Boltzmann-type distribution of side-chain conformation in proteins

We analyze packing imperfections in globular proteins as reflected in deviations of torsion angles from the equilibrium values for the isolated side chains. The distribution of conformations of methionine and lysine residues in a database of high-resolution structures is compared with energies of model compounds calculated with high-level quantum-mechanics. The distribution of the C-C and C-S torsion angles (Χ₃) correlates well with the Boltzmann factor of the torsion energy, exp(-βE) of the model compounds C₂H₅-C₂H₅ and C₂H ₅-S-CH₃. An exponential relation was again found between the relative occurrence of g+, g- and t conformations for C^α -Cβ bonds in long side chains and the energy differences of rotamers of α-amino n-butyric acid, when dependence on backbone conformation was taken into account. The distribution of all 27 rotamers of methionine was correlated with the energy differences between the model's rotamers, corrected for clashes with nearby residues, the correlation being good for a set with backbone in the β-conformation, but less clear for backbone α-conformation. In all correlations, the value of the coefficient β corresponds to a temperature of circa 300 K. These results can be interpreted with a model that considers the structure of a folded protein as resulting from packing imperfectly complementary parts, with a requirement of an overall low energy. Compromises are required to optimize the fit of nonbonded contacts with surrounding groups, and side chains assume conformations away from the energy minimum. An exponential distribution is a most probable distribution, and this can be established easily under conditions other than thermal equilibrium.