Discrete models of biopolymers

Compact Handbook of Computational Biology

Volumn , Issue , 2004, Pages 187-221

  Schuster, Peter ^a   Stadler, Peter F ^b  

^a UNIVERSITY OF VIENNA   (Austria)

^b UNIVERSITY OF LEIPZIG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84963849303     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (137)

1. Lau KF, Dill KA. Theory for protein mutability and biogenesis. Proc Natl Acad Sci USA 1990; 87:638–642.
2. Chan HS, Dill KA. Origins of structure in globular proteins. Proc Natl Acad Sci USA 1990; 87:6388–6392.
3. Chan HS, Dill KA. Sequence space soup. J Chem Phys 1991; 95:3775–3787.
4. Crippen GM. Prediction of protein folding from amino acid sequences of discrete conformation spaces. Biochemistry 1991; 30:4232–4237.
5. Lipman DJ, Wilbur WJ. Modelling neutral and selective evolution of protein folding. Proc Roy Soc Lond B 1991; 245:7–11.
6. Camacho CJ, Thirumalai D. Minimum energy compact structures of random sequences of heteropolymers. Phys Lett 1993; 71:2505–2508.
7. Sˇ ali A, Shakhnovich E, Karplus M. Kinetics of protein folding. A lattice model study on the requirements for folding of native states. J Mol Biol 1994; 253:1614–1636.
8. Dill KA, Bromberg S, Yue K, Fiebig KM, Yeo DP, Thomas PD, Chan HS. Principles of protein folding: a perspective from simple exact models. Protein Sci 1995; 4:561–602.
9. Chan HS, Dill KA. Comparing folding codes for proteins and polymers. Proteins 1996; 24:335–344.
10. Li H, Helling R, Tang C, Wingreen N. Emergence of preferred structures in a simple model of protein folding. Science 1996; 273:666–669.
11. Renner A, Bornberg-Bauer E. Exploring the fitness landscapes of lattice proteins. In: Altman R, Dunker AK, Hunter L, Klein TE, eds. Pacific Symposium on Biocomputing’97. Singapore: World Scientific, 1997: 361–372.
12. Hart WE, Istrail S. Lattice and off-lattice side chain models of protein folding: linear time structure prediction better than 86% of optimal. J Comput Biol 1997; 4:241–259.
13. Madras N, Sokal G. The Self-Avoiding Walk. Boston: Birkhäuser, 1993.
14. Flory PJ. Principles of PolymerChemistry. Ithaca, NY:CornellUniv. Press, 1971.
15. Whittington SG. Statistical mechanics of polymer solutions and polymer adsorption. Adv Chem Phys 1982; 51:1–48.
16. Chan HS, Dill KA. Interchain loops in polymers: effects of excluded volume. J Chem Phys 1988; 90:492–508.
17. Chen S-J, Dill KA. Statistical thermodynamics of double-stranded polymer molecules. J Chem Phys 1995; 103:5802–5808.
18. Touchard J. Sur une problème de configurations et sur les fractions continues. Can J Math 1952; 4:2–25.
19. Hsieh WN. Proportions of irreducible diagrams. Studies Appl Math 1973; 52:277–283.
20. Kleitman D. Proportions of irreducible diagrams. Studies Appl Math 1970; 49:297–299.
21. Stein PR. On a class of linked diagrams, I. Enumeration. J Comb Theory A 1978; 24:357–366.
22. Stein PR, Everett CJ. On a class of linked diagrams II. Asymptotics Disc Math 1978; 22:309–318.
23. Waterman MS. Secondary structure of single-stranded nucleic acids. Adv Math Suppl Studies 1978; 1:167–212.
24. Westhof E, Jaeger L. RNA pseudoknots. Cur Opin Struct Biol 1992; 2:327–333.
25. Loria A, Pan T. Domain structure of the ribozyme from eubacterial ribonuclease. P RNA 1996; 2:551–563.
26. Konings DAM, Gutell RR. A comparison of thermodynamic foldings with comparatively derived structures of 16S and 16S-like rRNAs. RNA 1995; 1:559–574.
27. Tang CK, Draper DE. Evidence for allosteric coupling between the ribosome and repressor binding sites of a translationally regulated mRNA. Biochemistry 1990; 29:4434–4439.
28. Stadler PF, Haslinger C. RNA structures with pseudo-knots: graph-theoretical and combinatorial properties. Bull Math Biol 1999; 61:437–467.
