Analysis of bacterial migration: I. Numerical solution of balance equation

Chemotaxis describes the ability of motile bacteria to bias their motion in the direction of increasing gradients of chemicals, usually energy sources, known as attractants. In experimental studies of the migration of chemotactic bacteria, 1-D phenomenological cell balance equations (Rivero et al., 1989) have been used to quantitatively analyze experimental observations (Ford et al., 1991; Ford and Lauffenburger, 1991). While attractive for their simplicity and the ease of solution, they are limited in the strict mathematical sense to the situation in which individual bacteria are confined to motion in one dimension and respond to attractant gradients in one dimension only. Recently, Ford and Cummings (1992) reduced the general 3-D cell balance equation of Alt (1980) to obtain an equation describing the migration of a bacterial population in response to a 1-D attractant gradient. Solutions of this equation for single gradients of attractants are compared to those of 1-D balance equations, results from cellular dynamics simulations (Frymier et al., 1993), and experimental data from our laboratory for E. coli responding to α-methylaspartate. We also investigate two aspects of the experimentally derived expression for the tumbling probability: the effect of different models for the down-gradient swimming behavior of the bacteria and the validity of ignoring the temporal derivative of the attractant concentration.