Subscribe
Issue No.04 - October-December (2010 vol.7)
pp: 763-767
István Miklós , Renyi Institute of Mathematical Sciences , Budapest
Bence Mélykúti , University of Oxford, Oxford
Krister Swenson , EPFL, Lausanne
ABSTRACT
Markov chain Monte Carlo has been the standard technique for inferring the posterior distribution of genome rearrangement scenarios under a Bayesian approach. We present here a negative result on the rate of convergence of the generally used Markov chains. We prove that the relaxation time of the Markov chains walking on the optimal reversal sorting scenarios might grow exponentially with the size of the signed permutations, namely, with the number of syntheny blocks.
INDEX TERMS
Stochastic programming, Markov processes, analysis of algorithms and problem complexity, biology and genetics.
CITATION
István Miklós, Bence Mélykúti, Krister Swenson, "The Metropolized Partial Importance Sampling MCMC Mixes Slowly on Minimum Reversal Rearrangement Paths", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.7, no. 4, pp. 763-767, October-December 2010, doi:10.1109/TCBB.2009.26
REFERENCES
 [1] Y. Ajana, J.-F. Lefebvre, E.R.M. Tillier, and N. El-Mabrouk, "Exploring the Set of All Minimal Sequences of Reversals—An Application to Test the Replication-Directed Reversal Hypothesis," Lecture Notes in Computer Science, vol. 2452, pp. 300-315, Springer-Verlag, 2002. [2] D.J. Aldous, "Some Inequalities for Reversible Markov Chains," J. London Math. Soc., vol. 2, no. 25, pp. 564-576, 1982. [3] 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, 2001. [4] M. Bader and E. Ohlebusch, "Sorting by Weighted Reversals, Transpositions and Inverted Transpositions," Proc. 10th Ann. Int'l Conf. Research in Computational Biology (RECOMB '06), A. Apostolico, C. Guerra, S. Istrail, P. Pevzner, and M. Waterman, eds., pp. 563-577, 2006 and J. Computational Biology, vol. 14, no. 5, pp. 615-636, June 2007, doi: 10.1089/cmb.2007.R006. [5] V. Bafna and A. Pevzner, "Sorting by Transpositions," SIAM J. Discrete Math., vol. 11, no. 2, pp. 224-240, 1998. [6] A. Bergeron, "A Very Elementary Presentation of the Hannenhalli-Pevzner Theory," Proc. 12th Ann. Symp. Combinatorial Pattern Matching (CPM '01), A. Amir and G.M. Landau, eds., pp. 106-117, 2001. [7] A. Bergeron, C. Chauve, T. Hartman, and K. St-Onge, "On the Properties of Sequences of Reversals that Sort a Signed Permutation," Proc. JOBIM '02, pp. 99-107, 2002. [8] A. Bergeron, J. Mixtacki, and J. Stoye, "A Unifying View of Genome Rearrangements," Proc. Workshop Algorithms in Bioinformatics (WABI '06), pp 163-173, 2006. [9] P. Berman, S. Hannenhalli, and M. Karpinski, "1.375-Approximation Algorithm for Sorting by Reversals," Proc. European Symp. Algorithms (ESA '02), pp. 200-210, 2002. [10] M. Blanchette, T. Kunisawa, and D. Sankoff, "Parametric Genome Rearrangement," Gene, vol. 172, pp. GC11-GC17, 1996. [11] A. Braga, M.F. Sagot, C. Scornavacca, and E. Tannier, "The Solution Space of Sorting by Reversals," Lecture Notes in Bioinformatics, vol. 4463, pp. 293-304, Springer-Verlag 2007. [12] A. Caprara, "Formulations and Hardness of Multiple Sorting by Reversals," Proc. Third Ann. Int'l. Conf. Research in Computational Molecular Biology, pp. 84-94, 1999. [13] A. Darling, I. Miklós, and M. Ragan, "Dynamics of Genome Rearrangement in Bacterial Populations," PLoS Genetics, vol. 4, no. 7., p. e1000128. [14] R. Durrett, R. Nielsen, and T.L. York, "Bayesian Estimation of Genomic Distance," Genetics, vol. 166, pp. 621-629, 2004. [15] N. Eriksen, "(1+$\varepsilon$ )-Approximation of Sorting by Reversals and Transpositions," Proc. Workshop Algorithms in Bioinformatics (WABI '01), pp. 227-237, 2001. [16] C.J. Geyer, "Markov Chain Monte Carlo Maximum Likelihood," Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, E.M. Keramidas and S.M. Selma, eds., pp. 156-163, Interface Foundation, 1991. [17] Q.