Supplementary material for Ash et al. (2000) Proc. Natl. Acad. Sci. USA 97 (19), 10371-10376.

In a study of ¹³C linewidth vs. pH for a sample of 90% C^e1-enriched a-lytic protease at 33ºC, rate and equilibrium constants were determined (4) for the protonation of imidazole (Im).

Im + H₃O⁺ = ImH⁺ + H₂O,        [1]

where the forward and backward reactions are described by rate constants k_on^H+ and k_off^H+, respectively. A recently refined analysis of these data yielded values of k_off^H+ = 3.4 ´ 10³ s^-1 and a pK_a value of 6.87. Using the following equality, we get a value for k_on^H+:

= 2.5 ´ 10¹⁰ M^-1s^-1.        [2]

Such a diffusion-controlled proton "on" rate underscores the free access of solvent to the active site (5). Assuming the continued absence of buffers and dominance of k_off^H+ over k_off^OH-, we can calculate DW, the extra linewidth due to fast exchange as a function of Dn, the "kinetic window" or chemical shift change in Hz upon protonation at various magnetic fields, p_a, the fraction of the protonated or imidazolium species at various pH values, and K_a as follows (3):

Fig. 4 A-C shows the calculated linewidth of the His C^e1-H proton of true a-lytic protease during a pH titration at various spectrometer fields, corresponding to 250, 500, and 360 MHz, respectively, assuming a "natural" linewidth of 25 Hz. The increased linewidths are evident, particularly in Fig. 4B at 500 MHz, which is similar to observed data points in Fig. 2, explaining the disappearance of the C^e1-H proton resonance in Fig. 2B and its reappearance (most downfield peak) in Fig. 2C. Fig. 4 C-E shows the calculated differences in linewidth vs. pH of true, "fresh," and "aged" a-lytic protease, all at 360 MHz, a spectrometer frequency employed for many of these early studies. The inactive forms are inherently easier to detect than the true form simply because of smaller Dn, the "kinetic window," and smaller pK_a values of the former. Though all forms, active and inactive, exhibit sharp lines at the high and low pH extremes, there could be a tendency to dismiss as spurious or impurity peaks any lines that disappear for a portion of the pH range.

+ C(1 - 3cos²q_Z)]      [4]

If the shielding cone is axially symmetric, only one of the three terms in Eq. 4 is required, otherwise any two (given two of the coefficients A, B, and C the third can be obtained by difference), with contributions from more than one shielding cone being summed. Table 3 lists the values of constants A, B, and C, along with the coordinates used herein for origins of the various shielding cones, which distinguish the models under consideration.

For all 29 serine hydrolases in the database of Derewenda et al. (12), the C^e1-H proton lies within or very close to the amide group O-C'-N plane, and in close proximity to the lone electron pair anti to the N atom. In the Ösapay & Case model (6), a single, axially symmetric shielding cone lies along the y axis, with its origin on the O-C'-N angle bisector at a distance of 0.7 Å from the C' atom (Table 3). As shown in Table 4, the Ösapay and Case model was derived from 17 protein structures, fitting over 5,000 carbon-bound protons with an R value of 0.88, but correlating poorly (R = 0.58) some 1,600 amide N-H protons.

Subsequent models have attempted to make better fits to both C-H and amide N-H protons. However, amide N-H proton chemical shifts are known to be dominated by H-bonding, and by ignoring completely any peptide anisotropic shielding contribution, various researchers have obtained remarkably good correlations of chemical shift vs. various functions of r_OH, the O^...H distance [Wagner et al. (15), using r_OH^-3; Wishart et al. (16), using r_OH^-1; and Tjandra & Bax (17), using 1.3 - r_OH^-2]. Thus, the fact that the Ösapay & Case (6) or any other model fits N-H protons poorly may be more justification than criticism of the model. Instead of lumping anisotropic shielding and H-bonding together in the same treatment, we would argue that it is more useful to try to identify separately the most significant structural element, i.e. H-bonding, distinguished by its associated molecular forces. "Electric field" contributions to chemical shift [Buckingham (18)], included in some modeling efforts, yield an effect analogous to H-bonding, as will be shown later.

The Herranz model (7) employs two axially symmetric shielding cones, the first with its axis lying along the y axis "in the coordinate center of the OCN atoms" (which we interpret to mean the centroid or average of all coordinates), and the second lying along the z axis, with its origin 70% of the interval from the C' toward the O atom. The purpose of the second shielding cone is to improve the fit to amide N-H protons. With regard to this cone, Herranz et al. state "it gives rise to the largest proton shift values for hydrogen bond geometries." It does indeed fit some 239 N-H protons fairly well (R = 0.76), with somewhat different coefficients, A and C. The fit to 265 C^a -H protons, however, is sacrificed compared to Ösapay & Case (6) with an R value of 0.79 (Table 4).

