Abstract
Diffusion of nanoparticles in the cytoplasm of live cells has frequently been reported to exhibit an anomalous and even heterogeneous character, i.e. particles seem to switch gears during ...their journey. Here we show by means of a hidden Markov model that individual trajectories of quantum dots in the cytoplasm of living cultured cells feature a dichotomous switching between two distinct mobility states with an overall subdiffusive mode of motion of the fractional Brownian motion (FBM) type. Using the extracted features of experimental trajectories as input for simulations of different variants of a two-state FBM model, we show that the trajectory-intrinsic and the ensemble-wise heterogeneity in the experimental data is mostly due to variations in the (local) transport coefficients, with only minor contributions due to locally varying anomaly exponents. Altogether, our approach shows that diffusion heterogeneities can be faithfully extracted and quantified from fairly short trajectories obtained by single-particle tracking in highly complex media.
Stochastic motion on the surface of living cells is critical to promote molecular encounters that are necessary for multiple cellular processes. Often the complexity of the cell membranes leads to ...anomalous diffusion, which under certain conditions it is accompanied by non-ergodic dynamics. Here, we unravel two manifestations of ergodicity breaking in the dynamics of membrane proteins in the somatic surface of hippocampal neurons. Three different tagged molecules are studied on the surface of the soma: the voltage-gated potassium and sodium channels Kv1.4 and Nav1.6 and the glycoprotein CD4. In these three molecules ergodicity breaking is unveiled by the confidence interval of the mean square displacement and by the dynamical functional estimator. Ergodicity breaking is found to take place due to transient confinement effects since the molecules alternate between free diffusion and confined motion.
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.
•Experiments are often disturbed by a noise which source can be the instrument error.•We introduce a test for the fractional Brownian motion with added Gaussian noise.•We derive a distribution of the ...test statistic which allows to find critical regions.•We check the quality of the test by studying its power for different alternatives.•The introduced test can be adapted to an arbitrary Gaussian process.
Fractional Brownian motion (FBM) is related to the notions of self-similarity, ergodicity and long memory. These properties have made FBM important in modeling real-world phenomena in different experiments ranging from telecommunication to biology. However, these experiments are often disturbed by a noise which source can be, e.g., the instrument error. In this paper we propose a rigorous statistical test for FBM with added white Gaussian noise which is based on the autocovariance function. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power for alternatives being FBM’s with different self-similarity parameters and the scaled Brownian motion which is also Gaussian and self-similar. We note that the introduced test can be adapted to an arbitrary Gaussian process with a given covariance structure.
We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory ...exponent α(t) in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by α(t). We derive the essential statistical properties of MMFBM such as its response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but a reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a steplike change of α(t) the scaling exponent of the MSD after the α step may be determined by the value of α(t) before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment.
Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great ...variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.
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.
Abstract
The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, ...these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.
Anomalous diffusion phenomena are ubiquitous in many areas, including physics, biology, medicine, climate and also financial markets. The description and analysis of this phenomena are challenging ...tasks. To fully define the anomalous diffusion and properly understand its nature there is need to combine the skills in advanced sensing techniques, mathematical and statistical knowledge of the corresponding theoretical processes and the experience in physical interpretation of this phenomena. The paper reviews on the measurement instrumentation and signal processing used for detection and description of anomalous diffusion regimes. What is more, estimation of the anomalous diffusion exponent is commented. We present here four methods based on the statistics commonly used in the analyzed problem. Finally the fractional Brownian motion (FBM) is proposed as an analog proper for reliable depiction of properties valid for observed anomalous diffusion phenomena. Simulation studies exhibit the scenario of Hurst exponent estimation in the FBM model when the additive noise of various amplitudes disturbance is added to a simulated trajectory of individual particle. Proposed algorithms have different efficiency depending on the model parameters. This implicates a need to provide more advanced analysis and introduce new methods that could be less sensitive to the measurement noise and/or external forces of noise character. This paper is the starting point for the complete description of the anomalous diffusive processes. It presents the areas for prospective applications of the original sensing techniques but also highlights the needs in a range of statistical and intelligent signal processing suitable for description of the small scale complex systems during the SPT experiment.
Particulate matter (PM) is an important component of air. Nowadays, major attention is payed to fine dust. It has considerable environmental impact, including adverse effect on human health. One of ...important issues regarding PM is the temporal variation of its concentration. The variation contains information about factors influencing this quantity in time. The work focuses on the character of PM concentration dynamics indoors, in the vicinity of emission source. The objective was to recognize between the homogeneous or heterogeneous dynamics. The goal was achieved by detecting normal and anomalous diffusion in fluctuations of PM concentration. For this purpose we used anomalous diffusion exponent, β which was derived from Mean Square Displacement (MSD) analysis. The information about PM concentration dynamics may be used to design sampling strategy, which serves to attain representative information about PM behavior in time. The data analyzed in this work was collected from single-point PM concentration monitoring in the vicinity of seven emission sources in industrial environment. In majority of cases we observed heterogeneous character of PM concentration dynamics. It confirms the complexity of interactions between the emission sources and indoor environment. This result also votes against simplistic approach to PM concentration measurement indoors, namely their occasional character, short measurement periods and long term averaging.
•The dust concentration data is analyzed in the form of time series.•The analysis is based on anomalous diffusion exponent estimators.•Heterogeneous dynamics of dust is observed in the vicinity of emission sources.
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