  One of simplest examples of navigation found in nature is run-and-tumble
chemotaxis. Tumbles reorient cells randomly, and cells can drift toward
attractants or away from repellents by biasing the frequency of these events.
The post-tumble swimming directions are typically correlated with those prior,
as measured by the variance of the reorientation angle distribution. This
variance can range from large, in the case of bacteria, to so small that tumble
events are imperceptible, as observed in choanoflagellates. This raises the
question of optimality: why is such a range of persistence observed in nature?
Here we study persistent run-and-tumble dynamics, focusing first on the
optimisation of the linearised chemotactic response within the two-dimensional
parameter space of tumble frequency and angular persistence. Although an
optimal persistence does exist for a given tumble frequency, in the full
parameter space there is a continuum of optimal solutions. Introducing finite
tumble times that depend on the persistence can change this picture,
illuminating one possible method for selecting tumble persistence based on
species-specific reorientation dynamics. Moving beyond linear theory we find
that optimal chemotactic strengths exist, and that these maximise reaction when
swimming in a wrong direction, but have little or no reaction when swimming
with even the slightest projection along the chemoattractant gradient.
