33 lines
2.0 KiB
Markdown
33 lines
2.0 KiB
Markdown
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# Kmers and super-kmers
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## Kmers
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A **kmer** is a DNA subsequence of fixed length k. Two constraints govern the choice of k:
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- **k ∈ [11, 31]**: the range ensures the kmer is long enough to be specific and short enough to fit in a single machine word.
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- **k is odd**: an odd-length sequence cannot equal its own reverse complement (no palindromes). This guarantees that the canonical form `min(kmer, revcomp(kmer))` is always strictly defined — the two orientations are always distinct — which is required for strand-independent counting.
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## Super-kmers
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A **super-kmer** 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 **canonical minimizer**. The **canonical minimizer** of a kmer is the smallest value of `min(m-mer, revcomp(m-mer))` over all m-mers within the kmer (m < k, m odd).
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### Canonical super-kmers
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A **canonical super-kmer** is the lexicographic minimum of a super-kmer and its reverse complement:
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```
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canonical(super-kmer) = min(super-kmer, revcomp(super-kmer))
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```
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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.
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### Expected length of a super-kmer
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For a random minimizer of length m over k-mers of length k, the density of minimizer positions is approximately 2/(k−m+2) [@Zheng2020-ji; @Golan2025-xf], 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:
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$$L_{\text{nt}} = \frac{k-m+2}{2} + k - 1$$
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For k=31, m=13: expected ≈ 40 nt. In practice super-kmers rarely exceed a few dozen nucleotides.[^superkmer_length]
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[^superkmer_length]: The expected length formula and the density approximation 2/(k−m+2) should be verified against the values reported in [@Zheng2020-ji] and [@Golan2025-xf].
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