Mathematical models for chemotaxis and their applications in self-organisation phenomena

Abstract

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.

Swarm Hunting and Cluster Ejections in Chemically Communicating Active Mixtures

“…3(c) and right panel of Fig. 3(d) and movies 4,8). To see how these remarkable clusters emerge, let us explore the dynamics underlying their formation.…”

“…It is involved in many biological processes where microorganisms (or cells) coordinate their motion; these include wound healing, fertilization, pathogenic invasion of a host, and bacterial colonization [1, 2]. In such cases, microorganisms are attracted (or repelled) by certain substances (chemoattractants/ chemorepellents), but they are also attracted to chemicals produced by other microorganisms (or cells), such as cAMP in the case of Dictyostelium cells [3] or autoinducers in signaling Escherichia coli [4], which leads to chemical interactions (communication) among the microorganisms.While many existing models studying microbiological chemotaxis (or chemical interactions) focus on a single species [5][6][7][8][9][10][11][12], the typical situation in the microbiological habitat is that various different species simultaneously produce certain chemicals to which others respond via chemotaxis or based on quorum sensing mechanisms. One simple example involving chemical signaling across species is provided by macrophage-facilitated breast cancer cell invasion which has recently been modeled [13].…”

“…While many existing models studying microbiological chemotaxis (or chemical interactions) focus on a single species [5][6][7][8][9][10][11][12], the typical situation in the microbiological habitat is that various different species simultaneously produce certain chemicals to which others respond via chemotaxis or based on quorum sensing mechanisms. One simple example involving chemical signaling across species is provided by macrophage-facilitated breast cancer cell invasion which has recently been modeled [13].…”

Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments

“…Chemotaxis is also crucial in macroscopic process such as population dynamics and gravitational collapse. The reader is referred to [34] for applications of chemotaxis models in a wide range of biological phenomena. The reader is also referred to [15,16] for some detailed introduction into the mathematics of Keller-Segel chemotaxis models.…”

A chemotaxis-based explanation of spheroid formation in 3D cultures of breast cancer cells

“…In a deterministic setting, the latter phenomenon is commonly modeled thanks to the Keller-Segel system [16] or one of its numerous generalizations [13]. Originally employed to model spatial patterns in bacteria populations, this model has been shown to be a rich tool for the modeling of self-organization phenomena [24].…”