De novo determination of near-surface electrostatic potentials by NMR.

Electrostatic potentials computed from three-dimensional structures of biomolecules by solving the Poisson-Boltzmann equation are widely used in molecular biophysics, structural biology, and medicinal chemistry. Despite the approximate nature of the Poisson-Boltzmann theory, validation of the computed electrostatic potentials around biological macromolecules is rare and methodologically limited. Here, we present a unique and powerful NMR method that allows for straightforward and extensive comparison with electrostatic models for biomolecules and their complexes. This method utilizes paramagnetic relaxation enhancement arising from analogous cationic and anionic cosolutes whose spatial distributions around biological macromolecules reflect electrostatic potentials. We demonstrate that this NMR method enables de novo determination of near-surface electrostatic potentials for individual protein residues without using any structural information. We applied the method to ubiquitin and the Antp homeodomain-DNA complex. The experimental data agreed well with predictions from the Poisson-Boltzmann theory. Thus, our experimental results clearly support the validity of the theory for these systems. However, our experimental study also illuminates certain weaknesses of the Poisson-Boltzmann theory. For example, we found that the theory predicts stronger dependence of near-surface electrostatic potentials on ionic strength than observed in the experiments. Our data also suggest that conformational flexibility or structural uncertainties may cause large errors in theoretical predictions of electrostatic potentials, particularly for highly charged systems. This NMR-based method permits extensive assessment of near-surface electrostatic potentials for various regions around biological macromolecules and thereby may facilitate improvement of the computational approaches for electrostatic potentials.

Due to the fundamental importance of electrostatic interactions in chemistry and biology, electrostatic potentials are invaluable information for the understanding of molecular recognition, enzymatic catalysis, and other functions of proteins and nucleic acids (1–4). Quantification of electrostatics is also important for successful protein engineering (5) and structure-based drug design (6). Computational approaches based on the Poisson–Boltzmann theory are commonly used to calculate electrostatic potentials from three-dimensional (3D) molecular structures (1, 7). Owing to available software such as Adaptive Poisson-Boltzmann Solver (APBS) (8, 9) and DelPhi (10, 11), computation of the electrostatic potentials around biomolecules has gained widespread popularity in the fields of molecular biophysics, structural biology, and medicinal chemistry.

However, the computed electrostatic potentials may not necessarily be accurate even if the 3D structures are precisely and accurately determined. Importantly, the Poisson–Boltzmann theory is approximate with known limitations. The electrostatic models based on this theory are valid under assumptions, which simplify the calculations (12). The lack of consideration of correlations between ions can diminish accuracy in calculations of electrostatic potentials for systems at high ionic strength (13). Due to the assumption of a dielectric continuum, the electrostatic potentials predicted with the Poisson–Boltzmann theory may be inaccurate for zones near the first hydration layer. Electrostatic potentials predicted for regions near highly charged molecular surfaces may also be inaccurate due to the assumption of linear dielectric response. Nonetheless, the Poisson–Boltzmann theory can accurately predict electrostatic interactions at longer range (7). The extent of validity for such electrostatic potentials near molecular surfaces remains to be addressed more rigorously through experiments.

Despite the need, experimental validation of computed electrostatic potentials is rather rare and methodologically limited for biological macromolecules. The validity of electrostatic models has been examined using pK data on titratable side-chain moieties (14–16), redox potentials of redox-active groups (17, 18), and electron–electron double resonance (19). Among them, pK data have been most commonly used for the validation, but even fundamentally incorrect electrostatic models can reproduce pK data (20). Electrostatic fields can be experimentally determined by vibrational spectroscopy, for example, for nitrile groups that are conjugated to cysteine thiol moieties of proteins (21, 22). However, the approaches utilizing vibrational spectroscopy or electron–electron double resonance provide only limited information about the extrinsically introduced probes, which may perturb native systems.

In this paper, we present a unique and powerful method for de novo determination of near-surface electrostatic potentials for many protein residues, regardless of their side-chain types, and without using any chemical modifications. In this method, data of NMR paramagnetic relaxation enhancement (PRE) arising from analogous charged paramagnetic cosolutes are analyzed for 1H nuclear magnetizations of proteins (Fig. 1). The PRE data reflect the electrostatic biases in spatial distributions of charged paramagnetic cosolutes and permit the determination of near-surface electrostatic potentials around proteins without using any structural information. The de novo determination of near-surface electrostatic potentials can greatly facilitate the examination of theoretical models for electrostatics of biological macromolecules.

