The evolution of paramagnetic NMR as a tool in structural biology

Abstract

Paramagnetic NMR data contain extremely accurate long-range information on metalloprotein structures and, when used in the frame of integrative structural biology approaches, they allow for the retrieval of structural details to a resolution that is not achievable using other techniques. Paramagnetic data thus represent an extremely powerful tool to refine protein models in solution, especially when coupled to X-ray or cryoelectron microscopy data, to monitor the formation of complexes and determine the relative arrangements of their components, and to highlight the presence of conformational heterogeneity. More recently, theoretical and computational advancements in quantum chemical calculations of paramagnetic NMR observables are progressively opening new routes in structural biology, because they allow for the determination of the structure within the coordination sphere of the metal center, thus acting as a loupe on sites that are difficult to observe but very important for protein function.

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Introduction

NMR spectroscopy has long proven a powerful technique for determining the structure of biological macromolecules. X-ray diffraction structures account for 87% of the protein data bank (PDB), and to date, the number of electron microscopy and NMR structures has become comparable (5.6% vs. 7.2%). However, X-ray and cryo-electron microscopy require crystallization and rapid freeze treatment, respectively, whereas NMR is performed in solution or on sedimented samples and thus has the advantage to operate at higher temperatures and in close to biologically relevant conditions.¹ Furthermore, NMR remains the method of choice for the investigation of dynamics. The presence of a paramagnetic metal ion can provide an additional source of valuable information, at a level that is often hardly accessible with other techniques: paramagnetic centers affect the NMR spectra of proteins in ways that depend (besides on the type of paramagnetic metal) on the fine (sub-Å) detail of the coordination environment of the metal ion and of the distribution of protein nuclei, even if these are nanometers away from the metal.

The major paramagnetic NMR effects, due to the dipole–dipole interaction between nuclear and electron magnetic moments, are the changes with respect to the values measured for an analogue diamagnetic system in (i) nuclear relaxation rates, called paramagnetic relaxation enhancements (PREs), (ii) NMR shifts of the protein nuclei, called hyperfine shifts, and (iii) coupling constants between paired nuclei, called paramagnetic residual dipolar couplings (pRDCs). Hyperfine shifts are composed of pseudocontact shifts (PCSs) and, in the presence of unpaired electron spin density onto the observed nuclei, Fermi-contact shifts (FCSs). All these paramagnetic effects report on the structural details of the protein and on the variability thereof in the NMR time scale and are thus increasingly used as structural restraints.

The obtainment of the paramagnetic NMR restraints depends critically on the electronic structure of the paramagnetic center; they are in general measured as the difference of a given observable for the same protein with and without the paramagnetic metal that can be (i) replaced by a diamagnetic metal ion, (ii) reduced to a diamagnetic state, or (iii) removed from the protein.

The first protein structure calculations taking advantage of paramagnetic NMR restraints date back to the nineties. PREs were included in solution structure protocols in the first solution structures of paramagnetic proteins.² The introduction of PCSs in structural determination was demonstrated in 1996,³ and shortly afterwards pRDCs were introduced.^4,5 These two classes of restraints, combined with paramagnetic cross-correlated relaxation rates,^6,7 allowed us to obtain the first backbone structure of a protein only relying on paramagnetic restraints.^8,9 Also PREs measured through tailored experiments even for nuclei close to the paramagnetic center could be used to determine the structure of a small protein.¹⁰ These examples show the importance and the relevance of the structural information contained in these restraints. When dealing with paramagnetic proteins, paramagnetic NMR restraints should thus be always included in protein structure calculation protocols together with all other available restraints, mainly distance restraints determined from NOEs (nuclear Overhauser effects) and dihedral angle restraints from NMR chemical shifts.

Important advances have been made in the last few years regarding the protein structure refinement strategies through paramagnetic data, the theory of paramagnetic NMR shift and relaxation, the calculation from first-principles of quantum mechanics of hyperfine shifts, and the implementation of tools for their use as structural restraints. Some of the main achievements will be discussed in the following sections in light of the new perspectives they are disclosing.

The electron–nucleus interaction: the origin of relaxation and shift

The electron–nucleus or hyperfine interaction can be divided into two contributions, one arising from the interaction of the nuclear magnetic moment with the unpaired electron spin density (ρ_I) residing on it (called Fermi-contact interaction) and due to both direct electron spin delocalization and spin polarization, and another arising from the through-space interaction of the nuclear magnetic moment with the electron magnetic moment (called dipolar interaction).^11,12 This second term comprises the nuclear spin–electron spin interaction as well as the interaction between the nuclear spin and the electron orbital magnetic moment.^13,14

Given that electron relaxation occurs orders of magnitude faster than nuclear relaxation, the nucleus senses fluctuating fields originating from the hyperfine interaction, and further nuclear spin relaxation mechanisms become available (PRE, vide infra). Because in the presence of a magnetic field, the electron spin magnetic moment (for Fermi-contact interaction) and the overall electron magnetic moment (for dipolar interaction) are not null due to the different population of the electronic states, the nuclear resonances shifted (hyperfine shift, vide infra).

The paramagnetic relaxation enhancements

Nuclear relaxation rates in paramagnetic metalloproteins depend on several relaxation mechanisms all related to the presence of the paramagnetic center which add to the diamagnetic relaxation rates. One relaxation mechanism depends on the modulation of the dipole coupling with the electron spin according to the Solomon equations:¹⁵
where τ_r and τ_ei are the reorientation time and the longitudinal (i = 1) and transverse (i = 2) electron relaxation times, respectively. In the case of lanthanoids and actinoids, the J quantum number substitutes the S quantum number and g_J substitutes g_iso. These equations are derived from a number of assumptions, which include (but are not limited to): (i) the point-dipole approximation for the electron spin, (ii) isotropic molecular reorientations, (iii) isotropic g tensors and an electron Zeeman interaction much larger, at the field of interest, than zero-field splitting (ZFS), and hyperfine coupling between the electron spin and metal nuclear spin (for paramagnetic metals with I > 0). Recent experimental works have shown that in some cases these assumptions cannot be safely made even at high magnetic fields, and tools for managing experimental data have been proposed (see later).