The model of Williamson & Asakura (8) employs two non-axially symmetric shielding cones, the first along the C'-O axis centered at 1.1 Å from C'. The second cone lies along the C'-N axis, and for this term we reoriented the molecule in Fig. 5 with the C-N bond along the z axis. This model, employing more adjustable parameters, typically fits the somewhat smaller collection of C^a -H protons about as well as the Ösapay & Case (1991) model (Table 4), but Asakura & Williamson (9) were able to fit some 1,500 N-H protons to a degree only intermediate between Ösapay & Case (6) and Herranz (7) (Table 4).

Correlation of observed NMR chemical shifts of His C^e1-H protons to those calculated with the anisotropic shielding models from structural data is hampered, as shown in Table 5, by the fact that we have corresponding NMR and structural data for only six enzymes. These structures were obtained from Derewenda et al.'s (12) database of 29 protein x-ray crystal structures (named in our legend, Fig. 6), augmented by no. 30, chymotrypsinogenA [Wang et al.(24); PDB:2CGA] and no. 31, trypsinogen [Kossiakoff et al.(25); PDB:1TGN]. The anisotropic shielding contributions were calculated using the three models under discussion (Table 5), ranging from -0.05 to -0.44, and no correlation vs. observed shift was obtained with any model. Possible factors for the lack of correlation include: (i) the fact that five of the six chemical shifts arise from different enzymes, which may have intrinsically different starting or baseline values; (ii) insufficiently precise x-ray coordinates; and (iii) chemical shift contributions due to H-bonding.

The three anisotropic shielding models under consideration were examined for self-consistency by plotting their calculated shielding values vs. various functions of r_OH, to determine the order of magnitude of C^{e 1}-H proton anisotropic shielding by the single neighboring peptide group identified as its H-bonding acceptor by Derewenda et al. (12). The calculations were carried out using the distances and angles supplied by Derewenda et al. (12). The results of the regression analysis are summarized in Table 5 and depicted in Fig. 6. Fig. 6A shows a plot of calculated anisotropic shielding of the 31 structures in our combined database vs r_OH^-1 using the model of Ösapay & Case (6). The fit is excellent, yielding an R value of 0.88 and standard error of 0.01 (Table 5). The magnitudes of the calculated shieldings are small, less than 0.15 ppm in absolute value. Fig. 6 B and C shows comparable plot for the Herranz et al. (7) and Williamson & Asakura (8) models, differing in that they are plotted vs. r_OH^-3, which gave better correlations (Table 5). Although predicting larger deshielding (the Herranz model as much as -0.47 ppm), none of these models fits the structural data as well as the Ösapay and Case model. The Herranz model gives an overall R value of 0.77, with standard error of 0.07 (Table 5). Recall that the second shielding term arising from the cone that lies along the C-O bond was intended to help account for H-bonded protons. Table 5 shows how poorly this C-O term serves the model, making a substantial contribution that does not fit the data (R = 0.46 for the C-O term). The Williamson & Asakura (8) model gives an overall R value of 0.75 and standard error of 0.07, similar to the Herranz model both in shielding magnitude (as large as -0.42) and the quality of the correlation. The shielding cone along the C-N bond in the Williamson model fits the data better (R = 0.86) than the one along the C-O bond (R = 0.72), but generally contributes less.

The anisotropic shielding calculations shown in Table 5 and Fig. 6 are not corrected for ring currents because some structures [no. 18, no. 19, and no. 22 from Derewenda et al. (12)] are not available from the Protein Data Bank. However, from a subset of the 31 proteins considered in this paper, namely the six for which NMR data exist for His C^{e 1}-H protons, we calculated the contributions of all surrounding aromatic groups using software available from both Ösapay & Case (6) and Williamson & Asakura (8). The data shown in Table 6 include all His residues, with the active site His residues in boldface type. The ring current contributions are generally of small absolute magnitude (<0.1), and there is no indication of any systematic deshielding (negative s ) trend upon the His C^{e 1}-H protons.

Table 6 also shows the magnitudes of the "electric field" term [Buckingham (18)] included in some chemical shift models. Ösapay & Case (6) and Williamson & Asakura (8) found that inclusion of the electric field term yielded a modest but systematic improvement in the quality of their data fitting. However, the model of Herranz et al. (7) ignored the electric field term, as did subsequent work on N-H proton shifts by Asakura et al. (9). As shown in Table 6, electric field contributions as large in magnitude as -0.25 ppm may be expected. Described as a "polarization of the C-H bond by the electric field of neighboring atoms" (8), this term seems to us barely, if at all, distinguishable from H-bonding. If the electric field term is indeed synonymous with H-bonding, its magnitude has been underestimated in all models, and by a significant factor in the model of Ösapay & Case (6) favored by our analysis.

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