29. Berge C. Hypergraphs. Amsterdam: Elsevier, 1989.
30. Singh RK, Tropsha A, Vaisman II. Delaunay tessellation of proteins: four body nearest neighbor propensities of amino acid residues. J Comput Biol 1996; 3:213–221.
31. Barber CB, Dobkin DP, Huhdanpaa H. The quickhull algorithm for convex hulls. ACM Trans Math Software 1996; 22:469–483.
32. Zheng W, Cho SJ, Vaisman II, Tropsha A. A new approach to protein fold recognition based on Delaunay tessellation of protein structure. In: Altman R, Dunker AK, Hunter L, Klein TE, eds. Pacific Symposium on Biocomputing’97. Singapore: World Scientific, 1997:487–496.
33. Munson PJ, Singh RK. Statistical significance of hierarchical multi-body potentials based on Delaunay tessellation and their application in sequencestructure alignment. Protein Sci 1997; 6:1467–1481.
34. Weberndorfer G, Hofacker IL, Stadler PF. An efficient potential for protein sequence design. In: Bielefeld D, ed. Computer Science in Biology. Proceedings of the GCB’99. Hannover: Univ. Bielefeld, 1999:107–112.
35. Batey RT, Rambo RP, Doudna JA. Tertiary motifs in structure and folding of RNA. Angew Chem Int Ed 1999; 38:2326–2343.
36. Tacker M, Stadler PF, Bornberg-Bauer EG, Hofacker IL, Schuster P. Algorithm independent properties of RNA structure prediction. Eur Biophys J 1996; 25:115–130.
37. Hofacker IL, Schuster P, Stadler PF. Combinatorics of RNA secondary structures. Discr Appl Math 1998; 88:207–237.
38. Howell JA, Smith TF, Waterman MS. Computation of generating functions for biological molecules. SIAM J Appl Math 1980; 39:119–133.
39. Schmitt WR, Waterman MS. Linear trees and RNA secondary structure. Discr Appl Math 1994; 12:412–427.
40. Penner RC, Waterman MS. Spaces of RNA secondary structures. Adv Math 1993; 101:31–49.
41. Stein PR, Waterman MS. On some new sequences generalizing the Catalan and Motzkin numbers. Disc Math 1978; 26:261–272.
42. Waterman MS, Smith TF. Combinatorics of RNA hairpins and cloverleaves. Studies Appl Math 1978; 60:91–96.
43. Waterman MS, Smith TF. RNA secondary structure: a complete mathematical analysis. Math Biosci 1978; 42:257–266.
44. Waterman MS. Introduction to Computational Biology: Maps, Sequences, and Genomes. London: Chapman & Hall, 1995.
45. Bender EA. Asymptotic methods in enumeration. SIAMRev 1974; 16:485–515.
46. Canfield ER. Remarks on an asymptotic method in combinatoric. J Comb Theory A 1984; 37:348–352.
47. Meir A, Moon JW. On an asymptotic method in enumeration. J Comb Theory A 1989; 51:77–89.
48. Darboux G. Mémoir sur l’approximation des fonctions de trè s grande nombres, et sur une classe étendu de développements en sé rie. J Math Pures Appl 1878; 4:5–56.
49. Szegö G, Orthogonal Polynomials, volume XXIII of Am Math Soc Coll Publ Am Math Soc, New York, 1959.
50. Leydold J, Stadler PF, Minimal cycle basis of outerplanar graphs. Elec. J Comb., 5. 1998. R16. See http://www.combinatorics.org and Santa Fe Institute Preprint 98-01-011.
51. Mathews DH, Sabina J, Zuker M, Turner DH. Expanded sequence dependence of thermodynamic parameter improves prediction of RNA secondary structure. J Mol Biol 1999; 288:911–940.
52. Walter AE, Turner DH, Kim J, Lyttle MH, Müller P, Mathews DH, Zuker M. Co-axial stacking of helixes enhances binding of oligoribonucleotides and improves predictions of RNA folding. Proc Natl Acad Sci USA 1994; 91: 9218–9222.
53. Jaeger JA, Turner DH, Zuker M. Improved predictions of secondary structures for RNA. Proc Natl Acad Sci USA 1989; 86:7706–7710.
54. He L, Kierzek R, SantaLucia J, Walter AE, Turner DH. Nearest-neighbor parameters for GU mismatches. Biochemistry 1991; 30:11124–11132.
55. Peritz AE, Kierzek R, Sugimoto N, Turner DH. Thermodynamic study of internal loops in oligoribonucleotides: symmetric loops are more stable than asymmetric loops. Biochemistry 1991; 30:6428–6436.
