Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T15:09:15.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson Approximation for the Number of Repeats in a Stationary Markov Chain

Part of: Combinatorial probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

Narjiss Touyar ,
Sophie Schbath ,
Dominique Cellier  and
Hélène Dauchel
Narjiss Touyar*
Affiliation:
Université de Rouen
Sophie Schbath*
Affiliation:
Institut National de la Recherche Agronomique
Dominique Cellier*
Affiliation:
Université de Rouen
Hélène Dauchel*
Affiliation:
Université de Rouen
*
∗Postal address: Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes EA4051, Université de Rouen, 76821 Mont Saint Aignan Cedex, France.
∗∗∗Postal address: INRA, UR1077 Mathématique, Informatique et Génome, Domaine de Vilvert, 78352 Jouy-en-Josas, France. Email address: [email protected]
∗Postal address: Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes EA4051, Université de Rouen, 76821 Mont Saint Aignan Cedex, France.
∗∗∗∗∗Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Detection of repeated sequences within complete genomes is a powerful tool to help understanding genome dynamics and species evolutionary history. To distinguish significant repeats from those that can be obtained just by chance, statistical methods have to be developed. In this paper we show that the distribution of the number of long repeats in long sequences generated by stationary Markov chains can be approximated by a Poisson distribution with explicit parameter. Thanks to the Chen-Stein method we provide a bound for the approximation error; this bound converges to 0 as soon as the length n of the sequence tends to ∞ and the length t of the repeats satisfies n2ρt = O(1) for some 0 < ρ < 1. Using this Poisson approximation, p-values can then be easily calculated to determine if a given genome is significantly enriched in repeats of length t.

Keywords

Poisson approximationnumber of repeatsChen-Stein methodMarkov chainDNA sequence

MSC classification

Secondary: 62E17: Approximations to distributions (nonasymptotic) 60C05: Combinatorial probability
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 440 - 455
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Arratia, R., Martin, D., Reinert, G. and Waterman, M. (1996). Poisson process approximation for sequence repeats and sequencing by hybridization. J. Comput. Biol. 3, 425–463.Google Scholar
[2] Barbour, A., Holst, L. and Janson, S. (1992). Poisson Approximation. Clarendon Press, Oxford.Google Scholar
[3] Benson, G. (1999). Tandem repeats finder: a program to analyze DNA sequences. Nucleic Acids Res. 27, 573–580.Google Scholar
[4] Delcher, A. L. et al. (1999). Alignment of whole genomes. Nucleic Acids Res. 27, 2369–2376.Google Scholar
[5] Kolpakov, R., Bana, G. and Kucherov, G. (2003). Mreps: efficient and flexible detection of tandem repeats in DNA. Nucleic Acids Res. 31, 3672–3678.Google Scholar
[6] Kurtz, S. et al. (2001). Reputer: the manifold applications of repeat analysis on a genomic scale. Nucleic Acids Res. 29, 4633–4642.Google Scholar
[7] Lefèbvre, A., Lecroq, T., Dauchel, H. and Alexandre, J. (2003). FORRepeats: detects repeats on entire chromosomes and between genomes. Bioinformatics 19, 319–326.CrossRefGoogle ScholarPubMed
[8] Reinert, G., Schbath, S. and Waterman, M. (2000). Probabilistic and statistical properties of words: an overview. J. Comput. Biol. 7, 1–46.Google Scholar
[9] Robin, S., Rodolphe, F. and Schbath, S. (2005). DNA, Words and Models. Cambridge University Press.Google Scholar
[10] Taylor, J.S. and Raes, J. (2004). Duplication and divergence: the evolution of new genes and old ideas. Ann. Rev. Genet. 38, 615–643.Google Scholar