High-precision protein-tracking with interferometric scattering microscopy

Richard W. Taylor, Cornelia Holler, Reza Gholami Mahmoodabadi, Michelle Küppers, Houman Mirzaalian Dastjerdi, Vasily Zaburdaev, Alexandra Schambony, Vahid Sandoghdar

The mobility of proteins and lipids within the cell, sculpted oftentimes by the organisation of the membrane, reveals a great wealth of information on the function and interaction of these molecules as well as the membrane itself. Single particle tracking has proven to be a vital tool to study the mobility of individual molecules and unravel details of their behaviour. Interferometric scattering (iSCAT) microscopy is an emerging technique well suited for visualising the diffusion of gold nanoparticle-labelled membrane proteins to a spatial and temporal resolution beyond the means of traditional fluorescent labels. We discuss the applicability of interferometric single particle tracking (iSPT) microscopy to investigate the minutia in the motion of a protein through measurements visualising the mobility of the epidermal growth factor receptor in various biological scenarios on the live cell.

Liquid Phase Separation Controlled by pH

Omar Adame-Arana, Christoph A. Weber, Vasily Zaburdaev, Jacques Prost, Frank Julicher

We present a minimal model to study the effects of pH on liquid phase separation of macromolecules. Our model describes a mixture composed of water and macromolecules that exist in three different charge states and have a tendency to phase separate. This phase separation is affected by pH via a set of chemical reactions describing protonation and deprotonation of macromolecules, as well as self-ionization of water. We consider the simple case in which interactions are captured by Flory-Huggins interaction parameters corresponding to Debye screening lengths shorter than a nanometer, which is relevant to proteins inside biological cells under physiological conditions. We identify the conjugate thermodynamic variables at chemical equilibrium and discuss the effective free energy at fixed pH. First, we study phase diagrams as a function of macromolecule concentration and temperature at the isoelectric point of the macromolecules. We find a rich variety of phase diagram topologies, including multiple critical points, triple points, and first-order transition points. Second, we change the pH relative to the isoelectric point of the macromolecules and study how phase diagrams depend on pH. We find that these phase diagrams as a function of pH strongly depend on whether oppositely charged macromolecules or neutral macromolecules have a stronger tendency to phase separate. One key finding is that we predict the existence of a reentrant behavior as a function of pH. In addition, our model predicts that the region of phase separation is typically broader at the isoelectric point. This model could account for both in vitro phase separation of proteins as a function of pH and protein phase separation in yeast cells for pH values close to the isoelectric point of many cytosolic proteins.

Exogenous ethanol induces a metabolic switch that prolongs the survival of Caenorhabditis elegansdauer larva and enhances its resistance to desiccation

Damla Kaptan, Sider Penkov, Xingyu Zhang, Vamshidhar R. Gade, Bharath Kumar Raghuraman, Roberta Galli, Julio L. Sampaio, Robert Haase, Edmund Koch, et al.

The dauer larva ofCaenorhabditis elegans, destined to survive long periods of food scarcity and harsh environment, does not feed and has a very limited exchange of matter with the exterior. It was assumed that the survival time is determined by internal energy stores. Here, we show that ethanol can provide a potentially unlimited energy source for dauers by inducing a controlled metabolic shift that allows it to be metabolized into carbohydrates, amino acids, and lipids. Dauer larvae provided with ethanol survive much longer and have greater desiccation tolerance. On the cellular level, ethanol prevents the deterioration of mitochondria caused by energy depletion. By modeling the metabolism of dauers of wild-type and mutant strains with and without ethanol, we suggest that the mitochondrial health and survival of an organism provided with an unlimited source of carbon depends on the balance between energy production and toxic product(s) of lipid metabolism.

Ultrahigh-speed imaging of rotational diffusion on a lipid bilayer

Mahdi Mazaheri, Jens Ehrig, Alexey Shkarin, Vasily Zaburdaev, Vahid Sandoghdar

We studied the rotational and translational diffusion of a single gold nanorod linked to a supported lipid bilayer with ultrahigh temporal resolution of two microseconds. By using a home-built polarization-sensitive dark-field microscope, we recorded particle trajectories with lateral precision of three nanometers and rotational precision of four degrees. The large number of trajectory points in our measurements allows us to characterize the statistics of rotational diffusion with unprecedented detail. Our data show apparent signatures of anomalous diffusion such as sublinear scaling of the mean-squared angular displacement and negative values of angular correlation function at small lag times. However, a careful analysis reveals that these effect stem from the residual noise contributions and confirms normal diffusion. Our experimental approach and observations can be extended to investigate diffusive processes of anisotropic nanoparticles in other fundamental systems such as cellular membranes or other two-dimensional fluids.

Rectification of Bacterial Diffusion in Microfluidic Labyrinths

Ariane Weber, Marco Bahrs, Zahra Alirezai, Xingyu Zhang, Carsten Beta, Vasily Zaburdaev

In nature as well as in the context of infection and medical applications, bacteria often have to move in highly complex environments such as soil or tissues. Previous studies have shown that bacteria strongly interact with their surroundings and are often guided by confinements. Here, we investigate theoretically how the dispersal of swimming bacteria can be augmented by microfluidic environments and validate our theoretical predictions experimentally. We consider a system of bacteria performing the prototypical run-and-tumble motion inside a labyrinth with square lattice geometry. Narrow channels between the square obstacles limit the possibility of bacteria to reorient during tumbling events to an area where channels cross. Thus, by varying the geometry of the lattice it might be possible to control the dispersal of cells. We present a theoretical model quantifying diffusive spreading of a run-and-tumble random walker in a square lattice. Numerical simulations validate our theoretical predictions for the dependence of the diffusion coefficient on the lattice geometry. We show that bacteria moving in square labyrinths exhibit enhanced dispersal as compared to unconfined cells. Importantly, confinement significantly extends the duration of the phase with strongly non-Gaussian diffusion, when the geometry of channels is imprinted in the density profiles of spreading cells. Finally, in good agreement with our theoretical findings, we observe the predicted behaviors in experiments with E. coli bacteria swimming in a square lattice labyrinth created in a microfluidic device. Altogether, our comprehensive understanding of bacterial dispersal in a simple two-dimensional labyrinth makes the first step toward the analysis of more complex geometries relevant for real world applications.

