Improved design and analysis of practical minimizers

k-nonical space: sketching with reverse complements

MOTIVATION

Sequences 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).

RESULTS

The 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 IMPLEMENTATION

The code used in this analysis is available under a permissive license at https://github.com/Kingsford-Group/mdsscope.

When less is more: sketching with minimizers in genomics

The exponential increase in sequencing data calls for conceptual and computational advances to extract useful biological insights. One such advance, minimizers, allows for reducing the quantity of data handled while maintaining some of its key properties. We provide a basic introduction to minimizers, cover recent methodological developments, and review the diverse applications of minimizers to analyze genomic data, including de novo genome assembly, metagenomics, read alignment, read correction, and pangenomes. We also touch on alternative data sketching techniques including universal hitting sets, syncmers, or strobemers. Minimizers and their alternatives have rapidly become indispensable tools for handling vast amounts of data.

OCTOPUS: Disk-based, Multiplatform, Mobile-friendly Metagenomics Classifier

Portable genomic sequencers such as Oxford Nanopore's MinION enable real-time applications in clinical and environmental health. However, there is a bottleneck in the downstream analytics when bioinformatics pipelines are unavailable, e.g., when cloud processing is unreachable due to absence of Internet connection, or only low-end computing devices can be carried on site. Here we present a platform-friendly software for portable metagenomic analysis of Nanopore data, the Oligomer-based Classifier of Taxonomic Operational and Pan-genome Units via Singletons (OCTOPUS). OCTOPUS is written in Java, reimplements several features of the popular Kraken2 and KrakenUniq software, with original components for improving metagenomics classification on incomplete/sampled reference databases, making it ideal for running on smartphones or tablets. OCTOPUS obtains sensitivity and precision comparable to Kraken2, while dramatically decreasing (4- to 16-fold) the false positive rate, and yielding high correlation on real-word data. OCTOPUS is available along with customized databases at https://github.com/DataIntellSystLab/OCTOPUS and https://github.com/Ruiz-HCI-Lab/OctopusMobile.

Sketching Methods with Small Window Guarantee Using Minimum Decycling Sets

Most sequence sketching methods work by selecting specific k-mers from sequences so that the similarity between two sequences can be estimated using only the sketches. Because estimating sequence similarity is much faster using sketches than using sequence alignment, sketching methods are used to reduce the computational requirements of computational biology software. Applications using sketches often rely on properties of the k-mer selection procedure to ensure that using a sketch does not degrade the quality of the results compared with using sequence alignment. Two important examples of such properties are locality and window guarantees, the latter of which ensures that no long region of the sequence goes unrepresented in the sketch. A sketching method with a window guarantee, implicitly or explicitly, corresponds to a decycling set of the de Bruijn graph, which is a set of unavoidable k-mers. Any long enough sequence, by definition, must contain a k-mer from any decycling set (hence, the unavoidable property). Conversely, a decycling set also defines a sketching method by choosing the k-mers from the set as representatives. Although current methods use one of a small number of sketching method families, the space of decycling sets is much larger and largely unexplored. Finding decycling sets with desirable characteristics (e.g., small remaining path length) is a promising approach to discovering new sketching methods with improved performance (e.g., with small window guarantee). The Minimum Decycling Sets (MDSs) are of particular interest because of their minimum size. Only two algorithms, by Mykkeltveit and Champarnaud, are previously known to generate two particular MDSs, although there are typically a vast number of alternative MDSs. We provide a simple method to enumerate MDSs. This method allows one to explore the space of MDSs and to find MDSs optimized for desirable properties. We give evidence that the Mykkeltveit sets are close to optimal regarding one particular property, the remaining path length. A number of conjectures and computational and theoretical evidence to support them are presented. Code available at https://github.com/Kingsford-Group/mdsscope.

Conway-Bromage-Lyndon (CBL): an exact, dynamic representation of k-mer sets

SUMMARY

In this article, we introduce the Conway-Bromage-Lyndon (CBL) structure, a compressed, dynamic and exact method for representing k-mer sets. Originating from Conway and Bromage's concept, CBL innovatively employs the smallest cyclic rotations of k-mers, akin to Lyndon words, to leverage lexicographic redundancies. In order to support dynamic operations and set operations, we propose a dynamic bit vector structure that draws a parallel with Elias-Fano's scheme. This structure is encapsulated in a Rust library, demonstrating a balanced blend of construction efficiency, cache locality, and compression. Our findings suggest that CBL outperforms existing dynamic k-mer set methods. Unique to this work, CBL stands out as the only known exact k-mer structure offering in-place set operations. Its different combined abilities position it as a flexible Swiss knife structure for k-mer set management.

