Bulletin of Mathematical Biology

, Volume 80, Issue 12, pp 3227–3246 | Cite as

Position and Content Paradigms in Genome Rearrangements: The Wild and Crazy World of Permutations in Genomics

  • Sangeeta BhatiaEmail author
  • Pedro Feijão
  • Andrew R. Francis
Education Article


Modellers of large-scale genome rearrangement events, in which segments of DNA are inverted, moved, swapped, or even inserted or deleted, have found a natural syntax in the language of permutations. Despite this, there has been a wide range of modelling choices, assumptions and interpretations that make navigating the literature a significant challenge. Indeed, even authors of papers that use permutations to model genome rearrangement can struggle to interpret each others’ work, because of subtle differences in basic assumptions that are often deeply ingrained (and consequently sometimes not even mentioned). In this paper, we describe the different ways in which permutations have been used to model genomes and genome rearrangement events, presenting some features and limitations of each approach, and show how the various models are related. This paper will help researchers navigate the landscape of permutation-based genome rearrangement models and make it easier for authors to present clear and consistent models.


Genome rearrangement Permutation Symmetric group Double cut and join Inversion 



This research was partly undertaken during reciprocal visits of PF to Western Sydney University and ARF to Bielefeld University. The authors acknowledge the support of these institutions.


  1. Alekseyev MA, Pevzner PA (2008) Multi-break rearrangements and chromosomal evolution. Theor Comput Sci 395(2–3):193–202MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bafna V, Pevzner PA (1993) Genome rearrangements and sorting by reversals. In: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp 148–157Google Scholar
  3. Bafna V, Pevzner PA (1998) Sorting by transpositions. SIAM J Discrete Math 11(2):224–240MathSciNetzbMATHCrossRefGoogle Scholar
  4. Baudet C, Dias U, Dias Z (2015) Sorting by weighted inversions considering length and symmetry. BMC Bioinform 16(19):S3CrossRefGoogle Scholar
  5. Bergeron A, Mixtacki J, Stoye J (2006) A unifying view of genome rearrangements. In: Bücher P, Moret BME (eds) Algorithms in bioinformatics. Springer, Berlin, pp 163–173CrossRefGoogle Scholar
  6. Bhatia S, Egri-Nagy A, Francis AR (2015) Algebraic double cut and join. J Math Biol 71(5):1149–1178MathSciNetzbMATHCrossRefGoogle Scholar
  7. Caprara A (1997) Sorting by reversals is difficult. In: Proceedings of the first annual international conference on computational molecular biology. ACM, pp 75–83Google Scholar
  8. Chen T, Skiena SS (1996) Sorting with fixed-length reversals. Discrete Appl Math 71(1):269–295MathSciNetzbMATHCrossRefGoogle Scholar
  9. Darling ACE, Mau B, Blattner FR, Perna NT (2004) Mauve: multiple alignment of conserved genomic sequence with rearrangements. Genome Res 14(7):1394–1403CrossRefGoogle Scholar
  10. Dias Z, Meidanis J (2001) Genome rearrangements distance by fusion, fission, and transposition is easy. In: Proceedings of the 8th international symposium on string processing and information retrieval (SPIRE2001), SPIRE 2001. Citeseer, pp 250–253Google Scholar
  11. Dobzhansky T, Sturtevant AH (1938) Inversions in the chromosomes of Drosophila pseudoobscura. Genetics 23(1):28Google Scholar
  12. Doignon JP, Labarre A (2007) On Hultman numbers. J Integer Seq 10:1–13Google Scholar
  13. Egri-Nagy A, Francis AR, Gebhardt V (2014a) Bacterial genomics and computational group theory: the BioGAP package for GAP. In: International congress on mathematical software. Springer, Berlin pp 67–74Google Scholar
