  Scaling law for geometrical and dynamical quantities of biological molecules
is an interesting topic. According to Flory's theory, a power law between
radius of gyration and the length of homopolymer chain is found, with exponent
3/5 for good solvent and 1/3 for poor solvent. For protein in physiological
condition, a solvent condition in between, a power law with exponent ~2/5 is
obtained. In this paper, we present a unified formula to cover all above cases.
It shows that the scaling exponents are generally correlated with fractal
dimension of a chain under certain solvent condition. While applying our
formula to protein, the fractal dimension is found to depend on its
hydrophobicity. By turning a physical process-varying hydrophobicity of a chain
by amino acid mutation, to an equivalent chemical process-varying polarity of
solvent by adding polar or nonpolar molecules, we successfully deprive this
relation, with reasonable agreement to statistical data. And it will be helpful
for protein structure prediction. Our results indicate that the protein may
share the same basic principle with homopolymer, despite its specificity as a
heteropolymer.
