A survey of k-mer methods and applications in bioinformatics

Abstract

The rapid progression of genomics and proteomics has been driven by the advent of advanced sequencing technologies, large, diverse, and readily available omics datasets, and the evolution of computational data processing capabilities. The vast amount of data generated by these advancements necessitates efficient algorithms to extract meaningful information. K-mers serve as a valuable tool when working with large sequencing datasets, offering several advantages in computational speed and memory efficiency and carrying the potential for intrinsic biological functionality. This review provides an overview of the methods, applications, and significance of k-mers in genomic and proteomic data analyses, as well as the utility of absent sequences, including nullomers and nullpeptides, in disease detection, vaccine development, therapeutics, and forensic science. Therefore, the review highlights the pivotal role of k-mers in addressing current genomic and proteomic problems and underscores their potential for future breakthroughs in research.

Graphical Abstract

Fig. 1: Introduction to k-mers. A. All possible 2-mers, or k-mers with two nucleotides, are listed. In a specific DNA sequence, all 2-mers are recorded for frequency analysis. B. Nullomers, or possible 2-mers not in the genome, are counted by subtracting the observed 2-mers from all possible 2-mers. Nullpeptides are k-mers missing from proteomes. C. In a mutated sequence, neomers, or nullomers that resurface due to somatic mutations, can occur. AA is a neomer in this mutated sequence. D. When analyzing multiple genomes or sequences, primes, k-mers not present in any of the sequences, can be identified. There is one prime (CC) in these three sequences. Quasi-primes, or k-mers that only occur in one sequence (AA), can be identified.

1. Introduction

In recent years, the field of genomics has undergone a significant transformation due to advances in sequencing technologies and the generation of both large and diverse datasets [1], [2]. The advancements have enabled the efficient acquisition of vast amounts of genomic information within a short timeframe [3]. However, this also poses the challenge of deriving meaningful insights from complex, high-dimensional datasets to address sequence analysis problems. Conventional analytical approaches have struggled with the unprecedented scale and complexity of genomic datasets [4], [5], necessitating the development of efficient algorithms for extracting relevant information.

K-mers, defined as contiguous nucleotide or amino acid sequences of fixed length k (Table 1; Fig. 1A), have become integral in addressing these challenges. K-mer-based algorithms are widely utilized across genomic and proteomic applications and offer several advantages as fundamental units for analyses [6], [7]. First, they enable the fast retrieval of targeted sequences from next-generation sequencing data, facilitating efficient exploration and downstream analysis-based tasks [8], [9]. Moreover, k-mers can have intrinsic biological significance, and their distribution and frequency can provide insights into genomic characteristics such as repetitive elements, functional regions, genomic variation, and DNA damage and repair mechanisms [8], [9], [10], [11], [12], [13], [14], [15], [16].

Table 1.

Relevant definitions.

Term	Definition
Absent Word	Word that does not occur in a given genome or proteome; also see definitions for Nullomer and Nullpeptide[165]
First Order Nullomer	Nullomer where any single base substitution at any position along the k-mer still yields a nullomeric sequence[164]
Frequentmer	Short sequence that is specific and recurrently observed in either patient or healthy control samples, but not in both[146]
High Order Nullomer	Nullomer whose mutated sequence is still a nullomer[164]
K-mer	Contiguous subsequence of length k derived from a longer sequence
Minimal Absent Word	Absent word where removing the leftmost or rightmost nucleotide results in a sequence that is no longer an absent word[165]
Minimizer	Minimum value k-mer selected to represent a longer k-mer or group of k-mers to reduce memory consumption and run-time[199]
Neomer	Nullomer that resurfaces due to somatic mutations in cancer[29]
Nullomer	K-mer sequence absent from a specific genome[159], [164]
Nullpeptide	K-mer that does not exist within a proteome[163]
Prime	K-mer that does not exist in any genome or proteome[159]
Quasi-Prime	K-mer that is only found in one species or sequence[172]
Strobemers	Two or more linked shorter k-mers, where the combination of linked k-mers is determined by a hash function[200]
Syncmers	Set of k-mers defined by the position of the smallest-valued substring of length s < k within the k-mer[199]
Super K-mer	Substring of maximal length wherein all k-mers within it share the same minimizer[9], [45]

Fig. 1.

Introduction to k-mers. A. All possible 2-mers, or k-mers with two nucleotides, are listed. In a specific DNA sequence, all 2-mers are recorded for frequency analysis. B. Nullomers, or possible 2-mers not in the genome, are counted by subtracting the observed 2-mers from all possible 2-mers. Nullpeptides are k-mers missing from proteomes. C. In a mutated sequence, neomers, or nullomers that resurface due to somatic mutations, can occur. AA is a neomer in this mutated sequence. D. When analyzing multiple genomes or sequences, primes, k-mers not present in any of the sequences, can be identified. There is one prime (CC) in these three sequences. Quasi-primes, or k-mers that only occur in one sequence (AA), can be identified.

K-mers can serve as valuable clinical biomarkers for detecting pathogens [17], [18], [19], antimicrobial resistance [17], [20], [21], and human diseases [16], [22], [23], [24]. Additionally, k-mers identified from genomic or transcriptomic data from tumor samples or liquid biopsies prove valuable in cancer diagnosis, prognosis, and treatment [16], [22], [25], [26], [27]. K-mers can be categorized into various subtypes, each with distinct applications. Notably, nullomers and nullpeptides are k-mers missing from a genome or proteome, respectively (Fig. 1B). The emergence of nullomers and nullpeptides during cancer development can be used as a biomarker for cancer detection (Fig. 1C) [22], [28], [29], with certain nullpeptides even exhibiting cancer cell-killing properties [30], [31].

This review provides a comprehensive overview of the applications of k-mers in addressing analytic challenges across genomics and proteomics. It explores the development of k-mer-based tools and reviews research on the utilization of subsets of k-mers as informative indicators that reveal insights into genome variation, population genetics, and disease associations. As new k-mer-based algorithms are developed that broaden the applications for sequence analysis, the biological implications of k-mers will continue to grow.

2. The selection of k

Any genomic sequence can be fragmented into consecutive k-mers of length k. The selection of k can vary significantly depending on both the dataset and application. For example, in de novo genome assembly, shorter k-mer lengths may reduce the quality of the resulting contigs, whereas longer k-mers have a higher chance of including errors [32]. The complexity of the k-mer set, or all possible k-mers, increases exponentially with k and is equal to 4^k with the four base pair alphabet for DNA and RNA. For smaller values of k, the complexity may be insufficient to represent or distinguish long sequences because the possible k-mers may appear in a genome many times and be observed in a multitude of species.

