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.

Fractional Brownian motion with Hurst index less then 1/2 and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems. We propose a simple test, based on the analysis of the so-called p variations, which allows distinguishing between the two models on the basis of one realization of the unknown process. We apply the test to the data of Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)10.1103/PhysRevLett.96.098102], describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells. It is found that the data does not follow heavy tailed continuous-time random walk. The test shows that it is likely that fractional Brownian motion is the underlying process.

A computer algorithm for the visualization of sample paths of anomalous diffusion processes is developed. It is based on the stochastic representation of the fractional Fokker-Planck equation describing anomalous diffusion in a nonconstant potential. Monte Carlo methods employing the introduced algorithm will surely provide tools for studying many relevant statistical characteristics of the fractional Fokker-Planck dynamics.

A century after the celebrated Langevin paper [C.R. Seances Acad. Sci. 146, 530 (1908)] we study a Langevin-type approach to subdiffusion in the presence of time-dependent force fields. Using a subordination technique, we construct rigorously a stochastic Langevin process, whose probability density function is equal to the solution of the fractional Fokker-Planck equation with a time-dependent force. Our model provides physical insight into the nature of the corresponding process through the simulated trajectories. Moreover, the subordinated Langevin equation allows us to study subdiffusive dynamics both analytically and numerically via Monte Carlo methods.

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.

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.

We show in this paper that the sample (time average) mean-squared displacement (MSD) of the fractional Lévy α -stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for α<2 , the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, H -self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We show that the character of the process is controlled by a sign of the memory parameter d=H-1/α . We also introduce a sample p -variation dynamics test which allows to distinguish between two models of subdiffusive dynamics. Finally, we illustrate a subdiffusive behavior of the fractional Lévy stable motion on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics.

A canonical decomposition of H-self-similar Lévy symmetric alpha-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to the Lévy symmetric alpha-stable motion are shown to be related to the dissipative and conservative parts of the dynamics. This result provides stochastic analysis tools for study the anomalous diffusion phenomena in the Langevin equation framework. For example, a simple computer test for testing the origins of self-similarity is implemented for four real empirical time series recorded from different physical systems: an ionic current flow through a single channel in a biological membrane, an energy of solar flares, a seismic electric signal recorded during seismic Earth activity, and foreign exchange rate daily returns.