior. The present paper is the first step towards the developing of the mesoscopic models that lead to a diauxic growth at the macroscopic scale. We propose various nonlinear mesoscopic models conservative or not that lead directly to some diauxic growths. Full article

(This article belongs to the Special Issue Recent Advances in Single-Particle Tracking: Experiment and Analysis)

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