A Pathway-Based Mean-Field Model for E. coli Chemotaxis: Mathematical Derivation and Its Hyperbolic and Parabolic Limits

A pathway-based mean-field theory (PBMFT) that incorporated the most recent quantitatively measured signaling pathway was recently proposed for the E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we formally derive a new kinetic system of PBMFT under the assumption that the methylation level is locally concentrated, whose turning operator takes into account the dynamical intracellular pathway, and hence is more physically relevant. We recover the PBMFT proposed by Si et al. as the hyperbolic limit and connect to the Keller-Segel equation as the parabolic limit of this new model. We also present the numerical evidence to show the quantitative agreement of the kinetic system with the individual based E. coli chemotaxis simulator.