The Statistics of Parametrized Syncmers in a Simple Mutation Process Without Spurious Matches

Introduction: Often, bioinformatics uses summary sketches to analyze next-generation sequencing data, but most sketches are not well understood statistically. Under a simple mutation model, Blanca et al. analyzed complete sketches, that is, the complete set of unassembled k-mers, from two closely related sequences. The analysis extracted a point mutation parameter θ quantifying the evolutionary distance between the two sequences. Methods: We extend the results of Blanca et al. for complete sketches to parametrized syncmer sketches with downsampling. A syncmer sketch can sample k-mers much more sparsely than a complete sketch. Consider the following simple mutation model disallowing insertions or deletions. Consider a reference sequence A (e.g., a subsequence from a reference genome), and mutate each nucleotide in it independently with probability θ to produce a mutated sequence B (corresponding to, e.g., a set of reads or draft assembly of a related genome). Then, syncmer counts alone yield an approximate Gaussian distribution for estimating θ. The assumption disallowing insertions and deletions motivates a check on the lengths of A and B. The syncmer count from B yields an approximate Gaussian distribution for its length, and a p-value can test the length of B against the length of A using syncmer counts alone. Results: The Gaussian distributions permit syncmer counts alone to estimate θ and mutated sequence length with a known sampling error. Under some circumstances, the results provide the sampling error for the Mash containment index when applied to syncmer counts. Conclusions: The approximate Gaussian distributions provide hypothesis tests and confidence intervals for phylogenetic distance and sequence length. Our methods are likely to generalize to sketches other than syncmers and may be useful in assembling reads and related applications.