Statistical analysis of particle trajectories in living cells

Overstretching alveolar epithelial type II cells decreases surfactant secretion via actin polymerization and intracellular trafficking alteration

Pulmonary surfactant is essential for maintaining proper lung function. Alveolar epithelial type II (AE2) cells secrete surfactants via lamellar bodies (LBs). In tidal loading during each breath, the physiological cyclic stretching of AE2 cells promotes surfactant secretion. Excessive stretching inhibits surfactant secretion, which is considered to contribute to the development of lung damage. However, its precise mechanism remains unknown. This study tested whether actin polymerization and intracellular transport are required for pulmonary surfactant secretion and the association of actin polymerization and transport in identical human AE2-derived A549 cells using live-cell imaging, not in the bulk cells population. We found that overstretching approximately doubled actin polymerization into filaments (F-actin) and suppressed LB secretion by half in the fluorescent area ratio, compared with physiological stretching (F-actin: 1.495 vs 0.643 (P < 0.01); LB: 0.739 vs 0.332 (P < 0.01)). An inhibitor of actin polymerization increased LB secretion. Intracellular tracking using fluorescent particles revealed that cyclic stretching shifted the particle motion perpendicularly to the direction of stretching according to the orientation of the F-actin (proportion of perpendicular axis motion prior particle: 0h 40.12 % vs 2h 63.13 % (P < 0.01)), and particle motion was restricted over time in the cells subjected to overstretching, indicating that overstretching regulates intracellular transport dynamics (proportion of stop motion particle: 0h 1.01 % vs 2h 11.04 % (P < 0.01)). These findings suggest that overstretching changes secretion through the cytoskeleton: overstretching AE2 cells inhibits pulmonary surfactant secretion, at least through accelerating actin polymerization and decreasing intracellular trafficking, and the change in actin orientation would modulate intracellular trafficking.

Temporal analysis of relative distances (TARDIS) is a robust, parameter-free alternative to single-particle tracking

In single-particle tracking, individual particles are localized and tracked over time to probe their diffusion and molecular interactions. Temporal crossing of trajectories, blinking particles, and false-positive localizations present computational challenges that have remained difficult to overcome. Here we introduce a robust, parameter-free alternative to single-particle tracking: temporal analysis of relative distances (TARDIS). In TARDIS, an all-to-all distance analysis between localizations is performed with increasing temporal shifts. These pairwise distances represent either intraparticle distances originating from the same particle, or interparticle distances originating from unrelated particles, and are fitted analytically to obtain quantitative measures on particle dynamics. We showcase that TARDIS outperforms tracking algorithms, benchmarked on simulated and experimental data of varying complexity. We further show that TARDIS performs accurately in complex conditions characterized by high particle density, strong emitter blinking or false-positive localizations, and is in fact limited by the capabilities of localization algorithms. TARDIS' robustness enables fivefold shorter measurements without loss of information.

Changes in searching behaviour of CSL transcription complexes in Notch active conditions

During development cells receive a variety of signals, which are of crucial importance to their fate determination. One such source of signal is the Notch signalling pathway, where Notch activity regulates expression of target genes through the core transcription factor CSL. To understand changes in transcription factor behaviour that lead to transcriptional changes in Notch active cells, we have probed CSL behaviours in real time, using in vivo Single Molecule Localisation Microscopy. Trajectory analysis reveals that Notch-On conditions increase the fraction of bound CSL molecules, but also the proportion of molecules with exploratory behaviours. These properties are shared by the co-activator Mastermind. Furthermore, both CSL and Mastermind, exhibit characteristics of local exploration near a Notch target locus. A similar behaviour is observed for CSL molecules diffusing in the vicinity of other bound CSL clusters. We suggest therefore that CSL acquires an exploratory behaviour when part of the activation complex, favouring local searching and retention close to its target enhancers. This change explains how CSL can efficiently increase its occupancy at target sites in Notch-On conditions.

