Towards a theory of landscapes

  • Peter F. Stadler
  • Santa Fe Institute
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 461-461)


Travel Salesman Problem Cayley Graph Symmetry Class Simple Random Walk Transitive Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sewall Wright. The roles of mutation, inbreeding, crossbreeeding and selection in evolution. In D. F. Jones, editor, Int. Proceedings of the Sixth International Congress on Genetics, pages 356–366, 1932.Google Scholar
  2. 2.
    K. Binder and A. P. Young. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys., 58:801–976, 1986.ADSGoogle Scholar
  3. 3.
    M. Mézard, G. Parisi, and M.A. Virasoro. Spin Glass Theory and Beyond World Scientific, Singapore, 1987.zbMATHGoogle Scholar
  4. 4.
    M. R. Garey and D. S. Johnson. Computers and Intractability. A Guide to the Theory of NP Completeness. Freeman, San Francisco, 1979.zbMATHGoogle Scholar
  5. 5.
    M. Eigen. Selforganization of matter and the evolution of biological macromolecules. Die Naturwissenschaften, 10:465–523, 1971.ADSGoogle Scholar
  6. 6.
    W. Fontana and P. Schuster. A computer model of evolutionary optimization. Biophysical Chemistry, 26:123–147, 1987.Google Scholar
  7. 7.
    Walter Fontana, Wolfgang Schnabl, and Peter Schuster. Physical aspects of evolutionary optimization and adaption. Physical Review A, 40(6):3301–3321, 1989.ADSGoogle Scholar
  8. 8.
    C. Amitrano, L. Peliti, and M. Saber. Population dynamics in a spin-glass model of chemical evolution. J. Mol. Evol., 29:513–525, 1989.Google Scholar
  9. 9.
    M. Eigen, J. McCaskill, and P. Schuster. The molecular Quasispecies. Adv. Chem. Phys., 75:149–263, 1989.Google Scholar
  10. 10.
    Pete F. Stadler and Wolfgang Schnabl. The landscape of the traveling salesman problem. Phys. Letters A, 161:337–344, 1992.zbMATHADSMathSciNetGoogle Scholar
  11. 11.
    Peter F. Stadler Correlation in landscapes of combinatorial optimization problems. Europhys. Lett., 20:479–482, 1992.ADSGoogle Scholar
  12. 12.
    Peter F. Stadler and Robert Happel. Correlation structure of the landscape of the graph-bipartitioning-problem. J. Phys. A.: Math. Gen., 25:3103–3110, 1992.ADSMathSciNetGoogle Scholar
  13. 13.
    Catherine A. Macken and Alan S. Perelson. Protein evolution on rugged landscapes. Proc. Natl. Acad. Sci. USA, 86:6191–6195, 1989.ADSMathSciNetGoogle Scholar
  14. 14.
    C. A. Macken, P. S. Hagan, and A. S. Perelson. Evolutionary walks on rugged landscapes. SIAM J. Appl. Math., 51:799–827, 1991.zbMATHMathSciNetGoogle Scholar
  15. 15.
    Henrik Flyvbjerg and Benny Lautrup. Evolution in a rugged fitness landscape. Phys. Rev. A, 46:6714–6723, 1992.ADSGoogle Scholar
  16. 16.
    P. Bak, H. Flyvbjerg, and B. Lautrup. Coevolution in a rugged fitness landscape. Phys. Rev. A[15], 46:6724–6730, 1992.ADSGoogle Scholar
  17. 17.
    S. A. Kauffman and S. Levin. Towards a general theory of adaptive walks on rugged landscapes. J. Theor. Biol., 128:11, 1987.MathSciNetGoogle Scholar
  18. 18.
    S. A. Kauffman and E. D. Weinberger. The n-k model of rugged fitness landscapes and its application to maturation of the immune response. J. Theor. Biol., 141:211, 1989.Google Scholar
  19. 19.
    Edward D. Weinberger. Local properties of Kauffman's N-k model: a tunably rugged energy landscape. Phys. Rev. A, 44(10):6399–6413, 1991.ADSGoogle Scholar
  20. 20.