Alternatively, longer k-mers can lead to a sparser representation of the dataset in the k-mer space, allowing better differentiation between samples due to k-mers appearing in a very small number of species [33]. However, this also increases computational complexity and can result in too few common k-mers, rendering some applications, such as phylogenetic analysis or biomarker detection, impossible. For instance, in a three Gbp genome, the probability of observing a given 16-mer is 0.5. However, due to the exponential growth of the k-mer space, this probability drops to 0.01 at k = 19 [34].

This conundrum emphasizes one of the inherent limitations of some k-mer-based approaches. Although longer k-mers may more accurately reflect the biological function of sequences such as transcription factor binding sites, they can result in severe overfitting problems due to sparse k-mer counts. To address this issue, some methods using gapped k-mers as features have been introduced [13], [35].

3. Applications of k-mers

The introduction of k-mers has made it possible to process large and complex genetic data with reasonable time complexity. This improvement has allowed for the extraction of essential patterns and characteristics within both genomes and proteomes, tasks that were previously inefficient or unfeasible. As a result, the past decade has seen the development of numerous applications that utilize the capabilities of k-mers. These include methods for k-mer counting and frequency analysis, sequence alignment, genome assembly, comparative genomics, metagenomics, metaproteomics, and protein structure prediction (Table 2; Supplementary Table 1).

Table 2.

Current k-mer-based software methods and databases.

Application	Description	Tools(reference)
K-mer Counting	K-mer counting is preliminary to many applications such as genome assembly, error correction, sequence alignment, and classification. Common approaches include hash tables with lock-free-based tools, hash tables with lock-based tools, disk-based tools, bloom-filter tools, quotient filtering, and burst-tree tools.	BFCounter[201], CHTKC[202], Discount[203], DSK2[204], GECKO[205], Gerbil[206], Jellyfish2[8], KAnalyze[207], KCMBT[208], KCOSS[209], KHMer[210], KMC3[45], KmerAnalysis.jl[211], Kmerator[212], Kounta[213], Krust[214], Meryl[46], MSPKmerCounter[215], REINDEER[47], RNA-Skim[60], Salmon[59], SEEKR[134], SeqKit[216], Seqtrie[217], Squeakr[48], SWAPCounter[36], Tallymer[218], Turtle[219], VLmer[220]
Frequency Analysis	K-mer spectra can be used to estimate genome size and complexity prior to assembly. This information can then be used to optimize the assembly process, detect and correct errors in the sequencing reads, and evaluate the quality of a genome assembly.	GenomeScope[221], GenomeTester4[222], KAnalyze[207], KAT[223], KHMer[210] KmerGenie[224], KmerStream[225], ntCard[226], Squeakr[48]
Sequence Indexing	K-mer searches serve as effective proxies for both exact and approximate sequence searches in unassembled datasets. Current methods are able to determine the presence or absence of any k-mer within collections of up to ∼2500 datasets.	COBS[49], DiscoverY[227], HowDeSBT[228], KmerGO[229], Kmerind[230], Mantis[231], MetaGraph[232], PAC[233], RecoverY[234], SBT[235], SeqOthello[236], SSHash[237], VarGeno[238]
Genome Assembly	In de novo genome assembly, sequencing reads undergo fragmentation into k-mers, and their overlaps are employed to assemble longer contiguous sequences. This process commonly utilizes k-mers to build De Bruijn graphs or implements overlap-layout-consensus methods.	Allpaths-LG[239], Bifrost[240], Canu[241], Cortex[242], ELBA[243], KAT[223], MEGAHIT[244], Merqury[46], QUAST-LG[245], SKESA[246], SPAdes[75], TandemQUAST[247], TandemMapper[248]
Sequence Comparison	Alignment-free methods are increasingly used for DNA and protein sequence comparison since they are much faster than traditional alignment-based approaches. Most alignment-free algorithms are based on the word or k-mer composition of the sequences under study.[249]	BBMap[250], Bowtie2[251], [252], BWA[253], [254], [255], iMOKA[256], MiniMap2[71]
Taxonomic Classification	In sequence composition-based methods, the frequency and distribution of k-mers in metagenomic data are analyzed to assess genome similarity across various taxonomic ranks.[257], [258]	ARK[259], BinDash[122], Bracken[260], CDKAM[261], CLARK[262], Dashing[124], fmh-funprofiler[128], Genometa[263], Kaiju[138], KMCP[264], KmerFinder[265], Kraken2[136], KrakenUniq[19], LMAT[266], Mash[72], Mash Screen[34], Matchtigs[267], MetaCache[268], MetaPalette[269], MetaProFi[50], NIQKI[126], SEK[270], StrainSeeker[271], SuperSampler[127], TACOA[272], Taxonomer[273], TETRA[274], VirFinder[18], WGSQuikr[275]
Phylogeny Reconstruction	Pairwise evolutionary distances between protein or nucleic acid sequences and phylogenetic distances can be estimated from the number of k-mer matches between two sequences. Alignment-free sequence comparison quantifies distance using the decay of the number of k-mer matches between two sequences and compares the results to known phylogenetic trees.[249]	AAF[276], FSWM[277], PhyloPythia[278], Skmer[279], Slope-SpaM[280], SlopeTree[281]
Protein Sequence Searching and Alignment	Sequence match is determined by aligning translated DNA sequences to a reference protein database.	BLAT[65], BLAST[68], BLASTX[64], [282], DIAMOND[283], MMSeqs2[284], PAUDA[285], RAPSearch2[286], USEARCH & UBLAST[287]
Databases	K-mer databases provide k-mer sequences that are present or absent in each species.	kmerDB.com[171], nullomers.org[162]

3.1. K-mer counting

Counting the occurrence of all distinct k-mers in biological sequences is a crucial step in many bioinformatic applications such as genome assembly, sequence alignment, sequence clustering, error correction of sequencing reads, and genome size estimation (Fig. 1A; Fig. 2A; Table 2) [36]. Pattern recognition tasks in sequence analysis can be efficiently reduced to counting k-mers of length k. For example, k-mer counting tools are often used in the identification of repetitive elements, such as transposable elements and tandem repeats. Repetitive elements are characterized by high k-mer counts [37], [38]. In contrast, variable genomic regions, including single nucleotide polymorphisms [39] and structural variations associated with phenotypic differences or diseases, may exhibit differences in k-mer counts during alignment across various genomes (Fig. 2B) [25], [40]. Unique genomic regions are defined by lower k-mer counts [25], [41], [42], [43], [44].