A generative model to simulate spatiotemporal dynamics of biomolecules in cells

Generators of space-time dynamics in bioimaging have become essential to build ground truth datasets for image processing algorithm evaluation such as biomolecule detectors and trackers, as well as to generate training datasets for deep learning algorithms. In this contribution, we leverage a stochastic model, called birth-death-move (BDM) point process, in order to generate joint dynamics of biomolecules in cells. This particle-based stochastic simulation method is very flexible and can be seen as a generalization of well-established standard particle-based generators. In comparison, our approach allows us: (1) to model a system of particles in motion, possibly in interaction, that can each possibly switch from a motion regime (e.g., Brownian) to another (e.g., a directed motion); (2) to take into account finely the appearance over time of new trajectories and their disappearance, these events possibly depending on the cell regions but also on the current spatial configuration of all existing particles. This flexibility enables to generate more realistic dynamics than standard particle-based simulation procedures, by for example accounting for the colocalization phenomena often observed between intracellular vesicles. We explain how to specify all characteristics of a BDM model, with many practical examples that are relevant for bioimaging applications. As an illustration, based on real fluorescence microscopy datasets, we finally calibrate our model to mimic the joint dynamics of Langerin and Rab11 proteins near the plasma membrane, including the well-known colocalization occurrence between these two types of vesicles. We show that the resulting synthetic sequences exhibit comparable features as those observed in real microscopy image sequences.

Brain-localized CD4 and CD8 T cells perform correlated random walks and not Levy walks

Background. For survival of the organism, T cells must efficiently control pathogens invading different peripheral tissues. Whether or not such control is achieved by utilizing different movement strategies in different tissues remains poorly understood. Liver-localized CD8 T cells perform correlated random walks  --- a type of a Brownian walk -- in liver sinusoids but in some condition these T cells may also perform Levy flights -- rapid and large displacements by floating with the blood flow. CD8 T cells in lymph nodes or skin also undergo Brownian walks. A recent study suggested that brain-localized CD8 T cells, specific to Toxoplasma gondii, perform generalized Levy walks -- a walk type in which T cells alternate pausing and displacing long distances --- which may indicate that brain is a unique organ where T cells exhibit movement strategies different from other tissues. Methods.  We quantified movement patterns of brain-localized Plasmodium berghei-specific CD4 and CD8 T cells by using well-established statistical and computational methods. Results.  We found that T cells change their movement pattern with time since infection and that CD4 T cells move faster and turn less than CD8 T cells. Importantly, both CD4 and CD8 T cells move in the brain by correlated random walks without long displacements challenging previous observations. We have also re-analyzed the movement data of brain-localized CD8 T cells in T. gondii-infected mice and found no evidence of Levy walks. We hypothesize that the previous conclusion of Levy walks of T. gondii-specific CD8 T cells in the brain was reached due to missing time-frames in the data that create an impression of large movement lengths between assumed-to-be-sequential movements.  Conclusion. Our results suggests that movement strategies of CD8 T cells are largely similar between LNs, liver, and the brain and consistent with correlated random walks and not Levy walks.

Confined modes of single-particle trajectories induced by stochastic resetting

Random trajectories of single particles in living cells contain information about the interaction between particles, as well as with the cellular environment. However, precise consideration of the underlying stochastic properties, beyond normal diffusion, remains a challenge as applied to each particle trajectory separately. In this paper, we show how positions of confined particles in living cells can obey not only the Laplace distribution, but the Linnik one. This feature is detected in experimental data for the motion of G proteins and coupled receptors in cells, and its origin is explained in terms of stochastic resetting. This resetting process generates power-law waiting times, giving rise to the Linnik statistics in confined motion, and also includes exponentially distributed times as a limit case leading to the Laplace one. The stochastic process, which is affected by the resetting, can be Brownian motion commonly found in cells. Other possible models producing similar effects are discussed.

Tracking receptor motions at the plasma membrane reveals distinct effects of ligands on CCR5 dynamics depending on its dimerization status

G-protein-coupled receptors (GPCR) are present at the cell surface in different conformational and oligomeric states. However, how these states impact GPCRs biological function and therapeutic targeting remains incompletely known. Here, we investigated this issue in living cells for the CC chemokine receptor 5 (CCR5), a major receptor in inflammation and the principal entry co-receptor for Human Immunodeficiency Viruses type 1 (HIV-1). We used TIRF microscopy and a statistical method to track and classify the motion of different receptor subpopulations. We showed a diversity of ligand-free forms of CCR5 at the cell surface constituted of various oligomeric states and exhibiting transient Brownian and restricted motions. These forms were stabilized differently by distinct ligands. In particular, agonist stimulation restricted the mobility of CCR5 and led to its clustering, a feature depending on β-arrestin, while inverse agonist stimulation exhibited the opposite effect. These results suggest a link between receptor activation and immobilization. Applied to HIV-1 envelope glycoproteins gp120, our quantitative analysis revealed agonist-like properties of gp120s. Distinct gp120s influenced CCR5 dynamics differently, suggesting that they stabilize different CCR5 conformations. Then, using a dimerization-compromized mutant, we showed that dimerization (i) impacts CCR5 precoupling to G proteins, (ii) is a pre-requisite for the immobilization and clustering of receptors upon activation, and (iii) regulates receptor endocytosis, thereby impacting the fate of activated receptors. This study demonstrates that tracking the dynamic behavior of a GPCR is an efficient way to link GPCR conformations to their functions, therefore improving the development of drugs targeting specific receptor conformations.

