Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion. of larger molecules or tracers in living cells [3,[6][7][8][9][10][11][12][13][14][15][16]. While normal diffusion, by virtue of the central limit theorem, is characterized by the universal Gaussian probability density function and therefore uniquely determined by the first and second moments [17], anomalous diffusion of the form (1) is non-universal and may be caused by different stochastic mechanisms. These would give rise to vastly different behavior for diffusional mixing, diffusionlimited reactions, signaling, or regulatory processes. To better understand cellular dynamics, knowledge of the underlying stochastic mechanism is thus imperative.Here we report experimental evidence from extensive single trajectory time series of lipid granule motion in Schizosaccharomyces pombe (S. pombe) fission yeast cells obtained from tracking with optical tweezers (resolving 10 −6 sec to 1 sec) and video microscopy (10 −2 sec to 100 sec). Using complementary analysis tools we demonstrate that at short times the data are described best by continuous time random walk (CTRW) subdiffusion, revealing pronounced features of weak ergodicity breaking in the time averaged mean squared displacement. At longer times the stochastic mechanism is closest to subdiffusive fractional Brownian motion (FBM). The time scales over which this anomalous behavior persists is relevant for biological processes occurring in the cell. Anomalous diffusion may indeed be a good strategy for cellular signaling and reactions [7,8].CTRW and FBM both effect anomalous diffusion of the type (1) [18]. Subdiffusive CTRWs are random walks with finite variance δx 2 of jump lengths, while the waiting times between successive jumps are drawn from a density ψ(t) ≃ τ α /t 1+α with diverging characteristic time [4,17]. Such scale-free behavior results from multiple trapping events in, e.g., comb-like structures [19] or random energy landscapes [20]. Power-law waiting time distributions were also identified for tracer motion in reconstituted actin networks [21]. Subdiffusive FBM is a random process driven by Gaussian noise ξ with longrange correlations, ξ(0)ξ(t) ≃ α(α − 1)t α−2 [22], and it is related to fractional Langevin equations [23]. The finite characteristic time scales associated with FBM contrast the ageing property of the subdiffusive CTRW processes [Supplementary Material (SM)].Single particle tracking microscopy has become a standard tool to probe the motion of individual tracers, espe...

How long does it take a random walker to reach a given target point? This quantity, known as a first-passage time (FPT), has led to a growing number of theoretical investigations over the past decade. The importance of FPTs originates from the crucial role played by first encounter properties in various real situations, including transport in disordered media, neuron firing dynamics, spreading of diseases or target search processes. Most methods of determining FPT properties in confining domains have been limited to effectively one-dimensional geometries, or to higher spatial dimensions only in homogeneous media. Here we develop a general theory that allows accurate evaluation of the mean FPT in complex media. Our analytical approach provides a universal scaling dependence of the mean FPT on both the volume of the confining domain and the source-target distance. The analysis is applicable to a broad range of stochastic processes characterized by length-scale-invariant properties. Our theoretical predictions are confirmed by numerical simulations for several representative models of disordered media, fractals, anomalous diffusion and scale-free networks.

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss what we believe are new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions (MMEs), i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean-squared displacement we show that the MME analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the MME method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.