Histone H3K27 acetylation precedes active transcription during zebrafish zygotic genome activation as revealed by live-cell analysis

Yuko Sato, Lennart Hilbert, Haruka Oda, Yinan Wan, John M. Heddleston, Teng-Leong Chew, Vasily Zaburdaev, Philipp Keller, Timothee Lionnet, et al.

Histone post-translational modifications are key gene expression regulators, but their rapid dynamics during development remain difficult to capture. We applied a Fab-based live endogenous modification labeling technique to monitor the changes in histone modification levels during zygotic genome activation (ZGA) in living zebrafish embryos. Among various histone modifications, H3 Lys27 acetylation (H3K27ac) exhibited most drastic changes, accumulating in two nuclear foci in the 64- to 1k-cell-stage embryos. The elongating form of RNA polymerase II, which is phosphorylated at Ser2 in heptad repeats within the C-terminal domain (RNAP2 Ser2ph), and miR-430 transcripts were also concentrated in foci closely associated with H3K27ac. When treated with alpha-amanitin to inhibit transcription or JQ-1 to inhibit binding of acetyl-reader proteins, H3K27ac foci still appeared but RNAP2 Ser2ph and miR-430 morpholino were not concentrated in foci, suggesting that H3K27ac precedes active transcription during ZGA. We anticipate that the method presented here could be applied to a variety of developmental processes in any model and non-model organisms.

How bacterial cells and colonies move on solid substrates

Wolfram Poenisch, Christoph A. Weber, Vasily Zaburdaev

Many bacteria rely on active cell appendages, such as type IV pili, to move over substrates and interact with neighboring cells. Here, we study the motion of individual cells and bacterial colonies, mediated by the collective interactions of multiple pili. It was shown experimentally that the substrate motility of Neisseria gonorrhoeae cells can be described as a persistent random walk with a persistence length that exceeds the mean pili length. Moreover, the persistence length increases for a higher number of pili per cell. With the help of a simple, tractable stochastic model, we test whether a tug of war without directional memory can explain the persistent motion of single Neisseria gonorrhoeae cells. While persistent motion of single cells indeed emerges naturally in the model, a tug of war alone is not capable of explaining the motility of microcolonies, which becomes weaker with increasing colony size. We suggest sliding friction between the microcolonies and the substrate as the missing ingredient. While such friction almost does not affect the general mechanism of single cell motility, it has a strong effect on colony motility. We validate the theoretical predictions by using a three-dimensional computational model that includes explicit details of the pili dynamics, force generation, and geometry of cells.

The shape of pinned forced polymer loops

Wenwen Huang, Vasily Zaburdaev

Loop geometry is a frequent encounter in synthetic and biological polymers. Here we provide an analytical theory to characterize the shapes of polymer loops subjected to an external force field. We show how to calculate the polymer density, gyration radius and its distribution. Interestingly, the distribution of the gyration radius shows a non-monotonic behavior as a function of the external force. Furthermore, we analyzed the gyration tensor of the polymer loop characterizing its overall shape. Two parameters called asphericity and the nature of asphericity derived from the gyration tensor, along with the gyration radius, can be used to quantify the shape of polymer loops in theory and experiments.

Identifying the mechanism for superdiffusivity in mouse fibroblast motility

Giuseppe Passucci, Megan E. Brasch, James H. Henderson, Vasily Zaburdaev, M. Lisa Manning

We seek to characterize the motility of mouse fibroblasts on 2D substrates. Utilizing automated tracking techniques, we find that cell trajectories are super-diffusive, where displacements scale faster than t(1/2) in all directions. Two mechanisms have been proposed to explain such statistics in other cell types: run and tumble behavior with Levy-distributed run times, and ensembles of cells with heterogeneous speed and rotational noise. We develop an automated toolkit that directly compares cell trajectories to the predictions of each model and demonstrate that ensemble-averaged quantities such as the mean-squared displacements and velocity autocorrelation functions are equally well-fit by either model. However, neither model correctly captures the short-timescale behavior quantified by the displacement probability distribution or the turning angle distribution. We develop a hybrid model that includes both run and tumble behavior and heterogeneous noise during the runs, which correctly matches the short-timescale behaviors and indicates that the run times are not Levy distributed. The analysis tools developed here should be broadly useful for distinguishing between mechanisms for superdiffusivity in other cells types and environments.

Pili mediated intercellular forces shape heterogeneous bacterial microcolonies prior to multicellular differentiation

Wolfram Poenisch, Kelly B. Eckenrode, Khaled Alzurqa, Hadi Nasrollahi, Christoph Weber, Vasily Zaburdaev, Nicolas Biais

Microcolonies are aggregates of a few dozen to a few thousand cells exhibited by many bacteria. The formation of microcolonies is a crucial step towards the formation of more mature bacterial communities known as biofilms, but also marks a significant change in bacterial physiology. Within a microcolony, bacteria forgo a single cell lifestyle for a communal lifestyle hallmarked by high cell density and physical interactions between cells potentially altering their behaviour. It is thus crucial to understand how initially identical single cells start to behave differently while assembling in these tight communities. Here we show that cells in the microcolonies formed by the human pathogen Neisseria gonorrhoeae (Ng) present differential motility behaviors within an hour upon colony formation. Observation of merging microcolonies and tracking of single cells within microcolonies reveal a heterogeneous motility behavior: cells close to the surface of the microcolony exhibit a much higher motility compared to cells towards the center. Numerical simulations of a biophysical model for the microcolonies at the single cell level suggest that the emergence of differential behavior within a multicellular microcolony of otherwise identical cells is of mechanical origin. It could suggest a route toward further bacterial differentiation and ultimately mature biofilms.