Fig. 1.

NMR PRE arising from cationic amino-methyl-PROXYL or anionic carboxy-PROXYL reflects their spatial distribution bias due to near-surface electrostatic potentials around a biological macromolecule.

Results

De Novo Determination of Near-Surface Electrostatic Potentials.

To determine the near-surface electrostatic potentials, we used two derivatives of 2,2,5,5-tetramethylpyrrolidine-N-oxyl nitroxide (PROXYL): namely, amino-methyl-PROXYL and carboxy-PROXYL (Fig. 1). These paramagnetic compounds are analogous but differ in charge at neutral pH: amino-methyl-PROXYL is cationic (+1e) whereas carboxy-PROXYL is anionic (−1e). For PRE arising from paramagnetic cosolute molecules, the PRE rate for a transverse 1H magnetization (Γ2) is given as follows (23):where c is the concentration of the paramagnetic cosolute; τ is the correlation time for the dipole–dipole interaction between the macromolecular 1H nucleus and the unpaired electron of the cosolute; r is the distance between the unpaired electron and the 1H nucleus; U() is the potential of mean force for the cosolute as a function of the position vector for the unpaired electron with respect to the 1H nucleus; k is the Boltzmann constant; T is temperature; and ξ is a parameter reflecting other spin properties such as the gyromagnetic ratios. More details about Eq. and the relevant parameters are provided in . The integral in Eq. is conveniently discretized on a 3D lattice grid surrounding the macromolecule:where i is the index of a grid point, and v is the volume of the voxel each grid point represents. Eq. is similar to the empirical equation used for the so-called Otting–LeMaster approach (24–26) but differs in that each r−6 term involves a Boltzmann factor representing the bias in the spatial distribution of the cosolute molecule due to the effective potential U. When comparing PREs arising from the cationic and anionic PROXYL derivatives at the same concentration, the ratio of Γ2 rates will be as follows:where the annotations “+” and “−” are for the cationic and anionic PROXYL derivatives, respectively. The cancellation of τ in Eq. requires identical diffusional properties for the cationic and anionic PROXYL derivatives, which we confirmed through NMR-based diffusion measurements (). Due to the strong distance dependence imposed by r−6, each Γ2 rate is dominated by terms for a near-surface zone close to the observed 1H nucleus. We referred to it as the effective near-surface (ENS) zone. As shown in Fig. 2, the ENS zone for each observed protein 1H nucleus is relatively narrow in the total space surrounding a macromolecule.

Fig. 2.

Electrostatic potentials and ENS zones for PROXYL-induced PRE for the T22 and L71 1HN nuclei of ubiquitin (PDB: 1UBQ). (A) The ENS zones for the 1HN nuclei of T22 and L71. (Left) Two-dimensional (2D) plane containing the HN atoms of T22 and L71 (shown in dark green and dark orange, respectively). Possible positions of the paramagnetic center of PROXYL derivatives around a ubiquitin molecule are shown in lattice representations (with a spacing of 0.5 Å). The grid points within a zone making 68% contribution to , where r is the distance between the 1H nucleus and a grid point i in the exterior space, are colored in green (T22) or orange (L71). These zones are also shown in the 3D structure. (B) Exterior electrostatic potentials at the grid points on the 2D plane shown in A. Electrostatic potentials at 30 mM ionic strength were computed using the APBS software. The location of the 2D plane is also indicated by a smaller rectangle in a 3D surface electrostatic potential map. The average values of electrostatic potentials at grid points within the ENS zones () were −21.2 mV for T22 1HN and 14.9 mV for L71 1HN. The corresponding potentials calculated using all grid points (both inside and outside the ENS zones) along with Eqs. and were −20.1 mV for T22 1HN and 13.5 mV for L71 1HN. The threshold (68%) for the ENS zones was chosen as a value at which becomes close to for the majority of 1HN nuclei (see Results). Although the physical meaning of is clearer than that of , we use potentials to compare with experimental data because does not depend on any arbitrary threshold.

