Incorporation of residual chemical shift anisotropy into the treatment of ¹⁵N pseudo contact shifts for structural refinement

Abstract

Paramagnetic NMR experiments, including the pseudocontact shift experiment, have seen increasing use due to recently developed probes and labeling strategies. The pseudocontact shift experiment can provide valuable intra- or inter-molecular distance and orientation information. However, the use of ¹H/¹³C or ¹H/¹⁵N PCS data in structure calculations is currently complicated by the contribution of residual chemical shift anisotropy to the ¹³C or ¹⁵N datasets. Here, we present a corrected PCS energy term for the software package Xplor-NIH with the appropriate residual chemical shift anisotropy correction and show its suitability for model refinements of ubiquitin labeled at residue 57 with a Tm-M8-SPy tag. For data taken at 800 MHz, the improvement with the corrected energy term is sufficient to make the quality of the fit for the ¹⁵N dataset comparable to that of the ¹H dataset, for which no correction is needed. The corrected energy term is expected to become more relevant with increased use of higher field instruments and as new paramagnetic probes with larger magnetic susceptibility tensors continue to be developed.

Graphical Abstract

Introduction:

Recent developments in paramagnetic probes and labeling strategies have greatly increased the use and efficacy of paramagnetic experiments in solution NMR. Among these, the pseudocontact shift (PCS) experiment has the potential to provide distance and orientational information between nuclei and paramagnetic centers, whether present endogenously or incorporated via engineered tags. In the PCS experiment, the hyperfine shift is obtained as the difference between the observed paramagnetic and diamagnetic chemical shifts. The hyperfine shift is comprised of both Fermi contact and through-space dipolar components, where the contact contribution is dependent on localized electron density on the relevant nucleus and is thus not applicable to atoms outside of the paramagnetic coordination sphere. The rotational average of the latter, dipolar, contribution defines the PCS and is given by equation 1:

$$PCS=\frac{1}{12\pi r^3}\left[\Delta {\chi }_a\left(3{\mathit{\text{cos}}}^2\theta -1\right)+\frac{3}{2}\Delta {\chi }_r{\mathit{\text{sin}}}^2\theta \mathit{\text{cos}}2\phi \right]$$ (1)

where χ_a and χ_r are the axial and rhombic components of the magnetic susceptibility tensor and r, θ and φ are the polar coordinates of the nucleus in a reference frame centered on the metal¹. Anisotropy in the magnetic susceptibility manifests as partial self-alignment, resulting in nonzero values of the chemical shielding anisotropy (CSA) tensor. This residual CSA is additive with the pseudocontact shift and is given by equation 2^{1, 2}:

$$\text{rCSA}=\frac{{\text{B}}_0^2}{15{\mu }_0\text{kT}}{\sum }_{\text{i},\text{j}}{\sigma }_{\text{ii}}^{\text{CSA}}{\text{cos}}^2{\theta }_{\text{ij}}\Delta {\chi }_{\text{ij}}$$ (2)

where the rCSA is dependent on the square of the external magnetic field, B₀, the temperature, T, and the sum over all directional cosines, θ_ij, of the magnetic susceptibility tensor, χ_jj, with respect to the nuclear CSA tensor, ${\sigma }_{ii}^{CSA}$ The rCSA is negligible for ¹H due to the small magnitude of the CSA tensor. However, for nuclei with larger CSA tensor magnitudes, including nitrogen and aromatic or carbonyl carbons, the rCSA can become significant at high magnetic fields and for sites far from the paramagnetic center where the PCS contribution becomes small².

¹H/¹⁵N PCS data are seeing increasing use in solution structure determination, leveraging CYANA, Xplor-NIH, ROSETTA and other software packages^3–5, but current implementations do not account for the rCSA, necessitating that the ¹⁵N data is taken at low fields where the rCSA effect is assumed to be negligible, or that it is discarded^{2, 6, 7}. Several standalone software packages can account for the rCSA effect in the back-calculation of χ tensors, aiding in structure validation, but here we present a refinement solution using a combined PCS energy term implemented in Xplor-NIH which allows the rCSA effect to be used during refinement^{8, 9}.