Exactly solvable dynamics of forced polymer loops

Wenwen Huang, Yen Ting Lin, Daniela Froemberg, Jaeoh Shin, Frank Juelicher, Vasily Zaburdaev

Here, we show that a problem of forced polymer loops can be mapped to an asymmetric simple exclusion process with reflecting boundary conditions. The dynamics of the particle system can be solved exactly using the Bethe ansatz. We thus can fully describe the relaxation dynamics of forced polymer loops. In the steady state, the conformation of the loop can be approximated by a combination of Fermi-Dirac and Brownian bridge statistics, while the exact solution is found by using the fermion integer partition theory. With the theoretical framework presented here we establish a link between the physics of polymers and statistics of many-particle systems opening new paths of exploration in both research fields. Our result can be applied to the dynamics of the biopolymers which form closed loops. One such example is the active pulling of chromosomal loops during meiosis in yeast cells which helps to align chromosomes for recombination in the viscous environment of the cell nucleus.

Intracellular Mass Density Increase Is Accompanying but Not Sufficient for Stiffening and Growth Arrest of Yeast Cells

Shada Abuhattum, Kyoohyun Kim, Titus M. Franzmann, Anne Esslinger, Daniel Midtvedt, Raimund Schluessler, Stephanie Mollmert, Hui-Shun Kuan, Simon Alberti, et al.

Many organisms, including yeast cells, bacteria, nematodes, and tardigrades, endure harsh environmental conditions, such as nutrient scarcity, or lack of water and energy for a remarkably long time. The rescue programs that these organisms launch upon encountering these adverse conditions include reprogramming their metabolism in order to enter a quiescent or dormant state in a controlled fashion. Reprogramming coincides with changes in the macromolecular architecture and changes in the physical and mechanical properties of the cells. However, the cellular mechanisms underlying the physical-mechanical changes remain enigmatic. Here, we induce metabolic arrest of yeast cells by lowering their intracellular pH. We then determine the differences in the intracellular mass density and stiffness of active and metabolically arrested cells using optical diffraction tomography (ODT) and atomic force microscopy (AFM). We show that an increased intracellular mass density is associated with an increase in stiffness when the growth of yeast is arrested. However, increasing the intracellular mass density alone is not sufficient for maintenance of the growth-arrested state in yeast cells. Our data suggest that the cytoplasm of metabolically arrested yeast displays characteristics of a solid. Our findings constitute a bridge between the mechanical behavior of the cytoplasm and the physical and chemical mechanisms of metabolically arrested cells with the ultimate aim of understanding dormant organisms.

Live cell X-ray imaging of autophagic vacuoles formation and chromatin dynamics in fission yeast

Natalja Strelnikova, Nora Sauter, Manuel Guizar-Sicairos, Michael Gollner, Ana Diaz, Petrina Delivani, Mariola Chacon, Iva M. Tolic, Vasily Zaburdaev, et al.

Seeing physiological processes at the nanoscale in living organisms without labeling is an ultimate goal in life sciences. Using X-ray ptychography, we explored in situ the dynamics of unstained, living fission yeast Schizosaccharomyces pombe cells in natural, aqueous environment at the nanoscale. In contrast to previous X-ray imaging studies on biological matter, in this work the eukaryotic cells were alive even after several ptychographic X-ray scans, which allowed us to visualize the chromatin motion as well as the autophagic cell death induced by the ionizing radiation. The accumulated radiation of the sequential scans allowed for the determination of a characteristic dose of autophagic vacuole formation and the lethal dose for fission yeast. The presented results demonstrate a practical method that opens another way of looking at living biological specimens and processes in a time-resolved label-free setting.

Elasticity-based polymer sorting in active fluids: a Brownian dynamics study

Jaeoh Shin, Andrey G. Cherstvy, Won Kyu Kim, Vasily Zaburdaev

While the dynamics of polymer chains in equilibrium media is well understood by now, the polymer dynamics in active non-equilibrium environments can be very different. Here we study the dynamics of polymers in a viscous medium containing self-propelled particles in two dimensions by using Brownian dynamics simulations. We find that the polymer center of mass exhibits a superdiffusive motion at short to intermediate times and the motion turns normal at long times, but with a greatly enhanced diffusivity. Interestingly, the long time diffusivity shows a non-monotonic behavior as a function of chain length and stiffness. We analyze how the polymer conformation and the accumulation of self-propelled particles, and therefore the directed motion of the polymer, are correlated. At the point of maximal polymer diffusivity, the polymer has preferentially bent conformations maintained by the balance between the chain elasticity and the propelling force generated by the active particles. We also consider the barrier crossing dynamics of actively-driven polymers in a double-well potential. The barrier crossing times are demonstrated to have a peculiar non-monotonic dependence, related to that of the diffusivity. This effect can be potentially utilized for sorting polymers from solutions in in vitro experiments.

A tunable refractive index matching medium for live imaging cells, tissues and model organisms

Tobias Boothe, Lennart Hilbert, Michael Heide, Lea Berninger, Wieland B. Huttner, Vasily Zaburdaev, Nadine L. Vastenhouw, Eugene W. Myers, David N. Drechsel, et al.

In light microscopy, refractive index mismatches between media and sample cause spherical aberrations that often limit penetration depth and resolution. Optical clearing techniques can alleviate these mismatches, but they are so far limited to fixed samples. We present Iodixanol as a non-toxic medium supplement that allows refractive index matching in live specimens and thus substantially improves image quality in live-imaged primary cell cultures, planarians, zebrafish and human cerebral organoids.

Competition between histone and transcription factor binding regulates the onset of transcription in zebrafish embryos

Shai R. Joseph, Mate Palfy, Lennart Hilbert, Mukesh Kumar, Jens Karschau, Vasily Zaburdaev, Andrej Shevchenko, Nadine L. Vastenhouw

Upon fertilization, the genome of animal embryos remains transcriptionally inactive until the maternal-to-zygotic transition. At this time, the embryo takes control of its development and transcription begins. How the onset of zygotic transcription is regulated remains unclear. Here, we show that a dynamic competition for DNA binding between nucleosome-forming histones and transcription factors regulates zebrafish genome activation. Taking a quantitative approach, we found that the concentration of non-DNA-bound core histones sets the time for the onset of transcription. The reduction in nuclear histone concentration that coincides with genome activation does not affect nucleosome density on DNA, but allows transcription factors to compete successfully for DNA binding. In agreement with this, transcription factor binding is sensitive to histone levels and the concentration of transcription factors also affects the time of transcription. Our results demonstrate that the relative levels of histones and transcription factors regulate the onset of transcription in the embryo.