AVAILABILITY AND IMPLEMENTATION

https://github.com/imartayan/CBL.

On Minimizers and Convolutional Filters: Theoretical Connections and Applications to Genome Analysis

Minimizers and convolutional neural networks (CNNs) are two quite distinct popular techniques that have both been employed to analyze categorical biological sequences. At face value, the methods seem entirely dissimilar. Minimizers use min-wise hashing on a rolling window to extract a single important k-mer feature per window. CNNs start with a wide array of randomly initialized convolutional filters, paired with a pooling operation, and then multiple additional neural layers to learn both the filters themselves and how they can be used to classify the sequence. In this study, our main result is a careful mathematical analysis of hash function properties showing that for sequences over a categorical alphabet, random Gaussian initialization of convolutional filters with max-pooling is equivalent to choosing a minimizer ordering such that selected k-mers are (in Hamming distance) far from the k-mers within the sequence but close to other minimizers. In empirical experiments, we find that this property manifests as decreased density in repetitive regions, both in simulation and on real human telomeres. We additionally train from scratch a CNN embedding of synthetic short-reads from the SARS-CoV-2 genome into 3D Euclidean space that locally recapitulates the linear sequence distance of the read origins, a modest step toward building a deep learning assembler, although it is at present too slow to be practical. In total, this article provides a partial explanation for the effectiveness of CNNs in categorical sequence analysis.

Highly efficient clustering of long-read transcriptomic data with GeLuster

MOTIVATION

The advancement of long-read RNA sequencing technologies leads to a bright future for transcriptome analysis, in which clustering long reads according to their gene family of origin is of great importance. However, existing de novo clustering algorithms require plenty of computing resources.

RESULTS

We developed a new algorithm GeLuster for clustering long RNA-seq reads. Based on our tests on one simulated dataset and nine real datasets, GeLuster exhibited superior performance. On the tested Nanopore datasets it ran 2.9-17.5 times as fast as the second-fastest method with less than one-seventh of memory consumption, while achieving higher clustering accuracy. And on the PacBio data, GeLuster also had a similar performance. It sets the stage for large-scale transcriptome study in future.

AVAILABILITY AND IMPLEMENTATION

GeLuster is freely available at https://github.com/yutingsdu/GeLuster.

Density and Conservation Optimization of the Generalized Masked-Minimizer Sketching Scheme

Minimizers and syncmers are sketching methods that sample representative k-mer seeds from a long string. The minimizer scheme guarantees a well-spread k-mer sketch (high coverage) while seeking to minimize the sketch size (low density). The syncmer scheme yields sketches that are more robust to base substitutions (high conservation) on random sequences, but do not have the coverage guarantee of minimizers. These sketching metrics are generally adversarial to one another, especially in the context of sketch optimization for a specific sequence, and thus are difficult to be simultaneously achieved. The parameterized syncmer scheme was recently introduced as a generalization of syncmers with more flexible sampling rules and empirically better coverage than the original syncmer variants. However, no approach exists to optimize parameterized syncmers. To address this shortcoming, we introduce a new scheme called masked minimizers that generalizes minimizers in manner analogous to how parameterized syncmers generalize syncmers and allows us to extend existing optimization techniques developed for minimizers. This results in a practical algorithm to optimize the masked minimizer scheme with respect to both density and conservation. We evaluate the optimization algorithm on various benchmark genomes and show that our algorithm finds sketches that are overall more compact, well-spread, and robust to substitutions than those found by previous methods. Our implementation is released at https://github.com/Kingsford-Group/maskedminimizer. This new technique will enable more efficient and robust genomic analyses in the many settings where minimizers and syncmers are used.

