Cellular and Molecular Life Sciences

, Volume 68, Issue 16, pp 2711–2737 | Cite as

Fractal symmetry of protein interior: what have we learned?

Review
First Online:
Received:
Revised:
Accepted:

Abstract

The application of fractal dimension-based constructs to probe the protein interior dates back to the development of the concept of fractal dimension itself. Numerous approaches have been tried and tested over a course of (almost) 30 years with the aim of elucidating the various facets of symmetry of self-similarity prevalent in the protein interior. In the last 5 years especially, there has been a startling upsurge of research that innovatively stretches the limits of fractal-based studies to present an array of unexpected results on the biophysical properties of protein interior. In this article, we introduce readers to the fundamentals of fractals, reviewing the commonality (and the lack of it) between these approaches before exploring the patterns in the results that they produced. Clustering the approaches in major schools of protein self-similarity studies, we describe the evolution of fractal dimension-based methodologies. The genealogy of approaches (and results) presented here portrays a clear picture of the contemporary state of fractal-based studies in the context of the protein interior. To underline the utility of fractal dimension-based measures further, we have performed a correlation dimension analysis on all of the available non-redundant protein structures, both at the level of an individual protein and at the level of structural domains. In this investigation, we were able to separately quantify the self-similar symmetries in spatial correlation patterns amongst peptide–dipole units, charged amino acids, residues with the π-electron cloud and hydrophobic amino acids. The results revealed that electrostatic environments in the interiors of proteins belonging to ‘α/α toroid’ (all-α class) and ‘PLP-dependent transferase-like’ domains (α/β class) are highly conducive. In contrast, the interiors of ‘zinc finger design’ (‘designed proteins’) and ‘knottins’ (‘small proteins’) were identified as folds with the least conducive electrostatic environments. The fold ‘conotoxins’ (peptides) could be unambiguously identified as one type with the least stability. The same analyses revealed that peptide–dipoles in the α/β class of proteins, in general, are more correlated to each other than are the peptide–dipoles in proteins belonging to the all-α class. Highly favorable electrostatic milieu in the interiors of TIM-barrel, α/β-hydrolase structures could explain their remarkably conserved (evolutionary) stability from a new light. Finally, we point out certain inherent limitations of fractal constructs before attempting to identify the areas and problems where the implementation of fractal dimension-based constructs can be of paramount help to unearth latent information on protein structural properties.

Keywords

Fractal Dimension Aromatic Amino Acid Correlation Dimension Hydrophobic Amino Acid Fractal Kinetic 
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.
This is a preview of subscription content, log in to check access

Notes

Acknowledgments

One of the authors, Anirban, would like to thank the COE-DBT scheme, Govt. of India, for supporting him during the tenure of this work. He would also like to thank Charudatta Navare for various technical help and Professor Paul Bourke for permitting him to use the self-similarity figures.

Supplementary material

18_2011_722_MOESM1_ESM.xls (24 kb)
Supplementary material 1 (XLS 24 kb)

