Genotype-Phenotype Maps

Biological Theory

Volumn 1, Issue 3, 2006, Pages 268-279

  Stadler, Peter F ^a,b,c   Stadler, Bärbel M R ^d  

^a UNIVERSITY OF LEIPZIG   (Germany)

^b Inst Theor Chem Molec S   (Austria)

^c SANTA FE INSTITUTE   (United States)

^d MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

generalized topology; phenotypic characters; quasi independence

Indexed keywords

EID: 37249050647     PISSN: 15555542     EISSN: 15555550     Source Type: Journal    
DOI: 10.1162/biot.2006.1.3.268     Document Type: Article

References (74)

1. Alexandroff P (1937) Diskrete Räume, Mathematica Sbornik (N.S.) 2: 501–518.
2. Ancel L, Fontana W (2000) Plasticity, evolvability and modularity in RNA. Journal of Experimental Zoology (Molecular and Developmental Evolution) 288: 242–283. DOI: 10.1002/1097-010X(20001015)288:3<242::AID-JEZ5>3.0.CO;2-O
3. Arenas FG (1999) Alexandroff spaces. Acta Mathematica Universitatis Comenianae 68: 17–25.
4. Babajide A, Farber R, Hofacker IL, Inman J, Lapedes AS, Stadler PF (2001) Exploring protein sequence space using knowledge based potentials. Journal of Theoretical Biology 212: 35–46. DOI: 10.1006/jtbi.2001.2343
5. Babajide A, Hofacker IL, Sippl MJ, Stadler PF (1997) Neutral networks in protein space: A computational study based on knowledge-based potentials of mean force. Folding and Design 2: 261–269. DOI: 10.1016/S1359-0278(97)00037-0
6. Biebricher CK, Gardiner WC (1997) Molecular evolution of RNA in vitro. Biophysical Chemistry 66: 179–192. DOI: 10.1016/S0301-4622(97)00059-8
7. Brinn LW (1985) Computing topologies. Mathematical Journal 58: 67–77.
8. Calude C, Căzănescu VE (1979) On topologies generated by Mosil resemblance relations. Discrete Mathematics 25: 109–115. DOI: 10.1016/0012-365X(79)90012-8
9. Čech E (1966) Topological Spaces. London: Wiley.
10. Changat M, Klavžar S, Mulder HM (2001) The all-path transit function of a graph. Czechoslovak Mathematical Journal 51: 439–148. DOI: 10.1023/A:1013715518448
11. Cupal J, Kopp S, Stadler PF (2000) RNA shape space topology. Artificial Life 6: 3–23. DOI: 10.1162/106454600568294
12. Dalal S, Balasubramanian S, Regan L (1997) Protein alchemy: Changing β-sheet into α-helix. Nature Structural and Molecular Biology 4(7): 548–552. DOI: 10.1038/nsb0797-548
13. Derrida B, Peliti L (1991) Evolution in a flat fitness landscape. Bulletin of Mathematical Biology 53: 355–382. DOI: 10.1007/BF02460723
14. Dörfler W, Imrich W (1970) Über das starke Produkt von endlichen Graphen. Österreichische Akademie der Wissenschaften, Mathematischnaturwissenschaftliche Klasse S.-B. II 178: 247–262.
15. Eigen M, McCaskill J, Schuster P (1989) The molecular quasispecies. Advances in Chemical Physics 75: 149–263.
16. Feigenbaum J, Schäffer AA (1992) Finding the prime factors of strong direct products of graphs in polynomial time. Discrete Mathematics 109: 77–102. DOI: 10.1016/0012-365X(92)90280-S
17. Flamm C, Hofacker IL, Stadler PF (1999) RNA in silico: The computational biology of RNA secondary structures. Advances in Complex Systems 2: 65–90. DOI: 10.1142/S0219525999000059
18. Fontana W, Schuster P (1998a) Continuity in evolution: On the nature of transitions. Science 280: 1451–1455. DOI: 10.1126/science.280.5368.1451
19. Fontana W, Schuster P (1998b) Shaping space: The possible and the attainable in RNA genotype-phenotype mapping. Journal of Theoretical Biology 194: 491–515. DOI: 10.1006/jtbi.1998.0771
20. Gavrilets S (1997) Evolution and speciation on holey adaptive landscapes. Trends in Ecology and Evolution 12: 307–312. DOI: 10.1016/S0169-5347(97)01098-7
21. Gitchoff P, Wagner GP (1996) Recombination induced hypergraphs: A new approach to mutation-recombination isomorphism. Complexity 2: 37–43. DOI: 10.1002/(SICI)1099-0526(199609/10)2:1<37::AID-CPLX9>3.0.CO;2-C
22. Gnilka S (1994) On extended topologies. I. Closure operators. Annales de la Societe Polonaise de Mathematique, Seria I, Commentations Mathematical 34: 81–94.
23. Gnilka S (1997) On continuity in extended topologies. Annales de la Societé Polonaise de Mathematique., Seria I, Commentations Mathematical 37: 99–108.