    Walter Fontana, Peter F. Stadler, Erich G. Bornberg-Bauer, Thomas Griesmacher, Ivo L. Hofacker, Manfred Tacker, Pedro Tarazona, Edward D. Weinberger, and Peter Schuster. RNA folding and combinatory landscapes. Phys. Rev. E, 47(3):2083–2099, 1993.ADSGoogle Scholar
  21. 21.
    E. D. Weinberger and P. F. Stadler. Why some fitness landscapes are fractal. J. Theor. Biol., 163:255–275, 1993.Google Scholar
  22. 22.
    Sebastian Bonhoeffer and Peter F. Stadler. Errortreshold on complex fitness landscapes. J. Theor. Biol., 164:359–372, 1993.Google Scholar
  23. 23.
    W. Fontana, T. Griesmacher, W. Schnabl, P. F. Stadler, and P. Schuster. Statistics of landscapes based on free energies, replication and degredation rate constants of RNA secondary structures. Monatsh. Chemie, 122:795–819, 1991.Google Scholar
  24. 24.
    W. Fontana, D. A. M. Konings, P. F. Stadler, and P. Schuster. Statistics of rna secondary structures. Biochemistry, 33:1389–1404, 1993.Google Scholar
  25. 25.
    Ivo L. Hofacker, Walter Fontana, Peter F. Stadler, Sebastian Bonhoeffer, Manfred Tacker, and Peter Schuster. Fast folding and comparison of RNA secondary structures. Monatsh. Chemie, 125(2):167–188, 1994.Google Scholar
  26. 26.
    M. A. Huynen and P. Hogeweg. Pattern generation in molecular evolution. Exploitation of the variation in RNA landscapes. J. Mol. Evol., 39:71–79, 1994.Google Scholar
  27. 27.
    Martijn A. Huynen, Peter F. Stadler, and Walter Fontana. Evolution of RNA and the Neutral Theory. 1995. SFI Preprint #95-01-006.Google Scholar
  28. 28.
    P. Schuster. Complex optimization in an artificial RNA world. In D. Farmer, C. Langton, S. Rasmussen, and C. Taylor, editors, Artificial Life II, pages 277–291, Addison-Wesley, 1992.Google Scholar
  29. 29.
    Peter Schuster, Walter Fontana, Peter F Stadler, and Ivo L Hofacker. From sequences to shapes and back: a case study in RNA secondary structures. Proc. Roy. Soc. Lond. B, 255:279–284, 1994.ADSGoogle Scholar
  30. 30.
    Peter Schuster and Peter F Stadler. Landscapes: complex optimization problems and biopolymer structures. Computers Chem., 18:295–314, 1994.zbMATHGoogle Scholar
  31. 31.
    Peter F. Stadler and Walter Grüner. Anisotropy in fitness landscapes. J. Theor. Biol., 165:373–388, 1993.Google Scholar
  32. 32.
    Manfred Tacker, Walter Fontana, Peter Stadler, and Peter Schuster. Statistics of RNA melting kinetics. Eur. J. Biophys., 23:29–38, 1994.Google Scholar
  33. 33.
    Manfred Tacker, Peter F. Stadler, Erich G. Bornberg-Bauer, Ivo L. Hofacker, and Peter Schuster. Robust properties of RNA secondary structure folding algorithms. 1995. In preparation.Google Scholar
  34. 34.
    E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan, and D. B. Shmoys. The Traveling Salesman Problem. A Guided Tour of Combinatorial Optimization. John Wiley & Sons, 1985.Google Scholar
  35. 35.
    R. G. Bland and D. F. Shallcross. Large traveling salesmen problems arising from experiments in x-ray crystallography. Oper. Res. Lett., 8:125–128, 1988.MathSciNetGoogle Scholar
  36. 36.
    D. Chan and D. Mercier. IC insertion: an application of the TSP. Int. J. Prod. Res., 3:9–28, 1989.Google Scholar
  37. 37.
    H. Bohr and S. Brunak. Travelling salesman approach to protein conformation. Complex Systems, 3:9–28, 1990.Google Scholar
  38. 38.
    D.L. Miller and J. F. Pekny. Exact solution of large asymmetric traveling salesman problems. Science, 251:754–761, 1991.ADSGoogle Scholar
  39. 39.
