The pK_a values of acidic and basic residues buried at the same internal location in a protein are governed by different factors

Summary

The pK_a values of internal ionizable groups are usually very different than the normal pK_a values of ionizable groups in water. To examine the molecular determinants of pK_a values of internal groups, we compared the properties of Lys, Asp and Glu at internal position 38 in staphylococcal nuclease. Lys-38 titrates with a normal or elevated pK_a whereas Asp-38 and Glu-38 titrate with elevated pK_a values of 7.0 and 7.2, respectively. In the structure of the L38K variant, the buried amino group of the Lys-38 side chain makes an ion pair with Glu-122; whereas, in the structure of the L38E variant, the buried carboxyl group of Glu-38 interacts with two backbone amides and has several nearby carboxyl oxygen atoms. Previously we showed that the pK_a of Lys-38 is normal owing to structural reorganization and water penetration concomitant with ionization of the Lys side chain. In contrast, the pK_a of Asp-38 and Glu-38 are perturbed significantly owing to an imbalance between favorable polar interactions and unfavorable contributions from dehydration and from Coulomb interactions with surface carboxylic groups. Their ionization is also coupled to subtle structural reorganization. These results illustrate the complex interplay between local polarity, Coulomb interactions and structural reorganization as determinants of pK_a values of internal groups in proteins. This study suggests that improvements to computational methods for pK_a calculations will require explicit treatment of the conformational reorganization that can occur when internal groups ionize.

Introduction

A small fraction of ionizable residues in proteins are sequestered from water and buried in the protein interior.^1-3 These internal ionizable groups are essential for catalysis,^4-6 H⁺/e^- transport,^7-10 and molecular recognition.¹¹ The pK_a values of internal ionizable groups are usually different than the normal pK_a values in water,^12-19 and are often tuned for specific biological purposes.⁴ Understanding the determinants of these pK_a values is important for quantitative description of the structural basis of function in a large variety of biological processes.

The shift in the pK_a of an internal group relative to the normal pK_a in water is governed by differences in the polarity and polarizability experienced by the charge in the two environments (ΔG_self), and by Coulomb interactions with the charges of other ionizable groups. Structural reorganization of the protein coupled to the ionization of internal groups can also influence their pK_a. One of the goals of this study was to examine the relative magnitude of these three determinants of the pK_a values of internal groups.

The polarity and polarizability in the protein interior are usually lower than in bulk water; therefore, ΔG_self is generally unfavorable for buried ionizable groups. For this reason, the pK_a values of internal ionizable groups are usually shifted in the direction that favors the neutral state (i.e. increase in pK_a for acidic groups and depression for basic ones).^12-18 Surprisingly, the apparent polarity and polarizability in the protein interior reported by internal ionizable groups is not as low as previously thought. ^{15-17; 20-25} In some cases, hydrogen bonds (i.e. high polarity) can actually compensate fully for the loss of hydration experienced by a charged atom inside a protein.^{6; 19; 26}

Coulomb interactions between surface charges are usually weak because charges are screened effectively by water.^27-31 In contrast, the Coulomb interaction of ion pairs sequestered from bulk solvent at protein-protein interfaces can be quite strong (3-5 kcal/mol).³² Coulomb interactions between surface and internal groups in protein active sites have never been studied directly. Surface ionizable group have been shown to have small, but observable effects on enzyme activity.^33-35 Even if the effects are small, the sum of many small interactions could lead potentially to a large effect.^{36; 37} A complete understanding of interactions between internal and surface charges is necessary to understand contributions of surface ionizable residues to the properties of internal groups at active sites and interfaces.⁴

Staphylococcal nuclease (SNase) is an excellent model system for studying properties of internal ionizable groups systematically, and for dissecting molecular determinants of their pK_a values. It has been shown that hyperstable variants of SNase can tolerate substitutions of 25 internal positions with Lys, Asp, Glu and Arg.³⁸ The majority of these internal ionizable groups titrate with pK_a values shifted in the direction that promotes the neutral state, some by as much as 5.7 pK_a units.^{15; 16; 22; 23; 25; 26; 39} We have shown previously that, although Lys-38 in staphylococcal nuclease is internal, it titrates with a normal or possibly elevated pK_a value. The pK_a is not depressed despite the amino group being secluded from bulk water in the crystal structure; water penetration facilitated by structural relaxation ensures hydration of the charged group.³⁹ In contrast, we show here that the pK_a of Glu-38 and Asp-38 are shifted significantly. The differences in the ionization behavior of Lys, Glu, and Asp at position 38 in SNase offer opportunities to examine contributions by the reaction field of bulk solvent, local polarity and polarizability, conformational reorganization, and Coulomb interactions to the pK_a values of these internal ionizable groups.

Results

Crystal structure of the L38E variant

Two hyperstable variants of SNase were used in this study: PHS and Δ+PHS. The structure of the PHS/L38E variant was solved to 2.0 Å and compared to the structures of PHS nuclease and the PHS/L38K variant.^{39; 40} Refinement statistics are shown in Table 1 in the Supplementary Information. PHS nuclease was used for crystallographic studies instead of the Δ+PHS form of nuclease that was used for equilibrium thermodynamic and NMR spectroscopy experiments because the PHS/L38E crystallized and Δ+PHS/L38E did not. PHS nuclease contains six residues (44-49) in a dynamic loop and two point mutations (F50G and N51V) that are not present in Δ+PHS nuclease. The structures of Δ+PHS and PHS variants are superimposable.^{40; 41}

The overall structure of the PHS/L38E variant is comparable to the structures of PHS nuclease (C_α RMSD = 0.7 Å) and of the PHS/L38K variant (C_α RMSD = 0.4 Å), even in the region surrounding Glu-38 (Figure 1A). The primary difference between the PHS/L38E and PHS/L38K structures is the position of Glu-122. This residue is in the same position in the structure of PHS and PHS/L38E, whereas, the C_δ of Glu-122 is shifted by 1.6 Å to establish a Lys-38/Glu-122 ion pair in the structure of PHS/L38K.³⁹ Residues 113-115 in the structure of PHS/L38E are in a slightly different conformation than in the other structures owing to the presence of an inhibitor (thymidine-3′,5′-diphosphate) that is present in the structure of PHS/L38E nuclease and absent in the other structures.

Figure 1.

Crystal structure of PHS/L38E (pink, PDB accession code 3D6C) overlaid on the structures of PHS/L38K (blue, PDB accession code 2RKS) and PHS nuclease (white, PDB accession code 1EY8). (A) The global fold of the protein is not perturbed. The C_α atom of Asp and Glu residues are shown as red spheres. Glu-38, Lys-38, and Glu-122 are shown as sticks. (B) Microenvironment of Glu-38 and Lys-38. Ionizable residues within 8.4 Å of Glu-38 are shown in stick, hydrogen bonds as black dashes.