  14. Egri-Nagy A, Gebhardt V, Tanaka MM, Francis AR (2014b) Group-theoretic models of the inversion process in bacterial genomes. J Math Biol 69(1):243–265MathSciNetzbMATHCrossRefGoogle Scholar
  15. Feijão P, Meidanis J (2013) Extending the algebraic formalism for genome rearrangements to include linear chromosomes. IEEE/ACM Trans Comput Biol Bioinform 10(4):819–831CrossRefGoogle Scholar
  16. Hannenhalli S, Pevzner PA (1995) Transforming men into mice (polynomial algorithm for genomic distance problem). In: Proceedings of 1995 IEEE 36th annual foundations of computer science, pp 581–592Google Scholar
  17. Hannenhalli S, Pevzner PA (1999) Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. JACM 46(1):1–27MathSciNetzbMATHCrossRefGoogle Scholar
  18. Huang Y-L, Huang C-C, Tang CY, Lu CL (2010) An improved algorithm for sorting by block-interchanges based on permutation groups. Inf Process Lett 110(8–9):345–350. ISSN 00200190
  19. Kececioglu JD, Ravi R (1995) Of mice and men: algorithms for evolutionary distances between genomes with translocation. In: Symposium on discrete algorithms, vol 604Google Scholar
  20. Labarre A (2013) Lower bounding edit distances between permutations. SIAM J Discrete Math 27(3):1410–1428. ISSN 0895-4801
  21. Labarre A, Cibulka J (2011) Polynomial-time sortable stacks of burnt pancakes. Theor Comput Sci 412(8–10):695–702. ISSN 03043975
  22. Meidanis J, Dias Z (2000) An alternative algebraic formalism for genome rearrangements. In: Sankoff D, Nadeau JH (eds) Comparative genomics. Springer, Berlin, pp 213–223CrossRefGoogle Scholar
  23. Meyer M, Munzner T, Pfister H (2009) MizBee: a multiscale synteny browser. IEEE Trans Vis Comput Graphics 15(6):897–904CrossRefGoogle Scholar
  24. Moulton V, Steel M (2012) The ‘Butterfly effect’ in Cayley graphs with applications to genomics. J Math Biol 65(6–7):1267-84. ISSN 1432-1416
  25. Revanna KV, Munro D, Gao A, Chiu C-C, Pathak A, Dong Q (2012) A web-based multi-genome synteny viewer for customized data. BMC Bioinform 13(1):190CrossRefGoogle Scholar
  26. Sankoff D, Leduc G, Antoine N, Paquin B, Lang BF, Cedergren R (1992) Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome. Proc Natl Acad Sci 89(14):6575–6579CrossRefGoogle Scholar
  27. Serdoz S, Egri-Nagy A, Sumner J, Holland BR, Jarvis PD, Tanaka MM, Francis AR (2017) Maximum likelihood estimates of pairwise rearrangement distances. J Theor Biol 423:31–40MathSciNetzbMATHCrossRefGoogle Scholar
  28. Solomon A, Sutcliffe P, Lister R (2003) Sorting circular permutations by reversal. In: Workshop on algorithms and data structures, pp 319–328. Springer, BerlinGoogle Scholar
  29. Sumner JG, Jarvis PD, Francis AR (2017) A representation-theoretic approach to the calculation of evolutionary distance in bacteria. J Phys A: Math Theor 50(33):335601MathSciNetzbMATHCrossRefGoogle Scholar
  30. Swenson KM, Simonaitis P, Blanchette M (2016) Models and algorithms for genome rearrangement with positional constraints. Algorithms Mol Biol 11(1):13CrossRefGoogle Scholar
  31. Tannier E, Zheng C, Sankoff D (2009) Multichromosomal median and halving problems under different genomic distances. BMC Bioinform 10:120. ISSN 1471-2105
  32. Watterson GA, Ewens WJ, Hall TE, Morgan A (1982) The chromosome inversion problem. J Theor Biol 99(1):1–7CrossRefGoogle Scholar
  33. Yancopoulos S, Attie O, Friedberg R (2005) Efficient sorting of genomic permutations by translocation, inversion and block interchange. Bioinformatics 21(16):3340–3346CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Centre for Research in MathematicsWestern Sydney UniversitySydneyAustralia
  2. 2.School of Public HealthImperial CollegeLondonUK
  3. 3.Faculty of TechnologyUniversität BielefeldBielefeldGermany

Personalised recommendations