Projection methods for the analysis of complex motions in macromolecules

In studies of macromolecular dynamics it is often desirable to analyze comp lex motions in terms of a small number of coordinates. Only for simple type s of motion, e.g., rigid-body motions, these coordinates can be easily cons tructed from the Cartesian atomic coordinates. This article presents an app roach that is applicable to infinitesimal or approximately infinitesimal mo tions, e.g., Cartesian velocities, normal modes, or atomic fluctuations. Th e basic idea is to characterize the subspace of interesting motions by a se t of (possibly linearly dependent) vectors describing elementary displaceme nts, 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 per mits 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 use ful for the interpretation of spectroscopic measurements on proteins, e.g., inelastic neutron scattering experiments.