(593r) Investigation of Some Characters of Proteins Using Three Different Fractal Approaches

Like other biological macromolecules such as
polysaccharides and nucleic acids, proteins are essential parts of organisms
and participate in virtually every process within cells. So that the researches
of protein structure and function are extremely important. However, the spatial
structures of proteins are so complicated and changeable that it is unrealistic
to use the basic units (spheres, cubes and other regular shapes) to describe
the complexity of an protein, i.e, protein molecules cannot be simply described
in terms of the Euclidean geometry. Actually, protein has been described as a
'complex mesoscopic system' and characterized by self-similarity [Banerji,
2011]. On the other hand, fractal theory, as a very active mathematic branch of
modern nonlinear science, has been used widely to describe irregular and
non-differentiable geometric shapes existing in both natural world and man-made
objects. So we can use the fractal method to characterize the complicated
spatial and dynamical strctures of proteins.

In this paper we calculated
the fractal dimensions of four proteins, chymotrypsin, elastase, trypsin and
subtilisin, which are made up of about 220-275 amino acids and belong to the
family of serine proteinase [Kraut, 1977; Hedstrom, 2002] by using three
definitions of fractal dimension i.e. the chain fractal dimension (D_L),
the mass fractal dimension (D_m) and the correlation fractal
dimension (D_c). We analyzed the relationship between fractal
dimension and space structure or secondary structure contents of proteins. The
results showed that the more similar structures, the more equal fractal
dimensions, and if the fractal dimensions of proteins are different from each
other, the three dimensional structures should not be similar. Furthermore, we
analyzed the differences between the fractal dimensions of complete proteins
and that of their active sites. It is found that the values of fractal dimensions
are almost same for the global mammalian enzymes (chymotrypsin, elastase and
trypsin), but different for the global subtilisin. On the other hand, the
detailed structures and fractal dimensions of the active sites of four enzymes
are extraordinarily similar. So the multifractal theory is a very useful tool
to solve this problem. To sum up, the fractal method can be applied to the elucidation
of protein evolution and the
fractal analysis can be used to depict some intrinsic characteristics of
proteins. It is also demonstrated that proteins are a kind of fractal object
with self-affinity and self-similarity.

(1)  
Banerji,
A., Ghosh, I.. (2011) Fractal
symmetry of protein interior: what have we learned? Cel. Mol. Life Sci. 68,
2711-2737.

(2)   Kraut,
J.. (1977) Serine Proteases: Structure and Mechanism of Catalysis. Annu. Rev.
Biochem. 46, 331-358.

(3)   Hedstrom,
L.. (2002)
Serine Protease Mechanism and Specificity.
Chem.
Rev. 102,
4501-4523.

This work was supported by the Program for
New Century Excellent Talents in Chinese University (NCET-08-0386), the 863
Program of China (2008AA10Z318), the Natural Science Foundation of China
(20976125; 31071509; 51173128) and Tianjin (10JCYBJC05100), and the Program of
Introducing Talents of Discipline to Universities of China (No. B06006).

Figure (A) The chain fractal dimension, (B) The mass fractal dimension, (C) The correlation fractal dimension.
The three dimensional structure of chymotrypsin (D) and subtilisin (E)