PKAN hiPS-Derived Astrocytes Show Impairment of Endosomal Trafficking: A Potential Mechanism Underlying Iron Accumulation

PKAN disease is caused by mutations in the PANK2 gene, encoding the mitochondrial enzyme pantothenate kinase 2, catalyzing the first and key reaction in Coenzyme A (CoA) biosynthetic process. This disorder is characterized by progressive neurodegeneration and excessive iron deposition in the brain. The pathogenic mechanisms of PKAN are still unclear, and the available therapies are only symptomatic. Although iron accumulation is a hallmark of PKAN, its relationship with CoA dysfunction is not clear. We have previously developed hiPS-derived astrocytes from PKAN patients showing iron overload, thus recapitulating the human phenotype. In this work, we demonstrated that PKAN astrocytes presented an increase in transferrin uptake, a key route for cellular iron intake via transferrin receptor-mediated endocytosis of transferrin-bound iron. Investigation of constitutive exo-endocytosis and vesicular dynamics, exploiting the activity-enriching biosensor SynaptoZip, led to the finding of a general impairment in the constitutive endosomal trafficking in PKAN astrocytes. CoA and 4-phenylbutyric acid treatments were found to be effective in partially rescuing the aberrant vesicular behavior and iron intake. Our results demonstrate that the impairment of CoA biosynthesis could interfere with pivotal intracellular mechanisms involved in membrane fusions and vesicular trafficking, leading to an aberrant transferrin receptor-mediated iron uptake.

Impact of neurite alignment on organelle motion

Intracellular transport is pivotal for cell growth and survival. Malfunctions in this process have been associated with devastating neurodegenerative diseases, highlighting the need for a deeper understanding of the mechanisms involved. Here, we use an experimental methodology that leads neurites of differentiated PC12 cells into either one of two configurations: a one-dimensional configuration, where the neurites align along lines, or a two-dimensional configuration, where the neurites adopt a random orientation and shape on a flat substrate. We subsequently monitored the motion of functional organelles, the lysosomes, inside the neurites. Implementing a time-resolved analysis of the mean-squared displacement, we quantitatively characterized distinct motion modes of the lysosomes. Our results indicate that neurite alignment gives rise to faster diffusive and super-diffusive lysosomal motion than the situation in which the neurites are randomly oriented. After inducing lysosome swelling through an osmotic challenge by sucrose, we confirmed the predicted slowdown in diffusive mobility. Surprisingly, we found that the swelling-induced mobility change affected each of the (sub-/super-)diffusive motion modes differently and depended on the alignment configuration of the neurites. Our findings imply that intracellular transport is significantly and robustly dependent on cell morphology, which might in part be controlled by the extracellular matrix.

An overview of diffusion models for intracellular dynamics analysis

We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g. biomolecules) inside eukaryotic living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g. connectivity for signal transduction). Another dynamical process that has been heavily studied in the past decade is the motor-mediated transport (e.g. dynein, kinesin and myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in eukariotic cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport and membrane diffusion.

Detecting Transient Trapping from a Single Trajectory: A Structural Approach

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 β2-adrenergic receptors, revealing that receptor stimulation results in increased trapping of receptors in defined domains, without changing the diffusion of free receptors.