When electron relaxation times are very short, nuclear relaxation caused by the Solomon equations is modest, and significant contributions to nuclear relaxation can arise from the modulation of the dipole–dipole interaction between the nuclear magnetic moment and the averaged electron magnetic moment (Curie spin relaxation). This contribution depends on the molecular reorientation time and on the magnetic susceptibility, and, in the case of isotropic χ tensors, Curie spin PREs are given by:¹⁶

For paramagnetic transition metal ions and for gadolinium(III), when excited states are far from the ground state, (spin-only approximation). In lanthanoids, because the spin–orbit coupling energy is generally larger than the effect of the crystal field, S is substituted by the spin–orbit coupled quantum number J, g_e by the Landé g-factor g_J, and the isotropic χ becomes .

The anisotropy of χ affects the Curie spin PREs,¹⁷ which can be calculated, as recently shown,¹⁸ with the following expression:

with

where σ is the dipolar shielding tensor:and the isotropic average of which corresponds to the PCS δ^pc (see below eqn (5)).

As already indicated, eqn (1) for PREs is derived from the isotropic g tensor and from the absence of ZFS and hyperfine coupling between unpaired electron(s) and metal nuclear magnetic moment. The latter effects can significantly affect the energy of electronic spin transitions, in such a way that sizably different rates are calculated, depending, for instance, on the magnitude of the ZFS energy with respect to the Zeeman energy, as well as on the position of the nucleus with respect to the axes of the ZFS tensor, with origin on the metal ion. Therefore, for systems with large ZFS, PREs have a structural dependence not limited to the metal–nuclear distance (as in eqn (1)), but comprising also the spherical angles defining the metal–nucleus direction in the ZFS frame.^19–22

The evidence of the angular dependence of PREs has been obtained experimentally for a paramagnetic lanthanoid(III) complex with a large static ZFS.¹⁸ PRE data, measured at 1 T, could in fact not be satisfactorily fitted using the Solomon equation (eqn (1a)) and showed a dependence on the direction of the r vector. PRE values were found to correlate with the sign of the PCSs, and thus to depend on the sign and orientation of the Δχ tensor, and could be nicely fitted using the parametric equation:

where the six independent components of the symmetric spectral density tensor G(ω_I) are treated as fitting parameters. The angular dependence of PREs is thus ascribed to the effect of the ZFS energy much larger than the electron Zeeman energy, as the Curie contribution and the effect of the magnetic susceptibility anisotropy on it (eqn (3a)) were calculated to be negligible at 1.0 Tesla with respect to the Solomon contribution.

The hyperfine shift

The analysis of the NMR spectra of a paramagnetic system passes through the evaluation of the hyperfine shifts, which correspond to the differences in the NMR shifts collected for the paramagnetic system and its diamagnetic analogue.

In solution, both the Fermi-contact and the dipolar contributions are averaged by molecular tumbling, which causes sampling of all molecular orientations with respect to the applied magnetic field. The rotational average of the dipolar shift is called PCS and is commonly analysed in terms of its dependence on a symmetric, traceless tensor (Δχ), corresponding to the anisotropic part of the magnetic susceptibility tensor, χ.²³

In the point-dipole approximation, the equations for PCSs and FCSs in isotropic solutions are:

where r is the distance vector of the nucleus I from the paramagnetic metal nucleus, and the Fermi contact coupling constant is defined as:and other symbols have the usual meaning. These equations for the pseudocontact and Fermi-contact shifts (written without using tensor notation) were derived for the first time by Kurland and McGarvey²³ in 1970.

Because the term (3rr^T/r² − 1) appearing in eqn (5) is traceless, δ^pc is only determined by the susceptibility anisotropy Δχ = χ − χ_iso1, where χ_iso = Tr(χ)/3, whereas δ^Fc depends on the overall χ tensor. In the reference frame where the Δχ tensor is diagonal, eqn (5) becomes

whereand r, θ and φ are the spherical coordinates of the nucleus in the principal frame of the χ tensor, with the origin in the position of the unpaired electron(s), and χ_xx, χ_yy and χ_zz corresponding to the eigenvalues of the χ tensor (the z and x directions being defined so that |Δχ_rh| ≤ 2 |Δχ_ax|/3 and |χ_xx − χ_yy| < |χ_xx − χ_zz|). PCSs thus contain valuable structural information, and hence they have been shown useful as restraints for protein structure determination.^24–31

Using a modern quantum chemistry (QC) formalism, in the spin Hamiltonian approximation, at high temperature (μ_BB₀ ≪ kT) and for a system with a singly-populated non-degenerate ground multiplet, the paramagnetic susceptibility tensor can be written in the form

where 〈SS^T〉 is the effective electron spin dyadic equal to
and Ẽ⁽⁰⁾_i is the energy of the state |ψ_i〉 at zero magnetic field. In the approximation to the first order in D/(kT), this provides³²where D is the ZFS tensor.

A question recently was raised on the correctness of eqn (5) and (10), i.e. in the derivation of the relationship between the tensor used to describe the PCSs, and the g and D tensors.^33–41 PCSs can in fact be calculated using QC approaches as a part of total hyperfine shifts.^{34,39,40,42–48} Using the effective spin Hamiltonian framework, it was first clarified that if the contribution of the orbital angular momenta of electrons to the PCSs is not correctly included in the first-principles QC treatment, a different expression for the PCSs is obtained.⁴⁹ The validity of eqn (5) and (10) was finally demonstrated^50,51 using the QC formulation of hyperfine shifts to be proportional to the second derivative of the thermally averaged Helmholtz free energy F with respect to the magnetic field and the nuclear magnetic moment, calculated at zero magnetic field and zero magnetic moment.⁵² In the effective spin Hamiltonian framework, the PCSs can in fact be written as

where the hyperfine coupling tensor A is composed of the spin dipolar, paramagnetic spin–orbit and gauge contributions. The gauge contribution is necessary to preserve gauge invariance in the presence of spin–orbit coupling.^50,51 At the level of second-order degenerate perturbation theory, in the long-range limit, eventually it was shown thatin agreement with eqn (5) and (10), because and . This derivation confirms the validity of the classical equations for PCSs and thus paves the way for their predictions from the χ tensors which can possibly be calculated through QC tools.

