2.1.1 Types of paramagnetic observables

Unpaired electrons have dramatic effects on NMR spectra, and these paramagnetic effects are useful both as long-distance structural restraints and as probes of sparsely populated species. Here we will discuss two fundamental types of paramagnetic phenomena in NMR, PRE and hyperfine shifts (i.e. PCSs), as well as additional NMR observables that are influenced by the presence of a paramagnetic center, including RDCs, relaxation dispersion, and CEST.

PRE arises from dipolar coupling between a nucleus (primarily ¹H in solution NMR) and an unpaired electron (or collection of unpaired electrons). The PRE is proportional to the population-averaged 〈r ⁻⁶〉 distance between the paramagnetic center and the nucleus. This distance dependence, coupled with the large magnitude of the PRE (owing to the large magnetic moment of an electron), makes it a useful long-distance structural restraint, providing distance information of up to ~20 Å for a nitroxide tag and ~35 Å for some metal ions (such as Mn²⁺ or Gd³⁺) with multiple unpaired electrons. The magnitude of the PRE is proportional to the spin quantum number s ( = 1/2 for each unpaired electron). Whereas PRE is an isotropic effect, PCSs, paramagnetic RDCs, and paramagnetic contributions to relaxation dispersion and CEST arise from a paramagnetic center with an anisotropic g-tensor. For measuring PRE data, it is thus desirable to employ a paramagnetic group with an isotropic g-tensor, eliminating these other paramagnetic effects. An isotropic g-tensor results from a symmetric arrangement of unpaired electrons; examples include nitroxides (s = 1/2), Mn²⁺ (s = 5/2, i.e. half-filled 3d-subshell), and Gd³⁺ (s = 7/2, i.e. half-filled 4f-subshell).

Owing to the large magnitude of the paramagnetic effect and its 〈r ⁻⁶〉 distance dependence, PRE values are extremely large at short distances. Because of this, under the right circumstances, the PRE profile from a major (visible) species will include contributions from minor (otherwise invisible) species, making PRE a powerful tool for characterizing such sparsely populated states (Anthis & Clore, Reference Anthis and Clore2013; Anthis et al. Reference Anthis, Doucleff and Clore2011; Bashir et al. Reference Bashir, Volkov, Ullmann and Ubbink2009; Bertini et al. Reference Bertini, Luchinat, Nagulapalli, Parigi and Ravera2012b; Fawzi et al. Reference Fawzi, Doucleff, Suh and Clore2010a; Iwahara & Clore, Reference Iwahara and Clore2006a; Iwahara et al. Reference Iwahara, Schwieters and Clore2004a; Tang et al. Reference Tang, Iwahara and Clore2006, Reference Tang, Schwieters and Clore2007, Reference Tang, Louis, Aniana, Suh and Clore2008b; Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006). Although the NOE, which is the basis of conventional NMR structure determination, also has an r ⁻⁶ distance dependence, the magnitude of the NOE is much smaller (limited to within ~6 Å); thus, NOE data do not sufficiently amplify minor state information, and sparsely populated states (<30% population) are not apparent in NOE data. The quantitative theoretical and practical aspects of PRE and its use as a probe of minor states are described further in Section 3.

In the presence of a paramagnetic center, the chemical shift of a nucleus may undergo a hyperfine shift, which is the sum of Fermi contact shift (FCS) and PCS contributions. The FCS is a through-bond effect resulting from the delocalization of unpaired electrons, and it occurs through direct contact (within a small number of chemical bonds) with the paramagnetic center (Bertini et al. Reference Bertini, Luchinat and Parigi2001a). Thus, the FCS is analogous to diamagnetic J coupling between nuclei. PCSs are much longer-range through-space dipolar effects that arise from paramagnetic centers with an anisotropic g-tensor. The PCS has an 〈r ⁻³〉 distance dependence and a strong orientational dependence. Thus, additional orientational information can be derived from PCSs, although the analysis of PCSs adds additional complications over the analysis of PRE data. PCSs can also be used in the same manner as PRE data to characterize minor states, although PCS data require a paramagnetic center with a fixed orientation relative to the observed nucleus due to the orientational dependence of the PCS. Whereas PCSs are only observed in the presence of a paramagnetic center with an anisotropic g-tensor, FCSs are also observed for isotropic g-tensors. However, due to their short range, FCSs are generally not useful for protein NMR.

The magnitude of the PCS is given by (Bertini et al. Reference Bertini, Luchinat and Parigi2002):

(2)$$\delta _{{\rm PCS}} = \displaystyle{1 \over {12}}r^{ - 3} \left\{ {{\rm \Delta} \chi _{{\rm ax}} (3\cos ^2 \theta {\rm -1) +} \displaystyle{3 \over 2}{\rm \Delta} \chi _{{\rm rh}} \sin ^2 \theta \;{\rm cos\; 2}{\varphi}} \right\},$$

where r is the distance between the paramagnetic metal and the observed nucleus, θ the angle between the metal/nucleus vector and the z-axis of the magnetic susceptibility tensor (χ, centered at the paramagnetic metal), and φ the angle between the projection of the metal/nucleus vector on the xy-plane and the x-axis of the χ-tensor. Δχ _ax and Δχ _rh are the axial and rhombic components, respectively, of the χ-tensor, which are defined as:

(3)$${\rm \Delta} \chi _{{\rm ax}} = \chi _{zz} - \displaystyle{1 \over 2}(\chi _{xx} + \chi _{yy} ),$$

(4)$${\rm \Delta} \chi _{{\rm rh}} = \chi _{xx} - \,\chi _{yy}. $$

The χ-tensor is related to the g-tensor by (Bertini et al. Reference Bertini, Luchinat and Parigi2002):

(5)$$\chi _{kk} = \displaystyle{{\mu _0 N_{\rm A} \mu _0 S(S + 1)} \over {3k_B T}}g_{kk}^2, $$

where s is the electron-spin quantum number, μ ₀ the permeability of free space, μ _B the Bohr magneton (the magnetic moment of a free electron), N _A Avogadro's number, k _B the Boltzmann constant, T the temperature, and g _kk an element of the g-tensor (k = x, y, or z).

The presence of a paramagnetic center with an anisotropic g-tensor will also cause partial alignment with the magnetic field. This gives rise to RDCs, given by (Bertini et al. Reference Bertini, Luchinat and Parigi2002):

(6)$$D_{{\rm AB}} = \displaystyle{{hB_0^2 \gamma _{\rm A} \gamma _{\rm B}} \over {240\pi ^3 kT}}r_{{\rm AB}}^{ - 3} \left\{ {{\rm \Delta} \chi _{{\rm ax}} (3\cos ^2 \theta - 1) + \displaystyle{3 \over 2}{\rm \Delta} \chi _{{\rm rh}} \sin ^2 \theta \cos 2\varphi} \right\},$$

where D _AB is the RDC between nuclei A and B, r _AB the distance between the two nuclei, θ the angle between the internuclear vector and the z-axis of the χ-tensor, φ the angle between the projection of the internuclear vector on the xy-plane and the x-axis of the χ-tensor, B ₀ the static magnetic field strength, and h the Planck constant. Note that both the PCS and the paramagnetic RDC have an orientational dependence, whereas the PRE does not. Also, the PCS has an r ⁻³ dependence on the distance between the observed nucleus and the paramagnetic center (compared with r ⁻⁶ for PRE). Thus use of a combination of PCS, RDC, and PRE data (in addition to data from different paramagnetic metals) can help resolve any degeneracy in paramagnetic data (Fragai et al. Reference Fragai, Luchinat, Parigi and Ravera2013).

Owing to the large magnitude of hyperfine chemical shifts induced by paramagnetic groups (up to ~10 ppm for ¹H PCSs in proteins, >100 ppm for atoms directly bound to paramagnetic metal ions), they can facilitate the measurement of other chemical shift-dependent observables. In the case of intermediate chemical exchange (see the next section), PCSs can lead to exchange broadening, which can be characterized by relaxation dispersion (Eichmuller & Skrynnikov, Reference Eichmuller and Skrynnikov2007; Hass et al. Reference Hass, Keizers, Blok, Hiruma and Ubbink2010; Wang et al. Reference Wang, Srisailam, Yee, Lemak, Arrowsmith, Prestegard and Tian2007). Hyperfine shifts also facilitate the measurement of CEST (Vinogradov et al. Reference Vinogradov, Sherry and Lenkinski2013), an effect that has proven useful in magnetic resonance imaging (MRI), where compounds have been developed that cause hyperfine shifts and can be used as contrast agents in paramagnetic CEST (paraCEST) imaging (Hancu et al. Reference Hancu, Dixon, Woods, Vinogradov, Sherry and Lenkinski2010; Vinogradov et al. Reference Vinogradov, Sherry and Lenkinski2013). These CEST and paraCEST agents exist in addition to the more extensively used collection of PRE-based MRI contrast agents, where the PRE from a small soluble paramagnetic molecule or ion (e.g. chelated Gd³⁺) locally enhances signal (most commonly, in T ₁-weighted measurements; or, the PRE decreases the signal in T ₂-weighted measurements) (Caravan et al. Reference Caravan, Ellison, Mcmurry and Lauffer1999).

2.1.2 Practical aspects of PRE

The use of paramagnetic restraints in biomolecular NMR was previously restricted primarily to metalloproteins, where a paramagnetic cation can easily replace the natural metal ligand (Bertini et al. Reference Bertini, Luchinat and Piccioli2001b; Cheng & Markley, Reference Cheng and Markley1995; Ubbink et al. Reference Ubbink, Worrall, Canters, Groenen and Huber2002). Although the underlying theory of PRE was originally developed in the 1950s (Bloembergen & Morgan, Reference Bloembergen and Morgan1961; Solomon, Reference Solomon1955) and the utility of PRE restraints for macromolecular structure determination was demonstrated in the 1980s (Kosen et al. Reference Kosen, Scheek, Naderi, Basus, Manogaran, Schmidt, Oppenheimer and Kuntz1986; Schmidt & Kuntz, Reference Schmidt and Kuntz1984), only recently have PRE measurements become practically useful and commonly performed due to the development of straightforward biochemical methods for site-specific introduction of paramagnetic labels in proteins (Battiste & Wagner, Reference Battiste and Wagner2000; Donaldson et al. Reference Donaldson, Skrynnikov, Choy, Muhandiram, Sarkar, Forman-Kay and Kay2001; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000). A wide range of chemistries are available for conjugating a paramagnetic tag (either a nitroxide or a metal ion-bound chelator) to a protein (Hubbell et al. Reference Hubbell, Lopez, Altenbach and Yang2013; Jeschke, Reference Jeschke2013; Keizers & Ubbink, Reference Keizers and Ubbink2011); most common are those that involve a nitroxide group or a paramagnetic metal ion chelated by EDTA and are conjugated to the protein via an exposed cysteine (Kosen, Reference Kosen1989). The inherent flexibility of these tags was previously a hurdle for the quantitative analysis of PRE data, although these issues have largely been overcome through improved computational methods (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). More rigid tags have also been developed (Fawzi et al. Reference Fawzi, Fleissner, Anthis, Kalai, Hideg, Hubbell and Clore2011a; Keizers et al. Reference Keizers, Desreux, Overhand and Ubbink2007), and the use of such rigid tags is essential for measuring PCSs. Although PRE values are also affected by internal motion, they do not, like PCSs, have an orientational dependence, thus simplifying analysis and enabling the use of tags with flexible linkers.