The oxygen atoms of the Glu-38 side chain are completely solvent inaccessible in the structure of PHS/L38E. The nearest crystallographic water molecule is 5.4 Å from the Glu-38 O_ε1 atom. Thus far, this is the only crystal structure of a SNase variant with an internal oxygen atom in which the atom is not hydrated by an internal water molecule.^{16; 42} Interactions with internal water molecules might be precluded by hydrogen bonds between the carboxylic group of Glu-38 and the backbone amides of Thr-120 and His-121, and the hydroxyl group of Tyr-91 (Figure 1B). The hydrogen bond to Tyr-91 directly links Glu-38 into an extensive hydrogen bond network.^{30; 31; 43; 44}

Although SNase is a basic protein, the ionizable residues nearest to Glu-38 in the crystal structure of the PHS/L38E variant are acidic: Asp-77 (4.0 Å) and Glu-122 (5.1 Å). The next nearest ionizable residues are basic: His-121 (6.5 Å) and Arg-126 (6.6 Å) (Figure 1B). The proximity of these residues makes them ideal for direct measurement of Coulomb interactions between surface charges and the carboxylic groups of Asp-38 and Glu-38.

pK_a values of Glu-38 and Asp-38

The pK_a values of Glu-38 and Asp-38 were measured by analysis of the pH-dependence of protein stability. This method takes advantage of the thermodynamic linkage between proton binding and stability.⁴⁵ Measurement of the unfolding free energy (ΔG°_H2O) of a protein as a function of pH reports on the pK_a values of all ionizable residues in the protein. The pK_a value of a single group introduced by mutagenesis can be measured by subtracting ΔG°_H2O of the background protein (i.e. Δ+PHS nuclease) from ΔG°_H2O of the variant protein (i.e. Δ+PHS/L38E). Shifts in the pK_a are reflected in the characteristic shape of the pH-dependence of ΔΔG°_H2O.^{15; 23; 25} It was shown previously that the pK_a of Lys-38 was ≥ 10.4, comparable to the normal pK_a of a Lys in water.³⁹ In contrast, the pK_a values of Glu-38 and Asp-38 were 7.0 +/- 0.3 and 6.8 +/-0.3 pH units, respectively (Figure 2). Relative to the normal pK_a values of 4.4 and 4.0 for Glu and Asp in water, respectively, this corresponds to shifts in pK_a of 2.6 and 2.8 pH units.

Figure 2.

pH-dependence of ΔΔG°_H2O for the Δ+PHS/L38D ( ), Δ+PHS/L38E ( ), and Δ+PHS/L38K ( ) variants. Solid lines are fits to the data (see methods). Error bars are propagated from GdnHCl denaturation fit errors.

The measurement of pK_a values by analysis of the pH dependence of stability is too imprecise for detailed investigation of the contribution of Coulomb interactions to the observed pK_a value. An attempt was made to measure the pK_a of Glu-38 with NMR spectroscopy using the pH-dependence of the Glu-38 C_δ resonance.⁴¹ Although the C_δ resonance could be assigned at low pH, the peak entered intermediate exchange above pH 5.6 and could not be followed at higher pH values. Resonances corresponding to the C_γ/C_δ atoms of Glu-73, Glu-75, Asp-77, Asp-83, and Glu-122 all showed a secondary apparent titration centered at pH 7.0. At positions 75 and 77, the magnitude of the secondary transition was greater than 0.5 ppm. All of the data from NMR spectroscopy are consistent with a pK_a of 7.0 for Glu-38.

The pK_a of Glu-38 was also obtained by performing a global fit to the pH titrations of multiple resonances.^{46; 47} Specifically, the pK_a was obtained by analysis of the pH-dependence of the ¹H chemical shift of six backbone amides (Thr-33, Phe-34, Arg-35, Glu-75, Gly-88, and Leu-89). The titration events monitored by these amide backbone atoms in Δ+PHS nuclease are shown in Figure 3A. No changes larger than 0.06 ppm were observed over the pH range studied. A small transition centered at pH 6.3 +/- 0.3 is visible for positions 34, 35, 75, and 89, most likely reflecting the titration of His-8 or Asp-21, whose pK_a values are both 6.5 in Δ+PHS nuclease.⁴¹ In contrast, the pH-dependence of the ¹H chemical shift of the same six amides in the Δ+PHS/L38E variant (Figure 3B) reflects a large transition. A global fit of the modified Hill equation to this transition yielded a pK_a value of 7.0 +/- 0.1, in excellent agreement with the pK_a of Glu-38 determined using linkage thermodynamics and the value inferred from the titration of carboxylic acids. A similar analysis of the L38D variant showed that Asp-38 has a pK_a of 7.2 +/- 0.1, which is also in agreement with the value of 6.8 +/- 0.3 obtained by analysis of the pH dependence of stability of the Δ+PHS/L38D variant. The pK_a values extracted by global fit of NMR spectroscopy data and by linkage analysis are summarized in Table 1.

Figure 3.

Apparent titration of ¹H backbone resonances in Δ+PHS nuclease (A) and the Δ+PHS/L38E variant (B). Series are Thr-33 (•), Phe-34 (+), Arg-35 (×), Glu-75 ( ), Gly-88 (+), and Leu-89 (×). Lines indicate a global fit to the apparent titration of all residues.

Table 1. Comparison of pK_a values determined by linkage analysis and NMR spectroscopy.

		linkage analysis		NMR
Variant	residue	pKa	+/-	pKa	+/-
Δ+PHS/L38E	Glu-38	7.0	0.3	7.0	0.1
Δ+PHS/L38E/E122Q	Glu-38	---	---	6.2	0.1
Δ+PHS/L38D	Asp-38	6.8	0.3	7.2	0.1
Δ+PHS/L38D/E122Q	Asp-38	6.9	0.3	6.6	0.1

^*pK_a could not be determined using linkage analysis because variant unfolds in an apparent 3-state manner

The agreement between the pK_a values measured from equilibrium thermodynamic data and from the global fit of titrations of backbone amide resonances suggests that the values obtained by NMR are accurate. However, the NMR experiment does not follow the amino acid of interest directly. Other groups could be responsible for the observed transition. To examine this possibility, the pK_a values of all residues that titrate between pH 4.6 and 8.5 in Δ+PHS nuclease were measured in the Δ+PHS/L38E variant. The pK_a values of His-8, His-121, and Asp-21 were found to be 6.5, 5.7, and 6.5, respectively in the Δ+PHS/L38E variant (Tables 2 and 3). This demonstrates that none of these groups are responsible for the apparent titration near pH 7 monitored with NMR spectroscopy. The ¹H chemical shifts of multiple groups appear to be reporting on the proton titration of Asp-38 or Glu-38.

