  Escherichia coli is a motile bacterium that moves up a chemoattractant
gradient by performing a biased random walk composed of alternating runs and
tumbles. This paper presents calculations of the chemotactic drift velocity vd
(the mean velocity up the chemoattractant gradient) of an E. coli cell
performing chemotaxis in a uniform, steady shear flow, with a weak
chemoattractant gradient at right angles to the flow. Extending earlier models,
a combined analytic and numerical approach is used to assess the effect of
several complications, namely (i) a cell cannot detect a chemoattractant
gradient directly but rather makes temporal comparisons of chemoattractant
concentration, (ii) the tumbles exhibit persistence of direction, meaning that
the swimming directions before and after a tumble are correlated, (iii) the
cell suffers random re-orientations due to rotational Brownian motion, and (iv)
the non-spherical shape of the cell affects the way that it is rotated by the
shear flow. These complications influence the dependence of vd on the shear
rate gamma. When they are all included, it is found that (a) shear disrupts
chemotaxis and shear rates beyond gamma = 2/second cause the cell to swim down
the chemoattractant gradient rather than up it, (b) in terms of maximising
drift velocity, persistence of direction is advantageous in a quiescent fluid
but disadvantageous in a shear flow, and (c) a more elongated body shape is
advantageous in performing chemotaxis in a strong shear flow.