Data for S57C-M8-DOTA-SPy labeled ubiquitin (UB) at 800 MHz is used as an example for refinement, where the combined energy term increases refinement quality of the ¹⁵N data to match that seen for ¹H and allows for joint refinement using both datasets across different magnetic fields. The uses for such refinements are likely to expand as instruments with higher magnetic fields are used more frequently, and as new probes are developed capable of inducing PCS effects at increasing distances from the paramagnetic center.

Methods:

Description of the Energy Term:

The corrected energy term, nPCSPot, is implemented in the python interface of Xplor-NIH^{10, 11} and is derived from the pyPot base class. Uncorrected PCS data can already be represented with the Xplor-NIH RDC energy term, rdcPot, since the angular dependence of the PCS and RDC relative to the susceptibility tensor is the same (square bracketed portion of equation (1)). Additionally, the dipolar distance contribution is already implemented as an option in the rdcPot energy term for the treatment of through-space RDCs, such as those between protons, leaving only the 12π portion of the prefactor unaccounted for in the conversion from the RDC to PCS implementation. This approach has been employed for ¹H datasets, and for ¹³C or ¹⁵N datasets taken at low magnetic field values; typically 600 MHz^{6, 7}.

Similarly, for the rCSA correction to the PCS we utilized the existing csaPot energy term, which is already setup to share the magnetic susceptibility tensor from the RDC energy term through a scaling factor. For the current purpose, this scaling factor is set to the product of the remaining 12π constant from the PCS prefactor and the rCSA prefactor in equation 2. The combined energy term thus creates instances of the Xplor-NIH RDC (for PCS) and CSA (for rCSA) energy term and sets the scaling factor for the CSA energy term so the two can share a magnetic susceptibility tensor. At each step, the target value for each energy term is adjusted by the calculated value of the other term, and the energies and derivatives of the source energy terms are then computed on the corrected values. Since both energy terms are corrected at each step, the relative scales of the two terms do not strongly impact the result, and we partitioned them equally here. Wrapping the existing RDC and CSA energy terms with the combined nPCSPot allows all existing rdcPot and csaPot methods to still be utilized. We have also included options to specify whether the input units of the PCS data are in parts per million (ppm) or per billion (ppb), to specify whether the input data is for carbonyl carbon, aromatic carbon, or nitrogen nuclei, and to account for the negative gyromagnetic ratio of nitrogen.

Experimental:

The expression and purification of the ¹⁵N labeled Ub-S57C samples, as well as the preparation of the thulium (Tm-DOTA-M8-SPy) and lutetium (Lu-DOTA-M8-SPy) complexes^{12, 13} for paramagnetic and diamagnetic measurements, respectively, were described previously^6,14. To obtain PCS data, ¹H/¹⁵N-HSQC spectra were taken on a Bruker Avance 800 MHz spectrometer using a cryoprobe at 310 K for samples of Ub-S57C with the Tm-DOTA-M8-SPy or Lu-DOTA-M8-SPy conjugated tags. Paramagnetic 1H/15N RDCs were also collected at 310 K using the IPAP experiment (ref). NMRPipe was used for initial data processing, peaks were picked in Xipp 1.19.6, and remaining analysis used CCPN Analysis v3 and python. Spectra were recorded in duplicate or triplicate, and the standard deviations were determined from the difference of the contour averaged peak assignments (to up to 2 contour levels) from the CAPP algorithm in Xipp, with standard deviations of 1.2 ppb (¹H) and 1.2 ppb (¹⁵N) for the Tm spectra, and 1.4 ppb (¹H) and 1.9 ppb (¹⁵N) for the Lu spectra. Full assignment of the ¹⁵N labeled Ub-S57C spectrum at 600 MHz was obtained previously. Overlapping resonances were excluded from the analysis. Structure vizualizations were produced with Pymol. Backbone rmsd values were computed in Pymol for the residue region for which PCS restraints were obtained (residues 2–72).

