Description of atomic burials in compact globular proteins by Fermi-Dirac probability distributions

We perform a statistical analysis of atomic distributions as a function of the distance R from the molecular geometrical center in a nonredundant set of compact globular proteins. The number of atoms increases quadratically for small R, indicating a constant average density inside the core, reaches a maximum at a size-dependent distance Rmax, and falls rapidly for larger R. The empirical curves turn out to be consistent with the volume increase of spherical concentric solid shells and a Fermi-Dirac distribution in which the distance R plays the role of an effective atomic energy ϵ(R) = R. The effective chemical potential μ governing the distribution increases with the number of residues, reflecting the size of the protein globule, while the temperature parameter β decreases. Interestingly, βμ is not as strongly dependent on protein size and appears to be tuned to maintain approximately half of the atoms in the high density interior and the other half in the exterior region of rapidly decreasing density. A normalized size-independent distribution was obtained for the atomic probability as a function of the reduced distance, r = R/Rg, where Rg is the radius of gyration. The global normalized Fermi distribution, F(r), can be reasonably decomposed in Fermi-like subdistributions for different atomic types τ, Fτ(r), with ΣτFτ(r) = F(r), which depend on two additional parameters μτ and hτ. The chemical potential μτ affects a scaling prefactor and depends on the overall frequency of the corresponding atomic type, while the maximum position of the subdistribution is determined by hτ, which appears in a type-dependent atomic effective energy, eτ(r) = hτr, and is strongly correlated to available hydrophobicity scales. Better adjustments are obtained when the effective energy is not assumed to be necessarily linear, or eτ*(r) = hτ*rα,, in which case a correlation with hydrophobicity scales is found for the product ατhτ*. These results indicate that compact globular proteins are consistent with a thermodynamic system governed by hydrophobic-like energy functions, with reduced distances from the geometrical center, reflecting atomic burials, and provide a conceptual framework for the eventual prediction from sequence of a few parameters from which whole atomic probability distributions and potentials of mean force can be reconstructed. Proteins 2007. © 2006 Wiley-Liss, Inc.