  The bacterium Escherichia coli (E. coli) moves in its natural environment in
a series of straight runs, interrupted by tumbles which cause change of
direction. It performs chemotaxis towards chemo-attractants by extending the
duration of runs in the direction of the source. When there is a spatial
gradient in the attractant concentration, this bias produces a drift velocity
directed towards its source, whereas in a uniform concentration, E.coli adapts,
almost perfectly in case of methyl aspartate. Recently, microfluidic
experiments have measured the drift velocity of E.coli in precisely controlled
attractant gradients, but no general theoretical expression for the same
exists. With this motivation, we study an analytically soluble model here,
based on the Barkai-Leibler model, originally introduced to explain the perfect
adaptation. Rigorous mathematical expressions are obtained for the chemotactic
response function and the drift velocity in the limit of weak gradients and
under the assumption of completely random tumbles. The theoretical predictions
compare favorably with experimental results, especially at high concentrations.
We further show that the signal transduction network weakens the dependence of
the drift on concentration, thus enhancing the range of sensitivity.
