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Fig. 1. The definitions of a local coordinate system affixed to each residue. The origin O of the local coordinate system is located at the C position of each residue. The Y and Z axes are ones formed by the vector product and the sum of the unit vectors from N to C and from C to C, respectively. The X axis is taken to form a right-handed coordinate system. The relative direction and rotation of one residue to the other in contacting residues are represented by polar angles (,) and Euler angles (,,), respectively. First citation in article

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Fig. 2. Dependencies of orientational entropies on parameters in the estimation of the orientational potentials. The orientational entropies averaged over all types of residue pairs with the weight of the number of contacts N_aa for each type of residue pair are plotted against the cutoff values for the expansion coefficients. Triplets of digits near solid lines indicate the values of (l,l,k); for the non-solid lines, l = l = k = 6 is used. The other parameters are = 0.2 for all lines, and O_cutoff = O₃₃₃₃₃ = 1792 for solid lines. The dotted line shows the case of O_cutoff = O₀₀₇₇₇ = 960, the dotted broken line is for O_cutoff = O₁₁₅₅₅ = 1584, and the broken line is for O_cutoff = O₂₂₄₄₄ = 2025. First citation in article

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Fig. 3. Dependencies of the number of significant expansion terms on estimation parameters for the orientational potentials. The numbers of significant terms averaged over all types of residue pairs with the weight of the number of contacts N_aa for each type of residue pair are plotted against the cutoff values for expansion coefficients. Triplets of digits near curves indicate the values of (l,l,k); for the non-solid lines, l = l = k = 6 is used. The other parameters are = 0.2 for all lines, and O_cutoff = O₃₃₃₃₃ = 1792 for solid lines. The dotted line shows the case of O_cutoff = O₀₀₇₇₇ = 960, the dotted broken line is for O_cutoff = O₁₁₅₅₅ = 1584, and the broken line is for O_cutoff = O₂₂₄₄₄ = 2025. First citation in article

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Fig. 4. Correlation between the number of significant expansion terms and orientational entropy. Those values for 210 different types of residue pairs, which are averaged over residue pairs (a,a) and (a,a), are plotted here. The orientational potentials are evaluated with l = l = k = 6, O_cutoff = 1792, = 0.2, and c_cutoff = 0.025. First citation in article

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Fig. 5. Histograms of the numbers of significant expansion terms for the 210 types of residue pairs; the numbers of significant expansion terms are averaged over residue pairs (a,a) and (a,a). The size of a bin is 200. These data are those for l = l = k = 6, O_cutoff = 1792, = 0.2, and c_cutoff = 0.025. First citation in article

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Fig. 6. Orientational entropies, –ln f_aa, for three types of distributions are plotted against the identification number of amino acid pair (a,a). Amino acid types are numbered in the order of amino acids written along the abscissa; see text for details. The broken line shows the entropy, 6.900, for a uniform distribution. The lowest solid line shows the distribution with polar and Euler angle dependencies, l = l = k = 6. The highest solid line shows the distribution with l = 6,l = k = 0 that depends on polar angles only. The middle solid line shows the distribution that depends on polar angles with l = 6, and on Euler angles with l = k = 6, but ignores any correlation between polar and Euler angles. The values of other parameters are O_cutoff = 1792, = 0.2, and c_cutoff = 0.025. First citation in article

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Fig. 7. The effects of Euler angle dependencies in the orientational potentials on the performance for fold recognition. The value of logarithm of rank probability P_e in the energy scale for each decoy set is plotted against the identification number of the decoy set that is listed in Table V and tables in the auxiliary material (Ref. 52). The left figure (a) corresponds to the decoy set group of monomeric proteins in "Decoys'R'Us" (Ref. 39), and the right figure (b) to the immunoglobulin decoy set group. The potential function used here consists of orientational potentials e^o only. Cross marks and solid lines show the case for the orientational potential with l = 7, l = k = 0, O_cutoff = , and c_cutoff = 0.025. Open circles and broken lines show the case for the orientational potential with l = l = k = 6, O_cutoff = 1792, and c_cutoff = 0.025. First citation in article

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Fig. 8. The effects of the orientational potentials on performance for fold recognition. The value of logarithm of rank probability P_e in the energy scale for each decoy set is compared between two types of potential functions, one of which includes the orientational potential. The abscissa shows the identification number of each decoy set that is listed in Table V and tables in the auxiliary material (Ref. 52). (a) The potentials for monomeric protein decoy sets consist of e + e^c for cross marks and solid lines, and e + e^c + e^o for open circles and broken lines. (b) The potentials for immunoglobulin decoy sets consist of e^c + e^r for cross marks and solid lines, and e^o + e^r for open circles and broken lines. The orientational energies are evaluated with l = l = k = 6, O_cutoff = 1792, = 0.2, c_cutoff = 0.025. First citation in article