Protein-DNA interactions activate many biochemical transitions used by living systems for successful functioning. The starting point in this process is a protein search and recognition of specific target sequences on DNA among millions of non-specific sites. One of the most striking observations is the fact that some proteins can find their targets on DNA much faster than expected from 3D bulk diffusional estimates which follow from the famous theory developed by M.Smoluchovski. This phenomenon is known as facilitated diffusion. After extensive investigation via theoretical and experimental methods, it has been argued that the facilitated diffusion of proteins (also known as accelerated search) is possible due to combining 3D bulk motion with 1D sliding of non-specifically bound protein molecules along the DNA chain. The recent development of single-molecule techniques has allowed for direct visualization of protein sliding during its binding site search. Thus, although we now have a better understanding of protein search on DNA, there are still many unanswered questions and many features related to this complex biological phenomenon that are still not resolved at the molecular level.

In this work we develop a theoretical method to explain the molecular picture behind these complex processes. Our approach is based on a discrete-state stochastic framework and utilizes a first-passage probability method that takes into account the most relevant physical-chemical processes. This allows us to explicitly evaluate the protein search for the targets on DNA for different conditions. Theoretical predictions are supported by Monte Carlo computer simulations, and obtained results are utilized for analysis of available experimental data.