Single-particle diffusional fingerprinting: A machine-learning framework for quantitative analysis of heterogeneous diffusion

Single-particle tracking (SPT) analysis of individual biomolecules is an indispensable tool for extracting quantitative information from dynamic biological processes, but often requires some a priori knowledge of the system. Here we present “single-particle diffusional fingerprinting,” a more general approach for extraction of diffusional patterns in SPT independently of the biological system. This method extracts a set of descriptive features for each SPT trajectory, which are ranked upon classification to yield mechanistic insights for the species under comparison. We demonstrate its capacity to yield a dictionary of diffusional traits across multiple systems (e.g., lipases hydrolyzing fat, transcription factors diffusing in cells, and nanoparticles in mucus), supporting its use on multiple biological phenomena (e.g., drug delivery, receptor dynamics, and virology).

Single-particle tracking (SPT) is a key tool for quantitative analysis of dynamic biological processes and has provided unprecedented insights into a wide range of systems such as receptor localization, enzyme propulsion, bacteria motility, and drug nanocarrier delivery. The inherently complex diffusion in such biological systems can vary drastically both in time and across systems, consequently imposing considerable analytical challenges, and currently requires an a priori knowledge of the system. Here we introduce a method for SPT data analysis, processing, and classification, which we term “diffusional fingerprinting.” This method allows for dissecting the features that underlie diffusional behavior and establishing molecular identity, regardless of the underlying diffusion type. The method operates by isolating 17 descriptive features for each observed motion trajectory and generating a diffusional map of all features for each type of particle. Precise classification of the diffusing particle identity is then obtained by training a simple logistic regression model. A linear discriminant analysis generates a feature ranking that outputs the main differences among diffusional features, providing key mechanistic insights. Fingerprinting operates by both training on and predicting experimental data, without the need for pretraining on simulated data. We found this approach to work across a wide range of simulated and experimentally diverse systems, such as tracked lipases on fat substrates, transcription factors diffusing in cells, and nanoparticles diffusing in mucus. This flexibility ultimately supports diffusional fingerprinting’s utility as a universal paradigm for SPT diffusional analysis and prediction.

Classification of particle trajectories in living cells: Machine learning versus statistical testing hypothesis for fractional anomalous diffusion

Single-particle tracking (SPT) has become a popular tool to study the intracellular transport of molecules in living cells. Inferring the character of their dynamics is important, because it determines the organization and functions of the cells. For this reason, one of the first steps in the analysis of SPT data is the identification of the diffusion type of the observed particles. The most popular method to identify the class of a trajectory is based on the mean-square displacement (MSD). However, due to its known limitations, several other approaches have been already proposed. With the recent advances in algorithms and the developments of modern hardware, the classification attempts rooted in machine learning (ML) are of particular interest. In this work, we adopt two ML ensemble algorithms, i.e., random forest and gradient boosting, to the problem of trajectory classification. We present a new set of features used to transform the raw trajectories data into input vectors required by the classifiers. The resulting models are then applied to real data for G protein-coupled receptors and G proteins. The classification results are compared to recent statistical methods going beyond MSD.

Detection of Anomalous Diffusion with Deep Residual Networks

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.

Bioimage Analysis and Cell Motility

Bioimage analysis (BIA) has historically helped study how and why cells move; biological experiments evolved in intimate feedback with the most classical image processing techniques because they contribute objectivity and reproducibility to an eminently qualitative science. Cell segmentation, tracking, and morphology descriptors are all discussed here. Using ameboid motility as a case study, these methods help us illustrate how proper quantification can augment biological data, for example, by choosing mathematical representations that amplify initially subtle differences, by statistically uncovering general laws or by integrating physical insight. More recently, the non-invasive nature of quantitative imaging is fertilizing two blooming fields: mechanobiology, where many biophysical measurements remain inaccessible, and microenvironments, where the quest for physiological relevance has exploded data size. From relief to remedy, this trend indicates that BIA is to become a main vector of biological discovery as human visual analysis struggles against ever more complex data.

Time-dependent classification of protein diffusion types: A statistical detection of mean-squared-displacement exponent transitions

In this paper, we have proposed a statistical procedure for detecting transitions of the mean-square-displacement exponent value within a single trajectory. With this procedure, we have identified three regimes of proteins dynamics on a cell membrane, namely, subdiffusion, free diffusion, and immobility. The fourth considered dynamics type, namely, superdiffusion was not detected. We show that the analyzed protein trajectories are not stationary and not ergodic. Moreover, classification of the dynamics type performed without prior detection of transitions may lead to the overestimation of the proportion of subdiffusive trajectories.

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

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.

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

The advent of single molecule microscopy has revolutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions.

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

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.