References

  1. 1.
    Karplus M (2000) Aspects of protein reaction dynamics: deviations from simple behavior. J Phys Chem B 104:11–27CrossRefGoogle Scholar
  2. 2.
    De Leeuw M, Reuveni S, Klafter J, Granek R (2009) Coexistence of flexibility and stability of proteins: an equation of state. PLoS One 4(10):e7296PubMedCrossRefGoogle Scholar
  3. 3.
    Reuveni S (2008) Proteins: unraveling universality in a realm of specificity. PhD thesis. Tel Aviv University, Tel AvivGoogle Scholar
  4. 4.
    Goetze T, Brickmann J (1992) Self similarity of protein surfaces. Biophys J 61:109–118PubMedCrossRefGoogle Scholar
  5. 5.
    Tissen J, Fraaije J, Drenth J, Berendsen H (1994) Mesoscopic theories for protein crystal growth. Acta Cryst D 50:569–571CrossRefGoogle Scholar
  6. 6.
    Banerji A, Ghosh I (2009) A new computational model to study mass inhomogeneity and hydrophobicity inhomogeneity in proteins. Eur Biophys J 38:577–587PubMedCrossRefGoogle Scholar
  7. 7.
    Liang J, Dill KA (2001) Are proteins well-packed? Biophys J 81:751–766PubMedCrossRefGoogle Scholar
  8. 8.
    Havlin S, Ben-Avraham D (1982) New approach to self-avoiding walks as a critical phenomenon. J Phys A 15:L321–L328CrossRefGoogle Scholar
  9. 9.
    Havlin S, Ben-Avraham D (1982) Fractal dimensionality of polymer chains. J Phys A 15:L311–L316CrossRefGoogle Scholar
  10. 10.
    Havlin S, Ben-Avraham D (1982) New method of analysing self-avoiding walks in four dimensions. J Phys A 15:L317–L320CrossRefGoogle Scholar
  11. 11.
    Mandelbrot BB (1982) The fractal geometry of nature. W. H. Freeman and Co, San FranciscoGoogle Scholar
  12. 12.
    Hausdorff F (1919) Dimension und ¨ausseres Mass. Math Ann 79:157–179CrossRefGoogle Scholar
  13. 13.
    Barnsley M (1988) Fractals everywhere. Academic Press, San DiegoGoogle Scholar
  14. 14.
    Falconer K (1990) Fractal geometry: mathematical foundations and applications. Wiley, New YorkGoogle Scholar
  15. 15.
    Meakin P (1998) Fractals scaling, and growth far from equilibrium. Cambridge University Press, CambridgeGoogle Scholar
  16. 16.
    Isogai Y, Itoh T (1984) Fractal analysis of tertiary structure of protein molecule. J Phys Soc Jpn 53:2162CrossRefGoogle Scholar
  17. 17.
    Wagner GC, Colvin JT, Allen JP, Stapleton HJ (1985) Fractal models of protein structure, dynamics, and magnetic relaxation. J Am Chem Soc 107:20CrossRefGoogle Scholar
  18. 18.
    Colvin JT, Stapleton HJ (1985) Fractal and spectral dimensions of biopolymer chains: solvent studies of electron spin relaxation rates in myoglobin azide. J Chem Phys 82:10CrossRefGoogle Scholar
  19. 19.
    Tanford C (1961) Physical chemistry of macromolecules. Wiley, New YorkGoogle Scholar
  20. 20.
    Wang CX, Shi YY, Huang FH (1990) Fractal study of tertiary structure of proteins. Phys Rev A 41:7043–7048PubMedCrossRefGoogle Scholar
  21. 21.
    Xiao Y (1994) Comment on fractal study of tertiary structure of proteins. Phys Rev E 46:6Google Scholar
  22. 22.
    Bytautas L, Klein DJ, Randic M, Pisanski T (2000) Foldedness in linear polymers: a difference between graphical and Euclidean distances, DIMACS. Ser Discr Math Theor Comput Sci 51:39–61Google Scholar
  23. 23.
    Aszódi A, Taylor WR (1993) Connection topology of proteins. Bioinformatics 9:523–529CrossRefGoogle Scholar
  24. 24.
    Gennes P (1996) Scaling concepts in polymer physics. Cornell University Press, IthacaGoogle Scholar
  25. 25.
    Elber R (1989) Fractal analysis of protein. In: Avnir D (ed) The fractal approach to heterogeneous chemistry. Wiley, New YorkGoogle Scholar
  26. 26.