Fig. 2.

Applications of k-mers. A. K-mer counting and frequency analysis are crucial steps in various bioinformatic applications, including detecting sample contamination. B. K-mers are used in graph-based genome assembly and identification of genetic variants. C. In sequence assembly, k-mers are utilized for sequence alignment. Sequencing reads are fragmented into k-mers, and overlaps between k-mers are identified to reconstruct the original sequence. D. In adaptive sequencing, k-mer counting allows for the identification of unique sequences. K-mer based variant-filtering methods are used for improving accuracy in genome assembly; algorithms will filter false positives from alignments. E. K-mers are utilized in genome editing to identify suitable target sites and design guide RNAs, and efficient k-mer indexing enables primer candidates to be identified with low off-target site potential. F. K-mers are used for taxonomic profiling and classification in comparative genomics and metagenomics.

In 2018, Manekar and Sathe performed a benchmark study comparing several common k-mer counting methods [10]. Bioinformatic tools such as Jellyfish [8], KMC3 [45], Meryl [46], REINDEER [47], and Squeakr [48] leverage optimized data structures and algorithms and are commonly written in low-level programming languages to process large-scale genomic datasets and accurately determine k-mer abundance (Supplementary Table 1) [9]. Jellyfish is a simple k-mer counter toolkit for lengths up to 31 bases that utilizes a combination of a Bloom filter and lock-free hash table, but is slower than more recent applications [8]. Squeakr uses a counting quotient filter, providing memory efficiency, faster processing for larger k-mers, and also the option to obtain approximate counts at a much lower computational cost for applications where exact counts are not required [48]. REINDEER was constructed to efficiently store and query exact k-mer abundances across multiple datasets, utilizing De Bruijn graphs for optimal indexing. Other tools include COBS, an inverted index based on Bloom filters [49], and MetaProFi, which utilizes k-mers to profile and query protein and nucleotide sequence data based on k-mer signatures [50]. MetaProFi uses Bloom filters for rapid searches in large sequence datasets and builds indexes of nucleic acid or amino acid sequences, enabling protein-level sequence comparison.

3.2. K-mer distribution and frequency analysis

A sequencing dataset can be characterized using the distribution of k-mers, and this is referred to as the k-mer spectrum or histogram. Different taxonomic domains are known to have differing distributions of genomic k-mer spectra [51]. Chor et al. discovered that archaeal and bacterial species display a unimodal spectra, while tetrapods are represented by multimodal spectra [51]. Tetrapods’ spectra are characterized by specific GC content and CpG suppression. Notably, the protozoa Entamoeba histolytica exhibits CpG suppression but lacks multimodal k-mer spectra, suggesting CpG suppression alone does not determine modality [51]. Typically, a moderately heterozygous diploid organism has a k-mer spectrum with four apparent coverage peaks: sequencing errors (low coverage), unique genomic sequences from heterozygous loci, all homozygous loci in the genome, and genomic duplications, which is usually a smaller peak. In terms of the first peak, k-mer spectrum-based algorithms have been developed to identify and remove sequencing read errors based on the infrequency of these k-mers [52], [53], [54].

The frequency of specific k-mers can reveal various aspects of the genomic structure and complexity of a biological sample (Fig. 1A; Fig. 2A) [9], [10], [51], [55]. By examining the empirical frequencies of k-mers, the GC-content, CpG suppression, repeat content, heterozygosity, and sequencing coverage of the sample can be inferred [51], [56], [57]. Bussi et al. analyzed k-mer frequency patterns in over 5000 archaea, bacteria, and eukaryotic complete genomes and found that sequence space coverage (SSC) depends heavily on both genome length and GC-content [56], a result that was consistent with findings by Liu et al. [56], [57]. Bussi et al. also found that maintaining a non-0.5 GC-content was shown to influence the prevalence of specific k-mers, leading to decreased sequence entropy and SSC [56].

Analyzing transcriptomic data, especially in non-model organisms or meta-transcriptomes, can be challenging due to the levels of complexity, variability, and noise. In such cases, k-mer frequency analysis can be utilized to quantify and detect subtle differences in gene expression or organism composition that may be missed by other methods [58]. K-mer-based software such as Salmon [59] and RNA-Skim [60] can accurately quantify transcript abundance levels from RNA-Seq data. Importantly, Salmon, the successor of Sailfish [61], corrects for fragment GC-content bias to improve the accuracy of abundance estimates [59].

In a proteomic analysis, Poznanski et al. utilized k-mer frequency analysis to analyze five-mer peptide frequencies in proteins and highlight deviations from expected patterns [62]. Enrichment was noted in the majority of permutation groups with numerous outlier sequences; this aligned across protein families and evolutionary lineages, suggesting non-random variations. Interestingly, the identified outlier sequences often contained known motifs, and over-represented five-mer peptides were significantly related to known functional motifs. Therefore, k-mer frequency analysis has numerous advantages for interpreting biologically relevant information from sequence analysis across the genome, transcriptome, and proteome.

Despite varied methodologies for analyzing k-mer frequency, approaches were limited in that they did not provide a measure of the rarity of nucleic or peptide sequences. In response, Chantzi et al. estimated the rarity of nucleic and peptide k-mers across organismal genomes and proteomes and introduced an index of k-mer rarity across organisms [63]. Ultimately, the study found that the rarity of k-mers can be inferred by their amino acid and nucleotide composition.

3.3. Sequence alignment and genome assembly

3.3.1. Sequence alignment

One of the challenges of analyzing biological sequences is finding the optimal alignment between two or more sequences that may have variations such as insertions, deletions, or substitutions. Programs such as Basic Local Alignment Search Tool (BLAST) are sequence similarity search programs that compare combinations of nucleotide or peptide sequence queries with genomic, transcriptomic, or protein databases [64]. Subsequently, they find short matches between two sequences and attempt to start alignments based on a measure of local similarity, the maximal segment pair (MSP) score.

