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The statistics of k-mers from a sequence undergoing a simple mutation process without spurious matches

Antonio Blanca, Robert S. Harris, View ORCID ProfileDavid Koslicki, View ORCID ProfilePaul Medvedev
doi: https://doi.org/10.1101/2021.01.15.426881
Antonio Blanca
1Department of Computer Science and Engineering, The Pennsylvania State University
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Robert S. Harris
2Department of Biology, The Pennsylvania State University
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David Koslicki
1Department of Computer Science and Engineering, The Pennsylvania State University
2Department of Biology, The Pennsylvania State University
3Huck Institutes of the Life Sciences, The Pennsylvania State University
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  • ORCID record for David Koslicki
Paul Medvedev
1Department of Computer Science and Engineering, The Pennsylvania State University
3Huck Institutes of the Life Sciences, The Pennsylvania State University
4Department of Biochemistry and Molecular Biology, The Pennsylvania State University
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  • ORCID record for Paul Medvedev
  • For correspondence: pzm11@psu.edu
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Abstract

K-mer-based methods are widely used in bioinformatics, but there are many gaps in our understanding of their statistical properties. Here, we consider the simple model where a sequence S (e.g. a genome or a read) undergoes a simple mutation process whereby each nucleotide is mutated independently with some probability r, under the assumption that there are no spurious k-mer matches. How does this process affect the k-mers of S? We derive the expectation and variance of the number of mutated k-mers and of the number of islands (a maximal interval of mutated k-mers) and oceans (a maximal interval of non-mutated k-mers). We then derive hypothesis tests and confidence intervals for r given an observed number of mutated k-mers, or, alternatively, given the Jaccard similarity (with or without minhash). We demonstrate the usefulness of our results using a few select applications: obtaining a confidence interval to supplement the Mash distance point estimate, filtering out reads during alignment by Minimap2, and rating long read alignments to a de Bruijn graph by Jabba.

Competing Interest Statement

The authors have declared no competing interest.

Footnotes

  • * Authors are listed in alphabetical order

  • † This is the full version of the paper of the same title appearing in the proceedings of RECOMB 2021.

  • Corrected typos in Lemma 1 and added acknowledgments.

Copyright 
The copyright holder for this preprint is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY 4.0 International license.
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Posted February 09, 2021.
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The statistics of k-mers from a sequence undergoing a simple mutation process without spurious matches
Antonio Blanca, Robert S. Harris, David Koslicki, Paul Medvedev
bioRxiv 2021.01.15.426881; doi: https://doi.org/10.1101/2021.01.15.426881
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The statistics of k-mers from a sequence undergoing a simple mutation process without spurious matches
Antonio Blanca, Robert S. Harris, David Koslicki, Paul Medvedev
bioRxiv 2021.01.15.426881; doi: https://doi.org/10.1101/2021.01.15.426881

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