Limit theorems for Levy walks in d dimensions: rare and bulk fluctuations

Itzhak Fouxon, Sergey Denisov, Vasily Zaburdaev, Eli Barkai

We consider super-diffusive Levy walks in d >= 2 dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Levy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Levy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Levy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Levy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Levy walk. We briefly discuss the consequences of non-universality for the d > 1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

Multiscale modeling of bacterial colonies: how pili mediate the dynamics of single cells and cellular aggregates

Wolfram Poenisch, Christoph A. Weber, Guido Juckeland, Nicolas Biais, Vasily Zaburdaev

Neisseria gonorrhoeae is the causative agent of one of the most common sexually transmitted diseases, gonorrhea. Over the past two decades there has been an alarming increase of reported gonorrhea cases where the bacteria were resistant to the most commonly used antibiotics thus prompting for alternative antimicrobial treatment strategies. The crucial step in this and many other bacterial infections is the formation of microcolonies, agglomerates consisting of up to several thousands of cells. The attachment and motility of cells on solid substrates as well as the cell-cell interactions are primarily mediated by type IV pili, long polymeric filaments protruding from the surface of cells. While the crucial role of pili in the assembly of microcolonies has been well recognized, the exact mechanisms of how they govern the formation and dynamics of microcolonies are still poorly understood. Here, we present a computational model of individual cells with explicit pili dynamics, force generation and pili-pili interactions. Weemploy the model to study a wide range of biological processes, such as the motility of individual cells on a surface, the heterogeneous cell motility within the large cell aggregates, and the merging dynamics and the self-assembly of microcolonies. The results of numerical simulations highlight the central role of pili generated forces in the formation of bacterial colonies and are in agreement with the available experimental observations. The model can quantify the behavior of multicellular bacterial colonies on biologically relevant temporal and spatial scales and can be easily adjusted to include the geometry and pili characteristics of various bacterial species. Ultimately, the combination of the microbiological experimental approach with the in silico model of bacterial colonies might provide new qualitative and quantitative insights on the development of bacterial infections and thus pave the way to new antimicrobial treatments.

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

Vasily Zaburdaev, I. Fouxon, S. Denisov, E. Barkai

It is recognized now that a variety of real-life phenomena ranging from diffusion of cold atoms to the motion of humans exhibit dispersal faster than normal diffusion. Levy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Levy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

Nucleosomal arrangement affects single-molecule transcription dynamics

Veronika Fitz, Jaeoh Shin, Christoph Ehrlich, Lucas Farnung, Patrick Cramer, Vasily Zaburdaev, Stephan W. Grill

In eukaryotes, gene expression depends on chromatin organization. However, how chromatin affects the transcription dynamics of individual RNA polymerases has remained elusive. Here, we use dual trap optical tweezers to study single yeast RNA polymerase II (Pol II) molecules transcribing along a DNA template with two nucleosomes. The slowdown and the changes in pausing behavior within the nucleosomal region allow us to determine a drift coefficient,., which characterizes the ability of the enzyme to recover from a nucleosomal backtrack. Notably, chi can be used to predict the probability to pass the first nucleosome. Importantly, the presence of a second nucleosome changes chi in a manner that depends on the spacing between the two nucleosomes, as well as on their rotational arrangement on the helical DNA molecule. Our results indicate that the ability of Pol II to pass the first nucleosome is increased when the next nucleosome is turned away from the first one to face the opposite side of the DNA template. These findings help to rationalize how chromatin arrangement affects Pol II transcription dynamics.

Reaction front propagation of actin polymerization in a comb-reaction system

A. Iomin, Vasily Zaburdaev, T. Pfohl

We develop a theoretical model of anomalous transport with polymerization-reaction dynamics. We are motivated by the experimental problem of actin polymerization occurring in a microfluidic device with a comb-like geometry. Depending on the concentration of reagents, two limiting regimes for the propagation of reaction are recovered: the failure of the reaction front propagation and a finite speed of the reaction front corresponding to the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) at the long time asymptotic regime. To predict the relevance of these regimes we obtain an explicit expression for the transient time as a function of geometry and parameters of the experimental setup. Explicit analytical expressions of the reaction front velocity are obtained as functions of the experimental setup. (C) 2016 Elsevier Ltd. All rights reserved.

A pH-driven transition of the cytoplasm from a fluid- to a solid-like state promotes entry into dormancy

Matthias Christoph Munder, Daniel Midtvedt, Titus Franzmann, Elisabeth Nueske, Oliver Otto, Maik Herbig, Elke Ulbricht, Paul Mueller, Anna Taubenberger, et al.

Cells can enter into a dormant state when faced with unfavorable conditions. However, how cells enter into and recover from this state is still poorly understood. Here, we study dormancy in different eukaryotic organisms and find it to be associated with a significant decrease in the mobility of organelles and foreign tracer particles. We show that this reduced mobility is caused by an influx of protons and a marked acidification of the cytoplasm, which leads to widespread macromolecular assembly of proteins and triggers a transition of the cytoplasm to a solid-like state with increased mechanical stability. We further demonstrate that this transition is required for cellular survival under conditions of starvation. Our findings have broad implications for understanding alternative physiological states, such as quiescence and dormancy, and create a new view of the cytoplasm as an adaptable fluid that can reversibly transition into a protective solid-like state.

Pulled Polymer Loops as a Model for the Alignment of Meiotic Chromosomes

Yen Ting Lin, Daniela Froemberg, Wenwen Huang, Petrina Delivani, Mariola Chacon, Iva M. Tolic, Frank Juelicher, Vasily Zaburdaev

During recombination, the DNA of parents exchange their genetic information to give rise to a genetically unique offspring. For recombination to occur, homologous chromosomes need to find each other and align with high precision. Fission yeast solves this problem by folding chromosomes in loops and pulling them through the viscous nucleoplasm. We propose a theory of pulled polymer loops to quantify the effect of drag forces on the alignment of chromosomes. We introduce an external force field to the concept of a Brownian bridge and thus solve for the statistics of loop configurations in space.

