Model for chemotaxis

A user’s guide to PDE models for chemotaxis

“…The review by Horstmann [40] considers five methods in detail and we refer to this work for further details of this significant area. Briefly, these are (i) arguments based on Fourier's law and Fick's law [45], (ii) biased random walk approaches [78], (iii) interacting particle systems [97], (iv) transport equations [2] or [35], and (v) stochastic processes [86]. A more recent derivation from multi-phase flow modelling has been proposed by Byrne and Owen [15].…”

“…In by far the majority of applications a constant diffusion coefficient is assumed, yet it is far more likely that this term should depend nonlinearly on the signal concentration and/or the cell density, as can be seen from derivations of Keller-Segel type systems through the various approaches mentioned in the introduction [15,45,80,92,95,96]. An explicit example is given in the formulation of the density-dependent chemotactic sensitivity models above, where a diffusion coefficient of the form D(u) = D (q − uq u ) was derived.…”

“…Theoretical and mathematical modelling of chemotaxis dates to the pioneering works of Patlak in the 1950s [86] and Keller and Segel in the 1970s [44,45]. The review article by Horstmann [40] provides a detailed introduction into the mathematics of the Keller-Segel (KS) model for chemotaxis.…”

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

“…Novick-Cohen and Segel (1984) later analyzed the original model of Keller and Segel (1971a) following the work of Scribner et al (1974), Keller and Odell (1975) and Alt (1980) and focused on the issue of whether the chemotactic coefficient is required to be singular in order to predict traveling wave solutions in the bacterial density profile. Their traveling wave and asymptotic analysis resolves issues of χ(s) needing to be singular and decreasing non-linearly to zero in order for the models to produce the experimentally observed bands of migrating bacteria.…”

“…Reynolds et al, 1989 andHornberger et al, 1992) or that related specifically to the study of biofilms lies outside our remit and is not considered here. We begin by considering continuum models of bacterial chemotaxis, in particular, the application of the Keller and Segel (1971a) model of chemotaxis, originally developed in the context of modeling slime molds, in describing the movement and behavior of chemotactic bacteria. Section 2.2 focuses on the various models developed over the past 30 years to understand Adler's observed bands of migrating bacteria within the capillary assay.…”