The Fractal Dimension of Protein Chain Conformation

Proteins are made up of 20 different amino acids linked together by peptide bonds [Petsko, 2004]. From a microscopic scale, a protein is determined by its amino acid sequence which is called primary structure via the process of protein folding. Since a protein's curved shape does not change with the change of observation scale, its backbone conformation obeys statistical laws. From a macroscopic scale, a protein is a three dimensional object which is called tertiary or quaternary structure. On the surface of proteins, there exist a great number of irregular "caves" and "gaps", and with further observation the micro-rugged, very irregular structures are also found. Namely proteins have a strong similarity between local structure and overall structure, which is an obvious characteristic of fractal geometry. The fractal geometry is a fascinating conceptual framework [Mandelbrot, 1982] because of its possibilities to characterize nature irregularities with a single number, a really tempting idea per se. Moreover, proteins in nature are irregular, and the most fascinating thing is that irregular objects are the norm in the fractal geometry. So the fractal method can be used to describe the complicated spatial and dynamical structures of proteins [Mandelbrot, 1982].

In this paper we examined the fractal properties of 750 folded proteins from four different structural classes [Mount, 2001], namely (1) the alpha-class (dominated by alpha-helices), (2) the beta-class (dominated by beta-pleated sheets), (3) the (alpha/beta)-class (alpha-helices and beta-sheets alternately mixed) and (4) the (alpha+beta)-class (alpha-helices and beta-sheets largely segregated) by using two fractal dimension methods, i.e. "the local fractal dimension" and "the backbone fractal dimension" (a new and useful quantitative parameter). The results showed that the protein molecules exhibited a fractal character in the range of 1≤N≤15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension was distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of alpha>alpha/beta>alpha+beta>beta.

All in all, the present results suggest that a protein can be regarded as a fractal object with self-similarity and self-affinity, and the concept of fractal may serve as a useful tool for description of the intrinsic characteristics of protein molecules.

This research was supported by the NSF of China (20976125, 31071509, 51173128) and Tianjin (10JCYBJC05100), the Program for New Century Excellent Talents in Chinese University (NCET-08-0386), and Beiyang Young Scholar Program (2012).

References

(1)   Petsko G., Ringe D.. (2004) Protein structure and function. London: Wiley Blackwell Press.

(2)   Mandelbrot B. B.. (1982) The fractal geometry of nature. New York: Freeman Press.

(3)   Mount D. W.. (2001) Bioinformatics: Sequence and Genome Analysis. New York: Cold Spring Harbor Laboratory Press.

Figure (A) A simple schematic representation of amino acid chain (a) and the bond angle in a dipeptide chain (b). In this protein model the amino acid is denoted by the Calpha-atom of amino acid residue. The red thick line represent the length between two adjacent amino acid residues. (B) The scatter diagram of fractal dimensions of proteins.