56. SantaLucia J Jr, Allawi HT, Seneviratne PA. Improved nearest-neighbor parameters for predicting DNA duplex stability. Biochemistry 1996; 35:3555-3562.
57. SantaLucia J Jr. A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. Proc Natl Acad Sci USA 1998; 95:1460–1465.
58. Nussinov R, Jacobson AB. Fast algorithm for predicting the secondary structure of single-stranded RNA. Proc Natl Acad Sci USA 1980; 77 (11):6309-6313.
59. Nussinov R, Piecznik G, Griggs JR, Kleitman DJ. Algorithms for loop matching. SIAM J Appl Math 1978; 35:68–82.
60. Zuker M, Stiegler P. Optimal computer folding of larger RNA sequences using thermodynamics and auxiliary information. Nucleic Acids Res 1981; 9:133–148.
61. Zuker M, Sankoff D. RNA secondary structures and their prediction. Bull Math Biol 1984; 46(4):591–621.
62. Zuker M. On finding all suboptimal foldings of an RNA molecule. Science 1989; 244:48–52.
63. Schmitz M, Steger G. Base-pair probability profiles of RNA secondary structures. Comput Appl Biosci 1992; 8:389–399.
64. McCaskill JS. The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers 1990; 29:1105–1119.
65. Cupal J, Hofacker IL, Stadler PF. Dynamic programming algorithm for the density of states of RNA secondary structures. In: Hofstädt R, Lengauer T, Lö ffler M, Schomburg D, eds. Computer Science and Biology 96 (Proceedings of the German Conference on Bioinformatics). Leipzig, Germany: Universität Leipzig, 1996:184–186.
66. Cupal J. The density of states of RNA secondary structures. Master’s Thesis, Univ Vienna, 1997.
67. Higgs PG. RNA secondary structure: a comparison of real and random sequences. J Phys I (France) 1995; 3:43–59.
68. Wuchty S, Fontana W, Hofacker IL, Schuster P. Complete suboptimal folding ofRNAand the stability of secondary structures. Biopolymers 1999; 49:145–165.
69. Wuchty S. Suboptimal secondary structures of RNA. Master’s Thesis, Univ Vienna, 1998.
70. Hofacker IL, Fontana W, Stadler PF, Bonhoeffer S, Tacker M, Schuster P. Fast folding and comparison of RNA secondary structures. Monatsh Chem 1994; 125:167–188.
71. Morgan SR, Higgs PG. Evidence for kinetic effects in the folding of large RNA molecules. J Chem Phys 1996; 105:7152–7157.
72. Martinez HM. An RNA folding rule. Nucleic Acid Res 1984; 12:323–335.
73. Mironov AA, Dyakonova LP, Kister AE. A kinetic approach to the prediction of RNA secondary structures. J Biomol Struct Dynam 1985; 2:953.
74. Abrahams JP, van den Berg M, van Batenburg E, Pleij C. Prediction of RNA secondary structure, including pseudoknotting, by computer simulation. Nucleic Acids Res 1990; 18:3035–3044.
75. Gultyaev AP. The computer simulation of RNA folding involving pseudoknot formation. Nucleic Acids Res 1991; 19:2489–2493.
76. Tacker M, Fontana W, Stadler PF, Schuster P. Statistics of RNA melting kinetics. Eur Biophys J 1994; 23:29–38.
77. Higgs PG. Thermodynamic properties of transfer RNA: a computational study. J Chem Soc Faraday Trans 1995; 91:2531–2540.
78. Gultyaev AP, van Batenburg Pleij CWA. The computer simulation of RNA folding pathways using a genetic algorithm. J Mol Biol 1995; 250:37–51.
79. Suvernev AA, Frantsuzov PA. Statistical description of nucleic acid secondary structure folding. J Biomol Struct Dynam 1995; 13:135–144.
80. Flamm C. Kinetic folding of RNA. PhDThesis, Universität Wien, 1998.
81. Flamm C, Fontana W, Hofacker IL, Schuster P. Elementary step dynamics of RNA folding. RNA 2000; 6:325–338.
82. Gutell RR. Evolutionary characteristics of RNA: inferring higher-order structure from patterns of sequence variation. Curr Opin Struct Biol 1993; 3:313–322.
83. Chaves C, Riera R. Correction to scaling for the self-avoiding walk in d=2: result based on a cell renormalization group. Phys Rev B 1993; 48:16084–16087.
