Abstract
Statistical properties of RNA folding landscapes obtained by the partition function algorithm (McCaskill 1990) are investigated in detail. The pair correlation of free energies as a function of the Hamming distance is used as a measure for the ruggedness of the landscape. The calculation of the partition function contains information about the entire ensemble of secondary structures as a function of temperature and opens the door to all quantities of thermodynamic interest, in contrast with the conventional minimal free energy approach. A metric distance of structure ensembles is introduced and pair correlations at the level of the structures themselves are computed. Just as with landscapes based on most stable secondary structure prediction, the landscapes defined on the full biophysical GCAU alphabet are much smoother than the landscapes restricted to pure GC sequences and the correlation lengths are almost constant fractions of the chain lengths. Correlation functions for multi-structure landscapes exhibit an increased correlation length, especially near the melting temperature. However, the main effect on evolution is rather an effective increase in sampling for finite populations where each sequence explores multiple structures.
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References
Cech TR (1988) Conserved sequences and structures of group I introns: building an active site for RNA catalysis. Gene 73:259–271
Cech TR (1990) Self-splicing of group I introns. Annu Rev Biochem 59:543–568
Eigen M (1971) Self-organization of matter and the evolution of macromolecules. Naturwissenschaften 10:465–523
Eigen M, Schuster P (1977) The hypercycle: a principle of natural self-organization A. Naturwissenschaften 64:541–565
Eigen M, McCaskill JS, Schuster P (1988) The molecular quasispecies. J Phys Chem 92:6881–6891
Eigen M, McCaskill JS, Schuster P (1989) The molecular quasispecies. Adv Chem Phys 75:149–263
Fontana W, Schuster P (1987) A computer model of evolutionary optimization. Biophys Chem 26:123–147
Fontana W, Schnabl W, Schuster P (1989) Physical aspects of evolutionary optimization and adaptation. Phys Rev A 40:3301–3321
Fontana W, Griesmacher T, Schnabl W, Stadler PF Schuster P (1991) Statistics of landscapes based on free energies replication and degradation rate constants of RNA secondary structures. Mh Chem 122:795–819
Fontana W, Konings DAM, Stadler PF, Schuster P (1992a) Statistics of RNA secondary structures. (Santa Fe Institute Preprint No. 92-02-008). Biopolymers (in press)
Fontana W, Stadler PF, Bornberg-Bauer EG, Griesmacher T, Hofacker IL, Tacker M, Tarazona P, Weinberger ED, Schuster P (1992 b) RNA folding and combinatory landscapes. Phys Rev E (in press)
Freier SM, Kierzek R, Jaeger JA, Sugimoto N, Caruthers MH, Neilson T, Turner DH (1986) Improved free-energy parameters for predictions of RNA duplex stability. Biochemistry 83:9373–9377
Hamming RW (1986) Coding and Information Theory (2nd ed.) Prentice Hall, Englewood Cliffs, NJ
Hogeweg P, Hesper P (1984) Energy directed folding of RNA sequences. Nucl Acid Res 12:67–74
Jaeger JA, Turner DH, Zuker M (1989) Improved predictions of secondary structures for RNA. Biochemistry 86:7706–7710
Kauffman SA, Levin S (1987) Towards a general theory of adaptive walks on rugged landscapes. J Theor Biol 128:11–45
Kauffman SA, Weinberger ED, Perelson AS (1988) Maturation of the immune response via adaptive walks on affinity landscapes. In: Theoretical Immunology, Part I (Santa Fe Institute Studies in the Sciences of Complexity) Perelson AS (ed). Addison-Wesley, Reading, Mass
Konings DAM (1989) Pattern analysis of RNA secondary structure. Proefschrift, Rijksuniversiteit to Utrecht
Konings DAM, Hogeweg P (1989) Pattern analysis of RNA secondary structure, similarity and consensus of minimal-energy folding. J Mol Biol 207:597–614
Le SY, Zuker M (1990) Common structures of the 5′ non-coding RNA in enteroviruses and rhinoviruses: thermodynamical stability and statistical significance. J Mol Biol 261:729–741
Macken CA, Perelson AS (1989) Protein evolution on rugged landscapes. Proc Natl Acad Sci 86:6191–6195
