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Efficient Sampling of Transpositions and Inverted Transpositions for Bayesian MCMC

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Algorithms in Bioinformatics (WABI 2006)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 4175))

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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.

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References

  1. Sturtevant, A.H., Novitski, E.: The homologies of chromosome elements in the genus Drosophila. Genetics 26, 517–541 (1941)

    Google Scholar 

  2. Nadau, J.H., Taylor, B.A.: Lengths of chromosome segments conserved since divergence of man and mouse. PNAS 81, 814–818 (1984)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Siepel, A.: An algorithm to find all sorting reversals. In: Proc. RECOMB 2002, pp. 281–290 (2002)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Bafna, V., Pevzner, A.: Sorting by transpositions. SIAM J. Disc. Math. 11(2), 224–240 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Blanchette, M., Kunisawa, T., Sankoff, D.: Parametric genome rearrangement. Gene. 172, GC11–GC17 (1996)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  18. Larget, B., Simon, D.L., Kadane, B.J.: Bayesian phylogenetic inference from animal mitochondrial genome arrangements. J. Royal Stat. Soc. B 64(4), 681–695

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  21. Durrett, R., Nielsen, R., York, T.L.: Bayesian estimation of genomic distance. Genetics 166, 621–629 (2004)

    Article  Google Scholar 

  22. Miklós, I.: MCMC Genome Rearrangement. Bioinformatics 19, ii130–ii137 (2003)

    Article  Google Scholar 

  23. Miklós, I., Ittzés, P., Hein, J.: ParIS genome rearrangement server. Bioinformatics 21(6), 817–820 (2005)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  26. Liu, J.S.: Monte Carlo strategies in scientific computing. Springer Series in Statistics, New-York (2001)

    Google Scholar 

  27. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)

    Article  MATH  Google Scholar 

  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 

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Miklós, I., Paige, T.B., Ligeti, P. (2006). Efficient Sampling of Transpositions and Inverted Transpositions for Bayesian MCMC. In: Bücher, P., Moret, B.M.E. (eds) Algorithms in Bioinformatics. WABI 2006. Lecture Notes in Computer Science(), vol 4175. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11851561_17

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  • DOI: https://doi.org/10.1007/11851561_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39583-6

  • Online ISBN: 978-3-540-39584-3

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