Coarse graining Escherichia coli chemotaxis: from multi-flagella propulsion to logarithmic sensing

Various sensing mechanisms in nature can be described by the Weber-Fechner law stating that the response to varying stimuli is proportional to their relative rather than absolute changes. The chemotaxis of bacteria Escherichia coli is an example where such logarithmic sensing enables sensitivity over large range of concentrations. It has recently been experimentally demonstrated that under certain conditions E. coli indeed respond to relative gradients of ligands. We use numerical simulations of bacteria in food gradients to investigate the limits of validity of the logarithmic behavior. We model the chemotactic signaling pathway reactions, couple them to a multi-flagella model for propelling and take the effects of rotational diffusion into account to accurately reproduce the experimental observations of single cell swimming. Using this simulation scheme we analyze the type of response of bacteria subject to exponential ligand profiles and identify the regimes of absolute gradient sensing, relative gradient sensing, and a rotational diffusion dominated regime. We explore dependance of the swimming speed, average run time and the clockwise (CW) bias on ligand variation and derive a small set of relations that define a coarse grained model for bacterial chemotaxis. Simulations based on this coarse grained model compare well with microfluidic experiments on E. coli diffusion in linear and exponential gradients of aspartate.