Table 2. pK_a values and hill coefficients of His residues measured using NMR spectroscopy.

	His-8		His-121
variant	pKa*	n	pKa*	n
Δ+PHS	6.6	1.0	5.4	0.8
Δ+PHS/E122Q	6.5	1.0	5.3	0.9
Δ+PHS/L38K	6.5	1.0	5.6	1.0
Δ+PHS/L38E	6.5	1.0	5.7	0.9
Δ+PHS/L38E/E122Q	6.4	1.1	5.7	0.9
Δ+PHS/L38D	6.5	1.0	5.7	0.9
Δ+PHS/L38D/E122Q	6.5	1.1	5.8	0.9

^*uncertainty in pK_a value is +/- 0.1

Table 3. pK_a values of Asp and Glu residues measured using NMR spectroscopy.

	Δ+PHS ‡	Δ+PHS/L38E		Δ+PHS/L38K
position	pKa*	pKa*	ΔpKa†	pKa*	ΔpKa†
D19	2.2	< 2.5	---	< 2.5	---
D21	6.5	6.6	0.0	6.5	0.0
D40	3.9	3.9	0.0	3.8	-0.1
D77	< 2.5	< 2.5	---	---	---
D83	< 2.5	< 2.5	---	< 2.5	---
D95	2.2	2.3	0.1	2.1$	0.0
D143	3.8	3.8	0.0	4.0	0.2
D146	3.9	3.9	0.0	4.0	0.1
E10	2.8	3.1	0.2	3.1$	0.2
E43	4.3	4.4	0.0	4.5	0.2
E52	3.9	4.0	0.1	4.1	0.2
E57	3.5	3.6	0.1	3.6	0.1
E67	3.8	3.8	0.1	3.9	0.1
E73	3.3	3.4	0.1	3.3$	0.0
E75	3.3	3.3	0.1	---	---
E101	3.8	3.9	0.1	3.9	0.0
E122	3.9	3.8	-0.1	3.9	0.0
E129	3.8	3.9	0.1	3.8	0.0
E135	3.8	3.8	0.1	3.9	0.1
E142	4.5	4.5	0.0	4.6	0.1

^*uncertainty in pK_a value is +/- 0.1

^†uncertainty in ΔpK_a value is +/- 0.1

^‡C. Castañeda et al.⁴¹

^$Acid baseline fixed to Δ+PHS value

Spectroscopic probes of structural rearrangement

Structural reorganization associated with the substitution of Leu-38 with Asp or Glu, or with the ionization of Asp, Glu and Lys at position 38 was probed by CD, Trp fluorescence, and NMR spectroscopy. The intrinsic fluorescence of Trp-140, which caps the C-terminal end of helix 3, has been shown to be a robust reporter of the global integrity of SNase.⁴⁸ Trp fluorescence at neutral pH was insensitive to the substitution of Leu-38 with ionizable groups (data not shown). Circular dichroism spectra of all variants in the far-UV range were also indistinguishable from one another at pH 4, 7, and 10 (Figure 4), suggesting that none of the substitutions altered the secondary structure of the protein significantly, even when the internal Lys, Asp or Glu groups were charged. Similarly, 70 to 90% of the 131 peaks in the ¹⁵N-¹H HSQC spectrum of Δ+PHS nuclease at pH 4.5 were identifiable by visual inspection on the spectrum of each variant (Figure 5).⁴¹ Overall, the spectroscopic probes suggest that the substitutions did not affect the structure of the protein over a wide range of pH.

Figure 4.

Far-UV CD spectra of at pH 4 (A) and pH 7 (B). Series are Δ+PHS (▾), Δ+PHS/L38E ( ), Δ+PHS/L38E/D77N ( ), Δ+PHS/L38E/E122Q ( ), Δ+PHS/L38E/E122D ( ), Δ+PHS/L38E/R126Q ( ), Δ+PHS/L38D ( ), and Δ+PHS/L38D/E122Q ( ).

Figure 5.

HSQC spectra of the Δ+PHS (black), Δ+PHS/L38D (red), Δ+PHS/L38E (green), and Δ+PHS/L38K variants (blue) at pH 4.6-4.7. The four spectra are overall quite similar. Arrows highlight 4 of the 12 residues that are seen in every spectrum except that of Δ+PHS/L38K. Boxes identify residues used in global fits to extract pK_a values of groups at position 38.

Although the global structure of the proteins remained intact, the previous investigation of the Δ+PHS/L38K variant suggested that flexibility of the loop containing residues 113-119 allowed water to penetrate the protein to solvate the charged moiety of the side chain of Lys-38. To probe the conformation of this loop in these variants, an HNN experiment was used to assign all backbone ¹⁵N-¹H peaks in the HSQC spectra of the Δ+PHS/L38E and Δ+PHS/L38K variants at pH 4.6.⁴⁹ These spectra were compared to the spectra of Δ+PHS collected previously,⁴¹ at pH values between 2.8 and 9.0 in steps of ∼0.4 pH units.

At pH 4.5, 137 peaks were evident for Δ+PHS nuclease, 138 for the Δ+PHS/L38E variant, and only 125 for the Δ+PHS/L38K variant. In the spectrum of Δ+PHS nuclease, only three peaks in the region of interest entered intermediate exchange (i.e. millisecond timescale) with increasing pH: Tyr-113 disappeared above pH 5.7; Lys-116 and Gly-117 disappeared above pH 7.2. Larger changes were observed in the spectrum of the Δ+PHS/L38E variant. The peaks for seven residues (78, 80, 114, 117-120) all entered exchange above pH 6.3, concomitant with ionization of Glu-38. Without further information, it cannot be determined if this is due to structural relaxation or due to the change in the electrostatic environment of the groups owing to the ionization of Glu-38. Changes in proton chemical shift as large as 0.4 ppm were also observed in helix 3 (residues 123-130), around Tyr-91 (residues 88-92), and in the residues adjacent in sequence to position 38 (residues 34-39).

Unlike Glu-38 and Asp-38, Lys-38 was ionized over the entire pH-range under investigation. Twelve peaks are missing in this spectrum at all pH values studied. Of these, 9 peaks correspond to a contiguous stretch from Tyr-113 to Glu-122 (Figure 4). The remaining three missing peaks are Lys-38, Lys-78, and Gln-80. Changes in ¹⁵N-¹H chemical shifts should not be overinterpreted;⁵⁰ however, the absence of these peaks is consistent with increased exchange of the amide protons with solvent. This interpretation is also consistent with the previous investigation of the L38K variant and with the measured pK_a value of His-121 (see next section).³⁹ Overall, the changes in chemical shift are smaller in the Lys-38 variant than in the Glu-38 variant, having a maximum shift of 0.2 ppm. The largest changes are limited to the N-terminal end of helix 3.

