Traveling waves in response to a diffusing quorum sensing signal in spatially-extended bacterial colonies

In the behavior known as quorum sensing (QS), bacteria release diffusible signal molecules known as autoinducers, which by accumulating in the environment induce population-wide changes in gene expression. Although QS has been extensively studied in well-mixed systems, the ability of diffusing QS signals to synchronize gene expression in spatially extended colonies is not well understood. Here we investigate the one-dimensional spatial propagation of QS-circuit activation in a simple, analytically tractable reaction-diffusion model for the LuxR-LuxI circuit, which regulates bioluminescence of the marine bacterium Aliivibrio fischeri. The quorum activation loop is modeled by a Hill function with a cooperativity exponent (m=2.2). The model is parameterized from laboratory data and captures the major empirical properties of the LuxR-LuxI system and its QS regulation of A. fischeri bioluminescence. Our simulations of the model show propagating waves of activation or deactivation of the QS circuit in a spatially extended colony. We further prove analytically that the model equations possess a traveling wave solution. This mathematical proof yields the rate of autoinducer degradation that is compatible with a traveling wave of gene expression as well as the critical degradation rate at which the nature of the wave switches from activation to deactivation. Our results can be used to predict the direction and activating or deactivating nature of a wave of gene expression in experimentally controlled bacterial populations subject to a diffusing autoinducer signal.