The Statistics of k -mers from a Sequence Undergoing a Simple Mutation Process Without Spurious Matches.

k -mer-based methods are widely used in bioinformatics, but there are many gaps in our understanding of their statistical properties. Here, we consider the simple model where a sequence S (e.g., a genome or a read) undergoes a simple mutation process through which each nucleotide is mutated independently with some probability r , under the assumption that there are no spurious k -mer matches. How does this process affect the k -mers of S ? We derive the expectation and variance of the number of mutated k -mers and of the number of islands (a maximal interval of mutated k -mers) and oceans (a maximal interval of nonmutated k -mers). We then derive hypothesis tests and confidence intervals (CIs) for r given an observed number of mutated k -mers, or, alternatively, given the Jaccard similarity (with or without MinHash). We demonstrate the usefulness of our results using a few select applications: obtaining a CI to supplement the Mash distance point estimate, filtering out reads during alignment by Minimap2, and rating long-read alignments to a de Bruijn graph by Jabba.