The effect of stereochemical constraints on the radius of gyration of folded proteins

Proteins are composed of linear chains of amino acids that fold into complex three-dimensional structures. A general feature of folded proteins is that they are compact with a radius of gyration $R_g(N) \sim N^{\nu}$ that obeys power-law scaling with the number of amino acids $N$ in each protein and $\nu \sim 1/3$. In this study, we investigate the {\it internal} scaling of the radius of gyration $R_g(n)$ versus the chemical separation $n$ between amino acids for all subchains of length $n$. We show that for globular proteins $R_g(n) \sim n^{\nu_{1,2}}$ with a larger exponent $\nu_1 > 1/3$ for small $n$ and a smaller exponent $\nu_{2} < 1/3$ for large $n$, such that $R_g(N) \propto N^\nu$. To describe this scaling behavior for $R_g(n)$, we carry out folding simulations for a series of coarse-grained models for proteins beginning with the freely-jointed and freely-rotating chain models composed of spherical monomers and varying degrees of stereochemical constraints. We show that a minimal model, which coarse-grains amino acids into a single spherical backbone atom and one variable-sized spherical side-chain atom, and enforces bend- and dihedral-angle constraints, can recapitulate $R_g(n)$ for x-ray crystal structures of globular proteins. In addition, this model predicts the correct average packing fraction and size of the hydrophobic core, which are two key physical features that can be used to distinguish between computational `decoys' and correctly folded proteins in protein design applications.