Discrete Haar Transform and Protein Structure

Abstract

The discrete Haar transform of the sequence of the backbone dihedral angles (ϕ and ψ) was performed over a set of X-ray protein structures of high resolution from the Brookhaven Protein Data Bank. Afterwards, the new dihedral angles were calculated by the inverse transform, using a growing number of Haar functions, from the lower to the higher degree. New structures were obtained using these dihedral angles, with standard values for bond lengths and angles, and with ω = 0°. The reconstructed structures were compared with the experimental ones, and analyzed by visual inspection and statistical analysis. When half of the Haar coefficients were used, all the reconstructed structures were not yet collapsed to a tertiary folding, but they showed yet realized most of the secondary motifs. These results indicate a substantial separation of structural information in the space of Haar transform, with the secondary structural information mainly present in the Haar coefficients of lower degrees, and the tertiary one present in the higher degree coefficients. Because of this separation, the representation of the folded structures in the space of Haar transform seems a promising candidate to encompass the problem of premature convergence in genetic algorithms.