# The independent loss model with ordered insertions for the evolution of CRISPR spacers

## TL;DR

This paper models the evolution of CRISPR spacer arrays using an independent loss model with ordered insertions, deriving spacer distance distributions to estimate spacer loss rates from data.

## Contribution

It introduces a mathematical model for CRISPR spacer evolution with ordered insertions and losses, providing explicit spacer distance distributions for small sample sizes.

## Key findings

- Derived spacer distance distributions for n=2 and n=3 arrays.
- Provided a method to estimate spacer loss rate from array data.

## Abstract

Today, the CRISPR (clustered regularly interspaced short palindromic repeats) region within bacterial and archaeal genomes is known to encode an adaptive immune system. We rely on previous results on the evolution of the CRISPR arrays, which led to the ordered independent loss model, introduced by Kupczok and Bollback (2013). When focusing on the spacers (between the repeats), new elements enter a CRISPR array at rate $\theta$ at the leader end of the array, while all spacers present are lost at rate $\rho$ along the phylogeny relating the sample. Within this model, we compute the distribution of distances of spacers which are present in all arrays in samples of size $n=2$ and $n=3$. We use these results to estimate the loss rate $\rho$ from spacer array data.