Pili-Induced Clustering of N-gonorrhoeae Bacteria

Johannes Taktikos, Yen Ting Lin, Holger Stark, Nicolas Biais, Vasily Zaburdaev

Type IV pili (Tfp) are prokaryotic retractable appendages known to mediate surface attachment, motility, and subsequent clustering of cells. Tfp are the main means of motility for Neisseria gonorrhoeae, the causative agent of gonorrhea. Tfp are also involved in formation of the microcolonies, which play a crucial role in the progression of the disease. While motility of individual cells is relatively well understood, little is known about the dynamics of N. gonorrhoeae aggregation. We investigate how individual N. gonorrhoeae cells, initially uniformly dispersed on flat plastic or glass surfaces, agglomerate into spherical microcolonies within hours. We quantify the clustering process by measuring the area fraction covered by the cells, number of cell aggregates, and their average size as a function of time. We observe that the microcolonies are also able to move but their mobility rapidly vanishes as the size of the colony increases. After a certain critical size they become immobile. We propose a simple theoretical model which assumes a pili-pili interaction of cells as the main clustering mechanism. Numerical simulations of the model quantitatively reproduce the experimental data on clustering and thus suggest that the agglomeration process can be entirely explained by the Tfp-mediated interactions. In agreement with this hypothesis mutants lacking pili are not able to form colonies. Moreover, cells with deficient quorum sensing mechanism show similar aggregation as the wild-type bacteria. Therefore, our results demonstrate that pili provide an essential mechanism for colony formation, while additional chemical cues, for example quorum sensing, might be of secondary importance.

Formation and dissolution of bacterial colonies

Christoph A. Weber, Yen Ting Lin, Nicolas Biais, Vasily Zaburdaev

Many organisms form colonies for a transient period of time to withstand environmental pressure. Bacterial biofilms are a prototypical example of such behavior. Despite significant interest across disciplines, physical mechanisms governing the formation and dissolution of bacterial colonies are still poorly understood. Starting from a kinetic description of motile and interacting cells we derive a hydrodynamic equation for their density on a surface, where most of the kinetic coefficients are estimated from experimental data for N. gonorrhoeae bacteria. We use it to describe the formation of multiple colonies with sizes consistent with experimental observations. Finally, we show how the changes in the cell-to-cell interactions lead to the dissolution of the bacterial colonies. The successful application of kinetic theory to a complex far from equilibrium system such as formation and dissolution of living bacterial colonies potentially paves the way for the physical quantification of the initial stages of biofilm formation.

Levy walks

Vasily Zaburdaev, S. Denisov, J. Klafter

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Levy-walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, biophysics, and behavioral science demonstrate that this particular type of random walk provides significant insight into complex transport phenomena. This review gives a self-consistent introduction to Levy walks, surveys their existing applications, including latest advances, and outlines further perspectives.

Asymptotic densities of ballistic Levy walks

D. Froemberg, M. Schmiedeberg, E. Barkai, Vasily Zaburdaev

We propose an analytical method to determine the shape of density profiles in the asymptotic long-time limit for a broad class of coupled continuous-time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random-walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard nonballistic random walks.

Random walk patterns of a soil bacterium in open and confined environments

M. Theves, J. Taktikos, Vasily Zaburdaev, H. Stark, C. Beta

We used microfluidic tools and high-speed time-lapse microscopy to record trajectories of the soil bacterium Pseudomonas putida in a confined environment with cells swimming in close proximity to a glass-liquid interface. While the general swimming pattern is preserved, when compared to swimming in the bulk fluid, our results show that cells in the presence of two solid boundaries display more frequent reversals in swimming direction and swim faster. Additionally, we observe that run segments are no longer straight and that cells swim on circular trajectories, which can be attributed to the hydrodynamic wall effect. Using the experimentally observed parameters together with a recently presented analytic model for a run-reverse random walker, we obtained additional insight on how the spreading behavior of a cell population is affected under confinement. While on short time scales, the mean square displacement of confined swimmers grows faster as compared to the bulk fluid case, our model predicts that for large times the situation reverses due to the strong increase in effective rotational diffusion. Copyright (C) EPLA, 2015

Uncovering the Mechanism of Trapping and Cell Orientation during Neisseria gonorrhoeae Twitching Motility

Vasily Zaburdaev, Nicolas Biais, Michael Schmiedeberg, Jens Eriksson, Ann-Beth Jonsson, Michael P. Sheetz, David A. Weitz

Neisseria gonorrheae bacteria are the causative agent of the second most common sexually transmitted infection in the world. The bacteria move on a surface by means of twitching motility. Their movement is mediated by multiple long and flexible filaments, called type IV pili, that extend from the cell body, attach to the surface, and retract, thus generating a pulling force. Moving cells also use pili to aggregate and form microcolonies. However, the mechanism by which the pili surrounding the cell body work together to propel bacteria remains unclear. Understanding this process will help describe the motility of N. gonorrheae bacteria, and thus the dissemination of the disease which they cause. In this article we track individual twitching cells and observe that their trajectories consist of alternating moving and pausing intervals, while the cell body is preferably oriented with its wide side toward the direction of motion. Based on these data, we propose a model for the collective pili operation of N. gonorrheae bacteria that explains the experimentally observed behavior. Individual pili function independently but can lead to coordinated motion or pausing via the force balance. The geometry of the cell defines its orientation during motion. We show that by changing pili substrate interactions, the motility pattern can be altered in a predictable way. Although the model proposed is tangibly simple, it still has sufficient robustness to incorporate further advanced pili features and various cell geometries to describe other bacteria that employ pili to move on surfaces.

How the Motility Pattern of Bacteria Affects Their Dispersal and Chemotaxis (vol 8, e81936, 2013)

J. Taktikos, H. Stark, Vasily Zaburdaev

How the Motility Pattern of Bacteria Affects Their Dispersal and Chemotaxis

Johannes Taktikos, Holger Stark, Vasily Zaburdaev

Most bacteria at certain stages of their life cycle are able to move actively; they can swim in a liquid or crawl on various surfaces. A typical path of the moving cell often resembles the trajectory of a random walk. However, bacteria are capable of modifying their apparently random motion in response to changing environmental conditions. As a result, bacteria can migrate towards the source of nutrients or away from harmful chemicals. Surprisingly, many bacterial species that were studied have several distinct motility patterns, which can be theoretically modeled by a unifying random walk approach. We use this approach to quantify the process of cell dispersal in a homogeneous environment and show how the bacterial drift velocity towards the source of attracting chemicals is affected by the motility pattern of the bacteria. Our results open up the possibility of accessing additional information about the intrinsic response of the cells using macroscopic observations of bacteria moving in inhomogeneous environments.

