Swarming bacteria migrate by Lévy Walk

Guided run-and-tumble active particles: wall accumulation and preferential deposition

Bacterial biofilms cost an enormous amount of resources in the health, medical, and industrial sectors. To understand early biofilm formation, beginning from planktonic states of active suspensions (such as Escherichia coli) to micro-colonization, it is vital to study the mechanics of cell accumulation near surfaces and subsequent deposition. Variability in bacterial motion strategies and the presence of taxis fields make the problem even more multifaceted. In this study, analytical expressions for the density and angular distributions, mean orientation, and deposition rates in such bacterial suspensions are derived, with and without the effects of external guiding or taxis fields. The derived results are closely verified by simulations of confined active particles using run-and-tumble statistics from multiple past experiments and utilizing a preferential sticking probability model for deposition. The behavioral changes in cell running strategies are modeled by varying the run-time distribution from an exponential to a heavy-tailed one. It is found that the deposition rates can be altered significantly by a guiding torque but are less affected by a change in the cell running behavior. However, both the mechanisms alter the pair correlation function of the deposited structures. The factor behind the changes in the architecture of deposited biomass under a torque generating guiding field turns out to be an asymmetrical rotational drift of planktonic cells, which can be an important physical mechanism behind the organization in confined active particle suspensions.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Anomalous Diffusion and Lévy Walks Distinguish Active from Inertial Turbulence

Bacterial swarms display intriguing dynamical states like active turbulence. Now, using a hydrodynamic model, we show that such dense active suspensions manifest superdiffusion, via Lévy walks, which masquerades as a crossover from ballistic to diffusive scaling in measurements of mean-squared displacements, and is tied to the emergence of hitherto undetected oscillatory streaks in the flow. Thus, while laying the theoretical framework of an emergent advantageous strategy in the collective behavior of microorganisms, our Letter underlines the essential differences between active and inertial turbulence.

Time-squeezing and time-expanding transformations in harmonic force fields

A variety of real life phenomena exhibit complex time-inhomogeneous nonlinear diffusive motion in the presence of an external harmonic force. In capturing the characteristics of such dynamics, the class of Ornstein-Uhlenbeck processes, with its physical time appropriately modulated, has long been regarded as the most relevant model on the basis of its mean reversion property. In this paper, we contrast two similar, yet definitely different, time-changing mechanisms in harmonic force fields by systematically deriving and presenting a variety of key properties all at once, that is, whether or not and how those time-changing mechanisms affect the characteristics of mean-reverting diffusion through sample path properties, the marginal probability density function, the asymptotic degeneracy of increments, the stationary law, the second-order structure, and the ensemble- and time-averaged mean square displacements. Some of those properties turn out rather counter-intuitive due to, or irrespective of, possible degeneracy of time-changing mechanisms in the long run. In light of those illustrative comparisons, we examine whether such time-changing operations are worth the additional operational cost, relative to physically relevant characteristics induced, and deduce practical implications and precautions from modeling and inference perspectives, for instance, on the experimental setup involving those anomalous diffusion processes, such as the observation start time and stepsize when measuring mean square displacements, so as to preclude potentially misleading results and paradoxical interpretations.

Mechanistic underpinning of cell aspect ratio-dependent emergent collective motions in swarming bacteria

Self-propelled bacteria can exhibit a large variety of non-equilibrium self-organized phenomena. Swarming is one such fascinating dynamical scenario where a number of motile individuals group into dynamical clusters and move in synchronized flows and vortices. While precedent investigations into rod-like particles confirm that an increased aspect-ratio promotes alignment and order, recent experimental studies in bacteria Bacillus subtilis show a non-monotonic dependence of the cell-aspect ratio on their swarming motion. Here, by computer simulations of an agent-based model of self-propelled, mechanically interacting, rod-shaped bacteria under overdamped conditions, we explore the collective dynamics of a bacterial swarm subjected to a variety of cell-aspect ratios. When modeled with an identical self-propulsion speed across a diverse range of cell aspect ratios, simulations demonstrate that both shorter and longer bacteria exhibit slow dynamics whereas the fastest speed is obtained at an intermediate aspect ratio. Our investigation highlights that the origin of this observed non-monotonic trend of bacterial speed and vorticity with the cell-aspect ratio is rooted in the cell-size dependence of motility force. The swarming features remain robust for a wide range of surface density of the cells, whereas asymmetry in friction attributes a distinct effect. Our analysis identifies that at an intermediate aspect ratio, an optimum cell size and motility force promote alignment, which reinforces the mechanical interactions among neighboring cells leading to the overall fastest motion. Mechanistic underpinning of the collective motions reveals that it is a joint venture of the short-range repulsive and the size-dependent motility forces, which determines the characteristics of swarming.

