Swarming bacteria migrate by Lévy Walk

Dynamic motility selection drives population segregation in a bacterial swarm

Population expansion in space, or range expansion, is widespread in nature and in clinical settings. Space competition among heterogeneous subpopulations during range expansion is essential to population ecology, and it may involve the interplay of multiple factors, primarily growth and motility of individuals. Structured microbial communities provide model systems to study space competition during range expansion. Here we use bacterial swarms to investigate how single-cell motility contributes to space competition among heterogeneous bacterial populations during range expansion. Our results revealed that motility heterogeneity can promote the spatial segregation of subpopulations via a dynamic motility selection process. The dynamic motility selection is enabled by speed-dependent persistence time bias of single-cell motion, which presumably arises from physical interaction between cells in a densely packed swarm. We further showed that the dynamic motility selection may contribute to collective drug tolerance of swarming colonies by segregating subpopulations with transient drug tolerance to the colony edge. Our results illustrate that motility heterogeneity, or "motility fitness," can play a greater role than growth rate fitness in determining the short-term spatial structure of expanding populations.

The nucleus does not significantly affect the migratory trajectories of amoeba in two-dimensional environments

For a wide range of cells, from bacteria to mammals, locomotion movements are a crucial systemic behavior for cellular life. Despite its importance in a plethora of fundamental physiological processes and human pathologies, how unicellular organisms efficiently regulate their locomotion system is an unresolved question. Here, to understand the dynamic characteristics of the locomotion movements and to quantitatively study the role of the nucleus in the migration of Amoeba proteus we have analyzed the movement trajectories of enucleated and non-enucleated amoebas on flat two-dimensional (2D) surfaces using advanced non-linear physical-mathematical tools and computational methods. Our analysis shows that both non-enucleated and enucleated amoebas display the same kind of dynamic migration structure characterized by highly organized data sequences, super-diffusion, non-trivial long-range positive correlations, persistent dynamics with trend-reinforcing behavior, and move-step fluctuations with scale invariant properties. Our results suggest that the presence of the nucleus does not significantly affect the locomotion of amoeba in 2D environments.

Single Nanoparticle Tracking Reveals Efficient Long-Distance Undercurrent Transport in Upper Fluid of Bacterial Swarms

Flagellated bacteria move collectively in a swirling pattern on agar surfaces immersed in a thin layer of viscous "swarm fluid," but the role of this fluid in mediating the cooperation of the bacterial population is not well understood. Herein, we use gold nanorods (AuNRs) as single particle tracers to explore the spatiotemporal structure of the swarm fluid. Individual AuNRs are moving in a plane of ∼2 μm above swarms, traveling for long distances in high speed without interferences from bacterial movements. The particles are lifted and transported by collective mixing of small vortices around bacteria during localized clustering and de-clustering of motile cells. Their motions fit the Lévy walk model, revealing efficient fluidic flows above the swarms. These flows provide obstacle-free highways for long-range material transportations, allow swarming bacteria to perform population-level communications, and imply the essential role of the fluid phase on the emergence of large-scale synergy.

On Modeling Ensemble Transport of Metal Reducing Motile Bacteria

Many metal reducing bacteria are motile with their run-and-tumble behavior exhibiting series of flights and waiting-time spanning multiple orders of magnitude. While several models of bacterial processes do not consider their ensemble motion, some models treat motility using an advection diffusion equation (ADE). In this study, Geobacter and Pelosinus, two metal reducing species, are used in micromodel experiments for study of their motility characteristics. Trajectories of individual cells on the order of several seconds to few minutes in duration are analyzed to provide information on (1) the length of runs, and (2) time needed to complete a run (waiting or residence time). A Continuous Time Random Walk (CTRW) model to predict ensemble breakthrough plots is developed based on the motility statistics. The results of the CTRW model and an ADE model are compared with the real breakthrough plots obtained directly from the trajectories. The ADE model is shown to be insufficient, whereas a coupled CTRW model is found to be good at predicting breakthroughs at short distances and at early times, but not at late time and long distances. The inadequacies of the simple CTRW model can possibly be improved by accounting for correlation in run length and waiting time.

Single Cells Exhibit Differing Behavioral Phases during Early Stages of Pseudomonas aeruginosa Swarming

Pseudomonas aeruginosa is among the many bacteria that swarm, where groups of cells coordinate to move over surfaces. It has been challenging to determine the behavior of single cells within these high-cell-density swarms. To track individual cells within P. aeruginosa swarms, we imaged a fluorescently labeled subset of the larger population. Single cells at the advancing swarm edge varied in their motility dynamics as a function of time. From these data, we delineated four phases of early swarming prior to the formation of the tendril fractals characteristic of P. aeruginosa swarming by collectively considering both micro- and macroscale data. We determined that the period of greatest single-cell motility does not coincide with the period of greatest collective swarm expansion. We also noted that flagellar, rhamnolipid, and type IV pilus motility mutants exhibit substantially less single-cell motility than the wild type.IMPORTANCE Numerous bacteria exhibit coordinated swarming motion over surfaces. It is often challenging to assess the behavior of single cells within swarming communities due to the limitations of identifying, tracking, and analyzing the traits of swarming cells over time. Here, we show that the behavior of Pseudomonas aeruginosa swarming cells can vary substantially in the earliest phases of swarming. This is important to establish that dynamic behaviors should not be assumed to be constant over long periods when predicting and simulating the actions of swarming bacteria.

