Ergodicity convergence test suggests telomere motion obeys fractional dynamics

Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics

Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.

Occupation time statistics of the fractional Brownian motion in a finite domain

We study statistics of occupation times for a fractional Brownian motion (fBm), which is a typical model of a non-Markov process. Due to the non-Markovian nature, recurrence times to the origin depend on the history. Numerical simulations indicate that dependence on the sum of successive recurrence times becomes weak. As a result, the distribution of the occupation time in a finite domain follows the Mittag-Leffler distribution when the Hurst exponent of the fBm is close to 1/2. We show this distributional behavior of a time-averaged observable by renewal theory. This result is an extension of the distributional limit theorem known as the Darling-Kac theorem in general Markov processes to non-Markov processes.

Probing local chromatin dynamics by tracking telomeres

Chromatin dynamics is key for cell viability and replication. In interphase, chromatin is decondensed, allowing the transcription machinery to access a plethora of DNA loci. Yet, decondensed chromatin occupies almost the entire nucleus, suggesting that DNA molecules can hardly move. Recent reports have even indicated that interphase chromatin behaves like a solid body on mesoscopic scales. To explore the local chromatin dynamics, we have performed single-particle tracking on telomeres under varying conditions. We find that mobile telomeres feature in all conditions a strongly subdiffusive, anti-persistent motion that is consistent with the monomer motion of a Rouse polymer in viscoelastic media. In addition, telomere trajectories show intermittent accumulations in local niches at physiological conditions, suggesting the surrounding chromatin to reorganize on these time scales. Reducing the temperature or exposing cells to osmotic stress resulted in a significant reduction of mobile telomeres and the number of visited niches. Altogether, our data indicate a vivid local chromatin dynamics, akin to a semi-dilute polymer solution, unless perturbations enforce a more rigid or entangled state of chromatin.

Fractal and Knot-Free Chromosomes Facilitate Nucleoplasmic Transport

Chromosomes in the nucleus assemble into hierarchies of 3D domains that, during interphase, share essential features with a knot-free condensed polymer known as the fractal globule (FG). The FG-like chromosome likely affects macromolecular transport, yet its characteristics remain poorly understood. Using computer simulations and scaling analysis, we show that the 3D folding and macromolecular size of the chromosomes determine their transport characteristics. Large-scale subdiffusion occurs at a critical particle size where the network of accessible volumes is critically connected. Condensed chromosomes have connectivity networks akin to simple Bernoulli bond percolation clusters, regardless of the polymer models. However, even if the network structures are similar, the tracer's walk dimension varies. It turns out that the walk dimension depends on the network topology of the accessible volume and dynamic heterogeneity of the tracer's hopping rate. We find that the FG structure has a smaller walk dimension than other random geometries, suggesting that the FG-like chromosome structure accelerates macromolecular diffusion and target-search.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...

Chromatin network retards nucleoli coalescence

Nuclear bodies are membraneless condensates that may form via liquid-liquid phase separation. The viscoelastic chromatin network could impact their stability and may hold the key for understanding experimental observations that defy predictions of classical theories. However, quantitative studies on the role of the chromatin network in phase separation have remained challenging. Using a diploid human genome model parameterized with chromosome conformation capture (Hi-C) data, we study the thermodynamics and kinetics of nucleoli formation. Dynamical simulations predict the formation of multiple droplets for nucleolar particles that experience specific interactions with nucleolus-associated domains (NADs). Coarsening dynamics, surface tension, and coalescence kinetics of the simulated droplets are all in quantitative agreement with experimental measurements for nucleoli. Free energy calculations further support that a two-droplet state, often observed for nucleoli in somatic cells, is metastable and separated from the single-droplet state with an entropic barrier. Our study suggests that nucleoli-chromatin interactions facilitate droplets' nucleation but hinder their coarsening due to the coupled motion between droplets and the chromatin network: as droplets coalesce, the chromatin network becomes increasingly constrained. Therefore, the chromatin network supports a nucleation and arrest mechanism to stabilize the multi-droplet state for nucleoli and possibly for other nuclear bodies.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Multiscale modeling of genome organization with maximum entropy optimization