Individual based models exhibiting Lévy-flight type movement induced by intracellular noise

We use an individual-based model and its associated kinetic equation to study the generation of long jumps in the motion of E. coli. These models relate the run-and-tumble process to the intracellular reaction where the intrinsic noise plays a central role. Compared with previous work in Perthame et al. (Z Angew Math Phys 69(3):1-15, 2018), in which the parametric assumptions are mainly targeted for mathematical convenience but not well-suited for numerical simulations or comparison with experimental results, our current paper makes use of biologically meaningful pathways and tumbling kernels. The main contribution of this current work is bridging the gap between the theoretical results and experimentally available data. Some particular forms of how the tumbling frequency depends on the internal variable are proposed. Moreover, we propose two individual-based models, one for the tumbling frequency and the other for the receptor activity, and perform numerical simulations. Power-law decay of the run length distribution, which corresponds to Lévy-type motions, is observed in our numerical results. The particular decay rate agrees quantitatively with the analytical result.

Lévy Walks and Path Chaos in the Dispersal of Elongated Structures Moving across Cellular Vortical Flows

In cellular vortical flows, namely arrays of counterrotating vortices, short but flexible filaments can show simple random walks through their stretch-coil interactions with flow stagnation points. Here, we study the dynamics of semirigid filaments long enough to broadly sample the vortical field. Using simulation, we find a surprising variety of long-time transport behavior-random walks, ballistic transport, and trapping-depending upon the filament's relative length and effective flexibility. Moreover, we find that filaments execute Lévy walks whose diffusion exponents generally decrease with increasing filament length, until transitioning to Brownian walks. Lyapunov exponents likewise increase with length. Even completely rigid filaments, whose dynamics is finite dimensional, show a surprising variety of transport states and chaos. Fast filament dispersal is related to an underlying geometry of "conveyor belts." Evidence for these various transport states is found in experiments using arrays of counterrotating rollers, immersed in a fluid and transporting a flexible ribbon.

Non-Genetic Diversity in Chemosensing and Chemotactic Behavior

Non-genetic phenotypic diversity plays a significant role in the chemotactic behavior of bacteria, influencing how populations sense and respond to chemical stimuli. First, we review the molecular mechanisms that generate phenotypic diversity in bacterial chemotaxis. Next, we discuss the functional consequences of phenotypic diversity for the chemosensing and chemotactic performance of single cells and populations. Finally, we discuss mechanisms that modulate the amount of phenotypic diversity in chemosensory parameters in response to changes in the environment.

Large-Scale Vortices with Dynamic Rotation Emerged from Monolayer Collective Motion of Gliding Flavobacteria

A collective motion of self-driven particles has been a fascinating subject in physics and biology. Sophisticated macroscopic behavior emerges through a population of thousands or millions of bacterial cells propelling itself by flagellar rotation and chemotactic responses. Here, we found a series of collective motions accompanying successive phase transitions for a nonflagellated rod-shaped soil bacterium, Flavobacterium johnsoniae, which was driven by a surface cell movement known as gliding motility. When we spotted the cells on an agar plate with a low level of nutrients, the bacterial community exhibited vortex patterns that spontaneously appeared as lattice and integrated into a large-scale circular plate. All patterns were exhibited with a monolayer of bacteria, which enabled us to two-dimensionally visualize an individual cell with high resolution within a wide-range pattern. The single cells moved with random orientation, but the cells that were connected with one another showed left-turn-biased trajectories in a starved environment. This feature is possibly due to the collision of cells inducing a nematic alignment of dense cells as self-propelled rods. Subsequently, each vortex oscillated independently and then transformed to the rotating mode as an independent circular plate. Notably, the rotational direction of the circular plate was counterclockwise without exception. The plates developed accompanying rotation with constant angular velocity, suggesting that the mode is an efficient strategy for bacterial survival. IMPORTANCE Self-propelled bacteria propelled by flagellar rotation often display highly organized dynamic patterns at high cell densities. Here, we found a new mode of collective motion in nonflagellated bacteria; vortex patterns spontaneously appeared as lattice and were integrated into a large-scale circular plate, comprising hundreds of thousands of cells, which exhibited unidirectional rotation in a counterclockwise manner and expanded in size on agar. A series of collective motions was driven by gliding motility of the rod-shaped soil bacterium Flavobacterium johnsoniae. In a low-nutrient environment, single cells moved with random orientation, while cells at high density moved together as a unitary cluster. This might be an efficient strategy for cells of this species to find nutrients.

Lévy walk dynamics explain gamma burst patterns in primate cerebral cortex

Lévy walks describe patterns of intermittent motion with variable step sizes. In complex biological systems, Lévy walks (non-Brownian, superdiffusive random walks) are associated with behaviors such as search patterns of animals foraging for food. Here we show that Lévy walks also describe patterns of oscillatory activity in primate cerebral cortex. We used a combination of empirical observation and modeling to investigate high-frequency (gamma band) local field potential activity in visual motion-processing cortical area MT of marmoset monkeys. We found that gamma activity is organized as localized burst patterns that propagate across the cortical surface with Lévy walk dynamics. Lévy walks are fundamentally different from either global synchronization, or regular propagating waves, because they include large steps that enable activity patterns to move rapidly over cortical modules. The presence of Lévy walk dynamics therefore represents a previously undiscovered mode of brain activity, and implies a novel way for the cortex to compute. We apply a biophysically realistic circuit model to explain that the Lévy walk dynamics arise from critical-state transitions between asynchronous and localized propagating wave states, and that these dynamics yield optimal spatial sampling of the cortical sheet. We hypothesise that Lévy walk dynamics could help the cortex to efficiently process variable inputs, and to find links in patterns of activity among sparsely spiking populations of neurons.

