Subscribe
Issue No.05 - September/October (2011 vol.8)
pp: 1411-1416
Simona Grusea , Université de Toulouse, Toulouse
ABSTRACT
We use the finite Markov chain embedding technique to obtain the distribution of the number of cycles in the breakpoint graph of a random uniform signed permutation. This further gives a very good approximation of the distribution of the reversal distance between two random genomes.
INDEX TERMS
Markov processes, probabilistic algorithms, distribution functions, biology and genetics.
CITATION
Simona Grusea, "On the Distribution of the Number of Cycles in the Breakpoint Graph of a Random Signed Permutation", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.8, no. 5, pp. 1411-1416, September/October 2011, doi:10.1109/TCBB.2010.123
REFERENCES
 [1] D.A. Bader, B.M.E. Moret, and M. Yan, “A Linear-Time Algorithm for Computing Inversion Distance between Signed Permutations with an Experimental Study,” J. Computational Biology, vol. 8, no. 5, pp. 483-491, Oct. 2001. [2] V. Bafna and P. Pevzner, “Genome Rearrangements and Sorting by Reversals,” SIAM J. Computing, vol. 25, no. 2, pp. 272-289, Feb. 1996, doi:10.1137/S0097539793250627. [3] A. Bergeron, C. Chauve, and Y. Gingras, “Formal Models of Gene Clusters,” Bioinformatics Algorithms: Techniques and Applications, I. Mandoiu and A. Zelikovsky, eds., Wiley Series of Bioinformatics, 2008. [4] A. Bergeron, S. Corteel, and M. Raffinot, “The Algorithmic of Gene Teams,” Proc. Int'l Workshop Algorithms in Bioinformatics, pp. 464-476, Jan. 2002, doi:10.1007/3-540-45784-4_36. [5] A. Bergeron, J. Mixtacki, and J. Stoye, “A Unifying View of Genome Rearrangements,” Proc. Int'l Workshop Algorithms in Bioinformatics, vol. 4175, pp. 163-173, Sept. 2006, doi:10.1007/11851561_16. [6] G. Blin and J. Stoye, “Finding Nested Common Intervals Efficiently,” Proc. Int'l Workshop Comparative Genomics, pp. 59-69, Sept. 2009, doi:10.1007/978-3-642-04744-2_6. [7] S. Bocker, K. Jahn, J. Mixtacki, and J. Stoye, “Computation of Median Gene Clusters,” J. Computational Biology, vol. 16, no. 8, pp. 1085-1099, Aug. 2009, doi:10.1089/cmb.2009.0098. [8] M. Bona and R. Flynn, “The Average Number of Block Interchanges Needed to Sort a Permutation and a Recent Result of Stanley,” Information Processing Letters, vol. 109, no. 16, pp. 927-931, July 2009, doi:10.1016/j.ipl.2009.04.019. [9] A. Caprara, “On the Tightness of the Alternating-Cycle Lower Bound for Sorting by Reversals,” J. Combinatorial Optimization, vol. 3, nos. 2/3, pp. 149-182, July 1999, doi:10.1023/A:1009838309166. [10] V. Choi, C. Zheng, Q. Zhu, and D. Sankoff, “Algorithms for the Extraction of Synteny Blocks from Comparative Maps,” Proc. Int'l Workshop Algorithms in Bioinformatics, pp. 277-288, Aug. 2007, doi:10.1007/978-3-540-74126-8_26. [11] D.A. Christie, “Sorting Permutations by Block-Interchanges,” Information Processing Letters, vol. 60, no. 4, pp. 165-169, Nov. 1996, doi:10.1016/S0020-0190(96)00155-X. [12] S. Corteel, G. Louchard, and R. Pemantle, “Common Intervals in Permutations,” Discrete Math. and Theoretical Computer Science, vol. 8, no. 1, pp. 189-216, 2006. [13] E. Danchin and P. Pontarotti, “Statistical Evidence for a More than 800-Million-Year-Old Evolutionarily Conserved Genomic Region in Our Genome,” J. Molecular Evolution, vol. 59, no. 5, pp. 587-597, Nov. 2004. [14] J-P. Doignon and A. Labarre, “On Hultman Numbers,” J. Integer Sequences, vol. 10, article no. 07.6.2, 2007. [15] D. Durand and D. Sankoff, “Tests for Gene Clustering,” J. Computational Biology, vol. 10, nos. 3/4, pp. 453-482, June 2003, doi:10.1089/10665270360688129. [16] J.C. Fu and