where more information on the data can be used to evaluate the restraint (e.g., by inspecting the peak shape).

B. Automated Assignment of Ambiguities in the NOE Data

Assigning ambiguous NOEs is one of the major bottlenecks in NMR solution structure determination, comparable to map fitting in X-ray crystallography. In principle, ambiguous NOEs need not be explicitly assigned if they are used as ADRs, because the assignment is done implicitly in the structure calculation. This is because the summed distance D is strongly weighted toward the shortest of the contributing distances. If ADRs are used in the refinement of an already reasonably well determined structure, this weighting is expected to, in most cases, favor the really dominating contribution to the ambiguous NOE. The implicit assignments, achieved through weighting with the distances in the structure, will be mostly correct, and the path to the final structure satisfying all data will be relatively smooth.

The case is very different when ADRs are used for calculating structures ab initio, starting from random structures. Obviously, most of the initial implicit assignments from the random structures are incorrect, and the path toward the final structure is much more difficult owing to additional local minima in the energy. During the calculation, the interproton distances, and with them the weighting on different assignment possibilities, need to change. However, convergence can be achieved by treating the amiguous NOEs appropriately in generalized simulated annealing protocols (see Fig. 5).

C. Iterative Explicit NOE Assignment

The main difficulties with a fully automated method lie in defining rules for explicit assignment based on an ensemble of structures with possibly incorrect features and providing mechanisms for correcting incorrect assignments. Two fully automated iterative assignment methods have been proposed, one based on ADRs (ARIA: ambiguous restraints for iterative assignment) [20,23] and the other on self-correcting distance geometry (NOAH [22]).

Methods to assign ambiguous NOEs follow a sequence of steps (see Fig. 6). Structures are calculated and NOEs are assigned in an iterative way. In the first iteration, the assignments have to be based on the frequencies alone. In the following iterations, the assignments of ambiguous NOEs are derived from the structures by comparing interproton distances corresponding to each assignment possibility. In the ‘‘traditional’’ approach, one possibility would be chosen by hand, and a peak would not be used if an unambiguous assignment were impossible. In contrast, the automated methods use ambiguous peaks during the structure calculation. The key difference between these methods is in how ambiguous peaks are converted into distance restraints. The program ARIA generates one ADR for each ambiguous peak, whereas the program NOAH creates an unambiguous restraint for each assignment possibility.

In general, the advantages of using an automated method may be comparable to those of SA refinement in X-ray crystallography [68], where many of the operations necessary to refine a structure can be done automatically and the remaining manual interventions are easier because the SA refinement usually results in a more easily interpreted electron density map. Automated methods are usually used in combination with manual assignment. However, fully automated assignment of the NOEs is possible (see Fig. 7) [69].

Figure 6 Steps in automated assignment. (1) Select the Sconv lowest energy structures from iteration i 1 that are used to interpret the spectra. (2) For each peak, list all possible assignments compatible with the resonances within a frequency range. (3) Extract a distance for each assignment possibility from the ensemble of structures. (4) Use the distances to assign ambiguous NOEs. (5) Calibrate the peak volumes to obtain distance restraints. (6) Calculate structures based on the new restraints.

D. Symmetrical Oligomers

Symmetrical oligomers present a special difficulty for NMR spectroscopy, because all symmetry-related hydrogens will have equivalent magnetic environments and therefore will be degenerate in chemical shift. Only one monomer is ‘‘seen’’ in the spectra. In principle, every single NOE peak in the spectrum is therefore ambiguous. This ambiguity that arises from the symmetry can be treated with the same concept as ambiguities due to limited spectral resolution with ambiguous distance restraints. Structure determinations of symmetrical oligomers were reviewed in detail recently [70].

The same principal ideas are incorporated in the calculation protocols for symmetrical oligomers as with asymmetrical systems, i.e., the weight is reduced specifically for ADRs or all distance restraints. In addition, the symmetry of the system restricts conforma-

Figure 7 Example of a fully automated assignment. The structure ensemble of the seven lowest energy structures at each iteration is shown. These structures are used for the violation analysis and for a partial assignment of ambiguous NOEs. In the first iteration the structures are calculated with all restraints, where each restraint has all assignment possibilities. In each subsequent iteration, consistently violated restraints are removed, and assignment possibilities are selected with increasingly tight criteria such that at the end of the eight iterations most NOEs are unambiguously assigned.

tional space and is maintained during the calculation by additional restraints. An attractive potential between the monomers can be used in the beginning of the protocol to prevent them from drifting apart. The special difficulties with symmetrical oligomer calculations arise for two reasons: First, all NOEs are ambiguous a priori; second, the assignments of neighboring residues are strongly correlated. A minimization method such as simulated annealing that moves single atoms (or rigid parts of amino acids) may not be optimal for moving larger parts of the structure coherently if a whole set of NOEs needs to be implicitly reassigned. As a result, the structure calculation has a lower convergence rate than for asymmetrical systems. A combination of annealing calculations with other optimization approaches (e.g., a recent branch-and-bound algorithm [36]) may be a more efficient approach. However, the present approaches based on annealing alone have been successful in several cases for quite complex systems (up to a tetramer and a hexamer [70]).

V.TREATMENT OF SPIN DIFFUSION

Depending on experimental parameters, NOE intensities will be affected by spin diffusion (Fig. 8). Magnetization can be transferred between two protons via third protons such that the NOE between the two protons is increased and may be observed even when the distance between the two protons is above the usual experimental limit. This is a consequence of the d 6 distance dependence of the NOE. Depending on the conformation, it can be more efficient to move magnetization over intermediate protons than directly. The treatment of spin diffusion during structure refinement is reviewed in more detail in Refs. 31, and 71– 73.

From a given structure, the NOE effect can be calculated more realistically by complete relaxation matrix analysis. Instead of considering only the distance between two protons, the complete network of interactions is considered (Fig. 8). Approximately, the

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, τm, and the complete relaxation matrix, (Rij), which is a function of all interproton distances dij and functions describing the motion of the protons. γ is the gyromagnetic ratio of the proton, is the Planck constant, τ is the rotational correlation time, and ω is the Larmor frequency of the proton in the magnetic field. The expression for (Rij) is an approximation assuming an internally rigid molecule.

relaxation matrix (Rij) depends on all interproton distances and on parameters describing overall and local motion. The long mixing times often necessary in heteronuclear NOE experiments would make spin diffusion estimates especially valuable. To calculate heteronuclear NOEs realistically, the transfer efficiencies between protons and heteronuclei should be incorporated into the equations [74].

One approach to include spin diffusion corrections in a structure calculation is a direct refinement against NOE intensities, analogous to X-ray crystal structure refinement. In this approach, forces are calculated directly from the difference between the experimental NOE intensities and those calculated from the structure via the relaxation matrix. This necessitates, however, an expensive evaluation of derivatives of the simulated NOE spectra with respect to coordinates at every minimization step. Approximations and faster methods to evaluate the gradients [75] make this direct approach more feasible. Various pseudo-energy functions have been proposed. The simplest form (harmonic in the difference between experimental and calculated NOE) places a predominant weight on the largest intensities (shortest distances), which are most often due to intraresidue interactions and will therefore contribute little to determining the conformation of the molecule. By using pseudo-energy functions depending on the sixth root of the difference between calculated and experimental NOEs, the weight is distributed more equally.