The effect of stereochemical constraints on the structural properties of folded proteins

Proteins are composed of chains of amino acids that fold into complex three-dimensional structures. Several key features, such as the radius of gyration, fraction of core amino acids $f_{\rm core}$, packing fraction $\langle \phi\rangle$ of core amino acids, and structure factor $S(q)$ define the structure of folded proteins. It is well-known that folded proteins are compact with a radius of gyration $R_g(N) \sim N^{\nu}$ that obeys power-law scaling with the number of amino acids $N$ and $\nu \sim 1/3$, $f_{\rm core} \approx 0.09$, and $\langle \phi \rangle \approx 0.55$. We also investigate the {\it internal} scaling of the radius of gyration $R_g(n)$ versus the chemical separation $n$ between amino acids for subchains of length $n$ and show that it does not obey simple power-law scaling with $\nu \sim 1/3$. Instead, $R_g(n) \sim n^{\nu_{1,2}}$ with a larger exponent $\nu_1 > 1/3$ for small $n$ and smaller exponent $\nu_{2} < 1/3$ for large $n$. To develop a minimal model for proteins that recapitulates these defining structural features, we carry out collapse simulations for a series of coarse-grained models with increasing complexity. We show that a model, which coarse-grains amino acids into a single spherical backbone bead and several variable-sized side-chain beads and enforces bend- and dihedral-angle constraints for the backbone, recapitulates $R_g(n)$, $f_{\rm core}$, $\langle \phi \rangle$, and $S(q)$ for more than $2500$ x-ray crystal structures of proteins.