  The Brownian dynamics of a single microorganism coupled by chemotaxis to a
diffusing concentration field which is secreted by the microorganism itself is
studied by computer simulations in spatial dimensions $d=1,2,3$. Both cases of
a chemoattractant and a chemorepellent are discussed. For a chemoattractant, we
find a transient dynamical arrest until the microorganism diffuses for long
times. For a chemorepellent, there is a transient ballistic motion in all
dimensions and a long-time diffusion. These results are interpreted with the
help of a theoretical analysis.