The PRE is measured by taking the difference in relaxation rate between a paramagnetic state and a control diamagnetic state. PRE can be measured as either an enhancement of the longitudinal (Γ₁) or transverse (Γ₂) relaxation rate, although the measurement of Γ₂ is advantageous (Clore & Iwahara, Reference Clore and Iwahara2009; Clore et al. Reference Clore, Tang and Iwahara2007) because its larger magnitude makes it a more sensitive probe and because it exhibits a simple dependence of the order parameter (S²) and is largely independent of the correlation time (τ _i) of internal motion (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). Moreover, ¹H transverse relaxation is a single-exponential process, whereas ¹H longitudinal relaxation is multi-exponential due to cross-relaxation and, in the case of ¹H_N, water exchange, which results in perturbation of the observed ¹H–Γ₁ values. PRE values in a dynamic system cannot be directly interpreted as a single distance r, since both Γ₁ and Γ₂ depend upon the population-averaged value of 〈r ⁻⁶〉. This difficulty with the quantitative analysis of PRE data has been addressed by the development of new theoretical and computational methods that use an ensemble approach to take into account the different conformations sampled by a paramagnetic label on the surface of a protein, calculating PRE order parameters on the fly from atomic coordinates as a simulation proceeds (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b).

PRE data provide useful measurements of long distances in macromolecular structures (up to ~35 Å), and hundreds of individual distances can be measured from a single spin-label site (compared with FRET or DEER, where each doubly labeled sample only yields a single pairwise distance). As such, PRE data have been used to characterize the structures of a variety of macromolecules (Clore & Iwahara, Reference Clore and Iwahara2009), including soluble proteins (Battiste & Wagner, Reference Battiste and Wagner2000; Donaldson et al. Reference Donaldson, Skrynnikov, Choy, Muhandiram, Sarkar, Forman-Kay and Kay2001; Dvoretsky et al. Reference Dvoretsky, Gaponenko and Rosevear2002; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000), membrane proteins (Liang et al. Reference Liang, Bushweller and Tamm2006; Roosild et al. Reference Roosild, Greenwald, Vega, Castronovo, Riek and Choe2005), and proteins in complex with other proteins (Card et al. Reference Card, Erbel and Gardner2005; Gross et al. Reference Gross, Moerke, Von Der Haar, Lugovskoy, Sachs, Mccarthy and Wagner2003; Mahoney et al. Reference Mahoney, Rastogi, Cahill, Girvin and Almo2000; Mal et al. Reference Mal, Ikura and Kay2002; Rumpel et al. Reference Rumpel, Becker and Zweckstetter2008), nucleic acids (Iwahara et al. Reference Iwahara, Anderson, Murphy and Clore2003, Reference Iwahara, Schwieters and Clore2004b; Ramos & Varani, Reference Ramos and Varani1998; Ueda et al. Reference Ueda, Kato, Ogawa, Torizawa, Kuramitsu, Iwai, Terasawa and Shimada2004; Varani et al. Reference Varani, Gunderson, Mattaj, Kay, Neuhaus and Varani2000), and oligosaccharides (Jain et al. Reference Jain, Venot, Umemoto, Leffler and Prestegard2001; Johnson et al. Reference Johnson, Brun, Mackenzie, Withers and Mcintosh1999; Macnaughtan et al. Reference Macnaughtan, Kamar, Alvarez-Manilla, Venot, Glushka, Pierce and Prestegard2007). Long-distance PRE restraints do not have the same power to constrain a molecular structure as a series of short-distance NOE measurements between atoms near in three-dimensional (3D) space (but not linear sequence), but these long-distance measurements complement these NOE-based short-distance measurements, and they are particularly important in scenarios where measuring many short-range distances by NOEs is difficult (e.g. membrane proteins) (Gottstein et al. Reference Gottstein, Reckel, Dotsch and Guntert2012).

PRE has also found applications in solid-state NMR, where additional challenges have had to be overcome, but where successes have included the de novo calculation of the fold of the protein GB1 (Sengupta et al. Reference Sengupta, Nadaud, Helmus, Schwieters and Jaroniec2012). Owing to extremely rapid transverse relaxation rates in the solid state, it is technically difficult to measure Γ₂, so Γ₁ is generally measured, requiring a paramagnetic center that maximizes the Γ₁/Γ₂ ratio. Thus, while paramagnetic centers with a very slow electron relaxation times (large values of τ _s; e.g. nitroxides), are unsuitable due to a small Γ₁/Γ₂ ratio, paramagnetic centers with a more intermediate value of τ _s give a more favorable Γ₁/Γ₂ ratio and are more appropriate for PRE measurements in the solid state (Sengupta et al. Reference Sengupta, Nadaud and Jaroniec2013). The large magnitude of intermolecular PREs due to molecular packing is another concern that arises in the solid state and must be considered (Nadaud et al. Reference Nadaud, Sengupta, Helmus and Jaroniec2011). Beyond their utility for generating distance restraints, the paramagnetic groups can be used as doping agents to increase the signal-to-noise (S/N) ratio and decrease the total acquisition time needed for a solid-state NMR experiment (Wickramasinghe et al. Reference Wickramasinghe, Kotecha, Samoson, Past and Ishii2007).

2.1.3 PRE and sparsely populated states

PRE is uniquely able to characterize otherwise invisible sparsely populated states (Anthis & Clore, Reference Anthis and Clore2013; Anthis et al. Reference Anthis, Doucleff and Clore2011; Bashir et al. Reference Bashir, Volkov, Ullmann and Ubbink2009; Bertini et al. Reference Bertini, Luchinat, Nagulapalli, Parigi and Ravera2012b; Fawzi et al. Reference Fawzi, Doucleff, Suh and Clore2010a; Iwahara & Clore, Reference Iwahara and Clore2006a; Iwahara et al. Reference Iwahara, Schwieters and Clore2004a; Tang et al. Reference Tang, Iwahara and Clore2006, Reference Tang, Schwieters and Clore2007, Reference Tang, Louis, Aniana, Suh and Clore2008b; Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006) if two criteria are met: (1) the minor state must undergo rapid exchange; and (2) the distance between the paramagnetic center and some of the nuclei being measured must be shorter in the minor state than in the major state (Clore, Reference Clore2008; Clore & Iwahara, Reference Clore and Iwahara2009; Clore et al. Reference Clore, Tang and Iwahara2007). This is possible because a nucleus in close proximity to a paramagnetic center will exhibit an extremely large PRE (>1000 s⁻¹), and the contribution from the close-proximity state will dominate the overall PRE measured for that nucleus, even if it constitutes a minor percentage of the total population (Fig. 2).

Fig. 2.

Characterizing a sparsely populated state by PRE. If a major state (A) is in exchange with a minor state (B), the effect of the minor state can be amplified by PRE if two points (corresponding to the paramagnetic center and observed nucleus) are in closer proximity in the minor state than in the major state. (a) In this example, the protein is in exchange between an extended state (A, r = 40 Å) and a compact state (B, r = 8 Å). If a nitroxide spin label is attached to one point, the PRE measured for a proton at the other point would be virtually zero in state A (Γ_2,A = 0·12 s⁻¹), but very large in state B (Γ_2,B = 1900 s⁻¹) causing broadening of the observed NMR signal. PRE values were calculated assuming a correlation time of τ _c = 10 ns (corresponding to a protein at room temperature with a molecular weight of ~20 kDa). Spectra were calculated by solving the McConnell equations (Eqs. (16)–(20) and (57)) in MATLAB, assuming R _2,dia = 10 s⁻¹ in each state (the height of the spectrum for state B [green] has been magnified 10⁷ times for the purposes of visualization). If states A and B are in fast exchange on the PRE timescale (in this example: k _ex ≫ Γ_2,A − Γ_2,B = 1900 s⁻¹; τ _ex ≪ 500 μs), the apparent PRE will be the population-averaged PRE of the two states (Γ₂^app = 0·99 × 0·12 s⁻¹ + 0·01 × 1900 s⁻¹ = 19·12 s⁻¹). Thus, even although state B comprises only 1% of the total population, Γ₂^app arises almost exclusively from this minor state. (b) If the exchange rate is slower, the apparent PRE measured for A will be less than the population-weighted averaged PRE. Shown in black are simulated spectra for the system undergoing exchange with a rate of k _ex = 10², 10³, or 10⁴ s⁻¹. The dashed blue spectrum corresponds to the paramagnetic spectrum of A alone, and the dashed red spectrum corresponds to the population-weighted averaged paramagnetic spectrum under very fast exchange conditions (10⁶ s⁻¹). (c) Apparent PRE plotted against k _ex. The PRE increases from 0·12 s⁻¹ for a system undergoing very slow exchange (the value for state A alone) to 19·12 s⁻¹ for a system undergoing very fast exchange.

With respect to the first (kinetic) requirement, the exchange must be fast on the PRE timescale, the latter meaning that the minor state exchanges with the major state at a rate that is much greater than the difference in PRE rates between the two states (k _ex ≫ Γ₂^minor − Γ₂^major). In practice, this generally requires a timescale of τ _ex < 250 μs (k _ex > 1000 s⁻¹). With respect to the second (structural) requirement, a shorter distance in the minor state could arise from two molecules forming a transient complex in the minor state or a conformational change that brings two parts of a molecule closer together in the dark state. In such a system, the overall PRE observed is an 〈r ⁻⁶〉 population-weighted average of the PREs for all conformations in exchange. Because the PRE can be extremely large at close distances, even a very small population can cause a PRE to be much greater than the very small near-zero value that would be observed for a nucleus located far from a paramagnetic center. Under these circumstances, one can not only detect, but also characterize, the structural details of any minor state(s) giving rise to PREs, even those with populations of <1%. In fact, PREs are the most robust and comprehensive method for providing structural information on minor states. No other technique provides direct structural information on minor states, although others can provide some indirect information (through chemical shifts or RDCs) and can provide more kinetic information than PRE. A limitation, however, is that PRE data offer less insight into the kinetics of an exchange process, due to the difficulty of deconvoluting the contribution of different parameters (distance, exchange rate, and population) in systems undergoing exchange on the intermediate PRE timescale.

PRE studies have been used to gain insight into the transient states underlying a range of biomolecular processes, including protein/DNA interactions, protein/protein interactions, and interdomain interactions within proteins. For example, PRE data have revealed that the A-box of HMGB-1 interacts non-specifically with DNA, relevant to its role as a DNA-bending protein (Iwahara et al. Reference Iwahara, Schwieters and Clore2004a). PRE studies have characterized the inter-DNA-strand and intra-DNA-strand (‘sliding’) translocation of the transcription factors HoxD9 (Iwahara & Clore, Reference Iwahara and Clore2006a; Iwahara et al. Reference Iwahara, Zweckstetter and Clore2006), Oct1 (Takayama & Clore, Reference Takayama and Clore2011, Reference Takayama and Clore2012), and Sox2 (Takayama & Clore, Reference Takayama and Clore2012), revealing transient interactions with non-specific DNA sequences that facilitate the formation of the specific complex. The molecular details of the encounter complex between Enzyme I (EI) and the histidine-containing phosphocarrier protein (HPr), which plays a central role in the bacterial phosphotransferase system, have been elucidated (Fawzi et al. Reference Fawzi, Doucleff, Suh and Clore2010a; Tang et al. Reference Tang, Iwahara and Clore2006). PRE measurements have also demonstrated that cytochrome c and cytochrome c peroxidase form an encounter complex that samples conformational space near, although not identical to, the major species observed by X-ray crystallography, offering fundamental insight into the structural basis of electron transport (Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006, Reference Volkov, Ubbink and Van Nuland2010). PRE studies have demonstrated that in the absence of a sugar ligand, maltose-binding protein (MBP) samples a partially closed state ~5% of the time that is similar – although not identical – to its closed sugar-bound state (Tang et al. Reference Tang, Schwieters and Clore2007) and in the absence of a peptide target, the calcium-sensing protein calmodulin (CaM) adopts a wide range of conformations, including structures similar in conformation to the more rigid peptide-bound complex (Anthis et al. Reference Anthis, Doucleff and Clore2011). These studies have offered a novel view into otherwise invisible states, providing fundamental insights into the process of molecular recognition and the complementary roles of conformational selection and induced fit.