    S. Lin and B.W. Kernighan. An effective heuristic algorithm for the traveling salesman problem. Oper. Res., 21:498–516, 1965.MathSciNetGoogle Scholar
  40. 40.
    H. Wielandt. Finite Permutation Groups. Academic Press, New York, 1964.zbMATHGoogle Scholar
  41. 41.
    D. G. Higman. Intersection matrices for finite permutation groups. J. Algebra, 6:22–42, 1967.zbMATHMathSciNetGoogle Scholar
  42. 42.
    Norman Biggs. Finite Groups of Automorphisms. Volume 6 of London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge UK, 1971.Google Scholar
  43. 43.
    Bela Bollobás. Graph Theory—An Introductory Course. Springer-Verlag, New York, 1979.zbMATHGoogle Scholar
  44. 44.
    C. D. Godsil. Algebraic Combinatorics. Chapman & Hall, New York, 1993.zbMATHGoogle Scholar
  45. 45.
    P. Delsarte. An algebraic approach to association schemes of coding theory. Volume 10 of Phillips Research Reports Supplements, Phillips, 1973.Google Scholar
  46. 46.
    Norman L. Biggs. Algebraic Graph Theory. Cambridge University Press, Cambridge UK, 2nd edition, 1994.zbMATHGoogle Scholar
  47. 47.
    F. Spitzer. Principles of Random Walks. Springer-Verlag, New York, 1976.Google Scholar
  48. 48.
    D.M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs—Theory and Applications. Academic Press, New York, 1980.Google Scholar
  49. 49.
    Peter F. Stadler and Robert Happel. Canonical approximation of landscapes. 1994. Submitted to J. Stat. Phys.Google Scholar
  50. 50.
    Paolo M. Soardi. Potential Theory on Infinite Networks. Volume 1590 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.Google Scholar
  51. 51.
    Mark Kac. Can you hear the shape of a drum. Am. Math. Monthly, 73(4):1–23, 1966.Google Scholar
  52. 52.
    G.A. Baker. Drum shapes and isospectral graphs. J. Math. Phys., 7:2238, 1966.zbMATHADSGoogle Scholar
  53. 53.
    M.E. Fisher. On hearing the shape of a drum. J. Comb. Theory, 1:105–125, 1966.zbMATHGoogle Scholar
  54. 54.
    C.T. Benson and J.B. Jacobs. On hearing the shape of combinatorial drums. J. Comb. Theory(B), 13:170–178, 1972.zbMATHMathSciNetGoogle Scholar
  55. 55.
    Gert Sabidussi. Vertex transitive graphs. Mh. Math, 68:426–438, 1964.zbMATHMathSciNetGoogle Scholar
  56. 56.
    Norman Biggs. Algebraic Graph Theory. Cambridge University Press, Cambridge UK, 1st edition, 1974.zbMATHGoogle Scholar
  57. 57.
    Peter F. Stadler. Random walks and orthogonal functions associated with highly symmetric graphs. Disc. Math., 1994. in press.Google Scholar
  58. 58.
    G.M. Adel'son-Velskii et al. Example of a graph without a transitive automorphism group. Soviet. Math. Dokl., 10:440–441, 1969. Russian.Google Scholar
  59. 59.
    Edward D. Weinberger. Fourier and Taylor series on fitness landscapes. Biological Cybernetics, 65:321–330, 1991.zbMATHGoogle Scholar
  60. 60.
    L. Lovász. Spectra of graphs with transitive groups. Periodica Math. Hung., 6:191–195, 1975.zbMATHGoogle Scholar
  61. 61.
    J.-P. Serre. Linear Representations of Finite Groups. Springer-Verlag, New York, Heidelberg, Berlin, 1977.zbMATHGoogle Scholar
  62. 62.
    A.J. Schwenk. Computing the characteristic polynomial of a graph. In Graphs and Combinatorics, pages 153–162. Springer-Verlag, Berlin, 1974.Google Scholar
  63. 63.
    D.L. Powers and M.M. Sulaiman. The walk partition and colorations of a graph. Linear Algebra Appl., 48:145–159, 1982.zbMATHMathSciNetGoogle Scholar
  64. 64.
