The effect of bond-angle constraints on the radius of gyration for collapsed polymers

Proteins are composed of linear chains of amino acids that fold into complex three-dimensional structures. When a protein folds, stereochemical, steric, and hydrophobic interactions are largely responsible for determining its native conformation. Coarse-grained polymer models that include these interactions are often used to understand important features of protein structure. Polymer size can be characterized by the power-law scaling of the radius of gyration R_g with the number of monomers N: R_g ∼ N^ν, where ν = ⅓ for collapsed polymers. We study the structure of collapsed freely-jointed and freely-rotating chains (with fixed bend angles) composed of spherical monomers. We calculate the scaling of R_g(n) as a function of the chemical separation between monomers n for all possible subchains of length n. We show that log R_g versus log n possesses a steep slope ν' > ⅓ at small n and a shallow slope ν'' < ⅓ for large n, such that R_g(N) ∼ N^ν with ν = ⅓ for the end-to-end separation. We further show that the values of ν’ and ν’’ for collapsed freely-rotating chains are similar to those for protein x-ray crystal structures when they possess a similar distribution of bend angles. These results emphasize that we can understand the scaling of the radius of gyration with chemical distance for proteins using coarse-grained polymer models.