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Abstract MotivationSampling k-mers is a ubiquitous task in sequence analysis algorithms. Sampling schemes such as the often-used random minimizer scheme are particularly appealing as they guarantee at least one k-mer is selected out of every w consecutive k-mers. Sampling fewer k-mers often leads to an increase in efficiency of downstream methods. Thus, developing schemes that have low density, i.e. have a small proportion of sampled k-mers, is an active area of research. After over a decade of consistent efforts in both decreasing the density of practical schemes and increasing the lower bound on the best possible density, there is still a large gap between the two. ResultsWe prove a near-tight lower bound on the density of forward sampling schemes, a class of schemes that generalizes minimizer schemes. For small w and k, we observe that our bound is tight when k≡1(mod w). For large w and k, the bound can be approximated by 1w+k⌈w+kw⌉. Importantly, our lower bound implies that existing schemes are much closer to achieving optimal density than previously known. For example, with the current default minimap2 HiFi settings w = 19 and k = 19, we show that the best known scheme for these parameters, the double decycling-set-based minimizer of Pellow et al. is at most 3% denser than optimal, compared to the previous gap of at most 50%. Furthermore, when k≡1(mod w) and the alphabet size σ goes to ∞, we show that mod-minimizers introduced by Groot Koerkamp and Pibiri achieve optimal density matching our lower bound. Availability and implementationMinimizer implementations: github.com/RagnarGrootKoerkamp/minimizers ILP and analysis: github.com/treangenlab/sampling-scheme-analysis.
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DeepMinimizer: A Differentiable Framework for Optimizing Sequence-Specific Minimizer Schemes
Minimizers are k-mer sampling schemes designed to generate sketches for large sequences that preserve sufficiently long matches between sequences. Despite their widespread application, learning an effective minimizer scheme with optimal sketch size is still an open question. Most work in this direction focuses on designing schemes that work well on expectation over random sequences, which have limited applicability to many practical tools. On the other hand, several methods have been proposed to construct minimizer schemes for a specific target sequence. These methods, however, require greedy approximations to solve an intractable discrete optimization problem on the permutation space of k-mer orderings. To address this challenge, we propose: (a) a reformulation of the combinatorial solution space using a deep neural network re-parameterization; and (b) a fully differentiable approximation of the discrete objective. We demonstrate that our framework, DEEPMINIMIZER, discovers minimizer schemes that significantly outperform state-of-the-art constructions on genomic sequences.
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- Award ID(s):
- 1937540
- PAR ID:
- 10328097
- Editor(s):
- Pe'er, I.
- Date Published:
- Journal Name:
- RECOMB 2022: Research in Computational Molecular Biology
- Volume:
- 13278
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract MotivationSequences equivalent to their reverse complements (i.e. double-stranded DNA) have no analogue in text analysis and non-biological string algorithms. Despite this striking difference, algorithms designed for computational biology (e.g. sketching algorithms) are designed and tested in the same way as classical string algorithms. Then, as a post-processing step, these algorithms are adapted to work with genomic sequences by folding a k-mer and its reverse complement into a single sequence: The canonical representation (k-nonical space). ResultsThe effect of using the canonical representation with sketching methods is understudied and not understood. As a first step, we use context-free sketching methods to illustrate the potentially detrimental effects of using canonical k-mers with string algorithms not designed to accommodate for them. In particular, we show that large stretches of the genome (“sketching deserts”) are undersampled or entirely skipped by context-free sketching methods, effectively making these genomic regions invisible to subsequent algorithms using these sketches. We provide empirical data showing these effects and develop a theoretical framework explaining the appearance of sketching deserts. Finally, we propose two schemes to accommodate for these effects: (i) a new procedure that adapts existing sketching methods to k-nonical space and (ii) an optimization procedure to directly design new sketching methods for k-nonical space. Availability and implementationThe code used in this analysis is available under a permissive license at https://github.com/Kingsford-Group/mdsscope.more » « less
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In computational biology, k-mers and edit distance are fundamental concepts. However, little is known about the metric space of all k-mers equipped with the edit distance. In this work, we explore the structure of the k-mer space by studying its maximal independent sets (MISs). An MIS is a sparse sketch of all k-mers with nice theoretical properties, and therefore admits critical applications in clustering, indexing, hashing, and sketching large-scale sequencing data, particularly those with high error-rates. Finding an MIS is a challenging problem, as the size of a k-mer space grows geometrically with respect to k. We propose three algorithms for this problem. The first and the most intuitive one uses a greedy strategy. The second method implements two techniques to avoid redundant comparisons by taking advantage of the locality-property of the k-mer space and the estimated bounds on the edit distance. The last algorithm avoids expensive calculations of the edit distance by translating the edit distance into the shortest path in a specifically designed graph. These algorithms are implemented and the calculated MISs of k-mer spaces and their statistical properties are reported and analyzed for k up to 15. Source code is freely available at https://github.com/Shao-Group/kmerspace.more » « less
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Robinson, Peter (Ed.)Abstract MotivationThe Jaccard similarity on k-mer sets has shown to be a convenient proxy for sequence identity. By avoiding expensive base-level alignments and comparing reduced sequence representations, tools such as MashMap can scale to massive numbers of pairwise comparisons while still providing useful similarity estimates. However, due to their reliance on minimizer winnowing, previous versions of MashMap were shown to be biased and inconsistent estimators of Jaccard similarity. This directly impacts downstream tools that rely on the accuracy of these estimates. ResultsTo address this, we propose the minmer winnowing scheme, which generalizes the minimizer scheme by use of a rolling minhash with multiple sampled k-mers per window. We show both theoretically and empirically that minmers yield an unbiased estimator of local Jaccard similarity, and we implement this scheme in an updated version of MashMap. The minmer-based implementation is over 10 times faster than the minimizer-based version under the default ANI threshold, making it well-suited for large-scale comparative genomics applications. Availability and implementationMashMap3 is available at https://github.com/marbl/MashMap.more » « less
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Boucher, Christina; Rahmann, Sven (Ed.)Many bioinformatics applications involve bucketing a set of sequences where each sequence is allowed to be assigned into multiple buckets. To achieve both high sensitivity and precision, bucketing methods are desired to assign similar sequences into the same bucket while assigning dissimilar sequences into distinct buckets. Existing k-mer-based bucketing methods have been efficient in processing sequencing data with low error rate, but encounter much reduced sensitivity on data with high error rate. Locality-sensitive hashing (LSH) schemes are able to mitigate this issue through tolerating the edits in similar sequences, but state-of-the-art methods still have large gaps. Here we generalize the LSH function by allowing it to hash one sequence into multiple buckets. Formally, a bucketing function, which maps a sequence (of fixed length) into a subset of buckets, is defined to be (d₁, d₂)-sensitive if any two sequences within an edit distance of d₁ are mapped into at least one shared bucket, and any two sequences with distance at least d₂ are mapped into disjoint subsets of buckets. We construct locality-sensitive bucketing (LSB) functions with a variety of values of (d₁,d₂) and analyze their efficiency with respect to the total number of buckets needed as well as the number of buckets that a specific sequence is mapped to. We also prove lower bounds of these two parameters in different settings and show that some of our constructed LSB functions are optimal. These results provide theoretical foundations for their practical use in analyzing sequences with high error rate while also providing insights for the hardness of designing ungapped LSH functions.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-031-04749-7_4
