Theory of local k-mer selection with applications to long-read alignment

Abstract

Motivation Selecting a subset of k-mers in a string in a local manner is a common task in bioinformatics tools for speeding up computation. Arguably the most well-known and common method is the minimizer technique, which selects the ‘lowest-ordered’ k-mer in a sliding window. Recently, it has been shown that minimizers may be a sub-optimal method for selecting subsets of k-mers when mutations are present. There is, however, a lack of understanding behind the theory of why certain methods perform well. Results We first theoretically investigate the conservation metric for k-mer selection methods. We derive an exact expression for calculating the conservation of a k-mer selection method. This turns out to be tractable enough for us to prove closed-form expressions for a variety of methods, including (open and closed) syncmers, (a, b, n)-words, and an upper bound for minimizers. As a demonstration of our results, we modified the minimap2 read aligner to use a more conserved k-mer selection method and demonstrate that there is up to an 8.2% relative increase in number of mapped reads. However, we found that the k-mers selected by more conserved methods are also more repetitive, leading to a runtime increase during alignment. We give new insight into how one might use new k-mer selection methods as a reparameterization to optimize for speed and alignment quality. Availability and implementation Simulations and supplementary methods are available at https://github.com/bluenote-1577/local-kmer-selection-results. os-minimap2 is a modified version of minimap2 and available at https://github.com/bluenote-1577/os-minimap2. Supplementary information Supplementary data are available at Bioinformatics online.

Fast and robust metagenomic sequence comparison through sparse chaining with skani

“…These new k-mers are called seeds instead of markers because we actually use them as seeds for k-mer matching and alignment. We note that while we could have used other "context-independent" k-mer seeding methods that are more "conserved" than FracMinHash [28], we found that FracMinHash works well enough for relatively sparse seeds when c ≫ k. By default, k = 15 and c = 125. We note that the small value of k = 15 used by default leads to too many repetitive anchors on larger genomes, so we mask the top seeds that occur more than 2500/c times by default.…”

“…These new k-mers are called seeds instead of markers because we actually use them as seeds for k-mer matching and alignment. We note that while we could have used other “context-independent” k-mer seeding methods that are more “conserved” than FracMinHash [28], we found that FracMinHash works well enough for relatively sparse seeds when c ≫ k . By default, k = 15 and c = 125.…”

Sequence aligners can guarantee accuracy in almostO(mlogn) time: a rigorous average-case analysis of the seed-chain-extend heuristic

“…The original open syncmer definition in (Edgar, 2021) had a parameter t where a k-mer was selected if the smallest s-mer was in the t−th position; we proved in (Shaw and Yu, 2022) that the optimal t is ⌈ k−s+1 2 ⌉ with respect to maximizing the number of conserved bases from k-mer matching. The reason we choose open syncmers is primarily to the following fact which was shown in (Edgar, 2021): Theorem 7 follows by examining the smallest s-mer in a k-mer and noticing that in the next overlapping k-mer, the locations for the new smallest s-mer are restricted.…”

“…) which we will ignore; see (Zheng et al, 2020) or (Shaw and Yu, 2022)). We will let c be the reciprocal of the density, so c = (k − s + 1).…”

How to optimally sample a sequence for rapid analysis

“…The main way of analyzing genetic sequences is by comparing them to each other. For large data, this is usually done via ‘seeds’, by which we mean simple similarities that can be found quickly ( Shaw and Yu, 2022 ). The simplest seeds are fixed-length exact matches, but they can also be inexact ( Altschul et al , 1990 ; Ma et al , 2002 ; Noé and Kucherov, 2004 ; Sahlin, 2021 ) and/or variable length ( Csűrös, 2004 ).…”