Communications in Mathematical Sciences

Volume 14 (2016)

Number 3

An augmented Keller–Segel model for E. coli chemotaxis in fast-varying environments

Pages: 883 – 891

(Fast Communication)



Tong Li (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

Min Tang (Institute of Natural Sciences and Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Xu Yang (Department of Mathematics, University of California at Santa Barbara)


This is a continuous study on E. coli chemotaxis under the framework of pathway-based mean-field theory (PBMFT) proposed in [G. Si, M. Tang, and X. Yang, 12, 907–926, 2014], following the physical studies in [G. Si, T.Wu, Q. Quyang, and Y. Tu, 109, 048101, 2012]. In this paper, we derive an augmented Keller–Segel system with macroscopic intercellular signaling pathway dynamics. It can explain the experimental observation of phase-shift between the maxima of ligand concentration and density of E. coli in fast-varying environments at the population level. This is a necessary complement to the original PBMFT where the phase-shift can only be modeled by moment systems. Formal analysis are given for the system in the cases of fast and slow adaption rates. Numerical simulations show the quantitative agreement of the augmented Keller–Segel model with the individual-based E. coli chemotaxis simulator.


chemotaxis, Keller–Segel equation, pathway-based mean field model, fast-varying environments

2010 Mathematics Subject Classification

35K55, 35Q92, 92B99

Full Text (PDF format)