Abstract
Most bacteria are motile, moving toward suitable environments for subsistence and reproduction. In isotropic chemical environments, they move randomly, changing direction at regular time intervals. In the presence of chemical gradients, they modulate the frequency of the direction change. Collectively, this modulation constitutes the chemotactic response toward a desirable chemical. This study investigates a one-dimensional discrete biased random walk model based on bacterial chemotaxis; a modified version of the classical random walk model. Each cell in the group moves along a uniformly spaced number line at the rate of one interval per time step. A chemical attractant is placed at the origin of the number line. When a cell has receded from the origin in the previous time step, it changes its direction with a probability of 1/2 in the current time step. On the other hand, when a cell has approached the origin in the previous time step, its direction changes with probability (1－α)/ 2 , where α denotes the intensity of the bias toward the origin. In numerical simulations, the cells establish a steady distribution from the origin. This distribution is expressed using a geometric progression whose common ratio depends on α. We provide an analytical explanation of this distribution, which actually constitutes two steady distributions alternating at odd and even positions. Next, the results of the discrete model are compared with those of the corresponding continuum model, namely, a diffusion-advection equation wherein α determines the advection speed. The theoretical solution of the diffusion-advection equation is an exponential decay function, consistent with the distribution obtained by the discrete model.