Here, for the ENS zone of each observed 1H atom, we define a potential as follows:in which e is the elementary charge. The physical meaning of the ϕ potential becomes apparent by considering a hypothetical case in which the potentials U and U are uniform within the ENS zone. Under this condition, Eq. is reduced to the following:where ΔU = U – U. Tetramethyl nitroxide compounds such as PROXYL or TEMPOL are known to preferentially interact with hydrophobic patches on protein surface (23, 26, 27). Because the hydrophobic moiety of PROXYL is identical for the two PROXYL derivatives, it is likely that the hydrophobic contribution in each U term is approximately cancelled in ΔU. Considering that the major difference between the cationic and anionic PROXYL derivatives is the charge (+1e versus −1e), it is reasonable to assume that the difference ΔU is dominated by the electrostatic interaction and given by the following:where ϕ is the electrostatic potential within the ENS zone. From Eqs. –, it is obvious that ϕ = ϕ in this hypothetical case with a uniform electrostatic potential within the ENS zone. In general cases with nonuniform electrostatic potentials, ϕ represents an effective electrostatic potential for the ENS zone. As demonstrated in the next subsection, ϕ is close to the average of the electrostatic potential within the ENS zone. Importantly, Eq. enables determination of the ENS electrostatic potentials ϕ from Γ2,+ and Γ2, data without using any structural information. Experimental ϕ data allow us to examine theoretical electrostatic models through direct comparison with theoretical predictions using Eqs. and .

Comparison with the Poisson–Boltzmann Equation-Based Electrostatic Potentials.

We applied this method to ubiquitin, a 76-residue protein containing 11 positively charged and 11 negatively charged side chains. We measured PRE Γ2,+ and Γ2, rates at 25 °C for backbone 1HN nuclei of 15N ubiquitin in solutions containing 10 mM amino-methyl- or carboxy-PROXYL at pH 7.5 and the ionic strength of 30 mM (Fig. 3 and ). We identified 63 1HN nuclei, which exhibited statistically significant PRE rates and Γ2,+/Γ2, ratios, using the criteria described in the section. For each of these 1HN nuclei, we determined the ENS electrostatic potential ϕ using Eq. and the measured Γ2,+ and Γ2, rates. Fig. 3 shows the ENS electrostatic potentials ϕ (red) determined from Γ2,+/Γ2, data for individual residues of ubiquitin at the ionic strength of 30 mM. The range of the ϕ potentials determined for ubiquitin was from −21 to +26 mV.

Fig. 3.

ENS electrostatic potentials ϕ determined from NMR PRE data for backbone HN atoms of ubiquitin at pH 7.5 and the ionic strength of 30 mM. (A) 1H PRE rates Γ2 measured with the cationic and anionic PROXYL derivatives (10 mM). (B) Experimental ENS electrostatic potentials ϕ (red) determined from the PRE data for individual residues. Theoretical ENS electrostatic potentials (blue) and (green) predicted from the 1.8-Å resolution crystal structure (PDB: 1UBQ) are also plotted. (C) Correlation plots for the experimental ϕ data and the theoretical data. Data points for the secondary structure regions are shown in black and those for loop regions are shown in gray. The corresponding data for ubiquitin at the ionic strength of 130 mM are shown in .

Using the APBS software and a 1.8-Å resolution crystal structure of ubiquitin (Protein Data Bank [PDB]: 1UBQ), we also performed the Poisson–Boltzmann theory-based calculations of electrostatic potentials. From the electrostatic potentials ϕ computed at all grid points in the exterior space, we predicted the ϕ potential () for each 1HN nucleus using Eqs. and along with U = eϕ and U = −eϕ. The accessibility of each voxel was assessed using van der Waals radii of macromolecular atoms and an empirically determined radius of 3.5 Å for the paramagnetic center of the PROXYL derivatives (). We also calculated the average () of the electrostatic potentials at grid points only within a proximal zone making a 68% contribution to for each 1HN nucleus. At this threshold (68%), the average potentials within these zones were in excellent agreement with the potentials calculated for all grid points (Fig. 3). The experimental ϕ data agreed remarkably well with the theoretical data, particularly for secondary structure regions. A correlation plot comparing the experimental ϕ and theoretical potentials is shown in Fig. 3. The rmsd between them was 5.4 mV, and the correlation coefficient was 0.94 for secondary structure regions of ubiquitin at the ionic strength of 30 mM. Experimental ϕ data were also consistent with , supporting the notion that the ϕ potential represents an effective electrostatic potential in the ENS zone. These results clearly indicate that the Poisson–Boltzmann theory predicts the near-surface electrostatic potentials for this system well and that our current experimental approach allows for straightforward comparison with electrostatic models.

