Adaptive Immunity Constrains the Composition and Fate of Large Bacterial Populations*

Many bacteria possess an adaptive immune system called CRISPR-Cas which allows them to 'remember' phage attacks by inserting pieces of phage DNA, called spacers, into their genome. Experiments have shown that the abundance of spacer types in a bacterial population under phage attack is both highly dynamic and can vary by orders of magnitude between spacer types, yet in our analysis we observed stability in the overall spacer abundance distribution. Bacteria can also up-regulate Cas protein expression through the quorum-sensing pathway.
We propose and analyze a simple model of bacteria-phage interactions with CRISPR. We address these two observed features: stable spacer abundance distributions despite rapid turnover of individual spacer types, and density-dependent regulation of CRISPR. We show that functionally identical spacers can have different abundances simply because of randomness in the underlying processes. We also find spacer abundance distributions that are broad and stable over time. The distribution shape depends on parameters and may provide a natural observable for many systems. We also show that CRISPR regulation can give rise to bistability, a surprising result that gives insight into the fate of natural populations of bacteria interacting with phages.