Power-laws in dog behavior may pave the way to predictive models: A pattern analysis study

Apparently random events in nature often reveal hidden patterns when analyzed using diverse and robust statistical tools. Power law distributions, for example, project diverse natural phenomenon, ranging from earthquakes to heartbeat dynamics into a common platform of self-similarity. Animal behavior in specific contexts has been shown to follow power law distributions. However, the behavioral repertoire of a species in its entirety has never been analyzed for the existence of such underlying patterns. Here we show that the frequency-rank data of randomly sighted behaviors at the population level of free-ranging dogs follow a scale-invariant power law behavior. It suggests that irrespective of changes in location of sightings, seasonal variations and observer bias, datasets exhibit a conserved trend of scale invariance. The data also exhibits robust self-similarity patterns at different scales which we extract using multifractal detrended fluctuation analysis. We observe that the probability of consecutive occurrence of behaviors of adjacent ranks is much higher than behaviors widely separated in rank. The findings open up the possibility of designing predictive models of behavior from correlations existing in true time series of behavioral data and exploring the general behavioral repertoire of a species for the presence of syntax.

DNA Gold Nanoparticle Motors Demonstrate Processive Motion with Bursts of Speed Up to 50 nm Per Second

Synthetic motors that consume chemical energy to produce mechanical work offer potential applications in many fields that span from computing to drug delivery and diagnostics. Among the various synthetic motors studied thus far, DNA-based machines offer the greatest programmability and have shown the ability to translocate micrometer-distances in an autonomous manner. DNA motors move by employing a burnt-bridge Brownian ratchet mechanism, where the DNA "legs" hybridize and then destroy complementary nucleic acids immobilized on a surface. We have previously shown that highly multivalent DNA motors that roll offer improved performance compared to bipedal walkers. Here, we use DNA-gold nanoparticle conjugates to investigate and enhance DNA nanomotor performance. Specifically, we tune structural parameters such as DNA leg density, leg span, and nanoparticle anisotropy as well as buffer conditions to enhance motor performance. Both modeling and experiments demonstrate that increasing DNA leg density boosts the speed and processivity of motors, whereas DNA leg span increases processivity and directionality. By taking advantage of label-free imaging of nanomotors, we also uncover Lévy-type motion where motors exhibit bursts of translocation that are punctuated with transient stalling. Dimerized particles also demonstrate more ballistic trajectories confirming a rolling mechanism. Our work shows the fundamental properties that control DNA motor performance and demonstrates optimized motors that can travel multiple micrometers within minutes with speeds of up to 50 nm/s. The performance of these nanoscale motors approaches that of motor proteins that travel at speeds of 100-1000 nm/s, and hence this work can be important in developing protocellular systems as well next generation sensors and diagnostics.

Scale-free movement patterns in termites emerge from social interactions and preferential attachments

As the number or density of interacting individuals in a social group increases, a transition can develop from uncorrelated and disordered behavior of the individuals to a collective coherent pattern. We expand this observation by exploring the fine details of termite movement patterns to demonstrate that the value of the scaling exponent μ of a power law describing the Lévy walk of an individual is modified collectively as the density of animals in the group changes. This effect is absent when termites interact with inert obstacles. We also show that the network of encounters and interactions among specific individuals is selective, resembling a preferential attachment mechanism that is important for social networking. Our data strongly suggest that preferential attachments, a phenomenon not reported previously, and favorite interactions with a limited number of acquaintances are responsible for the generation of Lévy movement patterns in these social insects.

Power Laws Derived from a Bayesian Decision-Making Model in Non-Stationary Environments

The frequency of occurrence of step length in the migratory behaviour of various organisms, including humans, is characterized by the power law distribution. This pattern of behaviour is known as the Lévy walk, and the reason for this phenomenon has been investigated extensively. Especially in humans, one possibility might be that this pattern reflects the change in self-confidence in one’s chosen behaviour. We used simulations to demonstrate that active assumptions cause changes in the confidence level in one’s choice under a situation of lack of information. More specifically, we presented an algorithm that introduced the effects of learning and forgetting into Bayesian inference, and simulated an imitation game in which two decision-making agents incorporating the algorithm estimated each other’s internal models. For forgetting without learning, each agents’ confidence levels in their own estimation remained low owing to a lack of information about the counterpart, and the agents changed their hypotheses about the opponent frequently, and the frequency distribution of the duration of the hypotheses followed an exponential distribution for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and exponential distributions universally turned into power law distribution.

Stochastic modeling of Lévy-like human eye movements

The standard model of visual search dynamics is Brownian motion. However, recent research in cognitive science reveals that standard diffusion processes seem not to be the appropriate models of human looking behavior. In particular, experimental results confirm that the superdiffusive Lévy-type dynamics appears in this context. In this paper, we analyze the diffusive properties of human eye movement in a language comprehension task. We propose a model that is a combination of a Markov chain with a finite number of states and a Lévy walk. Our model fits well the experimental data and allows one to investigate the properties of the visual search dynamics using numerical simulations.

Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes

Lévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks.

