Eric Coissac
931 lines
20 KiB
HTML
931 lines
20 KiB
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obikmer
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Kmers and super-kmers
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Theory
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Theory
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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Kmers and super-kmers
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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<span class="md-ellipsis">
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Kmers and super-kmers
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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Table of contents
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#kmers" class="md-nav__link">
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<span class="md-ellipsis">
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Kmers
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#super-kmers" class="md-nav__link">
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<span class="md-ellipsis">
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Super-kmers
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</span>
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</a>
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<nav class="md-nav" aria-label="Super-kmers">
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<ul class="md-nav__list">
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<li class="md-nav__item">
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<a href="#canonical-super-kmers" class="md-nav__link">
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<span class="md-ellipsis">
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Canonical super-kmers
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#expected-length-of-a-super-kmer" class="md-nav__link">
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<span class="md-ellipsis">
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Expected length of a super-kmer
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="../encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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DNA encoding
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../entropy/" class="md-nav__link">
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<span class="md-ellipsis">
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Entropy filter
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../indexing/" class="md-nav__link">
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<span class="md-ellipsis">
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Partitioning architecture
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" >
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
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<span class="md-ellipsis">
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Implementation
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_3">
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<span class="md-nav__icon md-icon"></span>
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Implementation
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../implementation/superkmer/" class="md-nav__link">
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<span class="md-ellipsis">
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SuperKmer
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../implementation/kmer/" class="md-nav__link">
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<span class="md-ellipsis">
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Kmer
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../implementation/chunkreader/" class="md-nav__link">
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<span class="md-ellipsis">
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Chunk reader
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../implementation/pipeline/" class="md-nav__link">
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<span class="md-ellipsis">
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Construction pipeline
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||
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../implementation/storage/" class="md-nav__link">
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<span class="md-ellipsis">
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||
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On-disk storage
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||
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||
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</span>
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||
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</a>
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</li>
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<li class="md-nav__item">
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||
<a href="../../implementation/mphf/" class="md-nav__link">
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||
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<span class="md-ellipsis">
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||
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MPHF selection
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||
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||
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</span>
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||
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||
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||
</a>
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||
</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
|
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||
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||
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||
<span class="md-ellipsis">
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||
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||
Architecture
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||
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||
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||
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||
</span>
|
||
|
||
|
||
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||
<span class="md-nav__icon md-icon"></span>
|
||
</label>
|
||
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||
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
|
||
<label class="md-nav__title" for="__nav_4">
|
||
<span class="md-nav__icon md-icon"></span>
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||
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||
Architecture
|
||
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||
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||
</label>
|
||
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
||
<a href="../../architecture/sequences/invariant/" class="md-nav__link">
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||
|
||
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<span class="md-ellipsis">
|
||
|
||
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Sequences
|
||
|
||
|
||
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||
</span>
|
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|
||
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||
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</a>
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</li>
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</ul>
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</nav>
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</li>
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</ul>
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Kmers
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Super-kmers
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Canonical super-kmers
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Expected length of a super-kmer
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<h1 id="kmers-and-super-kmers">Kmers and super-kmers</h1>
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<h2 id="kmers">Kmers</h2>
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<p>A <strong>kmer</strong> is a DNA subsequence of fixed length k. Two constraints govern the choice of k:</p>
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<ul>
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<li><strong>k ∈ [11, 31]</strong>: the range ensures the kmer is long enough to be specific and short enough to fit in a single machine word.</li>
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<li><strong>k is odd</strong>: an odd-length sequence cannot equal its own reverse complement (no palindromes). This guarantees that the canonical form <code>min(kmer, revcomp(kmer))</code> is always strictly defined — the two orientations are always distinct — which is required for strand-independent counting.</li>
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<h2 id="super-kmers">Super-kmers</h2>
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<p>A <strong>super-kmer</strong> is a maximal run of consecutive kmers from a DNA read, each overlapping the next by k−1 nucleotides. Each kmer of the run carries the same <strong>canonical minimizer</strong>. The <strong>canonical minimizer</strong> of a kmer is the smallest value of <code>min(m-mer, revcomp(m-mer))</code> over all m-mers within the kmer (m < k, m odd).</p>
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<h3 id="canonical-super-kmers">Canonical super-kmers</h3>
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<p>A <strong>canonical super-kmer</strong> is the lexicographic minimum of a super-kmer and its reverse complement:</p>
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<div class="highlight"><pre><span></span><code>canonical(super-kmer) = min(super-kmer, revcomp(super-kmer))
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</code></pre></div>
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<p>When a read and its reverse-complement are both sequenced, they produce super-kmers that are reverse complements of each other. Both map to the same canonical form: the same genomic region is represented by a single canonical super-kmer regardless of which strand was read.</p>
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<h3 id="expected-length-of-a-super-kmer">Expected length of a super-kmer</h3>
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<p>For a random minimizer of length m over k-mers of length k, the density of minimizer positions is approximately 2/(k−m+2) (Golan & Shur 2025; Zheng <em>et al.</em> 2020)<sup id="fnref:Zheng2020-ji"><a class="footnote-ref" href="#fn:Zheng2020-ji">2</a></sup> <sup id="fnref:Golan2025-xf"><a class="footnote-ref" href="#fn:Golan2025-xf">3</a></sup>, so the expected number of consecutive k-mers per super-kmer is (k−m+2)/2. A run of n k-mers spans n + k − 1 nucleotides, giving:</p>
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<div class="arithmatex">\[L_{\text{nt}} = \frac{k-m+2}{2} + k - 1\]</div>
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<p>For k=31, m=13: expected ≈ 40 nt. In practice super-kmers rarely exceed a few dozen nucleotides.<sup id="fnref:superkmer_length"><a class="footnote-ref" href="#fn:superkmer_length">1</a></sup></p>
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<hr />
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<ol>
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<li id="fn:superkmer_length">
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<p>The expected length formula and the density approximation 2/(k−m+2) should be verified against the values reported in (Zheng <em>et al.</em> 2020)<sup id="fnref2:Zheng2020-ji"><a class="footnote-ref" href="#fn:Zheng2020-ji">2</a></sup> and (Golan & Shur 2025)<sup id="fnref2:Golan2025-xf"><a class="footnote-ref" href="#fn:Golan2025-xf">3</a></sup>. <a class="footnote-backref" href="#fnref:superkmer_length" title="Jump back to footnote 1 in the text">↩</a></p>
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</li>
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<li id="fn:Zheng2020-ji">
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<p>Zheng, H., Kingsford, C. & Marçais, G. (2020). <a href="https://doi.org/10.1093/bioinformatics/btaa472">Improved design and analysis of practical minimizers</a>. <em>Bioinformatics (Oxford, England)</em>, 36, i119--i127. <a class="footnote-backref" href="#fnref:Zheng2020-ji" title="Jump back to footnote 2 in the text">↩</a><a class="footnote-backref" href="#fnref2:Zheng2020-ji" title="Jump back to footnote 2 in the text">↩</a></p>
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</li>
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<li id="fn:Golan2025-xf">
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<p>Golan, S. & Shur, A.M. (2025). <a href="https://doi.org/10.1007/978-3-031-82670-2\_25">Expected density of random minimizers</a>. In: <em>Lecture notes in computer science</em>, Lecture notes in computer science. Springer Nature Switzerland, Cham, pp. 347--360. <a class="footnote-backref" href="#fnref:Golan2025-xf" title="Jump back to footnote 3 in the text">↩</a><a class="footnote-backref" href="#fnref2:Golan2025-xf" title="Jump back to footnote 3 in the text">↩</a></p>
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