    D.L. Powers. Eigenvectors of distance-regular graphs. SIAM J. Matrix Anal. Appl., 9:399–407, 1988.zbMATHMathSciNetGoogle Scholar
  65. 65.
    D.M. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs—Theory and Applications. Volume New York, Academic Press, 1980.Google Scholar
  66. 66.
    R. W. Hamming. Error detecting and error correcting codes. Bell Syst. Tech. J., 29:147–160, 1950.MathSciNetGoogle Scholar
  67. 67.
    A.W.M. Dress and D.S. Rumschitzki. Evolution on sequence space and tensor products of representation spaces. Acta Appl. Math., 11:103–111, 1988.zbMATHMathSciNetGoogle Scholar
  68. 68.
    A.E. Brouwer, A.M. Cohen, and A. Neumaier. Distance-regular Graphs. Springer Verlag, Berlin, New York, 1989.zbMATHGoogle Scholar
  69. 69.
    C.F. Dunkl. A Krawtchouk polynomial addition theorem and wreath products of symmetric groups. Indiana Univ. Math. J., 25:335–358, 1976.zbMATHMathSciNetGoogle Scholar
  70. 70.
    T.H. Koornwinder. Krawtchouk polynomials. A unification of two different group theoretic interpretations. SIAM J. Math. Anal., 13:1011–1023, 1982.zbMATHMathSciNetGoogle Scholar
  71. 71.
    M. Krawtchouk. Sur une gènèralisation des polynomes d'Hermite. Comptes Rendus, 189:620–622, 1929.zbMATHGoogle Scholar
  72. 72.
    F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Correcting Codes. North-Holland, Amsterdam, New York, Oxford, Tokyo, 1991.Google Scholar
  73. 73.
    D. Rumschitzky. Spectral properties of eigen's evolution matrices. J. Math. Biol., 24:667–680, 1987.MathSciNetGoogle Scholar
  74. 74.
    J.H. vanLint. Introduction to Coding Theory. Springer-Verlag, New York, 1982.Google Scholar
  75. 75.
    Christian Reidys, Peter Schuster, and Peter F Stadler. Generic properties of combinatory maps and application on RNA secondary structures. 1995. Preprint.Google Scholar
  76. 76.
    Christian Reidys and Christian Forst. Replication on neutral networks in rna induced by rna secondary structures. 1994. Preprint.Google Scholar
  77. 77.
    P.J. Cameron and J.H. vanLint. Designs, Graphs, Codes, and their Links. Volume 22 of London Math. Soc. Student Texts. Cambridge University Press, Cambridge UK, 1991.Google Scholar
  78. 78.
    C.F. Dunkl. Orthogonal functions on some permutation groups. In D.K. Ray-Chaudhuri, editor, Proceeding of Symposia in Pure Mathematic Vol. 34, American Mathematical Society, New York, 1979.Google Scholar
  79. 79.
    Normal L. Biggs and A.T. White. Permutation Groups and Combinatorial Structures. Cambridge University Press, Cambridge UK, 1979.zbMATHGoogle Scholar
  80. 80.
    P. Diaconis and M. Shahshahani. Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitsth. verw. Gebiete, 57:159–179, 1981.zbMATHMathSciNetGoogle Scholar
  81. 81.
    R.E. Ingram. Some characters of the symmetric group. Proc. Amer. Math. Soc., 1:358–369, 1950.zbMATHMathSciNetGoogle Scholar
  82. 82.
    I.G. MacDonald. Symmetric Functions and Hall Polynomials. Oxford Univ. Press, Oxford UK, 1979.zbMATHGoogle Scholar
  83. 83.
    Julian Besag. Spatial interactions and the statistical analysis of lattice systems. Amer. Math. Monthly, 81:192–236, 1974.MathSciNetGoogle Scholar
  84. 84.
    M.J.E. Golay. Sieves for low-autocorrelation binary sequences. IEEE Trans. Inform. Th., IT-23:43–51, 1977.zbMATHGoogle Scholar
  85. 85.
    H. Hotelling. Analysis of a complex of statistical variables into principal components. J. Educ. Psych., 24:417–441 and 498–520, 1933.Google Scholar
  86. 86.