Statistics of pathogenic bacteria in the search of host cells

A crucial phase in the infection process, which remains poorly understood, is the localization of suitable host cells by bacteria. It is often assumed that chemotaxis plays a key role during this phase. Here, we report a quantitative study on how Salmonella Typhimurium search for T84 human colonic epithelial cells. Combining time-lapse microscopy and mathematical modeling, we show that bacteria can be described as chiral active particles with strong active speed fluctuations, which are of biological, as opposed to thermal, origin. We observe that there exists a giant range of inter-individual variability of the bacterial exploring capacity. Furthermore, we find Salmonella Typhimurium does not exhibit biased motion towards the cells and show that the search time statistics is consistent with a random search strategy. Our results indicate that in vitro localization of host cells, and also cell infection, are random processes, not involving chemotaxis, that strongly depend on bacterial motility parameters.

Self-reinforcing directionality generates truncated Lévy walks without the power-law assumption

We introduce a persistent random walk model with finite velocity and self-reinforcing directionality, which explains how exponentially distributed runs self-organize into truncated Lévy walks observed in active intracellular transport by Chen et al. [Nature Mater., 14, 589 (2015)10.1038/nmat4239]. We derive the nonhomogeneous in space and time, hyperbolic partial differential equation for the probability density function (PDF) of particle position. This PDF exhibits a bimodal density (aggregation phenomena) in the superdiffusive regime, which is not observed in classical linear hyperbolic and Lévy walk models. We find the exact solutions for the first and second moments and criteria for the transition to superdiffusion.

An expanding bacterial colony forms a depletion zone with growing droplets

Many species of bacteria have developed effective means to spread on solid surfaces. This study focuses on the expansion of Pseudomonas aeruginosa on an agar gel surface under conditions of minimal evaporation. We report the occurrence and spread of a depletion zone within an expanded colony, where the bacteria laden film becomes thinner. The depletion zone is colocalized with a higher concentration of rhamnolipids, the biosurfactants that are produced by the bacteria and accumulate in the older region of the colony. With continued growth in population, dense bacterial droplets occur and coalesce in the depletion zone, displaying remarkable fluid dynamic behavior. Whereas expansion of a central depletion zone requires activities of live bacteria, new zones can be seeded elsewhere by adding rhamnolipids. These depletion zones due to the added surfactants expand quickly, even on plates covered by bacteria that have been killed by ultraviolet light. We explain the observed properties based on considerations of bacterial growth and secretion, osmotic swelling, fluid volume expansion, interfacial fluid dynamics involving Marangoni and capillary flows, and cell-cell cohesion.

Swimming Escherichia coli Cells Explore the Environment by Lévy Walk

Escherichia coli cells swim in aqueous environment in a random walk of alternating runs and tumbles. The diffusion characteristics of this random walk remains unclear. In this study, by tracking the swimming of wild-type cells in a three-dimensional (3D) homogeneous environment, we found that their trajectories are superdiffusive, consistent with Lévy walk behavior. For comparison, we tracked the swimming of mutant cells that lack the chemotaxis signaling noise (the steady-state fluctuation of the concentration of the chemotaxis response regulator CheY-P) and found that their trajectories are normal diffusive. Therefore, wild-type E. coli cells explore the environment by Lévy walk, which originates from the chemotaxis signaling noise. This Lévy walk pattern enhances their efficiency in environmental exploration.IMPORTANCE E. coli cells explore the environment in a random walk of alternating runs and tumbles. By tracking the 3D trajectories of E. coli cells in an aqueous environment, we found that their trajectories are superdiffusive, with a power-law shape for the distribution of run lengths, which is characteristics of Lévy walk. We further show that this Lévy walk behavior is due to the random fluctuation of the output level of the bacterial chemotaxis pathway, and it enhances the efficiency of the bacteria in exploring the environment.

Collective motion enhances chemotaxis in a two-dimensional bacterial swarm

In a dilute liquid environment in which cell-cell interaction is negligible, flagellated bacteria, such as Escherichia coli, perform chemotaxis by biased random walks alternating between run-and-tumble. In a two-dimensional crowded environment, such as a bacterial swarm, the typical behavior of run-and-tumble is absent, and this raises the question whether and how bacteria can perform chemotaxis in a swarm. Here, by examining the chemotactic behavior as a function of the cell density, we showed that chemotaxis is surprisingly enhanced because of cell crowding in a bacterial swarm, and this enhancement is correlated with increase in the degree of cell body alignment. Cells tend to form clusters that move collectively in a swarm with increased effective run length, and we showed analytically that this resulted in increased drift velocity toward attractants. We also explained the enhancement by stochastically simulating bacterial chemotaxis in a swarm. We found that cell crowding in a swarm enhances chemotaxis if the cell-cell interactions used in the simulation induce cell-cell alignment, but it impedes chemotaxis if the interactions are collisions that randomize cell moving direction. Therefore, collective motion in a bacterial swarm enhances chemotaxis.