Impact of Conformational Variations.

The majority of large differences between the experimental and theoretical ϕ data for ubiquitin were found in loop regions. To examine the extent to which conformational variations affect the prediction of near-surface potentials, we conducted the same analysis of the ensemble of 10 NMR structures determined by Bax and coworkers (PDB: 1D3Z) (28). The potentials calculated for the NMR structures were largely consistent with those calculated for the crystal structure, and larger variations were observed for loop regions (). For further examination of the impact of conformational variations, we also generated 100 structures with varied conformations. The backbone atoms of the original crystal structure were kept fixed while the side-chain conformations were varied through high temperature dynamics under the influence of the conformational database potentials (29) using the Xplor-NIH software (30). The structural variations caused changes in potentials (). For the secondary structures, the mean of SDs of potentials for the 100 structures were 4.1 mV, which is comparable to the rmsd (5.4 mV) between the data for 1UBQ and the experimental ϕ data. The SDs of due to the structural variations were larger for loop regions, which also exhibited the large differences between the experimental and theoretical ϕ data. These results collectively suggest that the larger discrepancy for the residues in the loop regions of ubiquitin is mainly attributed to the conformational flexibility and/or structural uncertainties, which represents an inherent limitation of structure-based electrostatic potential predictions.

Examination at a Higher Ionic Strength.

Typical Poisson–Boltzmann and Debye–Hückel models do not take ionic correlation effects into consideration and are known to overestimate the dependence of the activity coefficients on the ionic strength (I) when I > ∼100 mM (31). To examine the impacts of an increase in I on the experimental and theoretical near-surface electrostatic potential data, we conducted the same experiments for ubiquitin at I = 130 mM. The PRE Γ2,+ and Γ2, rates at 25 °C are shown in . For many of the residues, the magnitudes of PRE were smaller at I = 130 mM, implying weaker electrostatic interactions between the protein and the charged cosolutes at the higher ionic strength. Due to the decreased magnitude of PRE and the lower sensitivity of the cryogenic NMR probe at higher ionic strength, the number of 1HN nuclei that met the criteria for ϕ determination was smaller at I = 130 mM (50 1HN nuclei) than at I = 30 mM (63 1HN nuclei). Fig. 4 compares experimental ϕ data for the secondary structure regions of ubiquitin at I = 30 and 130 mM. There was a clear trend that the magnitudes of ϕ potentials at I = 130 mM were smaller than those at I = 30 mM. The linear regression for the correlation plot gave a slope of 0.74 ± 0.07 (with the uncertainty estimated at a confidence interval [CI] of 95%). Fig. 4 shows the corresponding data predicted by the Poisson–Boltzmann theory. The slope for the linear regression for the correlation between data at I = 30 and 130 mM was 0.64 ± 0.01, showing that the Poisson–Boltzmann theory predicts a stronger dependence on ionic strength. Nonetheless, the agreement between the experimental ϕ and theoretical data at I = 130 mM (with an rmsd of 5.1 mV and a correlation coefficient of 0.89 for the secondary structure regions; reference ) was as good as at I = 30 mM. Thus, the Poisson–Boltzmann theory predicts the ENS electrostatic potentials at the ionic strengths of 30 and 130 mM almost equally well, though it appears to predict a slightly stronger dependence on I in the current range.

Fig. 4.

Comparison of ENS electrostatic potentials for backbone HN atoms in the secondary structures of ubiquitin at ion strengths of 30 and 130 mM. (A) Comparison of experimental ϕ data. (B) Comparison of computational data predicted by the Poisson–Boltzmann theory. The red lines represent linear regression. The uncertainties in the slopes were estimated at a CI of 95%. Diagonals are indicated in broken lines as a guide to the eye.

Examination for a Highly Charged System: The Antp Homeodomain–DNA Complex.