    Dewey TG (1993) Protein structure and polymer collapse. J Chem Phys 98:2250–2257CrossRefGoogle Scholar
  27. 27.
    Dewey TG (1995) Fractal dimensions of proteins: what are we learning? Het Chem Rev 2:91–101Google Scholar
  28. 28.
    Dewey TG (1997) Fractals in molecular biophysics. Oxford University Press, New YorkGoogle Scholar
  29. 29.
    Enright MB, Leitner DM (2005) Mass fractal dimension and the compactness of proteins. Phys Rev E 71:011912Google Scholar
  30. 30.
    Reuveni S, Granek R, Klafter J (2008) Proteins: coexistence of stability and flexibility. Phys Rev Lett 100:208101Google Scholar
  31. 31.
    Banerji A, Ghosh I (2009) Revisiting the myths of protein interior: studying proteins with mass-fractal hydrophobicity-fractal and polarizability-fractal dimensions. PLoS One 4(10):e7361PubMedCrossRefGoogle Scholar
  32. 32.
    Moret MA, Miranda JG, Nogueira E Jr, Santana MC, Zebende GF (2005) Self-similarity and protein chains. Phys Rev E 71:012901CrossRefGoogle Scholar
  33. 33.
    Lee CY (2006) Mass fractal dimension of the ribosome and implication of its dynamic characteristics. Phys Rev E 73:042901CrossRefGoogle Scholar
  34. 34.
    Figueirêdo PH, Moret MA, Nogueira E Jr, Coutinho S (2008) Dihedral-angle Gaussian distribution driving protein folding. Phys A 387:2019–2024CrossRefGoogle Scholar
  35. 35.
    Hong L, Jinzhi L (2009) Scaling law for the radius of gyration of proteins and its dependence on hydrophobicity. J Polym Sci Part B 47:207–214CrossRefGoogle Scholar
  36. 36.
    Novikov VU, Kozlov GV (2000) Structure and properties of polymers in terms of the fractal approach. Russ Chem Rev 69:523–549CrossRefGoogle Scholar
  37. 37.
    Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Phys D 9:183–208CrossRefGoogle Scholar
  38. 38.
    Lee CY (2008) Self-similarity of biopolymer backbones in the ribosome. Phys A 387:4871–4880CrossRefGoogle Scholar
  39. 39.
    Tejera E, Machadoa A, Rebelo I, Nieto-Villar J (2009) Fractal protein structure revisited: topological kinetic and thermodynamic relationships. Phys A 388:4600–4608CrossRefGoogle Scholar
  40. 40.
    Takens F (1985) On the numerical determination of the dimension of an attractor. In: Braaksma B, Broer H, Takens F (eds) Lecture notes in mathematics, vol 1125. Springer, Berlin, pp 99–106Google Scholar
  41. 41.
    Steinbach PJ, Ansari A, Berendzen J, Braunstein D, Chu K, Cowen BR, Ehrenstein D, Frauenfelder H, Johnson JB, Lamb DC (1991) Ligand binding to heme proteins: connection between dynamics and function. Biochemistry 30:3988–4001PubMedCrossRefGoogle Scholar
  42. 42.
    Rasmussen BF, Stock AM, Ringe D, Petsko GA (1992) Crystalline ribonuclease A loses function below the dynamical transition at 220 K. Nature 357:423–424PubMedCrossRefGoogle Scholar
  43. 43.
    Wilson KG (1975) The renormalization group: critical phenomena and the Kondo problem. Rev Mod Phys 47:773CrossRefGoogle Scholar
  44. 44.
    Wilson KG (1979) Problems in physics with many scales of length. Sci Am 241:140–157CrossRefGoogle Scholar
  45. 45.
    Goldenfeld N (1992) Lectures on phase transitions and the renormalization group. Addison-Wesley, ReadingGoogle Scholar
  46. 46.
    Goldstein RA, Luthey-Schulten ZA, Wolynes PG (1992) Protein tertiary structure recognition using optimized Hamiltonians with local interactions. Proc Natl Acad Sci USA 89:9029–9033PubMedCrossRefGoogle Scholar
  47. 47.
    Goldstein RA, Luthey-Schulten ZA, Wolynes PG (1992) Optimal protein-folding codes from spin-glass theory. Proc Natl Acad Sci USA 89:4918–4922PubMedCrossRefGoogle Scholar
  48. 48.