Structure Calculation:

Briefly, the first models of two Ubiquitin PDB entries, 1D3Z and 2MJB were labeled in silico with one, two, three, or four copies of the DOTA-M8-SPy tag, which in Xplor-NIH is represented by the CTSA residue. To fit experimental (PCS and RDC) restraints, two sets of calculations were performed, one allowing only motion of the M8-DOTA-Spy tag, and a second allowing motion of the entire protein. For rigid refinements, the non-tag residues were held fixed, allowing motion of only 7 sidechain torsion angles of the tag, with the atoms of the DOTA cage moving as a rigid body. A 5000 step molecular dynamics stage at 3000 K was used to randomize the tag linker orientation, after which simulated annealing in 30K steps from 3000 K to 30 K was performed, where at each temperature molecular dynamics was performed for 200 ps. The varTensor object which represents the magnetic susceptibility tensor was shared for both ¹H and ¹⁵N datasets, and was reoptimized at each annealing step. Final minimization in torsion-angle space was then performed on the tag position using a 3000 step Powell-Minimization stage.

For non-rigid refinements, the initial steps to provide a rigid optimization of the tag location and tensor configuration were repeated as in the rigid case, but the final gradient minimization steps included a 1000 step Powell Minimization stage where all of the protein’s torsion angle degrees of freedom were permitted to move, and where additional restraints (dihedral angles, NOEs, and hbond restraints) were included from the original 1D3Z structure¹⁵. This was followed by another 2000 step Powell Minimization with all restraints in Cartesian Space.

In cases that included the paramagnetic RDC restraints, the RDC data were represented by a second varTensor object. Non-rigid refinements also used a radius of gyration energy term to reduce motion of unrestrained atoms. During refinement, the orientations of the two tensors were held together while the Da and Rh values varied independently. Tensor orientations were then allowed to deviate to independent minima at the end of each run. All refinements also used the standard Xplor-NIH, bond, angle, improper, and repel energy terms. For comparability, energy scales and ramps were shared across conditions and runs with the exception of the RDC energy scale, where the maximum RDC energy scale was set to 1 for rigid refinement and increased to 5 for non-rigid refinement to ensure that RDC violations were minimized.

Results and Discussion:

As an example application of the combined rCSA and PCS energy term, we collected ¹H and ¹⁵N PCS data for ubiquitin S57C labeled with a 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid (DOTA) derivative Tm-M8-SPy tag (Fig. 1A), and obtained datasets for both nuclei with the tag in the twisted square anti-prism (TSAP) conformation. Data are shown in Fig. 1B for ¹H (blue) and ¹⁵N (black) and follow our previously recorded data at 600 MHz⁶. In order to test the data, we first performed refinements using either ¹⁵N data or ¹H data alone, where in the latter case the ¹⁵N values were backcalculated from the computed ¹H tensor and tag position. This was repeated with both uncorrected and corrected terms, where the system was held rigid except for the atoms of the tag linker, which were allowed to vary as in [6].

Figure 1:

Refinement of ubiquitin S57C-DOTA-M8-SPY ¹H and ¹⁵N illustrates improvements in ¹⁵N agreement with the corrected PCS energy term. (A) Cartoon view of ubiquitin S57C labeled in silico with the DOTA-M8-SPy tag. (B) Experimental data obtained at 800 MHz for ¹H (black) and ¹⁵N (blue). (C) Representative absolute value residual plot for the ¹⁵N dataset and 1D3Z structure. There is minimal impact on the ¹H results during joint refinement from the use of the corrected (red) or uncorrected (black) ¹⁵N PCS energy terms (red). Conversely, there are significant reductions in the residual for the ¹⁵N data, particularly for residues relatively far from the metal center in the N-terminal region and in the region from residues 35–50.

