Scaling Law for Radius of Gyration and Its Dependence on Hydrophobicity

TL;DR

This paper introduces a unified scaling law linking the radius of gyration of biological molecules to their solvent conditions and hydrophobicity, providing insights into protein structure and polymer physics.

Contribution

It presents a unified formula connecting scaling exponents with fractal dimension and demonstrates how hydrophobicity influences protein structure, bridging physical chemistry and biological modeling.

Findings

Scaling exponents correlate with fractal dimension under different solvent conditions.

Hydrophobicity affects the fractal dimension of proteins.

The proposed model aligns with statistical data and aids in protein structure prediction.

Abstract

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.