Optimal number of spacers in CRISPR arrays

Prokaryotic organisms survive under constant pressure of viruses. CRISPR-Cas system provides its prokaryotic host with an adaptive immune defense against viruses that have been previously encountered. It consists of two components: Cas-proteins that cleave the foreign DNA and CRISPR array that suits as a virus recognition key. CRISPR array consists of a series of spacers, short pieces of DNA that originate from and match the corresponding parts of viral DNA called protospacers. Here we estimate the number of spacers in a CRISPR array of a prokaryotic cell which maximizes its protection against a viral attack. The optimality follows from a competition between two trends: too few distinct spacers make host vulnerable to an attack by a virus with mutated corresponding protospacers, while an excessive variety of spacers dilutes the number of the CRISPR complexes armed with the most recent and thus most useful spacers. We first evaluate the optimal number of spacers in a simple scenario of an infection by a single viral species and later consider a more general case of multiple viral species. We find that depending on such parameters as the concentration of CRISPR-Cas interference complexes and its preference to arm with more recently acquired spacers, the rate of viral mutation, and the number of viral species, the predicted optimal number of spacers lies within a range that agrees with experimentally-observed values.