    Family F (1982) Direct renormalization group study of loops in polymer. Phys Lett 92A:341–344Google Scholar
  49. 49.
    Pierri CL, Grassi AD, Turi A (2008) Lattices for ab initio protein structure prediction. Protein Struct Funct Bioinf 73:351–361CrossRefGoogle Scholar
  50. 50.
    Böhm G (1991) Protein folding and deterministic chaos: limits of protein folding simulations and calculations. Chaos Solitons Fractals 1:375–382CrossRefGoogle Scholar
  51. 51.
    Li HQ, Chen SH, Zhao HM (1990) Fractal structure and conformational entropy of protein chain. Int J Biol Macromol 12:374–378PubMedCrossRefGoogle Scholar
  52. 52.
    Nonnenmacher TF (1989) Fractal scaling mechanisms in biomembranes. Eur Biophys J 16:375–379PubMedCrossRefGoogle Scholar
  53. 53.
    Bohr HG, Wolynes PG (1992) Initial events of protein folding from an information-processing viewpoint. Phys Rev A 46:5242–5248PubMedCrossRefGoogle Scholar
  54. 54.
    Chan HS, Dill KA (1991) Polymer principles in protein structure and stability. Annu Rev Biophys Chem 20:447–490CrossRefGoogle Scholar
  55. 55.
    Coveney PV, Fowler PW (2005) Modelling biological complexity: a physical scientist’s perspective. J R Soc Interf 2:267–280CrossRefGoogle Scholar
  56. 56.
    Heath AP, Kavraki LE, Clementi C (2007) From coarse-grain to all-atom: toward multiscale analysis of protein landscapes. Proteins 68:646–661PubMedCrossRefGoogle Scholar
  57. 57.
    Song C, Havlin S, Makse HA (2006) Origins of fractality in the growth of complex networks. Nat Phys 2:275–281CrossRefGoogle Scholar
  58. 58.
    Kitsak M, Havlin S, Paul G, Riccaboni M, Pammolli F, Stanley HE (2007) Betweenness centrality of fractal and nonfractal scale-free model networks and tests on real networks. Phys Rev E 75:056115CrossRefGoogle Scholar
  59. 59.
    Rozenfeld HD, Song C, Makse HA (2010) The small world-fractal transition in complex networks through renormalization group. Phys Rev Lett 104:025701PubMedCrossRefGoogle Scholar
  60. 60.
    Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New YorkGoogle Scholar
  61. 61.
    Karplus M, McCammon J (1983) Dynamics of proteins: elements and functions. Annu Rev Biochem 53:263–300CrossRefGoogle Scholar
  62. 62.
    Levitt M, Sanders C, Stern PS (1985) Protein normal-mode dynamics; trypsin inhibitor, crambin, ribonuclease, and lysozyme. J Mol Biol 181:423–447PubMedCrossRefGoogle Scholar
  63. 63.
    Alexander S, Orbach RL (1982) Density of states on fractals: fractons. J Phys Lett 43:L625–L631CrossRefGoogle Scholar
  64. 64.
    Elber R, Karplus M (1986) Low frequency modes in proteins: use of effective-medium approximation to interpret fractal dimension observed in electron-spin relaxation meaurements. Phys Rev Lett 56:394–397PubMedCrossRefGoogle Scholar
  65. 65.
    Burioni R, Cassi D (1996) Universal properties of spectral dimension. Phys Rev Lett 76:1091–1093PubMedCrossRefGoogle Scholar
  66. 66.
    Burioni R, Cassi D, Fontana MP, Vulpiani A (2002) Vibrational thermodynamic instability of recursive networks. Europhys Lett 58:806–810CrossRefGoogle Scholar
  67. 67.
    Burioni R, Cassi D, Cecconi F, Vulpiani A (2004) Topological thermal instability and length of proteins. Proteins Struct Funct Bioinf 55:529–535CrossRefGoogle Scholar
  68. 68.
    Leitner DM (2009) Frequency-resolved communication maps for proteins and other nanoscale materials. J Chem Phys 130:195101PubMedCrossRefGoogle Scholar
  69. 69.
    Chen J, Bryngelson JD, Thirumalai D (2008) Estimations of the size of nucleation regions in globular proteins. J Phys Chem B 112:16115–16120PubMedCrossRefGoogle Scholar