Mittag–Leffler Memory Kernel in Lévy Flights

In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.

Burst statistics in an early biofilm quorum sensing model: the role of spatial colony-growth heterogeneity

Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called “autoinducers”) and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a “cluster” of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get “induced” into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.

Rivalry in Bacillus subtilis colonies: enemy or family?

Two colonies of Bacillus subtilis of identical strains growing adjacent to each other on an agar plate exhibit two distinct types of interactions: they either merge as they grow or demarcation occurs leading to formation of a line of demarcation at the colony fronts. The nature of this interaction depends on the agar concentration in the growth medium and the initial separation between the colonies. When the agar concentration was 0.67% or lower, the two sibling colonies were found to always merge. At 1% or higher concentrations, the colonies formed a demarcation line only when their initial separation was 20 mm or higher. Interactions of a colony with solid structures and liquid drops have indicated that biochemical factors rather than the presence of physical obstacles are responsible for the demarcation line formation. A reaction diffusion model has been formulated to predict if two sibling colonies will form a demarcation line under given agar concentration and initial separation. The model prediction agrees well with experimental findings and generates a dimensionless phase diagram containing merging and demarcation regimes. The phase diagram is in terms of a dimensionless initial separation, d[combining macron], and a dimensionless diffusion coefficient, D[combining macron], of the colonies. The phase boundary between the two interaction regimes can be described by a power law relation between d[combining macron] and D[combining macron].

Recurrence time correlations in random walks with preferential relocation to visited places

Random walks with memory usually involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the transport properties as well as on the statistics of first-passage and subsequent recurrence times through a site. A preference for revisiting sites is thus expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in Phys. Rev. Lett. 112, 240601 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.240601 to explore this effect, together with the main transport properties induced by the long-range memory. Despite the long-range memory effects governing the process, our analytical treatment allows us to (i) observe the existence of an asymptotic logarithmic (ultraslow) growth for the mean square displacement, in accordance to the results found for the original discrete-time model, and (ii) confirm the existence of positive correlations between first-passage and subsequent recurrence times. This analysis is completed with a comprehensive numerical study which reveals, among other results, that these correlations between first-passage and recurrence times also exhibit clear signatures of this ultraslow relaxation dynamics.

Biased Lévy-walk pattern in the exploratory behavior of the Physarum plasmodium

The plasmodium of Physarum polycephalum is a unicellular and multinuclear giant amoeba. The plasmodium has the ability to sense and adapt to many kinds of environmental stimuli, and its optimization behavior in closed spaces has been analyzed extensively. However, few studies have tested the behavior of the plasmodium in an open spaces, despite the biological importance of the adaptability of biological entities in such conditions. Accordingly, we established an experimental setup with a very large and strictly homogeneous substrate and observed the long-term exploratory behavior of the plasmodium. As a result, we found that the movement of the plasmodium was consistent with a Lévy-walk, but with anisotropic bias.

A Lévy expansion strategy optimizes early dune building by beach grasses

Lifeforms ranging from bacteria to humans employ specialized random movement patterns. Although effective as optimization strategies in many scientific fields, random walk application in biology has remained focused on search optimization by mobile organisms. Here, we report on the discovery that heavy-tailed random walks underlie the ability of clonally expanding plants to self-organize and dictate the formation of biogeomorphic landscapes. Using cross-Atlantic surveys, we show that congeneric beach grasses adopt distinct heavy-tailed clonal expansion strategies. Next, we demonstrate with a spatially explicit model and a field experiment that the Lévy-type strategy of the species building the highest dunes worldwide generates a clonal network with a patchy shoot organization that optimizes sand trapping efficiency. Our findings demonstrate Lévy-like movement in plants, and emphasize the role of species-specific expansion strategies in landscape formation. This mechanistic understanding paves the way for tailor-made planting designs to successfully construct and restore biogeomorphic landscapes and their services.

