Abstract
Suppose a finite set X is repeatedly transformed by a sequence of permutations of a certain type acting on an initial element x to produce a final state y. For example, in genomics applications, X could be a set of genomes and the permutations certain genome ‘rearrangements’ or, in group theory, X could be the set of configurations of a Rubik’s cube and the permutations certain specified moves. We investigate how ‘different’ the resulting state y′ to y can be if a slight change is made to the sequence, either by deleting one permutation, or replacing it with another. Here the ‘difference’ between y and y′ might be measured by the minimum number of permutations of the permitted type required to transform y to y′, or by some other metric. We discuss this first in the general setting of sensitivity to perturbation of walks in Cayley graphs of groups with a specified set of generators. We then investigate some permutation groups and generators arising in computational genomics, and the statistical implications of the findings.
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References
Alon N, Spencer J (1992) The probabilistic method. Wiley, New York
Bafna V, Pevzner PA (1996) Genome rearrangements and sorting by reversals. SIAM J Comput 25(2): 272–289
Bergeron A, Mixtacki J, Stoye J (2009) A new linear time algorithm to compute the genomic distance via the double cut and join distance. Theor Comput Sci 410(51): 5300–5316
Chen T, Skiena S (1996) Sorting with fixed-length reversals. Discret Appl Math 71: 269–295
Chin LL, Ying CL, Yen LH, Chuan YT (2007) Analysis of genome rearrangement by block-interchanges. Methods Mol Biol 396: 121–134
Daskalakis C, Mossel E, Roch S (2010) Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture. Probab Theor Relat Fields 149: 149–189
Eppstein DBA (1992) Word processing in groups. A K Peters/CRC Press, New York
Erdös PL, Steel MA, Székely LA, Warnow T (1999) A few logs suffice to build (almost) all trees (part 1). Rand Struct Alg 14(2): 153–184
Evans SN, Speed TP (1993) Invariants of some probability models used in phylogenetic inference. Ann Stat 21: 355–377
Fertin G, Labarre A, Rusu I, Tannier E, Vialette S (2009) Combinatorics of genome rearrangements. The MIT Press, Cambridge
Gronau I, Moran S, Snir S (2008) Fast and reliable reconstruction of phylogenetic trees with very short edges. In: SODA: ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, Philadelphia, pp. 379–388
Hannenhalli S, Pevzner PA (1999) Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations via reversals. J Assoc Comput Mach 46(1): 1–27
Hilborn RC (2004) Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics. Am J Phys 72(4): 425–427
Holmgren R (1994) A first course in discrete dynamical systems, 2nd edn. Springer, New York
Kececioglu JD, Sankoff D (1995) Exact and approximate algorithms for sorting by reversals with application to genome rearrangement. Algorithmica 13: 180–210
Kimura M (1981) Estimation of evolutionary distances between homologous nucleotide sequences. Proc Natl Acad Sci USA 78: 454–458
Kostantinova E (2008) Some problems on Cayley graphs. Linear Algebra Appl. 429: 2754–2769
Kunkle D, Cooperman G (2009) Harnessing parallel disks to solve Rubik’s cube. J Symb Comput 44(7): 872–890
Labarre L (2006) New bounds and tractable instances for the transposition distance. IEEE/ACM Trans Comput Biol Bioinf 3(4): 380–394
Mossel E, Steel M (2005) How much can evolved characters tell us about the tree that generated them? In: Gascuel O (ed) Mathematics of evolution and phylogeny. Oxford University Press, Oxford, pp 384–412
Pevzner P (2000) Computational molecular biology. MIT Press, Cambridge
Rotman JJ (1995) An introduction to the theory of groups. Springer, New York
Saitou N, Nei M (1987) The neighbor-joining method: a new method for reconstructing phylogenetic trees. Mol Biol Evol 4(4): 406–425
Sankoff D, Blanchette M (1997) The median problem for breakpoints in comparative genomics. Computing and Combinatorics, Shanghai, pp 251–263
Sankoff D, Blanchette M (1998) Multiple genome rearrangement and breakpoint phylogeny. J Comput Biol 5: 555–570
Setubal J, Meidanis M (1997) Introduction to computational molecular biology. PWS Publishing Company, Boston
Semple C, Steel M (2003) Phylogenetics. Oxford University Press, Oxford
Sinha A, Meller J (2008) Sensitivity analysis for reversal distance and breakpoint re-use in genome rearrangements. Pac J Biocomput 13: 37–48
Steele J.M. (1986) An Efron-Stein inequality for nonsymmetric statistics. Ann Stat 14(2): 753–758
Trifonov V, Rabadan R (2010) Frequency analysis techniques for identification of viral genetic data. mBio 1(3): e00156-10
Wang L-S (2002) Genome rearrangement phylogeny using weighbor. In: Lecture Notes for Computer Sciences No. 2452. Proceedings for the second workshop on algorithms in bioinformatics (WABI’02), Rome, pp 112–125
Wang L-S, Warnow T (2005) Distance-based genome rearrangement phylogeny. In: Gascuel O (eds) Mathematics of evolution and phylogeny. Oxford University Press, Oxford, pp 353–380
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VM thanks the Royal Society for supporting his visit to University of Canterbury, where most of this work was undertaken. MS thanks the Royal Society of New Zealand under its James Cook Fellowship scheme.
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Moulton, V., Steel, M. The ‘Butterfly effect’ in Cayley graphs with applications to genomics. J. Math. Biol. 65, 1267–1284 (2012). https://doi.org/10.1007/s00285-011-0498-1
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DOI: https://doi.org/10.1007/s00285-011-0498-1