Target search of a protein on DNA in the presence of position-dependent bias

We study the target searching on the DNA for proteins in the presence of non-constant drift and non-Gaussian $\alpha$-stable Levy 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.