Enhanced transport of nutrients powered by microscale flows of the self-spinning dinoflagellate Symbiodinium sp

The metabolism of a living organism (e.g. bacteria, algae, zooplankton) requires a continuous uptake of nutrients from the surrounding environment. However, within local spatial scales, nutrients are quickly used up under dense concentrations of organisms. Here, we report that self-spinning dinoflagellates Symbiodinium sp. (clade E) generate a microscale flow that mitigates competition and enhances the uptake of nutrients from the surrounding environment. Our experimental and theoretical results reveal that this incessant active behavior enhances transport by approximately 80-fold when compared with Brownian motion in living fluids. We found that the tracer ensemble probability density function for displacement is time-dependent, but consists of a Gaussian core and robust exponential tails (so-called non-Gaussian diffusion). This can be explained by interactions of far-field Brownian motions and a near-field entrainment effect along with microscale flows. The contribution of exponential tails sharply increases with algal density, and saturates at a critical density, implying a trade-off between aggregated benefit and negative competition for the spatially self-organized cells. Our work thus shows that active motion and migration of aquatic algae play key roles in diffusive transport and should be included in theoretical and numerical models of physical and biogeochemical ecosystems.

Curvature dynamics and long-range effects on fluid-fluid interfaces with colloids

We investigate the dynamics of a phase-separating binary fluid, containing colloidal dumbbells anchored to the fluid-fluid interface. Extensive lattice Boltzmann-immersed boundary method simulations reveal that the presence of soft dumbbells can significantly affect the curvature dynamics of the interface between phase-separating fluids, even though the coarsening dynamics is left nearly unchanged. In addition, our results show that the curvature dynamics exhibits distinct non-local effects, which might be exploited for the design of new soft mesoscale materials. We point out that the inspection of the statistical dynamics of the curvature can disclose new insights into local inhomogeneities of the binary fluid configuration, as a function of the volume fraction and aspect ratio of the dumbbells.

Escherichia coli Remodels the Chemotaxis Pathway for Swarming

The fundamental motile behavior of E. coli is a random walk, where straight “runs” are punctuated by “tumbles.” This behavior, conferred by the chemotaxis signaling system, is used to track chemical gradients in liquid. Our study results show that when migrating collectively on surfaces, E. coli modifies its chemosensory physiology to decrease its tumble bias (and hence to increase run durations) by post-transcriptional changes that alter the levels of a key signaling protein. We speculate that the low tumble bias may contribute to the observed Lévy walk (LW) trajectories within the swarm, where run durations have a power law distribution. In animals, LW patterns are hypothesized to maximize searches in unpredictable environments. Swarming bacteria face several challenges while moving collectively over a surface—maintaining cohesion, overcoming constraints imposed by a physical substrate, searching for nutrients as a group, and surviving lethal levels of antimicrobials. The altered chemosensory behavior that we describe in this report may help with these challenges.

Many flagellated bacteria “swarm” over a solid surface as a dense consortium. In different bacteria, swarming is facilitated by several alterations such as those corresponding to increased flagellum numbers, special stator proteins, or secreted surfactants. We report here a change in the chemosensory physiology of swarming Escherichia coli which alters its normal “run tumble” bias. E. coli bacteria taken from a swarm exhibit more highly extended runs (low tumble bias) and higher speeds than E. coli bacteria swimming individually in a liquid medium. The stability of the signaling protein CheZ is higher in swarmers, consistent with the observed elevation of CheZ levels and with the low tumble bias. We show that the tumble bias displayed by wild-type swarmers is the optimal bias for maximizing swarm expansion. In assays performed in liquid, swarm cells have reduced chemotactic performance. This behavior is specific to swarming, is not specific to growth on surfaces, and persists for a generation. Therefore, the chemotaxis signaling pathway is reprogrammed for swarming.

Lévy like patterns in the small-scale movements of marsupials in an unfamiliar and risky environment

We investigate the movement patterns of three different Neotropical marsupials in an unfamiliar and risky environment. Animals are released in a matrix from which they try to reach a patch of forest. Their movements, performed on a small spacial scale, are best approximated by Lévy flights. Patterns of oriented and non-oriented individuals - with forest patches within or beyond their perceptual range - differ only slightly in the value of their exponents. These facts suggest that, for these species, the appearance of Lévy flights is the product of animals innate behaviour that emerges spontaneously, as a neutral characteristic proper of a default movement mode for alerted animals.

Run-and-tumble motion with steplike responses to a stochastic input

We study a simple run-and-tumble random walk whose switching frequencies between run mode and tumble mode depend on a stochastic signal. We consider a particularly sharp, steplike dependence, where the run-to-tumble switching probability jumps from zero to one as the signal crosses a particular value (say y_{1}) from below. Similarly, tumble-to-run switching probability also shows a jump like this as the signal crosses another value (y_{2}<y_{1}) from above. We are interested in characterizing the effect of signaling noise on the long-time behavior of the random walker. We consider two different time-evolutions of the stochastic signal. In one case, the signal dynamics is an independent stochastic process and does not depend on the run-and-tumble motion. In this case we can analytically calculate the mean value and the complete distribution function of the run duration and tumble duration. In the second case, we assume that the signal dynamics is influenced by the spatial location of the random walker. For this system, we numerically measure the steady state position distribution of the random walker. We discuss some similarities and differences between our system and Escherichia coli chemotaxis, which is another well-known run-and-tumble motion encountered in nature.