    C.R. Rao. Linear Statistical Interference and Its Applications. Wiley, New York, 2nd edition, 1973.Google Scholar
  87. 87.
    M.S. Bartlett. An Introduction to Stochastic Processes. Cambridge University Press, Cambridge UK, 1955.zbMATHGoogle Scholar
  88. 88.
    P. Whittle. Stochastic processes in several dimensions. Bull. Int. Statist. Inst., 40:974–994, 1963.MathSciNetGoogle Scholar
  89. 89.
    G.R. Grimmet. A theorem about random fields. Bull. London Math. Soc., 5:81–85, 1973.MathSciNetGoogle Scholar
  90. 90.
    John Moussouris. Gibbs and Markov systems with constraints. J. Stat. Phys., 10:11–33, 1974.MathSciNetADSGoogle Scholar
  91. 91.
    M.B. Averintsev. On a method of describing complete parameter fields. Problemy Peredaci Informatsii, 6:100–109, 1970.Google Scholar
  92. 92.
    R.L. Dobrushin. The description of a random field by means of its conditional probabilities, and conditions of its regularities. Th. Prob. & Appl., 13:197–224, 1968.Google Scholar
  93. 93.
    Frank Spitzer. Markov random fields and gibbs ensembles. Amer. Math. Monthly, 78:142–154, 1971.zbMATHMathSciNetGoogle Scholar
  94. 94.
    S. Karlin and H.M. Taylor. A first course in stochastic processes. Academic Press, New York, 1975.zbMATHGoogle Scholar
  95. 95.
    Peter F. Stadler. Linear operators on correlated landscapes. J. Physique, 4:681–696, 1994.ADSGoogle Scholar
  96. 96.
    B. Derrida. Random energy model: limit of a family of disordered models. Phys. Rev. Lett., 45:79–82, 1980.ADSMathSciNetGoogle Scholar
  97. 97.
    David Sherrington and Scott Kirkpatrick. Solvable model of a spin-glass. Physical Review Letters, 35(26):1792–1795, 1975.ADSGoogle Scholar
  98. 98.
    B. Derrida. The random energy model. Phys. Rep., 67:29–35, 1980.ADSMathSciNetGoogle Scholar
  99. 99.
    Bernard Derrida. Random-energy model: an exactly solvable model of disorderes systems. Phys. Rev. B, 24(5):2613–2626, 1981.ADSMathSciNetGoogle Scholar
  100. 100.
    E. Gardner and B. Derrida. The probability distribution of the partition function of the random energy model. J. Phys. A, 22:1975–1982, 1989.ADSMathSciNetGoogle Scholar
  101. 101.
    Edward D. Weinberger. Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern., 63:325–336, 1990.zbMATHGoogle Scholar
  102. 102.
    C.W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, Berlin, 2nd edition, 1990.zbMATHGoogle Scholar
  103. 103.
    S.A. Kauffman. The Origin of Order. Oxford University Press, New York, Oxford, 1993.Google Scholar
  104. 104.
    K.J. Laidler. Chemical Kinetics. Harper, New York, 3rd edition, 1992.Google Scholar
  105. 105.
    L.K. Grover. Local search and the local structure of NP-complete problems. Oper. Res. Lett., 12:235–243, 1992.zbMATHMathSciNetGoogle Scholar
  106. 106.
    W. Miller Jr. Symmetry and Separation of Variable. Volume 4 of Enceyclopedia of Mathematics and its Applications, Cambridge Univ. Press, Cambridge, UK, 1984.Google Scholar
  107. 107.
    R.A. Brualdi and H.J. Ryser. Combinatorial Matrix Theory. Cambridge Univ. Press, Cambridge UK, 1991.zbMATHGoogle Scholar
  108. 108.
    I. Chavel. Eigenvalues in Riemannian Geometry. Academic Press, Orlando Fl., 1984.zbMATHGoogle Scholar
  109. 109.
    R. Palmer. Optimization on rugged landscapes. In A. S. Perelson and S. A. Kauffman, editors, Molecular Evolution on Rugged Landscapes: Proteins, RNA, and the Immune Systems, pages 3–25, Addison Wesley, Redwood City, CA, 1991.Google Scholar
  110. 110.
