  Transcription factors (TFs) are key regulators of gene expression. Based on
the classical scenario in which the TF search process switches between
one-dimensional motion along the DNA molecule and free Brownian motion in the
nucleus, we study the arrival time of several TFs to multiple binding sites and
derive, in the presence of competitive binding ligands, the probability that
several target sites are bound. We then apply our results to the hunchback
regulation by bicoid in the fly embryo and we propose a general mechanism that
allows cells to read a morphogenetic gradient and specialize according to their
position in the embryo.