  70. 70.
    Frauenfelder H, Parak F, Young RD (1988) Conformational substates in proteins. Annu Rev Biophys Chem 17:451–479CrossRefGoogle Scholar
  71. 71.
    Allen JP, Colvin JT, Stinson DG, Flynn CP, Stapleton HJ (1982) Protein conformation from electron spin relaxation data. Biophys J 38:299–310PubMedCrossRefGoogle Scholar
  72. 72.
    Stapleton HJ, Allen JP, Flynn CP, Stinson DG, Kurtz SR (1980) Fractal form of proteins. Phys Rev Lett 45:1456–1459CrossRefGoogle Scholar
  73. 73.
    Helman JS, Coniglio A, Tsallis C (1984) Fractons and the fractal structure of proteins. Phys Rev Lett 53:1195–1197CrossRefGoogle Scholar
  74. 74.
    Herrmann HJ (1986) Comment on fractons and the fractal structure of proteins. Phys Rev Lett 56:2432PubMedCrossRefGoogle Scholar
  75. 75.
    Stapleton HJ (1985) Comment on fractons and the fractal structure of proteins. Phys Rev Lett 54:1734PubMedCrossRefGoogle Scholar
  76. 76.
    Liebovitch LS, Fischbary J, Koniarek JP, Todorova I, Wang M (1987) Fractal model of ion-channel kinetics. Biochim Biophys Acta 869:173–180Google Scholar
  77. 77.
    Liebovitch LS, Fischbary J, Koniarek J (1987) Ion channel kinetics: a model based on fractal scaling rather than multistate markov processes. Math Biosci 84:37–68CrossRefGoogle Scholar
  78. 78.
    Liebovitch LS, Sullivan JM (1987) Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons. Biophys J 52:979–988PubMedCrossRefGoogle Scholar
  79. 79.
    Korn SJ, Horn R (1988) Statistical discrimination of fractal and Markov models of single channel gating. Biophys J 54:871–877PubMedCrossRefGoogle Scholar
  80. 80.
    French AS, Stockbridge LL (1988) Fractal and Markov behavior in ion channel kinetics. Can J Physiol Pharm 66:967–970CrossRefGoogle Scholar
  81. 81.
    Millhauser G, Salpeter L, Oswald RE (1988) Diffusion models of ion-channel gating and the origin of the power-law distributions from single-channel recording. Proc Natl Acad Sci USA 85:1503–1507PubMedCrossRefGoogle Scholar
  82. 82.
    Lauger P (1988) Internal motions in proteins and gating kinetics of ion channels. Biophys J 53:877–884PubMedCrossRefGoogle Scholar
  83. 83.
    Liebovitch LS, Toth TI (1990) Using fractals to understand the opening and closing of ion channels. Ann Biomed Eng 18:177–194PubMedCrossRefGoogle Scholar
  84. 84.
    Liebovitch LS, Toth TI (1991) A model of ion channel kinetics using deterministic chaotic rather than stochastic processes. J Theor Biol 148:243–267PubMedCrossRefGoogle Scholar
  85. 85.
    Lowen SB, Teich MC (1993) Fractal renewal processes. IEEE Trans Info Theory 39:1669–1671CrossRefGoogle Scholar
  86. 86.
    Churilla AM, Gottschalke WA, Liebovitch LS, Selector LY, Todorov AT, Yeandle S (1995) Membrane potential fluctuations of human T-lymphocytes have fractal characteristics of fractional Brownian motion. Ann Biomed Eng 24:99–108CrossRefGoogle Scholar
  87. 87.
    Lowen SB, Liebovitch LS, White JA (1999) Fractal ion-channel behavior generates fractal firing patterns in neuronal models. Phys Rev E 59:5970–5980CrossRefGoogle Scholar
  88. 88.
    Rodriguez M, Pereda E, Gonzalez J, Abdala P, Obeso JA (2003) Neuronal activity in the substantia nigra in the anaesthetized rat has fractal characteristics. Evidence for firing-code patterns in the basal ganglia. Exp Brain Res 151:167–172PubMedCrossRefGoogle Scholar
  89. 89.
    Kim S, Jeong J, Kim YKY, Jung SH, Lee KJ (2005) Fractal stochastic modeling of spiking activity in suprachiasmatic nucleus neurons. J Comp Neurosci 19:39–51CrossRefGoogle Scholar
  90. 90.