2.2.1 Chemical exchange

If a molecule is in exchange between two states:

(7)$$A {\mathop{\rightleftharpoons}_{k_{{\rm BA}}}^{k_{{\rm AB}}}} B,$$

then the number of NMR peaks (as well as their height and linewidth) observed for a given nucleus will depend on the chemical shift difference (Δω) between the two states and on the exchange rate constant (k _ex = k _AB + k _BA; the kinetics of exchange can also be described by the exchange lifetime, defined by τ _ex = 1/k _ex.) (Fig. 1). If Δω (given in angular frequency (rad·s⁻¹), related to Δν (Hz) by Δω = 2πΔν) is much (>10 times) larger than k _ex (the slow chemical shift exchange regime), two distinct peaks will be observed for the resonance, at the chemical shifts of the individual states, with peak volumes proportional to the populations of the individual states. If k _ex is instead much (>10×) larger than Δω, a single peak will be observed at the population-weighted mean chemical shift; this is the fast chemical shift exchange regime. In the intermediate chemical shift exchange regime, however, the ratio of k _ex/Δω takes a value close to 1 (±one order of magnitude), and line broadening is observed due to an increase in the observed transverse relaxation rate (R ₂^obs), which is the sum of the intrinsic transverse relaxation rate (R ₂⁰) and an exchange contribution (R _ex) that depends upon k _ex and Δω (Palmer & Massi, Reference Palmer and Massi2006; Palmer et al. Reference Palmer, Kroenke and Loria2001, Reference Palmer, Grey and Wang2005):

(8)$$R_2^{{\rm obs}} = R_2^{\rm 0} + R_{{\rm ex}}. $$

The value of R _ex becomes insignificant in the extreme limits of the fast and slow exchange regimes. For certain conditions, simple expressions of R _ex can be constructed for systems undergoing two-state exchange. In the slow exchange regime (k _ex ≪ Δω), R _ex is given by (Palmer et al. Reference Palmer, Kroenke and Loria2001):

(9)$$R_{{\rm ex,A}} = k_{{\rm AB}} = (1 - p_{\rm A} )k_{{\rm ex}} = p_{\rm B} k_{{\rm ex}}, $$

(10)$$R_{{\rm ex,B}} = k_{{\rm BA}} = (1 - p_{\rm B} )k_{{\rm ex}} = p_{\rm A} k_{{\rm ex}}, $$

where R _ex,A is R _ex measured for state A, R _ex,B is R _ex measured for state B, p _A the population of state A (the observed state), and p _B = 1 − p _A the population of state B (the state in exchange with state A). There are two different values of R _ex in the slow exchange regime because a separate peak is seen for each state. In the fast exchange regime (k _ex ≫ Δω), R _ex is given by (Palmer et al. Reference Palmer, Kroenke and Loria2001):

(11)$$R_{{\rm ex}} = \displaystyle{{\,p_{\rm A} p_{\rm B} {\rm \Delta} \omega ^2} \over {k_{{\rm ex}}}}. $$

Thus, while R _ex is independent of magnetic field strength (B ₀) in the slow exchange regime, R _ex depends on the square of the magnetic field strength (i.e. Δω ²) in the fast exchange regime.

If the population of the minor state is very small (p _A ≫ p _B), the following holds for all timescales (Swift & Connick, Reference Swift and Connick1962):

(12)$$R_{{\rm ex}} = \displaystyle{{\,p_{\rm A} p_{\rm B} k_{{\rm ex}}} \over {{\rm 1 + (}k_{{\rm ex}} /{\rm \Delta} \omega {\rm )}^2}} = \displaystyle{{\,p_{\rm A} p_{\rm B} k_{{\rm ex}} {\rm \Delta} \omega ^2} \over {{\rm \Delta} \omega ^2 + k_{{\rm ex}}^2}}. $$

Under these conditions, the exchange-induced chemical shift (ω _ex) of state A – which is the difference in position of the peak in the exchanging system compared with its position in the absence of exchange – is given by:

(13)$$\omega _{{\rm ex}} = \displaystyle{{\,p_{\rm B} k_{{\rm ex}}^2 {\rm \Delta} \omega} \over {{\rm \Delta} \omega ^2 + k_{{\rm ex}}^2}}. $$

If states A and B have different intrinsic transverse relaxation rates (ΔR ₂⁰ = R _2,B⁰ − R _2,A⁰ > 0), then, for the case where p _A ≫ p _B, R _ex is given by (Skrynnikov et al. Reference Skrynnikov, Dahlquist and Kay2002):

(14)$$R_{{\rm ex}} = k_{{\rm AB}} \displaystyle{{{\rm \Delta} R_2^0 \left( {k_{{\rm BA}} + {\rm \Delta} R_2^0} \right) + {\rm \Delta} \omega ^2} \over {\left( {k_{{\rm BA}} + {\rm \Delta} R_2^0} \right)^2 + {\rm \Delta} \omega ^2}}. $$

Under these same conditions, the exchange-induced chemical shift is (Skrynnikov et al. Reference Skrynnikov, Dahlquist and Kay2002):

(15)$$\omega _{{\rm ex}} = k_{{\rm AB}} \displaystyle{{k_{{\rm BA}} {\rm \Delta} \omega} \over {\left( {k_{{\rm BA}} + {\rm \Delta} R_2^0} \right)^2 + {\rm \Delta} \omega ^2}}. $$

For a complete description of an exchanging system, it is necessary to invoke the McConnell equations (McConnell, Reference Mcconnell1958), a modified form of the Bloch equations (Bloch, Reference Bloch1946) extended to incorporate chemical exchange. Transverse relaxation in an exchanging system can be described by the following matrix equation:

(16)$$\displaystyle{{\rm d} \over {{\rm d}t}}{\bf M =} {\rm - (}{\bf R + K}-i{\bf \Omega} {\rm )}{\bf M}{\rm,} $$

where M is the vector describing the transverse magnetization of each state, R the matrix describing the transverse relaxation rates of each state, K the matrix describing the kinetic rates of exchange between each state, and Ω the matrix describing the chemical shift of each state. For a system undergoing two-site exchange between states A and B, these are given by:

(17)$${\bf M =} \left[ {\matrix{ {M_{\rm A}^ +} \cr {M_{\rm B}^ +} \cr}} \right],$$

(18)$${\bf R =} \left[ {\matrix{ {R_{2,{\rm A}}} & 0 \cr 0 & {R_{2,{\rm B}}} \cr}} \right],$$

(19)$${\bf K =} \left[ {\matrix{ {k_{{\rm AB}}} & { - k_{{\rm BA}}} \cr { - k_{{\rm AB}}} & {k_{{\rm BA}}} \cr}} \right],$$

(20)$${\bf \Omega} = \left[ {\matrix{ {{\rm \Omega} _{\rm A}} & 0 \cr 0 & {{\rm \Omega} _{\rm B}} \cr}} \right],$$

where M ⁺ is the transverse magnetization of each state (represented by a complex variable Mx + iMy) and Ω/2π is the chemical shift of each state. The McConnell equations can be solved (analytically for simple two-state cases (Palmer et al. Reference Palmer, Kroenke and Loria2001) or numerically for all cases by integration or by calculating the matrix exponential) to provide a time course of the magnetization (i.e. free induction decay):

(21)$${\bf M}{\rm (}t{\rm )} = {\bf A}\;{\rm \times}\; {\bf M}{\rm (0),}$$

where M(t) is the transverse magnetization vector M at time t, M(0) is the initial value of M at t = 0, and A(t) is a matrix of time-dependent elements determined by solving the McConnell equations. Fourier transformation of Eq. (21) yields a frequency-domain NMR spectrum, from which apparent relaxation rates can be extracted by line shape fitting.

2.2.2 CPMG and rotating frame relaxation dispersion

Even a very small population of an otherwise invisible second state (as long as p _b ≥ 0.5%) can cause significant broadening of NMR peaks under conditions of intermediate exchange (in general τ _ex ~100 μs–10 ms), allowing characterization by relaxation dispersion NMR and insight into the otherwise invisible minor state (Baldwin & Kay, Reference Baldwin and Kay2009; Korzhnev & Kay, Reference Korzhnev and Kay2008; Loria et al. Reference Loria, Berlow and Watt2008; Mittermaier & Kay, Reference Mittermaier and Kay2009; Neudecker et al. Reference Neudecker, Lundstrom and Kay2009; Palmer et al. Reference Palmer, Kroenke and Loria2001), using CPMG or R _1ρ experiments. These experiments are performed by measuring the apparent R ₂^obs of a nucleus (usually ¹⁵N or ¹³C, but also ¹H) while varying the repetition rate of 180° pulses (CPMG) or the strength of a spin lock field (R _1ρ) to attenuate the contribution of R _ex to the overall measured R ₂ rate (Palmer et al. Reference Palmer, Grey and Wang2005).

In a CPMG relaxation dispersion experiment (Fig. 3), the effective observed R ₂ (R _2,CP^obs) is measured as a function of the CPMG field (ν _CP [Hz]), which is the frequency of repetition pairs of 180° CPMG pulses and is related to τ _CP [s], the delay between 180° CPMG pulses, by:

(22)$$\nu _{{\rm CP}} = \displaystyle{1 \over {2\tau _{{\rm CP}}}}. $$

Fig. 3.

Characterizing exchange with a ‘dark’ state by CPMG relaxation dispersion. Relaxation dispersion curves were simulated in MATLAB, showing the apparent observed transverse relaxation rate (R ₂^obs) as a function of the repetition rate of 180° CPMG pulses (ν _CPMG, 0–1 kHz). Curves were calculated by solving the McConnell equations in MATLAB using Eq. (31). Curves are shown for range of exchange rates (k _ex = 10²–10⁴ s⁻¹) for a system where a visible species (p _A = 95%) is in exchange with a dark state (p _B = 5%), analogous to a ¹⁵N resonance with a chemical shift difference between states A and B of Δδ = 2 ppm. The blue curves correspond to a static B ₀ field strength of 600 MHz (Δω/2π = 120 Hz), and the red curves correspond to 900 MHz (Δω/2π = 180 Hz). At a large ν _CPMG, R ₂^obs approaches the intrinsic transverse relaxation rate R ₂⁰ (set here for both states to 10 s⁻¹ at 600 MHz and 13 s⁻¹ at 900 MHz). CPMG relaxation dispersion is most useful for characterizing exchange with a minor species when R _ex (= R ₂^obs[ν _CPMG = 0] − R ₂^obs[ν _CPMG → ∞]) is largest, which occurs when k _ex = Δω (~1000 s⁻¹, middle panel).

At each CPMG field, the observed R ₂ is given by:

(23)$$R_{2,{\rm CP}}^{{\rm obs}} = R_2^0 + R_{{\rm ex,CP}}, $$

where R ₂⁰ is the intrinsic transverse relaxation rate in the absence of exchange and R _ex,CP is the contribution to transverse relaxation from chemical exchange broadening, which has been attenuated to some degree by CPMG pulses.

