Biological functions of low-frequency vibrations (phonons). III. Helical structures and microenvironment

Low-frequency vibrations in biomacromolecules possess significant biological functions. In this paper, the alpha-helix element is compared with a mass-distributed spring. Based on this, a set of intuitive and easily handled equations are derived for predicting the fundamental frequencies of helical structures in protein molecules. As shown in the equations, the fundamental frequency depends not only on the constituents of a helix itself but also on its microenvironment. The calculated results agree with the observations. The calculations also demonstrate that the low-frequency vibrations with wave number of approximately 30 cm-1 do not necessarily arise from motions that involve either all or very large portions of the protein molecule as previously thought; a piece of helix containing more than 10 residues and surrounded by a proper microenvironment can also generate such low-frequency motions. Furthermore , we illustrate that the low-frequency motions are closely related to the native state of a protein molecule. Upon denaturation, which is accompanied by a radical change of the relevant microenvironment, the original fundamental frequency also disappears. Consequently, this kind of special frequency termed activating low frequency can serve as a dynamic criterion in identifying whether a biomacromolecule is in its native state. The energy of a phonon excited by this kind of low-frequency vibration is of the same order of magnitude as the average enthalpy value per residue measured during conformational change in some protein molecules. Therefore, there must be some intrinsic relation between the allosteric transitions of protein molecules and their low-frequency motions.