The relationships between PCSs, the Δχ tensor and the g tensor were also experimentally demonstrated to be correctly described by eqn (5) and (10) for some copper(II) proteins, where the g tensor could be measured through EPR under the same conditions of the NMR experiments.⁵³

Eqn (5) for PCSs is valid in the point-dipole approximation, which assumes that the unpaired electron(s) are localized onto the paramagnetic atom. If, on the contrary, the unpaired electron spin(s) are spread over the ligand atoms through molecular orbital overlap, PCS values should be calculated from integration performed over space by considering the actual electron density distribution.²⁵ In the assumption that the χ tensor is independent of the electron spin positions, a relationship – called the Inverse Kuprov Equation – was recently found that permits to recover the electron density distribution, under some regularization conditions, from the experimental PCSs and the atomic coordinates of the molecule.^25,54

The paramagnetic residual dipolar couplings

The anisotropy of the magnetic susceptibility causes partial self-orientation of paramagnetic molecules. In fact, the probabilities for different molecular orientations with respect to the magnetic field are not equal and depend on the anisotropy of the χ tensor. The principal components of the self-alignment tensor are given by .^12,55 This partial self-alignment of the molecules causes a non-null average of the energy of the dipole–dipole interaction between the nuclear magnetic moments, and thus this results in a paramagnetic contribution to their ¹J coupling. Therefore, pRDCs are usually calculated from the differences of the J-couplings measured for paramagnetic and diamagnetic samples (assuming negligible dynamic frequency shifts¹²). As a result, pRDCs and PCSs depend on the same magnetic susceptibility anisotropy tensor Δχ of the paramagnetic metal ion, and are thus often analyzed jointly. Nevertheless, as the dependences on the structural parameters are different, PCSs and pRDCs bear complementary information. In the reference frame where the Δχ tensor is diagonal, pRDCs are equal towhere α and β are the spherical angles describing the orientation of the internuclear r₁₂ vector in the main frame of the χ tensor. Note that pRDCs depend on the nuclear magnetogyric ratio (PCSs do not depend on the observed nuclei) and on the square of the applied magnetic field (PCSs are field independent, until approaching saturation conditions), whereas they are independent of the nuclear distance from the paramagnetic metal ion (PCSs depend on r⁻³). The term S_LS is a Lipari-Szabo order parameter⁵⁶ which amounts to 1 for completely rigid systems, but is smaller than 1 in the presence of some local mobility.

In the presence of paramagnetic tags attached to proteins with some degrees of flexibility, the pRDCs are significantly smaller than the values that are calculated for rigid systems using the Δχ tensor. In turn, when pRDCs are analysed using eqn (15) with S_LS = 1, as routinely done, “effective Δχ tensors” are obtained that may be significantly smaller than the real Δχ tensors. The different sizes of PCS-derived and pRDC-derived Δχ tensors can be in fact used to pinpoint the occurrence of tag mobility.

Relation between PCSs and RDCs

PCSs and pRDCs can be fitted against the same molecular structural models to obtain the magnetic susceptibility anisotropy tensors, Δχ, on which they depend (see eqn (5) and (15)), using programs like FANTEN⁵⁷ or NUMBAT.⁵⁸ Because PCSs and pRDCs depend on the same tensor, they can fit simultaneously, provided that the biomolecular structure is rigid, and a slight possible reduction in the orienting tensor (S_LS < 1) is allowed for, taking into account some local mobility of the coupled nuclei vectors.

Magnetic susceptibility anisotropy and electron relaxation rates of paramagnetic metals

As seen in the previous sections, the magnitudes of PCSs and pRDCs increase with the anisotropy of magnetic susceptibility, whereas PREs are the functions of the electron relaxation time and of the overall magnetic susceptibility (besides of the molecular reorientation time). Magnetic susceptibility anisotropies and electron relaxation times in turn depend on the presence of excited states close in energy (with respect to kT) to the ground state, which makes spin–orbit coupling efficient. Metal ions with low-lying excited states thus have (i) very efficient electron relaxation mechanisms, so that their electron relaxation times are very short, and (ii) very anisotropic g tensors, so that their magnetic susceptibility anisotropies are very large. Therefore, these systems originate from small Solomon contributions to PREs and large PCSs and pRDCs. Conversely, systems with excited states far in energy from the ground state have long electron relaxation times and small magnetic susceptibility anisotropies, so that Solomon contributions to PREs are large and PCSs and pRDCs are small. Fig. 1 and 2 show the typical values of τ_e, χ and Δχ for the first row transition metal ions and lanthanoids. The figures also show the maximum distances from the nuclei of the paramagnetic metal (those in axial position) experiencing a PCS as small as 0.05 ppm. These distances are calculated from the expression .

Fig. 1 Typical values of electron relaxation time (τ_e), magnetic susceptibility (χ, spin-only values, equal to and effective values, reported in parenthesis) and its axial anisotropy (Δχ_ax) of transition metal ions. The typical radii of blind spheres (black) and spheres with ¹H PREs larger than 0.1 s⁻¹ (green), and the maximum distances of nuclei in axial position with PCSs of 0.05 ppm (blue) are also shown for a protein with the reorientation time of 10 ns at 700 MHz (HS = high spin, LS = low spin).

Fig. 2 Typical values of electron relaxation time (τ_e), magnetic susceptibility (χ) and axial and rhombic anisotropy (Δχ_ax and Δχ_rh) of lanthanoid ions. The typical radii of blind spheres (black) and spheres with ¹H PREs larger than 0.1 s⁻¹ (green), and the maximum distances of nuclei in axial position with PCSs of 0.05 ppm (blue) are also shown for a protein with the reorientation time of 10 ns at 700 MHz.

Large PREs determine the presence of a blind sphere around the paramagnetic metal, where nuclear signals are undetectable in standard NMR experiments due to the excessive paramagnetic line broadening. Curie relaxation can sizably contribute to the PREs in large molecules (due to the large τ_r), as in metalloproteins, especially in the case of late lanthanoids, when χ is quite large. Fig. 3 and 4 show typical R₂¹H PREs for a proton at 10 Å from the metal, in a molecule with the τ_r of 10 ns, at 900 MHz, and the radius of the ¹H blind spheres for the different metal ions. Blind sphere radii are also shown in Fig. 1 and 2 together with the radii of the spheres containing nuclei with R₁¹H PREs larger than 0.1 s⁻¹ (taken as detection limit).