Stronger environmental adaptation of rare rather than abundant bacterioplankton in response to dredging in eutrophic Lake Nanhu (Wuhan, China)

Deciphering responses of rare versus abundant bacterioplankton to environmental change, crucial for understanding and mitigating of cyanobacterial blooms, is an important but poorly investigated subject. Using MiSeq sequencing, we investigated the taxonomic and phylogenetic diversity of rare and abundant bacterioplankton in eutrophic Lake Nanhu before and after dredging. We estimated environmental breadths and phylogenetic signals of ecological preferences of rare and abundant bacterioplankton, and investigated community function and bacterioplankton assembly processes. Both taxonomic and phylogenic distances of rare and abundant bacterioplankton communities were significantly positively correlated with the dissimilarity of environmental factors. Threshold indicator taxa analysis and Blomberg's K statistic indicated that rare taxa held broader environmental thresholds and stronger phylogenetic signals for ecological traits than abundant taxa. Environmental adaptations of both rare and abundant taxa exhibited distinct changes after dredging. Higher functional redundancy occurred in the abundant compared to the rare bacterioplankton, with functions of rare bacterioplankton decreasing and for the abundant ones increasing after dredging. The null model revealed that dispersal limitation belonging to stochastic processes determined the abundant bacterioplankton community assembly, whereas variable selection belonging to deterministic processes drove the rare one. Rare bacterioplankton was more environmentally constrained than the abundant one. Dissolved oxygen was the decisive factor in determining the balance between stochasticity and determinism in both rare and abundant bacterioplankton. Our study extends our knowledge of environmental adaptation of rare versus abundant bacterioplankton to massive disturbing measures, i.e. dredging, and allows to estimate dredging performance for mitigating cyanobacterial blooms from a molecular ecology viewpoint.

Doppler imaging detects bacterial infection of living tissue

Living 3D in vitro tissue cultures, grown from immortalized cell lines, act as living sentinels as pathogenic bacteria invade the tissue. The infection is reported through changes in the intracellular dynamics of the sentinel cells caused by the disruption of normal cellular function by the infecting bacteria. Here, the Doppler imaging of infected sentinels shows the dynamic characteristics of infections. Invasive Salmonella enterica serovar Enteritidis and Listeria monocytogenes penetrate through multicellular tumor spheroids, while non-invasive strains of Escherichia coli and Listeria innocua remain isolated outside the cells, generating different Doppler signatures. Phase distributions caused by intracellular transport display Lévy statistics, introducing a Lévy-alpha spectroscopy of bacterial invasion. Antibiotic treatment of infected spheroids, monitored through time-dependent Doppler shifts, can distinguish drug-resistant relative to non-resistant strains. This use of intracellular Doppler spectroscopy of living tissue sentinels opens a new class of microbial assay with potential importance for studying the emergence of antibiotic resistance.

Honggu Choi et al. use biodynamic Doppler imaging to monitor bacterial infection of 3D living tissue and describe changes in the intracellular motions of living host tissue induced by early-stage infection. This work demonstrates the potential for the clinical use of this method to test for antibiotic-resistant infections.

Sediment availability provokes a shift from Brownian to Lévy-like clonal expansion in a dune building grass

In biogeomorphic landscapes, plant traits can steer landscape development through plant-mediated feedback interactions. Interspecific differences in clonal expansion strategy can therefore lead to the emergence of different landscape organisations. Yet, whether landscape-forming plants adopt different clonal expansion strategies depending on their physical environment remains to be tested. Here, we use a field survey and a complementary mesocosm approach to investigate whether sediment deposition affects the clonal expansion strategy employed by dune-building marram grass individuals. Our results reveal a consistent shift in expansion pattern from more clumped, Brownian-like, movement in sediment-poor conditions, to patchier, Lévy-like, movement under high sediment supply rates. Additional model simulations illustrate that the sediment-dependent shift in movement strategies induces a shift in optimisation of the cost-benefit relation between landscape engineering (i.e. dune formation) and expansion. Plasticity in expansion strategy may therefore allow landscape-forming plants to optimise their engineering ability depending on their physical landscape.

Biofilm and swarming emergent behaviours controlled through the aid of biophysical understanding and tools

Bacteria can organise themselves into communities in the forms of biofilms and swarms. Through chemical and physical interactions between cells, these communities exhibit emergent properties that individual cells alone do not have. While bacterial communities have been mainly studied in the context of biochemistry and molecular biology, recent years have seen rapid advancements in the biophysical understanding of emergent phenomena through physical interactions in biofilms and swarms. Moreover, new technologies to control bacterial emergent behaviours by physical means are emerging in synthetic biology. Such technologies are particularly promising for developing engineered living materials (ELM) and devices and controlling contamination and biofouling. In this minireview, we overview recent studies unveiling physical and mechanical cues that trigger and affect swarming and biofilm development. In particular, we focus on cell shape, motion and density as the key parameters for mechanical cell-cell interactions within a community. We then showcase recent studies that use physical stimuli for patterning bacterial communities, altering collective behaviours and preventing biofilm formation. Finally, we discuss the future potential extension of biophysical and bioengineering research on microbial communities through computational modelling and deeper investigation of mechano-electrophysiological coupling.

Large deviations of the ballistic Lévy walk model

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a "light" cone -v_{0}t<x<v_{0}t. In particular we study this density close to its maximum in the vicinity of the light cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations. However, this law blows up in the vicinity of the spreading horizon, which is nonphysical in the sense that any finite-time observation will never diverge. We claim that one can find two laws for the spatial density: The first one is the mentioned Lamperti-arcsine law describing the central part of the distribution, and the second is an infinite density illustrating the dynamics for x≃v_{0}t. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short "time" using a rate formalism.

