Nonlinear properties of the atomic vibrations in protein backbones

The temperature factors series of protein backbone atoms resulting from the X-ray diffraction analysis of crystal data were analyzed by fast Fourier transform, detrended fluctuation analysis (DFA) and correlation dimension. The spectral analysis of 50 randomly selected proteins revealed the presence of a long-range correlation characterized on average by a power law P≈1/f1.79. The DFA showed evidence for long-range correlation extending over an average 48 atoms length, equivalent to 16 amino acid residues. The mean value of the scaling exponent was about 1.35. Beyond this range, the scaling exponent decreased significantly. The correlation dimension analysis revealed the presence of a strange attractor having an average dimension of 4.34. A similar analysis of the temperature factor for proteins in solution, as resulting from the NMR analysis, revealed that the spectral scaling exponent was lower in solution than in crystal form. This was expected from our model described in Coza and Morariu (Generating 1/fβ noise with a low-dimensional attractor characteristic: its significance for atomic vibration in proteins and cognitive data, Physica A). The attractor dimension was lower by about one unit for proteins in solution compared to the crystal form. Therefore, the nature of at least one of the independent variables controlling the vibrations is dependent on the form of aggregation. Higher values of the scaling exponents were associated to the inactive or tense state of the protein. Lower values of the scaling exponent were associated to higher enzyme activity regardless of higher or lower flexibility. While the molecular biologist's view stresses the relation of particular sites in the protein to its function, the present analysis underlines the unity of the protein molecule.