84. Beretti A, Sokal A. New Monte Carlo method for the self-avoiding walk. J Stat Phys 1985; 40:483–531.
85. Watts MG. Application of the method of pad e approximants to the excluded volume problem. J Phys A 1975; 8:61–66.
86. Guttmann AJ. The high-temperature susceptibility and spin-spin correlation function of the three-dimensional sing model. J Phys A 1987; 20:1855.
87. Enting IG, Guttmann AJ. Self-avoiding polygons on the square, 1 and manhatten lattice. J Phys A 1985; 18:1007.
88. Ishinabe T, Chikahisa Y. Exact enumeration of self-avoiding lattice walks with different nearest-neighbor contacts. J Chem Phys 1986; 85(2):1009–1017.
89. Guttmann AJ, Osborn TR, Sokal AD. Connective constant of the self-avoiding walk on the triangular lattice. J Phys A: Math Gen 19: 2591–2598.
90. Kremer K, Baumgärtner A, Binder K. Collapse transition and crossover scaling for self-avoiding walks on the diamond lattice. J Phys A: Math Gen 1981; 15:2879–2897.
91. Guttmann AJ. On the critical behaviour of self-avoiding walks. J Phys A July 1986; 20:1839–1854.
92. Rapaport DC. On three-dimensional self-avoiding walks. J Phys A 1985; 18:113–126.
93. JD et al. Scaling exponents of the self-avoiding-walk problem in three dimensions. J. Dayantis and J. Palierne Phys Rev B 1994; 49:3217–3225.
94. Guttmann A, Ninham B, Thompson C. Determination of critical behaviour in lattice statistics from series expansions. I Phys Rev 1968; 172:554–558.
95. Kolinski A, Milik M, Skolnick J. Static and dynamic properties of a new lattice model of polypeptide chains. J Chem Phys 1991; 94(5):3978–3985.
96. Dill KA. Theory for the folding and stability of globular proteins. Biochemistry 1985; 24:1501–1509.
97. Flory PJ. Statistical Mechanics of Chain Molecules. New York: Wiley, 1969.
98. Sun S, Brem R, Chan HS, Dill KA. Designing amino-acid sequences to fold with good hydrophobic cores. Protein Eng 1996; 8:1205–1213.
99. Buchler NEG, Goldstein RA. The effect of alphabet size and foldability requirements on protein structure designability. Proteins 1999; 34:113–124.
100. Miyazawa S, Jernigan RL. Estimation of effective interresidue contact energies from protein crystal structures: quasi-chemical approximation. Macromolecules 1985; 18:534–552.
101. Sippl MJ. Calculation of conformational ensembles from potentials of mean force—an approach to the knowledge-based prediction of local structures in globular proteins. J Mol Biol 1990; 213:859–883.
102. Casari G, Sippl MJ. Structure-derived hydrophobic potentials—hydrophobic potentials derived from X-ray structures of globular proteins are able to identify native folds. J Mol Biol 1992; 224:725–732.
103. Sippl MJ. Recognition of errors in three-dimensional structures of proteins. Proteins 1993; 17:355–362.
104. Ngo JT, Marks J. Computational complexity of a problem in molecular structure prediction. Protein Eng 1992; 5:313–321.
105. Unger R, Moult J. Finding the lowest free energy conformation of a protein is an NP-hard problem: proof and implications. Bull Math Biol 1993; 55:1183–1198.
106. Hart WE, Istrail S. Robust proofs of NP-hardness for protein folding: general lattices and energy potentials. J Comput Biol 1997; 4:1–22.
107. Li H, Helling R, Tang C, Wingreen N. Emergence of preferred structures in a simple model of protein folding. Science 1996; 273:666–669.
108. Yue K, Dill KA. Sequence structure relationships in proteins and copolymers. Phys Rev E 1993; 48:2267–2278.
109. Hart WE, Istrail S. Fast protein folding in the hydrophilic-hydrophobic model within three-eights optimal. Extended Abstract published in the Proceedings of the 27th AnnualACMSymposium on Theory of Computation, May 1995, 1994.
110. Bollobás B. Random Graphs. London: Academic Press, 1985.
111. Reidys CM. Random induced subgraphs of generalized n-cubes. Adv Appl Math 1997; 19:360–377.
112. Reidys CM, Stadler PF, Schuster P. Generic properties of combinatory maps: neutral networks of RNA secondary structures. Bull Math Biol 1997; 59:339-397.
