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Issue No.06 - November/December (2011 vol.8)
pp: 1468-1482
Markus E. Nebel , University of Kaiserslautern, Kaiserslautern
Anika Scheid , University of Kaiserslautern, Kaiserslautern
ABSTRACT
There are two custom ways for predicting RNA secondary structures: minimizing the free energy of a conformation according to a thermodynamic model and maximizing the probability of a folding according to a stochastic model. In most cases, stochastic grammars are used for the latter alternative applying the maximum likelihood principle for determining a grammar's probabilities. In this paper, building on such a stochastic model, we will analyze the expected minimum free energy of an RNA molecule according to Turner's energy rules. Even if the parameters of our grammar are chosen with respect to structural properties of native molecules only (and therefore, independent of molecules' free energy), we prove formulae for the expected minimum free energy and the corresponding variance as functions of the molecule's size which perfectly fit the native behavior of free energies. This gives proof for a high quality of our stochastic model making it a handy tool for further investigations. In fact, the stochastic model for RNA secondary structures presented in this work has, for example, been used as the basis of a new algorithm for the (nonuniform) generation of random RNA secondary structures.
INDEX TERMS
RNA folding, RNA secondary structure, RNA structure prediction, free energy, generating functions.
CITATION
Markus E. Nebel, Anika Scheid, "Analysis of the Free Energy in a Stochastic RNA Secondary Structure Model", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.8, no. 6, pp. 1468-1482, November/December 2011, doi:10.1109/TCBB.2010.126
REFERENCES
[1] M.S. Waterman, “Secondary Structure of Single-Stranded Nucleic Acids,” Advances in Math. Supplementary Studies, vol. 1, pp. 167-212, 1978.
[2] P.R. Stein and M.S. Waterman, “On Some New Sequences Generalizing the Catalan and Motzkin Numbers,” Discrete Math., vol. 26, pp. 216-272, 1978.
[3] G. Viennot and M. Vauchaussade de Chaumont, “Enumeration of RNA Secondary Structures by Complexity,” Mathematics in Medicine and Biology, vol. 57, pp. 360-365, 1985.
[4] M.E. Nebel, “Combinatorial Properties of RNA Secondary Structures,” J. Computational Biology, vol. 9, no. 3, pp. 541-574, 2002.
[5] I.L. Hofacker, P. Schuster, and P.F. Stadler, “Combinatorics of RNA Secondary Structures,” Discrete Applied Math., vol. 88, pp. 207-237, 1998.
[6] M.E. Nebel, “Investigation of the Bernoulli-Model of RNA Secondary Structures,” Bull. Math. Biology, vol. 66, pp. 925-964, 2004.
[7] M. Zuker and D. Sankoff, “RNA Secondary Structures and Their Prediction,” Bull. Math. Biology, vol. 46, pp. 591-621, 1984.
[8] M.E. Nebel, “On a Statistical Filter for RNA Secondary Structures,” technical report, Frankfurter Informatik-Berichte, 2002.
[9] M.E. Nebel, “Identifying Good Predictions of RNA Secondary Structure,” Proc. Pacific Symp. Biocomputing, pp. 423-434, 2004.
[10] M. Zuker, D.H. Mathews, and D.H. Turner, “Algorithms and Thermodynamics for RNA Secondary Structure Prediction: A Practical Guide,” RNA Biochemistry and Biotechnology, J. Barciszewski and B.F.C. Clark, eds., pp. 11-43, Kluwer Academic Publishers, 1999.
[11] M. Zuker, “RNA Folding Prediction: The Continued Need for Interaction between Biologists and Mathematicians,” Lectures on Mathematics in the Life Sciences, vol. 17, pp. 87-124, 1986.
[12] D. Sankoff, J.B. Kruskal, S. Mainville, and R.J. Cedergren, “Fast Algorithms to Determine RNA Secondary Structures Containing Multiple Loops,” Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, ch. 3, pp. 93-120, Addison-Wesley, 1983.
[13] B. Knudsen and J. Hein, “RNA Secondary Structure Prediction Using Stochastic Context-Free Grammars and Evolutionary History,” Bioinformatics, vol. 15, no. 6, pp. 446-454, 1999.
[14] B. Knudsen and J. Hein, “Pfold: RNA Secondary Structure Prediction Using Stochastic Context-Free Grammars,” Nucleic Acids Research, vol. 31, no. 13, pp. 3423-3428, 2003.
[15] R. Nussinov, G. Pieczenik, J.R. Griggs, and D.J. Kleitman, “Algorithms for Loop Matchings,” SIAM J. Applied Math., vol. 35, pp. 68-82, 1978.
[16] M. Zuker and P. Stiegler, “Optimal Computer Folding of Large RNA Sequences Using Thermodynamics and Auxiliary Information,” Nucleic Acids Research, vol. 9, pp. 133-148, 1981.
