Computationally efficient gradients for relaxation matrix-based structure refinement including the accommodation of internal motions.

A general method for deriving analytical gradients based on NOESY cross-peak intensities is presented. This method allows for very rapid calculation of exact gradients of cross-peak intensities with respect to parameters directly related to the pairwise dipole-dipole interaction giving rise to the cross peak. In the simplest case, gradients with respect to internuclear separation can be calculated, which allows for the rational modification of distance constraints used in structural refinement. In the general case, any arbitrary level of knowledge of the internal dynamics of the molecule can be introduced, thereby providing a pathway for the experimental determination of motional parameters. The motional characteristics of the internuclear vectors defining dipole-dipole interactions are cast in model-free terms. The form of the gradient circumvents many of the limitations of gradients expressed in terms of Cartesian or dihedral variables. The gradients presented are simple, direct, and exact and require little computational effort to calculate.