Long-distance Transport in Bacterial Swarms Revealed by Single Nanoparticle Tracking

During swarming, high density flagella-driven bacteria migrate collectively in a swirling pattern on wet agar surfaces, immersed in a thin viscous fluid layer called "swarm fluid". Though the fluid environment has essential role in the emergence of swarming behavior, the microscopic mechanisms of it in mediating the cooperation of bacteria populations are not fully understood. Here, instead of micro-sized tracers used in previous research, we use gold nanorods as single particle tracers to probe the dynamics of the swarm fluid. This protocol includes five major parts: (1) the culture of swarming bacterial colony; (2) the preparations of gold nanorod tracers and the micro-spraying technique which are used to put the nanotracers into the upper fluid of bacterial swarms; (3) imaging and tracking; (4) other necessary control experiments; (5) data analysis and fitting of physical models. With this method, the nano-sized tracers could move long distances above motile cells without direct collisions with the bacteria bodies. In this way, the microscopic dynamics of the swarm fluid could be tracked with high spatiotemporal resolution. Moreover, the comprehensive analysis of multi-particle trajectories provides systematic visualization of the fluid dynamics. The method is promising to probe the fluid dynamics of other natural or artificial active matter systems.

Bacterial biomechanics-From individual behaviors to biofilm and the gut flora

Bacteria inhabit a variety of locations and play important roles in the environment and health. Our understanding of bacterial biomechanics has improved markedly in the last decade and has revealed that biomechanics play a significant role in microbial biology. The obtained knowledge has enabled investigation of complex phenomena, such as biofilm formation and the dynamics of the gut flora. A bottom-up strategy, i.e., from the cellular to the macroscale, facilitates understanding of macroscopic bacterial phenomena. In this Review, we first cover the biomechanics of individual bacteria in the bulk liquid and on surfaces as the base of complex phenomena. The collective behaviors of bacteria in simple environments are next introduced. We then introduce recent advances in biofilm biomechanics, in which adhesion force and the flow environment play crucial roles. We also review transport phenomena in the intestine and the dynamics of the gut flora, focusing on that in zebrafish. Finally, we provide an overview of the future prospects for the field.

Swimming statistics of cargo-loaded single bacteria

Burgeoning interest in the area of bacteria-powered micro robotic systems prompted us to study the dynamics of cargo transport by single bacteria. In this paper, we have studied the swimming behaviour of oil-droplets attached as cargo to the cell bodies of single bacteria. The oil-droplet loaded bacteria exhibit super-diffusive motion which is characterised by a high degree of directional persistence. Interestingly, bacteria could navigate with cargo size as large as 8 μm resulting in an increased rotational drag of more than 2 orders when compared to the free bacteria. The directional change of cargo loaded bacterial trajectories seems to be enhanced by steric hindrance from other oil-droplets present in the environment.

Critical patch size reduction by heterogeneous diffusion

Population survival depends on a large set of factors and on how they are distributed in space. Due to landscape heterogeneity, species can occupy particular regions that provide the ideal scenario for development, working as a refuge from harmful environmental conditions. Survival occurs if population growth overcomes the losses caused by adventurous individuals that cross the patch edge. In this work, we consider a single species dynamics in a patch with a space-dependent diffusion coefficient. We show analytically, within the Stratonovich framework, that heterogeneous diffusion reduces the minimal patch size for population survival when contrasted with the homogeneous case with the same average diffusivity. Furthermore, this result is robust regardless of the particular choice of the diffusion coefficient profile. We also discuss how this picture changes beyond the Stratonovich framework. Particularly, the Itô case, which is nonanticipative, can promote the opposite effect, while Hänggi-Klimontovich interpretation reinforces the reduction effect.

Universality in incompressible active fluid: Effect of nonlocal shear stress

Phase transitions in active fluids attracted significant attention within the last decades. Recent results show [L. Chen et al., New J. Phys. 17, 042002 (2015)10.1088/1367-2630/17/4/042002] that an order-disorder phase transition in incompressible active fluids belongs to a new universality class. In this work, we further investigate this type of phase transition and focus on the effect of long-range interactions. This is achieved by introducing a nonlocal shear stress into the hydrodynamic description, which leads to superdiffusion of the velocity field, and can be viewed as a result of the active particles performing Lévy walks. The universal properties in the critical region are derived by performing a perturbative renormalization group analysis of the corresponding response functional within the one-loop approximation. We show that the effect of nonlocal shear stress decreases the upper critical dimension of the model, and can lead to the irrelevance of the active fluid self-advection with the resulting model belonging to an unusual long-range Model A universality class not reported before, to our knowledge. Moreover, when the degree of nonlocality is sufficiently high all nonlinearities become irrelevant and the mean-field description is valid in any spatial dimension.