Fig. 3 (a) Typical radii of ¹H blind spheres at 900 MHz (paramagnetic linewidths ≥100 Hz), for molecules with the τ_r of 10 ns and typical values of the electron relaxation times of paramagnetic metal ions. (b) ¹H R₂ PREs and their Solomon (c) and Curie (d) contributions at 900 MHz, for a proton at 10 Å from the metal, in a molecule with the τ_r of 10 ns. Besides transition metal ions, only gadolinium(III) is shown in this figure; other lanthanoids, differing in their g-values, are reported in Fig. 4. Metals are labeled as LS: low spin; HS: high spin; t1: type-1; t2: type-2; Td: tetrahedral coordination; Oh: octahedral coordination.

Fig. 4 Typical radii of ¹H blind spheres of lanthanoid(III) ions (paramagnetic linewidths ≥100 Hz, left panel), and ¹H R₂ PRE (right panel), due to Curie relaxation, for a proton at 10 Å from the lanthanoid ion, at 900 MHz. The electron relaxation times τ_e are set to 1 ps and τ_r to 10 ns. Gadolinium(III), with τ_e several orders of magnitude larger, is shown in Fig. 3.

Lanthanoid binding tags rigidly attached to diamagnetic systems^59–62 or specific lanthanoid-binding sites introduced in diamagnetic proteins⁶³ are conveniently used for the generation of paramagnetic data to be used as structural restraints in the study of diamagnetic biomolecules. It was recently made clear that the choice of the lanthanoid ligand can significantly affect the magnitude of the magnetic susceptibility anisotropy, and thus of the PCSs, even in the assumption of a completely rigid attachment to the biomolecule. The need for a rigid attachment of the paramagnetic site is mandatory for avoiding conformational averaging effects at the origin of a reduction in the “effective” Δχ tensor (see above – although an average tensor may even not be defined when the position of the metal with respect to the molecule changes), and thus on the measured PCSs and pRDCs.⁶⁴

The dependence of the Δχ tensor on the coordination environment of lanthanoid ions, usually neglected due to the assumption that the ligand field splitting is small compared to the large spin–orbit coupling of lanthanoids, originates from the observation that in lanthanoid complexes, ligand field splittings are commonly larger than kT at room temperature,⁶⁵ and the electron density distribution is not spherical in f orbitals, except that for gadolinium(III). The Δχ tensor has been shown to depend on the compliance of the single-ion electron density with the crystal field environment in which it is placed. The (2J + 1)-fold degeneracy of the ground state can in fact be removed by the surrounding crystal field, so that the electronic structure of the ground state is linked to the strength and symmetry of the ligand field.^66,67 In turn, a proper choice of the coordination environment of the lanthanoid ion can increase the performance of lanthanoid tags.

As described in ref. 66 and 67, there are two general optimum ligand architectures providing a highly anisotropic ground state depending on whether the basic overall shape of the free-ion electron density is oblate, as for Ce³⁺, Pr³⁺, Nd³⁺, Tb³⁺, Dy³⁺ and Ho³⁺, or prolate, as for Pm³⁺, Sm³⁺, Er³⁺, Tm³⁺ and Yb³⁺. The magnetic susceptibility anisotropy of an oblate ion is maximized by a crystal field for which the ligand electron density is concentrated above and below the xy plane (see Fig. 5), so that the ground state corresponds to the configurations that minimize repulsive charge contacts between ligand and f-electron charge clouds. This configuration corresponds to the m_J state with the highest value. At the same time, states with low m_J values force the f-electron charge cloud into direct contact with the ligands, thus having a high energy. Conversely, the magnetic susceptibility anisotropy of a prolate ion is maximized by an equatorial coordination geometry so that the ground state, minimizing the charge contact with the axially located f-element electron density, corresponds to the m_J state with the highest value. Also the main direction (z axis) of the Δχ tensor depends on the non-spherical electron density distribution of the lanthanoid ion and on the ligand atom positions,^68,69 so as it may be oriented differently for the different lanthanoids, even if they are bound by the same ligand and, likewise, it may be oriented differently for different ligands, even if bound to the same lanthanoid ion. Careful engineering can lead to ligands preserving an almost completely axial susceptibility even at room temperature in solution. For example, a dysprosium(III) complex is available with Δχ_ax = 2.16 × 10⁻³⁰ m³, which is more than 220% of its isotropic magnetic susceptibility, and Δχ_rh = 2.17 × 10⁻³² m³, which is less than 3% of its isotropic value.⁷⁰

Fig. 5 Charge density angular distribution for lanthanoid(III) 4f shells for the different m_J states. Reproduced from ref. 66 with permission from the Royal Society of Chemistry.

The magnetic susceptibility anisotropy values obtained for lanthanoids in calbindin D_9k, reported in Fig. 2, can be compared to the values obtained for some DOTA-based complexes, for tris(dipicolinato)lanthanoids (dipic = pyridine-2,6-dicarboxylate), and for some widely used and efficient lanthanoid-binding tags, shown in Fig. 6. The anisotropy values were determined from the shifts of the ligand nuclei in the case of the DOTA,⁷¹ DOTA-M8,⁷² DOTA-M7FPy⁷² and tris(dipicolinato)^73,74 complexes, and from the nuclear shifts of the attached proteins in the case of the tags CLaNP-5,^28,75 BrPSPy-DO3M(S)A⁷⁶ and Ln-M7-Nitro.⁷⁷ In the case of the proteins with paramagnetic tags, the shifts may be reduced to some extent by a non-completely rigid attachment of the tag.

Fig. 6 Magnetic susceptibility anisotropy tensors (Δχ_ax, Δχ_rh, in 10⁻³² m³) and isosurfaces of PCS = 0.05 ppm (in blue) or −0.05 ppm (in red) for some lanthanoid ions in different coordination geometries. Axis values are in Å.

The data show that if the DOTA-based tags providing the largest anisotropies could be rigidly attached to proteins, PCSs could be detected for nuclei at larger distances than expected from the Δχ of the same ions in calbindin D_9k, and higher, and thus more accurate, pRDCs could be measured, thus extending the perspectives for their use in the structural determination of biomolecular systems.

