Physical Mechanisms for Chemotactic Pattern Formation by Bacteria
Authors: Michael P. Brenner, Leonid S. Levitov, and Elena O. Budrene
Publication: Brenner et al. Physical Mechanisms for Chemotactic Pattern Formation by Bacteria. Biophysical Journal, (1998) 74: 1677-1693
Key words： Chemotaxis， pattern formation, bacteria, model
The general goal of this paper is to study the motion of E. coli cells in response to external conditions, i.e. chemicals. E. coli cell moves due to its flagella, each of which is propelled by a rotary motor. The flagellum motor has two modes of operation: clockwise (CW) and counter-clockwise (CCW). In CW mode, the flagella form a bundle and propels the bacterium forward ("runs"), while in CCW mode, the flagella motions are independent to each other and thus make the movement of bacterium random ("tumble"). The E. coli cells performs a random walk with diffusion constant D=v2t, where v is the propulsion speed during "run" and t is mean running time.
It has been long since people found the chemotaxis of E. coli, in which cells move up an external chemical gradient. When the cell finds it is running in the directions of increasing attractant gradient, the probability of tumbling decreases. And certain experiments have demonstrated the E. coli cells can form patterns given particular attractant gradient, where the gradient can also be created by bacteria execretion or consumption. This paper focuses on theoretically understanding these experimental results.
In Budrene and Berg's experiment, the bacteria form either swarm ring or aggregates, due to different succinate (energy source) concentration. This paper presents a theory to describe the bacteria pattern formation, swarm ring migration and aggregate formatio based on two simple ideas. 1) Swarm ring migration results from the depletion of a chemical used in the process of attractant production. In general, solutions for swarm rings can exist whenever the rate of chemotactic chemical production (or depeletion) depends on the concentration of another external field. 2) Aggregate formation is caused by singular collapse of a cloud of bacteria into more compact structures of lower dimensionality.
Fig. 1 shows the structure of the swarm ring immediately before and after collapse visualized by scattered light, from a traveling band-like swarm ring to a cylindrical structure and finally into aggregates.
It is a good question whether the Brownian motion, or the concept of random walk, can be applied to aspects other than particles, e.g. cell motion, organism motion and organelle motion. In this paper, the authors described the modelling of E. coli bacteria motion, which has both running and tumbling modes. The running mode is active by flagella motion, which is different from particle motion. The latter mode of course has similar behavior like Brownian motion. By the model in this paper, the combined cellular motion is analogous to the particle motion in concentration gradient. Thus in the derivation of this paper, the authors not only use the diffusion constant for chemicals, but also uses the concept of diffusion constant for bacteria cells.