Lévy walk dynamics in mixed potentials from the perspective of random walk theory

Cell migration through extracellular matrices is critical to many physiological processes, such as tissue development, immunological response and cancer metastasis. Previous models including persistent random walk (PRW) and Lévy walk only explain the migratory dynamics of some cell types in a homogeneous environment. Recently, it was discovered that the intracellular actin flow can robustly ensure a universal coupling between cell migratory speed and persistence for a variety of cell types migrating in the in vitro assays and live tissues. However, effects of the correlation between speed and persistence on the macroscopic cell migration dynamics and patterns in complex environments are largely unknown. In this study, we developed a Monte Carlo random walk simulation to investigate the motility, the search ability and the search efficiency of a cell moving in both homogeneous and porous environments. The cell is simplified as a dimensionless particle, moving according to PRW, Lévy walk, random walk with linear speed-persistence correlation (linear RWSP) and random walk with nonlinear speed-persistence correlation (nonlinear RWSP). The coarse-grained analysis showed that the nonlinear RWSP achieved the largest motility in both homogeneous and porous environments. When a particle searches for targets, the nonlinear coupling of speed and persistence improves the search ability (i.e. find more targets in a fixed time period), but sacrifices the search efficiency (i.e. find less targets per unit distance). Moreover, both the convex and concave pores restrict particle motion, especially for the nonlinear RWSP and Lévy walk. Overall, our results demonstrate that the nonlinear correlation of speed and persistence has the potential to enhance the motility and searching properties in complex environments, and could serve as a starting point for more detailed studies of active particles in biological, engineering and social science fields.

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Swimming Escherichia coli explore the environment by Lévy walk

Applied and Environmental Microbiology ◽

10.1128/aem.02429-20 ◽

2021 ◽

Author(s):

Haiyan Huo ◽

Rui He ◽

Rongjing Zhang ◽

Junhua Yuan

Keyword(s):

Random Walk ◽

Response Regulator ◽

Bacterial Chemotaxis ◽

Random Fluctuation ◽

Aqueous Environment ◽

E Coli ◽

Lévy Walk ◽

Diffusion Characteristics ◽

Levy Walk ◽

Run Lengths

E. coli cells swim in aqueous environment in a random walk of alternating runs and tumbles. The diffusion characteristics of this random walk remains unclear. Here, by tracking the swimming of wildtype cells in a 3d homogeneous environment, we found that their trajectories are super diffusive, 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, wildtype 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 aqueous environment, we find that their trajectories are super diffusive, 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.

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

Science Advances ◽

10.1126/sciadv.abe8211 ◽

2021 ◽

Vol 7 (15) ◽

pp. eabe8211

Author(s):

Brieuc Guinard ◽

Amos Korman

Keyword(s):

Random Walk ◽

Mathematical Proof ◽

Higher Dimensions ◽

Two Dimensional ◽

Power Law Distribution ◽

Arbitrary Size ◽

Lévy Walk ◽

Lévy Walks ◽

Levy Walk ◽

Biological Organisms

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.

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Diffusion

10.1093/oso/9780198753919.003.0014 ◽

2018 ◽

Author(s):

Ginestra Bianconi

Keyword(s):

Random Walks ◽

Spectral Properties ◽

Diffusion Constant ◽

Network Structures ◽

Multilayer Networks ◽

Multilayer Network ◽

Lévy Walk ◽

Number Of Layers ◽

Levy Walk ◽

Speed Up

This chapter addresses diffusion, random walks and congestion in multilayer networks. Here it is revealed that diffusion on a multilayer network can be significantly speed up with respect to diffusion taking place on its single layers taken in isolation, and that sometimes it is possible also to observe super-diffusion. Diffusion is here characterized on multilayer network structures by studying the spectral properties of the supra-Laplacian and the dependence on the diffusion constant among different layers. Random walks and its variations including the Lévy Walk are shown to reflect the improved navigability of multilayer networks with more layers. These results are here compared with the results of traffic on multilayer networks that, on the contrary, point out that increasing the number of layers could be detrimental and could lead to congestion.

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Amoebic Foraging Model of Metastatic Cancer Cells

Symmetry ◽

10.3390/sym13071140 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1140

Author(s):

Daiki Andoh ◽

Yukio-Pegio Gunji

Keyword(s):

Bayesian Inference ◽

Cancer Cells ◽

Metastatic Cancer ◽

Gait Patterns ◽

Lévy Walk ◽

Levy Walk ◽

Unicellular Organisms ◽

Living Organisms ◽

Walk Algorithm ◽

Metastatic Cancer Cells

The Lévy walk is a pattern that is often seen in the movement of living organisms; it has both ballistic and random features and is a behavior that has been recognized in various animals and unicellular organisms, such as amoebae, in recent years. We proposed an amoeba locomotion model that implements Bayesian and inverse Bayesian inference as a Lévy walk algorithm that balances exploration and exploitation, and through a comparison with general random walks, we confirmed its effectiveness. While Bayesian inference is expressed only by P(h) = P(h|d), we introduce inverse Bayesian inference expressed as P(d|h) = P(d) in a symmetry fashion. That symmetry contributes to balancing contracting and expanding the probability space. Additionally, the conditions of various environments were set, and experimental results were obtained that corresponded to changes in gait patterns with respect to changes in the conditions of actual metastatic cancer cells.

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