The parameters of interest are determined by fitting the CPMG data to a series of equations. Although the equations describing complicated systems involving three or more states can only be solved numerically, a system undergoing two-state exchange can be solved analytically (Carver & Richards, Reference Carver and Richards1972; Davis et al. Reference Davis, Perlman and London1994; Jen, Reference Jen1978; Palmer et al. Reference Palmer, Kroenke and Loria2001; Rule & Hitchens, Reference Rule and Hitchens2006):

(24)$$R_{2,{\rm CP}}^{{\rm obs}} = \displaystyle{1 \over 2}\left( {R_{2,{\rm A}}^0 + R_{2,{\rm B}}^0 + k_{{\rm ex}} - [2\nu _{{\rm CP}} ]\,\,\cosh ^{ - 1} \left[ {D_ + \cosh \,(\eta _ + ) - D_ - \cos \,(\eta _ - )} \right]} \right),$$

where

(25)$$D_{\rm \pm} = \displaystyle{1 \over 2}\left( {{\rm \pm} 1 + \displaystyle{{\psi + 2{\rm \Delta} \omega ^2} \over {[\psi ^2 + \zeta ^2 ]^{1/2}}}} \right),$$

(26)$$\eta _{\rm \pm} = \displaystyle{1 \over {\left( {2\nu _{{\rm CP}}} \right)\sqrt 2}} \left( {{\rm \pm} \psi + \left[ {\psi ^2 + \zeta ^2} \right]^{1/2}} \right)^{1/2}, $$

(27)$$\psi = \left( {R_{2,{\rm A}}^0 - R_{2,{\rm B}}^0 - p_{\rm A} k_{{\rm ex}} + p_{\rm B} k_{{\rm ex}}} \right)^2 - {\rm \Delta} \omega ^2 + 4p_{\rm A} p_{\rm B} k_{{\rm ex}}^2, $$

(28)$$\zeta = 2{\rm \Delta} \omega \left( {R_{2,{\rm A}}^0 - R_{2,{\rm B}}^0 - p_{\rm A} k_{{\rm ex}} + p_{\rm B} k_{{\rm ex}}} \right),$$

and where R _2,A⁰ is the R ₂ of state A in the absence of chemical exchange (i.e. ν _CP = ∞), R _2,B⁰ is the R ₂ of state B in the absence of chemical exchange, p _A is the population of state A, p _B is the population of state B, and Δω is the difference in chemical shift between states A and B.

Simplified expressions can be written for R _2,CP^obs under certain scenarios. In the fast exchange regime, the expression for R _2,CP^obs can be simplified to (Palmer et al. Reference Palmer, Kroenke and Loria2001):

(29)$$R_{2,{\rm CP}}^{{\rm obs}} = p_{\rm A} R_{2,{\rm A}}^0 + p_{\rm B} R_{2,{\rm B}}^0 + \left( {\displaystyle{{\,p_{\rm A} p_{\rm B} {\rm \Delta} \omega ^2} \over {k_{{\rm ex}}}}} \right)\left( {1 - \displaystyle{{2[2\nu _{{\rm CP}} ]} \over {k_{{\rm ex}}}} \tanh \left[ {\displaystyle{{k_{{\rm ex}}} \over {2\left( {2\nu _{{\rm CP}}} \right)}}} \right]} \right).$$

If p _A ≫ p _B, regardless of the timescale, the expression for R ₂^obs can be simplified to (Ishima & Torchia, Reference Ishima and Torchia1999):

(30)$$R_{2,{\rm CP}}^{{\rm obs}} = R_2^0 + \displaystyle{{\,p_{\rm A} p_{\rm B} {\rm \Delta} \omega ^2 k_{{\rm ex}}} \over {k_{{\rm ex}}^2 + \left( {\,p_{\rm A}^2 {\rm \Delta} \omega ^4 + 48\left[ {2\nu _{{\rm CP}}} \right]^4} \right)^{1/2}}}, $$

where R ₂⁰ is equal to R _2,A⁰ in the slow exchange regime and p _AR _2,A⁰ + p _BR _2,B⁰ in the fast exchange regime.

For systems more complex than a two-state system, R ₂^obs is not given by an analytical expression, but can only be calculated by numerical solution of the McConnell equations. For any system, the transverse magnetization at the end of a CPMG period (T _CP) consisting of an even number (2n) of 180° pulses, each separated by time τ _CP, is given by (Palmer et al. Reference Palmer, Kroenke and Loria2001):

(31)$${\bf M}{\rm (2}n\tau _{{\rm CP}} ) = \left( {{\bf AA}^* {\bf A}^* {\bf A}} \right)^n \;{\bf M}{\rm (0),}$$

where M(2nτ _CP) is the transverse magnetization vector at time T _CP = 2nτ _CP and A* is the complex conjugate of A (Eq. (21)) when the McConnell equations have been solved for t = τ _CP/2 (Palmer et al. Reference Palmer, Kroenke and Loria2001).

The CPMG relaxation dispersion curve has a complex shape that is a function of k _ex, the populations of the states undergoing exchange, and the chemical shift difference between these states. If the system is in exchange between two states, a simple model can be used to fit the data; more complicated multi-state models can then be employed if the two-site fit is not adequate, although one is generally limited to three-site exchange models to improve the fit without overfitting the data (Korzhnev & Kay, Reference Korzhnev and Kay2008; Li et al. Reference Li, Martins and Rosen2011; Mittermaier & Kay, Reference Mittermaier and Kay2009; Neudecker et al. Reference Neudecker, Lundstrom and Kay2009; Trott & Palmer, Reference Trott and Palmer2004). The contribution of the populations and chemical shift differences to R _ex cannot be separated a priori for data recorded at a single magnetic field. Thus, unless the populations or chemical shift differences are already known, one must record data at different B ₀ magnetic field strengths, because the populations of the species are independent of the magnetic field, while the differences in frequency are linearly proportional to the magnetic field.

A system in intermediate chemical exchange can also be characterized by rotating frame R _1ρ relaxation dispersion, fitting the effective R _1ρ ( = 1/T _1ρ) as a function of the effective spin-lock field strength (ω _e). The observed R _1ρ for a given spin-lock field (R _1ρ^obs) is given by the sum of R _1ρ in the absence of exchange (R _1ρ⁰) plus the contribution to R _1ρ from chemical exchange broadening (R _ex,1ρ), which has been attenuated to some degree by the spin-lock field:

(32)$$R_{1\rho} ^{{\rm obs}} = R_{1\rho} ^{\rm 0} + R_{{\rm ex,1}\rho}. $$

For two-state exchange in the fast exchange regime, the expression for R _1ρ^obs is given by (Davis et al. Reference Davis, Perlman and London1994; Deverell et al. Reference Deverell, Morgan and Strange1970; Meiboom, Reference Meiboom1961; Palmer et al. Reference Palmer, Kroenke and Loria2001; Rule & Hitchens, Reference Rule and Hitchens2006):

(33)$$R_{1\rho} ^{{\rm obs}} = R_{1\rho} ^0 + \displaystyle{{\,p_{\rm A} p_{\rm B} {\rm \Delta} \omega ^2 k_{{\rm ex}}} \over {k_{{\rm ex}}^2 + \omega _{\rm e}^2}} \sin ^2 \theta, $$

where R _1ρ⁰ is R _1ρ in the absence of chemical exchange (i.e. ω _e = ∞) and the angle between the B ₁ spin-lock field and the static B ₀ field is given by:

(34)$$\theta = \tan ^{ - 1} \displaystyle{{\omega _1} \over {\delta \omega}}, $$

where ω ₁ (rad·s⁻¹) is the strength of the spin-lock field and δω (rad·s⁻¹) is the frequency offset (the difference between the resonance frequency of the nucleus and the transmitter frequency). The effective spin-lock field strength is given by:

(35)$$\omega _{\rm e} = \sqrt {\omega _1^2 + \delta \omega ^2}. $$

If p _A ≫ p _B and the spin-lock field is on resonance (δω = 0), R _1ρ^obs can be expressed regardless of the timescale as (Meiboom, Reference Meiboom1961):

(36)$$R_{1\rho} ^{{\rm obs}} = R_{1\rho} ^0 + \displaystyle{{\,p_{\rm A}^2 p_{\rm B} {\rm \Delta} \omega ^2 k_{{\rm ex}}} \over {k_{{\rm ex}}^2 + p_{\rm A}^2 {\rm \Delta} \omega ^2 + \omega _1^2}}, $$

where R _1ρ in the absence of chemical exchange in the slow exchange regime is given by:

(37)$$R_{1\rho} ^{\rm 0} = R_{1,{\rm A}}^0 \cos ^2 \theta + R_{2,{\rm A}}^0 \sin ^2 \theta $$

and in the fast exchange regime is given by:

(38)$$R_{1\rho} ^0 = \left( {\,p_{\rm A} R_{1,{\rm A}}^0 + p_{\rm B} R_{1,{\rm B}}^0} \right)\cos ^2 \theta + \left( {\,p_{\rm A} R_{2,{\rm A}}^0 + p_{\rm B} R_{2,{\rm B}}^0} \right)\sin ^2 \theta. $$

For off-resonance spin-lock fields, as long as the population of the minor state is very small (p _A ≫ p _B), R _1ρ^obs can be expressed in a form analogous to Eq. (30) (Palmer, Reference Palmer2014):

(39)$$R_{1\rho} ^{{\rm obs}} = R_{1\rho} ^{\rm 0} + \displaystyle{{\,p_{\rm A} p_{\rm B} {\rm \Delta} \omega ^2 k_{{\rm ex}}} \over {k_{{\rm ex}}^2 + \omega _{{\rm e,A}}^2 \omega _{{\rm e,B}}^2 /\omega _{{\rm e,ave}}^2}}, $$

where ω _e,A is the effective spin-lock field (Eq. (35)) for the chemical shift of state A, ω _e,B for state B, and ω _e,ave is the effective spin-lock field for the population-averaged chemical shift of states A + B. If the intrinsic transverse relaxation rates differ between the two states (ΔR ₂⁰ = R _2,A − R _2,B > 0), R _1ρ^obs can be given by the following expression, which is accurate as long as p _A ≫ p _B (Palmer, Reference Palmer2014):

(40)$$R_{1\rho} ^{{\rm obs}} = R_{1\rho} ^{\rm 0} + k_{{\rm AB}} \displaystyle{{k_{{\rm BA}} \left( {{\rm \Delta} \omega ^2 + \left[ {{\rm \Delta} R_2^0} \right]^2} \right) + {\rm \Delta} R_2^0 \left( {\omega _{{\rm e,A}}^2 + k_{{\rm BA}}^2} \right)} \over {k_{{\rm BA}} \left( {{\rm \Delta} \omega _{{\rm e,B}}^2 + k_{{\rm BA}}^2 + \left[ {{\rm \Delta} R_2^0} \right]^2} \right) + {\rm \Delta} R_2^0 \omega _1^2}}. $$

A more comprehensive analytical expression is given by Baldwin & Kay (Reference Baldwin and Kay2013). Under all circumstances, though, values of R _1ρ^obs can be calculated by solving the McConnell equations using the same matrices as for a saturation transfer experiment (cf. Eqs. (82)–(84) below).

An advantage of R _1ρ experiments is that they can characterize faster exchange processes than CPMG experiments. Whereas CPMG data alone can be used to characterize exchange on a timescale of roughly ~500 μs–5 ms, R _1ρ relaxation dispersion experiments can measure exchange processes as fast as k _ex < 40 000 s⁻¹ (τ _ex > 25 μs) (Ban et al. Reference Ban, Gossert, Giller, Becker, Griesinger and Lee2012; Hansen et al. Reference Hansen, Nikolova, Casiano-Negroni and Al-Hashimi2009; Lundstrom & Akke, Reference Lundstrom and Akke2005; Weininger et al. Reference Weininger, Blissing, Hennig, Ahlner, Liu, Vogel, Akke and Lundstrom2013). The remainder of this section, however, focuses on CPMG relaxation dispersion.

