Skip to main content

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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  • Alekseyev MA, Pevzner PA (2008) Multi-break rearrangements and chromosomal evolution. Theor Comput Sci 395(2–3):193–202

    MathSciNet  MATH  Article  Google Scholar 

  • Bafna V, Pevzner PA (1993) Genome rearrangements and sorting by reversals. In: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp 148–157

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

    MathSciNet  MATH  Article  Google Scholar 

  • Baudet C, Dias U, Dias Z (2015) Sorting by weighted inversions considering length and symmetry. BMC Bioinform 16(19):S3

    Article  Google Scholar 

  • 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 163–173

    Chapter  Google Scholar 

  • Bhatia S, Egri-Nagy A, Francis AR (2015) Algebraic double cut and join. J Math Biol 71(5):1149–1178

    MathSciNet  MATH  Article  Google Scholar 

  • Caprara A (1997) Sorting by reversals is difficult. In: Proceedings of the first annual international conference on computational molecular biology. ACM, pp 75–83

  • Chen T, Skiena SS (1996) Sorting with fixed-length reversals. Discrete Appl Math 71(1):269–295

    MathSciNet  MATH  Article  Google Scholar 

  • Darling ACE, Mau B, Blattner FR, Perna NT (2004) Mauve: multiple alignment of conserved genomic sequence with rearrangements. Genome Res 14(7):1394–1403

    Article  Google Scholar 

  • 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 250–253

  • Dobzhansky T, Sturtevant AH (1938) Inversions in the chromosomes of Drosophila pseudoobscura. Genetics 23(1):28

    Google Scholar 

  • Doignon JP, Labarre A (2007) On Hultman numbers. J Integer Seq 10:1–13

    Google Scholar 

  • 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 67–74

  • 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):243–265

    MathSciNet  MATH  Article  Google Scholar 

  • Feijão P, Meidanis J (2013) Extending the algebraic formalism for genome rearrangements to include linear chromosomes. IEEE/ACM Trans Comput Biol Bioinform 10(4):819–831

    Article  Google Scholar 

  • 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 581–592

  • Hannenhalli S, Pevzner PA (1999) Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. JACM 46(1):1–27

    MathSciNet  MATH  Article  Google Scholar 

  • 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. https://doi.org/10.1016/j.ipl.2010.03.003. ISSN 00200190

  • Kececioglu JD, Ravi R (1995) Of mice and men: algorithms for evolutionary distances between genomes with translocation. In: Symposium on discrete algorithms, vol 604

  • Labarre A (2013) Lower bounding edit distances between permutations. SIAM J Discrete Math 27(3):1410–1428. https://doi.org/10.1137/13090897X. ISSN 0895-4801

  • Labarre A, Cibulka J (2011) Polynomial-time sortable stacks of burnt pancakes. Theor Comput Sci 412(8–10):695–702. https://doi.org/10.1016/j.tcs.2010.11.004. ISSN 03043975

  • Meidanis J, Dias Z (2000) An alternative algebraic formalism for genome rearrangements. In: Sankoff D, Nadeau JH (eds) Comparative genomics. Springer, Berlin, pp 213–223

    Chapter  Google Scholar 

  • Meyer M, Munzner T, Pfister H (2009) MizBee: a multiscale synteny browser. IEEE Trans Vis Comput Graphics 15(6):897–904

    Article  Google Scholar 

  • Moulton V, Steel M (2012) The ‘Butterfly effect’ in Cayley graphs with applications to genomics. J Math Biol 65(6–7):1267-84. https://doi.org/10.1007/s00285-011-0498-1. ISSN 1432-1416

  • 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):190

    Article  Google Scholar 

  • 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):6575–6579

    Article  Google Scholar 

  • 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 423:31–40

    MathSciNet  MATH  Article  Google Scholar 

  • Solomon A, Sutcliffe P, Lister R (2003) Sorting circular permutations by reversal. In: Workshop on algorithms and data structures, pp 319–328. Springer, Berlin

  • 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):335601

    MathSciNet  MATH  Article  Google Scholar 

  • Swenson KM, Simonaitis P, Blanchette M (2016) Models and algorithms for genome rearrangement with positional constraints. Algorithms Mol Biol 11(1):13

    Article  Google Scholar 

  • Tannier E, Zheng C, Sankoff D (2009) Multichromosomal median and halving problems under different genomic distances. BMC Bioinform 10:120. https://doi.org/10.1186/1471-2105-10-120. ISSN 1471-2105

  • Watterson GA, Ewens WJ, Hall TE, Morgan A (1982) The chromosome inversion problem. J Theor Biol 99(1):1–7

    Article  Google Scholar 

  • Yancopoulos S, Attie O, Friedberg R (2005) Efficient sorting of genomic permutations by translocation, inversion and block interchange. Bioinformatics 21(16):3340–3346

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sangeeta Bhatia.

Additional information

Research supported in part by Australian Research Council Discovery Grant DP130100248.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bhatia, S., Feijão, P. & Francis, A.R. Position and Content Paradigms in Genome Rearrangements: The Wild and Crazy World of Permutations in Genomics. Bull Math Biol 80, 3227–3246 (2018). https://doi.org/10.1007/s11538-018-0514-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-018-0514-3

Keywords

  • Genome rearrangement
  • Permutation
  • Symmetric group
  • Double cut and join
  • Inversion