    A.J. Bray and M.A. Moore. Metastable states in spin glasses. J. Phys. C: Solid St. Phys., 13:L469–L476, 1980.ADSGoogle Scholar
  111. 111.
    A.J. Bray and M.A. Moore. Metastable states in spin glasses with short-ranged interactions J. Phys. C: Solid St. Phys., 14:1313–1327, 1981.ADSMathSciNetGoogle Scholar
  112. 112.
    C. De Dominicis, M. Gabay, T. Garel, and H. Orland. White and weighted averages over solutions of the Thouless Anderson Palmer equations for the Sherrington Kirkpatrick spin glass. J. Physique, 41:923–930, 1980.MathSciNetGoogle Scholar
  113. 113.
    Bernard Derrida and E. Gardner. Metastable states of a spin glass chain at 0 temperature. J. Physique, 47:959–965, 1986.Google Scholar
  114. 114.
    F. Tanaka and S.F. Edwards. Analytic theory of the ground state properties of a spin glass: I. Ising spin glass. J. Phys. F: Metal Phys., 10:2769–2778, 1980.ADSGoogle Scholar
  115. 115.
    Alan S. Perelson and Catherine A. Macken. Protein evolution on partially correlated landscapes. Santa Fe Institute Preprint 94-11-060.Google Scholar
  116. 116.
    D.J. Thouless, P.W. Anderson, and R.G. Palmer. Phil. Mag., 35:593, 1977.ADSGoogle Scholar
  117. 117.
    Catherine A. Macken and Peter F. Stadler. Rugged landscapes. 1995. To appear in SFI summerschool volume 1993.Google Scholar
  118. 118.
    E.H.L. Aarts and J. Korst. Simulated Annealing and Boltzman Machines. J. Wiley & Sons, New York, 1990.Google Scholar
  119. 119.
    R.H.J.M. Otten and L.P.P.P. vanGinneken. The Annealing Algorith. Kluwer Acad. Publ., Boston, 1989.Google Scholar
  120. 120.
    Y. Fu and P. W. Anderson. Application of statistical mechanics to NP-complete problems in combinatorial optimization. J. Phys. A: Math. Gen., 19:1605–1620, 1986.ADSMathSciNetzbMATHGoogle Scholar
  121. 121.
    J. Bernasconi. Low autocorrelation binary sequences: statistical mechanics and configuration space analysis. J. Physique, 48:559–567, 1987.Google Scholar
  122. 122.
    Bärbel Krakhofer. Local Optima in Landscapes of Combinatorial Optimization Problems. Master's thesis, University of Vienna, Dept. of Theoretical Chemistry, 1995.Google Scholar
  123. 123.
    G. B. Sorkin. Combinatorial optimization, simulated annealing, and fractals. Technical Report RC13674 (No. 61253), IBM Research Report, 1988.Google Scholar
  124. 124.
    R. Voss. Characterization and measurement of random fractals. Physical Scripta, T13:257–260, 1986.Google Scholar
  125. 125.
    M. Mézard and G. Parisi. Replicas and optimization. J. Physique Lett., 46:L771–L778, 1986.Google Scholar
  126. 126.
    M. Zuker. The use of dynamic programming algorithms in RNA secondary structure prediction. In Michael S. Waterman, editor, Mathematical Methods for DNA Sequences, pages 159–184, CRC Press, 1989.Google Scholar
  127. 127.
    M. Zuker and D. Sankoff. RNA secondary structures and their prediction. Bull. Math. Biol., 46(4):591–621, 1984.zbMATHGoogle Scholar
  128. 128.
    H. M. Martinez. An RNA folding rule. Nucl. Acid. Res., 12:323–335, 1984.Google Scholar
  129. 129.
    Manfred Tacker. Robust Properties of RNA Secondary Structure Folding Algorithms. PhD thesis, University of Vienna, 1993.Google Scholar
  130. 130.
    John S. McCaskill. The equilibrium partition function and base pair binding probabilities for RNA secondary structure. Biopolymers, 29:1105–1119, 1990.Google Scholar
  131. 131.