Similar in many ways to BLAST, BLAT is a mRNA/DNA and translated protein alignment algorithm, and its speed originates from an index of all nonoverlapping k-mers in the genome [65]. BLAT utilizes the index to identify potentially homologous regions, performs an alignment between them, assembles these aligned segments into larger alignments, and then revisits small internal exons while adjusting large gap boundaries with canonical splice sites when possible. Recently, k-mer-based alignment algorithms have been utilized in detecting and filtering out contamination by comparing observed k-mers against a reference database of potential contaminants (Fig. 2D) [66], [67].

In sequence alignment algorithms (Fig. 2B) [68], k-mers can serve as effective minimizers to reduce memory and computational requirements for genomic data analysis (Table 1) [69]. The basic idea behind minimizers, or k-mers that represent groups of k-mers, is that it is not necessary to consider every possible k-mer in a sequence but that it is often sufficient to focus on a smaller, representative subset for an analysis. The selection of the minimizer k-mer is done through a deterministic procedure; traditionally, the smallest k-mer in lexicographical ordering is chosen, but more recently, other alternatives such as ordering based on universal hitting sets have also been proposed [70]. This approach allows indexing a single minimizer as a representative of multiple k-mer sequences, significantly reducing memory requirements as well as the time taken to compare subsets. Currently, minimizers are used to accelerate aligners such as Minimap2 [71] and assist algorithms such as Mash [72] in estimating the similarity between genomes.

3.3.2. Genome assembly

Genome assembly is the process of reconstructing the original DNA sequence of an organism from sequencing reads. It is a crucial step in bioinformatic analyses, but accurate assembly is often challenging due to sequencing errors, repetitive regions, and structural variations in DNA. There are two forms of genomic assembly: reference-based and de novo, which does not utilize a reference genome for alignment.

Reference-based genome assembly can be utilized when a high quality reference genome is available [73]. Tools include RaGOO [74], which utilizes Minimap2 [71] alignments to a closely related reference genome with a k-mer size of 19 base pairs to cluster, order, and orient genome assembly contigs into pseudomolecules. While reference-based methods might introduce a bias towards the reference genome, it typically offers a significantly quicker and more cost-effective alternative to de novo methods.

In de novo genome assemblers such as SPAdes [75] and multiple others [76], a common approach is to first divide sequencing reads into k-mers and use each unique k-mer to represent a node in a De Brujin Graph [77]. K-mers with an overlap of k-1 bases are represented as adjacent nodes. Valid paths through the graph are then used to identify longer contiguous sequences or contigs, which are then arranged and oriented to create larger units called scaffolds. Ultimately, the scaffolds are utilized to reconstruct the original genomic sequence (Fig. 2C) (Fig. 2C) [78], [79], [80], [81].

In genome assembly, pangenome graphs, which model direct relationships between all genomes in the analysis, can be utilized to capture genetic diversity and understand genetic variations across different populations and species [82]. K-mers are utilized for genotyping known variants directly from raw sequencing reads without alignment. Algorithms such as Pangenome-based Genome Inference or PanGenie use haplotype-resolved pangenome references and k-mer counts from short-read sequencing data to analyze genetic variation [83].

3.3.3. Error correction

Short-read sequencing technologies such as Illumina allow for the high-throughput production of short reads at a low cost, but produced reads can contain various errors [84]. However, k-mers can play a significant role in error correction for de novo genome assembly and short-read mapping. Short-read error correction algorithms include RECKONER [85], [86] and RACER [87], which correct substitution errors, are dedicated to Illumina reads, and rely on k-mer counting. The workflow of RECKONER, whose new version also corrects indel errors, specifically includes k-mer counting, determining the threshold of number of k-mer appearances, removing untrusted k-mers from the database, and correcting the reads [85], [86]. Other algorithms for short-read error correction include the k-mer spectrum-based error correctors Musket [52] and BLESS [53], which utilize both k-mer counting and a Bloom filter. Interestingly, the tool Lighter fully avoids k-mer counting and relies instead on a pair of Bloom filters [88], while Karect [89], SAMDUDE [90], and CARE [91] utilize alignment-based approaches. Lastly, Rcorrector is a k-mer-based method that relies on De Brujin graphs for correcting random sequencing errors in Illumina RNA-seq reads specifically [92].

Long-read sequencing technologies such as those developed by Pacific Biosciences (PacBio) and Oxford Nanopore provide long reads, which, unlike short-reads, are able to span repeated elements or repetitive regions in de novo genome assembly and structural variant calling [93]. One caveat of these methods is that they suffer from significantly higher error rates, necessitating novel methods of error correction [94], [95], [96]. In response to these issues, multiple hybrid approaches based on k-mers were proposed, utilizing long reads for scaffolding and short Illumina reads for correcting errors [97], [98], [99], [100]. For a detailed comparison, readers can also refer to a number of publications evaluating the strengths of individual approaches [101], [102], [103].

3.4. Genome editing

CRISPR-Cas9 (Clustered Regularly Interspaced Short Palindromic Repeats) enables precise and efficient genome editing, but encounters various analytical challenges such as understanding off-target effects and improving delivery efficiency [104], [105]. K-mers have been employed as a strategy to improve the performance and accuracy of CRISPR-Cas9 technology [106], [107], [108], [109]. This is achieved by refining the specificity of guide RNA design, reducing false positives of target detection, and identifying genomic variation (Fig. 2E). For example, k-mer-based technologies like Redk-mer and BoostMEC analyze k-mer distributions to identify unique target sites with a low likelihood of off-target effects [106], [107]. JACKIE, a k-mer-based tool, was developed to improve the creation of guide RNAs (gRNAs) complementary to specific target sequences [108]. In addition, a web application named KmerKeys, which facilitates rapid querying of k-mers in genome assemblies, holds potential for CRISPR/Cas9 target design [110]. Its efficient k-mer indexing enables the identification of primer candidates with off-target site potential and allows for the representation of variants at the population level.

3.5. Comparative genomics

The study of genomic differences and similarities across and within taxonomic levels has benefited from k-mer-based approaches (Fig. 2D) [56]. These methods enable comparative analysis of k-mer occurrences and distributions across genomes, facilitating the identification of conserved regions, gene families, regulatory motifs, and genomic rearrangements to better understand evolutionary relationships [51], [111], [112], [113]. One primary application of k-mers in comparative genomics is constructing phylogenetic trees by quantifying the genetic distance between species based on k-mer frequencies [114], [115], [116], [117], [118]. Recent phylogenetic analyses have led to discoveries like the novel picorna-like viral sequences found in gut metagenomes [119], [120].

