Distance-constraint approach to protein folding. I. Statistical analysis of protein conformations in terms of distances between residues

A statistical analysis of protein conformations in terms of the distance between residues, represented by their C^α atoms, is presented. We consider four factors that contribute to the determination of the distance d_i,i+k between a given pair of ith and (i+k)th residues in the native conformation of a globular protein: (1) the distance k along the chain, (2) the size of the protein, (3) the conformational states of the ith to (i+k)th residues, and (4) the amino acid types of the and (i+k)th residues. In order to account for the dependence on the distance k along the chain, the statistics are taken for three ranges, viz., short, medium, and long ranges (k≤8; 9≤k≤20; and k≥21; respectively). In the statistics of short-range distances, a mean distance D_k and its standard deviation S_k are calculated for each value of k, with and without taking into account the conformational states of all residues from i to i+k (factors 1 and 3). As an Appendix, the relations for converting from the distances between residues into other conformational parameters are discussed. In the statistics of long-range distances, a reduced distance d*_ij (the actual distance divided by the radius of gyration) is used to scale the data so that they become independent of protein size, and then a mean reduced distance D_l (a_μ, a_ν) and its standard deviation σ_l (a_μ, a_ν) are calculated for each amino acid pair (a_μ, a_ν) (factors 2 and 4). The effect of the neighboring residues along the chain on the value of the distance d*_ij is explored by a linear regression analysis between the actual reduced distance d*_ij and the mean value over the D_l for all possible pairs of residues in the two segments of the (i-2)th to the (i+2)th and the (j-2)th to the (j+2)th residues. The effect is assessed in terms of the tangent A_l (a_μ, a_ν) of the calculated regression line for each amino acid pair (a_μ, a_ν). In the statistics of medium-range distances, only factors 1 and 4 are considered, to simplify the analysis. The scaled distance d_i,i+k^†=(d_i,i+k-D_k)/S_k is used to eliminate the dependence on k, the distance along the chain. The properties D_m (a_μ, a_ν), σ_m (a_μ, a_ν) and A_m (a_μ, a_ν) corresponding to D_l (a_μ, a_ν), σ_l (a_μ, a_ν), and A_l (a_μ, a_ν), and also calculated for each amino acid pair (a_μ, a_ν). The results are interpreted as follows: the smaller values of D_l (a_μ, a_ν) and D_m (a_μ, a_ν) indicate a preference of the pair (a_μ, a_ν) for a contact (e.g., pairs between hydrophobic amino acids, and pairs of Cys with aromatic amino acids), and the larger values of these quantities indicate a preference for distant mutual location (e.g., pairs between strong hydrophilic amino acids); the smaller values of σ_l (a_μ, a_ν) and σ_m (a_μ, a_ν) indicate a strong preference for either contact or noncontact (e.g., pairs between hydrophobic amino acids, and pairs between strong hydrophobic and hydrophilic amino acids, respectively), and the larger values of these quantities indicate the ambivalent/neutral nature of the preference for contact and noncontact (e.g., pairs containing Ser or Thr); the smaller values of A_l (a_μ, a_ν) and A_m (a_μ, a_ν) indicate that the distance of an (a_μ, a_ν) pair is determined independently of the amino acid character of the neighboring residues along the chain (e.g., some pairs of Cys or Met with other amino acids) and the larger values of these quantities indicare that such amino acid character contributes strongly to the determination of the distance (e.g., pairs containing Ser or Thr, and pairs between amino acids with small side chains). The difference between the statistics for the long- and medium-range distances is also discussed; the former reflect the difference between the hydrophobic and hydrophilic character of the residues, but the latter cannot be easily interpretable only in terms of hydrophobicity and hydrophilicity. The data analyzed here are used in the optimization of an object function to compute protein conformation in a subsequent paper.