Identifying the mechanism for superdiffusivity in mouse fibroblast motility

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.

Eggshell spheres protect brown widow spider ( Latrodectus geometricus) eggs from bacterial infection

Eggs provide a rich source of nutrients for the developing embryo, making them a favoured food source for other organisms as well. Several defence mechanisms have evolved to protect the developing embryos against microbial threats. In this article, we elucidate the defence strategy of brown widow spider ( Latrodectus geometricus) eggs against bacteria. Antibacterial activity was shown by inhibition of bacterial growth on agar plate, liquid culture and retarded biofilm formation. The defence strategy against bacterial invasion was demonstrated in the whole egg, whole egg extract, egg surface extract, eggshell and eggshell extract. The source and characteristics of this antibacterial activity are distinctive and stem in part from a dense layer of spheres covering the egg surface, likely originated from the oviposition fluid. The spheres are rich in low-molecular-weight proteins, yet their exact composition remains unknown. In this study, we demonstrate that the egg surface is hydrophobic, while the spheres are superhydrophilic. Egg surface roughness and hydrophobicity combined with its antibacterial chemical properties reduce the ability of bacteria to grow on the egg surface. Understanding the properties of these unique structures may contribute significantly to our knowledge of how nature deals with bacterial infections.

Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System

In flat space, changing a system's velocity requires the presence of an external force. However, an isolated nonrigid system can freely change its orientation due to the nonholonomic nature of the angular momentum conservation law. Such nonrigid isolated systems may thus manifest their internal dynamics as rotations. In this work, we show that for such systems chaotic internal dynamics may lead to macroscopic rotational random walk resembling thermally induced motion. We do so by studying the classical harmonic three-mass system in the strongly nonlinear regime, the simplest physical model capable of zero angular momentum rotation as well as chaotic dynamics. At low energies, the dynamics are regular and the system rotates at a constant rate with zero angular momentum. For sufficiently high energies a rotational random walk is observed. For intermediate energies the system performs ballistic bouts of constant rotation rates interrupted by unpredictable orientation reversal events, and the system constitutes a simple physical model for Lévy walks. The orientation reversal statistics in this regime lead to a fractional rotational diffusion that interpolates smoothly between the ballistic and regular diffusive regimes.

Learning the space-time phase diagram of bacterial swarm expansion

Most living systems, from individual cells to tissues and swarms, display collective self-organization on length scales that are much larger than those of the individual units that drive this organization. A fundamental challenge is to understand how properties of microscopic components determine macroscopic, multicellular biological function. Our study connects intracellular physiology to macroscale collective behaviors during multicellular development, spanning five orders of magnitude in length and six orders of magnitude in time, using bacterial swarming as a model system. This work is enabled by a high-throughput adaptive microscopy technique, which we combined with genetics, machine learning, and mathematical modeling to reveal the phase diagram of bacterial swarming and that cell–cell interactions within each swarming phase are dominated by mechanical interactions.

Coordinated dynamics of individual components in active matter are an essential aspect of life on all scales. Establishing a comprehensive, causal connection between intracellular, intercellular, and macroscopic behaviors has remained a major challenge due to limitations in data acquisition and analysis techniques suitable for multiscale dynamics. Here, we combine a high-throughput adaptive microscopy approach with machine learning, to identify key biological and physical mechanisms that determine distinct microscopic and macroscopic collective behavior phases which develop as Bacillus subtilis swarms expand over five orders of magnitude in space. Our experiments, continuum modeling, and particle-based simulations reveal that macroscopic swarm expansion is primarily driven by cellular growth kinetics, whereas the microscopic swarming motility phases are dominated by physical cell–cell interactions. These results provide a unified understanding of bacterial multiscale behavioral complexity in swarms.

Reaction fronts in persistent random walks with demographic stochasticity

Standard reaction-diffusion systems are characterized by infinite velocities and no persistence in the movement of individuals, two conditions that are violated when considering living organisms. Here we consider a discrete particle model in which individuals move following a persistent random walk with finite speed and grow with logistic dynamics. We show that, when the number of individuals is very large, the individual-based model is well described by the continuous reactive Cattaneo equation (RCE), but for smaller values of the carrying capacity important finite-population effects arise. The effects of fluctuations on the propagation speed are investigated both considering the RCE with a cutoff in the reaction term and by means of numerical simulations of the individual-based model. Finally, a more general Lévy walk process for the transport of individuals is examined and an expression for the front speed of the resulting traveling wave is proposed.

