Heterogeneous Diversity of Spacers within CRISPR (Clustered Regularly Interspaced Short Palindromic Repeats)

Clustered regularly interspaced short palindromic repeats (CRISPR) in bacterial and archaeal DNA have recently been shown to be a new type of antiviral immune system in these organisms. We here study the diversity of spacers in CRISPR under selective pressure. We propose a population dynamics model that explains the biological observation that the leader-proximal end of CRISPR is more diversified and the leader-distal end of CRISPR is more conserved. This result is shown to be in agreement with recent experiments. Our results show that the CRISPR spacer structure is influenced by and provides a record of the viral challenges that bacteria face.

Figure 1

A schematic representation to describe CRISPR-phage dynamics. Spacers are shown in numbers, and repeats are shown in dark squares. The leader sequence is directly adjacent to the short spacer-repeat units and possibly involved in promoting transcription towards the repeats. The virus DNA that is recognized by CRISPR is represented by the letter “$i$.” Only the CRISPR of the bacterial genome are shown; other parts of genome are assumed to be identical in all bacteria strains.

Figure 2

Diversity of two spacers of CRISPR with time. The differential equation solution and simulation are based on the parameter values $c=0.15$, $\beta =2\times {10}^{-6}$, $\gamma =0.1$, and $r=0.01$. The viruses have four strains (length of string $n=2$) with an initial population ratio $6:2:1:1$. In the stochastic simulation, the maximal population size is ${10}^6$ for virus and ${10}^5$ for bacteria. Diversity is measured by Shannon entropy. Other measures of diversity, such as Simpson’s index of diversity, give similar results. Error bars are one standard error. The inset shows solutions of differential equations with 200 different parameter combinations using Latin hypercube sampling. The parameter ranges we used are $c\in (0.01,0.15)$, $\beta \in ({10}^{-5},2\times {10}^{-5})$, $\gamma \in (0.01,0.1)$, $r\in (0.01,0.1)$. We used 200 samplings of this parameter space. The up branches are the first spacer, and the down branches are the second spacers.

Figure 3

Diversity of spacers at different positions of CRISPR, when the system reaches steady state. The positions with a small number in the $x$ axis are leader proximal. In this extended simulation, we use the parameters $c=0.15$, $\beta =2\times {10}^{-5}$, $\gamma =0.1$, $r=0.05$, mutation rate per sequence of $\epsilon =0.01$, size of virus bit string $n=10$. Initially, there are 150 phage strains with a logarithmic population distribution [10]. Other parameter settings give similar results. Error bars are one standard error.

Figure 4

Diversity of spacers of CRISPR loci 1 of S. thermophilus strains [14]. The positions with a small number in the $x$ axis are leader proximal.

Figure 5

Diversity of spacers of CRISPR loci of Leptospirillum species. The data are noisy because the CRISPR loci sequence data of Leptospirillum are fragmented.