Three-dimensional (3D) organization of the human genome plays an essential role in all DNA-templated processes, including gene transcription, gene regulation, and DNA replication. Computational modeling can be an effective way of building high-resolution genome structures and improving our understanding of these molecular processes. However, it faces significant challenges as the human genome consists of over 6 × 109 base pairs, a system size that exceeds the capacity of traditional modeling approaches. In this perspective, we review the progress that has been made in modeling the human genome. Coarse-grained models parameterized to reproduce experimental data via the maximum entropy optimization algorithm serve as effective means to study genome organization at various length scales. They have provided insight into the principles of whole-genome organization and enabled de novo predictions of chromosome structures from epigenetic modifications. Applications of these models at a near-atomistic resolution further revealed physicochemical interactions that drive the phase separation of disordered proteins and dictate chromatin stability in situ. We conclude with an outlook on the opportunities and challenges in studying chromosome dynamics.

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.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells

Diffusion of tracer particles in the cytoplasm of mammalian cells is often anomalous with a marked heterogeneity even within individual particle trajectories. Despite considerable efforts, the mechanisms behind these observations have remained largely elusive. To tackle this problem, we performed extensive single-particle tracking experiments on quantum dots in the cytoplasm of living mammalian cells at varying conditions. Analyses of the trajectories reveal a strong, microtubule-dependent subdiffusion with antipersistent increments and a substantial heterogeneity. Furthermore, particles stochastically switch between different mobility states, most likely due to transient associations with the cytoskeleton-shaken endoplasmic reticulum network. Comparison to simulations highlight that all experimental observations can be fully described by an intermittent fractional Brownian motion, alternating between two states of different mobility.

Diffusion of DNA-Binding Species in the Nucleus: A Transient Anomalous Subdiffusion Model

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

Chromatin Viscoelasticity Measured by Local Dynamic Analysis

The nucleus in eukaryotic cells is a crowded environment that consists of genetic code along the DNA, together with a condensed solution of proteins, RNA, and other molecules. It is subjected to highly dynamic processes, including cell division, transcription, and DNA repair. In addition, the genome in the nucleus is subjected to external forces applied by the cytoplasmic skeleton and neighboring cells, as well as to internal nuclear forces. These forces oppose the need to maintain the genome order, which may be compensated by the internal nuclear viscoelastic properties that can restrain these forces. The structural and mechanical properties of chromatin inside the nucleus are still not fully clear; however, their importance for the proper functioning of the cells is unquestionable. Different approaches have been developed for this aim, ranging from directly measuring the dynamic and elastic properties of chromatin to studying the interactions of chromatin with the surrounding envelope and nuclear bodies. Although the elasticity of naked DNA in vitro is well characterized, the elasticity of chromatin in live cells is more complex and is still not fully understood. Here, we studied the elastic properties of chromatin by dynamic measurements in live cells, followed by viscoelastic modeling. We measured the trajectories of single chromatin loci, centromeres, and telomeres in live cells and analyzed their dynamics using the Langevin formalism. We assumed that the overall effect of the chromatin network forces can be modeled for each locus by a local harmonic potential and calculated the effective force constant. In addition, we assumed that this harmonic force results from the chromatin network formed by the internal polymer structure together with cross-links formed by the protein complex. We show that lamin A has the greatest effect on chromatin viscoelasticity and that its removal leads to a significant reduction in the local harmonic force.

