On the optimal contact potential of proteins

Properties of contact matrices induced by pairwise interactions in proteins

The properties of contact matrices ( C matrices) needed for native proteins to be the lowest-energy conformations are considered in relation to a contact energy matrix ( E matrix). The total conformational energy is assumed to consist of pairwise interaction energies between atoms or residues, each of which is expressed as a product of a conformation-dependent function (an element of the C matrix) and a sequence-dependent energy parameter (an element of the E matrix). Such pairwise interactions in proteins force native C matrices to be in a relationship as if the interactions are a Go-like potential [N. Go, Annu. Rev. Biophys. Bioeng. 12, 183 (1983)] for the native C matrix, because the lowest bound of the total energy function is equal to the total energy of the native conformation interacting in a Go-like pairwise potential. This relationship between C and E matrices corresponds to (a) a parallel relationship between the eigenvectors of the C and E matrices and a linear relationship between their eigenvalues and (b) a parallel relationship between a contact number vector and the principal eigenvectors of the C and E matrices, where the E matrix is expanded in a series of eigenspaces with an additional constant term. The additional constant term in the spectral expansion of the E matrix is indicated by the lowest bound of the total energy function to correspond to a threshold of contact energy that approximately separates native contacts from non-native ones. Inner products between the principal eigenvector of the C matrix, that of the E matrix, and a contact number vector have been examined for 182 proteins, each of which is a representative from each family of the SCOP database [Murzin, J. Mol. Biol. 247, 536 (1995)], and the results indicate the parallel tendencies between those vectors. A statistical contact potential [S. Miyazawa and R. L. Jernigan, Proteins 34, 49 (1999); S. Miyazawa and R. L. Jernigan, Proteins50, 35 (2003)] estimated from protein crystal structures was used to evaluate pairwise residue-residue interactions in the proteins. In addition, the spectral representation of C and E matrices reveals that pairwise residue-residue interactions, which depend only on the types of interacting amino acids, but not on other residues in a protein, are insufficient and other interactions including residue connectivities and steric hindrance are needed to make native structures unique lowest-energy conformations.

Nature of protein family signatures: insights from singular value analysis of position-specific scoring matrices

Position-specific scoring matrices (PSSMs) are useful for detecting weak homology in protein sequence analysis, and they are thought to contain some essential signatures of the protein families. In order to elucidate what kind of ingredients constitute such family-specific signatures, we apply singular value decomposition to a set of PSSMs and examine the properties of dominant right and left singular vectors. The first right singular vectors were correlated with various amino acid indices including relative mutability, amino acid composition in protein interior, hydropathy, or turn propensity, depending on proteins. A significant correlation between the first left singular vector and a measure of site conservation was observed. It is shown that the contribution of the first singular component to the PSSMs act to disfavor potentially but falsely functionally important residues at conserved sites. The second right singular vectors were highly correlated with hydrophobicity scales, and the corresponding left singular vectors with contact numbers of protein structures. It is suggested that sequence alignment with a PSSM is essentially equivalent to threading supplemented with functional information. In addition, singular vectors may be useful for analyzing and annotating the characteristics of conserved sites in protein families.