Creating and Using Minimizer Sketches in Computational Genomics

Processing large data sets has become an essential part of computational genomics. Greatly increased availability of sequence data from multiple sources has fueled breakthroughs in genomics and related fields but has led to computational challenges processing large sequencing experiments. The minimizer sketch is a popular method for sequence sketching that underlies core steps in computational genomics such as read mapping, sequence assembling, k-mer counting, and more. In most applications, minimizer sketches are constructed using one of few classical approaches. More recently, efforts have been put into building minimizer sketches with desirable properties compared with the classical constructions. In this survey, we review the history of the minimizer sketch, the theories developed around the concept, and the plethora of applications taking advantage of such sketches. We aim to provide the readers a comprehensive picture of the research landscape involving minimizer sketches, in anticipation of better fusion of theory and application in the future.

Sketching methods with small window guarantee using minimum decycling sets

Most sequence sketching methods work by selecting specific k -mers from sequences so that the similarity between two sequences can be estimated using only the sketches. Because estimating sequence similarity is much faster using sketches than using sequence alignment, sketching methods are used to reduce the computational requirements of computational biology software packages. Applications using sketches often rely on properties of the k -mer selection procedure to ensure that using a sketch does not degrade the quality of the results compared with using sequence alignment. Two important examples of such properties are locality and window guarantees, the latter of which ensures that no long region of the sequence goes unrepresented in the sketch. A sketching method with a window guarantee, implicitly or explicitly, corresponds to a Decycling Set, an unavoidable sets of k -mers. Any long enough sequence, by definition, must contain a k -mer from any decycling set (hence, it is unavoidable). Conversely, a decycling set also defines a sketching method by choosing the k -mers from the set as representatives. Although current methods use one of a small number of sketching method families, the space of decycling sets is much larger, and largely unexplored. Finding decycling sets with desirable characteristics (e.g., small remaining path length) is a promising approach to discovering new sketching methods with improved performance (e.g., with small window guarantee). The Minimum Decycling Sets (MDSs) are of particular interest because of their minimum size. Only two algorithms, by Mykkeltveit and Champarnaud, are previously known to generate two particular MDSs, although there are typically a vast number of alternative MDSs. We provide a simple method to enumerate MDSs. This method allows one to explore the space of MDSs and to find MDSs optimized for desirable properties. We give evidence that the Mykkeltveit sets are close to optimal regarding one particular property, the remaining path length. A number of conjectures and computational and theoretical evidence to support them are presented. Code available at https://github.com/Kingsford-Group/mdsscope.

LexicHash: sequence similarity estimation via lexicographic comparison of hashes

MOTIVATION

Pairwise sequence alignment is a heavy computational burden, particularly in the context of third-generation sequencing technologies. This issue is commonly addressed by approximately estimating sequence similarities using a hash-based method such as MinHash. In MinHash, all k-mers in a read are hashed and the minimum hash value, the min-hash, is stored. Pairwise similarities can then be estimated by counting the number of min-hash matches between a pair of reads, across many distinct hash functions. The choice of the parameter k controls an important tradeoff in the task of identifying alignments: larger k-values give greater confidence in the identification of alignments (high precision) but can lead to many missing alignments (low recall), particularly in the presence of significant noise.

RESULTS

In this work, we introduce LexicHash, a new similarity estimation method that is effectively independent of the choice of k and attains the high precision of large-k and the high sensitivity of small-k MinHash. LexicHash is a variant of MinHash with a carefully designed hash function. When estimating the similarity between two reads, instead of simply checking whether min-hashes match (as in standard MinHash), one checks how "lexicographically similar" the LexicHash min-hashes are. In our experiments on 40 PacBio datasets, the area under the precision-recall curves obtained by LexicHash had an average improvement of 20.9% over MinHash. Additionally, the LexicHash framework lends itself naturally to an efficient search of the largest alignments, yielding an O(n) time algorithm, and circumventing the seemingly fundamental O(n2) scaling associated with pairwise similarity search.

AVAILABILITY AND IMPLEMENTATION

LexicHash is available on GitHub at https://github.com/gcgreenberg/LexicHash.

Minmers are a generalization of minimizers that enable unbiased local Jaccard estimation

MOTIVATION

The 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.

RESULTS

To 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 IMPLEMENTATION

MashMap3 is available at https://github.com/marbl/MashMap.

