## 1. Introduction

With the development of NMR techniques that enabled the spectroscopic characterisation of larger biomolecules, conventional structural approaches that relied mainly on NOE-based distance information reached their limits due to increasing signal overlap and the need for deuteration, which reduced the number of observable ^{1}H–^{1}H NOE cross-peaks. Recent achievements in advanced sample preparation strategies, and the incorporation of alternative restraints, such as residual dipolar couplings,1 and paramagnetic data2 have overcome some of these limitations and provide a toolbox that can complement and, in some favourable cases, replace sparse NOE-based distance data. However, these approaches require sample modifications, such as the addition of external alignment media, in the case of residual dipolar couplings (RDCs), or the placement of covalently attached tags, in the case of paramagnetic data. In contrast, paramagnetic relaxation enhancements (PREs) obtained from soluble and freely diffusing agents [solvent PREs (sPREs)], such as ions, organic radicals or metal chelates yield long-range distance information that can be used in structural and dynamic characterisation of biomolecules and biomolecular complexes. Additionally, sPREs can be tuned by variation of probes and/or concentration. Although this technique has been known for several decades, a rapidly growing number of developments and applications have been published in recent years. Herein, we provide an overview of available sPRE probes, their application in structural and dynamic studies of biomolecules and their complexes, and potential future applications in biomolecular science.

Nuclear paramagnetism is mediated by the magnetic moment of unpaired electron spins. This electron gyromagnetic ratio is approximately 660 times larger than the gyromagnetic ratio of protons. Various paramagnetic effects have been exploited for NMR spectroscopy studies, of which the PRE, the pseudo-contact shift (PCS) and the RDC are the most commonly used paramagnetic data.2 The possibility of obtaining any of the data depends on the magnetic susceptibility tensor (*χ*) of the paramagnetic centre, reflecting the variation of its magnetic moment with different orientations of the molecule in the magnetic field2 (i.e. isotropic, no variation; anisotropic, variation) and the nature of the interaction between the paramagnetic agent and the co-solute diamagnetic molecule.3 In the case of soluble paramagnetic probes, the probe can either form a transient, non-specific, yet rotationally correlated, complex with the diamagnetic molecule (i.e. the biomolecule) or freely diffuse in solution. Depending on which of these interactions applies, either the inner- (rotationally correlated) or outer-sphere (purely diffusive) relaxation model is used to quantitatively describe the sPRE. Whereas the outer-sphere model has to be applied to certain small molecules, sPREs of biomolecules and their complexes are best described by the inner-sphere model.4

The PRE is governed by two mechanisms: the pure dipole–dipole Solomon–Bloembergen contribution5 and the Curie spin contribution.2, 6 The degree to which the relaxation enhancement is affected by either of these contributions depends on the relative magnitudes of the spin relaxation time of the electron, *T*_{1e}, the lifetime of the intermolecular complex, *τ*_{M}, and the rotational tumbling of the solute molecule, *τ*_{R} [Eq. (1)].(1)

The Curie relaxation component, also known as *χ* relaxation, only becomes important if *T*_{1e} is at least four orders of magnitude shorter than the rotational correlation time, *τ*_{R}.7 In the case of most ideal solvent paramagnetic probes, the Solomon–Bloembergen contribution predominates, in part, due to a long lifetime of the electronic spin state. In this case, assuming that the Curie spin contribution can be ignored, the PRE is defined as an additional contribution to relaxation [Eq. (2)]:

in which and are the longitudinal (*i*=1) and transverse (*i*=2) relaxation rates for the biomolecule in the diamagnetic and paramagnetic states, respectively.

The overall Γ_{1} relaxation rate of the nucleus can be written as (inner-sphere relaxation model) shown in Equation (3):3b(3)

and the Γ_{2} relaxation rate of the nucleus can be written as shown in Equation (4):

in which *μ*_{0} is the magnetic permeability of a vacuum, *γ*_{I} is the gyromagnetic ratio of the spin of interest, *g _{J}* is the Landé factor,

*μ*

_{B}is the Bohr magneton,

*J*is the total angular momentum of the paramagnetic centre (e.g.

