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Bulletin of Mathematical Biology

, Volume 80, Issue 6, pp 1514–1538 | Cite as

The Rainbow Spectrum of RNA Secondary Structures

  • Thomas J. X. Li
  • Christian M. Reidys
Original Article

Abstract

In this paper, we analyze the length spectrum of rainbows in RNA secondary structures. A rainbow in a secondary structure is a maximal arc with respect to the partial order induced by nesting. We show that there is a significant gap in this length spectrum. We shall prove that there asymptotically almost surely exists a unique longest rainbow of length at least \(n-O(n^{1/2})\) and that with high probability any other rainbow has finite length. We show that the distribution of the length of the longest rainbow converges to a discrete limit law and that, for finite k, the distribution of rainbows of length k becomes for large n a negative binomial distribution. We then put the results of this paper into context, comparing the analytical results with those observed in RNA minimum free energy structures, biological RNA structures and relate our findings to the sparsification of folding algorithms.

Keywords

Secondary structure Rainbow Length spectrum Gap Arc Generating function Singularity analysis 

Mathematics Subject Classification

05A16 92E10 92B05 

Notes

Acknowledgements

We would like to thank the reviewers for their comments and suggestions and specifically for pointing out a gap in the proof of Lemma 1. We gratefully acknowledge the help of Kevin Shinpaugh and the computational support team at BI. Many thanks to Christopher L. Barrett and Henning Mortveit for discussions. The second author is a Thermo Fisher Scientific Fellow in Advanced Systems for Information Biology and acknowledges their support of this work.

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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Biocomplexity Institute of Virginia TechBlacksburgUSA

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