Directional persistence and the optimality of run-and-tumble chemotaxis

E. coli does chemotaxis by performing a biased random walk composed of alternating periods of swimming (runs) and reorientations (tumbles). Tumbles are typically modelled as complete directional randomisations but it is known that in wild type E. coli, successive run directions are actually weakly correlated, with a mean directional difference of approximately 63 degrees. We recently presented a model of the evolution of chemotactic swimming strategies in bacteria which is able to quantitatively reproduce the emergence of this correlation. The agreement between model and experiments suggests that directional persistence may serve some function, a hypothesis supported by the results of an earlier model. Here we investigate the effect of persistence on chemotactic efficiency, using a spatial Monte Carlo model of bacterial swimming in a gradient, combined with simulations of natural selection based on chemotactic efficiency. A direct search of the parameter space reveals two attractant gradient regimes, (a) a low-gradient regime, in which efficiency is unaffected by directional persistence and (b) a high-gradient regime, in which persistence can improve chemotactic efficiency. The value of the persistence parameter that maximises this effect corresponds very closely with the value observed experimentally. This result is matched by independent simulations of the evolution of directional memory in a population of model bacteria, which also predict the emergence of persistence in high-gradient conditions. The relationship between optimality and persistence in different environments may reflect a universal property of random-walk foraging algorithms, which must strike a compromise between two competing aims: exploration and exploitation. We also present a new graphical way to generally illustrate the evolution of a particular trait in a population, in terms of variations in an evolvable parameter.