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The fractal nature of folds and the Walsh copolymers

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Abstract

We discuss the fractal nature of protein folds and some mathematical apparatus to describe them. In particular, the scaling symmetry of such selfsimilar objects is described using semigroup theory. As is also shown, purely mathematical considerations may open the way toward a possible rational design of a wide class of synthetic folds of diverse chemical nature (for potential nanotechnological applications). In this regard, a special role is given to the Walsh functions and associated with them molecular constructions called Walsh copolymers.

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References

  1. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, 1979)

    Google Scholar 

  2. D.A. Klein, W.A. Seitz, J.E. Kilpatrick, Branched polymer models. J. Appl. Phys. 53(10), 6599–6603 (1982)

    Article  CAS  Google Scholar 

  3. D.J. Klein, W.A. Seitz, Self-similar self-avoiding structures: models for polymers. PNAS 80(10), 3125–3128 (1983)

    Article  CAS  Google Scholar 

  4. D.J. Klein, W.A. Seitz, Graphs, polymer models, excluded volume, and chemical reality, in Topology and Graph Theory in Chemistry, ed. by R.B. King (Elsevier, Amsterdam, 1983), pp. 430–445

    Google Scholar 

  5. L. Bytautas, D.J. Klein, M. Randić, T. Pisanski, Foldedness in linear polymers: a difference between graphical and Euclidean distances. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 51, 39–61 (2000)

    Google Scholar 

  6. Y. Almirantis, A. Provata, An evolutionary model for the origin of non-random long-range order and fractality in the genome. BioEssays 23, 647–656 (2001)

    Article  CAS  Google Scholar 

  7. N.N. Oiwa, J.A. Glazier, The fractal structure of the mitochondrial genomes. Phys. A 311, 221–230 (2002)

    Article  CAS  Google Scholar 

  8. M.A. Moret, J.G. Miranda, E. Noqueira Jr, M.C. Santana, G.F. Zebende, Self-similarity and protein chains. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 71(1 Pt 1), 012901 (2005)

    Article  CAS  Google Scholar 

  9. C. Cattani, Fractals and hidden symmetries in DNA. Math. Probl. Eng. 2010, Article ID 507056, 1–31

  10. N. Todoroff, J. Kunze, H. Schreuder, K.-H. Baringhaus, G. Schneider, Fractal dimensions of macromolecular structures. Mol. Inf. 33, 588–596 (2014)

    Article  CAS  Google Scholar 

  11. R. Hancock, Structures and functions in the crowded nucleus: new biophysical insights. Front. Phys. 2(53), 1–7 (2014). doi:10.3389/fphy.2014.00053

    Google Scholar 

  12. R.P. Bywater, Protein folding: a problem with multiple solutions. J. Biomol. Struct. Dyn. 31(4), 351–362 (2013)

    Article  CAS  Google Scholar 

  13. A. Ben-Naim, Levinthal’s question revisited, and answered. J. Biomol. Struct. Dyn. 31(4), 113–124 (2013)

    Google Scholar 

  14. I.N. Berezovsky, V.M. Kirzhner, A.Z. Kirzhner, V.R. Rosenfeld, E.N. Trifonov, Closed loops: persistence of the protein chain returns. Protein Eng. 15(12), 955–957 (2002)

    Article  CAS  Google Scholar 

  15. I.N. Berezovsky, A.Z. Kirzhner, V.R. Rosenfeld, E.N. Trifonov, Protein sequences yield a proteomic code. J. Biomol. Struct. Dynam. 21(3), 317–326 (2003)

    Article  CAS  Google Scholar 

  16. N. Papandreou, I.N. Berezovsky, A. Lopes, E. Eliopoulos, J. Chomilier, Universal positions in globular proteins. From observation to simulation. Eur. J. Biochem. 271, 4762–4768 (2004)

    Article  CAS  Google Scholar 

  17. V.R. Rosenfeld, Using semigroups in modeling of genomic sequences. MATCH Commun. Math. Comput. Chem. 56(2), 281–290 (2006)

    CAS  Google Scholar 

  18. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, 2nd edn. (American Mathematical Society, Providence, 1967)

