Kinetics of target site localization of a protein on DNA: a stochastic approach

Abstract

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 one-dimensional diffusions of a restriction enzyme on nonspecific DNA sequences interrupted by three-dimensional 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 et al. in 1981, 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 EcoRV-DNA interaction, we quantify: 1), the mean residence time per binding event of EcoRV on DNA for a representative one-dimensional diffusion coefficient; 2), the average lengths of DNA scanned during the one-dimensional diffusion (during one binding event and during the overall process); and 3), 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 one-dimensional diffusion on the same DNA substrate following a three-dimensional excursion.

FIGURE 1

A representative path of the restriction enzyme which reaches the target site. Excursions in the solution are represented by dashed lines, one-dimensional diffusion by continuous lines. The solid square is the target site.

FIGURE 2

Representative view of the model. Here the protein executes three excursions before finding the target site.

FIGURE 3

Extended target site.

FIGURE 4

The mean search time plotted against the one-dimensional residence frequency λ. The length of DNA is 5000 bp, the three-dimensional residence frequency is 10 s⁻¹, and the one-dimensional diffusion coefficient is 5 × 10⁵ bp²/s.

FIGURE 5

The average number of distinct DNA sites visited by the enzyme against the one-dimensional residence frequency λ. The half-length of DNA is 100 bp which allows one to also read this number as a percentage.

FIGURE 6

Schematic representation of the three substrates of length 690 bp. The position of the second target site relative to the first target equals 54 bp, 200 bp, and 387 bp, respectively.

FIGURE 7

The preference of the protein for the target site 2 over the target site 1. The solid line represents the fitted solution which gives The two dashed lines correspond to the limit cases when there is no sliding (straight line, λ = ∞) and when there is only sliding (upper line, λ = 0). The other parameters were drawn from experimental data (ℓ = 690 bp).

FIGURE 8

The processive action of the restriction enzyme. Dashed lines represent two fitted solutions of the model of Stanford, et al. (2000) with pure sliding. The two solid lines represent the solutions of our model for and p_init = 0.5: one for π_r = 0, and the other one which passes near experimental points for π_r = 0.85.