Crispr Detection from Short Reads Using Partial Overlap Graph Ilan Ben-bassat and Benny Chor

Notations: Let n be the number of `-long reads in a dataset sampled from a bacterial genome G, with a coverage of c. Let K be the number of frequent k-mers in the data set, which are not part of a CRISPR repeat (also referred to as irrelevant frequent k-mers). For every frequent k-mer, u, in G, let Fu be the number of reads containing u. Let F be the total number of reads containing some frequent k-mer. We denote by m the maximum number of spacers in a CRISPR array of G, and by p the number of CRISPR arrays in G. We also define several constants to be used in the analysis. Two of these express our current knowledge of the attributes of CRISPR loci: R is the maximum length of a CRISPR repeat, and S is the maximum length of a CRISPR spacer. In addition, we assume that k-mers from repeats are not frequent outside of CRISPR arrays. We also assume that sequences of at least σ bases (where σ is the minimum overlap length), that are a combination of a spacer subsequence and a repeat subsequence, are also not frequent in rest of the bacterial genome, G. We, therefore, denote by I the upper bound on the number of times that a k-mer, whose origin is either a repeat, or a spacer, or a consecutive combination of them, can appear in the genome outside of a CRISPR array. This bound also takes into account isolated copies of such a k-mer indside a CRISPR array (due to degenerated or truncated repeats). For simplicity, we assume that different CRISPR arrays share no common repeats.