Self Trapping of a Single Bacterium in its Own Chemoattractant

Bacteria (e.g. E. Coli) are very sensitive to certain chemoattractants (e.g. asparate) which they themselves produce. This leads to chemical instabilities in a uniform population. We discuss here the different case of a single bacterium, following the general scheme of Brenner, Levitov and Budrene. We show that in one and two dimensions (in a capillary or in a thin film) the bacterium can become self-trapped in its cloud of attractant. This should occur if a certain coupling constant $g$ is larger than unity. We then estimate the reduced diffusion D_eff of the bacterium in the strong coupling limit, and find D_eff ~ 1/g.