Competing active and passive interactions drive amoebalike crystallites and ordered bands in active colloids

Swimmers and self-propelled particles are physical models for the collective behavior and motility of a wide variety of living systems, such as bacteria colonies, bird flocks, and fish schools. Such artificial active materials are amenable to physical models which reveal the microscopic mechanisms underlying the collective behavior. Here we study colloids in a dc electric field. Our quasi-two-dimensional system of electrically driven particles exhibits a rich and exotic phase behavior exhibiting passive crystallites, motile crystallites, an active gas, and banding. Amongst these are two mesophases, reminiscent of systems with competing interactions. At low field strengths activity suppresses demixing, leading to motile crystallites. Meanwhile, at high field strengths, activity drives partial demixing to traveling bands. We parametrize a particulate simulation model which reproduces the experimentally observed phases.

Mobility in immersed granular materials upon cyclic loading

We study the mobility of objects embedded in an immersed granular packing and subjected to cyclic loadings. With this aim, we conducted uplift experiments whereby a horizontal plate is embedded in the packing and subjected to a vertical cyclic force oscillating between zero and a maximum amplitude. Tests performed at different cyclic force frequencies and amplitudes evidence the development of three mobility regimes whereby the plate stays virtually immobile, moves up steadily, or slowly creeps upwards. Results show that steady plate uplift can occur at lower force magnitudes when the frequency is increased. We propose an interpretation of this frequency-weakening behavior based on force relaxation experiments and on the analysis of the mobility response of theoretical viscoelastoplastic mechanical analog. These results and analysis point out inherent differences in mobility response between steady and cyclic loadings in immersed granular materials.

Rules of collective migration: from the wildebeest to the neural crest

Collective migration, the movement of groups in which individuals affect the behaviour of one another, occurs at practically every scale, from bacteria up to whole species' populations. Universal principles of collective movement can be applied at all levels. In this review, we will describe the rules governing collective motility, with a specific focus on the neural crest, an embryonic stem cell population that undergoes extensive collective migration during development. We will discuss how the underlying principles of individual cell behaviour, and those that emerge from a supracellular scale, can explain collective migration. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Hydrodynamics, superfluidity, and giant number fluctuations in a model of self-propelled particles

We derive hydrodynamics of a prototypical one-dimensional model, having variable-range hopping, which mimics passive diffusion and ballistic motion of active, or self-propelled, particles. The model has two main ingredients-the hardcore interaction and the competing mechanisms of short- and long-range hopping. We calculate two density-dependent transport coefficients-the bulk-diffusion coefficient and the conductivity, the ratio of which, despite violation of detailed balance, is connected to particle-number fluctuation by an Einstein relation. In the limit of infinite-range hopping, the model exhibits, upon tuning density ρ (or activity), a "superfluidlike" transition from a finitely conducting fluid phase to an infinitely conducting "superfluid" phase, characterized by a divergence in conductivity χ(ρ)∼(ρ-ρ_{c})^{-1} with ρ_{c} being the critical density. The diverging conductivity greatly increases particle (or vacancy) mobility and thus induces "giant" number fluctuations in the system.

Tumble Suppression Is a Conserved Feature of Swarming Motility

Bacteria within a swarm move characteristically in packs, displaying an intricate swirling motion in which hundreds of dynamic rafts continuously form and dissociate as the swarm colonizes an increasing expanse of territory. The demonstrated property of E. coli to reduce its tumble bias and hence increase its run duration during swarming is expected to maintain and promote side-by-side alignment and cohesion within the bacterial packs. In this study, we observed a similar low tumble bias in five different bacterial species, both Gram positive and Gram negative, each inhabiting a unique habitat and posing unique problems to our health. The unanimous display of an altered run-tumble bias in swarms of all species examined in this investigation suggests that this behavioral adaptation is crucial for swarming.

Many bacteria use flagellum-driven motility to swarm or move collectively over a surface terrain. Bacterial adaptations for swarming can include cell elongation, hyperflagellation, recruitment of special stator proteins, and surfactant secretion, among others. We recently demonstrated another swarming adaptation in Escherichia coli, wherein the chemotaxis pathway is remodeled to decrease tumble bias (increase run durations), with running speeds increased as well. We show here that the modification of motility parameters during swarming is not unique to E. coli but is shared by a diverse group of bacteria we examined—Proteus mirabilis, Serratia marcescens, Salmonella enterica, Bacillus subtilis, and Pseudomonas aeruginosa—suggesting that increasing run durations and speeds are a cornerstone of swarming.

Trapping of swimmers in a vortex lattice

We examine the motion of rigid, ellipsoidal swimmers subjected to a steady vortex flow in two dimensions. Numerical simulations of swimmers in a spatially periodic array of vortices reveal a range of possible behaviors, including trapping inside a single vortex and motility-induced diffusion across many vortices. While the trapping probability vanishes at a sufficiently high swimming speed, we find that it exhibits surprisingly large oscillations as this critical swimming speed is approached. Strikingly, at even higher swimming speeds, we find swimmers that swim perpendicular to their elongation direction can again become trapped. To explain this complex behavior, we investigate the underlying swimmer phase-space geometry. We identify the fixed points and periodic orbits of the swimmer equations of motion that regulate swimmer trapping inside a single vortex cell. For low to intermediate swimming speeds, we find that a stable periodic orbit surrounded by invariant tori forms a transport barrier to swimmers and can trap them inside individual vortices. For swimming speeds approaching the maximum fluid speed, we find instead that perpendicular swimmers can be trapped by asymptotically stable fixed points. A bifurcation analysis of the stable periodic orbit and the fixed points explains the complex and non-monotonic breakdown and re-emergence of swimmer trapping as the swimmer speed and shape are varied.

