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 Deff of the bacterium in the strong coupling limit, and find Deff ∼ g . Introduction. – Budrene and Berg [1] studied an initially homogeneous population of Eshrechia Coli (E. Coli) bacteria on an agar plate, in conditions where food (succinate) is available. Depending on the food content they discovered various patterns such as moving rings or aggregates. These patterns were lucidly interpreted by Brenner, Levitov and Budrene [2]. They observed that in the (usual) conditions of rapid diffusion, the bacteria produce a concentration field of the chemoattractant, c(r), which has the form of a gravitational field (c(r) ∼ 1/r in three dimensions). The bacteria attract each other and cluster by a “gravitational” instability. However, when a cluster (“star”) is formed, food is depleted and the star then becomes dark in the center, a ring is created, etc. Brenner et. al. also discussed the spontaneous aggregation of a small group of bacteria, and found that they shall indeed aggregate if their number N is larger than a certain limit N∗. Because of the (rough) similarity with astrophysics, they called N∗ the Chandrasekhar limit. Our aim here is to discuss some properties of these small clusters and in particular the limit of a single bacterium. We point out that it may be trapped in its own cloud of asparate. This question has some weak similarity with a classical problem of solid-state physics and field theory, the polaron problem, defined first by Frolich [3] and analyzed by many theorists [4–6]. A polaron is an electron coupled to a phonon field in a solid. In the strong coupling limit analyzed by Pekar [5], the electron builds up a distorted region, and is essentially occupying the lower bound state in the resulting (self-consistent) potential. However, the electron moves slowly: it has a large effective mass.