2.2.3 Practical aspects of CPMG

From a series of CPMG experiments, one can determine the populations of the major and minor states, the rate of exchange between the states, and the magnitude of the chemical shift difference between the states, which offers indirect structural insight into the minor state. However, CPMG data alone only directly provide information on the magnitude, but not direction, of chemical shift differences. The direction of the chemical shift differences can be determined by either comparing the chemical shift of the resonance between heteronuclear single (HSQC) and multiple (HMQC) quantum coherence spectra or between spectra acquired at different B ₀ fields (Bouvignies et al. Reference Bouvignies, Korzhnev, Neudecker, Hansen, Cordes and Kay2010; Skrynnikov et al. Reference Skrynnikov, Dahlquist and Kay2002). As Δω is directly proportional to B ₀, moving from a lower to a higher B ₀ field has the effect of decreasing k _ex/Δω, thus slowing the exchange regime and moving the frequency of the resonance closer to its intrinsic chemical shift (slow exchange) and farther from the population-averaged chemical shift (fast exchange). The difference in chemical shift observed between HSQC and HMQC spectra arises from the fact that, for ¹⁵N for example, the exchange regime in the HSQC experiment is characterized by k _ex/Δω _N, but in the HMQC, exchange of zero-quantum and double-quantum magnetization are characterized by k _ex/(Δω _N − Δω _H) and k _ex/(Δω _N + Δω _H), respectively, which do not average linearly, thus altering the exchange regime and giving rise to a small difference in chemical shift between the HSQC and HMQC spectra (Skrynnikov et al. Reference Skrynnikov, Dahlquist and Kay2002). Likewise, if there is any direct knowledge of the chemical shift of the minor state from other measurements (e.g. ligand-binding studies, protein-folding/unfolding studies, concentration dependence, etc.), such information on the magnitude of chemical shift change can assist the CPMG fitting process.

The fitting of CPMG data is prone to difficulties, and distinguishing between k _ex, Δω, and p _B requires resonances undergoing exchange in the intermediate regime, with at least some satisfying the condition k _ex/Δω < 2; this limits the range of k _ex values measurable from CPMG alone to between ~200 and ~2000 s⁻¹. However, the incorporation of additional chemical shift information (e.g. from HSQC versus HMQC spectra) can extend this range further into the fast timescale, allowing fitting of data in the fast-intermediate exchange regime, up to k _ex/Δω ~ 4 (in practice, exchange rates up to k _ex ~ 6000 s⁻¹; τ _ex ~ 170 μs) (Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2011). The study of fast exchange processes by CPMG relaxation dispersion can be problematic in general, leading to large errors when the repetition rate (ν _CP = 1/τ _CP) is not much greater than k _ex. However, it has been demonstrated that if this condition is not satisfied, these errors can be reduced dramatically (from ~50 to ~5%) by measuring the intrinsic relaxation rate R ₂⁰ using an R _1ρ experiment with a high spin-lock field (Ban et al. Reference Ban, Mazur, Carneiro, Sabo, Giller, Koharudin, Becker, Gronenborn, Griesinger and Lee2013).

Relaxation dispersion experiments on proteins are most commonly performed on the backbone ¹⁵N nucleus, although measurements on ¹H_N, ¹³C_α, ¹H_α, ¹³C_β, and ¹³C^′ backbone nuclei and side chain methyl ¹H and ¹³C nuclei (Hansen et al. Reference Hansen, Vallurupalli, Lundstrom, Neudecker and Kay2008b; Ishima & Torchia, Reference Ishima and Torchia2003; Lundstrom et al. Reference Lundstrom, Vallurupalli, Religa, Dahlquist and Kay2007, Reference Lundstrom, Hansen, Vallurupalli and Kay2009a, Reference Lundstrom, Lin and Kay2009b) are also feasible. Minor state chemical shifts derived from CPMG studies can be used to calculate backbone torsion angle restraints for the minor state using a program such as TALOS-N (Shen & Bax, Reference Shen and Bax2013), and it is possible to calculate the structure of a small protein from backbone chemical shifts alone, using programs such as CHESHIRE (Cavalli et al. Reference Cavalli, Salvatella, Dobson and Vendruscolo2007) or CS-Rosetta (Shen et al. Reference Shen, Lange, Delaglio, Rossi, Aramini, Liu, Eletsky, Wu, Singarapu, Lemak, Ignatchenko, Arrowsmith, Szyperski, Montelione, Baker and Bax2008, Reference Shen, Vernon, Baker and Bax2009), which rely upon matching the empirical chemical shifts with a peptide fragment database. Such an approach can provide direct insight into the structure of a minor state (Korzhnev et al. Reference Korzhnev, Religa, Banachewicz, Fersht and Kay2010), although complications can arise if the minor state is partially unfolded, as the databases used for calculation of structures from chemical shifts are derived from folded proteins. It has also been demonstrated that the orientation of bond vectors in minor states can be measured through relaxation dispersion experiments (Hansen et al. Reference Hansen, Vallurupalli and Kay2008a; Vallurupalli et al. Reference Vallurupalli, Hansen, Stollar, Meirovitch and Kay2007, Reference Vallurupalli, Hansen and Kay2008a, Reference Vallurupalli, Hansen and Kayb). By performing relaxation dispersion experiments in weakly aligned media, one can measure RDCs and residual chemical shift anisotropy (RCSA) for the minor state, although highly accurate measurements are required to detect the small effect of the minor state on these observables. Additionally, the introduction of a paramagnetic group with an anisotropic g-tensor will magnify Δω if different PCSs are observed between the different states, potentially causing additional exchange broadening and facilitating CPMG relaxation dispersion studies. This strategy has been applied to intrinsic metalloproteins (Wang et al. Reference Wang, Srisailam, Yee, Lemak, Arrowsmith, Prestegard and Tian2007) and to otherwise diamagnetic proteins by incorporating a rigid metal-chelating paramagnetic tag to induce relaxation dispersion when none was apparent in the diamagnetic state of the protein (Eichmuller & Skrynnikov, Reference Eichmuller and Skrynnikov2007; Hass et al. Reference Hass, Keizers, Blok, Hiruma and Ubbink2010).

For larger proteins, the favorable spectral properties of methyl groups make methyl ¹³C, ¹H, and ²H (D) nuclei useful probes of dynamics (Igumenova et al. Reference Igumenova, Frederick and Wand2006; Ruschak & Kay, Reference Ruschak and Kay2010). Methods have been developed to express recombinant proteins in Escherichia coli with various methyl isotopic labeling schemes (Tugarinov et al. Reference Tugarinov, Kanelis and Kay2006). Favorable relaxation properties are seen in ¹³CDH₂-labeled methyl groups, and, for ¹³CH₃-labeled methyl groups on a perdeuterated background, acquiring data with an HMQC experiment instead of an HSQC experiment allows one to take advantage of the slower multiple quantum relaxation due to cancellation of dipolar interactions within the methyl group (i.e. the ‘methyl-TROSY’ effect), helping make large systems accessible to analysis by NMR (Tugarinov et al. Reference Tugarinov, Hwang, Ollerenshaw and Kay2003). Using these methods, ¹³C CPMG relaxation dispersion experiments have been performed to characterize intermediate dynamics in systems as large as the 82-kDa enzyme malate synthase G (Korzhnev et al. Reference Korzhnev, Kloiber, Kanelis, Tugarinov and Kay2004a), the 230-kDa HsIV component of the bacterial proteasome (Shi & Kay, Reference Shi and Kay2014), and the 670-kDa 20S core particle of the archaeal proteasome (Sprangers & Kay, Reference Sprangers and Kay2007). Using a labeling scheme that isolates ¹H spins, ¹H CPMG relaxation dispersion experiments can also be performed on methyl groups (Hansen et al. Reference Hansen, Lundstrom, Velyvis and Kay2012; Otten et al. Reference Otten, Villali, Kern and Mulder2010).

2.2.4 Applications of CPMG

CPMG relaxation dispersion is well suited for studying protein folding, due to its ability to identify minor states (i.e. folding intermediates) in exchange with a visible (folded or unfolded) species on an intermediate timescale (Fig. 3). These folding intermediates occupy local minima (i.e. ‘frustrations’) (Ferreiro et al. Reference Ferreiro, Cho, Komives and Wolynes2005) on an otherwise smooth, minimally frustrated protein-folding energy landscape (Bryngelson & Wolynes, Reference Bryngelson and Wolynes1987; Onuchic et al. Reference Onuchic, Lutheyschulten and Wolynes1997). CPMG relaxation dispersion studies on the folding of the SH3 domain from Fyn have identified an intermediate in the folding pathway (populated at ~1–2% at equilibrium), which has evaded detection by other methods and contains both native (central β-sheet) and non-native structural elements (Korzhnev et al. Reference Korzhnev, Salvatella, Vendruscolo, Di Nardo, Davidson, Dobson and Kay2004b, Reference Korzhnev, Neudecker, Zarrine-Afsar, Davidson and Kay2006; Neudecker et al. Reference Neudecker, Zarrine-Afsar, Davidson and Kay2007, Reference Neudecker, Lundstrom and Kay2009). Backbone ¹H, ¹³C, and ¹⁵N chemical shifts and ¹H–¹⁵N RDCs from CPMG relaxation dispersion were used to determine the structure, using CS-Rosetta, of an intermediate in the folding pathway of the FF domain from HYPA/FBP11, revealing a folding core comprising helices H1 and H2, and also other non-native interactions that slow the kinetics of folding (Korzhnev et al. Reference Korzhnev, Religa, Banachewicz, Fersht and Kay2010). A similar approach was used to solve the structure of a folding intermediate of a mutant of the Fyn SH3 domain, revealing an exposed β-strand that could potentially participate in protein misfolding and aggregation (Neudecker et al. Reference Neudecker, Robustelli, Cavalli, Walsh, Lundstrom, Zarrine-Afsar, Sharpe, Vendruscolo and Kay2012). The structure of a transient state of T4 lysozyme with an occluded binding cavity was also revealed in this manner (Bouvignies et al. Reference Bouvignies, Vallurupalli, Hansen, Correia, Lange, Bah, Vernon, Dahlquist, Baker and Kay2011). ¹⁵N CPMG relaxation dispersion data, in conjunction with chemical shift titrations, have been used to explore how an intrinsically unstructured peptide (pKID) undergoes folding upon binding to its target, the KIX domain of the CREB-binding protein (Sugase et al. Reference Sugase, Dyson and Wright2007); binding proceeds through an intermediate state consisting of partially folded pKID bound to KIX, consistent with an induced fit mechanism of binding (Wright & Dyson, Reference Wright and Dyson2009).