    Google Scholar 

  19. P.M. Higgins, Techniques of Semigroup Theory (Oxford University Press, Oxford, 1992)

    Google Scholar 

  20. L.N. Shevrin, Semigroups, in General Algebra, vol. 2, ed. by L.A. Skornyakov (Nauka, Moscow, 1991), pp. 11–191. (in Russian)

    Google Scholar 

  21. G. Lallement, Semigroups and Combinatorial Applications (Wiley, New York, 1979)

    Google Scholar 

  22. T.S. Blyth, M.H. Almeida, Regular semigroups with skew pairs of idempotents. Semigroup Forum 65, 264–274 (2002)

    Article  Google Scholar 

  23. V.R. Rosenfeld, Emulating the function of introns in pre-mRNA. MATCH Commun. Math. Comput. Chem. 57(1), 135–142 (2007)

    CAS  Google Scholar 

  24. V.R. Rosenfeld, D.J. Klein, Implications of sense/antisense nucleic-acid codons on amino-acid counts. Stud. Univ. Babes-Bolyai Chem. 55(4), 167–176 (2010)

    CAS  Google Scholar 

  25. V.R. Rosenfeld, Color symmetry, semigroups, fractals. Croat. Chem. Acta 86(4), 555–559 (2013)

    Article  CAS  Google Scholar 

  26. V.R. Rosenfeld, D.J. Klein, Cyclic nucleotide sequences codonically invariant under frame shifting. Stud. Univ. Babes-Bolyai Chem. 55(4), 177–182 (2010)

    CAS  Google Scholar 

  27. V.R. Rosenfeld, Studying the polypeptide sequence (\(\alpha \)-code) of Escherichia coli. J. Theor. Chem. (2013). Article ID 961378

  28. B.B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Co., New York, 1982)

    Google Scholar 

  29. M.F. Barnsley, H. Rising, Fractals Everywhere (Academic Press Professional, Boston, 1993)

    Google Scholar 

  30. J.-F. Gouyet, Physics and Fractal Structures (foreword by B. Mandelbrot), Masson (Springer, New York, 1996)

  31. K. Falconer, Techniques in Fractal Geometry (Wiley, New York, 1997)

    Google Scholar 

  32. V.R. Rosenfeld, Equivalent genomic (proteomic) sequences and semigroups. J. Math. Chem. 53(6), 1488–1494 (2015)

    Article  CAS  Google Scholar 

  33. V.R. Rosenfeld, Selfcomplementary, selfreverse cyclic nucleotide sequences codonically invariant under frame shifting. J. Math. Chem. 51(10), 2644–2653 (2013)

    Article  CAS  Google Scholar 

  34. N.C. Seeman, H. Wang, X. Yang, F. Liu, C. Mao, W. Sun, L. Wenzler, Z. Shen, R. Sha, H. Yan, M.H. Wong, P. Sa-Ardyen, B. Liu, H. Qiu, X. Li, J. Qi, S.M. Du, Y. Zhang, J.E. Mueller, T.-J. Fu, Y. Wang, J. Chen, New motifs in DNA nanotechnology. Nanotechnology 9, 257–273 (1998)

    Article  CAS  Google Scholar 

  35. N.C. Seeman, At the crossroads of chemistry, biology, and materials: structural DNA nanotechnology. Chem. Biol. 10, 1151–1159 (2003)

    Article  CAS  Google Scholar 

  36. S.M. Douglas, H. Dietz, T. Liedl, B. Högberg, F. Graf, W.M. Shih, Self-assembly of DNA into nanoscale three-dimensional shapes. Nature 459, 414–418 (2009)

    Article  CAS  Google Scholar 

  37. H. Dietz, S.M. Douglas, W.M. Shih, Folding DNA into twisted and curved nanoscale shapes. Science 325, 725–730 (2009)

    Article  CAS  Google Scholar 

  38. Y. Ke, S.M. Douglas, M. Liu, J. Sharma, A. Cheng, A. Leung, Y. Liu, W.M. Shih, H. Yan, Multilayer DNA origami packed on a square lattice. J. Am. Chem. Soc. 131, 15903–15908 (2009)