Kmer-db is an efficient tool for estimating evolutionary relationships between pathogens based on derived k-mers [121]. This makes it well-suited for processing large and diverse bacterial genome datasets, enabling more accurate comparisons for phylogeny reconstruction even among distantly related genomes with limited shared k-mers. Mash, which extends the MinHash dimensionality-reduction technique, has been used for phylogeny reconstruction and allows for efficient clustering, search, and mutation distance estimation in large sequence collections [72]. However, Kmer-db is roughly 26 times faster than Mash and is subsequently better equipped to process larger datasets [121]. Additional sketch-based methods that utilize k-mers in comparative genomics include Bindash 1.0 [122] and 2.0 [123], Dashing 1.0 [124] and 2.0 [125], NIQKI [126], SuperSampler [127], and fmh-funprofiler [128] (Table 2).

3.6. Metagenomics

Determining the abundance of taxonomic communities within diverse environments has important implications for agriculture, wildlife conservation, and healthcare improvements [129]. However, metagenomic profiling, such as the analysis of entire microbial communities, encounters challenges associated with the diversity, complexity, and quality of the data (Fig. 2D) [119]. By efficiently characterizing genetic material with k-mer-based approaches, diverse taxonomic groups can be identified and profiled [130]. Low abundance species in metagenomics data containing tens of millions of reads can be detected (Fig. 2F) [19].

K-mer-based methods for profiling taxa consist of three steps: first, k-mer profiles are compared to reference datasets, the individual sequences are then labeled, and lastly, the abundance of microbial taxa in large-scale metagenomic datasets is estimated [118], [131], [132]. K-mers can be utilized to label these individual sequences, facilitating metabarcoding across various organisms or individual cells [133]. Additionally, to understand the functional potential of metagenomic data, the distribution of k-mers associated with functional genes or metabolic pathways can be analyzed [134], [135].

Several k-mer-based approaches have been developed for quantifying and classifying taxa in metagenomic samples. Notable examples include Kraken2 [136], KrakenUniq [19], YACHT [137], and Kaiju [138], which assign reads to taxa without alignment based on approximate matches to a reference database of genomic or proteomic sequences. Despite their efficiency and sensitivity, these methods are limited by the quality and completeness of the reference database, an inability to detect novel taxa, and susceptibility to false positives due to horizontal gene transfer or contamination. In large-scale prokaryote and virus projects, k-mer frequency-based tools like VirFinder are used to identify viral sequences in mixed metagenomic datasets containing both viral and host contigs [18]. Additionally, k-mer-based genome binning techniques, such as Phages from Metagenomics Binning (PHAMB), facilitate the extraction and categorization of thousands of viral genomes from bulk metagenomics datasets [139]. These tools cluster viral genomes into taxonomic viral populations and have been instrumental in understanding viral-microbial host interactions from the Human Microbiome Project 2 dataset [139]. Furthermore, k-mer-based approaches, such as metaSPades, prove valuable for reconstructing individual genomes from metagenomic datasets [140].

K-mer-based approaches in clinical metagenomics hold the potential for rapid diagnosis and monitoring of infectious diseases [141]. With the reduced cost of high-throughput or next generation sequencing, clinical applications of sequencing are on the rise [142]. Researchers can employ k-mers to directly characterize microbial pathogens from patient samples, infer antibiotic resistance profiles for treatment guidance, and identify novel or emerging pathogens not yet present in reference databases. The development of machine learning models for predicting antimicrobial resistance faces challenges stemming from the high-dimensional and strong correlations of k-mer-based representations [20], [143], [144].

To enhance the interpretability of genomic signatures in k-mer-based predictive models, Jaillard et al. introduced an adaptive cluster lasso strategy, which identified sparse, meaningful, and interpretable genomic signatures and improved clinical utility [145]. Additionally, Wang et al. developed MetaGO, which uses group-specific metagenomic sequences to highlight differences between patients affected by liver cirrhosis and healthy cohorts [23]. This method aimed to predict clinical disease outcomes using long k-mer (k ≥ 30 bps) sequence signatures.

Similarly, Mouratidis et al. introduced “frequentmers,” short sequences that are specific and recurrently observed in either patient or healthy control samples, but not in both [146]. Using metagenomic NGS data from liver cirrhosis patients and healthy controls, machine learning models trained with frequentmers (k = 16) outperformed previous models and achieved an AUC of 0.91 in liver cirrhosis detection. Interestingly, the authors also identified microbial organisms in liver cirrhosis samples associated with the most predictive frequentmer biomarkers, potentially offering insight into the role of the gut microbiome in disease.

3.7. Metaproteomics

Metaproteomic datasets contain a vast array of protein sequences from diverse and unlabeled organisms, posing significant challenges for accurate identification and functional interpretation [147], [148]. Similar to metagenomics, k-mers aid in identifying proteins and understanding the functional potential of microbial communities [149]. Protein sequences are broken down into overlapping k-mers, and the generated peptide fragments are utilized for protein database searching [150]. Subsequently, researchers can detect proteins from less-known organisms and identify closely related species with the closest matching and similar sequences. Quantifying shared and unique k-mers across metaproteomic samples helps identify similarities and differences in protein expression profiles, enabling comparisons of microbial communities under different environmental conditions or between healthy and diseased states [151]. The utilization of k-mers in metaproteomics has even led to the identification of virion-associated protein annotations in “viral dark matter” genomic sequences [152].

3.8. Modeling protein structure

Protein structures can be modeled and predicted using k-mer-based algorithms that analyze the distribution of specific k-mers within protein sequences. K-mers prove effective in capturing and predicting information about protein folding patterns [153], secondary structure elements [153], and tertiary structures [154]. Additionally, k-mers are utilized to identify antigenic regions and potential epitopes within protein sequences [155], [156]. Analyzing k-mer distribution in known epitopes aids in developing models to predict the likelihood of specific regions acting as antigenic determinants or antimicrobial peptides [157], [158]. This approach enhances our understanding of antibody-epitope interactions and proves valuable in the design of diagnostics, therapeutic antibodies, and vaccines [157].