Structural reorganization probed with His-121

The properties of His residues of SNase have been characterized extensively.^{30; 31} Changes in the microenvironments of His residues can be probed by measuring their pK_a values by 1D ¹H NMR, a method that has high accuracy and precision > 0.1 pH units. The Δ+PHS variant of nuclease only contains two of the four His residues normally present in wildtype SNase: His-8 and His-121. In Δ+PHS nuclease, His-8 and His-121 titrate with pK_a values of 6.6 and 5.4, respectively. The pK_a of His-8 was entirely insensitive to the presence and ionization of Asp, Glu, or Lys at position 38 (Table 2). This was expected because His-8 is 17 Å from position 38.

It has been shown previously that the pK_a of His-121 is depressed owing primarily to dehydration in a partially buried configuration.³¹ His-121 is 6 Å from the O_ε1 of Glu-38 and 9 Å from the N_ζ atom of Lys-38 (Figure 1B). In both the L38D and L38E variants, the pK_a value of His-121 was elevated from 5.4 to 5.7 (Table 2). This cannot be due to a Coulomb interaction because His-121 and the internal carboxylic groups do not ionize in the same range of pH. It has been observed previously that perturbations to the hydrogen bonding network centered around His-121 almost always cause the pK_a value of His-121 to increase relative to its pK_a in Δ+PHS.³¹ Thus, it is likely that Glu-38 and Asp-38 perturb the hydrogen-bond network, possibly allowing more water to reach His-121 than in Δ+PHS nuclease.

Lys-38 is fully charged in the range of pH where His-121 titrates; therefore, a Coulomb interaction between these residues is possible. An unfavorable Coulomb interaction would further depress the pK_a of His-121. This was not observed. Instead, as in the L38D and L38E variants, the pK_a of His-121 was elevated to 5.6 in the presence of Lys-38. This small change is consistent with slight structural relaxation induced by the substitution of Leu-38 with Lys, or by the ionization of Lys-38. This is fully consistent with the conclusion of our previous study of Lys-38 and with the evidence from NMR spectroscopy of slight structural reorganization in the L38K variant.³⁹

Coulomb interactions between surface and internal ionizable groups

Δ+PHS nuclease has 20 Asp and Glu residues. The pK_a values of these groups are known.⁴¹ With the exception of Asp-21, which has an elevated pK_a, all carboxylic groups titrate with depressed or normal pK_a values. The pK_a values of surface Asp and Glu residues in the Δ+PHS/L38E and Δ+PHS/L38K variants were measured with NMR spectroscopy (Table 3).

Substitution of Leu-38 with Glu did not alter the pK_a values of any of the Asp or Glu residues (Table 3). This does not imply that Glu-38 does not interact with these groups. Glu-38 titrates with a pK_a of 7.0, while most other acidic groups titrate with pK_a values near 4.0; therefore, Glu-38 is neutral during the titration of the other Asp and Glu residues and cannot affect their pK_a values by Coulomb interaction. These data also show that the L38E substitution has no detectable impact on the structure of the protein and thus on the electrostatic environments of the surface Asp and Glu residues. If Asp-21 had a significant Coulomb interaction with Glu-38, its pK_a of 6.5 would be affected. The lack of any detectable shift indicates that any Coulomb interaction between these two carboxylic groups is < 0.1 kcal/mol. Overall, the absence of any measurable impact of substitutions of Leu-38 on the pK_a values of surface residues corroborates the results from spectroscopic experiments showing that the structure of the variants are very similar if not identical to the structures of the background protein.

To probe interactions between Asp-38 and Glu-38 with neighboring ionizable residues directly, five double mutants were made: L38E/D77N, L38E/E122Q, L38E/E122D, L38E/R126Q, and L38D/E122Q. The pK_a value of the internal Glu-38 or Asp-38 was measured by global fit of the pH dependence of ¹H^N chemical shifts measured with NMR spectroscopy. The measured pK_a values are listed in Table 4. These pK_a values can be readily converted to apparent ΔG_ij values by multiplying the difference in pK_a between the background protein and the variants with neutral substitutions by RTln(10). To measure the interaction between Glu-38 and Asp-122, the ΔpK_a was measured using the L38E/E122Q variant as a background. The observed ΔG_ij values were strongly distance-dependent and ranged from 0 to 1.5 kcal/mol (Table 5).

Table 4. pK_a values of Asp-38 and Glu-38 measured using global fit to ¹H^N chemical shift data.

variant	residue	n	pKa*	ΔpKa
Δ+PHS/L38E	Glu-38	0.8	7.0	0.0
Δ+PHS/L38E/D77N	Glu-38	0.7	6.1	-0.9
Δ+PHS/L38E/E122Q	Glu-38	0.9	6.2	-0.8
Δ+PHS/L38E/E122D	Glu-38	0.8	7.4	0.4
Δ+PHS/L38E/R126Q	Glu-38	0.9	7.2	0.2
Δ+PHS/L38D	Asp-38	0.9	7.2	0.0
Δ+PHS/L38D/E122Q	Asp-38	1.0	6.6	-0.6

^*uncertainty in pK_a value is +/- 0.1

Table 5. Apparent Coulomb interactions between Glu-38 and other ionizable residues.

interaction	ΔGij * (kcal/mol)	rij (Å)	interaction	ΔGij* (kcal/mol)	rij (Å)
Glu-38/Asp-122‡	1.5	3.6†	---	---	--
Glu-38/Asp-77	1.2	4.0	---	---	---
Glu-38/Glu-122	1.1	5.1	Asp-38/Glu-122	0.8	5.3†
Glu-38/Arg-126	-0.3	6.6	---	---	---
Glu-38/Asp-21	0.0	13.1	---	---	---
Glu-38/His-8	0.0	15.2	Asp-38/His-8	0.0	18.9†

^*uncertainty is 0.1 kcal/mol

^†distance estimated using structural model

^‡ΔpK_a estimated using L38E/E122Q variant

Unlike Asp-38 or Glu-38, which are neutral at low pH, Lys-38 is ionized below pH 10. Any Coulomb interaction with Lys-38 should therefore alter the pK_a values of surface carboxylic groups. In particular, Lys-38 is involved in a 2.7 Å ion pair with Glu-122 in the crystal structure of the L38K variant. A conservative estimate of the possible pK_a shift can be made using Coulomb's law using the dielectric constant of pure water. This indicates that an interaction of 1.5 kcal/mol between Lys-38 and Glu-122 is possible. This would shift the pK_a of Glu-122 by 1.1 units. The measured pK_a values of the carboxylic groups in the Δ+PHS/L38K variant are shown in Table 3. Surprisingly, no shifts in pK_a value were observed. This experiment demonstrates conclusively that the ion pair between Lys-38 and Glu-122 observed in the crystal structure is not present in solution, and that Lys-38 does not interact with any carboxylic acid residues in the protein. This behavior is fully consistent with a structural rearrangement leading to hydration of the charged side chain of Lys-38.