Resampling single-particle tracking data eliminates localization errors and reveals proper diffusion anomalies

Single-particle tracking (SPT) is a versatile tool for quantifying diffusional motion in complex soft-matter systems, e.g., in biological specimen. Evaluating SPT data often invokes the fitting of a trajectory's time-averaged mean-square displacement (TA-MSD) with a simple power law, 〈r^{2}(τ)〉_{t}∼τ^{α}, where the scaling exponent α can yield important insights into the nature of the transport process. Biological specimen, for example, frequently feature a diffusion anomaly, i.e., an exponent α<1 ("subdiffusion"). However, due to SPT-inherent static and dynamic localization errors, in combination with typically short trajectories, it is often a real challenge to determine the value of α reliably by simply fitting TA-MSDs. Here a straightforward resampling approach is presented and tested that eliminates both localization errors in the TA-MSD of trajectories originating from subdiffusive fractional Brownian motion processes. As a result, the mean anomaly exponent 〈α〉_{E} of an ensemble of trajectories is revealed in a robust fashion.

Subdiffusive continuous-time random walks with stochastic resetting

We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.

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.

Langevin equation in complex media and anomalous diffusion

The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.

Mean-squared-displacement statistical test for fractional Brownian motion

Anomalous diffusion in crowded fluids, e.g., in cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous diffusion of a single particle can be classified is the time-averaged mean square displacement (TAMSD). A classical mechanism leading to the anomalous diffusion is the fractional Brownian motion (FBM). A validation of such process for single-particle tracking data is of great interest for experimentalists. In this paper we propose a rigorous statistical test for FBM based on TAMSD. To this end we analyze the distribution of the TAMSD statistic, which is given by the generalized chi-squared distribution. Next, we study the power of the test by means of Monte Carlo simulations. We show that the test is very sensitive for changes of the Hurst parameter. Moreover, it can easily distinguish between two models of subdiffusion: FBM and continuous-time random walk.

Genome organization in the nucleus: From dynamic measurements to a functional model

A biological system is by definition a dynamic environment encompassing kinetic processes that occur at different length scales and time ranges. To explore this type of system, spatial information needs to be acquired at different time scales. This means overcoming significant hurdles, including the need for stable and precise labeling of the required probes and the use of state of the art optical methods. However, to interpret the acquired data, biophysical models that can account for these biological mechanisms need to be developed. The structure and function of a biological system are closely related to its dynamic properties, thus further emphasizing the importance of identifying the rules governing the dynamics that cannot be directly deduced from information on the structure itself. In eukaryotic cells, tens of thousands of genes are packed in the small volume of the nucleus. The genome itself is organized in chromosomes that occupy specific volumes referred to as chromosome territories. This organization is preserved throughout the cell cycle, even though there are no sub-compartments in the nucleus itself. This organization, which is still not fully understood, is crucial for a large number of cellular functions such as gene regulation, DNA breakage repair and error-free cell division. Various techniques are in use today, including imaging, live cell imaging and molecular methods such as chromosome conformation capture (3C) methods to better understand these mechanisms. Live cell imaging methods are becoming well established. These include methods such as Single Particle Tracking (SPT), Continuous Photobleaching (CP), Fluorescence Recovery After Photobleaching (FRAP) and Fluorescence Correlation Spectroscopy (FCS) that are currently used for studying proteins, RNA, DNA, gene loci and nuclear bodies. They provide crucial information on its mobility, reorganization, interactions and binding properties. Here we describe how these dynamic methods can be used to gather information on genome organization, its stabilization mechanisms and the proteins that take part in it.

Rad4 recognition-at-a-distance: Physical basis of conformation-specific anomalous diffusion of DNA repair proteins

Since Robert Brown's first observations of random walks by pollen particles suspended in solution, the concept of diffusion has been subject to countless theoretical and experimental studies in diverse fields from finance and social sciences, to physics and biology. Diffusive transport of macromolecules in cells is intimately linked to essential cellular functions including nutrient uptake, signal transduction, gene expression, as well as DNA replication and repair. Advancement in experimental techniques has allowed precise measurements of these diffusion processes. Mathematical and physical descriptions and computer simulations have been applied to model complicated biological systems in which anomalous diffusion, in addition to simple Brownian motion, was observed. The purpose of this review is to provide an overview of the major physical models of anomalous diffusion and corresponding experimental evidence on the target search problem faced by DNA-binding proteins, with an emphasis on DNA repair proteins and the role of anomalous diffusion in DNA target recognition.