Proving sequence aligners can guarantee accuracy in almost O(m log n) time through an average-case analysis of the seed-chain-extend heuristic

Seed-chain-extend with k-mer seeds is a powerful heuristic technique for sequence alignment used by modern sequence aligners. Although effective in practice for both runtime and accuracy, theoretical guarantees on the resulting alignment do not exist for seed-chain-extend. In this work, we give the first rigorous bounds for the efficacy of seed-chain-extend with k-mers in expectation Assume we are given a random nucleotide sequence of length ∼n that is indexed (or seeded) and a mutated substring of length ∼m ≤ n with mutation rate θ < 0.206. We prove that we can find a k = Θ(log n) for the k-mer size such that the expected runtime of seed-chain-extend under optimal linear-gap cost chaining and quadratic time gap extension is O(mn f (θ) log n), where f(θ) < 2.43 · θ holds as a loose bound. The alignment also turns out to be good; we prove that more than [Formula: see text] fraction of the homologous bases is recoverable under an optimal chain. We also show that our bounds work when k-mers are sketched, that is, only a subset of all k-mers is selected, and that sketching reduces chaining time without increasing alignment time or decreasing accuracy too much, justifying the effectiveness of sketching as a practical speedup in sequence alignment. We verify our results in simulation and on real noisy long-read data and show that our theoretical runtimes can predict real runtimes accurately. We conjecture that our bounds can be improved further, and in particular, f(θ) can be further reduced.

Efficient minimizer orders for large values of k using minimum decycling sets

Minimizers are ubiquitously used in data structures and algorithms for efficient searching, mapping, and indexing of high-throughput DNA sequencing data. Minimizer schemes select a minimum k-mer in every L-long subsequence of the target sequence, where minimality is with respect to a predefined k-mer order. Commonly used minimizer orders select more k-mers than necessary and therefore provide limited improvement in runtime and memory usage of downstream analysis tasks. The recently introduced universal k-mer hitting sets produce minimizer orders with fewer selected k-mers. Generating compact universal k-mer hitting sets is currently infeasible for k > 13, and thus, they cannot help in the many applications that require minimizer orders for larger k Here, we close the gap of efficient minimizer orders for large values of k by introducing decycling-set-based minimizer orders: new minimizer orders based on minimum decycling sets. We show that in practice these new minimizer orders select a number of k-mers comparable to that of minimizer orders based on universal k-mer hitting sets and can also scale to a larger k Furthermore, we developed a method that computes the minimizers in a sequence on the fly without keeping the k-mers of a decycling set in memory. This enables the use of these minimizer orders for any value of k We expect the new orders to improve the runtime and memory usage of algorithms and data structures in high-throughput DNA sequencing analysis.

Locality-preserving minimal perfect hashing of k-mers

MOTIVATION

Minimal perfect hashing is the problem of mapping a static set of n distinct keys into the address space {1,…,n} bijectively. It is well-known that n log 2(e) bits are necessary to specify a minimal perfect hash function (MPHF) f, when no additional knowledge of the input keys is to be used. However, it is often the case in practice that the input keys have intrinsic relationships that we can exploit to lower the bit complexity of f. For example, consider a string and the set of all its distinct k-mers as input keys: since two consecutive k-mers share an overlap of k-1 symbols, it seems possible to beat the classic  log 2(e) bits/key barrier in this case. Moreover, we would like f to map consecutive k-mers to consecutive addresses, as to also preserve as much as possible their relationship in the codomain. This is a useful feature in practice as it guarantees a certain degree of locality of reference for f, resulting in a better evaluation time when querying consecutive k-mers.

RESULTS

Motivated by these premises, we initiate the study of a new type of locality-preserving MPHF designed for k-mers extracted consecutively from a collection of strings. We design a construction whose space usage decreases for growing k and discuss experiments with a practical implementation of the method: in practice, the functions built with our method can be several times smaller and even faster to query than the most efficient MPHFs in the literature.

Minmers are a generalization of minimizers that enable unbiased local Jaccard estimation

Motivation

The 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.

Results

To 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.