M.V. Koutras, “Distribution Theory of Runs: A Markov Chain Approach,” J. Am. Statistical Assoc., vol. 89, no. 427, pp. 1050-1058, Sept. 1994. [17] S. Grusea, “Measures for the Exceptionality of Gene Order in Conserved Genomic Regions,” Advances in Applied Math., vol. 45, no. 3, pp. 359-372, Sept. 2010, doi:10.1016/j.aam.2010.02.002. [18] S. Hannenhalli and P. Pevzner, “Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals,” J. ACM, vol. 46, no. 1, pp. 1-27, Jan. 1999. [19] R. Hoberman and D. Durand, “The Incompatible Desiderata of Gene Cluster Properties,” Proc. Int'l Workshop Comparative Genomics, pp. 73-87, Sept. 2005, doi:10.1007/11554714. [20] R. Hoberman, D. Sankoff, and D. Durand, “The Statistical Analysis of Spatially Clustered Genes under the Maximum Gap Criterion,” J. Computational Biology, vol. 12, no. 8, pp. 1083-1102, Oct. 2005, doi:10.1089/cmb.2005.12.1083. [21] J. Kececioglu and D. Sankoff, “Exact and Approximation Algorithms for Sorting by Reversals, with Application to Genome Rearrangement,” Algorithmica, vol. 13, nos. 1/2, pp. 180-210, Feb. 1995, doi:10.1007/BF01188586. [22] Z. Li, L. Wang, and K. Zhang, “Algorithmic Approaches for Genome Rearrangement: A Review,” IEEE Trans. Systems, Man and Cybernetics, Part C, vol. 36, no. 5, pp. 636-648, Sept. 2006. [23] J. Nadeau and B. Taylor, “Lengths of Chromosomal Segments Conserved since Divergence of Man and Mouse,” Proc. Nat'l Academy of Sciences USA, vol. 81, pp. 814-818, 1984. [24] P. Pevzner and G. Tesler, “Genome Rearrangements in Mammalian Evolution: Lessons from Human and Mouse Genomes,” Genome Research, vol. 13, no. 1, pp. 37-45, Jan. 2003. [25] N. Raghupathy and D. Durand, “Gene Cluster Statistics with Gene Families,” Molecular Biology and Evolution, vol. 26, no. 5, pp. 957-968, Jan. 2009, doi:10.1093/molbev/msp002. [26] D. Sankoff and L. Haque, “Power Boosts for Cluster Tests,” Proc. Int'l Workshop Comparative Genomics, pp. 121-130, Dec. 2005, doi:10.1007/11554714_11. [27] D. Sankoff and L. Haque, “The Distribution of Genomic Distance between Random Genomes,” J. Computational Biology, vol. 13, no. 5, pp. 1005-1012, June 2006, doi:10.1089/cmb.2006.13.1005. [28] K.M. Swenson, Y. Lin, V. Rajan, and B.M.E. Moret, “Hurdles Hardly Have to Be Heeded,” Proc. Int'l Workshop Comparative Genomics (RECOMB-CG '08), pp. 239-249, 2008, doi:10.1007/978-3-540-87989-3_18. [29] K.M. Swenson, V. Rajan, Y. Lin, and B.M.E. Moret, “Sorting Signed Permutations by Inversions in ${\cal O}(n \log n)$ Time,” Proc. Ann. Int'l Conf. Research in Computational Molecular Biology, pp. 386-399, May 2009, doi:10.1007/978-3-642-02008-7_28. [30] E. Tannier, A. Bergeron, and M-F. Sagot, “Advances on Sorting by Reversals,” Discrete Applied Math., vol. 155, nos. 6/7, pp. 881-888, Apr. 2007, doi:10.1016/j.dam.2005.02.033. [31] W. Xu, “The Distance between Randomly Constructed Genomes,” Proc. Fifth Asia-Pacific Bioinformatics Conf., pp. 227-236, Oct. 2006, doi:10.1142/9781860947995_0025. [32] W. Xu, C. Zheng, and D. Sankoff, “Paths and Cycles in Breakpoint Graph of Random Multichromosomal Genomes,” J. Computational Biology, vol. 14, no. 4, pp. 423-435, May 2007, doi:10.1089/cmb.2007.A004. [33] S. Yancopoulos, O. Attie, and R. Friedberg, “Efficient Sorting of Genomic Permutations by Translocation, Inversion and Block Interchange,” Bioinformatics, vol. 21, no. 16, pp. 3340-3346, Aug. 2005, doi:10.1093/bioinformatics/bti535. [34] Q. Zhu, Z. Adam, V. Choi, and D. Sankoff, “Generalized Gene Adjacencies, Graph Bandwidth, and Clusters in Yeast Evolution,” IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 6, no. 2, pp. 213-220, Apr.-June 2009.