    Ruth Nussinov, George Piecznik, Jerrold R. Griggs, and Daniel J. Kleitman. Algorithms for loop matching. SIAM J. Appl. Math., 35(1):68–82, 1978.zbMATHMathSciNetGoogle Scholar
  132. 132.
    A. A. Mironov, L. P. Dyakonova, and A. E. Kister. A kinetic approach to the prediction of RNA secondary structures. Journal of Biomolecular Structure and Dynamics, 2(5):953, 1985.Google Scholar
  133. 133.
    A. A. Mironov and A. E. Kister. RNA secondary structure formation during transcription. J. of Biomolecular Structure and Dynamics, 4:1–9, 1986.Google Scholar
  134. 134.
    M. Zuker. mfold-2.0. pub/mfold.tar.Z (Public Domain Software).Google Scholar
  135. 135.
    I.L. Hofacker, W. Fontana, P.F. Stadler, L.S. Bonhoeffer, M. Tacker, and P. Schuster. Vienna RNA Package. pub/RNA/ViennaRNA-1.03 (Public Domain Software).Google Scholar
  136. 136.
    W. Salser. Globin messenger RNA sequences—analysis of base-pairing and evolutionary implications. Cold Spring Harbour Symp. Quant. Biol., 42:985, 1977.Google Scholar
  137. 137.
    Susan M. Freier, Ryszard Kierzek, John A. Jaeger, Naoki Sugimoto, Marvin H. Caruthers Thomas Neilson, and Douglas H. Turner. Improved free-energy parameters for predictions of RNA duplex stability. Proc. Natl. Acad. Sci., USA, 83:9373–9377, 1986.ADSGoogle Scholar
  138. 138.
    A. S. Perelson and G. Oster. Theoretical studies of clonal selection: minimal antibody repertoire size and reliability of self/non-self discrimination. Journal of Theoretical Biology, 81:645–670, 1979.MathSciNetGoogle Scholar
  139. 139.
    L. A. Segel and A. P. Perelson. Computations in shape space: a new approach to immune network theory. In Theoretical Immunology. Part Two, pages 321–343, Addison-Wesley, Redwood City (Cal.), 1988.Google Scholar
  140. 140.
    Bruce A. Shapiro. An algorithm for comparing multiple RNA secondary stuctures. CABIOS, 4(3):387–393, 1988.Google Scholar
  141. 141.
    Bruce A. Shapiro and Khaizhong Zhang. Comparing multiple RNA secondary structures using tree comparisons. CABIOS, 6:309–318, 1990.Google Scholar
  142. 142.
    K. Tai. The tree-to-tree correction problem. J. ACM, 26:422–433, 1979.zbMATHMathSciNetGoogle Scholar
  143. 143.
    K. Ohmori and E. Tanaka. A unified view on tree metrics. In G. Ferrate, editor, Syntactic and Structural Pattern Recognition, pages 85–100, Springer-Verlag, Berlin, Heidelberg, 1988.Google Scholar
  144. 144.
    Pauline Hogeweg and B. Hesper. Energy directed folding of RNA sequences. Nucleic acids research, 12:67–74, 1984.Google Scholar
  145. 145.
    I.L. Hofacker, P. Schuster, and P.F. Stadler. Combinatorics of secondary structures. submitted to SIAM J. Disc. Math., 1993.Google Scholar
  146. 146.
    P.R. Stein and M.S. Waterman. On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Mathematics, 26:261–272, 1978.MathSciNetGoogle Scholar
  147. 147.
    M. S. Waterman. Secondary structure of single-stranded nucleic acids. Studies on foundations and combinatorics, Advances in mathematics supplementary studies, Academic Press N.Y., 1:167–212, 1978.MathSciNetGoogle Scholar
  148. 148.
    E. Szathmáry. Four letters in the genetic alphabet: a frozen evolutionary optimum? Proc. Roy. Soc. London B, 245:91–99, 1991.ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Peter F. Stadler
    • 1
  • Santa Fe Institute
    • 2
  1. 1.Institut für Theoretische ChemieUniversität WienWienAustria
  2. 2.Santa FeUSA

Personalised recommendations