Swarming Motility Without Flagellar Motor Switching by Reversal of Swimming Direction in E. coli

In a crowded environment such as a bacterial swarm, cells frequently got jammed and came to a stop, but were able to escape the traps by backing up in their moving course with a head-to-tail change (a reversal). Reversals are essential for the expansion of a bacterial swarm. Reversal for a wildtype cell usually involved polymorphic transformation of the flagellar filaments induced by directional switching of the flagellar motors. Here we discovered a new way of reversal in cells without motor switching and characterized its mechanisms. We further found that this type of reversal was not limited to swarmer cells, but also occurred for cells grown in a bulk solution. Therefore, reversal was a general way of escaping when cells got jammed in their natural complex habitats. The new way of reversal we discovered here offered a general strategy for cells to escape traps and explore their environment.

Investigating Flagella-Driven Motility in Escherichia coli by Applying Three Established Techniques in a Series

Motility is crucial to the survival and success of many bacterial species. Many methodologies exist to exploit motility to understand signaling pathways, to elucidate the function and assembly of flagellar parts, and to examine and understand patterns of movement. Here we demonstrate a combination of three of these methodologies. Motility in soft agar is the oldest, offering a strong selection for isolating gain-of-function suppressor mutations in motility-impaired strains, where motility is restored through a second mutation. The cell-tethering technique, first employed to demonstrate the rotary nature of the flagellar motor, can be used to assess the impact of signaling effectors on the motor speed and its ability to switch rotational direction. The "border-crossing" assay is more recent, where swimming bacteria can be primed to transition into moving collectively as a swarm. In combination, these protocols represent a systematic and powerful approach to identifying components of the motility machinery, and to characterizing their role in different facets of swimming and swarming. They can be easily adapted to study motility in other bacterial species.

Deciphering anomalous heterogeneous intracellular transport with neural networks

Intracellular transport is predominantly heterogeneous in both time and space, exhibiting varying non-Brownian behavior. Characterization of this movement through averaging methods over an ensemble of trajectories or over the course of a single trajectory often fails to capture this heterogeneity. Here, we developed a deep learning feedforward neural network trained on fractional Brownian motion, providing a novel, accurate and efficient method for resolving heterogeneous behavior of intracellular transport in space and time. The neural network requires significantly fewer data points compared to established methods. This enables robust estimation of Hurst exponents for very short time series data, making possible direct, dynamic segmentation and analysis of experimental tracks of rapidly moving cellular structures such as endosomes and lysosomes. By using this analysis, fractional Brownian motion with a stochastic Hurst exponent was used to interpret, for the first time, anomalous intracellular dynamics, revealing unexpected differences in behavior between closely related endocytic organelles.

E. coli "super-contaminates" narrow ducts fostered by broad run-time distribution

One notable feature of bacterial motion is their ability to swim upstream along corners and crevices, by leveraging hydrodynamic interactions. This motion through anatomic ducts or medical devices might be at the origin of serious infections. However, it remains unclear how bacteria can maintain persistent upstream motion while exhibiting run-and-tumble dynamics. Here, we demonstrate that Escherichia coli can travel upstream in microfluidic devices over distances of 15 mm in times as short as 15 min. Using a stochastic model relating the run times to the time that bacteria spend on surfaces, we quantitatively reproduce the evolution of the contamination profiles when considering a broad distribution of run times. The experimental data cannot be reproduced using the usually accepted exponential distribution of run times. Our study demonstrates that the run-and-tumble statistics determine macroscopic bacterial transport properties. This effect, which we name "super-contamination," could explain the fast onset of some life-threatening medical emergencies.

Lymphoidal chemokine CCL19 promoted the heterogeneity of the breast tumor cell motility within a 3D microenvironment revealed by a Lévy distribution analysis

Tumor cell heterogeneity, either at the genotypic or the phenotypic level, is a hallmark of cancer. Tumor cells exhibit large variations, even among cells derived from the same origin, including cell morphology, speed and motility type. However, current work for quantifying tumor cell behavior is largely population based and does not address the question of cell heterogeneity. In this article, we utilize Lévy distribution analysis, a method known in both social and physical sciences for quantifying rare events, to characterize the heterogeneity of tumor cell motility. Specifically, we studied the breast tumor cell (MDA-MB-231 cell line) velocity statistics when the cells were subject to well-defined lymphoid chemokine (CCL19) gradients using a microfluidic platform. Experimental results showed that the tail end of the velocity distribution of breast tumor cell was well described by a Lévy function. The measured Lévy exponent revealed that cell motility was more heterogeneous when CCL19 concentration was near the dynamic kinetic binding constant to its corresponding receptor CCR7. This work highlighted the importance of tumor microenvironment in modulating tumor cell heterogeneity and invasion.

Rare events in generalized Lévy Walks and the Big Jump principle

The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations and with complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.

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.