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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 t1/2 in all directions. Two mechanisms have been proposed to explain such statistics in other cell types: run and tumble behavior with Lévy-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 Lévy distributed. The analysis tools developed here should be broadly useful for distinguishing between mechanisms for superdiffusivity in other cells types and environments., Author summary Cells must move through their environment in many different biological processes, from wound healing to cancer invasion to the development of an embryo. There are different ways for cells to explore the physical space around them—ranging from moving along a straight path at constant speed to executing a random walk where the cell changes direction at every time point. Understanding what mechanisms are driving motility patterns in different cell types is important for identifying possible treatments for disease. We found that mouse fibroblast cells moving on a two-dimensional substrate were super-diffusive, meaning that they were able to cover distance faster than a random walk but not as fast as a straight walk. Traditional analysis of cell trajectories was not well-suited to distinguish between different possible mechanisms for super-diffusivity, so we developed a new tool to examine cell trajectories and distinguish between mechanisms. We found that mouse fibroblasts were super-diffusive due to a combination of large fluctuations in speed and “run-and-tumble” behavior, where cells move in a straight line for a while before changing direction rapidly. We expect this tool to be useful for analyzing motion in many other cell types.

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 the 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., Comment: 25 pages, 17 figures

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.

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. L\'evy 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 L\'evy 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.

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.

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 non-ballistic random walks.

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.

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.

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. We adopt here this concept for single-particle random walks and demonstrate that the corresponding space-time velocity auto-correlation 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 nano-scales., Phys. Rev. Lett., in press

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 L\'{e}vy 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.

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.

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. Anomalous diffusion is commonly characterized by an exponent in the power law of the mean square displacement as a function of time . On the other hand, it is known that the probability distribution function of diffusing particles can be approximated by (1/t?)?(r/t?). While for classical normal diffusion this scaling relation is exact, it may not be valid globally for anomalous diffusion. In general, the exponent ? obtained from the scaling of the central part of the probability distribution function differs from the exponent ? 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 ?, the scaling exponent ? and the profile of the scaling function ?. We also show that alternatively the scaling exponent ? can be determined by analyzing fractional moments |r|q with . All analytical results are obtained in the framework of continuous-time random walks. For a wide class of coupled random walks, including the famous L?vy 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.