24. Grüner W, Giegerich R, Strothmann D, Reidys C, Weber J, Hofacker IL, Stadler PF, Schuster P (1996a) Analysis of RNA sequence structure maps by exhaustive enumeration. I. Neutral networks. Monatshefte für Chemie 127: 355–374. DOI: 10.1007/BF00810881
25. Grüner W, Giegerich R, Strothmann D, Reidys C, Weber J, Hofacker IL, Stadler PF, Schuster P (1996b) Analysis of RNA sequence structure maps by exhaustive enumeration. II. Structures of neutral networks and shape space covering. Monatshefte für Chemie 127: 375–389. DOI: 10.1007/BF00810882
26. Hofacker IL, Fontana W, Stadler PF, Bonhoeffer LS, Tacker M, Schuster P (1994) Fast folding and comparison of RNA secondary structures. Monatshefte für Chemie 125: 167–188. DOI: 10.1007/BF00818163
27. Hofacker IL, Schuster P, Stadler PF (1998) Combinatorics of RNA secondary structures. Discrete Applied Mathematics 89: 177–207.
28. Huynen MA (1996) Exploring phenotype space through neutral evolution. Journal of Molecular Evolution 43: 165–169. DOI: 10.1007/BF02338823
29. Huynen MA, Stadler PF, Fontana W (1996) Smoothness within ruggedness: The role of neutrality in adaptation. Proceedings of the National Academy of Sciences (The USA) 93: 397–401. DOI: 10.1073/pnas.93.1.397
30. Imrich W (1998) Factoring cardinal product graphs in polynomial time. Discrete Mathematics 192: 119–144. DOI: 10.1016/S0012-365X(98)00069-7
31. Imrich W, Klavžar S (2000) Product Graphs: Structure and Recognition. New York: Wiley.
32. Imrich W, Stadler PF (2006) A prime factor theorem for a generalized direct product. Discussiones Mathematicae Graph Theory 26: 135–140. DOI: 10.7151/dmgt.1307
33. Keefe AD, Szostak JW (2001) Functional proteins from a random-sequence library. Nature 410: 715–718. DOI: 10.1038/35070613
34. Kent DC (1967) On convergence groups and convergence uniformities. Fun-damenta Mathematicae 60: 213–222.
35. Lewontin RC (1978) Adaptation. Scientific American 239: 156–169. DOI: 10.1038/scientificamerican0978-212
36. Liu YM, Luo MK (1998) Fuzzy Topology. Singapore: World Scientific.
37. Lovász L (1967) Operations with structures. Acta Mathematica Academiae Scientiarum Hungaricae 18: 321–328. DOI: 10.1007/BF02280291
38. Lovasz L (1971) Unique factorization in certain classes of structures. In: Mini-Conf. Univers. Algebra, Szeged 1971 24–25. János Bolyai Mathematical Society.
39. Malitza M (1975) Topology, binary relations, and internal operations. Revue Roumaine de Mathematiques Pures et Appliquees 4: 515–519.
40. Martinez MA, Pezo V, Marlière P, Wain-Hobson S (1996) Exploring the functional robustness of an enzyme by in vitro evolution. EMBO Journal 15: 1203–1210.
41. Mathews D, Sabina J, Zuker M, Turner H (1999) Expanded sequence dependence of thermodynamic parameters provides robust prediction of RNA secondary structure. Journal of Molecular Biology 288: 911–940. DOI: 10.1006/jmbi.1999.2700
42. Maynard-Smith J (1970) Natural selection and the concept of a protein space. Nature 225: 563–564. DOI: 10.1038/225563a0
43. McKenzie R (1971) Cardinal multiplication of structures with a reflexive multiplication. Fundamenta Mathematicae 70: 59–101.
44. Menger K (1942) Statistical metrics. Proceedings of the National Academy of Sciences USA 28: 535–537. DOI: 10.1073/pnas.28.12.535
45. Morgana MA, Mulder HM (2002) The induced path convexity, betweenness, and svelte graphs. Discrete Mathematics 254: 349–370. DOI: 10.1016/S0012-365X(01)00296-5
46. Reidys C, Stadler PF, Schuster P (1997) Generic properties of combinatory maps. Neutral networks of RNA secondary structure. Bulletin of Mathematical Biology 59: 339–397. DOI: 10.1007/BF02462007
47. Reidys CM (1997) Random induced subgraphs of generalized n-cubes. Advances in Applied Mathematics 19: 360–377. DOI: 10.1006/aama.1997.0553
48. Richardson G, Kent D (1996) Probabilistic convergence spaces. Journal of the Australian Mathematical Society (Series A) 61: 1–21. DOI: 10.1017/S1446788700000483
49. Sanin N (1943) On separation in topological space. Doklady Akademii Nauk SSSR 38: 110–113.
50. Schultes EA, Bartel DP (2000) One sequence, two ribozymes: Implications for the emergence of new ribozyme folds. Science 289: 448–452. DOI: 10.1126/science.289.5478.448
51. Schuster P (1997) Genotypes with phenotypes: Adventures in an RNA toy world. Biophysical Chemistry 66: 75–110. DOI: 10.1016/S0301-4622(97)00058-6