Major F, Turcotte M, Gautheret D, Lapalme G, Fillion E, Cedergren R (1991) The combination of symbolic and numerical computation for three-dimensional modeling of RNA. Science 253:1255–1260
Maynard-Smith J (1970) Natural selection and the concept of a protein space. Nature 225:563–564
McCaskill JS (1984a) A localization threshold for macromolecular quasispecies from continuously distributed replication rates. J Chem Phys 80:5194–5202
McCaskill JS (1984b) A stochastic theory of macromolecular evolution. Biol Cybern 50:63–73
McCaskill JS (1990) The equilibrium partition function and base pair binding probabilities for RNA secondary structures. Biopolymers 29:1105–1119
Noller HF, Hoffarth V, Zimniak L (1992) Unusual resistence of peptidyl transferase to protein extraction procedures. Science 256:1416–1419
Nowak M, Schuster P (1989) Error thresholds for replication in finite populations. Mutation frequencies and the onset of Muller's ratchet. J Theor Biol 137:375–395
Perelson AS, Oster GF (1979) Theoretical studies of clonal selection: minimal antibody and reliability of self- and non-self discrimination. J Theor Biol 81:645–670
Peritz AE, Kierzek R, Sugimoto N, Turner DH (1991) Thermodynamic study of internal loops in oligoribonucleotides: symmetric loops are more stable than asymmetric loops. Biochemistry 30:6428–6436
Piccirilli JA, McConnell TS, Zaug AJ, Noller HF, Cech TR (1992) Aminoacyl esterase activity of the tetrahymena ribozyme. Science 256:1420–1424
Sankoff D, Morin AM, Cedergren RJ (1988) The evolution of 5S RNA secondary structures. Can J Biochem 56:440–443
Schuster P, Swetina J (1988) Stationary mutant distributions and evolutionary optimization. Bull Math Biol 50:636–660
Shapiro BA (1988) An algorithm for comparing multiple RNA secondary structures. CABIOS 4:381–393
Shapiro BA, Zhang K (1990) Comparing multiple RNA secondary structures using tree comparisons. CABIOS 6:309–318
Stadler PF, Schnabl W (1992) The landscape of the travelling problem. Phys Lett A 161:337–344
Sugimoto N, Kierzek R, Turner DH (1987a) Sequence dependence for the energetics of dangling ends and terminal base pairs in ribonucleic acid. Biochemistry 26:4554–4558
Sugimoto N, Kierzek R, Turner DH (1987b) Sequence dependence for the energetics of terminal mismatches in ribooligonucleotides. Biochemistry 26:4559–4561
Swetina J, Schuster P (1982) Selfreplication with errors. A model for polynuclcotide replication. Biophys Chem 16:329–353
Tinoco I, Uhlenbeck OC, Levine MD (1971) Estimation of secondary structure in ribonucleic acids. Nature 230:362–367
Turner DH, Sugimoto N, Freier S (1988) RNA structure prediction. Ann Rev Biophys Chem 17:167–192
Waterman MS (1978) Secondary structure of single-stranded nucleic acids. Adv Math Suppl Studies 1:167–212
Waterman MS (1984) General methods of sequence comparison. Bull Math Biol 46:473–500
Waterman MS, Byers TH (1985) A dynamic programming algorithm to find all solutions in a neighborhood of the optimum. Math Biosc 77:179–188
Waterman MS, Smith TF (1978) RNA secondary structure: A complete mathematical analysis. Math Biosc 42:257–266
Weinberger ED (1990) Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol Cybern 63:325–336
Weinberger ED (1991) Local properties in the N-k model, a tuneably rugged energy landscape. Phys Rev A 44:6399–6413
Weinberger ED, Stadler PF (1992) Why some fitness landscapes are fractal. Submitted to J Theor Biol 1992
Wright S (1932) The role of mutation, inbreeding, cross-breeding, and selection in evolution. In: Proc 6th Int Congress on Genetics Vol. 21, pp. 356–366
Zuker M, Sankoff D (1984) RNA secondary structures and their prediction. Bull Math Biol 46:591–621
Zuker M, Stiegler P (1981) Optimal computer folding of large RNA sequences using thermodynamic and auxiliary information. Nucl Acid Res 9:133–148
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Correspondence to: P. Schuster
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Bonhoeffer, S., McCaskill, J.S., Stadler, P.F. et al. RNA multi-structure landscapes. Eur Biophys J 22, 13–24 (1993). https://doi.org/10.1007/BF00205808
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DOI: https://doi.org/10.1007/BF00205808