113. Ajtai M, Komlós J, Szemerédi E. Largest random component of a k-cube. Combinatorica 1982; 2:1–7.
114. Gavrilets S, Gravner J. Percolation on the fitness hypercube and the evolution of reproductive isolation. J Theor Biol 1997; 184:51–64.
115. Gavrilets S, Li H, Vose MD. Rapid parapatric speciation on holey adaptive landscapes. Proc Roy Soc Lond B 1998; 265:1483–1489.
116. Schuster P, Fontana W, Stadler PF, Hofacker IL. From sequences to shapes and back: a case study in RNA secondary structures. Proc Roy Soc Lond B 1994; 255:279–284.
117. Grüner W, Giegerich R, Strothmann D, Reidys C, Weber J, Hofacker IL, Stadler PF, Schuster P. Analysis of RNA sequence structure maps by exhaustive enumeration. I. Neutral networks. Monatsh Chem 1996; 127:355–374.
118. Grüner W, Giegerich R, Strothmann D, Reidys C, Weber J, Hofacker IL, Stadler PF, Schuster P. Analysis of RNA sequence structure maps by exhaustive enumeration. II. Structures of neutral networks and shape space covering. Monatsh Chem 1996; 127:375–389.
119. Reidys CM, Stadler PF. Neutrality in fitness landscapes. Appl Math & Comput 2001; 117:321–350.
120. Fontana W, Stadler PF, Bornberg-Bauer EG, Griesmacher T, Hofacker IL, Tacker M, Tarazona P, Weinberger ED, Schuster P. RNA folding and combinatory landscapes. Phys Rev E 1993; 47:2083–2099.
121. Huynen MA, Stadler PF, Fontana W. Smoothness within ruggedness: the role of neutrality in adaptation. Proc Natl Acad Sci USA 1996; 93:397–401.
122. van Nimwegen E, Crutchfield JP, Mitchell M. Finite populations induce metastability in evolutionary search. Phys Lett A 1997; 229:144–150.
123. Fontana W, Schuster P. Shaping space. The possible and the attainable in RNA genotype-phenotype mapping. J Theor Biol 1998; 194:491–515.
124. Fontana W, Schuster P. Continuity in evolution. On the nature of transitions. Science 1998; 280:1451–1455.
125. Schuster P. How to search for RNA secondary structures. Theoretical concepts in evolutionary biotechnology. J Biotechnol 1995; 41:239–257.
126. Göbel U. Neutral Networks of Minimum Free Energy RNA Secondary Structures. PhDthesis, Universität Wien, 2000. http://www.tbi.univie.ac.at/papers/PhD_theses.html.
127. Fontana W, Griesmacher T, Schnabl W, Stadler PF, Schuster P. Statistics of landscapes based on free energies, replication and degradation rate constants of RNA secondary structures. Monatsh Chem 1991; 122:795–819.
128. Fontana W, Konings DA, Stadler PF, Schuster P. Statistics of RNA secondary structures. Biopolymers 1993; 33:1389–1404.
129. Göbel U, Forst CV, Schuster P. Structural constraints and neutrality in RNA. In: Hofestädt R, Lengauer T, Lö ffler M, Schomburg D, eds. Bioinformatics. German Conference on Bioinformatics, GCB’96, volume 1278 of Lecture Notes Comput Sci. Berlin: Springer-Verlag, 1997:156–165.
130. Cupal J, Kopp S, Stadler PF. RNA shape space topology. Alife 2000; 6:3–23. In press. SFI Preprint 99-03-022.
131. Hausdorff F. Gestufte Räume. Fund Math 1935; 25:486–502.
132. Choquet G. Convergences. Ann Univ Grenoble 1947; 23:55–112.
133. Kent DC. Convergence functions and their related topologies. Fund Math 1964; 54:125–133.
134. Katetov M. On continuity structures and spaces of mappings. Commun Math Univ Carolinae 1965; 6:257–278.
135. Babajide A, Hofacker IL, Sippl MJ, Stadler PF. Neutral networks in protein space: a computational study based on knowledge-based potentials of mean force. Folding Des 1997; 2:261–269.
136. Babajide A, Farber R, Hofacker IL, Inman J, Lapedes AS, Stadler PF. Exploring protein sequence space using knowledge based potentials. J Comput Biol 1999. J Theor Biol 2001; 212:35–46. Santa Fe Preprint 98-11-103.
137. Plaxco K, Riddle D, Grantcharova V, Baker D. Simplified proteins: minimalist solutions to the “protein folding problem”. Curr Opin Struct Biol 1998; 8:80–85.