A Bacterial Swimmer with Two Alternating Speeds of Propagation

Matthias Theves, Johannes Taktikos, Vasily Zaburdaev, Holger Stark, Carsten Beta

We recorded large data sets of swimming trajectories of the soil bacterium Pseudomonas putida. Like other prokaryotic swimmers, P. putida exhibits a motion pattern dominated by persistent runs that are interrupted by turning events. An in-depth analysis of their swimming trajectories revealed that the majority of the turning events is characterized by an angle of phi(1) = 180 degrees (reversals). To a lesser extent, turning angles of phi(2 Sigma Sigma Sigma Sigma) = 00 are also found. Remarkably, we observed that, upon a reversal, the swimming speed changes by a factor of two on average a prominent feature of the motion pattern that, to our knowledge, has not been reported before. A theoretical model, based on the experimental values for the average run time and the rotational diffusion, recovers the mean-square displacement of P. putida if the two distinct swimming speeds are taken into account. Compared to a swimmer that moves with a constant intermediate speed, the mean-square displacement is strongly enhanced. We furthermore observed a negative dip in the directional autocorrelation at intermediate times, a feature that is only recovered in an extended model, where the nonexponential shape of the run-time distribution is taken into account.

Space-Time Velocity Correlation Function for Random Walks

Vasily Zaburdaev, S. Denisov, P. Haenggi

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. Here we adopt this concept for single-particle random walks and demonstrate that the corresponding space-time velocity autocorrelation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation functions. A generic feature of considered random-walk processes is an effect of velocity echo identified by the existence of time-dependent regions where most of the walkers are moving in the direction opposite to their initial motion. We discuss the relevance of the space-time velocity correlation functions for the experimental studies of cold atom dynamics in an optical potential and charge transport on micro- and nanoscales. DOI: 10.1103/PhysRevLett.110.170604

Liquid transport facilitated by channels in Bacillus subtilis biofilms

James N. Wilking, Vasily Zaburdaev, Michael De Volder, Richard Losick, Michael P. Brenner, David A. Weitz

Many bacteria on earth exist in surface-attached communities known as biofilms. These films are responsible for manifold problems, including hospital-acquired infections and biofouling, but they can also be beneficial. Biofilm growth depends on the transport of nutrients and waste, for which diffusion is thought to be the main source of transport. However, diffusion is ineffective for transport over large distances and thus should limit growth. Nevertheless, biofilms can grow to be very large. Here we report the presence of a remarkable network of well-defined channels that form in wildtype Bacillus subtilis biofilms and provide a system for enhanced transport. We observe that these channels have high permeability to liquid flow and facilitate the transport of liquid through the biofilm. In addition, we find that spatial variations in evaporative flux from the surface of these biofilms provide a driving force for the flow of liquid in the channels. These channels offer a remarkably simple system for liquid transport, and their discovery provides insight into the physiology and growth of biofilms.

Langevin description of superdiffusive Levy processes

S. Eule, Vasily Zaburdaev, R. Friedrich, T. Geisel

The description of diffusion processes is possible in different frameworks such as random walks or Fokker-Planck or Langevin equations. Whereas for classical diffusion the equivalence of these methods is well established, in the case of anomalous diffusion it often remains an open problem. In this paper we aim to bring three approaches describing anomalous superdiffusive behavior to a common footing. While each method clearly has its advantages it is crucial to understand how those methods relate and complement each other. In particular, by using the method of subordination, we show how the Langevin equation can describe anomalous diffusion exhibited by Levy-walk-type models and further show the equivalence of the random walk models and the generalized Kramers-Fokker-Planck equation. As a result a synergetic and complementary description of anomalous diffusion is obtained which provides a much more flexible tool for applications in real-world systems.

Collective dynamics of model microorganisms with chemotactic signaling

Johannes Taktikos, Vasily Zaburdaev, Holger Stark

Various microorganisms use chemotaxis for signaling among individuals-a common strategy for communication that is responsible for the formation of microcolonies. We model the microorganisms as autochemotactic active random walkers and describe them by an appropriate Langevin dynamics. It consists of rotational diffusion of the walker's velocity direction and a deterministic torque that aligns the velocity direction along the gradient of a self-generated chemical field. To account for finite size, each microorganism is treated as a soft disk. Its velocity is modified when it overlaps with other walkers according to a linear force-velocity relation and a harmonic repulsion force. We analyze two-walker collisions by presenting typical trajectories and by determining a state diagram that distinguishes between free walker, metastable, and bounded cluster states. We mention an analogy to Kramer's escape problem. Finally, we investigate relevant properties of many-walker systems and describe characteristics of cluster formation in unbounded geometry and in confinement.

Levy walks with velocity fluctuations

S. Denisov, Vasily Zaburdaev, P. Haenggi

The standard Levy walk is performed by a particle that moves ballistically between randomly occurring collisions when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events, the particle randomly changes the direction of motion but maintains the same constant speed. We generalize the standard model to incorporate velocity fluctuations into the process. Two types of models are considered, namely (i) with a walker changing the direction and absolute value of its velocity during collisions only, and (ii) with a walker whose velocity continuously fluctuates. We present a full analytic evaluation of both models and emphasize the importance of initial conditions. We show that, in the limit of weak velocity fluctuations, the integral diffusion characteristics and the bulk of diffusion profiles are identical to those for the standard Levy walk. However, the type of underlying velocity fluctuations can be identified by looking at the ballistic regions of the diffusion profiles. Our analytical results are corroborated by numerical simulations.

