Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

https://iopscience.iop.org/article/10.1088/0034-4885/76/4/046602/pdf

Anomalous transport in the crowded world of biological cells.

We track the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells. We find that the motion is subdiffusive, with an exponent that is robust to physiological changes, including the disruption of cytoskeletal elements. By modifying the parameters of the RNA molecule and the bacterial cell, we are able to examine the possible mechanisms that can lead to this unique type of motion, especially the effect of macromolecular crowding. We also examine the implications of anomalous diffusion on the kinetics of bacterial gene regulation, in particular, how transcription factors find their DNA targets.

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Physical nature of bacterial cytoplasm.

Tracking of fluorescently labeled chromosomal loci in live bacterial cells reveals a robust scaling of the mean square displacement (MSD) as τ(0.39). We propose that the observed motion arises from relaxation of the Rouse modes of the DNA polymer within the viscoelastic environment of the cytoplasm. The time-averaged and ensemble-averaged MSD of chromosomal loci exhibit ergodicity, and the velocity autocorrelation function is negative at short time lags. These observations are most consistent with fractional Langevin motion and rule out a continuous time random walk model as an explanation for anomalous motion in vivo.

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Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm.

Researchers use squeezed light to track the constituents of yeast cells with a performance that overcomes the quantum noise limit. This approach allows for the utilization of low optical power, which helps to minimize cell damage.

Biological measurement beyond the quantum limit

The viscoelastic properties of the cytoplasm of living yeast cells were investigated by studying the motion of lipid granules naturally occurring in the cytoplasm. A large frequency range of observation was obtained by a combination of video-based and laser-based tracking methods. At time scales from 10(-4) to 10(2) s, the granules typically perform subdiffusive motion with characteristics different from previous measurements in living cells. This subdiffusive behavior is thought to be due to the presence of polymer networks and membranous structures in the cytoplasm. Consistent with this hypothesis, we observe that the motion becomes less subdiffusive upon actin disruption.

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Anomalous diffusion in living yeast cells

Yeast and other walled cells possess high internal turgor pressure that allows them to grow and survive in the environment This turgor pressure, however, may oppose the invagination of the plasma membrane needed for endocytosis Here we study the effects of turgor pressure on endocytosis in the fission yeast Schizosaccharomyces pombe by time-lapse imaging of individual endocytic sites Decreasing effective turgor pressure by addition of sorbitol to the media significantly accelerates early steps in the endocytic process before actin assembly and membrane ingression but does not affect the velocity or depth of ingression of the endocytic pit in wild-type cells Sorbitol also rescues endocytic ingression defects of certain endocytic mutants and of cells treated with a low dose of the actin inhibitor latrunculin A Endocytosis proceeds after removal of the cell wall, suggesting that the cell wall does not contribute mechanically to this process These studies suggest that endocytosis is governed by a mechanical balance between local actin-dependent inward forces and opposing forces from high internal turgor pressure on the plasma membrane

Role of turgor pressure in endocytosis in fission yeast

The functions of the actin-myosin-based contractile ring in cytokinesis remain to be elucidated. Recent findings show that in the fission yeast Schizosaccharomyces pombe, cleavage furrow ingression is driven by polymerization of cell wall fibers outside the plasma membrane, not by the contractile ring. Here we show that one function of the ring is to spatially coordinate septum cell wall assembly. We develop an improved method for live-cell imaging of the division apparatus by orienting the rod-shaped cells vertically using microfabricated wells. We observe that the septum hole and ring are circular and centered in wild-type cells and that in the absence of a functional ring, the septum continues to ingress but in a disorganized and asymmetric manner. By manipulating the cleavage furrow into different shapes, we show that the ring promotes local septum growth in a curvature-dependent manner, allowing even a misshapen septum to grow into a more regular shape. This curvature-dependent growth suggests a model in which contractile forces of the ring shape the septum cell wall by stimulating the cell wall machinery in a mechanosensitive manner. Mechanical regulation of the cell wall assembly may have general relevance to the morphogenesis of walled cells.

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The contractile ring coordinates curvature-dependent septum assembly during fission yeast cytokinesis

During cytokinesis, fission yeast and other fungi and bacteria grow a septum that divides the cell in two. In fission yeast closure of the circular septum hole by the β-glucan synthases (Bgs) and other glucan synthases in the plasma membrane is tightly coupled to constriction of an actomyosin contractile ring attached to the membrane. It is unknown how septum growth is coordinated over scales of several microns to maintain septum circularity. Here, we documented the shapes of ingrowing septum edges by measuring the roughness of the edges, a measure of the deviation from circularity. The roughness was small, with spatial correlations indicative of spatially coordinated growth. We hypothesized that Bgs-mediated septum growth is mechanosensitive and coupled to contractile ring tension. A mathematical model showed that ring tension then generates almost circular septum edges by adjusting growth rates in a curvature-dependent fashion. The model reproduced experimental roughness statistics and showed that septum synthesis sets the mean closure rate. Our results suggest that the fission yeast cytokinetic ring tension does not set the constriction rate but regulates septum closure by suppressing roughness produced by inherently stochastic molecular growth processes.

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Emilia Laura Munteanu

Papers

Anomalous diffusion in living yeast cells

Anomalous diffusion in living yeast cells

Role of turgor pressure in endocytosis in fission yeast

The contractile ring coordinates curvature-dependent septum assembly during fission yeast cytokinesis

The fission yeast cytokinetic contractile ring regulates septum shape and closure