4. Absent sequences from genomes and proteomes

4.1. Overview of nullomers, nullpeptides, MAWs, primes, and quasi-primes

Nullomers, defined initially as the shortest k-mers absent from a specific genome (Table 1; Fig. 1B) [159], have gained attention across various fields due to their usefulness in different applications. They serve as distinctive markers for pathogens [160] and cancer [22], are used in barcoding [161], and have potential applications in drug discovery [31]. Their absence has been attributed to detrimental effects on an organism, higher mutation rates, and stochastic effects [162], [163], [164]. Research into the potentially deleterious effects of absent sequences has also included investigation into nullpeptides, which are peptide sequences that do not exist within a given proteome (Table 1; Fig. 1B) [159], [162], [163].

The potential applications of nullomers and nullpeptides have stimulated the development of various related k-mer concepts and algorithms to prioritize absent sequences. For example, Vergni & Santoni introduced a subset of nullomers termed high order nullomers, nullomers whose mutated sequences are still nullomers [164]. The nth order nullomer is one where any n base substitutions across each base in the sequence still yield a nullomer; first-order nullomers are the subset of nullomers where any single base substitution at any position along the k-mer still yields a nullomeric sequence (Table 1). In addition, Pinho et al. defined minimal absent words (MAWs) as absent sequences that lose the property of being absent from a genome (or sequence space by extension) if a character is removed from either end (Table 1) [162], [165]. The growth rate of the set of MAWs is linear with respect to the length of the string, unlike the exponential growth rate of nullomers. Therefore, the identification and classification of MAWs is more manageable when accounting for high computational costs in analysis [165]. MAWs have been proposed as potential tools for drug discovery, design, and delivery as they may have specific binding affinities and interactions with other molecules [162], [166], [167], [168], [169].

Recent work characterized significant MAWs, which were statistically expected to exist yet absent in the genomes and proteomes of different species [162]. The researchers found that substitution mutations in human protein coding genes can cause the appearance of MAWs and that over 25% of human proteins have the potential to generate a significant MAW through a single substitution mutation [162]. MAWs can also expand phylogenetic inference by providing novel insights into the evolutionary relationships and divergence times of different taxa [30], [167]. For example, the intra-species variation in the number and content of MAWs is generally less pronounced than inter-species variation, suggesting that MAWs are relatively conserved and stable within species [162], [169].

Extending on the derivation of nullomers and nullpeptides, which are absent from one genome or proteome, primes are designated as sequences that are universally absent from every sequenced genome or proteome [159]. Nucleic primes are k-mers absent from every genome, while peptide primes are the nullpeptides absent from every proteome (Table 1; Fig. 1D) [163]. In 2009, 417 five-amino acid primes not found in the universal proteome were identified [170]. An analysis across the genome of over twelve species (human, chimp, and ten other non-primates) identified 60,370 nullomers absent across them for 15 bp length. Work in 2021 expanded the identification and functional characterization of peptide primes; 140,308,851 nullpeptide primes absent from all known species were identified using the UniParc database (which had 1,030,456,800 proteins) [163]. Recently, kmerDB was made public containing 5,186,757 nucleic and 214,904,089 peptide primes isolated from the examination of 45,785 complete organismal genomes and 22,386 reference organismal proteomes [171].

The identification of organisms at the species level is often challenging due to the limitations of molecular markers that can discriminate between closely related taxa. A novel approach based on k-mers is to use quasi-primes, which are the shortest k-mers that are unique to a single species and absent from any other sequenced genome or proteome. Mouratidis et al. cataloged quasi-prime peptides (for k-mer lengths up to seven amino acids) from 21,875 species and identified quasi-primes (for k-mer lengths of six and seven amino acids) for 21 human pathogens, 8 model organisms, humans, Bonobo, and Gorilla [172]. The identification of quasi-primes may offer insights into species-specific traits, trait acquisition, evolutionary relationships, and mechanistic processes and may be applied for highly sensitive and specific real-time organismal detection (Table 1; Fig. 1D). For example, in the human genome, nucleic quasi-prime loci are predominantly linked to genes involved in brain development and cognitive function [173].

4.2. Nullomers, nullpeptides, and selection constraints

Nullomers have been viewed as potential signatures of natural selection against deleterious sequences [162], [163], [164]. Acquisti et al. challenged this idea and suggested that the mutational dynamics of the genome, namely the hypermutability of CpG dinucleotides in vertebrates, likely account for the emergence of certain short sequence motifs as nullomers (k = 11 bp) [174]. Their analysis revealed that many reported human nullomers differ by only one nucleotide, implying the significant role of mutations in the evolution of nullomers. Further emphasizing the insufficiency of the CpG hypermutability model, Vergini & Santoni found that nullomers from the human genome exhibit different statistical properties compared to those expected from random sequences [164]. The study demonstrated that CpG dinucleotide frequencies in genomes cluster into homogenous groups based on their CpG frequency profiles, with close species sharing similar patterns as depicted in phylogenetic trees. Lastly, the authors found that nullomers display higher mean helical rise values, suggesting a potential interaction with histone complexes that could explain their removal from DNA pending experimental validation [175], [176].

In contrast, many studies have suggested that the hypermutability of CpG dinucleotides is an insufficient explanation for nullomer origin [164], [167]. Garcia et al. examined 22 organisms from various domains and found that mutational biases are not uniform across different sets of MAWs of increasing length [169]. They suggested that MAWs may have been inherited from a common ancestor, in addition to lineage-specific inheritance. Subsequently, they advocated for the utility of MAWs in inferring genomic homology, understanding genome evolution, and addressing limitations in existing methods that often overlook non-protein-coding regions. Koulouras and Frith introduced an open-source software tool, Nullomers Assessor, to identify statistically significant MAWs in biological sequences [162]. Their analysis of over 147,000 viral sequences utilizing Nullomers Assessor revealed that the most frequent significant absent motifs in viral genomes corresponded to restriction recognition sites. This evidence supports the hypothesis that MAWs are absent due to negative selection and implies that they may have been replaced by specialized sequences with similar or optimized functions.

In 2021, Georgakopoulos-Soares et al. investigated the occurrence of both putative and germline nullomer-emerging mutations, which are genetic changes that introduce nullomers into a genome [163]. A significantly higher number of mutations than expected by chance for specific nullomer sequences within transposable elements were found, potentially due to their suppression and their role in the jumping activity of these elements. Genes with high-density nullomer-emerging mutations were involved in epigenetic regulation and DNA organization, while genes with low-density mutations were linked to processes like cell-to-cell contact and chemical stimuli detection. An enrichment of these mutations in promoters and enhancers suggested putative functions of nullomers in gene regulation. When examining nullomers across different species, nullomers were used for phylogenetic classification in vertebrate evolution. The human genome, in particular, was found to contain a higher number of nullomers than expected based on simulated genomes. This evidence supports that nullomers are typically under negative selection, with a subset of nullomers either not resurfacing or resurfacing with low probability through common variants, indicating their deleterious nature.

