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Efficient Sampling of Transpositions and Inverted Transpositions for Bayesian MCMC

  • István Miklós
  • Timothy Brooks Paige
  • Péter Ligeti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4175)

Abstract

The evolutionary distance between two organisms can be determined by comparing the order of appearance of orthologous genes in their genomes. Above the numerous parsimony approaches that try to obtain the shortest sequence of rearrangement operations sorting one genome into the other, Bayesian Markov chain Monte Carlo methods have been introduced a few years ago. The computational time for convergence in the Markov chain is the product of the number of needed steps in the Markov chain and the computational time needed to perform one MCMC step. Therefore faster methods for making one MCMC step can reduce the mixing time of an MCMC in terms of computer running time.

We introduce two efficient algorithms for characterizing and sampling transpositions and inverted transpositions for Bayesian MCMC. The first algorithm characterizes the transpositions and inverted transpositions by the number of breakpoints the mutations change in the breakpoint graph, the second algorithm characterizes the mutations by the change in the number of cycles. Both algorithms run in O(n) time, where n is the size of the genome. This is a significant improvement compared with the so far available brute force method with O(n 3) running time and memory usage.

Keywords

Markov Chain Monte Carlo Genome Rearrangement Markov Chain Monte Carlo Method Proposal Distribution Signed Permutation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Sturtevant, A.H., Novitski, E.: The homologies of chromosome elements in the genus Drosophila. Genetics 26, 517–541 (1941)Google Scholar
  2. 2.
    Nadau, J.H., Taylor, B.A.: Lengths of chromosome segments conserved since divergence of man and mouse. PNAS 81, 814–818 (1984)CrossRefGoogle Scholar
  3. 3.
    Palmer, J.D., Herbon, L.A.: Plant mitochondrial DNA evolves rapidly in structure, but slowly in sequence. J. Mol. Evol. 28, 87–97 (1988)CrossRefGoogle Scholar
  4. 4.
    Bader, D.A., Moret, B.M.E., Yan, M.: A linear-time algorithm for computing inversion distance between signed permutations with an experimental study. J. Comp. Biol. 8(5), 483–491 (2001)CrossRefGoogle Scholar
  5. 5.
    Bergeron, A.: A very elementary presentation of the Hannenhalli-Pevzner theory. In: Amir, A., Landau, G.M. (eds.) CPM 2001. LNCS, vol. 2089, pp. 106–117. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Hannenhalli, S., Pevzner, P.A.: Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals. J. ACM 46(1), 1–27 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kaplan, H., Shamir, R., Tarjan, R.: A faster and simpler algorithm for sorting signed permutations by reversals. SIAM J. Comput. 29(3), 880–892 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Siepel, A.: An algorithm to find all sorting reversals. In: Proc. RECOMB 2002, pp. 281–290 (2002)Google Scholar
  9. 9.
    Tannier, E., Sagot, M.-F.: Sorting by reversals in subquadratic time. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Hannenhalli, S.: Polynomial algorithm for computing translocation distance between genomes. In: Hirschberg, D.S., Meyers, G. (eds.) CPM 1996. LNCS, vol. 1075, pp. 168–185. Springer, Heidelberg (1996)Google Scholar
  11. 11.
    Bafna, V., Pevzner, A.: Sorting by transpositions. SIAM J. Disc. Math. 11(2), 224–240 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Berman, P., Hannenhalli, S., Karpinski, M.: 1.375-Approximation Algorithm for Sorting by Reversals. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 200–210. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Eriksen, N.: (1+ε)-approximation of sorting by reversals and transpositions. In: Gascuel, O., Moret, B.M.E. (eds.) WABI 2001. LNCS, vol. 2149, pp. 227–237. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Gu, Q.-P., Peng, S., Sudborough, H.I.: A 2-Approximation Algorithm for Genome Rearrangements by Reversals and Transpositions. Theor. Comp. Sci. 210(2), 327–339 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kececioglu, J.D., Sankoff, D.: Exact and Approximation Algorithms for Sorting by Reversals, with Application to Genome Rearrangement. Algorithmica 13(1/2), 180–210 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Blanchette, M., Kunisawa, T., Sankoff, D.: Parametric genome rearrangement. Gene. 172, GC11–GC17 (1996)CrossRefGoogle Scholar
  17. 17.
    Bader, M., Ohlebusch, E.: Sorting by weighted reversals, transpositions and inverted transpositions. In: Apostolico, A., Guerra, C., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2006. LNCS (LNBI), vol. 3909, pp. 563–577. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Larget, B., Simon, D.L., Kadane, B.J.: Bayesian phylogenetic inference from animal mitochondrial genome arrangements. J. Royal Stat. Soc. B 64(4), 681–695Google Scholar
  19. 19.
    York, T.L., Durrett, R., Nielsen, R.: Bayesian estimation of inversions in the history of two chromosomes. J. Comp. Biol. 9, 808–818 (2002)Google Scholar
  20. 20.
    Larget, B., Simon, D.L., Kadane, J.B., Sweet, D.: A Bayesian analysis of metazoan mitochondrial genome arrangements Mol. Biol. Evol. 22(3), 486–495 (2005)CrossRefGoogle Scholar
  21. 21.
    Durrett, R., Nielsen, R., York, T.L.: Bayesian estimation of genomic distance. Genetics 166, 621–629 (2004)CrossRefGoogle Scholar
  22. 22.
    Miklós, I.: MCMC Genome Rearrangement. Bioinformatics 19, ii130–ii137 (2003)CrossRefGoogle Scholar
  23. 23.
    Miklós, I., Ittzés, P., Hein, J.: ParIS genome rearrangement server. Bioinformatics 21(6), 817–820 (2005)CrossRefGoogle Scholar
  24. 24.
    Miklós, I., Hein, J.: Genome rearrangement in mitochondria and its computational biology. In: Lagergren, J. (ed.) RECOMB-WS 2004. LNCS (LNBI), vol. 3388, pp. 85–96. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  25. 25.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1091 (1953)CrossRefGoogle Scholar
  26. 26.
    Liu, J.S.: Monte Carlo strategies in scientific computing. Springer Series in Statistics, New-York (2001)Google Scholar
  27. 27.
    Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)zbMATHCrossRefGoogle Scholar
  28. 28.
    von Neumann, J.: Various techniques used in connection with random digits. National Bureau of Standards Applied Mathematics Series 12, 36–38 (1951)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • István Miklós
    • 1
    • 3
  • Timothy Brooks Paige
    • 2
  • Péter Ligeti
    • 3
    • 4
  1. 1.eScience Regional Knowledge CenterEötvös Loránd UniversityBudapestHungary
  2. 2.Amherst CollegeAmherstUSA
  3. 3.Bioinformatics group, Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  4. 4.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary

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