Modeling a self-propelled autochemotactic walker

Johannes Taktikos, Vasily Zaburdaev, Holger Stark

We develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis. This means that microorganisms, such as bacteria, amoebae, or cells, follow the gradient of a chemical that they produce themselves to attract or repel each other. A microorganism is represented as a self-propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle. We study the autochemotactic response of a single self-propelled walker whose dynamics is non-Markovian. We show that its long-time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak-and strong-coupling case. We confirm our findings by numerical simulations.

Langevin Dynamics Deciphers the Motility Pattern of Swimming Parasites

Vasily Zaburdaev, Sravanti Uppaluri, Thomas Pfohl, Markus Engstler, Rudolf Friedrich, Holger Stark

The parasite African trypanosome swims in the bloodstream of mammals and causes the highly dangerous human sleeping sickness. Cell motility is essential for the parasite's survival within the mammalian host. We present an analysis of the random-walk pattern of a swimming trypanosome. From experimental time-autocorrelation functions for the direction of motion we identify two relaxation times that differ by an order of magnitude. They originate from the rapid deformations of the cell body and a slower rotational diffusion of the average swimming direction. Velocity fluctuations are athermal and increase for faster cells whose trajectories are also straighter. We demonstrate that such a complex dynamics is captured by two decoupled Langevin equations that decipher the complex trajectory pattern by referring it to the microscopic details of cell behavior.

Perturbation Spreading in Many-Particle Systems: A Random Walk Approach

Vasily Zaburdaev, S. Denisov, P. Haenggi

The propagation of an initially localized perturbation via an interacting many-particle Hamiltonian dynamics is investigated. We argue that the propagation of the perturbation can be captured by the use of a continuous-time random walk where a single particle is traveling through an active, fluctuating medium. Employing two archetype ergodic many-particle systems, namely, (i) a hard-point gas composed of two unequal masses and (ii) a Fermi-Pasta-Ulam chain, we demonstrate that the corresponding perturbation profiles coincide with the diffusion profiles of the single-particle Levy walk approach. The parameters of the random walk can be related through elementary algebraic expressions to the physical parameters of the corresponding test many-body systems.

On moments and scaling regimes in anomalous random walks

Michael Schmiedeberg, Vasily Zaburdaev, Holger Stark

More and more stochastic transport phenomena in various real-world systems prove to belong to the class of anomalous diffusion. This paper is devoted to the scaling of diffusion-a very fundamental feature of this transport process. Our aim is to provide a comprehensive theoretical overview of scaling properties, but also to connect it to the analysis of experimental data.<br> Anomalous diffusion is commonly characterized by an exponent in the power law of the mean square displacement as a function of time < r(2)(t)> proportional to t(2 nu). On the other hand, it is known that the probability distribution function of diffusing particles can be approximated by (1/t(alpha))Phi(r/t(alpha)). While for classical normal diffusion this scaling relation is exact, it may not be valid globally for anomalous diffusion. In general, the exponent a obtained from the scaling of the central part of the probability distribution function differs from the exponent nu given by the mean square displacement. In this paper we systematically study how the scaling of different moments and parts of the probability distribution function can be determined and characterized even when no global scaling exists. We consider three rigorous methods for finding, respectively, the mean square displacement exponent nu, the scaling exponent alpha and the profile of the scaling function Phi. We also show that alternatively the scaling exponent alpha can be determined by analyzing fractional moments <vertical bar r vertical bar(q)> with q << 1. All analytical results are obtained in the framework of continuous-time random walks. For a wide class of coupled random walks, including the famous Levy walk model, we introduce a new unifying description which allows straightforward generalizations to other systems. Finally, we show how fractional moments help to analyze experimental or simulation data consistently.

Microscopic approach to random walks

Vasily Zaburdaev

In the present paper the microscopic approach to random walk models is introduced. For any particular model it provides a rigorous way to derive the transport equations for the macroscopic density of walking particles. Although it is not more complicated than the standard random walk framework it has virtually no limitations with respect to the initial distribution of particles. As a consequence, the transport equations derived with this method almost automatically give answers to such important problems as aging and two point probability distribution.

Random walks with random velocities

Vasily Zaburdaev, Michael Schmiedeberg, Holger Stark

We consider a random walk model that takes into account the velocity distribution of random walkers. Random motion with alternating velocities is inherent to various physical and biological systems. Moreover, the velocity distribution is often the first characteristic that is experimentally accessible. Here, we derive transport equations describing the dispersal process in the model and solve them analytically. The asymptotic properties of solutions are presented in the form of a phase diagram that shows all possible scaling regimes, including superdiffusive, ballistic, and superballistic motion. The theoretical results of this work are in excellent agreement with accompanying numerical simulations.

Stochastic transport through complex comb structures

Vasily Zaburdaev, P. V. Popov, A. S. Romanov, K. V. Chukbar

A unified rigorous approach is used to derive fractional differential equations describing subdiffusive transport through comb structures of various geometrical complexity. A general nontrivial effect of the initial particle distribution on the subsequent evolution is exposed. Solutions having qualitative features of practical importance are given for joined structures with widely different fractional exponents.

Chaoticity of the wet granular gas

A. Fingerle, S. Herminghaus, Vasily Zaburdaev

In this work we derive an analytic expression for the Kolmogorov-Sinai entropy of dilute wet granular matter, valid for any spatial dimension. The grains are modeled as hard spheres and the influence of the wetting liquid is described according to the capillary model, in which dissipation is due to the hysteretic cohesion force of capillary bridges. The Kolmogorov-Sinai entropy is expanded in a series with respect to density. We find a rapid increase of the leading term when liquid is added. This demonstrates the sensitivity of the granular dynamics to humidity, and shows that the liquid significantly increases the chaoticity of the granular gas.

Dry and wet granular shock waves

Vasily Zaburdaev, S. Herminghaus

The formation of a shock wave in one-dimensional granular gases is considered, for both the dry and the wet cases, and the results are compared with the analytical shock wave solution in a sticky gas. Numerical simulations show that the behavior of the shock wave in both cases tends asymptotically to the sticky limit. In the inelastic gas (dry case) there is a very close correspondence to the sticky gas, with one big cluster growing in the center of the shock wave, and a steplike stationary velocity profile. In the wet case, the shock wave has a nonzero width which is marked by two symmetric heavy clusters performing breathing oscillations with slowly increasing amplitude. All three models have the same asymptotic energy dissipation law, which is important in the context of the free cooling scenario. For the early stage of the shock formation and asymptotic oscillations we provide analytical results as well.