Paramagnetic structural restraints

Paramagnetic restraints efficiently provide structural information throughout the whole macromolecular system, or a large part of it because of their long-range nature. This feature becomes particularly relevant when paramagnetic restraints are used in integration with the data obtained using other techniques, and can help in removing assignment ambiguities.^11,78–82 Besides the use for de novo structural determination, the interest towards paramagnetic restraints has increased in the last decades,⁸³ thanks to other notable implications of their use in (e.g.) detecting transient interactions, conformational changes, etc.

Paramagnetic restraints in the protein structure calculation

Determination of the traditional structure by NMR suffers from the lack of long-range observables. The collected PCSs, pRDCs and PREs can thus be used for the calculation of de novo structure or for the refinement of structures together with dihedral angle restraints and other diamagnetic restraints in programs like PARAMAGNETIC-CYANA^84,85 or Xplor-NIH.^86,87 For instance, the solution structure of the protein calbindin D_9k could be refined using paramagnetic data from lanthanoid ions substituted in the second calcium binding site of the protein, with an increase in resolution not only in the close proximity of the paramagnetic center but along the overall protein backbone chain.⁸⁸ A further insight into the structure of calbindin D_9k with the thulium(III) ion substituting one calcium(II) ion arose from the application of the Inverse Kuprov Equation: a probability density distribution of the paramagnetic center was reconstructed, providing a better agreement with the experimental data than calculated from the point-dipole approximation.²⁵

PCSs can efficiently guide the reconstruction of protein conformations in the ROSETTA fragment assembly method,²⁴ especially when multiple paramagnetic metals positioned in different places on the protein surface are used.^89,90 Also PREs showed usefulness to recover protein structures in molecular fragment searching protocols, when used in conjunction with diamagnetic RDCs and/or backbone dihedral angles.^91,92 In microcrystalline protein samples, PCSs have been used to obtain unambiguous signal assignments in solid-state NMR spectra, allowing for determining high-resolution protein structure,^93,94 as well as for determining the relative positions of the protein molecules in the crystal.³¹

Interdomain position reorganization in solution and crystals

Also the solution structure of calmodulin in complex with the calmodulin-binding peptide of the death-associated protein kinase could be determined using PCSs and pRDCs from three different lanthanoids substituted into the second binding site of the N-terminal domain of the N60D mutant of the protein.⁹⁵ The calculated structure shows a different arrangement of the two domains with respect to the crystal structure: this is due to the absence of inter-protein contacts that are present in the crystal. Indeed, this solution structure resulted in better agreement with the crystal structure of calmodulin in complex with the whole kinase protein⁹⁶ than with the crystal structure of calmodulin bound to the peptide (Fig. 7a).⁹⁷ The steric hindrance of the whole protein in the crystal prevents the formation of H-bonds and salt bridges between different calmodulin molecules, thus allowing the latter to have a conformation more similar to the solution structure. The residual discrepancy between the paramagnetic data and the solution structure can be attributed to a limited but significant interdomain mobility.⁹⁸ In general, the achievement of a good fit of the experimental PCSs and pRDCs^30,57,97 is an indication of a good accuracy of the protein structure (Fig. 7b).

Fig. 7 (a) X-ray structure (PDB entry 1YR5, in red) and refined solution structure (PDB entry 2K61, in blue) of the peptide-bound calmodulin, and X-ray structure of calmodulin in complex with the whole kinase protein (PDB entry 2X0G, in green). (b) Best fit of PCSs and pRDCs for the refined solution structure, calculated using FANTEN⁵⁷ and a common Δχ tensor for each paramagnetic metal ion.

Paramagnetic data for protein structural refinement in integrative structural biology

The long-range nature of paramagnetic restraints is particularly appealing to validate, and possibly refine, structures obtained from X-ray data, cryo-electron microscopy, or homology modeling. X-ray crystallography, in particular, provides very precise protein structures, which can, however, suffer from crystal packing forces. Therefore, they can be inaccurate models, to some extent, in solution, where these interactions are abolished. Furthermore, crystal structures are affected by “structural noise”,⁹⁹ depending on the resolution of the X-ray reflections and on the structural refinement protocol employed, which may result in a lower accuracy of local structural details than that contained in NMR data. The high accuracy, despite their sparsity, of paramagnetic restraints makes them fully complementary to X-ray data for achieving a more accurate structural description of the biomolecules, also because NMR data provide direct information on local details, in the form of interatomic distances or orientations of vectors connecting chemically bound nuclei.³⁰

The information contained in PCSs and pRDCs are of different kinds. On the one hand, PCSs are barely affected by small structural inaccuracies, so that they are nicely fitted when the structural model is in overall agreement with the biomolecular structure in solution. Their fit provides a very robust estimation of the magnetic susceptibility anisotropy tensor, being determined by several tens, or hundreds, of PCS values. On the other hand, pRDCs, which are sensitive to even small local inaccuracies in the orientation of nuclear pairs, hardly agree with the susceptibility tensor obtained from PCSs unless the molecular structure is a very accurate model for the protein in solution. For instance, the pRDCs measured for the catalytic domain of matrix metalloproteinase-1 (MMP1), attached with the CLaNP-5 tag⁶² with either one of the three paramagnetic lanthanoids Tb³⁺, Tm³⁺ and Yb³⁺, provide an agreement with the crystallographic structure much poorer than that expected from the experimental errors of the data (Fig. 8), despite the very good fit of the PCS data.²⁹ This disagreement could be due to either some protein structural rearrangements on passing from the crystal state to the solution state, or to some inaccuracy in the nuclear coordinates determined from X-ray data (structural noise). Actually, the second case seems more plausible for this system because the disagreement of pRDCs could be solved by very small conformational adjustments, within the indeterminated X-ray data. Similarly, the disagreement between the diamagnetic RDCs measured for the protein ubiquitin and its crystal structure could be accommodated within the structural noise of the X-ray data,²⁹ and even for sub-atomic resolution structures of lysozyme¹⁰⁰ RDC restraints could be accommodated within the X-ray structural noise without the need for invoking conformational averaging.

Fig. 8 Best fit agreement of experimental pRDCs measured for the catalytic domain of matrix metalloproteinase-1 with the crystallographic structure (PDB entry 3SHI) (2.2 Å resolution) and superposition of original (in blue) and refined (in yellow) structures.