Results are summarized in Table 1 against state 1 of two Ub structures: (PDB IDs: 1D3Z and 2MJB)^{15, 16}. With just the ¹⁵N dataset, the corrected energy term results in rmsd and Qfactor improvements for both the 1D3Z and 2MJB models of 8 ppb and 0.5%. The improvements are still present for backcalculation of ¹⁵N values for a tensor and tag position computed with only the ¹H data. Plots of the absolute residuals for the ¹⁵N data are given in Figure 1C for the top ¹⁵N result with the 1D3Z starting structure. The ¹⁵N residuals (right) show improvement with the corrected energy term (red) near the N-terminus from residues 10–25, in the stretch from residues 40–50, and in parts of the C-terminus around residue 70.

Table 1:

Results of rigid refinements of Ub-S57C-M8-DOTA-SPy with ¹H and ¹⁵N PCS datasets against state 1 of PDB IDs: 1D3Z and 2MJB. Runs were performed for rigid refinements where only the tag linker atoms could move, and with only the ¹⁵N PCS data (¹⁵N), or with only the ¹H dataset followed by backcalculation of the ¹⁵N values for the determined tensor and tag conformation (¹H). In all cases, the runs were repeated with corrected and uncorrected ¹⁵N PCS energy terms. All error measurements are the standard deviations of the 10 best structures from a total of 100 structures in each run.

PDB	DATA	Corrected	Xa (10−31 m3)	Xr (10−31 m3)	1H rmsd (ppb)	1H Qfactor (%)	15N rmsd (ppb)	15N Qfactor (%)
1D3Z	15N	Yes	2.34 ± 0.012	0.96 ± 0.000			40.38 ± 0.024	2.75 ± 0.003
1D3Z	15N	No	2.36 ± 0.012	1.06 ± 0.000			48.71 ± 0.010	3.31 ± 0.000
2MJB	15N	Yes	2.46 ± 0.018	0.99 ± 0.000			37.39 ± 0.091	2.54 ± 0.008
2MJB	15N	No	2.5 ± 0.014	1.13 ± 0.000			45.36 ± 0.009	3.08 ± 0.000
1D3Z	1H	Yes	2.22 ± 0.065	1.04 ± 0.000	27.99 ± 0.127	1.89 ± 0.008	52.54 ± 0.520	3.57 ± 0.037
1D3Z	1H	No	2.24 ± 0.059	1.06 ± 0.000	27.99 ± 0.127	1.89 ± 0.008	58.96 ± 0.363	4.0 ± 0.026
2MJB	1H	Yes	2.38 ± 0.103	1.05 ± 0.000	25.53 ± 0.072	1.72 ± 0.006	48.15 ± 1.568	3.28 ± 0.107
2MJB	1H	No	2.41 ± 0.089	1.06 ± 0.000	25.53 ± 0.072	1.72 ± 0.006	53.2 ± 1.215	3.61 ± 0.082

To further examine the impact of the corrected ¹⁵N PCS energy term, we collected complementary paramagnetic ¹H/¹⁵N residual dipolar couplings (RDCs) (Fig. S1A) and incorporated them alongside the ¹H and ¹⁵N PCS datasets for joint refinement. The RDC and PCS data ostensibly both report on the same magnetic susceptibility tensor, and so we fixed the orientation of the RDC and PCS tensors together during refinement. The Axial and rhombic components of each tensor were allowed to vary independently, however, and the tensors were allowed to diverge from one another at the end of each run to assess the agreement between the datasets. With the additional complexity, we first screened the effect of introducing additional copies of the in vitro tag (Fig. S1B). Addition of a second tag significantly improved the ¹⁵N PCS rmsd and normalized scalar product (NSP) between the PCS and RDC tensors, with minimal change for the third and fourth tag copies. The size of the improvement from the corrected energy term increased from approximately 6 ppb to 10 ppb from one to two tag copies, and remained at 10 ppb for the three and four tag cases. Looking at the metal positions for the top structure of each refinement, the trend was for additional tags to shift towards the central Ub helix, likely to better represent the restraints for sites 18 and 21 in that region (Fig. S1C). For the refinements with the corrected term, the maximum displacements between metal centers for each representative structure were 5 Å with two tags, 10 Å with three tags, and 11 Å with four tags.

