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Invariants of Graphs Modeling Nucleotide Rearrangements

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Discrete and Topological Models in Molecular Biology

Part of the book series: Natural Computing Series ((NCS))

Abstract

Nucleotide rearrangements occur in many biological systems. These genome reorganisations are especially widespread in ciliates, making these protozoans an attractive model system for experimental, computational, and theoretical studies. Rearrangements of ciliate chromosomes are modeled by the so-called assembly graphs. Edges of these graphs represent double-stranded DNA molecules, while vertices correspond to DNA recombination sites. This work is an expository article in which we discuss topological and combinatorial invariants of assembly graphs. The topological invariant, called the genus range, gives information about possible spatial arrangement of the corresponding DNA molecule. The combinatorial invariant, called the assembly polynomial, is closely related to the possible products of the rearrangement modeled by the assembly graph.

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Notes

  1. 1.

    Locally at each non-rigid vertex there are two possible connectivities of the boundary components of a ribbon surface

    and

    . However, since each squared perturbation represents a rigid vertex of \(\mathcal{T}_{n}\), only the cases where all four vertices of a squared perturbation have the same connectivity of the boundary components are considered.

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Acknowledgements

We wish to thank F. Din-Houn Lau and Kylash Rajendran, Erica Flapan, Mauro Mauricio and Julian Gibbons for insightful discussions. We are greatful to Natasha Jonoska and Masahico Saito for many useful suggestions and help with preparing the manuscript. ED has been supported by the NSF grants DMS-0900671, KV is supported by EP/G0395851.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620, USA

    Egor Dolzhenko

  2. Department of Mathematics, Imperial College London, South Kensington Campus, Office: 640, London, SW7 2AZ, UK

    Karin Valencia

Authors
  1. Egor Dolzhenko
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  2. Karin Valencia
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Corresponding author

Correspondence to Egor Dolzhenko .

Editor information

Editors and Affiliations

  1. Dept. Mathematics & Statistics, University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

  2. Dept. Mathematics & Statistics, University of South Florida, Tampa, Florida, USA

    Masahico Saito

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Dolzhenko, E., Valencia, K. (2014). Invariants of Graphs Modeling Nucleotide Rearrangements. In: Jonoska, N., Saito, M. (eds) Discrete and Topological Models in Molecular Biology. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40193-0_14

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  • DOI: https://doi.org/10.1007/978-3-642-40193-0_14

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  • Print ISBN: 978-3-642-40192-3

  • Online ISBN: 978-3-642-40193-0

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