A scaling law in CRISPR repertoire sizes arises from avoidance of autoimmunity

Some bacteria and archaea possess an adaptive immune system that maintains a memory of past viral infections as DNA elements called spacers, stored in the CRISPR loci of their genomes. This memory is used to mount targeted responses against threats. However, cross-reactivity of CRISPR targeting mechanisms suggests that incorporation of foreign spacers can also lead to autoimmunity. We show that balancing antiviral defense against autoimmunity predicts a scaling law relating spacer length and CRISPR repertoire size. By analyzing a database of microbial CRISPR-Cas systems, we find that the predicted scaling law is realized empirically across prokaryotes, and arises through the proportionate use of different CRISPR types by species differing in the size of immune memory. In contrast, strains with nonfunctional CRISPR loci do not show this scaling. We also demonstrate that simple population-level selection mechanisms can generate the scaling, along with observed variations between strains of a given species.