The paramagnetic data can thus be used to assess how accurately crystal structures represent solution structures and to improve them by simultaneous refinements through NMR and X-ray data. PCSs and pRDCs can be fitted, using the same Δχ tensor for each metal ion, together with X-ray data using the program REFMAC-NMR²⁹ developed within REFMAC5 from CCP4.¹⁰¹ The program provides a refined structure, resulting from the combined use of the information on the heavy atom positions mainly contained in X-ray reflections, and on nuclear (mostly hydrogen) positions and bond orientations of nuclear pairs contained in PCSs and pRDCs. If this structure is in good agreement with both X-ray data (in terms of R and R-free values) and NMR data (in terms of Q-factor), it can be concluded that there are no inconsistencies between the crystal and the solution structures, and thus the refinement performed with both NMR and X-ray data provides a more reliable structural model.¹⁰² On the other hand, the presence of some NMR data which do not match the values predicted by the refined conformation within their error is an indication of minor inconsistencies, i.e., of local conformational rearrangements between the solution and the solid states (or the presence of conformational heterogeneity in solution) in specific regions of the protein.^30,102

In the case of the catalytic domain of MMP1, REFMAC-NMR provides a refined structure with a backbone root mean square deviation from the original model of only 0.04 Å (Fig. 8), in full agreement with the X-ray reflections and a significant overall improvement in the agreement of the NMR data, the Q-factor of the pRDCs decreasing from 0.41 to 0.16.²⁹ This example shows the high accuracy in defining structural details that these NMR data can provide.

For systems composed of multiple domains, the evaluation of the Δχ tensors determined from the fit of PCSs and/or pRDCs measured for each domain provides a simple but powerful way for assessing their relative arrangement in solution. Several situations can be encountered.

1. If both sizes and orientations of the tensors coincide, the crystal conformation is maintained in solution.

2. If only the sizes coincide, and not the orientations, the relative arrangements of the refined domains in solution can be retrieved by superimposing the anisotropy tensors.⁵⁷ As an example, a cobalt(II) binding tag was attached to the N-terminal domain of the homodimeric protein STAT4. The PCSs permitted to refine the monomeric structure⁸⁰ and to determine the arrangement of the monomeric subunits. Sub-stoichiometric addition of the paramagnetic metal in this case permitted to disrupt the symmetry in the NMR spectra of the protein.¹⁰³ Data from multiple paramagnetic ions or from ions located in multiple positions are often needed to identify the correct conformation among the possible degenerate solutions.^104,105

3. Finally, if the tensors differ both in size and in orientation, interdomain mobility occurs and thus conformational averaging takes place.⁶⁴ In these conditions, the measured PCSs, pRDCs and PREs are population-weighted averaged over all sampled conformations (PREs may also depend on the time of interconversion between the different conformations).^106,107 Several approaches have been developed to recover some information on the experienced conformations, although the reconstruction of the real structural ensemble is impossible due to the inherent presence of degenerate solutions.^64,108–113

PREs, providing very sensitive information on the distance between the observed nucleus and the paramagnetic metal ion (eqn (1) and (2)), have been found to be very useful to monitor the occurrence of structural mobility in solution. In particular, PREs can detect the presence of less (few percent) populated conformational states when the latter imply metal–nuclear distances much shorter than in the more populated conformational states.¹¹⁴ In the presence of two states, in fact, if their exchange rate is faster than the difference in relaxation rates and slower than the reorientation time of the system, the measured PREs are population weighted averages of the PREs in the two states. Along the same lines, PREs have been used to detect the presence of transient, sparsely populated, protein–protein encounter complexes, and to identify the patches of protein residues that come into short-lived close contact with one another.^115–117 Such sensitivity of PREs to lowly populated structural states is an extraordinary informative tool for the characterization of transient states in solution. Paramagnetic restraints can also be used in HADDOCK¹¹⁸ for driving docking calculations of protein complexes.

Coupling QC calculations to paramagnetic data analysis

Quantum chemical methods for the calculation of hyperfine shifts and relaxation rates are becoming increasingly accessible and can now be included in the toolkit for protein structural calculations. The hyperfine shifts of the two high-spin pentacoordinate nickel(II) complexes, NiSAL-MeDPT and NiSAL-HDPT (SAL = salicylaldiminate; DPT = dipropylenetriamine), composed of both PCSs and FCSs, could, for instance, be calculated¹¹⁹ using the ORCA quantum chemistry package^120,121 and the NiSAL-MeDPT structure after refinement at the density functional theory (DFT) level. PCSs were calculated from eqn (5) and the Δχ tensors obtained using state-averaged complete active space self-consistent field (SA-CASSCF),¹²² accounting for the dynamic correlations by N-electron valence perturbation theory to the second order (NEVPT2),¹²³ and FCSs were calculated using the Fermi contact coupling constant and the isotropic g values obtained from DFT calculations. The calculated hyperfine shifts agree very well with the experimental shifts of both complexes (Fig. 9). The accuracy of the calculations allowed for a swap in the assignment of the methylene signals previously reported,¹²⁴ and for a refinement of the NiSAL-HDPT structure using the structure of NiSAL-MeDPT as a starting template.¹¹⁹ The small discrepancies between the calculated shifts and the shifts measured for the HDPT derivative, for which a structure is not experimentally available, were finally rationalized in terms of minor structural changes.

Fig. 9 Agreement between experimental and QC calculated shifts of NiSAL-HDPT, with the assignment previously reported (a),¹²⁴ and the theory-based reassignment (b). Comparison between initial (beige) and DFT-optimized (cyan) structures (c) and agreement between the experimental and QC calculated shifts for the refined structure (d).

As it is clear from eqn (5), PCSs depend on the magnitude and orientation of the Δχ tensor, which in turn reflects the very fine details of the coordination geometry of the paramagnetic metal ion. For instance, in cobalt(II)-substituted human carbonic anhydrase II, the cobalt(II) ion is essentially tetracoordinated when the enzyme is free in water or bound to sulfonamides, and pentacoordinated in the presence of ligands such as oxalate, where two oxygen atoms of the ligand complete the coordination environment.¹²⁵ Indeed, the magnetic susceptibility anisotropy changes dramatically in the two cases.¹²⁶

It was recently figured out that PCSs can be used not only for refining protein structures at the positions of the observed nuclei but also around the paramagnetic center, thus recovering accurate conformational details about the metal ligand positions. In fact, recently, for the first time, it was shown that the paramagnetic Δχ tensor of a large metalloprotein can be calculated ab initio with advanced quantum-chemical approaches using only a structural model of the metal coordination site.³³ Using the effective spin Hamiltonian (valid when all orbital excited states are sufficiently more in energy from the ground state with respect to the energy of the thermal bath), the Δχ tensor can also be determined from the g and the ZFS (D) tensors (see eqn (10)–(12)).^12,23,41,127 From the knowledge of the Δχ tensor and of the protein coordinates, it is easy to predict the PCSs in the point-dipole approximation. A good matching between predicted and experimental PCSs is thus a clear indication of a good prediction of the Δχ tensor.