CPMG relaxation dispersion has proven to be a valuable tool for exploring the connection in enzymes between the catalytic rate and internal conformational dynamics (Villali & Kern, Reference Villali and Kern2010), an area that is the subject of a lively scientific debate (Kamerlin & Warshel, Reference Kamerlin and Warshel2010; Karplus, Reference Karplus2010; Pisliakov et al. Reference Pisliakov, Cao, Kamerlin and Warshel2009). Results from CPMG relaxation dispersion studies have demonstrated close correspondence between the k _ex of conformational exchange and catalytic turnover (k _cat) for a variety of enzymes, including HIV protease (Ishima et al. Reference Ishima, Freedberg, Wang, Louis and Torchia1999), RNase A (Cole & Loria, Reference Cole and Loria2002; Watt et al. Reference Watt, Shimada, Kovrigin and Loria2007), adenylate kinase (Adk) (Wolf-Watz et al. Reference Wolf-Watz, Thai, Henzler-Wildman, Hadjipavlou, Eisenmesser and Kern2004), cyclophilin A (Eisenmesser et al. Reference Eisenmesser, Millet, Labeikovsky, Korzhnev, Wolf-Watz, Bosco, Skalicky, Kay and Kern2005), dihydrofolate reductase (DHFR) (Boehr et al. Reference Boehr, Mcelheny, Dyson and Wright2006, Reference Boehr, Dyson and Wright2008, Reference Boehr, Mcelheny, Dyson and Wright2010), flavin oxidoreductase (Vallurupalli & Kay, Reference Vallurupalli and Kay2006), and triosphosphate isomerase (Kempf et al. Reference Kempf, Jung, Ragain, Sampson and Loria2007). DHFR undergoes exchange with a minor open conformation at a rate that is identical to product dissociation, which is the rate-limiting step in catalysis (Boehr et al. Reference Boehr, Mcelheny, Dyson and Wright2006); its hydride transfer step is also closely tied to the dynamics of conformational exchange (Boehr et al. Reference Boehr, Dyson and Wright2008). Likewise, it has been demonstrated by CPMG relaxation dispersion that the catalytic tunover (k _cat) of Adk is limited by the rate of opening of the nucleotide-binding lid (Wolf-Watz et al. Reference Wolf-Watz, Thai, Henzler-Wildman, Hadjipavlou, Eisenmesser and Kern2004). Thus, for processes that occur on an intermediate timescale, relaxation dispersion offers exceptional insight into molecular dynamics, and in some cases into the structures of otherwise invisible minor states.

2.3.1 Principles and applications of saturation transfer

States that are invisible because they are sparsely populated or because they involve large, slowly tumbling molecules can be detected by saturation transfer if there are resonances in the minor state that can be saturated by off-resonance radiofrequency (RF) radiation that does not saturate the resonances of the major state; this saturation can then be transferred to the visible species by chemical exchange. Resonances can be selectively saturated in the dark state if their chemical shifts are sufficiently different from those in the visible state or if the dark state resonances are broadened considerably compared with those in the visible state. The DEST experiment can be used to characterize larger assemblies and aggregates that are invisible because they tumble slowly (causing large transverse relaxation rates and significant line broadening), as long as the large invisible state is in exchange with a smaller visible state on an intermediate-to-slow timescale (500 μs–1 s) (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b, Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Ying, Torchia and Clore2012; Libich et al. Reference Libich, Fawzi, Ying and Clore2013). The CEST experiment, on the other hand, can be used to visualize a minor state in exchange with a major state, as long as the lifetime of the minor state is ~2–100 ms, its rotational correlation time is comparable to the major species, and it exhibits significantly different chemical shifts from the major state (Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012). Both experiments involve the application of off-resonance RF radiation to saturate a nucleus (usually ¹⁵N, but also ¹³C and ¹H) in the minor state (Fig. 4).

Fig. 4.

Characterizing ‘dark’ states by exchange saturation transfer. Saturation profiles are plotted for (a) CEST and (b) DEST experiments by solving the homogeneous McConnell equations (Eqs. (82)–(84)) in MATLAB. The y-axis gives the ratio of the peak intensity measured in the presence of a CW RF B ₁ field (I _sat) over the peak intensity measured for a reference experiment in the absence of the saturation field (I _ref). The x-axis gives the frequency offset of the applied B ₁ field. Note that the units are (a) in Hz for the CEST profiles, but are (b) in kHz for the DEST profiles, where the x-axis has been expanded almost 50 fold to accommodate the broader DEST profiles. B ₁ field strengths (applied in the x-dimension) are (a) 2·5 Hz (green), 5 Hz (cyan), 10 Hz (blue), and 25 Hz (purple) for CEST and (b) 25 Hz (green), 50 Hz (cyan), 100 Hz (blue), and 250 kHz (purple) for DEST (i.e. the DEST experiment is performed at about one order of magnitude higher B ₁ field strength than CEST). In this figure, a major visible state A (p _A = 95%) is separated from a minor dark state B (p _B = 5%) by a chemical shift difference of Δω/2π = 120 Hz (e.g. 2 ppm for ¹⁵N in a 600 MHz spectrometer). For the CEST profiles (a), R ₂⁰ = 10 s⁻¹ and R ₁ = 1 s⁻¹ for both states. For the DEST profiles (b), R _2,B⁰ = 10 000 s⁻¹. CW saturation was applied for 1 s (= 1 × T ₁). Results are shown for a system undergoing no exchange (top; i.e. the saturation profile of the major visible species A alone), and exchange at a rate of k _ex = 1, 10, and 100 s⁻¹ (bottom). Full saturation of the signal (I _sat/I _ref = 0) occurs when the B ₁ field is applied on resonance to state A, and saturation can be transferred from states B to A by exchange either from (a) on-resonance saturation of state B due to a different chemical shift (CEST) or (b) off-resonance saturation of the broader profile of state B due to its faster rate of transverse relaxation (DEST).

A DEST or CEST experiment yields a plot of signal intensity versus the chemical shift offset of the B ₁ saturating field for each resonance being monitored. These profiles can then be fit to a model (based on the McConnell equations (McConnell, Reference Mcconnell1958)) to determine the populations of and rates of exchange between each state. In addition to these kinetic and thermodynamic parameters, the DEST experiment also provides information on the transverse relaxation rates (R ₂) of the resonances in the dark state (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Ying, Torchia and Clore2012). The CEST experiment provides direct information on the chemical shifts of the minor state (Bouvignies & Kay, Reference Bouvignies and Kay2012a, Reference Bouvignies and Kayb; Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012), which are difficult to extract from DEST data due to the broadness of the saturation transfer profiles (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2012).

NMR saturation transfer experiments were originally developed in the 1960s (Forsén & Hoffman, Reference Forsén and Hoffman1963), and the DEST and CEST experiments are related to small molecule saturation transfer difference (STD) NMR spectroscopy, where the broader resonances in a larger macromolecule are saturated, and saturation is transferred to a bound ligand via cross-relaxation and ultimately to the NMR-visible unbound ligand via chemical exchange (Mayer & Meyer, Reference Mayer and Meyer1999). Saturation transfer found some early biomolecular applications (Cayley et al. Reference Cayley, Albrand, Feeney, Roberts, Piper and Burgen1979; Gupta & Redfield, Reference Gupta and Redfield1970), although CEST was first employed in MRI, where exchangeable protons with a chemical shift distinct from water (either in endogenous molecules or introduced CEST contrast agents) are saturated, and this saturation is transferred to bulk water via slow chemical exchange, thus amplifying these otherwise invisible signals to provide additional contrast (Van Zijl & Yadav, Reference Van Zijl and Yadav2011; Ward et al. Reference Ward, Aletras and Balaban2000). More recently, PCSs from paramagnetic groups (chelated lanthanides) have been used to generate even larger chemical shifts in paraCEST imaging, enabling faster timescales for which k _ex ≪ Δω remains true (Hancu et al. Reference Hancu, Dixon, Woods, Vinogradov, Sherry and Lenkinski2010; Vinogradov et al. Reference Vinogradov, Sherry and Lenkinski2013). Interestingly, one of the earliest biomolecular applications of saturation transfer involved using the hyperfine shifts in Fe³⁺-bound cytochrome c to study less-well-resolved Fe²⁺-bound cytochrome c via chemical exchange (Gupta & Redfield, Reference Gupta and Redfield1970). More recently, the applications of saturation transfer have been expanded to observe invisible states in macromolecules by CEST (Bouvignies & Kay, Reference Bouvignies and Kay2012a, Reference Bouvignies and Kayb; Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012) and DEST (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Ying, Torchia and Clore2012, Reference Fawzi, Libich, Ying, Tugarinov and Clore2014; Libich et al. Reference Libich, Fawzi, Ying and Clore2013).

2.3.2 The DEST experiment

The DEST experiment is used to characterize a visible species in exchange with a dark state that is invisible due to its large size (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b, Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Ying, Torchia and Clore2012, Reference Fawzi, Libich, Ying, Tugarinov and Clore2014; Libich et al. Reference Libich, Fawzi, Ying and Clore2013). Owing to the rapid transverse relaxation rate of the signals in a slowly tumbling large dark state, these resonances can be saturated far off resonance at frequencies that do not excite the narrow signals of the rapidly tumbling visible species in the absence of the larger dark state. If the invisible species converts to the visible species at a rate that is of the order of or faster than its longitudinal relaxation rate, saturation will be transferred to the NMR-visible species and observed as a loss of intensity of cross-peaks for the visible species. Thus, it becomes possible to detect and characterize large aggregates and macromolecular assemblies that would otherwise be invisible. The DEST experiment is explored in greater detail in Section 3.

DEST is complementary to the transferred nuclear Overhauser effect (TRNOE) (Clore & Gronenborn, Reference Clore and Gronenborn1982, Reference Clore and Gronenborn1983), which offers structural restraints (in the form of short interproton distances) for a bound state in fast exchange with the free state. The TRNOE, however, is really only applicable in the case of exchanging systems involving small (low molecular weight) ligands that exhibit positive (or negligibly small) NOEs in the free state and large negative NOEs in the bound (high-molecular weight) state. DEST is qualitatively similar to other saturation-based experiments, including CEST and STD (Post, Reference Post2003). However, unlike STD, DEST offers quantitative results when performed on a heteronucleus (e.g. ¹⁵N and ¹³C), as it is not complicated by spin diffusion, which occurs in ¹H saturation experiments (Bodner et al. Reference Bodner, Dobson and Bax2009; Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b).

The DEST experiment has revealed details of the aggregation of amyloid-β (Aβ) peptides into protofibrils (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b, Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Libich, Ying, Tugarinov and Clore2014), demonstrating that Aβ reaches a pseudo-equilibrium with protofibrils, with free peptides transiently binding to the surface of the protofibrils with a lifetime of ~20 ms. During this process, the more hydrophobic segments of the peptide (residues 16–25 and 28–37) remain more tightly associated with the surface of the protofibrils than the other portions. These protofibrils were shown to have R ₂ relaxation rates of ~19 000 s⁻¹, consistent with the size distribution of 1·8–85 MDa observed by AFM, EM, analytical ultracentrifugation, and dynamic light scattering (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b). DEST has also been used to characterize the interaction of the Aβ peptide with the 780-kDa GroEL chaperonin (Libich et al. Reference Libich, Fawzi, Ying and Clore2013), revealing that Aβ transiently interacts with the surface of the large accessible internal cavity of the chaperone on a faster (~700 μs) timescale, using the hydrophobic portions of its sequence.

DEST experiments are performed with relatively large (>100 Hz) B ₁ saturation field strengths to allow maximal sensitivity to the dark state. Unlike in CEST, where the need to resolve a difference in chemical shift between the visible and dark states puts an upper limit on the saturation field strength, no such upper limit exists for DEST. Higher saturation fields are needed to monitor slower exchange processes, and those involving a smaller dark state population or lower R ₂^dark. In the example shown in Fig. 4, DEST is effective at revealing the dark state at exchange rates of 10 and 100 s⁻¹. DEST could be measured on a more slowly exchanging system if the population of the dark state were larger. DEST can also be measured on a more rapidly exchanging system; however, measurement of fast exchanging systems (k _ex > 1000 s⁻¹) is only possible for smaller values R ₂^dark. Otherwise, R ₂^obs would be too fast and there would be no measurable NMR signal.

