As mentioned in Section A, analysis of NMR line shapes and estimation of peak width W can provide values of T2 via eq. (8.3). In this section we shall see an example of the estimation of correlation times of a deuteriumlabeled protein from the peak width.
If observed line shapes for the 2D NMR spectra are Lorentzian, the effective correlation time (t2jeff) for [2D6]isopropyl groups incorporated into the protein can be calculated directly. Under such conditions, effective correlation times are calculated from the relationship 
where (e2qQ/h) = quadrupole coupling constant and W is the line width at half-height in Hz. The quadrupole coupling constant in our example is 170 kHz.
Deuterium labeling was achieved by covalently binding [2D6]isopropyl groups to 80% of the lysine residues of /3-lactoglobulin . Figure 8.4A shows the 61.25 MHz spectrum of the deuterium-labeled /3-lactoglobulin in an aqueous buffer. Underlying the raw data envelope are the two Lo-rentzian peaks (see Section 3.B.2 for models) found to give the best fit by a program employing a Gauss-Newton algorithm. The resulting residual plot is shown at the bottom. The corresponding spectrum of deuteriumlabeled /3-lactoglobulin in 6 M guanidine hydrochloride is shown in Figure 8.4B. Models with one or two Gaussian peaks or with a single Lorentzian peak resulted in poor fits with clearly nonrandom residual plots.
Thus, a model composed of the sum of two Lorentzian peaks with nearly identical positions but different linewidths produced the best fit for spectra of the intact protein at 61.25 MHz in either solution. Using the results of nonlinear regression analyses, residual plots, and F tests confirmed that the two Lorentzian model gave the best fit.
Two populations of lysine residues were apparent in spectra obtained in either aqueous buffer or 6 M guanidine hydrochloride. Slow and fast correlation times were determined from the individual linewidths obtained from the nonlinear regression analysis of the spectra. The numbers of residues corresponding to each tcic[f (eq. (8.5)) were obtained from the ratio of the areas of the two Lorentzian curves. In an aqueous buffer, the equivalent of 9.7 modified lysine residues were observed. Here, 6.5 residues
Figure 8.4 (A) 61.25 MHz 2D NMR spectra (solid lines) of [2D6]isopropyl-/3-lactoglobulin in tris buffer pH 7.5 and (B) in 6 M guanidine hydrochloride, pH 7.5. Each spectrum gave the best fit by nonlinear regression to a model composed of the sum of two Lorentzian peaks (see Section 3.C.2), shown underlying the spectra. Traces representing residual plots are shown at the bottom, below each spectrum. (Reprinted with permission from , copyright by the American Chemical Society.)
had a fast average correlation time (tc eff = 70 ps), 3.2 residues had a slow, but observable average correlation time (rCjeff = 320 ps), and the remaining 2.3 residues had a tc>efl apparently so long that they could not be resolved from the baseline at this field strength. In 6 M guanidine hydrochloride, where all of the modified lysines were observed, the equivalent of 8.7 residues had a fast average correlation time (rceff = 20 ps) and 3.3 residues a slow average correlation time (rc>eff = 320 ps).
The best fit for the spectrum of deuterium-labeled /3-lactoglobulin after hydrolysis to individual amino acids (not shown) was a single Lorentzian. This result showed that in this hydrolyzed sample, all [2D6]isopropyl groups were equally mobile.
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