Revealing nonergodic dynamics in living cells from a single particle trajectory

We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.

Loss of lamin A function increases chromatin dynamics in the nuclear interior

Chromatin is organized in a highly ordered yet dynamic manner in the cell nucleus, but the principles governing this organization remain unclear. Similarly, it is unknown whether, and how, various proteins regulate chromatin motion and as a result influence nuclear organization. Here by studying the dynamics of different genomic regions in the nucleus of live cells, we show that the genome has highly constrained dynamics. Interestingly, depletion of lamin A strikingly alters genome dynamics, inducing a dramatic transition from slow anomalous diffusion to fast and normal diffusion. In contrast, depletion of LAP2α, a protein that interacts with lamin A and chromatin, has no such effect on genome dynamics. We speculate that chromosomal inter-chain interactions formed by lamin A throughout the nucleus contribute to chromatin dynamics, and suggest that the molecular regulation of chromatin diffusion by lamin A in the nuclear interior is critical for the maintenance of genome organization.

Analysis of Single Locus Trajectories for Extracting In Vivo Chromatin Tethering Interactions

Is it possible to extract tethering forces applied on chromatin from the statistics of a single locus trajectories imaged in vivo? Chromatin fragments interact with many partners such as the nuclear membrane, other chromosomes or nuclear bodies, but the resulting forces cannot be directly measured in vivo. However, they impact chromatin dynamics and should be reflected in particular in the motion of a single locus. We present here a method based on polymer models and statistics of single trajectories to extract the force characteristics and in particular when they are generated by the gradient of a quadratic potential well. Using numerical simulations of a Rouse polymer and live cell imaging of the MAT-locus located on the yeast Saccharomyces cerevisiae chromosome III, we recover the amplitude and the distance between the observed and the interacting monomer. To conclude, the confined trajectories we observed in vivo reflect local interaction on chromatin.

Chromosomal locus tracking with proper accounting of static and dynamic errors

The mean-squared displacement (MSD) and velocity autocorrelation (VAC) of tracked single particles or molecules are ubiquitous metrics for extracting parameters that describe the object's motion, but they are both corrupted by experimental errors that hinder the quantitative extraction of underlying parameters. For the simple case of pure Brownian motion, the effects of localization error due to photon statistics ("static error") and motion blur due to finite exposure time ("dynamic error") on the MSD and VAC are already routinely treated. However, particles moving through complex environments such as cells, nuclei, or polymers often exhibit anomalous diffusion, for which the effects of these errors are less often sufficiently treated. We present data from tracked chromosomal loci in yeast that demonstrate the necessity of properly accounting for both static and dynamic error in the context of an anomalous diffusion that is consistent with a fractional Brownian motion (FBM). We compare these data to analytical forms of the expected values of the MSD and VAC for a general FBM in the presence of these errors.

Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments

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 estimator.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.

Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion

We present a systematic statistical analysis of the recently measured individual trajectories of fluorescently labeled telomeres in the nucleus of living human cells. The experiments were performed in the U2OS cancer cell line. We propose an algorithm for identification of the telomere motion. By expanding the previously published data set, we are able to explore the dynamics in six time orders, a task not possible earlier. As a result, we establish a rigorous mathematical characterization of the stochastic process and identify the basic mathematical mechanisms behind the telomere motion. We find that the increments of the motion are stationary, Gaussian, ergodic, and even more chaotic--mixing. Moreover, the obtained memory parameter estimates, as well as the ensemble average mean square displacement reveal subdiffusive behavior at all time spans. All these findings statistically prove a fractional Brownian motion for the telomere trajectories, which is confirmed by a generalized p-variation test. Taking into account the biophysical nature of telomeres as monomers in the chromatin chain, we suggest polymer dynamics as a sufficient framework for their motion with no influence of other models. In addition, these results shed light on other studies of telomere motion and the alternative telomere lengthening mechanism. We hope that identification of these mechanisms will allow the development of a proper physical and biological model for telomere subdynamics. This array of tests can be easily implemented to other data sets to enable quick and accurate analysis of their statistical characteristics.

