Steady-state NOE

It can be shown (see Appendix VII) that the following relationship holds  

In order to define and pi(j) it is convenient to refer to Fig. 7.1 and to define w[ and w( as the transition probabilities between two states involving a single quantum transition either of spin 7 or J wo is the zero quantum transition and corresponds to the —I— + — transition and vice versa wj corresponds to 

It represents the total probability for nucleus 7 to change its spin component along z in a coupled two spin system. 

The transitions corresponding to w2 and u o involve simultaneous changes in spin state of both nuclei. The difference between W2 and wq tells us how the variation in population of J affects the equilibrium of /. In other words, one can think in terms of transfer of spin population from J to I when J is saturated. The larger the cross relaxation, the larger the dipolar coupling. The NOE on I is proportional to the cross relaxation from J and inversely proportional to the capability of / to return to equilibrium once its equilibrium is perturbed through cross relaxation. 

The transition probabilities depend on the mean squared interaction energy relative to the mechanism which causes the transition, times the value of the spectral density at the required frequencies (Eq. (3.14)). The square of the dipolar interaction energy is, as usual (see Eq. (1.4) and Appendix V), proportional to (p, 1 1x2/r3)2, where p and p2 are the magnetic moments of the two spins. The actual equations are 

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Besides measuring and T2 for nuclei such as C or N, relaxation studies for these nuclei also include measurements of the NOE factor, cf equation B 1.13.6. Knowing the (pj) and the steady-state NOE... 

Following is a linear three-spin system and the observed steady-state nOe s between three nuclei. The distance between nuclei A and B is... 

There are two types of NOE experiments that can be performed. These are referred to as the steady-state NOE and the transient NOE. The steady-state NOE experiment is exemplified by the classic NOE difference experiment [15]. Steady-state NOE experiments allow one to quantitate relative atomic distances. However, there are many issues that can complicate their measurement, and a qualitative interpretation is more reliable [16]. Spectral artifacts can be observed from imperfect subtraction of spectra. In addition, this experiment is extremely susceptible to inhomogeneity issues and temperature fluctuations. 

The measured spin relaxation parameters (longitudinal and transverse relaxation rates, Ri and P2> and heteronuclear steady-state NOE) are directly related to power spectral densities (SD). These spectral densities, J(w), are related via Fourier transformation with the corresponding correlation functions of reorientional motion. In the case of the backbone amide 15N nucleus, where the major sources of relaxation are dipolar interaction with directly bonded H and 15N CSA, the standard equations read [21] ... 

For experiments that involve intermolecular NOEs, the mathematical description is not straightforward. For the ranking presented at the end, we decided to show the steady-state NOE and the NOE build-up as a function of the MW of the target for a comparison. The different experiments in use are described in detail, together with examples, in the following sections. A ranking with respect to their practical applicability will be given at the end of this section. 

Fig. 14.13 Graphical representation of the effect of MW on T2 (dashed-dotted), on the translational diffusion rate D (solid), on the steady state NOE (dashed) and on the build-up of the NOE (dotted). All values are normalized to a 300 Da molecular weight molecule. For the calculation of the parameters involving dipolar relaxation we used a formula that can be found in the literature...

The second set of experiments, where only one sample is required, always involves an intermolecular NOE transfer. The STD experiment exploits the spin-diffusion mediated steady-state NOE. In our experience, this method yields reliable results and can be easily implemented. Ligand signals are easily distinguished from protein signals by inserting a relaxation filter prior to acquisition. Care has to be taken to avoid direct saturation of ligand resonances. 

TOPHAT-shaped 90° pulses are used in other cases as the best compromise with respect to the excitation profile, the phase homogeneity and length. Depending on the type of the detected spin-spin interaction - being either scalar or dipolar coupling - each selected spin is initially perturbed only once (ID TOCSY, ID INADEQUATE, ID C/H COSY, 2D TOCSY-COSY and 2D HMBC), or for several times (ID NOE). With each of the selected spins initially perturbed only once the inherently smaller transient NOEs would be detected in the latter case, whereas with the multiple excitation of a selected spin within the NOE build-up period the stronger steady-state NOEs are more or less approximated. 

In contrast to the 1D experiment, where steady-state NOEs may be obtained, only the less intense transient NOE.s are measured in the NOESY experiment. ROEs can only be obtained as transient effects in both the ID and the 2D experiment. Furthermore the intensities of the NOESY and ROESY cross peaks depend upon the molecular size as well as the length of the mixing period. In the case of large molecules, e.g. polypeptides, rather short mixing times are usually chosen to avoid spin diffusion. 

When measuring steady state NOE, the effect measured on saturating / or on saturating J may be different because p and pj are in general different. We say that steady state NOE is not symmetric. It follows that larger NOEs, and then more favorable cases, occur when the signal with larger p is saturated. 

Up to now steady state NOEs have been considered, i.e. when one signal is saturated for a long time with respect to T of the nucleus on which NOE is going to be measured. Let s consider here what happens when the saturation time is short and variable. The resulting NOE is called truncated NOE [17] because not enough time is left for full polarization transfer. These experiments are of fundamental importance for the measurement of pi, for evaluating cross relaxation, and to avoid or to measure spin diffusion. 

In Appendix VII, where the steady state NOE has been derived, the equation for the NOE as a function of the irradiation time is also derived. In the case of homonuclear NOE, it is... 

For irradiation times of J short with respect to the relaxation time of / the NOE extent is independent of the relaxation time of the nucleus and provides a direct measurement of time required to saturate signal J is not negligible compared with t, the response of the system is not linear [18]. The truncated NOE is independent of paramagnetism as it does not depend on p/, which contains the electron spin vector S in the R[m term, and only depends on gkj), which does not contain S. If then the steady state NOE is reached, the value of p/ can also be obtained. This is the correct way to measure p/ of a nucleus, provided saturation of J can be considered instantaneous. In general, measurements at short t values minimize spin diffusion effects. In fact, in the presence of short saturation times, the transfer of saturation affects mainly the nuclei directly coupled to the one whose signal is saturated. Secondary NOEs have no time to build substantially. As already said, this is more true in paramagnetic systems, the larger the R[m contribution to p/. 

The ROE dependence on the spin lock time has the same profile as that of transient NOE, with the difference that the limiting values are 0.385 and 0.675 at the condition that o>itc < 1 (see Fig. 7.10). It appears that the ROE is less convenient than the transient and steady state NOEs in the sense that the expected effect is smaller when all other conditions are the same. Another disadvantage in paramagnetic molecules is that it is difficult to spin lock all the signals in a... 

From comparison of Table 8.2 with Table 7.1 (or of Eq. (7.20) with Eq. (7.10)), i.e. of transient NOE or NOESY vs. steady state NOE intensities, it appears that the latter are superior under any circumstance. This superiority is striking if the intrinsic asymmetry of the steady state NOE with respect to the symmetry of transient NOE and NOESY experiments (Section 7.4) can be exploited, as in the case of irradiation of fast relaxing nuclei to detect NOE to slow relaxing nuclei. Of course, NOE experiments are advantageous over NOESY experiments if one is looking for dipolar connectivities from only a few specific signals. 

The calculations are performed for a tr/(7> value of —1 s-1. Steady state NOE is always superior to transient NOE, except when T of the irradiated signal is more than twice the 7/ of the responding signal (upper right part of the table). 

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