Journal
JOURNAL OF MATHEMATICAL BIOLOGY
Volume 65, Issue 3, Pages 581-599Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00285-011-0467-8
Keywords
RNA; Boltzmann partition function; Asymptotic combinatorics; Dynamic programming
Categories
Funding
- Digiteo Foundation
- National Science Foundation [DMS-0817971, DBI-0543506, DMS-1016618]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1016618] Funding Source: National Science Foundation
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In The ends of a large RNA molecule are necessarily close, Yoffe et al. (Nucleic Acids Res 39(1):292-299, 2011) used the programs [resp. ] from Vienna RNA Package to calculate the distance between 5' and 3' ends of the minimum free energy secondary structure [resp. thermal equilibrium structures] of viral and random RNA sequences. Here, the 5'aEuro3' distance is defined to be the length of the shortest path from 5' node to 3' node in the undirected graph, whose edge set consists of edges {i, i + 1} corresponding to covalent backbone bonds and of edges {i, j} corresponding to canonical base pairs. From repeated simulations and using a heuristic theoretical argument, Yoffe et al. conclude that the 5'aEuro3' distance is less than a fixed constant, independent of RNA sequence length. In this paper, we provide a rigorous, mathematical framework to study the expected distance from 5' to 3' ends of an RNA sequence. We present recurrence relations that precisely define the expected distance from 5' to 3' ends of an RNA sequence, both for the Turner nearest neighbor energy model, as well as for a simple homopolymer model first defined by Stein and Waterman. We implement dynamic programming algorithms to compute (rather than approximate by repeated application of Vienna RNA Package) the expected distance between 5' and 3' ends of a given RNA sequence, with respect to the Turner energy model. Using methods of analytical combinatorics, that depend on complex analysis, we prove that the asymptotic expected 5'aEuro3' distance of length n homopolymers is approximately equal to the constant 5.47211, while the asymptotic distance is 6.771096 if hairpins have a minimum of 3 unpaired bases and the probability that any two positions can form a base pair is 1/4. Finally, we analyze the 5'aEuro3' distance for secondary structures from the STRAND database, and conclude that the 5'aEuro3' distance is correlated with RNA sequence length.
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