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)], 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.

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 L\'evy $\alpha${}-stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for $\alpha${}<2, the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, H-self-similar L\'evy 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/$\alpha${}. 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 L\'evy 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.

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.