Structure-based pK_a calculations

Reproducing the shifts in pK_a values in multiple sites in a protein is still a difficult challenge for structure-based electrostatics calculations. To test the ability of computational methods to calculate Coulomb interactions accurately, five computational methods were used to calculate the pK_a of Glu-38 in the L38E, L38E/D77N, L38E/E122Q, L38E/E122D, and L38E/R126Q variants. The pK_a values of Asp-38 (in the L38D variant), His-8, and His-121 were also calculated. Overall, the calculations attempted to reproduce 12 unique pK_a values: Glu-38 in the five variants listed above (Table 4), Asp-38 in the L38D variant (Table 4), and His-8 and His-121 in the L38E, L38E/E122Q, and L38D variants (Table 2). A variety of different computational methods were tested: PROPKA,⁵¹ the single site (S/FDPB) and full-site PARSE (F/FDPB) implementation of finite difference Poisson-Boltzmann electrostatics,^52-54 the pH-adapted conformer FDPB method (PAC),^{55; 56} and multi-conformer continuum electrostatics (MCCE).^{57; 58}

PROPKA uses a set of empirical rules to estimate pK_a values from structure. The pK_a of Glu-38 calculated with PROPKA was 5.3. The group was classified as buried and experienced an unfavorable dehydration energy (2.1 kcal/mol) offset by favorable polar interactions with the Tyr-91 side chain and His-121/Glu-122 backbone (-1.3 kcal/mol). No Coulomb interactions were calculated because PROPKA only treats Coulomb interactions between two residues if they are both classified as buried and are within 7 Å of one another. Only His-121 meets both criteria, but no interaction energy was calculated because the calculated pK_a of His-121 (5.1) is below that of Glu-38; the PROPKA algorithm excludes possible interaction because the groups are never charged simultaneously. The lack of explicit Coulomb interactions causes PROPKA to be entirely insensitive to substitution of residues at positions 77, 122, or 126 with uncharged analogs. The overall RMS error for the calculated versus experimental pK_a values was 1.0 (Table 6).

Table 6. Comparison of calculated and measured pK_a values.

		Experimental		FDPB
residue	variant	By NMR	PROPKA	S	F	PAC	MCCE
	L38E	7.0	5.3	7.0	7.1	4.9	8.5
	L38E/D77N	6.1	5.3	5.9	5.7	0.0	5.4
Glu-38	L38E/E122Q	6.2	5.3	5.6	6.0	2.7	7.2
	L38E/E122D	7.4	5.2	8.2	7.5	6.5	9.0
	L38E/R126Q	7.2	5.2	7.5	8.1	5.3	9.0
Asp-38	L38D	7.2	5.0	6.7	7.4	16.2	8.7
	L38E	6.5	6.5	6.8	6.8	6.5	6.7
His-8	L38E/E122Q	6.4	6.5	6.7	6.8	6.6	6.7
	L38D	6.5	6.5	6.7	6.8	6.6	6.7
	L38E	5.7	5.1	8.0	10.2	8.3	7.6
His-121	L38E/E122Q	5.7	5.1	8.0	10.0	8.1	7.7
	L38D	5.7	5.3	7.9	9.6	5.9	7.7
	RMS	---	1.0	1.2	2.1	3.4	1.4

The results of FDPB calculations were highly dependent on the choice of protein dielectric constant. Using S/FDPB, the calculated pK_a of Glu-38 ranged from 19.2 when ε_p = 4, to 5.4 when ε_p = 20. Likewise, the pK_a of Glu-38 calculated using F/FDPB ranges from 19.2 when ε_p = 4, to 5.2 when ε_p = 20. The pK_a of 7.0 of Glu-38 was reproduced by dielectric constants of 12 and 11 with the S/FDPB and F/FDPB methods, respectively. The dielectric constant that reproduced the experimental pK_a value (ε_app) was used for all further calculations. In both types of calculations, the pK_a at ε_app was governed by unfavorable dehydration energy (4 kcal/mol), net favorable polar interactions (-3 kcal/mol), and net unfavorable Coulomb interactions (2.5 kcal/mol). The overall RMS for all 13 measured pK_a values was 1.2 and 2.1 for S/FDPB and F/FDPB respectively (Table 6). The relatively large RMS value is due to a large error in the calculated pK_a value of His-121 (8.0 and 10.2). If this is excluded from the RMS calculation, both methods have RMS errors of 0.4. S/FDPB overestimated the apparent Coulomb interactions by 75 % and F/FDPB by 25 %.

The PAC method generates ensembles of possible side chain positions at pH extremes, calculates electrostatic potentials of each configuration using FDPB, then Boltzmann weights these ensembles as a function of pH. To test the best-case scenario for PAC calculations, a variety of parameter combinations were used to maximize agreement with experiment. For PAC, it was found that minimizing side chain positions was the most important user-adjustable parameter, lowering the RMS error by 20 %. Even so, the calculations performed poorly. The calculated pK_a of Glu-38 was 4.9 and the overall RMS error the pK_a values was 3.4 pH units (Table 6). On average, the apparent coulomb interactions calculated using this method were 3.7 times higher than experiment.

MCCE couples conventional FDPB calculations to side chain rotamer sampling with a Monte Carlo method. MCCE does not allow for arbitrary adjustment of the protein dielectric constant without extensive reparameterization; therefore, these calculations used the default ε_p values of 4 or 8. The calculations with ε_p = 4 failed to converge, giving pK_a values for Glu-38 that were > 14. Only calculations using ε_p = 8 are reported here. The calculated pK_a value of Glu-38 was 8.5 and the overall RMS error was 1.4 pH units (Table 6). The apparent Coulomb interactions reported by MCCE were, on average, twice as strong as observed experimentally. A correlation plot of the experimental and calculated apparent Coulomb interactions for each calculation type is shown in Supplementary Figure 1.