A sequential algorithm to detect diffusion switching along intracellular particle trajectories

MOTIVATION

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of single molecules in living cells. Changes of dynamics can occur along a trajectory. Then, an issue is to estimate the temporal change-points that is the times at which a change of dynamics occurs. The number of points in the trajectory required to detect such changes will depend on both the magnitude and type of the motion changes. Here, the number of points per trajectory is of the order of 102, even if in practice dramatic motion changes can be detected with less points.

RESULTS

We propose a non-parametric procedure based on test statistics computed on local windows along the trajectory to detect the change-points. This algorithm controls the number of false change-point detections in the case where the trajectory is fully Brownian. We also develop a strategy for aggregating the detections obtained with different window sizes so that the window size is no longer a parameter to optimize. A Monte Carlo study is proposed to demonstrate the performances of the method and also to compare the procedure to two competitive algorithms. At the end, we illustrate the efficacy of the method on real data in 2D and 3D, depicting the motion of mRNA complexes-called mRNA-binding proteins-in neuronal dendrites, Galectin-3 endocytosis and trafficking within the cell.

AVAILABILITY AND IMPLEMENTATION

A user-friendly Matlab package containing examples and the code of the simulations used in the paper is available at http://serpico.rennes.inria.fr/doku.php? id=software:cpanalysis:index.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Biophysics of high density nanometer regions extracted from super-resolution single particle trajectories: application to voltage-gated calcium channels and phospholipids

The cellular membrane is very heterogenous and enriched with high-density regions forming microdomains, as revealed by single particle tracking experiments. However the organization of these regions remain unexplained. We determine here the biophysical properties of these regions, when described as a basin of attraction. We develop two methods to recover the dynamics and local potential wells (field of force and boundary). The first method is based on the local density of points distribution of trajectories, which differs inside and outside the wells. The second method focuses on recovering the drift field that is convergent inside wells and uses the transient field to determine the boundary. Finally, we apply these two methods to the distribution of trajectories recorded from voltage gated calcium channels and phospholipid anchored GFP in the cell membrane of hippocampal neurons and obtain the size and energy of high-density regions with a nanometer precision.

Statistical testing approach for fractional anomalous diffusion classification

Taking advantage of recent single-particle tracking data, we compare the popular standard mean-squared displacement method with a statistical testing hypothesis procedure for three testing statistics and for two particle types: membrane receptors and the G proteins coupled to them. Each method results in different classifications. For this reason, more rigorous statistical tests are analyzed here in detail. The main conclusion is that the statistical testing approaches might provide good results even for short trajectories, but none of the proposed methods is "the best" for all considered examples; in other words, one needs to combine different approaches to get a reliable classification.

Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models

In this paper we show that an autoregressive fractionally integrated moving average time-series model can identify two types of motion of membrane proteins on the surface of mammalian cells. Specifically we analyze the motion of the voltage-gated sodium channel Nav1.6 and beta-2 adrenergic receptors. We find that the autoregressive (AR) part models well the confined dynamics whereas the fractionally integrated moving average (FIMA) model describes the nonconfined periods of the trajectories. Since the Ornstein-Uhlenbeck process is a continuous counterpart of the AR model, we are also able to calculate its physical parameters and show their biological relevance. The fitted FIMA and AR parameters show marked differences in the dynamics of the two studied molecules.

Fluid heterogeneity detection based on the asymptotic distribution of the time-averaged mean squared displacement in single particle tracking experiments

A tracer particle is called anomalously diffusive if its mean squared displacement grows approximately as σ 2 t α as a function of time t for some constant σ 2, where the diffusion exponent satisfies α ≠ 1. In this article, we use recent results on the asymptotic distribution of the time-averaged mean squared displacement [20] to construct statistical tests for detecting physical heterogeneity in viscoelastic fluid samples starting from one or multiple observed anomalously diffusive paths. The methods are asymptotically valid for the range 0 < α < 3/2 and involve a mathematical characterization of time-averaged mean squared displacement bias and the effect of correlated disturbance errors. The assumptions on particle motion cover a broad family of fractional Gaussian processes, including fractional Brownian motion and many fractional instances of the generalized Langevin equation framework. We apply the proposed methods in experimental data from treated P. aeruginosa biofilms generated by the collaboration of the Hill and Schoenfisch Labs at UNC-Chapel Hill.

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?