    Brooks CL, Karplus M, Pettitt BM (1988) Proteins: a theoretical perspective of dynamics, structure and thermodynamics. Wiley, New YorkGoogle Scholar
  91. 91.
    Dewey TG, Spencer DB (1991) Are protein dynamics fractal? Commun Mol Cell Biophys 7:155–171Google Scholar
  92. 92.
    Bagchi B, Fleming GR (1990) Dynamics of activationless reactions in solution. J Phys Chem 94:9–20CrossRefGoogle Scholar
  93. 93.
    Zwanzig R (1990) Rate processes with dynamical disorder. Acc Chem Res 23:148–152CrossRefGoogle Scholar
  94. 94.
    Dewey TG, Bann JG (1992) Protein dynamics and noise. Biophys J 63:594–598PubMedCrossRefGoogle Scholar
  95. 95.
    Ramakrishnan A, Sadana A (1999) Analysis of analyte-receptor binding kinetics for biosensor applications: an overview of the influence of the fractal dimension on the surface on the binding rate coefficient. Biotech Appl Biochem 29:45–57Google Scholar
  96. 96.
    Goychuk I, Hänggi P (2002) Ion channel gating: a first-passage time analysis of the Kramers type. Proc Natl Acad Sci USA 99:3552–3556PubMedCrossRefGoogle Scholar
  97. 97.
    Carlini P, Bizzarri AR, Cannistraro S (2002) Temporal fluctuations in the potential energy of proteins: noise and diffusion. Phys D 165:242–250CrossRefGoogle Scholar
  98. 98.
    Kopelmann R (1986) Rate processes on fractals: theory, simulations, and experiments. J Stat Phys 42:185–200CrossRefGoogle Scholar
  99. 99.
    Kopelmann R (1988) Fractal reaction kinetics. Science (Washington DC) 241:1620–1625CrossRefGoogle Scholar
  100. 100.
    Li HQ, Chen SH, Zhao HM (1990) Fractal mechanisms for the allosteric effects of proteins and enzyme. Biophys J 58:1313–1320PubMedCrossRefGoogle Scholar
  101. 101.
    Lewis M, Rees DC (1985) Fractal surfaces of proteins. Science 230:1163–1165PubMedCrossRefGoogle Scholar
  102. 102.
    Argyrakis P, Kopelman R (1990) Nearest-neighbor distance distribution and self-ordering in diffusion-controlled reactions. Phys Rev A 41:2114–2126PubMedCrossRefGoogle Scholar
  103. 103.
    Turner TE, Schnell S, Burrage K (2004) Stochastic approaches for modelling in vivo reactions. Comp Biol Chem 28:165–178CrossRefGoogle Scholar
  104. 104.
    Berry H (2002) Monte Carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation. Biophys J 83:1891–1901PubMedCrossRefGoogle Scholar
  105. 105.
    Yuste SB, Acedo L, Lindenberg K (2004) Reaction front in an A + B -> C reaction-subdiffusion process. Phys Rev E 69:036126CrossRefGoogle Scholar
  106. 106.
    Kosmidis K, Argyrakis P, Macheras P (2003) Fractal kinetics in drug release from finite fractal matrices. J Chem Phys 119:63–73CrossRefGoogle Scholar
  107. 107.
    Grimaa R, Schnell S (2006) A systematic investigation of the rate laws valid in intracellular environments. Biophys Chem 124:1–10CrossRefGoogle Scholar
  108. 108.
    Shlesinger MF (1988) Fractal time in condensed matter. Annu Rev Phys Chem 39:269–290CrossRefGoogle Scholar
  109. 109.
    Yu X, Leitner DM (2003) Anomalous diffusion of vibrational energy in proteins. J Chem Phys 119:12673–12679CrossRefGoogle Scholar
  110. 110.
    Leitner DM (2008) Energy flow in proteins. Annu Rev Phys Chem 59:233–259PubMedCrossRefGoogle Scholar
  111. 111.
    Li MS, Klimov DK, Thirumalai D (2004) Finite size effects on thermal denaturation of globular proteins. Phys Rev Lett 93:268107PubMedCrossRefGoogle Scholar
  112. 112.
    Peierls RE (1934) Bemerkungüber Umwandlungstemperaturen. Helv Phys Acta 7:S81–S83Google Scholar
  113. 113.
    Reuveni S, Granek R, Klafter J (2010) Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure. Proc Natl Acad Sci USA 107:13696–13700PubMedCrossRefGoogle Scholar