Conversely, as the magnetic susceptibility tensor is very sensitive to the details of the coordination geometry of the paramagnetic metal ion, the PCSs, measured for nuclei far from the paramagnetic metal, can be used to refine the molecular structure around the metal ion.⁸² This approach offers the possibility to overcome one of the most important limitations of paramagnetic NMR, which is the large line broadening often affecting the signals of nuclei in the immediate vicinity of the paramagnetic center, responsible for the presence of a blind sphere. The knowledge of the metal coordination environment at the highest possible resolution is crucial to understand structure–activity relationships for metal ions in proteins and for the successful use of docking strategies for drug discovery.^128,129

This approach has been tested to refine the structure in solution of the catalytic domain of the human matrix metalloproteinase-12, where the catalytic zinc(II) ion was substituted with the paramagnetic high spin cobalt(II) ion (CoMMP12), coordinated by the N-isobutyl-N-[4-methoxyphenylsulphonyl]glycyl hydroxamic acid (NNGH) inhibitor. The cobalt(II) ion coordinates three histidines through their nitrogen atoms, and the inhibitor NNGH through the oxygen atoms of its hydroxamic moiety. A Δχ tensor was obtained using FANTEN⁵⁷ from the best fit of the PCS data measured for this protein and the X-ray structure of the NNGH-inhibited zinc(II) protein at 1.34 Å resolution (PDB entry 5LAB³³), with cobalt(II) replacing zinc(II). The high quality of this fit indicates the good accuracy of the crystallographic model in representing the overall structure of the protein in solution, as well as of the Δχ tensor because it is determined from several hundreds of ¹³C PCSs.¹³⁰ A magnetic susceptibility tensor was also calculated with ORCA^120,121 from the coordinates of the metal ligands taken from the same X-ray structure of the protein. The anisotropy of this QC calculated susceptibility tensor was not found in agreement with the Δχ tensor obtained from the fit of the PCSs. On the other hand, relativistic CASSCF calculations, with second-order perturbation theory corrections, implemented in ORCA, have been shown to be quite reliable in predicting the magnetic properties of transition metal complexes and their temperature dependencies in a number of cases.^119,131,132 Therefore, starting from the crystallographic model of the cobalt(II) coordination cage, a structural refinement procedure was implemented so as to obtain a susceptibility tensor from ORCA in good agreement with the PCSs-derived anisotropy tensor.¹³³ The structural refinement was performed by adjusting, through a steepest descent search, 19 degrees of freedom among coordination bond lengths, angles and dihedral angles, selected in a way to avoid altering the structure of the ligands themselves. The calculations also showed that an additional proton in the structure of NNGH should be included (implying that the hydroxamic acid moiety, and not the hydroxamate moiety, coordinates the cobalt(II) ion). This procedure permitted to refine the structure of the metal coordination site in solution to picometer precision (Fig. 10).

Fig. 10 Agreement between experimental and QC calculated PCSs for the CoMMP-12 protein structure (PDB entry 5LAB) with the hydroxamate (model A initial structure) or the hydroxamic acid moieties of the NNGH inhibitor, before (model B initial structure) and after (model B refined structure) geometry refinement. The refined structure (green) is superimposed to the initial one (grey) to highlight the subtle structural differences.

In summary, having clarified the QC terms to be included in the hyperfine coupling tensor, and thus having confirmed the dependency of PCSs on the χ tensor from first principles, QC tools can now be used for an accurate prediction of the χ tensor from a structural model of the metal coordination cage, and consequently for an accurate prediction of the PCSs of all protein nuclei from the knowledge of the protein structure, in the point-dipole approximation. The χ tensor can be predicted from relativistic CASSCF/NEVPT2 calculations for metal ions like cobalt(II) and nickel(II). Current challenges are now: (i) to extend the applicability of the QC tools to the robust and efficient prediction of the χ tensor of other paramagnetic metal ions, including lanthanoids(III),^{121,134–136} and (ii) to accurately predict PCSs outside the point-dipole approximation.⁵⁴

Also FCSs can be used for the refinement of the metal coordination site to a similar precision. In fact, FCSs depend on the electron spin densities delocalized onto the nuclei, and in turn on the coordination geometry of the paramagnetic metal ion. It has been recently shown that the FCSs measured for ¹H, ¹³C and ¹⁵N nuclei of the cobalt(II) ligands in solid-state NMR spectra of the human protein superoxide dismutase 1, in a microcrystalline form and containing a cobalt(II) ion, allowed for the determination of the high-resolution structure of the metal coordination sphere at picometer resolution in the precision of the bond lengths and ±1° resolution of bond angles.¹³⁷ A structural model could, in fact, be selected within an ensemble of X-ray structures so as to achieve the best agreement of the measured FCSs with those calculated from the molecular structures using DFT. The structural precision so achieved was enough to correlate the coordination geometry with the unreactive nature of the metal center in these proteins, where it plays a purely structural role.

In perspective, QC calculations of hyperfine shifts are expected to increase in importance and popularity with the increase in computational power, which will extend the applicability of the approach for the structural refinements of proteins even when starting from more distant initial models.