As mentioned in the introduction, it has been suggested that the Poisson–Boltzmann theory may inaccurately predict electrostatic potentials for highly charged systems (12). Previous studies showed that the Poisson–Boltzmann equation-based approaches can accurately predict the total number of ions accumulated around highly charged proteins (32) and nucleic acids (33–35). However, the total number of accumulated ions is more relevant to long-range electrostatic potentials. The validity of the Poisson–Boltzmann theory for near-surface zones remains to be examined for highly charged systems. For such examination, we applied our method to the complex of the Antp homeodomain and a 15-base pair (bp) DNA duplex. The total charge of this protein–DNA complex is −16e. Using the complex of 15N-labeled protein and unlabeled DNA, we measured the 1H PRE Γ2,+ rates for amino-methyl-PROXYL and Γ2, rates for carboxy-PROXYL (Fig. 5; see also ) and determined the ϕ potentials for 1HN atoms in the complex at 25 °C (Fig. 5). The determined ϕ potentials ranged from −54 to −8 mV. Unlike ubiquitin’s case, backbone 1HN nuclei of the Antp homeodomain–DNA complex exhibited only negative ϕ potentials, reflecting the strong influence from the large negative charge of DNA.

Fig. 5.

Near-surface electrostatic potentials determined from 1H PRE rates Γ2 for protein backbone HN atoms of the Antp homeodomain–DNA complex at pH 7.5 and the ionic strength of 30 mM. The data in panels A, B, and C are presented in the same manner as shown in Fig. 3. For straightforward comparison of the two systems, the axis scale is set identical for this figure and Fig. 3. Note that for the entire system, the near-surface electrostatic potentials are negative due to a strongly negatively charged DNA duplex.

We also computed and using the electrostatic potentials calculated with the Poisson–Boltzmann equation for a 2.4-Å resolution crystal structure (PDB: 9ANT). Fig. 5 shows the experimental ϕ data (red) and the theoretical (blue) and (green) data for the Antp homeodomain–DNA complex. For loop regions, differences between experimental ϕ and computational data for this complex were remarkably larger than those for ubiquitin. Conformational flexibility or structural uncertainties may influence predictions more strongly for highly charged systems. This trend may arise partly from the significant packing and orientational correlation of PROXYL-based ions, which are not well represented in Poisson–Boltzmann theory. Nonetheless, for the secondary structure regions of the Antp homeodomain bound to DNA, the rmsd between experimental and theoretical ϕ data were 5.0 mV, which is even lower than the corresponding rmsd for ubiquitin at 30 mM, and the correlation coefficient was 0.84. Thus, the Poisson–Boltzmann theory appears to produce reasonable predictions, at least for the structurally well-defined regions of this highly charged system.

Discussion

Our NMR-based method has several advantages over previous experimental approaches for validation of theoretical models of protein electrostatics. The most remarkable advantage is that the NMR method can directly provide the near-surface electrostatic potentials for many protein residues simultaneously regardless of residue types. Previous experimental approaches provide only limited information on particular types of groups. Although validations based on pK data have been common (14–16), the pK-based method does not provide near-surface electrostatic potentials. Experimental pK determination requires fitting to experimental data at various pH values and is impacted by secondary effects (e.g., conformational changes and deprotonation/protonation of nearby residues due to the variation in pH) (36). In our paramagnetic NMR method, experiments are performed at a single pH and do not require any fitting procedures. Although electron paramagnetic resonance or vibrational spectroscopic methods are able to provide electrostatic potentials or fields (19, 21, 22), information from each experiment with these methods is limited to a single extrinsic probe that is conjugated to a protein and may perturb its native molecular properties. Our paramagnetic NMR-based method provides electrostatic potentials for many residues in native molecules and does not require any protein modification. Although we measured the ϕ potentials only for 1HN nuclei of 15N-labeled proteins in our current study, the same method can in principle be applied to 1H nuclei of CH/CH2/CH3 groups in 13C/15N-labeled proteins as well. This method is relatively easy in sample preparation, experimental procedures, and data analysis but allows for de novo determination of near-surface electrostatic potentials for a large number of protein residues without using any structural information.