52. Schuster P (2001) Evolution in Silico and in Vitro: The RNA model. Biological Chemistry 382: 1301–1314. DOI: 10.1515/BC.2001.162
53. Schuster P (2002) A testable genotype-phenotype map: Modeling evolution of RNA molecules. In: Biological Evolution and Statistical Physics (Lässig M, Valleriani A, eds), 56–83. Berlin: Springer-Verlag.
54. Schuster P, Fontana W, Stadler PF, Hofacker IL (1994) From sequences to shapes and back: A case study in RNA secondary structures. Proceedings of the Royal Society London B 255: 279–284. DOI: 10.1098/rspb.1994.0040
55. Schuster P, Stadler PF, Renner A (1997) RNA structure and folding. From conventional to new issues in structure predictions. Current Opinion in Structural Biology 7: 229–235. DOI: 10.1016/S0959-440X(97)80030-9
56. Schweizer B, Sklar A (1983) Probabilistic Metric Spaces. New York: North Holland.
57. Shpak M, Wagner GP (2000) Asymmetry of configuration space induced by unequal crossover: implications for a mathematical theory of evolutionary innovation. Artificial Life 6: 25–43. DOI: 10.1162/106454600568302
58. Sippl MJ (1993) Boltzmann’s principle, knowledge-based mean fields and protein folding: An approach to the computational determination of protein structures. Journal of Computer-Aided Molecular Design 7: 473–501. DOI: 10.1007/BF02337562
59. Slapal J (1993) Relations and topologies. Czechoslovak Mathematical Journal 43: 141–150.
60. Stadler BMR (2002) Diffusion of a population of interacting replicators in sequence space. Advances in Complex Systems 5: 457–461. DOI: 10.1142/S0219525902000675
61. Stadler BMR, Stadler PF (2002) Generalized topological spaces in evolutionary theory and combinatorial chemistry. Journal of Chemical Information and Computer Sciences 42: 577–585.
62. Stadler BMR, Stadler PF (2004) The topology of evolutionary biology. In: Modeling in Molecular Biology (Ciobanu G, ed), Natural Computing Series 267–286. New York: Springer Verlag. DOI: 10.1007/978-3-642-18734-6_12
63. Stadler BMR, Stadler PF, Shpak M, Wagner GP (2002) Recombination spaces, metrics, and pretopologies. Zeitschrift für Physikalische Chemie 216: 217–234.
64. Stadler BMR, Stadler PF, Wagner G, Fontana W (2001) The topology of the possible: Formal spaces underlying patterns of evolutionary change. Journal of Theoretical Biology 213: 241–274. DOI: 10.1006/jtbi.2001.2423
65. Stadler PF, Seitz R, Wagner GP (2000) Evolvability of complex characters: Population dependent Fourier decomposition of fitness landscapes over recombination spaces. Bulletin of Mathematical Biology 62: 399–428. DOI: 10.1006/bulm.1999.0167
66. Stadler PF, Wagner GP (1998) The algebraic theory of recombination spaces. Evolutionary Computation 5: 241–275. DOI: 10.1162/evco.1997.5.3.241
67. Stephan-Otto Attolini C, Stadler PF (2005) Neutral networks of interacting RNA secondary structures. Advances in Complex Systems 8: 275–284. DOI: 10.1142/S0219525905000427
68. Stephan-Otto Attolini C, Stadler PF (2006) Evolving towards the hypercycle: A spatial model of molecular evolution. Physica D 217: 134–141. DOI: 10.1016/j.physd.2006.03.015
69. Wagner G, Stadler PF (2003) Quasi-independence, homology and the unity of type: A topological theory of characters. Journal of Theoretical Biology 220: 505–527. DOI: 10.1006/jtbi.2003.3150
70. Weberndorfer G, Hofacker IL, Stadler PF (1999) An efficient potential forprotein sequence design. In: Computer Science in Biology 107–112. Bielefeld, D: Univ. Bielefeld Proceedings of the GCB’99, Hannover, D.
71. Wilson DS, Szostak JW (1999) In Vitro selection of functional nucleic acids. Annual Review of Biochemistry 68: 611–647. DOI: 10.1146/annurev.biochem.68.1.611
72. Wright S (1932) The roles of mutation, inbreeding, crossbreeeding and selection in evolution. In: Proceedings of the Sixth International Congress on Genetics, vol. 1 (Jones DF, ed), 356–366.
73. Wright S (1967) “Surfaces” of selective value. Proceedings of the National Academy of Sciences (The USA) 58: 165–172. DOI: 10.1073/pnas.58.1.165
74. Zuker M, Sankoff D (1984) RNA secondary structures and their prediction. Bulletin of Mathematical Biology 46(4): 591–621. DOI: 10.1007/BF02459506