We chose to continue with two tag copies for the remaining analysis, and results of refinements with the ¹H+¹⁵N PCS data together with the paramagnetic RDCs are summarized in Table 2. With the additional data and second tag conformer, the difference in rmsd with the corrected ¹⁵N energy term is 10 ppb, with a Qfactor improvement of 0.7%. The improvement was slightly smaller for runs where each structure had 10% of the data removed at random (8ppb and 0.6%) (italic lines in Table 2). Next, we tested the corrected energy term by introducing a second, non-rigid refinement step, where the protein atoms, in addition to the tag, were permitted to move. Along with the PCS and RDC datasets, this non-rigid refinement used NOE and dihedral restraints taken from the restraints for the ubiquitin 1D3Z structure¹⁵. We wanted to evaluate how much change from the starting structure would be needed to satisfy the additional PCS data, and whether differences can be observed with the corrected energy term. Allowing backbone motion significantly improves the rmsd and Qfactor results overall, and starting from the 2MJB structures now resulted in a final ¹⁵N PCS rmsd of 18 ppb, with corresponding uncorrected rmsd of 28 ppb. For all tested cases, the final RDC and PCS tensors were in good agreement with NSPs of 0.94 or greater. For the RDC restraints, the resulting rmsd and Qfactors for the rigid refinements (0.777 Hz and 10.83%) were unaffected by the presence of the corrected ¹⁵N energy term, while for the non-rigid refinements, the difference was within 0.14 Hz for rmsd and 0.2% for Qfactor.

Table 2:

Results of rigid and non-rigid refinement of Ub-S57C-M8-DOTA-SPy with paramagnetic RDCs, ¹H PCS and ¹⁵N PCS datasets against state 1 of PDB ID: 2MJB. Runs were performed both for rigid refinements where only the tag linker atoms moved, and non-rigid refinements were the protein atoms were allowed to move and constrained by the PCS and RDC datasets together with the original restraints for the 1D3Z structure. Runs were repeated with and without the corrected ¹⁵N PCS energy term. Values marked in italics are the results of runs of 100 structures where for each structure, 10% of the data was left out at random. All error measurements are the standard deviations of the 10 best structures. NSP is the normalized scalar product of the obtained PCS and RDC magnetic susceptibility tensors.

Refine	Corrected	Xa (10–31 m3)	Xr (10–31 m3)	1H rmsd (ppb)	1H Qfactor (%)	15N rmsd (ppb)	15N Qfactor (%)	RDC rmsd (Hz)	RDC Qfactor (%)	NSP
Rigid	Yes	2.3 ± 0.081	0.83 ± 0.001	28.07 ± 1.223	1.89 ± 0.082	35.84 ± 1.586	2.44 ± 0.110	0.777 ± 0.000	10.830 ± 0.000	0.94 ± 0.03
		2.23 ± 0.041	0.81 ± 0.001	27.36 ± 0.854	1.77 ± 0.051	36.65 ± 2.207	2.39 ± 0.148	0.768 ± 0.000	11.370 ± 0.000	0.94 ± 0.02
Rigid	No	2.26 ± 0.166	0.89 ± 0.007	27.24 ± 1.102	1.84 ± 0.076	45.76 ± 0.778	3.11 ± 0.053	0.777 ± 0.000	10.830 ± 0.000	0.95 ± 0.03
		2.3 ± 0.098	0.81 ± 0.002	25.75 ± 0.543	1.76 ± 0.040	44.84 ± 1.097	3.21 ± 0.347	0.757 ± 0.000	10.650 ± 0.000	0.94 ± 0.03
Non-Rigid	Yes	1.94 ± 0.130	0.91 ± 0.006	17.24 ± 0.386	1.16 ± 0.025	18.43 ± 0.507	1.25 ± 0.035	0.182 ± 0.002	2.544 ± 0.031	0.99 ± 0.0
		1.94 ± 0.149	0.87 ± 0.011	15.72 ± 0.593	1.02 ± 0.056	17.66 ± 0.685	1.19 ± 0.123	0.143 ± 0.003	1.986 ± 0.042	0.99 ± 0.0
Non-Rigid	No	2.62 ± 0.195	0.54 ± 0.004	18.4 ± 0.501	1.24 ± 0.035	28.23 ± 0.274	1.92 ± 0.020	0.168 ± 0.006	2.338 ± 0.077	0.99 ± 0.0
		2.5 ± 0.329	0.62 ± 0.023	17.21 ± 0.447	1.11 ± 0.030	25.22 ± 1.064	1.67 ± 0.069	0.156 ± 0.003	2.144 ± 0.044	0.99 ± 0.0

