Kinetic theory modeling of bacteria diffusion and chemotaxis

The standard description for bacterial motion is the run-and-tumble model. At a given rate μ, bacteria stop their run mode to rapidly change direction to a new one, chosen at random from an angular distribution. This Poisson process gives rise, for times longer than μ^-1, to a diffusive motion. Chemotaxis is usually modeled by saying that the tumbling rate changes if the bacterium swims along or against the chemotactic signal, resulting in a biased random walk. Microscopically, the tumbling rate is controlled by the concentration of a protein whose production depends on the local values of the chemoattractant, responding with a characteristic time T.

Here, we build a kinetic model for the chemotactic bacterial motion that includes the fluctuations of the protein concentration. It is found that the run times follow a broad log-normal distribution, which implies that the decay times of the orientation correlation function should also follow this distribution. Using this result, by experimentally following hundreds of bacteria, we were able to fit the model parameters for E. coli.

The kinetic model is then used to quantify the bacterial diffusion in a homogeneous medium and the chemotactic response in presence of imposed external gradients. It is found that for high sensitivities of the tumbling rate to the protein concentration or for long memory times, which are indeed consistent with the experimental values, the times it takes to reach the diffusive regime or the stationary chemotactic current grows enormously. These long-living transient regimes are characterized by a strong deviation from a Gaussian distribution of displacements and by a non-monotonic dependence of the current on the chemoattractant gradient. Implications of the results for macroscopic modeling of bacterial contamination are finally given.