Bacterial chemotaxis considering memory effects: Derivation of the reaction-diffusion equations

Bacterial chemotaxis for the case of Escherichia coli is controlled by methylation of chemoreceptors, which in a biochemical pathway regulates the concentration of the CheY-P protein that finally controls the tumbling rate. As a consequence, the tumbling rate adjusts to changes in the concentration of relevant chemicals, such as to produce a biased random walk toward chemoattractants or against the repellers. Methylation is a slow process, implying that the internal concentration of CheY-P is not instantaneously adapted to the environment, and therefore the tumbling rate presents a memory. This implies that the Keller-Segel equations used to describe chemotaxis at the macroscopic scale, which assume a local relation between the bacterial flux and the chemical gradient, cannot be fully valid as memory and the associated nonlocal response are not taken into account. To derive the new equations that replace the Keller-Segel ones, we use a kinetic approach, in which a kinetic equation for the bacterial transport is written considering the dynamics of the protein concentration. When memory is large, the protein concentration field must be considered a relevant variable on equal foot as the bacterial density. Working out in detail the Chapman-Enskog method, the dynamical equations for these fields are obtained, which have the form of reaction-diffusion equations with flux and source terms depending on the gradients on the chemical signal. Also, the transport coefficients are obtained entirely in terms o the microscopic dynamics, showing important symmetry properties and giving their values of the case of E. coli. Solving the equations for an inhomogeneous signal it is shown that the response is nonlocal, with a smoothing length as large as 170µm for E. coli. The homogeneous response and the relaxational dynamics are also studied in detail. For completeness, the case of small memory is also studied, in which case the Chapman-Enskog method reproduces the Keller-Segel equations, with explicit expressions for the transport coefficients.