In conclusion, we have developed a unique and powerful experimental method that permits straightforward examination of theoretical models for biomolecular electrostatics. Experimental ϕ data can serve for benchmarks for testing electrostatic models. Comparison of the paramagnetic NMR-based data and theoretical data are straightforward and greatly facilitate validation of theoretical models. Our current work illuminates the effectiveness and weakness of the Poisson–Boltzmann theory. Further experiments with the de novo method can produce large data sets of near-surface electrostatic potentials for various regions of biomolecules under various conditions. Although some discrepancy may arise from structural uncertainties, comparison of the experimental and computational electrostatic potentials will promote the improvement of theoretical models on protein electrostatics. Such improvement of electrostatic models would have broad impacts on molecular biophysics, structural biology, protein engineering, and drug development.

Materials and Methods

Proteins, DNA, and Other Materials.

15N-labeled ubiquitin was purchased from Sigma-Isotec. 15N-labeled Antp homeodomain bearing the C39S mutation was expressed in Escherichia coli and purified as described by Zandarashvili et al. (37) The unlabeled 15-bp DNA duplex with the sequence of CTCCTTTCT (bold, the Antp recognition site) was prepared as described (37). The paramagnetic compounds, amino-methyl-PROXYL (Chemical Abstracts Service [CAS] number: 54606-49-4) and carboxy-PROXYL (CAS number: 2154-68-9), were purchased from Sigma-Aldrich. Other reagents were also purchased from Sigma-Aldrich.

Quantifications of PROXL Derivatives.

For the de novo determination of near-surface electrostatic potentials, it is crucial to precisely measure the concentrations of amino-methyl-PROXYL and carboxy-PROXYL in the PRE experiments. Since pure amino-methyl-PROXYL is a highly viscous gel-like substance, it is impractical to quantify this compound by weighing. To resolve this problem, an NMR-based approach was used. In this approach, the PROXYL nitroxide (paramagnetic) sampled from a stock solution was completely reduced to hydroxylamine (diamagnetic) by adding an excess amount of ascorbic acid (38) and then a one-dimensional 1H NMR spectrum was recorded in the presence of 10 mM dimethyl sulfoxide (DMSO) using a single scan (without any dummy scan) to excite 1H nuclei at the Boltzmann equilibrium. The integrals of methyl 1H NMR signals from reduced PROXYL and those from 10 mM DMSO as the internal standard were used to determine the concentration of the PROXYL derivative (an example is shown in ). Based on the concentrations of the PROXYL derivatives measured through this approach, the stock solutions of 20 mM carboxy- or amino-methyl-PROXYL in the NMR buffer were prepared.

Samples for NMR Experiments.

NMR experiments were performed for samples containing 0.4 mM 15N-labeled proteins. The NMR samples of 15N ubiquitin at the ionic strength of 30 mM were 520-μL solutions containing 0.4 mM protein, 20 mM acetate, 28 mM Tris, 10 mM amino-methyl- or carboxy-PROXYL, 10 mM DMSO, and 5% D2O at pH 7.5. For the ubiquitin samples at the ionic strength of 130 mM, the buffer also contained 100 mM KCl. The NMR samples of the Antp homeodomain–DNA complex contained 0.4 mM 15N Antp homeodomain, 0.51 mM 15-bp DNA, 20 mM acetate, 28 mM Tris, 10 mM amino-methyl- or carboxy-PROXYL, 10 mM DMSO, and 5% D2O at pH 7.5. Solutions containing no paramagnetic substance were prepared through buffer equilibration using an Amicon Ultra-4 centrifugal filter with a molecular weight cutoff at 3 kDa (Millipore EMD) and then were concentrated to ∼0.8 mM. The diamagnetic control samples containing no paramagnetic substance were prepared by diluting the high-concentration solution with the same buffer containing no PROXYL derivatives. The paramagnetic samples were prepared by mixing the high-concentration solution with the stock solution of 20 mM carboxy- or amino-methyl-PROXYL in the same buffer to the final concentration of 10 mM at the final step. This procedure ensures the exact final concentration of amino-methyl- or carboxy-PROXYL. It should be noted that buffer equilibration using a buffer containing a charged PROXYL derivative would cause a significant deviation of its final concentration from the original concentration due to ion accumulation or exclusion by proteins and DNA (32, 39). Each NMR sample was 0.5 mL and was sealed in a 5-mm tube (Norell).

NMR Experiments.