Paramagnetic restraints to monitor protein dynamics

A challenging objective of structural biology is the characterization of the conformational variability which can be exploited by biomolecular systems to perform their biological function. Experimental NMR data are averaged values reflecting this structural variability. Recovering information on the different conformations from these averaged data is a quite cumbersome, ill-posed, inverse problem, which can be tackled in a number of ways^112,113,138 but always without the possibility of reconstructing a single structural ensemble if a priori assumptions are not introduced. In the case of systems composed of multiple structured domains which can reorient with respect to one another, the most used approaches either rely on the Occam's razor¹³⁹ (or maximum parsimony) principle or on the maximum entropy principle, aiming at identifying the least populated structural ensembles and the broadest and flattest probability distribution, respectively.^112,113,138 Alternatively, one can rely on unbiased molecular dynamics simulations reweighted in order to achieve an agreement with the experimental data. From a yet different perspective, the maximum occurrence approach^{27,28,112,116,138,140} has been proposed to extract the information contained in the averaged data without any prior assumption or calculation. This approach provides the largest weight possible for each allowed conformation, independently from all other conformations present in the ensemble, and is achieved at the cost of giving up the reconstruction of structural ensembles.

Paramagnetic NMR data contain averaged structural information which can be very useful for monitoring the conformational variability of the biomolecular system.⁶⁴ For instance, in multidomain proteins, pRDCs are very sensitive to the mobility of a metal-free domain with respect to another domain containing a paramagnetic metal; PREs are very sensitive to even very lowly populated conformations with some nuclei close to the paramagnetic metal, and PCSs are sensitive to both reorientations and translations.

As an example, PCSs and RDCs, complemented by SAXS data, measured for the protein matrix metalloproteinase-1 allowed for the recovery of the mostly populated conformations within the wide structural ensemble experienced by the protein due to the long flexible linker connecting the catalytic and the hemopexin-like domain.²⁸ These conformations differ largely from the structures obtained by X-ray crystallography.

Conclusions and perspectives

Recent technological advancements, which comprise cryo-electron microscopy and computational methods for structure prediction,¹⁴¹ offer new tools for the structural characterization of biomolecules, in addition to NMR and X-ray crystallography. Nevertheless, only paramagnetic NMR can offer a spatial resolution on the picometer scale. Therefore, although other techniques can be more advantageous in terms of high-throughput and automation, structural validation and refinement in solution through NMR studies are still crucial for many applications. We thus envision that paramagnetic NMR data, rather than being used for ab initio structure calculations, will be increasingly combined with experimental data from different sources in integrative structural biology approaches.

Challenging applications of paramagnetic NMR in structural biology are mainly expected in the following fields:

(1) Refinement of molecular models obtained from other techniques (X-ray crystallography, cryoelectron microscopy, artificial intelligence –AI– predictions). Paramagnetic data offer precious information with an accuracy hardly available using the other techniques and an unprecedented level of detail. Paramagnetic data, in fact, despite their long range nature, provide immediate information about short-range inter-atom distances and bond orientations and can probe the dynamics experienced in solution. Therefore, they are highly suitable to refine conformational models determined using techniques not allowing for such accurate local structural details, and to evaluate the presence of conformational averaging. This approach can improve X-ray structures, which can be affected by structural noise and can be from slightly to significantly inaccurate in solution due to the presence of crystal packing forces, as well as models determined by cryoelectron microscopy and by computational methods for predicting the protein structure, like AlphaFold.¹⁴² Structures determined by cryoelectron microscopy can, in fact, suffer from low resolution, especially in the case of small biomolecules, and can be inaccurate in solution, and the accuracy of AlphaFold structures may need to be improved depending on the required level of structural details.

(2) Evaluation and characterization of the conformational dynamics relevant to functions in systems exploiting structural heterogeneity. In all cases, no or very limited insights into conformational dynamics are obtained using other structural techniques (X-ray crystallography, cryoelectron microscopy, AI predictions). NMR is an irreplaceable technique in this respect, and paramagnetic NMR, in particular, can provide very informative data.

(3) Monitoring of protein–protein interactions and protein–drug interactions in solution. Paramagnetic data can allow for monitoring the presence of interactions between proteins, even if transient and elusive,¹¹⁶ as well as between proteins and small paramagnetic molecules, even with a low binding affinity. These studies are crucial for the understanding of protein biological functions and for the development of drugs.

(4) Refinement of the metal coordination site of paramagnetic proteins, also taking advantage of the available QC tools. The improvement in QC predictions of paramagnetic NMR data and the development of computational tools for the exploitation of QC data are challenging fields of research. Thanks to QC computations, it can, in fact, be possible to improve the biomolecular structures to a resolution unachievable by any other techniques.¹⁴³ Such an accuracy may be needed, for instance, for a thorough understanding of structure–activity relationships and in drug discovery. For a routinary and reliable use of QC tools, much work is still needed for the improvement of the prediction tools, for the whole range of paramagnetic metal ions, in terms of accuracy and computational time, and for cross validation of the predicted data. In parallel, efficient protocols for structural refinement should be implemented. To be noted, the accuracy of X-ray atomic coordinates in the immediate vicinity of a metal ion can sometimes be compromised by phase problems, and AlphaFold is not yet optimized for the structural modeling of metalloproteins.

(5) High magnetic fields represent a challenge not only for relaxation (eqn (2) and (3)) but also because of the span in frequencies that are associated with large hyperfine shifts. Efforts are currently being devoted to the development of pulses that can efficiently cover the full spectral width,¹⁴⁴ as well as for methods that allow for reducing the massive phase distortion that occurs as a result of pulse imperfections and dead time.¹⁴⁵

(6) When discussing the use of high fields, it is mandatory to mention the use of solid-state NMR, where the absence/reduction of incoherent molecular tumbling yields an effective reduction of the Curie-spin relaxation^146,147 and, in parallel, the appearance of a “powder pattern” reflects the geometry and the anisotropy of the interaction of the nuclear spin with the “Curie Spin”, and encodes highly relevant structural/dynamical information.^{11,127,148–150}

The possibility of providing structural information in solution with an increasing level of detail, in conjunction with the possibility of characterizing biomolecular dynamics, is expected to further increase the number of users and applications of paramagnetic NMR, and to foster the development of computational tools and automated protocols for integrated data analysis. These advancements will keep paramagnetic NMR vital in the foreseeable future, contributing to solve challenging and important biological problems.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work has been supported by the Fondazione Cassa di Risparmio di Firenze and the Italian Ministero dell’Istruzione, dell’Università e della Ricerca through the “Progetto Dipartimenti di Eccellenza 2018–2022” to the Department of Chemistry “Ugo Schiff” of the University of Florence. The authors acknowledge the support and the use of resources of Instruct-ERIC, a landmark ESFRI project, and specifically the CERM/CIRMMP Italy center.

Notes and references

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