  Surviving in a diverse environment requires corresponding organism responses.
At the cellular level, such adjustment relies on the transcription factors
(TFs) which must rapidly find their target sequences amidst a vast amount of
non-relevant sequences on DNA molecules. Whether these transcription factors
locate their target sites through a 1D or 3D pathway is still a matter of
speculation. It has been suggested that the optimum search time is when the
protein equally shares its search time between 1D and 3D diffusions. In this
paper, we study the above problem using a Monte Carlo simulation by considering
a very simple physical model. A 1D strip, representing a DNA, with a number of
low affinity sites, corresponding to non-target sites, and high affinity sites,
corresponding to target sites, is considered and later extended to a 2D strip.
We study the 1D and 3D exploration pathways, and combinations of the two modes
by considering three different types of molecules: a walker that randomly walks
along the strip with no dissociation; a jumper that represents dissociation and
then re-association of a TF with the strip at later time at a distant site; and
a hopper that is similar to the jumper but it dissociates and then
re-associates at a faster rate than the jumper. We analyze the final
probability distribution of molecules for each case and find that TFs can
locate their targets fast enough even if they spend 15% of their search time
diffusing freely in the solution. This indeed agrees with recent experimental
results obtained by Elf et al. 2007 and is in contrast with theoretical
expectation.