Discussion

Difference in ionization properties of Lys, Glu, and Asp at position 38

Previous experimental and computational studies with the L38K variant suggested that the pK_a of Lys-38 was governed by structural reorganization and water penetration, with little or no contribution from the buried ion pair between Lys-38 and Glu-122 observed in the crystal structure.³⁹ We have now confirmed this with NMR spectroscopy. The HSQC spectrum of this variant shows no peaks for residues 113-123 at any pH value studied. Although chemical shifts are sensitive to factors other than structural relaxation, the data are consistent with reorganization of this segment of the protein and exchange with solvent. The shift in the pK_a of His-121 from a depressed value towards a more normal value is opposite from the shift expected from Coulomb interactions between His-121 and Lys-38. However, it is fully consistent with enhanced hydration of His-121 owing to structural relaxation in this region of the protein. Finally, Lys-38 does not shift the pK_a value of any acidic residue in the protein. This is particularly striking for Glu-122, which is part of a 2.7 Å ion pair in the crystal structure of the L38K variant, but apparently insensitive to the charge in Lys-38. This can only be explained if Lys-38 is fully hydrated when charged, and if the structure of the L38K variant in solution is different than the crystal structure.

In contrast to the behavior of Lys-38, the pK_a values of Glu-38 and Asp-38 are higher than normal pK_a values by 2.6 and 3.2 pH units, respectively. These two carboxylic groups experience substantial Coulomb interactions with multiple neighboring groups. There are several possible explanations for this difference in behavior between Lys-38 and the carboxylic groups at this same position. The first is proximity to solvent. Although Lys-38 and Glu-38 are internal in the crystal structures of the PHS/L38K and PHS/L38E variants, the side chains point towards the protein-water interface and they are much closer to bulk water than the internal ionizable groups of acidic and basic residues at internal positions 66 and 92 studied previously.^{15-17; 21-23; 59} The shorter side chain of Glu-38 is further from bulk water than the longer Lys-38 side chain, and therefore will experience the reaction field of bulk water more weakly. This would contribute to the shift in its pK_a. Consistent with expectation, the shifts in pK_a correlate inversely with side chain length. We also suspect that the short length of the side chain of Asp-38 is responsible for its not being fully dehydrated. A model of the L38D variant made from the crystal structure of the PHS/L38E variant (see Methods) showed that Asp-38 forms a single hydrogen bond with its own backbone, but is too short to reach any other hydrogen bonding partners without substantial structural rearrangement. Water molecules have been found in association with the carboxylic oxygen atoms of internal Asp and Glu in all previous structures of SNase variants with internal Glu or Asp.^{16; 42} To examine the possibility that internal water molecules can satisfy the hydrogen bonding potential of the side chain of Asp-38, the DOWSER algorithm⁶⁰, which places internal water molecules in polar cavities in crystal structures, was applied to the model of the L38D variant. DOWSER identified two potential water molecules in the region between Asp-38 and the 113-119 loop, which make hydrogen bonds with Asp-38 thereby linking this residue to the backbones of Asn-118 and Glu-122.

The difference in the pK_a of Lys-38 versus Glu-38 or Asp-38 is also consistent with the ability of carboxylic residues to form better hydrogen bonds than internal Lys residues. In the crystal structure, the amino group of Lys-38 makes an ion pair with the carboxylic side chain of Glu-122, and one bond to the backbone of Ala-112. In contrast, Glu-38 forms hydrogen bonds to the side chain of Tyr-91 and to the backbone amides of Thr-120 and His-121. If the predictions with the DOWSER method are correct, Asp-38 forms a hydrogen bond to its own backbone and two hydrogen bonds to internal water molecules. The polar network that Asp-38 and Glu-38 establish might rigidify this part of the protein and thereby prevent these groups from being solvated through water penetration. This would contribute to the shift in pK_a values in the direction that was observed.

The differences in the properties of Lys-38 and Glu-38 or Asp-38 probably also reflect differences in the pH dependence of stability of the different variant proteins. The stability of the L38K variant near pH 10, where Lys normally titrates, is lower than the stability of the L38D and L38E variants near pH 4, where Asp and Glu normally titrate. At pH 10, the local stability of the loop comprised of residues 113-119 does not appear to be sufficiently high to maintain Lys-38 buried in the neutral state. Instead, the loop appears to adopt alternative, more open conformations that allow Lys-38 to be charged, hydrated, and probably only partially buried. We showed previously that a subtle structural reorganization is sufficient to promote water penetration and solvation of the charged side chain of Lys-38.³⁹ Near pH 4, the local stability of the 113-119 loop is sufficiently high to maintain Glu-38 and Asp-38 buried in the neutral state; therefore, the pK_a values of these internal groups are shifted upwards. Because the stability of the L38E and L38D variants decrease with increasing pH, this loop might shift towards a more open, alternative conformation when Asp-38 and Glu-38 titrate near pH 7. This is analogous to what was observed in the L38K variant. Consistent with this hypothesis, several resonances in the HSQC spectra of the L38E variant corresponding to this loop disappear when Glu-38 ionizes. The disappearance of these peaks is somewhat difficult to interpret; it could reflect structural reorganization but it could also be due by the electrostatic effects of the charge of Glu-38 on the amides.⁵⁰

Coulomb interactions between internal and surface ionizable residues

How do the observed Coulomb interaction energies between internal and surface groups compare to previously measured interactions between surface ionizable moieties? The distance-dependence of ΔG_ij between internal and surface charges is plotted in Figure 6 and compared to previously measured long-range interactions between surface charges.^{30; 31} A single ΔG_ij vs r_ij plot is sufficient to describe all data. This indicates that the interactions between internal and surface groups are as weak as those between surface groups, at least for these internal groups, which are close to the protein-water interface. The effective dielectric constant obtained by fitting Coulomb's law with a Debye-Hückel parameter to the ΔG_ij obtained between Glu-38 and nearby surface residues is 35 +/- 3, which is almost as high as the value of 46 +/- 8 obtained by including ΔG_ij from interactions between surface groups, measured previously.^{30; 31} The high effective dielectric constant is fully consistent with the high local polarity in the microenvironment of Glu-38 and subtle structural reorganization coupled to its ionization.⁶¹

Figure 6.

Distance dependence of apparent Coulomb interactions for the current study (•), and for data published previously by Lee et al. ( ),³⁰ and Baran et al. ( ).³¹ The dashed line is a fit to the data from the current study only, the solid line a fit to all available data.

Another interesting feature of the measured Coulomb interactions is the asymmetry of the Glu-38/Asp-122 and Asp-38/Glu-122 interactions. The apparent Coulomb interaction of Glu-38/Asp-122 is 1.5 kcal/mol, whereas the apparent Coulomb interaction of Asp-38/Glu-122 is 0.8 kcal/mol. This is surprising: how could simply reversing the placement of the charges significantly alter the observed energy of interaction? Simple structural models of the L38D and L38E/E122D variants suggest an explanation. In these models, the nearest side chain oxygen atoms of the Glu-38/Asp-122 pair were 5.3 Å apart and the Asp-38/Glu-122 atoms were 3.6 Å apart. Due to their burial, neither Asp-38 nor Glu-38 can change position without a steric clash. Asp-122 and Glu-122, however, are on the surface and therefore not tethered in a fix position. Glu, with its extra methylene carbon, is simply longer and better able to respond to the repulsive interaction with Asp-38.

