Low frequency collective modes in proteins

Saravana Prakash and H. M. Urbassek

Introduction
The nature of the internal motions of proteins is a subject of considerable interest, particularly because some of these motions are known to play an important role in protein function such as protein dynamics and structural transitions. Various theoretical and experimental methods are now being employed to determine both the magnitudes and the time scales of the internal motions. The frequencies of the 3N-6 vibrational degrees of freedom existing in a protein are broadly distributed from approximately 3700 cm-1 (O-H stretching) to a few cm -1 in the mid- to far-infrared regions. Stretching vibrations of the O-H, N-H and C-H bonds of a protein contribute high frequencies due to their strong force constants and small masses, whereas low-frequency modes occur due to weak force-constant groups (such as the methyl group) as well as global collective motions of a large number of atoms. In particular these latter low-frequency modes manifest as bending, deformation, and twisting motions involving changes in bond angles. One of the central problems in the dynamics of protein action is to determine the lifetimes of the low-frequency collective modes of a protein molecule. The importance of these collective mode dynamics comes from two fundamental aspects of protein action. (i) Low-frequency modes of proteins are particularly interesting, because they are related to functional properties [1]. (ii) It is believed that low-frequency collective modes are responsible for the direct flow of conformational energy in many biological processes.
Low frequency modes
The two main approaches to understanding the low-frequency modes are normal mode dynamics and molecular dynamics simulations. Normal-mode analysis (NMA) is a direct way to analyze vibrational motion. This method has long been used as a tool for interpreting vibrational spectra of small molecules [3]. The frequencies obtained from NMA can be directly related to experimental infrared (IR) and/or Raman measurements. In recent years NMA has been extended to the study of large molecular systems such as proteins [2]. In normal mode calculations, the potential function is approximated by a sum of quadratic terms in the displacements around a minimum energy conformation.The calculation consists of a diagonalization of the mass-weighted second derivative matrix of the system; from this we could extract eigenmodes or normal modes. The other major technique to screen the low frequency is molecular dynamics (MD); it is based on solving Newton's equations of motion to yield a trajectory of atomic positions. We have used digital signal processing techniques to characterize the atom motion in MD simulations. Fourier transforming all the atomic trajectories yields the overall frequency distribution. We then choose the frequency ranges corresponding to motions of interest and eliminate the rest. Moreover, we are able to extend this approach to extract the vectors defining the characteristic motion for each frequency of interest in a MD simulation. These vectors are analogues to those obtained from NMA and provide a pictorial description of the motion as well as a means for comparing the results of the two methods. Fig.1 represents the results for protein hen-egg lysozyme.


Figure.1 Low-frequency modes of hen-egg lysozyme. Calculations are carried out through the CHARMM 22 empirical force field for both MD power spectra calculation (top) and NMA analysis (bottom).

References
  1. A. R. Leach, Molecular Modelling: Principles and Applications, 2001, ISBN 0-582-38210-6
  2. MacKerell, A.D., Jr., Empirical Force Fields for Biological Macromolecules: Overview and Issues, Journal of Computational Chemistry, 25: 1584-1604, 2004
  3. D. Steele, Theory of vibrational spectroscopy, 1971, ISBN 0-7216-85803