  We study the target searching on the DNA for proteins in the presence of
non-constant drift and non-Gaussian $\alpha$-stable L\'evy fluctuations. The
target searching is realized by the facilitated diffusion process. The existing
works are about this problem in the case of constant drift. Starting from a
non-local Fokker-Planck equation with a "sink" term, we obtain the possibility
density function for the protein occurring at position $x$ on time $t$. Based
on this, we further compute the survival probability and the first arrival
density in order to quantify the searching mechanisms. The numerical results
show that in the linear drift case, there is an optimal $\alpha$ index for the
search to be most likely successful (searching reliability reaches its
maximum). This optimal $\alpha$ index depends on initial position-target
separation. It is also found that the diffusion intensity plays a positive role
in improving the searching success. The nonlinear double-well drift could drive
the protein to reach the target with a larger possibility than the linear drag
at initial time period, but viewing at the long time evolution, the linear
drift is more beneficial for target searching success. In contrast to the
linear drift case, the search reliability and efficiency with nonlinear drift
have a monotonic relationship with the $\alpha$ index, that is, the smaller the
$\alpha$ index is, the more possibly a protein finds its target.

Statistical Mechanics