Implications for structure-based pK_a calculations

The pK_a values of internal ionizable residues are especially useful to test models for structure-based calculation of electrostatic energies and pK_a values.^{61; 62} In general, it is not sufficient to test the validity of these methods simply by reproducing pK_a values, especially when the methods include ad hoc modifications to improve their performance.^{52; 53; 63-65} In these cases the experimental pK_a values can be reproduced accurately but for entirely incorrect physical reasons.⁶² The utility of the data in this study is that it contributes additional constraints, such as the magnitude of pair-wise Coulomb interactions, that must be satisfied to ensure that pK_a values are reproduced for the right physical reasons. To reproduce these data, a computational method must correctly capture both the self-energy and Coulomb interaction energies experienced by ionizable residues at a single position, and it must also reproduce the differences in the ionization properties of Lys-38 and Asp-38 or Glu-38.

Our experimental data expose several weaknesses in current computational methods. For example, the attempts to reproduce the experimental pK_a values with PROPKA revealed the importance of medium-range electrostatics interactions, which can be an important determinant of pK_a values. These interactions are neglected in PROPKA, thus it fails to reproduce the experimental pK_a values (Table 6). Inclusion of this term would likely bring the calculated pK_a from 5.3 closer to the experimentally measured value. On the other hand, all of the continuum calculations overestimated the strength of Coulomb interactions, especially the PAC and MCCE methods. This reflects the fact that because the PAC and MCCE methods attempt to model side chain relaxation explicitly, lower protein dielectric constants are used. The failure of these methods demonstrates that these models either neglect important energetic terms in their Boltzmann-weighted ensembles, or that relaxation involving the protein backbone occurs on the timescale of the experiments, which is not treated by PAC or MCCE.

Conventional FDPB calculations did better than the PAC and MCCE methods, however, it is important to emphasize that in these calculations the protein dielectric constant was used as a tunable variable to maximize agreement between calculated and measured pK_a values. FDPB calculations with a static structure cannot predict pK_a values of internal ionizable groups. Both the S/FDPB and F/FDPB calculations slightly overestimated the strength of unfavorable Coulomb interactions. As the calculation reproduced the pK_a of Glu-38, the strength of the favorable polar interactions is either overestimated, the energetic penalty of dehydration is underestimated, or both. If slight relaxation of the structure occurs in solution, the strength of Coulomb interactions, polar interactions, and the energetic penalty of dehydration would all decrease. Such a process would explain the discrepancy between the experimental and computational results.

Our study demonstrates that very different factors can govern the pK_a values of different types of ionizable residues buried at the same location in a protein. Water penetration and structural reorganization are the primary determinants of the pK_a of Lys-38. In contrast, favorable polar interactions and unfavorable dehydration and Coulomb interactions determine the pK_a values of Asp-38 and Glu-38. Some amount of structural reorganization also influences the pK_a values of Asp-38 and Glu-38, as demonstrated by the fact that relatively high protein dielectric constants were needed to reproduce their pK_a values in structure-based calculations with static structures.

If structure-based electrostatics calculations are going to be useful predictors of structure/function relationships, they must be able to reproduce the pK_a values of internal ionizable groups. To this end, they must be able of identify a priori the cases in which the ionization of an internal group triggers structural reorganization. They must also be able to distinguish these cases from cases in which the ionization of an internal group does not promote structural reorganization. Because the probability that a protein undergoes local structural reorganization coupled to the ionization of an internal group is governed by the free energies of the folded protein and of the low lying excited states, improved methods for structure-based pK_a calculations will require methodology for improved conformational sampling and improved ability to estimate the stability of proteins.

Materials and Methods

Staphylococcal nuclease

Two hyperstable variants of SNase, known as PHS and Δ+PHS, were used for crystallography and equilibrium thermodynamic experiments, respectively. Both variants contain three substitutions: P117G, H124L and S128A. Δ+PHS nuclease has two additional mutations (G50F and V51N) and a short deletion (residues 44-49). All genes were engineered into the pET24A+ vector (Novagen, Madison, WI). Substitutions were introduced using QuikChange™ site-directed mutagenesis (Stratagene, La Jolla, CA). All proteins were expressed in BL21(DE3) cells from Stratagene, and purified using the protocol of Shortle and Meeker.⁶⁶ Uniformly ¹³C/¹⁵N labeled protein was made by growing E. coli in minimal media with ¹⁵NH₄Cl (1 g/L) and ¹³C₆-D-glucose (4 g/L) (Isotec, Inc.) Purity was determined to be greater than 98% by SDS-PAGE. Protein concentration was determined by absorbance at 280 nm using an extinction coefficient of 0.93 mL mg^-1 cm^-1.⁶⁷

Crystallography

PHS/L38E was crystallized in hanging drops using the vapor diffusion method. Three drops containing 4 μL: 4 μL mixture of protein and reservoir solution were suspended over 1 mL of reservoir solution and equilibrated at 4 °C. The protein solution consisted of 16.5 mg/mL protein, 2 milliequivalents of thymidine-3′,5′-diphosphate (THP), and 3 milliequivalents of CaCl₂. THP was synthesized in our laboratory, as described previously.⁶⁸ Reservoir solution consisted of 35 % (v/v) 2-methyl-2,4-pentanediol and 25 mM potassium phosphate at pH 7.0. Diffraction data were collected from a single crystal suspended in mother liquor in a cryoloop flash frozen in liquid nitrogen. Data were collected at 100 K using a Kappa ApexII diffractometer outfitted with a sealed copper tube, multilayer optics, and a CCD detector (Bruker/AXS, Madison, WI). Reflections were indexed and integrated using ApexII software, and merged using XPREP. Structure determination and refinement were performed using the ccp4i interface to the CCP4 program suite.^{69; 70} Molecular replacement was done with PHASER,⁷¹ using the PHS nuclease structure (PDB ID: 1EY8) as a search model.⁴⁰ Solvent atoms were removed, Leu-38 was truncated to Ala, and all B-factors were set to 20 Ų prior to molecular replacement. Iterative model building and refinement were performed using COOT and refmac5.^{72; 73} The sidechain for Glu-38 and the THP ligand were visible after the first round of refinement. Waters, phosphates, and ions were added in later rounds of refinement.

pK_a values measured using pH-dependence of stability

GdnHCl denaturation experiments were done using an Aviv Automatic Titrating Flourimeter 105 (Lakewood, NJ) using approach-to-equilibrium, as described previously.^{20; 22; 39; 74} pK_a values were extracted by the single-site linkage relationship to the pH-dependence of ΔΔG°_H2O, as described previously.^{15; 23; 39}

