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 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) similar to N-nu that obeys power-law scaling with the number of amino acids N and nu similar to 1/3, f(core) approximate to 0.09, and approximate to 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 nu similar to 1/3. Instead, R-g(n) similar to 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. 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), , and S( q) for more than 2500 x-ray crystal structures of proteins.