A statistical physics view of swarming bacteria

Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces, forming dynamic patterns of whirls and jets. This review presents a physical point of view of swarming bacteria, with an emphasis on the statistical properties of the swarm dynamics as observed in experiments. The basic physical principles underlying the swarm and their relation to contemporary theories of collective motion and active matter are reviewed and discussed in the context of the biological properties of swarming cells. We suggest a paradigm according to which bacteria have optimized some of their physical properties as a strategy for rapid surface translocation. In other words, cells take advantage of favorable physics, enabling efficient expansion that enhances survival under harsh conditions.

Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long "flights". Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for "reprogramming" the Lévy walks into either diffusive or ballistic movements.

The 3D architecture of a bacterial swarm has implications for antibiotic tolerance

Swarming bacteria are an example of a complex, active biological system, where high cell density and super-diffusive cell mobility confer survival advantages to the group as a whole. Previous studies on the dynamics of the swarm have been limited to easily observable regions at the advancing edge of the swarm where cells are restricted to a plane. In this study, using defocused epifluorescence video imaging, we have tracked the motion of fluorescently labeled individuals within the interior of a densely packed three-dimensional (3D) region of a swarm. Our analysis reveals a novel 3D architecture, where bacteria are constrained by inter-particle interactions, sandwiched between two distinct boundary conditions. We find that secreted biosurfactants keep bacteria away from the swarm-air upper boundary, and added antibiotics at the lower swarm-surface boundary lead to their migration away from this boundary. Formation of the antibiotic-avoidance zone is dependent on a functional chemotaxis signaling system, in the absence of which the swarm loses its high tolerance to the antibiotics.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α<2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

We consider the two- (2D) and three-dimensional (3D) Ising models on a square lattice at the critical temperature T_{c}, under Monte Carlo spin flip dynamics. The bulk magnetization and the magnetization of a tagged line in the 2D Ising model, and the bulk magnetization and the magnetization of a tagged plane in the 3D Ising model, exhibit anomalous diffusion. Specifically, their mean-square displacements increase as power laws in time, collectively denoted as ∼t^{c}, where c is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power laws in time-also with exponent c -in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based generalized Langevin equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.

Lévy foraging patterns of rural humans

Movement patterns resembling Lévy walks, often attributed to the execution of an advantageous probabilistic searching strategy, are found in a wide variety of organisms, from cells to human hunter-gatherers. It has been suggested that such movement patterns may be fundamental to how humans interact and experience the world and that they may have arisen early in our genus with the evolution of a hunting and gathering lifestyle. Here we show that Lévy walks are evident in the Me’Phaa of Mexico, in Brazilian Cariri farmers and in Amazonian farmers when gathering firewood, wild fruit and nuts. Around 50% of the search patterns resemble Lévy walks and these are characterized by Lévy exponents close to 1.7. The other search patterns more closely resemble bi-phasic walks. We suggest potential generative mechanisms for the occurrence of these ubiquitous Lévy walks which can be used to guide future studies on human mobility. We show that frequent excursions and meanderings from pre-existing trails can account for our observations.

Entangled time in flocking: Multi-time-scale interaction reveals emergence of inherent noise

Collective behaviors that seem highly ordered and result in collective alignment, such as schooling by fish and flocking by birds, arise from seamless shuffling (such as super-diffusion) and bustling inside groups (such as Lévy walks). However, such noisy behavior inside groups appears to preclude the collective behavior: intuitively, we expect that noisy behavior would lead to the group being destabilized and broken into small sub groups, and high alignment seems to preclude shuffling of neighbors. Although statistical modeling approaches with extrinsic noise, such as the maximum entropy approach, have provided some reasonable descriptions, they ignore the cognitive perspective of the individuals. In this paper, we try to explain how the group tendency, that is, high alignment, and highly noisy individual behavior can coexist in a single framework. The key aspect of our approach is multi-time-scale interaction emerging from the existence of an interaction radius that reflects short-term and long-term predictions. This multi-time-scale interaction is a natural extension of the attraction and alignment concept in many flocking models. When we apply this method in a two-dimensional model, various flocking behaviors, such as swarming, milling, and schooling, emerge. The approach also explains the appearance of super-diffusion, the Lévy walk in groups, and local equilibria. At the end of this paper, we discuss future developments, including extending our model to three dimensions.

Hydrodynamic Choreographies of Microswimmers

We unveil orbital topologies of two nearby swimming microorganisms using an artificial microswimmer, called Quadroar. Depending on the initial conditions of the microswimmers, we find diverse families of attractors including dynamical equilibria, bound orbits, braids, and pursuit–evasion games. We also observe a hydrodynamic slingshot effect: a system of two hydrodynamically interacting swimmers moving along braids can advance in space faster than non-interacting swimmers that have the same actuation parameters and initial conditions as the interacting ones. Our findings suggest the existence of complex collective behaviors of microswimmers, from equilibrium to rapidly streaming states.

