Krzysztof Burnecki

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.

Property insurance loss distributions

Property claim services (PCS) provides indices for losses resulting from catastrophic events in the US. In this paper, we study these indices and take a closer look at distributions underlying insurance claims. Surprisingly, the lognormal distribution seems to give a better fit than the Paretian one. Moreover, lagged autocorrelation study reveals a mean-reverting structure of indices returns.

Recognition of stable distribution with Lévy index α close to 2.

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

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.

Single-molecule imaging reveals receptor–G protein interactions at cell surface hot spots

G-protein-coupled receptors mediate the biological effects of many hormones and neurotransmitters and are important pharmacological targets. They transmit their signals to the cell interior by interacting with G proteins. However, it is unclear how receptors and G proteins meet, interact and couple. Here we analyse the concerted motion of G-protein-coupled receptors and G proteins on the plasma membrane and provide a quantitative model that reveals the key factors that underlie the high spatiotemporal complexity of their interactions. Using two-colour, single-molecule imaging we visualize interactions between individual receptors and G proteins at the surface of living cells. Under basal conditions, receptors and G proteins form activity-dependent complexes that last for around one second. Agonists specifically regulate the kinetics of receptor–G protein interactions, mainly by increasing their association rate. We find hot spots on the plasma membrane, at least partially defined by the cytoskeleton and clathrin-coated pits, in which receptors and G proteins are confined and preferentially couple. Imaging with the nanobody Nb37 suggests that signalling by G-protein-coupled receptors occurs preferentially at these hot spots. These findings shed new light on the dynamic interactions that control G-protein-coupled receptor signalling.

FARIMA MODELING OF SOLAR FLARE ACTIVITY FROM EMPIRICAL TIME SERIES OF SOFT X-RAY SOLAR EMISSION

A time series of soft X-ray emission observed by the Geostationary Operational Environment Satellites from 1974 to 2007 is analyzed. We show that in the solar-maximum periods the energy distribution of soft X-ray solar flares for C, M, and X classes is well described by a fractional autoregressive integrated moving average model with Pareto noise. The model incorporates two effects detected in our empirical studies. One effect is a long-term dependence (long-term memory), and another corresponds to heavy-tailed distributions. The parameters of the model: self-similarity exponent H, tail index α, and memory parameter d are statistically stable enough during the periods 1977-1981, 1988-1992, 1999-2003. However, when the solar activity tends to minimum, the parameters vary. We discuss the possible causes of this evolution and suggest a statistically justified model for predicting the solar flare activity.

Algorithms for testing of fractional dynamics: a practical guide to ARFIMA modelling

In this survey paper we present a systematic methodology which demonstrates how to identify the origins of fractional dynamics. We consider three mechanisms which lead to it, namely fractional Brownian motion, fractional Levy stable motion and an autoregressive fractionally integrated moving average (ARFIMA) process but we concentrate on the ARFIMA modelling. The methodology is based on statistical tools for identification and validation of the fractional dynamics, in particular on an ARFIMA parameter estimator, an ergodicity test, a self-similarity index estimator based on sample p-variation and a memory parameter estimator based on sample mean-squared displacement. A complete list of algorithms needed for this is provided in appendices A–F. Finally, we illustrate the methodology on various empirical data and show that ARFIMA can be considered as a universal model for fractional dynamics. Thus, we provide a practical guide for experimentalists on how to efficiently use ARFIMA modelling for a large class of anomalous diffusion data.

Simulation of risk processes

This paper is intended as a guide to simulation of risk processes. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk. In this paper we present efficient simulation algorithms for several classes of claim arrival processes.

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

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Statistical modelling of subdiffusive dynamics in the cytoplasm of living cells: A FARIMA approach

Golding and Cox (Phys. Rev. Lett., 96 (2006) 098102) tracked the motion of individual fluorescently labelled mRNA molecules inside live E. coli cells. They found that in the set of 23 trajectories from 3 different experiments, the automatically recognized motion is subdiffusive and published an intriguing microscopy video. Here, we extract the corresponding time series from this video by image segmentation method and present its detailed statistical analysis. We find that this trajectory was not included in the data set already studied and has different statistical properties. It is best fitted by a fractional autoregressive integrated moving average (FARIMA) process with the normal-inverse Gaussian (NIG) noise and the negative memory. In contrast to earlier studies, this shows that the fractional Brownian motion is not the best model for the dynamics documented in this video.