  Protein-DNA interactions are critical for the successful functioning of all
natural systems. The key role in these interactions is played by processes of
protein search for specific sites on DNA. Although it has been studied for many
years, only recently microscopic aspects of these processes became more clear.
In this work, we present a review on current theoretical understanding of the
molecular mechanisms of the protein target search. A comprehensive
discrete-state stochastic method to explain the dynamics of the protein search
phenomena is introduced and explained. Our theoretical approach utilizes a
first-passage analysis and it takes into account the most relevant
physical-chemical processes. It is able to describe many fascinating features
of the protein search, including unusually high effective association rates,
high selectivity and specificity, and the robustness in the presence of
crowders and sequence heterogeneity.