-P. Gu, S. Peng, and H.I. Sudborough, "A 2-Approximation Algorithm for Genome Rearrangements by Reversals and Transpositions," Theoretical Computer Science, vol. 210, no. 2, pp. 327-339, 1999. [18] S. Hannenhalli, "Polynomial Algorithm for Computing Translocation Distance between Genomes," Proc. Combinatorial Pattern Matching (CPM '96), pp. 168-185, 1996. [19] S. Hannenhalli and P.A. Pevzner, "Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals," J. Assoc. for Computing Machinery, vol. 46, no. 1, pp. 1-27, 1999. [20] W.K. Hastings, "Monte Carlo Sampling Methods using Markov Chains and Their Applications," Biometrika, vol. 57, no. 1, pp. 97-109, 1970. [21] H. Kaplan, R. Shamir, and R. Tarjan, "A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals," SIAM J. Computing, vol. 29, no. 3, pp. 880-892, 1999. [22] J.D. Kececioglu and D. Sankoff, "Exact and Approximation Algorithms for Sorting by Reversals, with Application to Genome Rearrangement," Algorithmica, vol. 13, pp. 180-210, 1995. [23] B. Larget, D.L. Simon, and B.J. Kadane, "Bayesian Phylogenetic Inference from Animal Mitochondrial Genome Arrangements," J. Royal Statistical Soc. Series B, vol. 64, no. 4, pp. 681-695, 2002. [24] B. Larget, D.L. Simon, J.B. Kadane, and D. Sweet, "A Bayesian Analysis of Metazoan Mitochondrial Genome Arrangements," Molecular Biology and Evolution, vol. 22, no. 3, pp. 486-495, 2005. [25] J.S. Liu, "Monte Carlo Strategies in Scientific Computing," Springer Series in Statistics. Springer 2001. [26] G.A. Lunter, I. Miklós, A.J. Drummond, J.L. Jensen, and J.J. Hein, "Bayesian Coestimation of Phylogeny and Sequence Alignment," BMC Bioinformatics, vol. 6, article no. 83, 2005. [27] G.F. Lawler and A.D. Sokal, "Bounds on the $L^2$ Spectrum for Markov Chains and Markov Processes: A Generalization of Cheeger's Inequality," Trans. Am. Math. Soc., vol. 309, no. 2, pp. 557-580, 1988. [28] B. Mélykúti, "The Mixing Rate of Markov Chain Monte Carlo Methods and Some Applications of MCMC Simulation in Bioinformatics," MSc thesis, Eötvös Loránd Univ., http://ramet.elte.hu/~miklosi/MScMelykuti_thesis.pdf , 2006. [29] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, "Equations of State Calculations by Fast Computing Machines," J. Chemical Physics, vol. 21, no. 6, pp. 1087-1091, 1953. [30] I. Miklós, "MCMC Genome Rearrangement," Bioinformatics, vol. 19, pp. ii130-ii137, 2003. [31] I. Miklós and A. Darling, "Efficient Sampling of Parsimonious Inversion Histories with Application to Genome Rearrangement in Yersinia," Genome Biology and Evolution, vol. 1, no. 1, pp. 153-164, 2009. [32] I. Miklós and J. Hein, "Genome Rearrangement in Mitochondria and its Computational Biology," Proc. Second RECOMB Satellite Workshop Computational Genomics, pp. 85-96, 2005. [33] I. Miklós, P. Ittzés, and J. Hein, "ParIS Genome Rearrangement Server," Bioinformatics, vol. 21, no. 6, pp. 817-820, 2005. [34] J.H. Nadau and B.A. Taylor, "Lengths of Chromosome Segments Conserved since Divergence of Man and Mouse," Proc. Nat'l Academy of Sciences USA, vol. 81, pp. 814-818, 1984. [35] J. von Neumann, "Various Techniques used in Connection with Random Digits," Nat'l. Bureau of Standards Applied Math. Series, vol. 12, pp. 36-38, 1951. [36] J.D. Palmer and L.A. Herbon, "Plant Mitochondrial DNA Evolves Rapidly in Structure, but Slowly in Sequence," J. Molecular Evolution, vol. 28, pp. 87-97, 1988. [37] F. Ronquist and J.P. Huelsenbeck, "MrBayes 3: Bayesian Phylogenetic Inference under Mixed Models," Bioinformatics, vol. 19, pp. 1572-1574, 2003. [38] A. Siepel, "An Algorithm to Find all Sorting Reversals," Proc. Sixth Ann. Int'l Conf. Research in Computational Biology (RECOMB '02), pp. 281-290, 2002. [39] D. Simon and B. Larget, Bayesian Analysis to Describe Genomic Evolution by Rearrangement (BADGER), Version 1.01 Beta, Dept. of Math. Computer Science, Duquesne Univ., 2004. [40] A.H. Sturtevant and E. Novitski, "The Homologies of Chromosome Elements in the Genus Drosophila," Genetics, vol. 26, pp. 517-541, 1941. [41] E. Tannier and M.-F. Sagot, "Sorting by Reversals in Subquadratic Time," Proc. 15th Ann. Symp. Combinatorial Pattern Matching (CPM '04), pp. 1-13, 2004. [42] T.L. York, R. Durrett, and R. Nielsen, "Bayesian Estimation of Inversions in the History of Two Chromosomes," J. Computational Biology, vol. 9, pp. 808-818, 2002. [43] J.D. Watson and F.H.C. Crick, "Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid," Nature, vol. 171, pp. 737-738, 1953.