  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.