4.3. Applications of absent sequences

4.3.1. Pathogen and cancer detection

Nullomers have been used in examining sequence composition differences between organisms, profiling the evolutionary pressures of prokaryotic and eukaryotic genomes, and constructing phylogenetic frameworks (Fig. 3A) [167], [177]. In the context of the 2014 Ebola virus outbreak, Silva et al. introduced a novel subset of MAWs termed minimal relative absent words (RAWs) that are derived from a pathogen genome, but absent in its host, to analyze distinctions among Ebola virus sequences associated with the outbreak, sequences from other Ebola virus species, historical outbreak sequences, and the human genome [160]. Three minimal RAWs were identified as conserved pathogen-specific signatures with implications for rapid diagnostics and targeted therapeutic interventions. The authors suggested using these specific RAWs to design primers for identifying Ebola virus infections or distinguishing between Ebola virus species or outbreaks, highlighting the potential use of k-mer-based solutions for enhancing the precision of pathogen detection. Similarly, Pratas & Silva analyzed SARS-CoV-2 genomes from the outbreak in late 2019 and revealed the presence of RAWs of 12 or 13 base pairs persistent across all SARS-CoV-2 genomes and absent from the human genome and transcriptome [178]. These are minimal signatures of the SARS-CoV-2 genome that distinguish it from other human coronavirus species.

Fig. 3.

Applications of absent sequences. A. The nullomer profile of individuals may be used to characterize populations. B. Nullomers have been detected in cell-free DNA and utilized for cancer detection. C. Immunogenicity is associated with nullomers and nullpeptides, and both have been used in vaccines to increase efficacy and decrease autoimmune reactions. D. Certain nullpeptides exhibit cytotoxic effects and have cancer-killing properties. E. Nullomers and primes can be used for barcoding purposes to label or differentiate samples, as they are absent from one or more organisms. F. Quasi-primes can serve as species-specific biomarkers. Their use as universal fingerprints has potential for real-time organismal detection, understanding evolution as well as for biosecurity.

Beyond pathogen detection, nullomers and nullpeptides have emerged as diagnostic tools for cancer detection and show promise as sensitive and highly specific biomarkers (Fig. 3B). A study by Georgakopoulos-Soares et al. introduced neomers, a subset of nullomers largely absent from healthy human genomes but emerging recurrently in the tumor genome due to somatic mutations, as a potential tool for early cancer detection using cell-free DNA (Table 1; Fig. 1C) [29]. The authors analyzed over 2500 whole-genome sequencing tumor samples and demonstrated that a neomer-based classifier can accurately distinguish the tissue of origin between twenty-one tumor types. Neomers were utilized to sensitively and specifically identify cancer from a limited amount of plasma in four cancer types (prostate, lung, ovarian, colorectal) and therefore offer a potential tool for liquid biopsies and early-stage cancer detection.

In an analysis of over 10,000 Whole Exome Sequencing tumor samples, Montgomery et al. identified nullomer emerging mutational hotspots within cancer genes and used cell-free RNA from peripheral blood samples to demonstrate the utility of nullomers in classifying diverse tumor types (AUC > 0.90) [22]. Tsiatsianis et al. examined the emergence of nullpeptides during cancer development, finding multiple recurrently emerging nullpeptides in cancer genes in patients [28]. The study identified specific hotspots for nullpeptides within the loci of oncogenes and tumor suppressors and observed that recurrent nullpeptides are more frequently associated with neoantigens, implicating them as potentially effective targets for immunotherapy. This suggests that nullpeptides could serve as valuable indicators for prioritizing candidates for immunotherapeutic interventions in cancer treatment.

4.3.2. Vaccine development

In light of the observed correlation between the strength of immune response and the prevalence of specific amino acid sequences in nature [179], [180], it is hypothesized that nullomers and nullpeptides display heightened immunogenicity and therefore reinforce host defenses against pathogens (Fig. 3C) [181], [182], [183]. The absence of specific five and six amino acid peptide combinations in publicly available proteome sequences has previously been identified [184], [185], and there is potential enhancement of immunomodulatory effects and immunogenicity when incorporating these rare sequences into the development of vaccines.

Patel et al. found that rare five-mer peptides induced stronger cellular responses than common sequences [186]. Integration of these peptides into an H5N1 hemagglutinin antigen in a DNA vaccine improved immune responses, with several five-mer peptides enhancing clinical outcomes against lethal influenza virus challenges in mice and ferrets. Also, combining a five-mer peptide with a commercial Hepatitis B vaccine significantly increased anti-HBsAg antibody production in mice. The study suggested that rare or non-existent oligopeptides could be developed as immunomodulators and encouraged further evaluation of specific five-mer peptides as potential vaccine adjuvants. In pursuit of potential peptides for a SARS-CoV-2 vaccine, Santoni et al. opted for third-order nullpeptides [187]. This decision aimed to address concerns related to cross-reactivity and mitigate the risks of autoimmunity, given their lower similarity to human self.

4.3.3. Cancer therapeutics

Recent clinical investigations highlight the potential of peptide-based drugs in cancer therapy, emphasizing advantages such as their small size, specificity, efficacy across a broad spectrum of cancers, potentially selective cytotoxicity, and cost-effectiveness [188], [189]. The rationale behind k-mer-based drug design approaches proposes that injecting the smallest absent sequences into hosts can induce toxic or immune-stimulatory reactions (Fig. 3D) [31]. Alileche et al. first demonstrated the anti-neoplastic benefits of nullpeptides, including 9R and 9S1R [31]. These nullpeptides proved lethal to cancer cells. 9R and 9S1R induced mitochondrial impairment as evidenced by increased reactive oxygen species production, ATP depletion, cell growth inhibition, and cell death. Remarkably, while normal cells exhibited diminishing sensitivity to nullpeptides over time, the impact on cancer cells intensified over the same duration. In a subsequent study, Alileche et al. investigated the effects of nullpeptides 9R, 9S1R, and 124R (five-amino acid) on the NCI-60 panel of cancer cells from nine organs and four normal cell lines. The research revealed that these nullpeptides demonstrated a broad lethal effect on cancer cells, including drug- and hormone-resistant cell lines and cancer stem cells [30]. 9R and 9S1R, unlike 124R, displayed a broader activity spectrum than several existing cancer drugs. Within a three-hour timeframe, they induced substantial cellular ATP depletion, while exhibiting lower toxicity towards normal cells. These findings highlight the potential of nullpeptides as cancer treatments capable of targeting diverse cancer types and addressing cellular heterogeneity.

