Kinetics of target site localization of a protein on DNA: a stochastic approach
It is widely recognized that the cleaving rate of a restriction enzyme on target DNA sequences is several orders of magnitude faster than the maximal one calculated from the diffusion--limited theory. It was therefore commonly assumed that the target site interaction of a restriction enzyme with DNA has to occur via two steps: one--dimensional diffusion along a DNA segment, and long--range jumps coming from association/dissociation events. We propose here a stochastic model for this reaction which comprises a series of 1D diffusions of a restriction enzyme on non-specific DNA sequences interrupted by 3D excursions in the solution until the target sequence is reached. This model provides an optimal finding strategy which explains the fast association rate. Modeling the excursions by uncorrelated random jumps, we recover the expression of the mean time required for target site association to occur given by Berg & al. \cite{berg81}, and we explicitly give several physical quantities describing the stochastic pathway of the enzyme. For competitive target sites we calculate two quantities: processivity and preference. By comparing these theoretical expressions to recent experimental data obtained for \textit{Eco}RV--DNA interaction, we quantify: i) the mean residence time per binding event of \textit{Eco}RV on DNA for a representative 1D diffusion coefficient, ii) the average lengths of DNA scanned during the 1D diffusion (during one binding event and during the overall process), iii) the mean time and the mean number of visits needed to go from one target site to the other. Further, we evaluate the dynamics of DNA cleavage with regard to the probability for the restriction enzyme to perform another 1D diffusion on the same DNA substrate following a 3D excursion.