    Article  CAS  Google Scholar 

  39. H. Gradišar, R. Jerala, Self-assembled bionanostructures: proteins following the lead of DNA nanostructures. J. Nanobiotechnol. 12(4), 1–9 (2014)

    Google Scholar 

  40. L. Jaeger, E. Westhof, N.B. Leontis, TectoRNA: modular assembly units for the construction of RNA nano-objects. Nucleic Acids Res. 29(2), 455–463 (2001)

    Article  CAS  Google Scholar 

  41. K.A. Afonin, M. Kireeva, W.W. Grabow, M. Kashlev, L. Jaeger, B.A. Shapiro, Co-transcriptional assembly of chemically modified RNA nanoparticles functionalized with siRNAs. Nano Lett. 12(10), 5192–5195 (2012)

    Article  CAS  Google Scholar 

  42. Y. Shu, F. Haque, D. Shu, W. Li, Z. Zhu, M. Kotb, Y. Lyubchenko, P. Guo, Fabrication of 14 different RNA nanoparticles for specific tumor targeting without accumulation in normal organs. RNA 19, 767–777 (2013)

    Article  CAS  Google Scholar 

  43. J.G. Heddle, Protein cages, rings and tubes: useful components of future nanodevices? Nanotechnol. Sci. Appl. 1, 67–78 (2008)

    CAS  Google Scholar 

  44. J.L. Walsh, A closed set of normal orthogonal functions. Am. J. Math. 45, 5–24 (1923)

    Article  Google Scholar 

  45. N.J. Fine, On the Walsh functions. Trans. Am. Math. Soc. 65, 372–414 (1949)

    Article  Google Scholar 

  46. K.G. Beauchamp, Walsh Functions and Their Applications (Academic Press, London, 1975)

    Google Scholar 

  47. S.G. Tzafestas, Walsh Functions in Signal and Systems Analysis and Design (Van Nostrand Reinhold, New York, 1985)

    Google Scholar 

  48. J. Hadamard, Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17, 240–246 (1893)

    Google Scholar 

  49. A. Hedayat, W.D. Wallis, Hadamard matrices and their applications. Ann. Stat. 6(6), 1184–1238 (1978)

    Article  Google Scholar 

  50. A.P. Bisson, F.J. Carver, D.S. Eggleston, R.C. Haltiwanger, C.A. Hunter, D.L. Livingstone, J.F. McCabe, C. Rotger, A.E. Rowan, Synthesis and recognition properties of aromatic amide oligomers: molecular zippers. J. Am. Chem. Soc. 122, 8856–8868 (2000)

    Article  CAS  Google Scholar 

  51. D.G. Allis, J.T. Spencer, Nanostructural architectures from molecular building blocks, in Handbook of Nanoscience, Engineering, and Technology, Ch. 18, 2nd edn., ed. by W.A. Goddard Iii, D.W. Brenner, S.E. Lyshevski, G.J. Iafrate (CRC Press LLC, Boca Raton, 2007)

    Google Scholar 

  52. A. Banerji, Studying protein interior with fractal dimension, in Fractal Symmetry of Protein Interior, Ch. 2, (SpringerBriefs in Biochemistry and Molecular Biology, Springer, Basel, 2013) pp. 84. doi: 10.1007/978-3-0348-0651-0_2

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Acknowledgments

The author is grateful to Prof. Douglas J. Klein (Galveston) and anonymous Referees for their helpful comments. This work was supported by the Welch Foundation of Houston, Texas (through Grant BD-0894) and the Ministry of Absorption of the State Israel (via fellowship “Shapiro”).

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  1. Vladimir R. Rosenfeld

    Present address: Department of Computer Science and Mathematics, Ariel University, 40700, Ariel, Israel

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  1. Mathematical Chemistry Group, Department of Marine Sciences, Texas A&M University at Galveston, Galveston, TX, 77553–1675, USA

    Vladimir R. Rosenfeld

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Correspondence to Vladimir R. Rosenfeld.

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Rosenfeld, V.R. The fractal nature of folds and the Walsh copolymers. J Math Chem 54, 559–571 (2016). https://doi.org/10.1007/s10910-015-0574-7

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  • DOI: https://doi.org/10.1007/s10910-015-0574-7

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