2.3.3 The CEST experiment

The CEST experiment (Bouvignies & Kay, Reference Bouvignies and Kay2012a, Reference Bouvignies and Kayb; Vallurupalli & Kay, Reference Vallurupalli and Kay2013; Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012) provides information similar to relaxation dispersion, but for processes on a slower timescale. If the visible species is in exchange with a dark state that has a significantly different chemical shift, application of a continuous-wave (CW) RF field at the invisible state resonance will cause a loss in intensity of the corresponding resonance in the NMR-visible major species (as long as the minor state has a lifetime of ~2–100 ms and is populated ≥1%; Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012). By varying the offset of the saturating CW RF field, one can acquire, effectively, a ¹H, ¹⁵N, or ¹³C spectrum of the minor state, depending on which nucleus is used for saturation. The CEST data can be fit to the McConnell equations, providing kinetic parameters and populations for each state (thus occupying a niche similar to that of CPMG relaxation dispersion but for processes undergoing slower exchange).

The observation of CEST is dependent upon a chemical shift difference between the two states, and the maximum field strength that can be used in CEST is limited by the chemical shift difference between the two states. A higher B ₁ field improves the sensitivity of CEST to a minor state, but decreases chemical shift resolution between the two states. In the example, in Fig. 4, a field strength of 10 Hz is best suited for observing the minor state B, which can be characterized if the exchange rate is 10 or 100 s⁻¹. However, the sensitivity is too low under conditions of slower exchange (k _ex = 1 s⁻¹) to accurately observe state B. Characterization of exchange much faster than k _ex = 100 s⁻¹ would not be possible because distinct chemical shifts are not observed for each state in the intermediate or fast exchange regimes. Higher saturation field strengths can be used in the presence of greater chemical shift differences between the major and minor states, and, in practice, most CEST studies have used field strengths of ~20–30 Hz (Bouvignies et al. Reference Bouvignies, Vallurupalli and Kay2014). Note that much larger field strengths can be used in DEST experiments because the results are not dependent upon resolving chemical shift differences between the light and dark states.

CEST experiments on proteins have been performed on ¹⁵N and ¹³C nuclei. If ¹³C-CEST is performed on uniformly ¹³C-labeled samples, the strong ¹³C–¹³C scalar couplings must be taken into account (Bouvignies et al. Reference Bouvignies, Vallurupalli and Kay2014; Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2013). Owing to the narrow linewidths of the peaks in the minor state, CEST experiments require a large number of saturation field offset frequencies (e.g. ¹⁵N CEST experiments have been performed with offsets at uniform increments of 15–25 Hz, requiring ≥60 total offsets to cover the ¹⁵N chemical shift range of ~30 ppm (Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012)). Owing to the broader linewidths, DEST experiments require a larger range of saturation frequencies, but a smaller total number (e.g. ¹⁵N-DEST experiments on Aβ were performed with non-uniform increments of 2–7 kHz between offsets requiring just 15 total offsets to cover a large frequency range of 70 kHz, which is >1000 ppm in a 600 MHz static ¹H field (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b)).

The CEST experiment is applicable to studying slow exchange processes, which include protein folding. Exchange between the visible folded species of a mutant of the HYPA/FBP11 FF domain and an intermediate along its folding pathway has been characterized by ¹⁵N-CEST (Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012) and carbonyl ¹³C-CEST (Vallurupalli & Kay, Reference Vallurupalli and Kay2013), yielding a best-fit exchange rate of k _ex ≈ 60 s⁻¹ and a minor state population of 1·4%. ¹⁵N-DEST and methyl ¹³C-DEST studies have also characterized the exchange between a mutant of the Fyn SH3 domain and a folding intermediate (Bouvignies & Kay, Reference Bouvignies and Kay2012a). FF domain and SH3 domain folding have also been characterized by aliphatic ¹³C-CEST, allowing the measurement of most side chain ¹³C chemical shifts in the sparsely populated folding intermediates (Bouvignies et al. Reference Bouvignies, Vallurupalli and Kay2014). Additionally, ¹⁵N-CEST has been used to characterize slow conformational dynamics in active Ras-GTPases under low-temperature conditions (used to slow intrinsic GTP hydrolysis), where exchange was too slow and the population of the minor state too small to give reliable CPMG relaxation dispersion data (Long et al. Reference Long, Marshall, Bouvignies, Mazhab-Jafari, Smith, Ikura and Kay2013).

3.1.1 Quantitative description of PRE

PRE is an enhancement of the relaxation rate of a nucleus caused by a dipolar interaction with a paramagnetic center consisting of one or more unpaired electrons. Two other NMR observables can yield long-range distances in paramagnetic systems: PCSs and RDCs. Whereas PCSs and paramagnetic RDCs require an anisotropic electron g-tensor, PRE can be measured in any paramagnetic system, though, as discussed below, it is advantageous to use a system with an isotropic g-tensor. Examples of paramagnetic groups with isotropic g-tensors include nitroxides, Mn²⁺, and Gd³⁺. Many other paramagnetic metal ions, including Fe³⁺ and Dy³⁺, have anisotropic g-tensors (Otting, Reference Otting2010).

The longitudinal (Γ₁) and transverse (Γ₂) PRE rates arising from direct dipole–dipole interactions are described by the following form of the Solomon–Bloembergen (SB) equations (Bloembergen & Morgan, Reference Bloembergen and Morgan1961; Solomon, Reference Solomon1955):

(42)$${\rm \Gamma} _1 = \displaystyle{2 \over {15}}\left( {\displaystyle{{\mu _0} \over {4\pi}}} \right)^2 \gamma _{\rm I}^2 g^2 \mu _{\rm B}^2 s(s + 1)\left\{ {3J_{{\rm SB}} \left( {\omega _{\rm I}} \right) + J_{{\rm SB}} \left( {\omega _{\rm I} - \omega _{\rm S}} \right) + 6J_{{\rm SB}} \left( {\omega _{\rm I} + \omega _{\rm S}} \right)} \right\},$$

(43)$$\eqalign{{\rm \Gamma} _2 & = \displaystyle{1 \over {15}}\Big( {\displaystyle{{\mu _0} \over {4\pi}}} \Big)^2 \gamma _{\rm I}^2 g^2 \mu _{\rm B}^2 s(s + 1)\big\{ {4J_{{\rm SB}} \lpar 0 \rpar + 3J_{{\rm SB}} \lpar {\omega _{\rm I}} \rpar + J_{{\rm SB}} \lpar {\omega _{\rm I} - \omega _{\rm S}} \rpar + 6J_{{\rm SB}} \lpar {\omega _{\rm S}} \rpar }\cr & \quad + 6J_{{\rm SB}} \lpar {\omega _{\rm I} + \omega _{\rm S}} \rpar \big\}},$$

where s is the electron spin quantum number, μ ₀ the permeability of free space (~1·257 × 10⁻⁶ m kg s⁻² Å⁻²), μ _B the Bohr magneton (the magnetic moment of a free electron; ~9·274 × 10⁻²⁴ J T⁻¹), g the electron g-factor (~2·0), γ _I the proton gyromagnetic ratio (~2·675 × 10⁸ rad s⁻¹ T⁻¹), ω _I/2π the Larmor frequency of the proton (e.g. 600 MHz in a 14·1-T magnetic field), ω _S/2π the Larmor frequency of the electron (e.g. 395 GHz in a 14·1-T magnetic field), and J _SB(ω) is the generalized spectral density function for the reduced correlation function, given by:

(44)$$J_{{\rm SB}} (\omega ) = r^{ - 6} \displaystyle{{\tau _{\rm c}} \over {1 + (\omega \tau _{\rm c} )^2}}, $$

where r is the distance between the paramagnetic center and the reporter nucleus. The overall correlation time, τ _c, is defined as:

(45)$$\tau _{\rm c} = \left( {\tau _{\rm r}^{ - 1} + \tau _{\rm s}^{ - 1}} \right)^{ - 1}, $$

where τ _r is the rotational correlation time of the macromolecule and τ _s is the effective electron relaxation time. Values of τ _s depend on the nature of the paramagnetic group. Paramagnetic metal ions with anisotropic g-tensors generally exhibit very small values (e.g. τ _s ~ 10⁻¹³ for anisotropic lanthanides; Otting, Reference Otting2010), thus making τ _c ≈ τ _s in this case. Nitroxides general exhibit much larger τ _s values (~10⁻⁶; Bernini et al. Reference Bernini, Venditti, Spiga and Niccolai2009), thus making τ _c ≈ τ _r in that case. Paramagnetic metal ions with isotropic g-tensors, however, exhibit intermediate values on the same order as τ _r (e.g. τ _s ~ 10⁻⁹ for Mn²⁺, τ _s ~ 10⁻⁸ for Gd³⁺ (Bertini et al. Reference Bertini, Luchinat, Parigi and Pierattelli2005; Iwahara et al. Reference Iwahara, Schwieters and Clore2004b; Pintacuda et al. Reference Pintacuda, Kaikkonen and Otting2004a)), preventing simplification of the expression for τ _c.

Under the conditions of biomolecular NMR (where ω _s²τ _c² ≫ 1), the high-frequency components of the spectral density function (those dependent on ω _S) can be ignored, and the longitudinal and transverse PREs can be expressed by the simplified formulas (Clore & Iwahara, Reference Clore and Iwahara2009):

(46)$${\rm \Gamma} _1 = \displaystyle{2 \over 5}\left( {\displaystyle{{\mu _0} \over {4\pi}}} \right)^2 \gamma _{\rm I}^2 g^2 \mu _{\rm B}^2 s(s + 1)J_{{\rm SB}} (\omega _{\rm I} ),$$

(47)$${\rm \Gamma} _2 = \displaystyle{1 \over {15}}\left( {\displaystyle{{\mu _0} \over {4\pi}}} \right)^2 \gamma _{\rm I}^2 g^2 \mu _{\rm B}^2 s(s + 1)\left\{ {4J_{{\rm SB}} (0) + 3J_{{\rm SB}} (\omega _{\rm I} )} \right\}.$$

A paramagnetic group can also cause enhanced relaxation through Curie spin relaxation, which arises from the interaction between the nucleus and the time-averaged magnetization of the electrons (due to the Boltzmann population difference in electron spins states; Bertini et al. Reference Bertini, Luchinat and Parigi2001a). The Γ₂ rate due to Curie spin relaxation is given by (Gueron, Reference Gueron1975):

(48)$${\rm \Gamma} _{2,{\rm Curie}} = \displaystyle{1 \over 5}\left( {\displaystyle{{\mu _0} \over {4\pi}}} \right)^2 \displaystyle{{\left( {\omega _{\rm I}^2 g^4 \mu _{\rm B}^4 s^2 (s + 1)^2} \right)} \over {(3k_{\rm B} T)^2 r^6}} \left\{ {4\tau _{\rm r} + \displaystyle{{3\tau _{\rm r}} \over {1 + (\omega _{\rm I} \tau _{\rm r} )^2}} - 4\tau _{\rm c} - \displaystyle{{3\tau _{\rm c}} \over {1 + (\omega _{\rm I} \tau _{\rm c} )^2}}} \right\}.$$

Note that Curie spin relaxation is independent of the rate of electron relaxation (τ _s).