*J*=7/2 for Gd

^{3+}),

*r*is the distance between the nucleus and the paramagnetic centre (Figure 1),

*τ*

_{1C}is the effective correlation time for longitudinal relaxation,

*τ*

_{2C}is the effective correlation time for transverse relaxation, and

*ω*

_{I}and

*ω*

_{S}are the nuclear and electronic Larmor frequencies, respectively, with the approximation that

*ω*

_{S}≫

*ω*

_{I}.

In cases for which the paramagnetic agent and the co-solute do not interact, a purely diffusive model, also referred to as the outer-sphere model,8 needs to be considered in which the observed PRE effects are only dependent on the steric accessibility of the observed nuclei and the molar concentration of the paramagnetic agent.3b

In this model, Γ_{1} and Γ_{2} can be written as shown in Equations (5) and (6):(6)

in which *j _{j}*(

*ω*) is defined by Equation (7):

*N*_{A} is the Avogadro number, [S] is the molar concentration of electron spins, *P* is a steric factor that accounts for the accessibility of the nuclear spin, *T*_{jS} is the electron spin relaxation time, *b* is the distance of closest approach between the electron and nuclear spin, *D* is the relative translational diffusion constant, *τ*=*b*^{2}/*D* is the diffusion correlation time, and Re represents the real component of the spectral density function *j _{j}*(

*ω*). Notably, in some cases, predictions based on the diffusional relaxation model might not correlate well with experimental data due to the approximation that treats molecules as rigid and spherical and neglects electrostatic interactions.4a

The dipole–dipole relaxation mechanism between the lone electron and the nucleus, as described in the Solomon–Bloembergen contribution, increases the relaxation rate of the nucleus. Because the resonance linewidth is proportional to the transverse relaxation rate *R*_{2}, resonances of nuclei in proximity to the paramagnetic centre will broaden. An increase in *R*_{1} leads to faster recovery of magnetisation. For a single paramagnetic probe, and assuming the inner-sphere relaxation model, this effect has a *r*^{−6} distance dependence [Eq. (8)]. The constant *a _{j}* accounts for the combination of all terms in either Equation (3) or (4), depending on the type of relaxation [Eq. (8)]:4b(8)

For sPREs, under conditions that approximate the inner-sphere model, the sPRE values are determined from the integral over the entire solvent volume. The sPRE can be analytically derived for a few special cases, such as for a planar surface, including, to a good approximation, the surface of a large spherical system, such as a micelle. By using a coordinate transformation to spherical coordinates, the volume integration yields Equation (9):

in which *θ* and *φ* are the angles in the spherical coordinate system, *r* is the distance between the nucleus and the paramagnetic centre, *d* is the sum of the distance of the nucleus from the surface of the sphere and the radius of the sphere, and *r*_{para} is the hydrodynamic radius of the paramagnetic compound. Although the sPRE drops off rapidly with increasing distance, much larger distances (>15 Å) can be extracted than those measurable by more conventionally used ^{1}H–^{1}H NOE.4b Because the sPRE scales linearly with the concentration of the paramagnetic probe, higher distances can be resolved with higher concentrations of the probe and are, in principle, only bound by the solubility limit of the probe itself.

Back-calculation (or prediction) of sPREs is an essential step in assessing the accuracy of a sPRE-derived model. Pintacuda and Otting predicted *R*_{1} values from Equation (2) by using a grid based approach, in which the effective distance, *r*, between each protein proton and all probe accessible sites within 10 Å of the protein was determined.4a This effective distance is derived from the average value of *r _{i}*

^{−6}in a given NMR conformer, where

*r*is the distance between a proton and a grid point

_{i}*i*(1 Å point spacing). This method can be implemented in a computationally fast way (Hartlmüller, Madl, unpublished data) and is generally applicable to a wide variety of applications, including structure validation, structure calculation and docking of biomolecular complexes. Varrazzo et al. found a weak correlation between the atom depth and its paramagnetic attenuation that was restricted mainly to the innermost atoms of a biomolecule.9 In the protocol used by Tjandra et al. (see below) the back-calculation of sPRE values (Γ) was obtained from an implicit energy function relating a metric for solvent accessibility of a nucleus to its sPRE value.10 This metric can be readily derived from each NMR conformer because it directly reflects the level of ‘crowdedness’ of a nucleus from neighbouring heavy atoms [see Eq. (11)]. The great advantage of this approach is that the formulae can be differentiated and used in molecular dynamics based structure calculations (see Section 3. Applications of sPREs).