Biomedical Chemistry: Research and Methods, 2018, 1(3), e00070
The 40th Anniversary of the Institute of Physiologically Active Compounds of the Russian Academy of Sciences

Calculation and Analysis of Fractal Descriptors for Protein Amino Acids
in Various Conformational States

V.Yu. Grigorev1*, L.D. Grigoreva2

1Institute of Physiologically Active Compounds of the Russian Academy of Sciences, 1 Severny proezd, Moscow region, Chernogolovka, 142432 Russia;*e-mail:
2Department of Fundamental Physical and Chemical Engineering, Moscow State University, Moscow, 119991 Russia

Key words: amino acids; fractal descriptors; α-helix; β-sheet

DOI: 10.18097/BMCRM00070

The whole version of this paper is available in Russian.

A series of 20 proteinogenic amino acids was studied. Four types of fractal descriptors for 2 conformational states are calculated: α-helix and 1-strand β-sheet. Based on the analysis of the results obtained, it is established that when the conformational state of the amino acids (α-helix→β-sheet) changes, significant changes in the fractal descriptor Dtot, in the calculation of which all the atoms of the molecule are used, are not observed. However, the more specific descriptors Dval, Dvdw and Dunb, which reflect the aggregate of valence-coupled, van der Waals contact and unbound atoms, respectively, are more sensitive to the conformational transition. The increase Dval, Dvdw and the  decrease  Dunb values  were  established for a series of 7 amino acids

Figure 1. Histogram of the interatomic distances for the alanine.

Figure 2. Change in fractal descriptors of amino acids depending on the conformation.

Table 1. Fractal descriptors (Dtot, Dval, Dvdw, Dunb and standard errors Δ) of amino acids (α-helix).

Table 2. Fractal descriptors (Dtot, Dval, Dvdw, Dunb and standard errors Δ) of amino acids (β-sheet).


The work was performed within the framework of the State Task for 2018 (topic number 0090-2017-0020).


  1. Anfinsen, C.B. (1973) Principles that govern the folding of protein chains. Science, 181(4096), 223-230. DOI
  2. Sweet, R.M., & Eisenberg, D. (1983) Correlation of sequence hydrophobicities measures similarity in three-dimensional protein structure. J. Mol. Biol., 171(4), 479-488. DOI
  3. Giuliani, A., Benigni, R., Zbilut, J.P., Webber, Jr., C.L., Sirabella, P., & Colosimo, A. (2002) Nonlinear signal analysis methods in the elucidation of protein sequence-structure relationships. Chem. Rev., 102(5), 1471-1492. DOI
  4. Huang, Y., Xiao, Y. (2003) Nonlinear deterministic structures and the randomness of protein sequences. Chaos, Solitons and Fractals, 17, 895-900. DOI
  5. Kanduc, D., Capone, G., Delfino, V.P., & Losa, G. (2010) The fractal dimension of protein information. Advanced Studies in Biology, 2(2), 53-62.
  6. Kornev, A.P., Taylor, S.S. (2017) Fractal nature of protein interior and its implications for protein function. Biophys. J., 112(3), 194A-195A. DOI
  7. Pavan, Y.S., Mitra, C.K. (2005) Fractal studies on the protein secondary structure elements. Ind. J. Biochem. Biophys., 42, 141-144.
  8. Yu, Z.G., Anh, V., & Lau, K.S. (2003) Multifractal and correlation analyses of protein sequences from complete genomes. Phys. Rev. E, 68, 021913. DOI
  9. Todoroff, N., Kunze, J., Schreuder, H., Hessler, G., Baringhaus, K.H., & Schneider, G. (2014) Fractal dimensions of macromolecular structures. Mol. Inf., 33, 588-596. DOI
  10. Andoyo, R., Lestari, V.D., Mardawati, E., & Nurhadi, B. (2018) Fractal dimension analysis of texture formation of whey protein-based foods. Int. J. Food Sci., 2018, article ID 7673259, 17 pages. DOI
  11. Grigor’ev, V.A., Raevskii, O.A. (2011) Fractal dimension of the interatomic distance histogram: new 3D descriptor of molecular structure. Russ. J. Gen. Chem., 81(3), 449-455. DOI
  12. HyperChem. Retrieved August 28, 2018, from
  13. Crownover, R.M. (1995) Introduction in fractals and chaos, Boston, Jones and Bartlett, 306.
  14. Grigorev, V.Yu., Grigoreva, L.D. (2016) Calculation and properties of fractal descriptors for C2–C9 alkanes. Moscow Univ. Chem. Bull., 71(3), 199-204. DOI
  15. Pande, V.S., Grosberg, A.Y., & Tanaka, T. (1994) Nonrandomness in protein sequences: evidence for a physically driven stage of evolution? Proc. Natl. Acad. Sci. USA, 91(26), 12972-12975.