Differentiable Learning of Sequence-Specific Minimizer Schemes with DeepMinimizer

Minimizers are widely used to sample representative k-mers from biological sequences in many applications, such as read mapping and taxonomy prediction. In most scenarios, having the minimizer scheme select as few k-mer positions as possible (i.e., having a low density) is desirable to reduce computation and memory cost. Despite the growing interest in minimizers, learning an effective scheme with optimal density is still an open question, as it requires solving an apparently challenging discrete optimization problem on the permutation space of k-mer orderings. Most existing schemes are designed to work well in 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, only approximate the original objective with likewise discrete surrogate tasks that are not able to significantly improve the density performance. This article introduces the first continuous relaxation of the density minimizing objective, DeepMinimizer, which employs a novel Deep Learning twin architecture to simultaneously ensure both validity and performance of the minimizer scheme. Our surrogate objective is fully differentiable and, therefore, amenable to efficient gradient-based optimization using GPU computing. Finally, we demonstrate that DeepMinimizer discovers minimizer schemes that significantly outperform state-of-the-art constructions on human genomic sequences.

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

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.

The minimizer Jaccard estimator is biased and inconsistent

Motivation

Sketching is now widely used in bioinformatics to reduce data size and increase data processing speed. Sketching approaches entice with improved scalability but also carry the danger of decreased accuracy and added bias. In this article, we investigate the minimizer sketch and its use to estimate the Jaccard similarity between two sequences.

Results

We show that the minimizer Jaccard estimator is biased and inconsistent, which means that the expected difference (i.e. the bias) between the estimator and the true value is not zero, even in the limit as the lengths of the sequences grow. We derive an analytical formula for the bias as a function of how the shared k-mers are laid out along the sequences. We show both theoretically and empirically that there are families of sequences where the bias can be substantial (e.g. the true Jaccard can be more than double the estimate). Finally, we demonstrate that this bias affects the accuracy of the widely used mashmap read mapping tool.

Availability and implementation

Scripts to reproduce our experiments are available at https://github.com/medvedevgroup/minimizer-jaccard-estimator/tree/main/reproduce.

Supplementary information

Supplementary data are available at Bioinformatics online.

Sparse and skew hashing of K-mers

MOTIVATION

A dictionary of k-mers is a data structure that stores a set of n distinct k-mers and supports membership queries. This data structure is at the hearth of many important tasks in computational biology. High-throughput sequencing of DNA can produce very large k-mer sets, in the size of billions of strings-in such cases, the memory consumption and query efficiency of the data structure is a concrete challenge.

RESULTS

To tackle this problem, we describe a compressed and associative dictionary for k-mers, that is: a data structure where strings are represented in compact form and each of them is associated to a unique integer identifier in the range [0,n). We show that some statistical properties of k-mer minimizers can be exploited by minimal perfect hashing to substantially improve the space/time trade-off of the dictionary compared to the best-known solutions.

AVAILABILITY AND IMPLEMENTATION

https://github.com/jermp/sshash.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Effective sequence similarity detection with strobemers

k-mer-based methods are widely used in bioinformatics for various types of sequence comparisons. However, a single mutation will mutate k consecutive k-mers and make most k-mer-based applications for sequence comparison sensitive to variable mutation rates. Many techniques have been studied to overcome this sensitivity, for example, spaced k-mers and k-mer permutation techniques, but these techniques do not handle indels well. For indels, pairs or groups of small k-mers are commonly used, but these methods first produce k-mer matches, and only in a second step, a pairing or grouping of k-mers is performed. Such techniques produce many redundant k-mer matches owing to the size of k Here, we propose strobemers as an alternative to k-mers for sequence comparison. Intuitively, strobemers consist of two or more linked shorter k-mers, where the combination of linked k-mers is decided by a hash function. We use simulated data to show that strobemers provide more evenly distributed sequence matches and are less sensitive to different mutation rates than k-mers and spaced k-mers. Strobemers also produce higher match coverage across sequences. We further implement a proof-of-concept sequence-matching tool StrobeMap and use synthetic and biological Oxford Nanopore sequencing data to show the utility of using strobemers for sequence comparison in different contexts such as sequence clustering and alignment scenarios.

Compact and evenly distributed k-mer binning for genomic sequences

MOTIVATION

The processing of k-mers (subsequences of length k) is at the foundation of many sequence processing algorithms in bioinformatics, including k-mer counting for genome size estimation, genome assembly, and taxonomic classification for metagenomics. Minimizers-ordered m-mers where m < k-are often used to group k-mers into bins as a first step in such processing. However, minimizers are known to generate bins of very different sizes, which can pose challenges for distributed and parallel processing, as well as generally increase memory requirements. Furthermore, although various minimizer orderings have been proposed, their practical value for improving tool efficiency has not yet been fully explored.