Residual plots for the top structure are shown in Figure 2A for rigid and non-rigid refinements against 2MJB with all datasets. The error is broken down by contribution, with the precorrection PCS values shown in black, the correction from the rCSA shown in green, and the combined values shown in red. For sites with significant improvement, the CSA contribution typically opposes the error in the calculated PCS and is frequently of comparable magnitude. This is even more apparent for the non-rigid case (Fig. 2B, bottom) where the region from residues 30–50 seems to be significantly improved by the addition of the rCSA term. Finally, the top structure for non-rigid refinements against the 2MJB reference structure (gray) are shown for the corrected (red) and uncorrected (blue) ¹⁵N cases in Figure 2B. The backbone rsmd for the restrained region (residues 2–72) was 0.543 Å for the corrected structure, and 0.598 Å without the correction. Deviations were spread throughout the structure without clear outlying regions.

Figure 2:

Effect of the corrected ¹⁵N PCS energy term on relative error in rigid and non-rigid refinements of ubiquitin S57C-DOTA-M8-SPy. (A) Breakdown of the contributions to relative error for the region from residues 20–60 in the rigid (top) or non-rigid (bottom) refinement cases. For many of the improved sites, which lie several nm from the metal center, the error in the PCS contribution (black) is of comparable scale and opposite sign to the rCSA correction (green), which gives rise to significant improvements for the combined value (red). This trend is accentuated with the non-rigid refinement case, where there are additional reductions in the 40–50 region. (B) Ribbon representations of the best structures resulting from non-rigid refinement with ¹H and corrected (red) or uncorrected (blue) ¹⁵N PCS energy terms plus paramagnetic RDCs relative to the PDB ID: 2MJB starting structure (gray). Backbone RMSD of residues 2–72 was 0.543 Å for the corrected structure and 0.598 Å for the uncorrected structure. Deviations from the starting structure are spread throughout.

Overall, the example results with ¹⁵N ubiquitin S57C-Tm-M8-SPy data at 800 MHz show a small but consistent improvement when the corrected ¹⁵N energy term is used. The improvement using the new energy term is sufficient to make the fit of the ¹⁵N data comparable to that of the ¹H data and we find that the corrected energy term reduces ¹⁵N rmsd by 35% in non-rigid refinement. Unsurprisingly, the impact of the corrected energy term is highest for sites near the termini and in the 35–50 region, which are the farthest from the paramagnetic center. This stems from the increase in relative contribution of the rCSA term, which lacks the inverse cubic distance dependence inherent to the PCS contribution. The corrected PCS results are consistent with ubiquitin solution NMR structures (PDB IDs: 1D3Z and 2MJB).

Conclusion:

Here, we present a corrected PCS energy term for the structural modeling package Xplor-NIH which can account for the rCSA effect. The energy term can be instantiated similarly to the existing PCS syntax in Xplor-NIH, is suitable to correct multiple types of PCS data, and allowed ¹⁵N PCS data to achieve comparable performance to the ¹H data for which this correction is unncessary. The need for rCSA correction to experimental PCS data is expected to become increasingly relevant at higher magnetic fields and improvements in lanthanide tags that result in greater susceptibility tensor magnitudes.

Highlights.

• Corrected PCS Energy term for Xplor-NIH improves ¹⁵N performance

• Corrected PCS Energy term suitable for joint refinement; tested on Ubiquitin S57C