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

We study the target search on DNA for proteins in the presence of non-constant drift. This search is realized by the facilitated diffusion process. Existing works on this problem focus on the case of constant drift. Starting from a non-local Fokker-Planck equation with the -order fractional Laplace operator and a ‘sink’ term, we obtain the possibility density function for a protein occurring at position x at 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 index for the search to be most likely successful (searching reliability reaches its maximum). This optimal index depends on the initial position-target separation. It is also found that the diffusion intensity plays a positive role in improving the search success. The nonlinear double-well drift could drive the protein to reach the target with a larger possibility than the linear drag at the initial time period, but viewed over a long time duration, the linear drift is more beneficial for target search success. In contrast to the linear drift case, the search reliability and efficiency with nonlinear double-well drift have a monotonic relationship with the index, that is, the smaller the index is, the higher the likelihood of a protein finding its target.