(584m) 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.