The PRE rates Γ2 for 1H transverse magnetizations were measured for protein backbone 1HN nuclei of 15N-labeled ubiquitin and the complex of 15N-labeled Antp homeodomain and unlabeled 15-bp DNA. The two time-point approach (40) with a 10-ms difference was used to measure PRE Γ2 rates. Uncertainties in Γ2 rates were estimated through the error propagation method, as previously described (40). For each molecular system, the PRE experiments were conducted with three samples: one without any paramagnetic cosolute (diamagnetic) and the others with 10 mM amino-methyl-PROXYL (cationic) or carboxy-PROXYL (anionic). All NMR experiments were performed at 25 °C using a Bruker Avance III spectrometer equipped with a QCI cryogenic probe operated at the 1H frequency of 600 MHz. The spectra were processed with the NMRPipe software (41). Analyses of the NMR spectra and quantification of signal intensities for PRE measurements were carried out using the NMRFAM-SPARKY software (42). For ubiquitin, the Biological Magnetic Resonance Bank entry 17769 was used for resonance assignment information. For the Antp homeodomain–DNA complex, the resonances assigned by Nguyen et al. (43) was used. Resonance assignment for each system was confirmed using 3D 15N-edited nuclear Overhauser effect spectroscopy heteronuclear single-quantum coherence (NOESY-HSQC) spectra.

Determination of Near-Surface Electrostatic Potentials.

ENS electrostatic potentials ϕ for individual backbone 1HN nuclei were determined from Γ2,+ and Γ2,- data using Eq. . The uncertainties in ϕ were estimated through error propagation (44) using the following:

where σ+ and σ are uncertainties in Γ2,+ and Γ2,, respectively. To use only statistically significant PRE rates and Γ2,+/Γ2, ratios, we chose 1H nuclei, which satisfied the following three criteria: Γ2,+ > 3σ+, Γ2, > 3σ, and Γ2,+/Γ2, > 3. The calculations were performed using the MATLAB software (MathWorks).

Poisson–Boltzmann Equation–Based Calculations of Electrostatic Potentials.

Theoretical electrostatic potentials were calculated with the APBS software (8, 9) by numerically solving the nonlinear Poisson–Boltzmann equation. Full-atom models for ubiquitin and the Antp–DNA complex were constructed from the crystal structures (PDB: 1UBQ and 9ANT) using the Xplor-NIH software (30). In this step, hydrogen and other atoms absent in the crystal structures but present in the actual molecules were added to the structure models. The PDB2PQR tool (45) was used to assign charges and radii to individual atoms. The AMBER ff99 force-field parameter set and the protonation states at pH 7.5 predicted with the PROPKA approach (46) were used. The grid used for the APBS calculations was 128 × 128 × 128 Å3 for ubiquitin. A larger space of 160 × 160 × 160 Å3 was used for the Antp homeodomain–DNA complex. The grid spacing was 0.5 Å in each case. The monovalent ion concentration was set to 30 or 130 mM, depending on the actual experimental ionic strength used. The solvent van der Waals radius and the ionic radius were set to 1.4 and 2.0 Å, respectively, for calculations of the electrostatic potential at each grid point. The dielectric constant of the solvent was set to 78.54, and the dielectric constant for the macromolecular interior (ε) was set to 2, which has been used for both proteins and DNA (33, 47). Other values (ε = 4, 10, 20) were also tested, which impacted with an rmsd ≤ 1.1 mV (). For each grid point, the accessibility of the paramagnetic moiety (i.e., nitroxide) was determined using a radius of 3.5 Å, which empirically gave the best agreement between experimental and computational ϕ data on amino-methyl-PROXYL and carboxy-PROXYL (). The ENS zone for each 1H nucleus was defined as the grid points that are located within the distance range making a 68% contribution to . The potential was calculated as the average of electrostatic potentials for the grid points within the ENS zone. The ENS electrostatic potentials () were computed with Eqs. and using all grid points, including those inside and outside the ENS zone, along with U = eϕ and U = −eϕ for individual grid points. When Eq. was used for calculating , the accessibility factor a (0 for inaccessible grids; 1 for accessible grids) was multiplied to each r−6 term. The calculations of and from the APBS outputs were conducted using the MATLAB software. Structures in Fig. 2 were drawn using ChimeraX (48).