Journal of Mathematical Biology

, Volume 69, Issue 2, pp 469–500 | Cite as

Robust \(k\)-mer frequency estimation using gapped \(k\)-mers

  • Mahmoud Ghandi
  • Morteza Mohammad-Noori
  • Michael A. BeerEmail author


Oligomers of fixed length, \(k\), commonly known as \(k\)-mers, are often used as fundamental elements in the description of DNA sequence features of diverse biological function, or as intermediate elements in the constuction of more complex descriptors of sequence features such as position weight matrices. \(k\)-mers are very useful as general sequence features because they constitute a complete and unbiased feature set, and do not require parameterization based on incomplete knowledge of biological mechanisms. However, a fundamental limitation in the use of \(k\)-mers as sequence features is that as \(k\) is increased, larger spatial correlations in DNA sequence elements can be described, but the frequency of observing any specific \(k\)-mer becomes very small, and rapidly approaches a sparse matrix of binary counts. Thus any statistical learning approach using \(k\)-mers will be susceptible to noisy estimation of \(k\)-mer frequencies once \(k\) becomes large. Because all molecular DNA interactions have limited spatial extent, gapped \(k\)-mers often carry the relevant biological signal. Here we use gapped \(k\)-mer counts to more robustly estimate the ungapped \(k\)-mer frequencies, by deriving an equation for the minimum norm estimate of \(k\)-mer frequencies given an observed set of gapped \(k\)-mer frequencies. We demonstrate that this approach provides a more accurate estimate of the \(k\)-mer frequencies in real biological sequences using a sample of CTCF binding sites in the human genome.


DNA sequence Oligomer \(k\)-mer Frequency estimation  Statistical learning 

Mathematics Subject Classification

92D20 Protein sequences DNA sequences 92-08 Computational Methods 15A09 Matrix inversion generalized inverses 



We thank the reviewers for their comments and suggestions which significantly improved the manuscript. We also thank users of online community, specifically users Joriki and Siva for their useful comments which helped us in the development of the proof. Dongwon Lee graciously provided the processed CTCF sequence data. The research of M.M. was in part supported by a grant from IPM (No. CS1390-4-07), and M.B. was supported by the Searle Scholars Program and in part by NIH grant NS062972.

Supplementary material

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Supplementary material 1 (r 7 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mahmoud Ghandi
    • 1
    • 2
  • Morteza Mohammad-Noori
    • 3
    • 4
  • Michael A. Beer
    • 1
    Email author
  1. 1.Department of Biomedical Engineering and McKusick-Nathans Institute of Genetic MedicineJohns Hopkins UniversityBaltimoreUSA
  2. 2.Broad InstituteCambridgeUSA
  3. 3.School of Mathematics, Statistics and Computer ScienceUniversity of TehranTehranIran
  4. 4.School of Computer ScienceInstitute for Research in Fundamental SciencesTehranIran

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