Neuronal messenger ribonucleoprotein transport follows an aging Lévy walk

Localization of messenger ribonucleoproteins (mRNPs) plays an essential role in the regulation of gene expression for long-term memory formation and neuronal development. Knowledge concerning the nature of neuronal mRNP transport is thus crucial for understanding how mRNPs are delivered to their target synapses. Here, we report experimental and theoretical evidence that the active transport dynamics of neuronal mRNPs, which is distinct from the previously reported motor-driven transport, follows an aging Lévy walk. Such nonergodic, transient superdiffusion occurs because of two competing dynamic phases: the motor-involved ballistic run and static localization of mRNPs. Our proposed Lévy walk model reproduces the experimentally extracted key dynamic characteristics of mRNPs with quantitative accuracy. Moreover, the aging status of mRNP particles in an experiment is inferred from the model. This study provides a predictive theoretical model for neuronal mRNP transport and offers insight into the active target search mechanism of mRNP particles in vivo.

Lévy walks in nonhomogeneous environments

The Lévy walk process with rests is discussed. The jumping time is governed by an α-stable distribution with α>1 while a waiting time distribution is Poissonian and involves a position-dependent rate which reflects a nonhomogeneous trap distribution. The master equation is derived and solved in the asymptotic limit for a power-law form of the jumping rate. The relative density of resting and flying particles appears time-dependent, and the asymptotic form of both distributions obeys a stretched-exponential shape at large time. The diffusion properties are discussed, and it is demonstrated that, due to the heterogeneous trap structure, the enhanced diffusion, observed for the homogeneous case, may turn to a subdiffusion. The density distributions and mean squared displacements are also evaluated from Monte Carlo simulations of individual trajectories.

Current status and future directions of Lévy walk research

Lévy walks are a mathematical construction useful for describing random patterns of movement with bizarre fractal properties that seem to have no place in biology. Nonetheless, movement patterns resembling Lévy walks have been observed at scales ranging from the microscopic to the ecological. They have been seen in the molecular machinery operating within cells during intracellular trafficking, in the movement patterns of T cells within the brain, in DNA, in some molluscs, insects, fish, birds and mammals, in the airborne flights of spores and seeds, and in the collective movements of some animal groups. Lévy walks are also evident in trace fossils (ichnofossils) – the preserved form of tracks made by organisms that occupied ancient sea beds about 252-66 million years ago. And they are utilised by algae that originated around two billion years ago, and still exist today. In September of 2017, leading researchers from across the life sciences, along with mathematicians and physicists, got together at a Company of Biologists' Workshop to discuss the origins and biological significance of these movement patterns. In this Review the essence of the technical and sometimes heated discussions is distilled and made accessible for all. In just a few pages, the reader is taken from a gentle introduction to the frontiers of a very active field of scientific enquiry. What emerges is a fascinating story of a truly inter-disciplinary scientific endeavour that is seeking to better understand movement patterns occurring across all biological scales.

Summary: Movement patterns resembling Lévy walks are found in a wide variety of organisms, from cells to humans. In this Review the latest research into their origins and biological significance is discussed.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Aging in mortal superdiffusive Lévy walkers

A growing body of literature examines the effects of superdiffusive subballistic movement premeasurement (aging or time lag) on observations arising from single-particle tracking. A neglected aspect is the finite lifetime of these Lévy walkers, be they proteins, cells, or larger structures. We examine the effects of aging on the motility of mortal walkers, and discuss the means by which permanent stopping of walkers may be categorized as arising from "natural" death or experimental artifacts such as low photostability or radiation damage. This is done by comparison of the walkers' mean squared displacement (MSD) with the front velocity of propagation of a group of walkers, which is found to be invariant under time lags. For any running time distribution of a mortal random walker, the MSD is tempered by the stopping rate θ. This provides a physical interpretation for truncated heavy-tailed diffusion processes and serves as a tool by which to better classify the underlying running time distributions of random walkers. Tempering of aged MSDs raises the issue of misinterpreting superdiffusive motion which appears Brownian or subdiffusive over certain time scales.

Emergence of Lévy Walks from Second-Order Stochastic Optimization

In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which Lévy flights or Lévy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required-instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios, the Lévy tail index is α=1, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.

Cellular automaton models for time-correlated random walks: derivation and analysis

Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits non-trivial temporal decay of velocity autocorrelation functions. This means that the corresponding dynamics is characterized by memory effects that slowly decay in time. Motivated by this we construct non-Markovian lattice-gas cellular automata models for moving agents with memory. For this purpose the reorientation probabilities are derived from velocity autocorrelation functions that are given a priori; in that respect our approach is “data-driven”. Particular examples we consider are velocity correlations that decay exponentially or as power laws, where the latter functions generate anomalous diffusion. The computational efficiency of cellular automata combined with our analytical results paves the way to explore the relevance of memory and anomalous diffusion for the dynamics of interacting cell populations, like confluent cell monolayers and cell clustering.

