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

Abstract

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 R(max), 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 epsilon(R) = R. The effective chemical potential mu governing the distribution increases with the number of residues, reflecting the size of the protein globule, while the temperature parameter beta decreases. Interestingly, betamu 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/R(g), where R(g) is the radius of gyration. The global normalized Fermi distribution, F(r), can be reasonably decomposed in Fermi-like subdistributions for different atomic types tau, F(tau)(r), with Sigma(tau)F(tau)(r) = F(r), which depend on two additional parameters mu(tau) and h(tau). The chemical potential mu(tau) 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 h(tau), which appears in a type-dependent atomic effective energy, epsilon(tau)(r) = h(tau)r, and is strongly correlated to available hydrophobicity scales. Better adjustments are obtained when the effective energy is not assumed to be necessarily linear, or epsilon(tau)*(r) = h(tau)*r(alpha,), in which case a correlation with hydrophobicity scales is found for the product alpha(tau)h(tau)*. 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.

Ab initio protein folding simulations using atomic burials as informational intermediates between sequence and structure

“…Intermediate layers (layer 2 and layer 3) are thinner than the internal layer 1 or external layer 4 in order to provide the same expected number of atoms in all layers. Limits for each layer were obtained from the expected radius of gyration as a function of the number N r of residues, , combined to a previously estimated probability density for the number of atoms as a function of normalized central distance, . ( b ) Nonstandard terms of the potential function used in the molecular dynamics simulations.…”

“…Additionally, burial predictions from sequence using simple statistical schemes such as Naive Bayesian Classifiers (NBC) and, particularly, Hidden Markov Models (HMM), were found to successfully extract most of this available sequence‐dependent burial information. Extracted information was shown to increase with the number of burial layers, up to at least four layers, and when the dependence of burial on side chain orientation was taken into consideration …”

“…Here we investigate this hypothesis directly. We combine discrete burial predictions from sequence, obtained by a statistical scheme based on a previously described HMM, to ab initio molecular dynamics simulations forcing each atom toward its predicted burial layer with geometric constraints imposed on covalent structure and hydrogen bond formation. We have obtained and distinguished correctly folded conformations for a selected group of globular proteins, comprising all three structural classes and good burial prediction quality, with correct burial assignment into four burial layers for of the atoms.…”

Prediction of functionally important residues in globular proteins from unusual central distances of amino acids

“…A continuous "mass" function derived by Gomes et al [39] to describe burials of whole residues was considered for fitting. The original function expresses the quadratic increase of the volume when moving away from the core of a protein and sigmoidal decrease (Fermi function) of the atomic density in the rim as dependent on the normalized radius, R :…”

“…This observation is fundamental to the current work, where we devised and validated a method for the identification of function-related residues based on the probabilistic description of atomic burials originating from the conceptual framework of Gomes et al [39]. We collected necessary statistics from a selection of globular proteins and, as opposed to the original application of the framework, we used a radial probability density function to describe preferred central distances of individual atoms of types defined within amino acids.…”