Improved estimation of anomalous diffusion exponents in single-particle tracking experiments

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.

Test for determining a subdiffusive model in ergodic systems from single trajectories

Experiments on particle motion show that it is often subdiffusive. This subdiffusion may be due to trapping, percolationlike structures, or viscoelastic behavior of the medium. While the models based on trapping (leading to continuous-time random walks) can easily be distinguished from the rest by testing their nonergodicity, the latter two cases are harder to distinguish. We propose a statistical test for distinguishing between these two based on the space-filling properties of trajectories, and prove its feasibility and specificity using synthetic data. We moreover present a flow chart for making a decision on a type of subdiffusion for a broader class of models.

Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional brownian motion.

Fractional process as a unified model for subdiffusive dynamics in experimental data

We show how to use a fractional autoregressive integrated moving average (FARIMA) model to a statistical analysis of the subdiffusive dynamics. The discrete time FARIMA(1,d,1) model is applied in this paper to the random motion of an individual fluorescently labeled mRNA molecule inside live E. coli cells in the experiment described in detail by Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)] as well as to the motion of fluorescently labeled telomeres in the nucleus of live human cells (U2OS cancer) in the experiment performed by Bronstein et al. [Phys. Rev. Lett. 103, 018102 (2009)]. It is found that only the memory parameter d of the FARIMA model completely detects an anomalous dynamics of the experimental data in both cases independently of the observed distribution of random noises.

Analytical tools to distinguish the effects of localization error, confinement, and medium elasticity on the velocity autocorrelation function

Single particle tracking is a powerful technique for investigating the dynamic behavior of biological molecules. However, many of the analytical tools are prone to generate results that can lead to mistaken interpretations of the underlying transport process. Here, we explore the effects of localization error and confinement on the velocity autocorrelation function, Cυ. We show that calculation of Cυ across a range of discretizations can distinguish the effects of localization error, confinement, and medium elasticity. Thus, under certain regimes, Cυ can be used as a diagnostic tool to identify the underlying mechanism of anomalous diffusion. Finally, we apply our analysis to experimental data sets of chromosomal loci and RNA-protein particles in Escherichia coli.

Obstructed diffusion propagator analysis for single-particle tracking

We describe a method for the analysis of the distribution of displacements, i.e., the propagators, of single-particle tracking measurements for the case of obstructed subdiffusion in two-dimensional membranes. The propagator for the percolation cluster is compared with a two-component mobility model against Monte Carlo simulations. To account for diffusion in the presence of obstacle concentrations below the percolation threshold, a propagator that includes the transient motion in finite percolation clusters and hopping between obstacle-induced compartments is derived. Finally, these models are shown to be effective in the analysis of Kv2.1 channel diffusive measurements in the membrane of living mammalian cells.

Anomalous diffusion: testing ergodicity breaking in experimental data

Recent advances in single-molecule experiments show that various complex systems display nonergodic behavior. In this paper, we show how to test ergodicity and ergodicity breaking in experimental data. Exploiting the so-called dynamical functional, we introduce a simple test which allows us to verify ergodic properties of a real-life process. The test can be applied to a large family of stationary infinitely divisible processes. We check the performance of the test for various simulated processes and apply it to experimental data describing the motion of mRNA molecules inside live Escherichia coli cells. We show that the data satisfy necessary conditions for mixing and ergodicity. The detailed analysis is presented in the supplementary material.