RESULTS

We present Discount, a distributed k-mer counting tool based on Apache Spark, which we use to investigate the behaviour of various minimizer orderings in practice when applied to metagenomics data. Using this tool, we then introduce the universal frequency ordering, a new combination of frequency-sampled minimizers and universal k-mer hitting sets, which yields both evenly distributed binning and small bin sizes. We show that this ordering allows Discount to perform distributed k-mer counting on a large dataset in as little as 1/8 of the memory of comparable approaches, making it the most efficient out-of-core distributed k-mer counting method available.

AVAILABILITY AND IMPLEMENTATION

Discount is GPL licensed and available at https://github.com/jtnystrom/discount. The data underlying this article are available in the article and in its online supplementary material.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Sequence-specific minimizers via polar sets

MOTIVATION

Minimizers are efficient methods to sample k-mers from genomic sequences that unconditionally preserve sufficiently long matches between sequences. Well-established methods to construct efficient minimizers focus on sampling fewer k-mers on a random sequence and use universal hitting sets (sets of k-mers that appear frequently enough) to upper bound the sketch size. In contrast, the problem of sequence-specific minimizers, which is to construct efficient minimizers to sample fewer k-mers on a specific sequence such as the reference genome, is less studied. Currently, the theoretical understanding of this problem is lacking, and existing methods do not specialize well to sketch specific sequences.

RESULTS

We propose the concept of polar sets, complementary to the existing idea of universal hitting sets. Polar sets are k-mer sets that are spread out enough on the reference, and provably specialize well to specific sequences. Link energy measures how well spread out a polar set is, and with it, the sketch size can be bounded from above and below in a theoretically sound way. This allows for direct optimization of sketch size. We propose efficient heuristics to construct polar sets, and via experiments on the human reference genome, show their practical superiority in designing efficient sequence-specific minimizers.

AVAILABILITY AND IMPLEMENTATION

A reference implementation and code for analyses under an open-source license are at https://github.com/kingsford-group/polarset.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Syncmers are more sensitive than minimizers for selecting conserved k‑mers in biological sequences

Minimizers are widely used to select subsets of fixed-length substrings (k-mers) from biological sequences in applications ranging from read mapping to taxonomy prediction and indexing of large datasets. The minimizer of a string of w consecutive k-mers is the k-mer with smallest value according to an ordering of all k-mers. Syncmers are defined here as a family of alternative methods which select k-mers by inspecting the position of the smallest-valued substring of length s < k within the k-mer. For example, a closed syncmer is selected if its smallest s-mer is at the start or end of the k-mer. At least one closed syncmer must be found in every window of length (k - s) k-mers. Unlike a minimizer, a syncmer is identified by its sequence alone, and is therefore synchronized in the following sense: if a given k-mer is selected from one sequence, it will also be selected from any other sequence. Also, minimizers can be deleted by mutations in flanking sequence, which cannot happen with syncmers. Experiments on minimizers with parameters used in the minimap2 read mapper and Kraken taxonomy prediction algorithm respectively show that syncmers can simultaneously achieve both lower density and higher conservation compared to minimizers.

Spectral Jaccard Similarity: A New Approach to Estimating Pairwise Sequence Alignments

Pairwise sequence alignment is often a computational bottleneck in genomic analysis pipelines, particularly in the context of third-generation sequencing technologies. To speed up this process, the pairwise k-mer Jaccard similarity is sometimes used as a proxy for alignment size in order to filter pairs of reads, and min-hashes are employed to efficiently estimate these similarities. However, when the k-mer distribution of a dataset is significantly non-uniform (e.g., due to GC biases and repeats), Jaccard similarity is no longer a good proxy for alignment size. In this work, we introduce a min-hash-based approach for estimating alignment sizes called Spectral Jaccard Similarity, which naturally accounts for uneven k-mer distributions. The Spectral Jaccard Similarity is computed by performing a singular value decomposition on a min-hash collision matrix. We empirically show that this new metric provides significantly better estimates for alignment sizes, and we provide a computationally efficient estimator for these spectral similarity scores.