Theory of continuum random walks and application to chemotaxis

Abstract

We formulate the general theory of random walks in continuum, essential for treating a collision rate which depends smoothly upon direction of motion. We also consider a smooth probability distribution of correlations between the directions of motion before and after collisions, as well as orientational Brownian motion between collisions. These features lead to an effective Smoluchowski equation. Such random walks involving an infinite number of distinct directions of motion cannot be treated on a lattice, which permits only a finite number of directions of motion, nor by Langevin methods, which make no reference to individual collisions. The effective Smoluchowski equation enables a description of the biased random walk of the bacterium Escherichia coli during chemotaxis, its search for food. The chemotactic responses of cells which perform temporal comparisons of the concentration of a chemical attractant are predicted to be strongly positive, whereas those of cells which measure averages of the ambient attractant concentration are predicted to be negative. The former prediction explains the observed behavior of wild-type (naturally occurring) cells; however, the latter behavior has yet to be observed, even in cells defective in adaption.