Projection methods for the analysis of complex motions in macromolecules

In studies of macromolecular dynamics it is often desirable to analyze complex motions in terms of a small number of coordinates. Only for simple types of motion, e.g., rigid-body motions, these coordinates can be easily constructed from the Cartesian atomic coordinates. This article presents an approach that is applicable to infinitesimal or approximately infinitesimal motions, e.g., Cartesian velocities, normal modes, or atomic fluctuations. The basic idea is to characterize the subspace of interesting motions by a set of (possibly linearly dependent) vectors describing elementary displacements, and then project the dynamics onto this subspace. Often the elementary displacements can be found by physical intuition. The restriction to small displacements facilitates the study of complicated coupled motions and permits the construction of collective-motion subspaces that do not correspond to any set of generalized coordinates. As an example for this technique, we analyze the low-frequency normal modes of proteins up to approximate to 20 THz (600 cm(-1)) in order to see what kinds of motions occupy which frequency range. This kind of analysis is useful for the interpretation of spectroscopic measurements on proteins, e.g., inelastic neutron scattering experiments.