Between two protons

In an isolated two-spin system, the NOE (or, more accurately, the slope of its buildup) depends simply on where d is the distance between two protons. The difficulties in the interpretation of the NOE originate in deviations from this simple distance dependence of the NOE buildup (due to spin diffusion caused by other nearby protons, and internal dynamics) and from possible ambiguities in its assignment to a specific proton pair. Mofec-ufar modeling methods to deaf with these difficulties are discussed further below. 

Depending on experimental parameters, NOE intensities will be affected by spin diffusion (Eig. 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 distance dependence of the NOE. Depending on the conformation, it can be more efficient to move magnetization over intennediate protons than directly. The treatment of spin diffusion during structure refinement is reviewed in more detail in Refs. 31, and 71-73. 

Erom 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 (Eig. 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, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

Now lei us turn to the problem of how the composition of a nucleus affects its stability. The forces that exist between the particles in the nucleus are very large. The most familiar of ihe intranuclear forces is the coulomb force of repulsion which the protons must exert on one another. In order to appreciate the magnitude of this repulsive force, let us compare the force between two protons when they are separated by 10 8 cm, as they are in the hydrogen molecule, with the force between two protons separated by 10-18 cm, as they are in a helium nucleus. In the first case we have... 

By comparing these two answers we can see that the repulsive force between two protons in the nucleus is about ten billion times as great as the repulsive force between two protons bound together in a hydrogen molecule. In order to overcome these enormous intranuclear coulomb repulsions and hold the nucleus together there must exist some very strong attractive forces between the nucleons. The nature of these forces is not understood and remains a very important problem in physics. 

From these studies of sugars and related molecules, it is possible to derive empirical rules whereby Ri(ns) values can be used as the basis for configurational assignments. These rules are illustrated graphically in Fig. 5, where the relative relaxation-efficiency between two protons is plotted as a function of the interproton distance. 

Fig. 5.—Plot of the Relative Relaxation Efficiency between Two Protons versus the Inverse Sixth Power of Their Internuclear Separation. (Derived from measurements of Dreiding stereomodels reproduced, with permission, from Ref. 43.)...

Because of the intrinsic flexibility and poor resonance dispersion of unfolded and partly folded proteins, long-range NOEs are generally very difficult to observe and assign. While observation of a long-range NOE between two protons provides a definitive indication that they are in close proximity in at least some structures in the conformational ensemble, determination of the nature of the folded structures is difficult unless an extensive network of NOEs can be observed. This has so far been achieved in only one case (Mok et al., 1999). 

Because the NOE falls off with the inverse sixth power of distance, an observed NOE between two protons implies that the two protons are reasonably close to each other. Thus, it is possible to determine their relative position using the NOE. The following examples illustrate the use of NOE for stereochemical assignments. 

In the absence of an electric field, E, there is a dynamic equilibrium between two proton boundary structures, but when E is present, the weight of boundary structure II is increased because the hydrogen bond is easily polarized. Furthermore, the... 

Q What is the repusion energy between two protons which are 106 pm apart, as they are in the ground state of the hydrogen molecule-ion ... 

FIGURE 3.27 Spin coupling between two protons with very different chemical shifts. 

The homonuclear NOE, almost always between two protons, is used to measure distances and determine stereochemical relationships. The NOE intensity (the percent increase or decrease of z magnetization observed at a nearby proton) is proportional to the inverse sixth power of distance between the two protons (1/r6), and is generally too weak to be observed for distances over 5 A. 

These spin-spin interactions called Coupling Constants can be experimentally determined even in very complex spectra. Coupling constants, e.g. between two protons or a 13C nucleus and a proton, are molecular parameters, independent of the applied external magnetic held. Expressed in Hertz, they can have positive or negative values depending on the numbers of bonds between the two nuclei. 

As the coupling constants, especially between two protons on adjacent carbon atoms, are very sensitive to rotational changes, one can also derive from them the preferred conformation of the polymer chain (see Diehl et al., 1971). 

The small 1.4 Hz coupling is a 2/hh coupling between two protons on the same carbon that are different because there is no rotation about the double bond. 2/hh coupling is called geminal coupling. 

The NOE interaction between two proton magnetic dipoles takes place predominantly directly through space rather than through bonds (as is the case in normal spin coupling). This through-space interaction is very sensitive to the distance... 

The Coulomb repulsion energy Ec between two protons is given by E... 

An important prerequisite for all these reactions is sufficiently high kinetic energy of the reacting particles. According to section 8.3, the Coulomb barrier for the reaction between two protons is x 0.5 MeV and that for the reaction of a proton with is 1.8 MeV. According to the kinetic theory of gases, the mean kinetic energy of gas molecules kin is proportional to the temperature T,... 

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