  Bacteria and archaea have evolved an adaptive, heritable immune system that
recognizes and protects against viruses or plasmids. This system, known as the
CRISPR-Cas system, allows the host to recognize and incorporate short foreign
DNA or RNA sequences, called `spacers' into its CRISPR system. Spacers in the
CRISPR system provide a record of the history of bacteria and phage
coevolution. We use a physical model to study the dynamics of this coevolution
as it evolves stochastically over time. We focus on the impact of mutation and
recombination on bacteria and phage evolution and evasion. We discuss the
effect of different spacer deletion mechanisms on the coevolutionary dynamics.
We make predictions about bacteria and phage population growth, spacer
diversity within the CRISPR locus, and spacer protection against the phage
population.