  114. 114.
    Lois G, Blawzdziewicz J, O’Hern CS (2010) Protein folding on rugged energy landscapes: conformational diffusion on fractal networks. Phys Rev E 81:051907CrossRefGoogle Scholar
  115. 115.
    Sangha AK, Keyes T (2009) Proteins fold by subdiffusion of the order parameter. J Phys Chem B 113:15886–15894PubMedCrossRefGoogle Scholar
  116. 116.
    Moret MA, Santana MC, Zebende GF, Pascutti PG (2009) Self-similarity and protein compactness. Phys Rev E 80:041908CrossRefGoogle Scholar
  117. 117.
    Morita H, Takano M (2009) Residue network in protein native structure belongs to the universality class of three dimensional critical percolation cluster. Phys Rev E79:020901Google Scholar
  118. 118.
    Murzin AG, Brenner SE, Hubbard T, Chothia C (1995) SCOP: a structural classification of proteins database for the investigation of sequences and structures. J Mol Biol 247:536–540PubMedGoogle Scholar
  119. 119.
    Bashford D, Karplus M (1990) pKas of ionization groups in proteins: atomic detail from a continuum electrostatic model. Biochemistry 29:10219–10225PubMedCrossRefGoogle Scholar
  120. 120.
    Warshel A, Papazyan A (1998) Electrostatic effects in macromolecules: fundamental concepts and practical modeling. Curr Opin Struct Biol 8:211–217PubMedCrossRefGoogle Scholar
  121. 121.
    Zhou Z, Payne P, Vasquez M, Kuhn N, Levitt M (1996) Finite-difference solution of the Poisson–Boltzmann equation: complete elimination of self-energy. J Comput Chem 17:1344–1351CrossRefGoogle Scholar
  122. 122.
    Lu B, Zhang D, McCammon J (2005) Computation of electrostatic forces between solvated molecules determined by the Poisson–Boltzmann equation using a boundary element method. J Chem Phys 122:214102–214108PubMedCrossRefGoogle Scholar
  123. 123.
    Feig M, Onufriev A, Lee M, Im W, Case E, Brooks C (2004) Performance comparison of generalized Born and Poisson methods in the calculation of electrostatic solvation energies for protein structures. J Comput Chem 25:265–284PubMedCrossRefGoogle Scholar
  124. 124.
    Spassov V, Ladenstein R, Karshikoff (1997) A optimization of the electrostatic interactions between ionized groups and peptide dipoles in proteins. Protein Sci 6:1190–1196PubMedCrossRefGoogle Scholar
  125. 125.
    Petrey D, Honig B (2000) Free energy determinants of tertiary structure and the evaluation of protein models. Protein Sci 9:2181–2191PubMedCrossRefGoogle Scholar
  126. 126.
    Shoemaker K, Kim P, York E, Stewart J, Baldwin R (1987) Tests of the helix dipole model for stabilization of alpha-helices. Nature 326:563–567PubMedCrossRefGoogle Scholar
  127. 127.
    Åqvist J, Luecke H, Quiocho F, Warshel A (1991) Dipoles localized at helix termini of proteins stabilize charges. Proc Natl Acad Sci USA 88:2026–2030PubMedCrossRefGoogle Scholar
  128. 128.
    Wada A (1976) The alpha-helix as an electric macro-dipole. Adv Biophys 9:1–63Google Scholar
  129. 129.
    Miyazawa S, Jernigan R (1999) Evaluation of short-range interactions as secondary structure energies for protein fold and sequence recognition. Proteins 36:347–356PubMedCrossRefGoogle Scholar
  130. 130.
    Selvaraj S, Gromiha MM (2003) Role of hydrophobic clusters and long-range contact networks in the folding of (alpha/beta)8 barrel proteins. Biophys J 84:1919–1925PubMedCrossRefGoogle Scholar
  131. 131.
    Berman H, Henrick K, Nakamura H (2003) Announcing the worldwide. Protein Data Bank Nat Struct Biol 10:980Google Scholar
  132. 132.
    Theiler J (1987) Efficient algorithm for estimating the correlation dimension from a set of discrete points. Phys Rev A 36:4456–4462PubMedCrossRefGoogle Scholar
