Abstract
Within this paper we investigate the Bernoulli model for random secondary structures of ribonucleic acid (RNA) molecules. Assuming that two random bases can form a hydrogen bond with probability p we prove asymptotic equivalents for the averaged number of hairpins and bulges, the averaged loop length, the expected order, the expected number of secondary structures of size n and order k and further parameters all depending on p. In this way we get an insight into the change of shape of a random structure during the process \(1\mathop \to \limits^p 0\). Afterwards we compare the computed parameters for random structures in the Bernoulli model to the corresponding quantities for real existing secondary structures of large subunit rRNA molecules found in the database of Wuyts et al. That is how it becomes possible to identify those parameters which behave (almost) randomly and those which do not and thus should be considered as interesting, e.g., with respect to the biological functions or the algorithmic prediction of RNA secondary structures.
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References
Abramowitz, M. and I. A. Stegun (1970). Handbook of Mathematical Functions, Dover.
Evers, D. J. and R. Giegerich (2001). Reducing the Conformation Space in RNA Structure Prediction, German Conference on Bioinformatics.
Flajolet, P., X. Gourdon and P. Dumas (1995). Mellin transforms and asymptotics: harmonic sums. Theor. Comput. Sci. 144, 3–58.
Flajolet, P. and A. Odlyzko (1990). Singularity analysis of generating functions. SIAM J. Disc. Math. 3, 216–240.
Flajolet, P. and R. Sedgewick (1993). The Average case analysis of algorithms: complex asymptotics and generating functions. INRIA rapport de recherche 2026.
Fontana, W., D. A. M. Konings, P. F. Stadler and P. Schuster (1993). Statistics of RNA secondary structures. Biopolymers 33, 1389–1404.
Higgs, P. G. (1993). RNA secondary structure: a comparison of real and random sequences. J. Phys. I 3, 43–59.
Hofacker, I. L., P. Schuster and P. F. Stadler (1998). Combinatorics of RNA secondary structures. Discrete Appl. Math. 88, 207–237.
Horton, R. E. (1945). Erosioned development of streams and their drainage basins, hydrophysical approach to quantitative morphology. Bull. Geol. Soc. Am. 56, 275–370.
Howell, J. A., T. F. Smith and M. S. Waterman (1980). Computation of generating functions for biological molecules. SIAM J. Appl. Math. 39, 119–133.
Lesk, A. M. (1974). A combinatorial study of the effects of admitting non-Watson-Crick base pairings and of base compositions on the helix-forming potential of polynucleotides of random sequences. J. Theor. Biol. 44, 7–17.
Mainville, S. (1981). Comparaisons et Auto-comparaisons de Chaînes Finies, PhD thesis, Université de Montréal, Canada.
McCaskill, J. S. (1990). The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers 29, 1105–1119.
Gô, M. (1967). Statistical mechanics of biopolymers and its application to the melting transition of polynucleotides. J. Phys. Soc. Japan 23, 597–608.
Nebel, M. E. (2002a). A unified approach to the analysis of Horton-Strahler parameters of binary tree structures. Random Struct. Algorithms 21, 252–277.
Nebel, M. E. (2002b). Combinatorial properties of RNA secondary structures. J. Comput. Biol. 9, 541–573.
Nebel, M. E. (2002c). On a statistical filter for RNA secondary structures. Frankfurter Informatik-Berichte 5.
Pipas, J. M. and J. E. McMahon (1975). Method for predicting RNA secondary structures. Proc. Natl. Acad. Sci., USA 72, 2017–2021.
Régnier, M., Generating Functions in Computational Biology, invited to MABS’97, 1998.
Schmitt, W. R. and M. S. Waterman (1994). Linear trees and RNA secondary structure. Discrete Appl. Math. 51, 317–323.
Schuster, P. and P. Stadler (2000). Discrete models of biopolymers, in Handbook of Computational Chemistry and Biology, J. Crabbe, A. Konopka and M. Drew (Eds), Marcel Dekker, Inc.
Sprinzl, M., K. S. Vassilenko, J. Emmerich, and F. Bauer (1999). Compilation of tRNA sequences and sequences of tRNA genes, (20 December, 1999), Available from http://www.uni-bayreuth.de/departments/biochemie/trna/.
Stein, P. R. and M. S. Waterman (1978). On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math. 26, 261–272.
Strahler, A. N. (1952). Hypsometric (area-altitude) analysis of erosonal topology. Bull. Geol. Soc. Am. 63, 1117–1142.
Viennot, G. and M. Vauchaussade de Chaumont (1985). Enumeration of RNA secondary structures by complexity, Mathematics in medicine and biology, Lecture Notes in Biomathematics, 57, pp. 360–365.
Waterman, M. S. (1978a). Secondary structure of single-stranded nucleic acids. Adv. Math. Suppl. Stud. 1, 167–212.
Waterman, M. S. (1978b). Combinatorics of RNA hairpins and cloverleaves. Stud. Appl. Math. 1, 91–96.
Waterman, M. S. and T. F. Smith (1978). RNA secondary structures: a complete mathematical analysis. Math. Biosci. 42, 257–266.
Wuyts, J., P. De Rijk, Y. Van de Peer, T. Winkelmans and R. De Wachter (2001). The European large subunit ribosomal RNA database. Nucleic Acids Res. 29, 175–177.
Zaks, S. (1980). Lexicographic generation of ordered trees. Theor. Comput. Sci. 10, 63–82.
Zeilberger, D. (1990). A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number. Discrete Math. 82, 89–92.
Zuker, M. and D. Sankoff (1984). RNA secondary structures and their prediction. Bull. Math. Biol. 46, 591–621.
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Nebel, M.E. Investigation of the Bernoulli model for RNA secondary structures. Bull. Math. Biol. 66, 925–964 (2004). https://doi.org/10.1016/j.bulm.2003.08.015
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DOI: https://doi.org/10.1016/j.bulm.2003.08.015