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)