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

Abstract

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.

Stable swarming using adaptive long-range interactions

“…When the density is low, the swarm is unstable and tends to collapse until it reaches the density that is given by Eq. (14). In other words starting from low density, we see that adaptive forces tend to pick a particular density ρ (n)…”

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

“…Also observe that for any τ >τ the density u corresponding to a = a * (τ ) satisfies u(0) = 2 a * (τ ) > 2 e. Finally, using v(z) ≥ v(r) in (18) and integrating the inequality on (0, ∞), one obtains e −v(0) − limr→∞ e −v(r) ≤ − σ I(τ ), for any τ > 0. As a consequence, using (21) and limr→∞ e −v(r) = 1, we obtain that…”

“…The Keller-Segel model is considered as a prototypical model for pattern formation in chemotaxis, and has attracted a lot of attention as a test case for more complex taxis phenomena driven by chemical substances. See [9,10,11,17,18] for further references.…”

Signal processing through a generalized module of adaptation and spatial sensing

“…The fact that this adaptation is extremely robust has important implications for the kind of chemical interactions and signal flow to achieve this in the face of variability in parameters. A considerable body of modelling efforts is aimed at describing this adaptation process, along with other response characteristics (Tindall et al, 2008). It is worth pointing that some models.…”