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_{core}, packing fraction 〈ϕ〉 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)∼N^{ν} that obeys power-law scaling with the number of amino acids N and ν∼1/3, f_{core}≈0.09, and 〈ϕ〉≈0.55. We also investigate the 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 ν∼1/3. Instead, R_{g}(n)∼n^{ν_{1,2}} with a larger exponent ν_{1}>1/3 for small n and a smaller exponent ν_{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_{core}, 〈ϕ〉, and S(q) for more than 2500 x-ray crystal structures of proteins.