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

Some prokaryotes possess CRISPR-Cas systems that use DNA segments called spacers, which are acquired from invading phages, to guide immune defense. Here, we propose that cross-reactive CRISPR targeting can, however, lead to "heterologous autoimmunity," whereby foreign spacers guide self-targeting in a spacer-length-dependent fashion. Balancing antiviral defense against autoimmunity predicts a scaling relation between spacer length and CRISPR repertoire size. We find evidence for this scaling through a comparative analysis of sequenced prokaryotic genomes and show that this association also holds at the level of CRISPR types. By contrast, the scaling is absent in strains with nonfunctional CRISPR loci. Finally, we demonstrate that stochastic spacer loss can explain variations around the scaling relation, even between strains of the same species. Our results suggest that heterologous autoimmunity is a selective factor shaping the evolution of CRISPR-Cas systems, analogous to the trade-offs between immune specificity, breadth, and autoimmunity that constrain the diversity of adaptive immune systems in vertebrates.