Ali et al. investigated the therapeutic potential of a ten amino acid nullpeptide, 9S1R-NulloPT [190]. The investigation revealed that mice treated with the peptide, in contrast to the untreated controls, demonstrated reduced tumor size during the initial treatment phase of triple-negative breast cancer. Subsequently, in later stages and in excised tumors, there was a reduction in in-vivo bioluminescence. This observation implied the presence of metabolically inactive tumors, while secondary metastasis in the lungs remained unaffected. Furthermore, the peptide treatment induced alterations in the tumor immune microenvironment, characterized by heightened immune cell infiltration and inflammation. These changes were accompanied by shifts in gene expression, marked by the downregulation of genes associated with cellular metabolism and translation machinery and the upregulation of genes linked to developmental pathways and extracellular matrix organization.

4.3.4. Engineering barcodes, PCR primers, and fingerprinting organisms

Synthetic DNA primers and molecular barcodes derived from nullomers have been employed to prevent forensic contamination (Fig. 3E) [161]. In order to mitigate the risks of cross-contamination and misidentification of reference samples in PCR analysis, Goswami et al. proposed a method that leverages nullomer barcodes as internal amplification controls to distinguish between reference and evidentiary samples in forensic DNA analysis [161]. The results demonstrated that nullomers could be integrated into the multiplex PCR reactions of forensic profiling kits and used jointly with PCR for sequencing. KeeSeek was developed to design distant non-existing k-mers for barcodes or PCR primers [191]. Unlike previously existing tools, the software produced longer sequences (up to 31 nt) and provided the distance from the reference genome in terms of the number of mismatches. With primer-like feature filters, it was optimized for targeted genomic manipulation experiments using techniques like zinc finger nucleases, TALEN, and CRISPR.

K-mers that appear in a single protein of a particular proteome have been identified and can be utilized for protein identification [192], [193]. Research on nucleic primes and quasi-primes has emphasized their potential for molecular barcodes (Fig. 3E-F) [172], [173]. Because quasi-primes are the shortest unique sequences in a species or sequence, their distinctiveness makes them ideal for applications requiring precise and unambiguous identification of a sample (Fig. 3F). However, quasi-prime identification may currently be limited by the availability of species-specific reference sequence information, variation, and sequencing errors. Future work will aim to address these limitations and explore their relationship to uniqueness and differentiation among organisms.

5. Conclusions

K-mers have influenced the fields of genomics, transcriptomics, proteomics, and biological data analysis. In this review, we explore various k-mer-based methods while highlighting their expansive applications in basic science research, translational medicine, healthcare, agriculture, biosecurity, and conservation. The discussed applications of k-mers include k-mer counting and frequency analysis, sequence alignment, genome assembly, comparative genomics, metagenomics, metaproteomics, and protein structure prediction. This broad spectrum of applications demonstrates the versatility of k-mer-based approaches in addressing biological problems, ranging from elucidating evolutionary relationships to serving as biomarkers. We emphasize the utilization of k-mer-based tools, delineating the integration of k-mers into bioinformatics methodologies as a means to enhance the analysis of vast genomic and proteomic datasets on a large scale.

Previous reviews have focused on several of the research areas covered in this review, including sequence alignment [194], [195], genome assembly [196], indexing and querying large datasets [197], and metagenomics-based methods and pipelines [6], [198]. However, we emphasize the utility of k-mer-based approaches across biological problems and review different k-mer concepts including nullomers, nullpeptides, minimal absent words, quasi-primes, and primes, as well as their applications. The review of nullomers, nullpeptides, and primes offers valuable insights into the implications of absent words for understanding genomic variation, evolution, and human diseases. Furthermore, the use of absent sequences in pathogen and cancer detection, vaccine development, cancer therapeutics, barcoding, and DNA fingerprinting accentuates their versatility in various applications. In light of recent global health crises like the SARS-CoV-2 pandemic, the significance of efficient diagnostic methods cannot be overstated. The ability to identify absent words in the genomes of pathogens such as SARS-CoV-2 not only facilitates accurate diagnosis but also paves the way for potential therapeutic interventions [178].

In conclusion, the versatility, scalability, and efficiency of k-mer-based analysis are evident across a spectrum of applications. Considering the increasing size and complexity of genomic and proteomic datasets, k-mer-based algorithms offer substantial potential to enhance biological information extraction from genetic material. Therefore, expanding k-mer-based methodologies with robust techniques may improve our understanding of biology and accelerate genetic translation into meaningful application.

Author Contributions

I.G.S. conceived and supervised the study. C.M., M.M., M.A.K., C.C., I.M., A.M., N.C., G.A.P., and I.G.S. wrote the manuscript. C.M., M.M., M.A.K., C.C. and I.G.S. generated the figures and tables.

Author Statement

All authors contributed to the work and have approved this version of the manuscript to be submitted. The authors have no conflicts or financial interests to disclose. This work has not been previously submitted and is not under consideration elsewhere. All ethical guidelines were upheld in the preparation of this manuscript. We are not using any copyrighted information, identifiers, or other protected health information in this paper. No text, text boxes, or tables in this article have been previously published or owned by another party.

CRediT authorship contribution statement

Manvita Mareboina: Writing – original draft. Camille Moeckel: Conceptualization, Supervision, Visualization, Writing – original draft, Writing – review & editing. Candace S.Y. Chan: Visualization, Writing – review & editing. Maxwell A. Konnaris: Writing – original draft, Writing – review & editing. Georgios A. Pavlopoulos: Writing – review & editing, Investigation, Supervision. Austin Montgomery: Writing – review & editing. Ioannis Mouratidis: Conceptualization, Supervision, Writing – original draft, Writing – review & editing. Ilias Georgakopoulos-Soares: Conceptualization, Supervision, Visualization, Writing – original draft, Writing – review & editing, Resources. Nikol Chantzi: Writing – review & editing.

Declaration of Competing Interest

All authors declare that they have no conflicts of interest.

Appendix A. Supplementary material

Supplementary material

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