  133. 133.
    Gallivan JP, Dougherty DA (1999) Cation–π interactions in structural biology. Proc Natl Acad Sci USA 96:9459–9464PubMedCrossRefGoogle Scholar
  134. 134.
    Glockle WG, Nonnenmacher TF (1995) A fractional calculus approach to self-similar protein dynamics. Biophys J 68:46–53PubMedCrossRefGoogle Scholar
  135. 135.
    Cserzöa M, Vicsek T (1991) Self-affine fractal analysis of protein structures. Chaos Solitons Fractals 1:431–438CrossRefGoogle Scholar
  136. 136.
    Isvoran A (2004) Describing some properties of adenylate kinase using fractal concepts. Chaos Solitons Fractals 19:141–145CrossRefGoogle Scholar
  137. 137.
    Mitra C, Rani M (1993) Protein sequences as random fractals. J Biosci 18:213–220CrossRefGoogle Scholar
  138. 138.
    Banerji A, Ghosh I (2010) Mathematical criteria to observe mesoscopic emergence of protein biochemical properties. J Math Chem 49(3):643–665. doi: 10.1007/s10910-010-9760-9 CrossRefGoogle Scholar
  139. 139.
    Grosberg Y, Khokhlov AR (1994) Statistical physics of macromolecules. American Institute of Physics, WoodburyGoogle Scholar
  140. 140.
    Røgen P, Fain B (2003) Automatic classification of protein structure by using Gauss integrals. Proc Natl Acad Sci USA 100:119–124PubMedCrossRefGoogle Scholar
  141. 141.
    Ramnarayan K, Bohr H, Jalkanen K (2008) Classification of protein fold classes by knot theory and prediction of folds by neural networks: a combined theoretical and experimental approach. Theor Chim Acta 119:265–274CrossRefGoogle Scholar
  142. 142.
    Kolaskar AS, Ramabrahmam V (1983) Conformational properties of pairs of amino acids. Int J Peptide Protein Res 22:83–91CrossRefGoogle Scholar
  143. 143.
    Alexander S (1989) Vibrations of fractals and scattering of light from aerogels. Phys Rev B 40(11):7953–7965CrossRefGoogle Scholar
  144. 144.
    Korb JP, Bryant RG (2005) Noise and functional protein dynamics. Biophys J 89:2685–2692PubMedCrossRefGoogle Scholar
  145. 145.
    Haliloglu T, Bahar I, Erman B (1997) Gaussian dynamics of folded proteins. Phys Rev Lett 79:3090–3093CrossRefGoogle Scholar
  146. 146.
    Aksimentiev A, Holyst R (1999) Single-chain statistics in polymer systems. Prog Polym Sci 24:1045–1093CrossRefGoogle Scholar
  147. 147.
    Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University, New YorkCrossRefGoogle Scholar
  148. 148.
    Arteca GA (1994) Scaling behaviour of some molecular shape descriptors of polymer chains and protein backbones. Phys Rev E 49:2417–2428CrossRefGoogle Scholar
  149. 149.
    Arteca GA (1995) Scaling regimes of molecular size and self-entanglements in very compact proteins. Phys Rev E 51:2600–2610CrossRefGoogle Scholar
  150. 150.
    Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, New YorkGoogle Scholar
  151. 151.
    Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon Press, OxfordGoogle Scholar
  152. 152.
    Rubinstein M, Colby RH (2003) Polymer physics. Oxford University Press, New YorkGoogle Scholar
  153. 153.
    Hypertext link. http://paulbourke.net/fractals/fracdim/. Accessed 16 May 2011
  154. 154.
    Granek R, Klafter J (2005) Fractons in proteins: can they lead to anomalously decaying time autocorrelations? Phys Rev Lett 95:098106PubMedCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Bioinformatics CentreUniversity of PunePuneIndia
  2. 2.School of Information TechnologyJawaharlal Nehru UniversityNew DelhiIndia
  3. 3.School of Computational & Integrative SciencesJawaharlal Nehru UniversityNew DelhiIndia

Personalised recommendations