The topography of the environment alters the optimal search strategy for active particles

In environments with scarce resources, adopting the right search strategy can make the difference between succeeding and failing, even between life and death. At different scales, this applies to molecular encounters in the cell cytoplasm, to animals looking for food or mates in natural landscapes, to rescuers during search and rescue operations in disaster zones, and to genetic computer algorithms exploring parameter spaces. When looking for sparse targets in a homogeneous environment, a combination of ballistic and diffusive steps is considered optimal; in particular, more ballistic Lévy flights with exponent [Formula: see text] are generally believed to optimize the search process. However, most search spaces present complex topographies. What is the best search strategy in these more realistic scenarios? Here, we show that the topography of the environment significantly alters the optimal search strategy toward less ballistic and more Brownian strategies. We consider an active particle performing a blind cruise search for nonregenerating sparse targets in a 2D space with steps drawn from a Lévy distribution with the exponent varying from [Formula: see text] to [Formula: see text] (Brownian). We show that, when boundaries, barriers, and obstacles are present, the optimal search strategy depends on the topography of the environment, with [Formula: see text] assuming intermediate values in the whole range under consideration. We interpret these findings using simple scaling arguments and discuss their robustness to varying searcher's size. Our results are relevant for search problems at different length scales from animal and human foraging to microswimmers' taxis to biochemical rates of reaction.

The evolutionary origins of Lévy walk foraging

We study through a reaction-diffusion algorithm the influence of landscape diversity on the efficiency of search dynamics. Remarkably, the identical optimal search strategy arises in a wide variety of environments, provided the target density is sparse and the searcher’s information is restricted to its close vicinity. Our results strongly impact the current debate on the emergentist vs. evolutionary origins of animal foraging. The inherent character of the optimal solution (i.e., independent on the landscape for the broad scenarios assumed here) suggests an interpretation favoring the evolutionary view, as originally implied by the Lévy flight foraging hypothesis. The latter states that, under conditions of scarcity of information and sparse resources, some organisms must have evolved to exploit optimal strategies characterized by heavy-tailed truncated power-law distributions of move lengths. These results strongly suggest that Lévy strategies—and hence the selection pressure for the relevant adaptations—are robust with respect to large changes in habitat. In contrast, the usual emergentist explanation seems not able to explain how very similar Lévy walks can emerge from all the distinct non-Lévy foraging strategies that are needed for the observed large variety of specific environments. We also report that deviations from Lévy can take place in plentiful ecosystems, where locomotion truncation is very frequent due to high encounter rates. So, in this case normal diffusion strategies—performing as effectively as the optimal one—can naturally emerge from Lévy. Our results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

How organisms improve the search for food, mates, etc., is a key factor to their survival. Mathematically, the best strategy to look for randomly distributed re-visitable resources—under scarce information and sparse conditions—results from Lévy distributions of move lengths (the probability of taking a step ℓ is proportional to 1/ℓ²). Today it is well established that many animal species in different habitats do perform Lévy foraging. This fact has raised a heated debate, viz., the emergent versus evolutionary hypotheses. For the former, a Lévy foraging is an emergent property, a consequence of searcher-environment interactions: certain landscapes induce Lévy patterns, but others not. In this view, the optimal strategy depends on the particular habitat. The evolutionary explanation, in contrast, is that Lévy foraging strategies are adaptations that evolved via natural selection. In this article, through simulations we exhaustively analyze the influence of distinct environments on the foraging efficiency. We find that the optimal procedure is the same in all situations, provided density is low and landscape information is scarce. So, the best search strategy is remarkably independent of details. These results constitute the strongest theoretical evidence to date supporting the evolutionary origins of experimentally observed Lévy walks.

Multi-fractal characterization of bacterial swimming dynamics: a case study on real and simulated Serratia marcescens

To add to the current state of knowledge about bacterial swimming dynamics, in this paper, we study the fractal swimming dynamics of populations of Serratia marcescens bacteria both in vitro and in silico, while accounting for realistic conditions like volume exclusion, chemical interactions, obstacles and distribution of chemoattractant in the environment. While previous research has shown that bacterial motion is non-ergodic, we demonstrate that, besides the non-ergodicity, the bacterial swimming dynamics is multi-fractal in nature. Finally, we demonstrate that the multi-fractal characteristic of bacterial dynamics is strongly affected by bacterial density and chemoattractant concentration.