NMR data collection

For NMR titration experiments, a 1.4 mL sample of 1 mM protein was prepared in a solution of 100 mM KCl, 0.5 mM NaN₃ and 10% D₂O (v/v) by buffer exchange using Amicon Ultra-4 centrifugal filter units with a 10,000 MWCO (Millipore Corp., Billerica, MA). This sample was split into two equal fractions, one for titration with acid and one for base. Titrations were performed as described elsewhere.⁴¹

NMR experiments were performed on a Bruker Avance II-600 equipped with a cryoprobe. All experiments were conducted at a calibrated temperature of 298 K. NMR spectra were referenced against the position of the HDO peak, which was determined relative to the DSS peak at 0.00 ppm.^{41; 75} All spectra were processed using NMRPipe⁷⁶ and peak assignments were completed using Sparky.⁷⁷

Backbone assignments (¹⁵N, ¹H^N) for the Δ+PHS/L38E and Δ+PHS/L38K variants were collected using the 3D HNN experiment.⁴⁹ The ¹⁵N carrier frequency was set to 117.0 ppm; a total of 42 complex points were collected in each of the t₁ and t₂ dimensions. Spectral widths of 10000 Hz, 1825 Hz and 1825 Hz were used for the ¹H, ¹⁵N, and ¹⁵N dimensions, respectively.

Assignments for the C_γ and C_δ resonances of Asp and Glu residues in the Δ+PHS/L38E and Δ+PHS/L38K variants were determined primarily by transferring assignments from the CBCGCO spectra of Δ+PHS⁴¹ at pH 4.7 to the spectra of the variants. The side chain aliphatic carbon (C_α, C_β, C_γ) assignments of Glu-38 were collected using the C-C TOCSY (CO)NH experiment.⁷⁸ Through modification of the ¹³C-detect HBHGCBCGCO pulse sequence to include a TOCSY element, the Glu-38 C_δ resonance was determined by correlation with its side chain aliphatic carbon assignments (Castañeda and Majumdar, unpublished).

The pK_a values of Asp and Glu residues were determined by monitoring the pH dependence of their C_γ and C_δ resonances, respectively, using a ¹³C-detect version of the 2D HBHGCBCGCO experiment (Castañeda and Majumdar, unpublished). The spectra were collected in 0.4 pH increments between pH 2 and pH 9. For indirect pK_a determination of carboxylic residues using ¹H^N chemical shifts, a series of ¹⁵N-¹H HSQC spectra were collected in a similar manner. These experiments were performed and processed as previously described.⁴¹

For pK_a determination of histidine residues in the Δ+PHS variants, samples were prepared in the buffer described above except in 100% D₂O. Samples were heated to 330 K for 15 minutes prior to experiments to promote exchange of amide protons. The H_ε1 resonance of the imidazole ring was monitored as a function of pH using 1D ¹H NMR experiments as described previously.²³ The His H_ε1 resonances were unambiguously assigned by comparison to the same resonances in Δ+PHS.

pKa values from pH-dependent chemical shifts

pK_a values of Asp, Glu and His residues were extracted using the modified Hill equation, as described previously.⁴¹ Fitting was done using the nls function within the R 2.7.1 Statistical environment.⁷⁹

A global fit to the pH-dependence of the ¹H chemical shift of six amides (Thr-33, Phe-34, Arg-35, Glu-75, Gly-88, and Leu-89) was used to determine the pK_a values of Glu-38 and Asp-38. Three criteria were used to select these resonances: 1) An apparent titration > 0.15 ppm in the variant spectra that was not evident in the Δ+PHS spectra; 2) The resonance could be unambiguously assigned in all variants and did not exhibit exchange over the course of the titration (Figures 2 and 4); 3) The residues were within close spatial proximity of position 38. The fit minimized the root mean squared (RMS) deviation between the calculated and measured chemical shifts:

$$\mathit{\text{RMS}}=\sum _{i=1}^N\sqrt{\sum _{pH}{\left({\delta }_{i,\mathit{\text{calc}}}\left(pH\right)-{\delta }_{i,\mathit{\text{meas}}}\left(pH\right)\right)}^2}$$ 1.

where N is the number of resonances included in the fit and δ_i,meas is the experimentally measured chemical shift. δ_i,calc is described by the modified Hill equation:

$${\delta }_{i,\mathit{\text{calc}}}=\frac{{\delta }_{i,\mathit{\text{acidic}}}+{\delta }_{i,\mathit{\text{basic}}}\cdot {10}^{n\left(pH-p{K}_a\right)}}{1+{10}^{n\left(pH-p{K}_a\right)}}$$ 2.

where δ_i,acidic and δ_i,basic are the acidic and basic baselines of the apparent titration. A single pK_a and Hill coefficient (n) were used for all resonances included in the fit. Minimization was done using the optim function inside the R 2.7.1 statistical environment.⁷⁹ To ensure that the pK_a and Hill coefficient did not depend on the resonances chosen, bootstrap sampling of resonances included in the fit was performed. The list of resonances was also expanded to include resonances that only met two of the three inclusion criteria. In all cases, the bootstrap error was less than 0.05 pH units for the pK_a value. The errors reported in the paper reflect the maximum possible systematic error in pH measurement rather than the fit error.

Structural models

Models of the PHS/L38D, PHS/L38E/D77N, PHS/L38E/E122Q, PHS/L38E/E122D, and PHS/L38E/R126Q variants were made using the structure of the PHS/L38E variant as a template. The positions of the atoms of the side chain that was mutated in silico were minimized using the CHARMM22 forcefield, as described previously.²³ All other atoms were held fixed.

Structure-based electrostatics calculations

PROPKA calculations were performed using PDB2PQR 1.1.2, downloaded from http://pdb2pqr.sourceforge.net/.^{51; 80} Conventional FDPB calculations were done using the University of Houston Brownian Dynamics (UHBD) package, version 5.1.⁸¹ FDPB calculations were done using two parameter sets: single-site (S/FDPB)^{52; 53} and full-site (F/FDPB) using the PARSE parameter set.⁵⁴ All calculations were performed at 100 mM ionic strength as described previously.²⁴ Multi-Conformer Continuum Electrostatics (MCCE) calculations were done using MCCE 2.2, downloaded from http://wwwsci.ccny.cuny.edu/∼mcce/.^{52 ; 58} The parameters distributed with the program were used for the ε_p = 4 and ε_p = 8 calculations. Calculations were done at 100 mM ionic strength. PAC calculations were done using the Karlsberg+ web interface (http://agknapp.chemie.fu-berlin.de/karlsberg/).^{55; 56} Side chain positions were minimized and salt bridges were randomized at high and low pH. Results were found to be insensitive to choice of solvation model. Values reported herein were calculated using the vacuum solvation model.