[17] M. Zuker, “Computer Prediction of RNA Structure,” RNA Processing, J.E. Dahlberg and J.N. Abelson, eds., vol. 180, pp. 262-288, Academic Press, 1989.
[18] S. Wuchty, W. Fontana, I. Hofacker, and P. Schuster, “Complete Suboptimal Folding of RNA and the Stability of Secondary Structures,” Biopolymers, vol. 49, pp. 145-165, 1999.
[19] M. Zuker, “On Finding All Suboptimal Foldings of an RNA Molecule,” Science, vol. 244, pp. 48-52, 1989.
[20] M. Zuker, “Mfold Web Server for Nucleic Acid Folding and Hybridization Prediction,” Nucleic Acids Research, vol. 31, no. 13, pp. 3406-3415, 2003.
[21] I.L. Hofacker, “The Vienna RNA Secondary Structure Server,” Nucleic Acids Research, vol. 31, no. 13, pp. 3429-3431, 2003.
[22] J. Gralla and D.M. Crothers, “Free Energy of Imperfect Nucleic Acid Helices: II. Small Hairpin Loops,” J. Molecular Biology, vol. 73, pp. 497-511, 1973.
[23] P.N. Borer, B. Dengler, I. Tinoco,Jr., and O.C. Uhlenbeck, “Stability of Ribonucleic Acid Double-Stranded Helices,” J. Molecular Biology, vol. 86, pp. 843-853, 1974.
[24] T. Xia, J. SantaLucia,Jr., M.E. Burkard, R. Kierzek, S.J. Schroeder, X. Jiao, C. Cox, and D.H. Turner, “Thermodynamic Parameters for an Expanded Nearest-Neighbor Model for Formation of RNA Duplexes with Watson-Crick Base Pairs,” Biochemistry, vol. 37, pp. 14719-14735, 1998.
[25] D.H. Mathews, J. Sabina, M. Zuker, and D.H. Turner, “Expanded Sequence Dependence of Thermodynamic Parameters Improves Prediction of RNA Secondary Structure,” J. Molecular Biology, vol. 288, pp. 911-940, 1999.
[26] M.J. Serra and D.H. Turner, “Predicting Thermodynamic Properties of RNA,” Methods in Enzymology, vol. 259, pp. 242-261, 1995.
[27] Y. Sakakibara, M. Brown, R. Hughey, I.S. Mian, K. Sjölander, R.C. Underwood, and D. Haussler, “Stochastic Context-Free Grammars for tRNA modeling,” Nucleic Acids Research, vol. 22, pp. 5112-5120, 1994.
[28] T. Huang and K.S. Fu, “On Stochastic Context-Free Languages,” Information Sciences, vol. 3, pp. 201-224, 1971.
[29] T. Chi and S. Geman, “Estimation of Probabilistic Context-Free Grammars,” Computational Linguistics, vol. 24, no. 2, pp. 299-305, 1998.
[30] N. Chomsky and M.P. Schützenberger, “The Algebraic Theory of Context-Free Languages,” Computer Programming and Formal Systems, P. Braffort and D. Hirschberg, eds., pp. 118-161, Amsterdam North-Holland, 1963.
[31] R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, second ed. Addison-Wesley Publishing Company, Inc., Sept. 2001.
[32] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, Jan. 2009.
[33] M. Sprinzl, C. Horn, M. Brown, A. Ioudovitch, and S. Steinberg, “Compilation of tRNA Sequences and Sequences of tRNA Genes,” Nucleic Acids Research, vol. 26, pp. 148-153, 1998.
[34] M. Szymanski, M.Z. Barciszewska, V.A. Erdmann, and J. Barciszewski, “5S Ribosomal RNA Database,” Nucleic Acids Research, vol. 30, pp. 176-178, 2002.
[35] J. Wuyts, Y.V. de Peer, T. Winkelmans, and R.D. Wachter, “The European Database on Small Subunit Ribosomal RNA,” Nucleic Acids Research, vol. 30, no. 1, pp. 183-185, 2002.
[36] J. Wuyts, P.D. Rijk, Y.V. de Peer, T. Winkelmans, and R.D. Wachter, “The European Large Subunit Ribosomal RNA Database,” Nucleic Acids Research, vol. 29, no. 1, pp. 175-177, 2001.
[37] F. Weinberg and M.E. Nebel, “Non Uniform Generation of Combinatorial Objects,” technical report, University of Kaiserslautern, 2010.
[38] M.E. Nebel and A. Scheid, “Random Generation of RNA Secondary Structures According to Native Distributions,” submitted for publication.
[39] M. Hofri, Analysis of Algorithms: Computational Methods and Mathematical Tools, Oxford Univ. Press, 1995.
[40] D.E. Knuth and H.S. Wilf, “A Short Proof of Darboux's Lemma,” Applied Math. Letters, vol. 2, pp. 139-140, 1989.
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