Free cooling of the one-dimensional wet granular gas

Vasily Zaburdaev, M. Brinkmann, S. Herminghaus

The free cooling behavior of a wet granular gas is studied in one dimension. We employ a particularly simple model system in which the interaction of wet grains is characterized by a fixed energy loss assigned to each collision. Macroscopic laws of energy dissipation and cluster formation are studied on the basis of numerical simulations and mean-field analytical calculations. We find a number of remarkable scaling properties which may shed light on earlier unexplained results for related systems.

Random walk model with waiting times depending on the preceding jump length

Vasily Zaburdaev

In the present paper, the generalized continuous time random walk model with a coupled transition kernel is considered. The coupling occurs through the dependence of the waiting time probability distribution on the preceding jump length. For the description of this model, a method is suggested that includes the details of the microscopic distribution over the waiting times and arrival distances at a given point. A close analogy to the problem of a random walk with finite velocity is demonstrated for the particular case of coupling, when a waiting time is a simple function of a preceding jump length. With its help an analytical solution for the generalized random walk model is found, including both effects ( finite velocity and jump dependent waiting times) simultaneously.

Theory of heat transport in a magnetized high-temperature plasma

Vasily Zaburdaev

The transport of charged particles across a strong magnetic field with a small random component is studied in the double diffusion approximation. It is shown that the density of the particles whose initial distribution is stretched along the field satisfies a subdiffusion equation with fractional derivatives. A more general initial particle distribution is also considered, and the applicability of the solutions obtained is discussed. (c) 2005 Pleiades Publishing, Inc.

Kolmogorov-Sinai entropy of the dilute wet granular gas

A Fingerle, S Herminghaus, Vasily Zaburdaev

We present an analytical expression for the Kolmogorov-Sinai entropy of a wet granular gas. The influence of the liquid is modeled by a hysteretic interaction force. For the dilute limit (two-particle collisions only), we find a simple expression accounting for the contribution of both the scattering states and the bound states in arbitrary dimensions. It is shown that the system is significantly more chaotic than a gas of (dry) hard spheres, as reflected by a pronounced increase of the Kolmogorov-Sinai entropy.

'Hermite' states in the quantum interaction of vortices

Vasily Zaburdaev, AS Romanov, KV Chukbar

Quantum effects in the dynamics of a pair of monopolar vortices, arbitrary in their nature, are described in both the Heisenberg and Schrodinger languages. This system proves to be very closely and universally related to a linear quantum oscillator, which allows us to call it a 'Hermite' system, in regard to the eigenfunctions of such an oscillator.

Subdiffusion in random compressible flows

K Chukbar, Vasily Zaburdaev

In this work, we study the diffusion of admixture particles in a one-dimensional velocity field given by a gradient of a random potential. This refers us to the case of random compressible flows, where previously only scaling estimates were available. We develop a general approach which allows to solve this problem analytically. With its help we derive the macroscopic transport equation and rigorously show in which cases transport can be subdiffusive. We find the Fourier-Laplace transform of the Green's function of this equation and prove that for some potential distributions it satisfies the subdiffusive equation with fractional derivative with respect to time.

Nonlinear dynamics of electron vortex lattices

Vasily Zaburdaev, VV Smirnov, KV Chukbar

Weak and strong nonlinearities that determine the evolution of regular ensembles of electron vortices in a magnetized plasma are analyzed. Qualitative differences in behavior between such a medium and standard nonlinear media are revealed. (C) 2004 MAIK "Nauka/Interpetiodica".

Comment on "Towards deterministic equations for Levy walks: The fractional material derivative"

KV Chukbar, Vasily Zaburdaev

The connection of problems considered in the paper by Sokolov and Metzler with stochastic transport in usual space and uniform medium is discussed.

Memory effects in stochastic transport

Vasily Zaburdaev, KV Chukbar

The memory effects in stochastic transport, namely, the dependence of the form of transport equations on the macroscopic time are considered. Equations explicitly taking into account the microscopic aspect of the problem, without which the transfer processes cannot be adequately described, are derived; the methods of their solution are suggested; and the asymptotic properties of the latter are analyzed. (C) 2003 MAIK "Nauka/Interperiodica".

Enhanced superdiffusion and finite velocity of Levy flights

Vasily Zaburdaev, KV Chukbar

A fractional differential equation is derived that describes the transformation of a stochastic transport from fast spreading ((x) over bar proportional to t (alpha), alpha > 1) to a pseudowave regime (alpha = 1) due to the finiteness of the velocities of individual particles. Qualitative features of the new regime are discussed. (C) 2002 MAIK "Nauka / Interperiodica".

Skin effects in a dusty plasma

Vasily Zaburdaev

A multifluid MHD model is applied to study the magnetic field dynamics in a dusty plasma. The motion of plasma electrons and ions is treated against the background of arbitrarily charged, immobile dust grains. When the dust density gradient is nonzero and when the inertia of the ions and electrons and the dissipation from their collisions with dust grains are neglected, we are dealing with a nonlinear convective penetration of the magnetic field into the plasma. When the dust density is uniform, the magnetic field dynamics is described by the nonlinear diffusion equations. The limiting cases of diffusion equations are analyzed for different parameter values of the problem (i.e., different rates of the collisions of ions and electrons with the dust grains and different ratios between the concentrations of the plasma components), and some of their solutions (including self-similar ones) are found. The results obtained can also be useful for research in solid-state physics, in which case the electrons and holes in a semiconductor may be analogues of plasma electrons and ions and the role of dust grains may be played by the crystal lattice and impurity atoms. (C) 2001 MAIK "Nauka/Interperiodica".

Theory of nondiffusive penetration of a magnetic field into a conducting medium

Vasily Zaburdaev

The penetration of a current and, accordingly, a magnetic field into the plasma of pulsed systems characterized by short temporal and spatial scales can be investigated in electron magnetohydrodynamics. A study is made of the rapid penetration of the magnetic field of an injected high-current ion beam into a plasma. (C) 2000 MAIK "Nauka/Interperiodica".