Role of tumbling in bacterial swarming

Typical wild-type bacteria swimming in sparse suspensions exhibit a movement pattern called "run and tumble," characterized by straight trajectories (runs) interspersed by shorter, random reorientation (tumbles). This is achieved by rotating their flagella counterclockwise, or clockwise, respectively. The chemotaxis signaling network operates in controlling the frequency of tumbles, enabling navigation toward or away from desired regions in the medium. In contrast, while in dense populations, flagellated bacteria exhibit collective motion and form large dynamic clusters, whirls, and jets, with intricate dynamics that is fundamentally different than trajectories of sparsely swimming cells. Although collectively swarming cells do change direction at the level of the individual cell, often exhibiting reversals, it has been suggested that chemotaxis does not play a role in multicellular colony expansion, but the change in direction stems from clockwise flagellar rotation. In this paper, the effects of cell rotor switching (i.e., the ability to tumble) and chemotaxis on the collective statistics of swarming bacteria are studied experimentally in wild-type Bacillus subtilis and two mutants-one that does not tumble and one that tumbles independently of the chemotaxis system. We show that while several of the parameters examined are similar between the strains, other collective and individual characteristics are significantly different. The results demonstrate that tumbling and/or flagellar directional rotor switching has an important role on the dynamics of swarming, and imply that swarming models of self-propelled rods that do not take tumbling and/or rotor switching into account may be oversimplified.

Noncontact Cohesive Swimming of Bacteria in Two-Dimensional Liquid Films

Bacterial swimming in confined two-dimensional environments is ubiquitous in nature and in clinical settings. Characterizing individual interactions between swimming bacteria in 2D confinement will help to understand diverse microbial processes, such as bacterial swarming and biofilm formation. Here we report a novel motion pattern displayed by flagellated bacteria in 2D confinement: When two nearby cells align their moving directions, they tend to engage in cohesive swimming without direct cell body contact, as a result of hydrodynamic interaction but not flagellar intertwining. We further found that cells in cohesive swimming move with higher directional persistence, which can increase the effective diffusivity of cells by ∼3 times as predicted by computational modeling. As a conserved behavior for peritrichously flagellated bacteria, cohesive swimming in 2D confinement may be key to collective motion and self-organization in bacterial swarms; it may also promote bacterial dispersal in unsaturated soils and in interstitial space during infections.

Morphologies of Bacillus subtilis communities responding to environmental variation

Bacterial communities exhibit a variety of growth morphologies in constructing robust systems under different environmental conditions. We review the diverse morphologies of Bacillus subtilis communities and their mechanisms of self-organization. B. subtilis uses different cell types to suit environmental conditions and cell density. The subpopulation of each cell type exhibits various environment-sensitive properties. Furthermore, division of labor among the subpopulations results in flexible development for the community as a whole. We review how B. subtilis community morphologies and growth strategies respond to environmental perturbations.

Generalized run-and-turn motions: From bacteria to Lévy walks

Swimming bacteria exhibit a repertoire of motility patterns, in which persistent motion is interrupted by turning events. What are the statistical properties of such random walks? If some particular instances have long been studied, the general case where turning times do not follow a Poisson process has remained unsolved. We present a generic extension of the continuous time random walks formalism relying on operators and noncommutative calculus. The approach is first applied to a unimodal model of bacterial motion. We examine the existence of a minimum in velocity correlation function and discuss the maximum of diffusivity at an optimal value of rotational diffusion. The model is then extended to bimodal patterns and includes as particular cases all swimming strategies: run-and-tumble, run-stop, run-reverse and run-reverse-flick. We characterize their velocity correlation functions and investigate how bimodality affects diffusivity. Finally, the wider applicability of the method is illustrated by considering curved trajectories and Lévy walks. Our results are relevant for intermittent motion of living beings, be they swimming micro-organisms or crawling cells.

Anomalous Fluctuations in the Orientation and Velocity of Swarming Bacteria

Simultaneous acquisition of phase-contrast light microscopy and fluorescently labeled bacteria, moving within a dense swarm, reveals the intricate interactions between cells and the collective flow around them. By comparing wild-type and immotile cells embedded in a dense wild-type swarm, the effect of the active thrust generated by the flagella can be singled out. It is shown that while the distribution of angles among cell velocity, cell orientation, and the local flow around it is Gaussian-like for immotile bacteria, wild-type cells exhibit anomalous non-Gaussian deviations and are able to move in trajectories perpendicular to the collective flow. Thus, cells can maneuver or switch between local streams and jets. A minimal model describing bacteria as hydrodynamic force dipoles shows that steric effects, hydrodynamics interactions, and local alignments all have to be taken into account to explain the observed dynamics. These findings shed light on the physical mechanisms underlying bacterial swarming and the balance between individual and collective dynamics.

Flagella bending affects macroscopic properties of bacterial suspensions

To survive in harsh conditions, motile bacteria swim in complex environments and respond to the surrounding flow. Here, we develop a mathematical model describing how flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease in the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random reorientation) as was previously done to explain the viscosity reduction. Second, we demonstrate how a bacterium escapes from wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.