Traveling bands of chemotactic bacteria: A theoretical analysis

Keller, Evelyn Fox; Segel, Lee A.

doi:10.1016/0022-5193(71)90051-8

Cited by 958 publications

(781 citation statements)

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“…A significant proportion of the mathematical and modelling literature has focused on other types of behaviour observed in the Keller-Segel equations, such as their ability to generate travelling wave solutions [41,46,53,73].…”

Section: Discussionmentioning

confidence: 99%

“…This model can be derived using a slight variation of the biased random walk approach above (see [78] for details) or through an argument from a "Weber-Fechner" law for cell behaviour [46]. Note that this model is again presented in nondimensional form with homogeneous steady state given by (1,1).…”

Section: (M2) Signal-dependent Sensitivitymentioning

confidence: 99%

“…With this version, we obtain the minimal model as β → ∞, while for β = 0 we obtain the "logistic" chemotactic sensitivity mentioned briefly above. Although the logistic sensitivity has inherent problems, amplified on below, its prominent employment both in specific applications [3,5,21,46] and mathematical analyses (for a review, see Sect. 6.1.1 of Horstmann's paper [40]) merits its consideration here.…”

Section: (M2) Signal-dependent Sensitivitymentioning

confidence: 99%

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A user’s guide to PDE models for chemotaxis

2008

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Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225-234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

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Section: Discussionmentioning

confidence: 99%

Section: (M2) Signal-dependent Sensitivitymentioning

confidence: 99%

Section: (M2) Signal-dependent Sensitivitymentioning

confidence: 99%

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A user’s guide to PDE models for chemotaxis

2008

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“…Keller and Segel (1971b) noted this form of the chemotactic coefficient allows the model to predict band behavior (traveling wave solutions) in the bacterial density profiles when the concentration of s is such that a continuum description is no longer adequate, the only condition being that χ/μ > 1. Holz and Chen (1979) demonstrated that the model of Keller and Segel (1971b) provides an adequate description of bacterial migration in the bacterial response of E. coli to serine. They derived estimates for μ and χ when the chemotactic coefficient is of the form in Eq.…”

Section: Modeling Chemotactic Bands Of Bacteriamentioning

confidence: 99%

“…In Lapidus and Schiller (1978), the authors extended their earlier model to include bacterial growth. They compared numerically determined model solutions with Adler's experimental results and those of Keller and Segel (1971b). The bacterial growth, while generally over a longer timescale than that of the experiment, was chosen to balance the loss of bacteria from the initial distribution due to the formation of bands.…”

Section: Modeling Chemotactic Bands Of Bacteriamentioning

confidence: 99%

Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis I: The Single Cell

Tindall¹,

et al. 2008

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We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller-Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated.

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Modeling microbial chemotaxis in a diffusion gradient chamber

Widman

Emerson

Chiu

et al. 1997

Biotechnol. Bioeng.

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The diffusion gradient chamber (DGC) has proven to be a useful experimental tool for studying population‐level microbial growth and chemotaxis. A mathematical model capable of reproducing the population‐level patterns formed as a result of cellular growth and chemotaxis in the DGC has been developed. The model consists of coupled partial differential balance equations for cells, chemoattractants, and a nutrient, which are solved simultaneously by the alternating direction implicit method. Modeling simulation results were compared with population‐level migration patterns of Escherichia coli growing on glycerol and responding to a gradient of the chemoattractant aspartate for two different initial conditions. To accurately reproduce the experimental results, a second chemoattractant equation was necessary. The second chemoattractant has been identified as oxygen by directly measuring oxygen gradients in the DGC. Important trends observed experimentally and reproduced by the model include the formation of a chemotactic wave, a reduction in the wave velocity as it encounters higher chemoattractant concentrations, and chemotaxis in response to two different chemoattractants simultaneously. The model was also used to study the relative magnitude of cell fluxes due to random motility and chemotaxis, and the suppression of chemotaxis due to receptor saturation. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 191–205, 1997.

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Traveling bands of chemotactic bacteria: A theoretical analysis

Cited by 958 publications

References 9 publications

A user’s guide to PDE models for chemotaxis

A user’s guide to PDE models for chemotaxis

Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis I: The Single Cell

Modeling microbial chemotaxis in a diffusion gradient chamber

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