  The mean time required by a transcription factor (TF) or an enzyme to find a
target in the nucleus is of prime importance for the initialization of
transcription, gene activation or the start of DNA repair. We obtain new
estimates for the mean search time when the TF or enzyme, confined to the cell
nucleus, can switch from a one dimensional motion along the DNA and a free
Brownian regime inside the crowded nucleus. We give analytical expressions for
the mean time the particle stays bound to the DNA, $\tau_{DNA}$, and the mean
time it diffuses freely, $\tau_{free}$. Contrary to previous results but in
agreement with experimental data, we find a factor $\tau_{DNA} \approx 3.7
\tau_{free}$ for the Lac-I TF. The formula obtained for the time required to
bind to a target site is found to be coherent with observed data. We also
conclude that a higher DNA density leads to a more efficient search process.
