A multiple regression analysis has established a non-linear relationship between the backbone dihedral angles and theca coordinates obtained from the X-ray crystal structures of fourteen proteins. The regression equations have been applied to predict specific dihedral angles of each residue in the backbone of twenty-four proteins. Overall this method (NLRDT) predicts values of ɸ and ψ within a ± 45° window of those found in the X-ray structure with an accuracy of 94% and 91% and within a ± 30° window of 88% and 81%.

Two methods for the assignment of motif from Cα coordinates are reported. For the first method motif is assigned from the dihedral angles predicted using the regression equations. If the predicted dihedral angles of a residue fall in the range of -15° > <ɸ> -90° and -10° >ψ> -70°, the residue is assigned as in an α-helix; and in the range of -90° > <ɸ> -150° and 95° <ψ< 170° as in a β-sheet. By the second method motif of the ith residue is assigned from the distance Ci-1α to Ci+2α (v6) and torsional angle Ci-1α, Ciα, Ci+1α, Ci+2α (v13). If these values for a residue fall in the range v6<6.0 Å and 100° > v13 > 0° the residue is assigned as in an α-helix. If the values are in a range V6 > 8.7 Å and ǀv13ǀ > 100° the residue is assigned as in a P-sheet. For the twenty four proteins 23.7% of the residues by the former method and 19.6% by the latter method are assigned differently from the PDB.

A Monte Carlo Protein Building (MCPB) method to construct the backbone and Cβ atomic coordinates of twenty-four proteins from known Cα coordinates is reported. The method selects values of dihedral angles from either ± 30° windows of the dihedral angle calculated for that amino acid by the Non Linear Regression Distance Torsion (NLRDT) method or from ranges established from a statistical analysis of the relationship between dihedral angles of the backbone and Cα coordinates for a protein data base. The averaged coordinates from ten backbone models of a protein were used to define a mean structure that was refined by energy minimisation using the AMBER force field (GB/SA). By the latter method the average atomic deviation and r.m.s.d. of the backbone and Cβ atoms is between 0.14 Å and 0.32 Å (average 0.22 Å) and 0.22 Å and 0.61 Å (average 0.43 Å) respectively. A comparison with other methods is made.

A model of nine proteins including side chain atoms have been built from the known Cα coordinates and amino acid sequences using a Monte Carlo Protein Building Annealing (MCPBA) method. The Cartesian coordinates for the side chain atoms were established with bond lengths and angles selected randomly from within ranges of values previously determined by analysis of fourteen protein crystal structures and with torsional angles randomly selected from -180° to 180°. A simulated annealing technique is used to generate some 300 structures with differing side chain conformations. The atomic coordinates of the backbone atoms are fixed during the simulated annealing process. The coordinates of the side-chain atoms of the 300 low energy conformations are averaged to obtain a mean structure which is minimisation with the Cα atoms constrained to their position in the X-ray structure using the OPLS/AMBER force field with the GB/SA water model. The r.m.s.d of the main-chain atoms (without Cβ) compared with the corresponding crystal structures is in the range 0.20 Å to 0.64 Å with a average value of 0.45 Å. The r.m.s.d of the side-chain atoms is from 1.72 Å to 2.71 Å with an average of 2.26 Å. The r.m.s.d of all atoms is from 1.19 Å to 1.99 Å with an average of 1.61 Å. The method is insensitive to random errors in the Cα positions and the computational requirement is modest. A full atomic model of 7c and 8c-1 coiled coil rod domain in wool protein has been established from the amino acid sequences using the MCPB/MCPBA method. For the particular knob-hole heptad repeat, for the single α-helix the rise per residue is 1.464 Å; the twist angle per residue 102.999°; the number of residues per turn is 3.524 and the pitch 5.171 Å. For the four coiled coil helical segments of the rod domain the pitch is in the range 124 Å to 192 Å (average 172 Å); the radius of the coiled coil varies between 5.24 Å to 5.92 Å; the average value of the radius is 5.56 Å; the average crossing angle of the helices in the coiled coil is 23.0°; the number of residues per major turn is 127.3 and there are 36.2 minor turns in a major turn. The interaction energy between the two α-helical chains in the coiled coil structure is evaluated from van der Waals non-bonded interactions, electrostatic and hydrogen bonding interactions. The optimum relationship of the α-helical chains to each other established the heptad repeat interaction; 34% of the leucine residues are located at the d position. Of the backbone hydrogen bonds in the protein a-helix between residues four apart, 18% have a distance between a donor NH nitrogen and acceptor carbonyl oxygen greater than 3.5 Å .The hydrogen bonds between the side chains of the two a-helices in the coiled coil structure are largely between Arg and Glu, Arg and Asp and Glu and Asp. The distances of the Cβ atoms of cysteine residues are > 4.5 Å. This distance is outside that required for formation of disulphide bonds. The interaction of charged residues with apolar, polar and charged residues in the a-a, a-d, d-d, and d-a heptad positions accounts for 70% of the interaction energy.