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

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