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 $f_{\mathrm{core}}$, packing fraction $\langle \varphi \rangle$ of core amino acids, and structure factor $S\left(q\right)$ define the structure of folded proteins. It is well-known that folded proteins are compact with a radius of gyration $R_g\left(N\right)\sim N^{\nu }$ that obeys power-law scaling with the number of amino acids $N$ and $\nu \sim 1/3$, $f_{\mathrm{core}}\approx 0.09$, and $\langle \varphi \rangle \approx 0.55$. We also investigate the internal scaling of the radius of gyration $R_g\left(n\right)$ 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, $R_g\left(n\right)\sim n^{{\nu }_{1,2}}$ with a larger exponent ${\nu }_1>1/3$ for small $n$ and a smaller exponent ${\nu }_2<1/3$ for large $n$. 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 $R_g\left(n\right)$, $f_{\mathrm{core}}$, $\langle \varphi \rangle$, and $S\left(q\right)$ for more than 2500 x-ray crystal structures of proteins.

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• Protein folding
• Proteins