Search and Localization Dynamics of the CRISPR-Cas9 System

The CRISPR-Cas9 system acts as the prokaryotic immune system and has important applications in gene editing. The protein Cas9 is one of its crucial components. The role of Cas9 is to search for specific target sequences on the DNA and cleave them. In this Letter, we introduce a model of facilitated diffusion for Cas9 and fit its parameters to single-molecule experiments. Our model confirms that Cas9 search for targets by sliding, but shows that its sliding length is rather short. We then investigate how Cas9 explores a long stretch of DNA containing randomly placed targets. We solve this problem by mapping it into the theory of Anderson localization in condensed matter physics. Our theoretical approach rationalizes experimental evidence on the distribution of Cas9 molecules along the DNA.

Figure 1

Scheme of the model. PAM sites and nonspecific sites are shown in yellow and blue, respectively. The second and third bases of PAM sequences are considered as nonspecific sites (light blue). Green arrows represent sliding rates and black arrows represent unbinding rates, see Eq. (2). Thicker arrows correspond to larger rates.

Figure 2

(a) Arrangements of PAM sites used in the experiments in [6]. Line colors correspond to the different curves in panel (b). The figure shows only the portion of the DNA sequence of length $N=98$ where the PAM sites are located. (b) Comparison of the prediction of our model (lines) with experiments [6] (points). Model parameters are determined by jointly fitting the experimental data for $j=0,\dots ,5$ PAM sites using maximum likelihood, see [18]. (c) Eigenvectors ${\mathit{\psi }}^{(1)}$ for $j=1,\dots ,5$.

Figure 3

Interference between $j=1,\dots ,5$ equally spaced PAM sites on an infinite DNA chain. Lowest eigenvalue ${\lambda }_1$ as a function of the interval between the PAM sites. Points are obtained by numerically diagonalizing the matrix $\widehat{A}$ corresponding to each case, with $N=220$. The horizontal line marks the value of ${\lambda }_1$ for a single PAM sequence, from the solution of Eq. (7).

Figure 4

(a) Cumulative density of states (DOS) and (b) localization length as function of $\lambda$ for the nearest neighbor model, Eq. (1), computed using Eq. (14). Results obtained by the transfer matrix method agree with those obtained by direct diagonalization. The DNA chain length is $N={10}^6$ for the transfer matrix method and $N=5000$ for the direct diagonalization. (c) Cumulative DOS and (d) localization length for the hopping model expressed by Eq. (16), computed using Eq. (17). In this case, the DNA chain length is $N=2000$.