Equations (46)–(48) show that the PRE has an r ⁻⁶ dependence on the distance between the observed nucleus and the paramagnetic center. The dependence on the magnetic field and on the size of the molecule, however, is slightly more complicated. The dipole–dipole Γ₂ rate is approximately independent of field, whereas the Curie Γ₂ rate is approximately proportional to the square of the magnetic field. (The dipole–dipole Γ₁ rate decreases with increasing field strength.) In cases where τ _c ≈ τ _r – due to a relatively long τ _s (e.g. nitroxide spin radicals) – Curie spin relaxation is negligible, and the Γ₂ rate is therefore independent of field. In these cases, the dipole–dipole Γ₂ rate is directly proportional to τ _r (and to molecular weight). On the other hand, for metal ions with a very short τ _s (especially those with an anisotropic g-tensor, e.g. Fe³⁺, Dy³⁺), the Γ₂ rate in macromolecules is dominated by Curie-spin relaxation. In these cases, the Curie-spin Γ₂ rate is also directly proportional to τ _r, while the dipole–dipole Γ₂ rate is independent of molecular weight because τ _c ≈ τ _s. For isotropic metal ions, though, such as Mn²⁺ and Gd³⁺, the contribution from Curie-spin relaxation is minor, and negligible for medium-sized macromolecules (<40 kDa). For example, in the 20 kDa complex of SRY and DNA–EDTA–Mn²⁺, Curie-spin relaxation contributed 2% of the Γ₂ rates at a ¹H-frequency of 800 MHz and a temperature of 308 K (Iwahara et al. Reference Iwahara, Tang and Clore2007). However, for a larger molecule (>100 kDa), Curie-spin relaxation can contribute more than 20% of the total value of Γ₂ under these conditions (Clore & Iwahara, Reference Clore and Iwahara2009).

Thus, for measuring PREs, it is advantageous to use a paramagnetic group with an isotropic g-tensor because the Curie spin contribution to the Γ₂ rate will be much less (often negligible) compared with one with an anisotropic g-tensor. In addition, for an anisotropic paramagnetic group, the generation of PCSs means that the chemical shifts (and therefore resonance assignments) are altered and that exchange contributions are different between the diamagnetic and paramagnetic samples, which complicates analysis, as described below (Clore & Iwahara, Reference Clore and Iwahara2009).

Artificially introduced paramagnetic groups are generally attached to the macromolecule of interest through linkers that have several rotatable bonds. Hence, these paramagnetic centers are intrinsically flexible. In addition, the observed nuclei may be located in mobile regions of the macromolecule. The effect of fast dynamics in the picosecond to nanosecond timescale on the PRE for paramagnetic systems with an isotropic g-tensor can be readily taken into account by modifying the SB equations described above by incorporating a ‘model-free’ formalism (SBMF) (Lipari & Szabo, Reference Lipari and Szabo1982a). This transforms the spectral density function in Eq. (44) into (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b):

(49)$$J_{{\rm SBMF}} (\omega ) = \left\langle {r^{ - 6}} \right\rangle \left\{ {\displaystyle{{{\bi S}^2 \tau _{\rm c}} \over {1 + (\omega \tau _{\rm c} )^2}} + \displaystyle{{(1 - {\bi S}^2 )\tau _{\rm t}} \over {1 + (\omega \tau _{\rm t} )^2}}} \right\},$$

where S² is the square of the generalized order parameter, 〈r ⁻⁶〉 is the population-averaged distance inverse sixth power of the distance between the paramagnetic center and the observed nucleus, and τ _t is the total correlation time, defined as:

(50)$$\tau _{\rm t} = \left( {\tau _{\rm r}^{ - 1} + \tau _{\rm s}^{ - 1} + \tau _{\rm i}^{ - 1}} \right)^{ - 1}, $$

where τ _i is the correlation time for internal motion. These equations are valid for isotropically tumbling systems; otherwise, the angles between the principal axes of the diffusion tensor and PRE vector must be taken into account (Benetis & Kowalewski, Reference Benetis and Kowalewski1985; Woessner, Reference Woessner1962).

The model-free formalism is most commonly employed for analyzing relaxation due to fixed-length dipole–dipole interactions, such as those in ¹H–¹⁵N or ¹H–¹³C bond vectors. However, it can also be applied to variable-length vectors (e.g. PRE; Iwahara et al. Reference Iwahara, Schwieters and Clore2004b), where it takes the form (Olejniczak et al. Reference Olejniczak, Dobson, Karplus and Levy1984):

(51)$${\bi S}^2 = \displaystyle{{4\pi} \over 5}\left\langle {r^{ - 6}} \right\rangle ^{ - 1} \sum\limits_{m = - 2}^2 {\left\vert {\left\langle {\displaystyle{{Y_2^m {\rm \Omega} ^{{\rm mol}}} \over {r^3}}} \right\rangle} \right\vert^2}, $$

where Y ₂^m(Ω) are second-order spherical harmonics and Ω^mol are Euler angles in the molecular frame. The order parameter can be approximated by decomposing it into its radial and angular components, as has been demonstrated for ¹H–¹H dipolar interactions (Bruschweiler et al. Reference Bruschweiler, Roux, Blackledge, Griesinger, Karplus and Ernst1992):

(52)$${\bi S}^2 \approx {\bi S}_{{\rm angular}}^2 {\bi S}_{{\rm radial}}^2, $$

where the angular (S _angular²) and radial (S _radial²) order parameters are defined as:

(53)$${\bi S}_{{\rm angular}}^2 = \displaystyle{{4\pi} \over 5}\sum\limits_{m = - 2}^2 {\left\vert {\left\langle {Y_2^m {\rm \Omega} ^{{\rm mol}}} \right\rangle} \right\vert^2}, $$

(54)$${\bi S}_{{\rm radial}}^2 = \left\langle {r^{ - 6}} \right\rangle ^{ - 1} \left\langle {r^{ - 3}} \right\rangle ^2. $$

These relationships are used directly for back-calculating PRE values from known structures (and for refining against PRE data) as described below. The PRE measured for a given nucleus has a population-weighted 〈r ⁻⁶〉 dependence on the distance between the nucleus and the paramagnetic center, hence its utility as a structural restraint (and as a probe of minor states) emerges. However, because most paramagnetic tags used for measuring PRE are highly flexible (and the molecules of interest are also often quite flexible), it is important in structure calculations to refine directly against the PRE data (e.g. using prePot in Xplor-NIH (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b) as discussed below) rather than converting PRE data into a single distance, which is generally incorrect (although PRE calculators have been made available online at spin.niddk.nih.gov/clore for qualitative purposes).

Both the transverse and longitudinal PREs are influenced by internal motion (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). However, while Γ₁ has a complicated dependence on the internal correlation time, the Γ₂ shows a simpler dependence. As the internal correlation time is increased from 0 (infinitely fast internal motion) to a value greater than the overall rotational correlation time, the ratio Γ_2,SBMF/Γ_2,SB (where Γ_2,SBMF is Γ₂ calculated from the SBMF equation, and Γ_2,SB is Γ₂ calculated from the SB equation) increases from the value of the order parameter (0 ≤ S² ≤ 1) to 1. For the longitudinal PRE, however, Γ_1,SBMF/Γ_1,SB first increases from the value of S² when τ _i = 0, up to a maximum value ≫ 1 when τ _i ≈ 1/ω (where ω is the resonance frequency of the observed nucleus in rad s⁻¹), before eventually approaching 1 (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). Thus, whereas a simple estimation of τ _i is sufficient for structural analysis of ¹H–Γ₂ data, such treatment of ¹H–Γ₁ data requires accurate information on τ _i for individual PRE vectors (Iwahara & Clore, Reference Iwahara and Clore2010). Another complication is that, due to cross-relaxation and water exchange, ¹H R ₁ relaxation does not follow a single-exponential curve. Thus, it is advantageous to measure the transverse PRE instead of the longitudinal PRE. For very fast motion (τ _i ≪ τ _r; i.e. picosecond timescale motion), the transverse PRE is virtually independent of the internal correlation time, and the order parameter acts as a scaling factor. In either case, though, the SBMF equations reduce to the SB equations under conditions where the internal motion is either very slow (τ _t ≈ τ _c) or highly restricted in space (S² ≈ 1).

3.1.2 PRE from minor states

In a system undergoing exchange between states on a fast timescale, the overall PRE profile observed for the ensemble of states is the population-weighted average of all the members of the ensemble. Owing to the 〈r ⁻⁶〉 distance dependence of the PRE, such a profile will be dominated by the states in which the paramagnetic center more closely approaches the observed nuclei. Thus, even if a state occupies a small (<10%) portion of the total population, it can be readily detected if it involves a shorter distance between a paramagnetic probe and reporter than the other state(s) (Fig. 2). Thus, an otherwise invisible minor state can be detected and structurally characterized using PRE (Clore & Iwahara, Reference Clore and Iwahara2009).

Under fast exchange conditions on the PRE timescale (i.e. the rate of exchange between the species is much greater than the difference in PRE rates between states) the overall PRE observed for a nucleus is given by:

(55)$${\rm \Gamma} _2^{{\rm app}} = \sum {\,p_i {\rm \Gamma} _{2,i}}, $$

where Γ₂^app is the apparent transverse PRE measured for the entire ensemble, Γ_2,i is the transverse PRE for state i, and p _i is the population of state i. Eq. (55) holds when the rate of exchange is slower than the overall correlation time (k _ex ≪ 1/τ _c); in the limit of ultrafast exchange, however, where k _ex ≫ 1/τ _c, in Eq. (55) would be replaced by ${\rm \Gamma} _2^{{\rm app}} = \left(\sum {\,p_i \sqrt{{\rm \Gamma} _{2,i}}\hskip 1pt}\right) ^2$ (i.e. under ultrafast exchange conditions the contribution of each state to ${\rm \Gamma} _2^{{\rm app}}$ is roughly proportional to the square of its population due to the fact that individual components of ${\rm \Gamma} _2^{{\rm app}}$ are averaged as <r ⁻³> as opposed to <r ⁻⁶> under the more usual conditions of Eq. (55)).

In the case of two-state exchange between a major state (A) and a minor state (B), the apparent observed PRE is given by:

(56)$${\rm \Gamma} _2^{{\rm app}} = p_{\rm A} {\rm \Gamma} _{2,{\rm A}} + p_{\rm B} {\rm \Gamma} _{2,{\rm B}}. $$

If the exchange between states A and B is not fast on the PRE timescale (i.e. if k _ex ≫ Γ_2,B − Γ_2,A is not true), the apparent PRE is not simply a population-averaged value, but is instead intermediate between Γ_2,A and the population-averaged value (Fig. 2c). Under these conditions of slower exchange, the apparent PRE can be calculated using the McConnell equations (McConnell, Reference Mcconnell1958). These necessary equations are identical to the general equations for chemical exchange (Eqs. (16)–(20)), except the relaxation matrix R (Eq. (18)) includes the PRE from each state, taking the form (for two-site exchange):

(57)$${\bf R =} \left[ {\matrix{ {R_{2,{\rm A}} + {\rm \Gamma} _{2,{\rm A}}} & 0 \cr 0 & {R_{2,{\rm B}} + {\rm \Gamma} _{2,{\rm B}}} \cr}} \right].$$

Apparent relaxation rates can be extracted by fitting the line shape of the Fourier-transformed solution of the McConnell equations. Under such cases, where exchange takes place on the intermediate PRE timescale (k _ex ≈ Γ_2,B − Γ_2,A), the apparent PRE measured for a system in exchange between two states, in addition to encoding distance and population information, also encodes kinetic information, although in practice it is difficult to extract this information from PRE data alone.

To visualize how PRE can be used to detect and characterize a minor state, consider a protein in exchange between two states (Fig. 2). In state A, the protein is in a more extended conformation, and the distance between the two selected points is 40 Å. In state B, however, the protein is in a more compact conformation, and the distance between these same two points is 8 Å. If one of the points is a paramagnetic center, and the other is a ¹H nucleus, we can calculate the PRE (after specifying the correlation times and the order parameter of the vector between the paramagnetic center and the observed nucleus). In this example, we will consider a single unpaired electron (e.g. a nitroxide spin label), and we will set τ _c = 10 ns (corresponding to a protein at room temperature with a molecular weight of ~20 kDa). Also, we will assume that the paramagnetic center/observed nucleus vector is rigid, although this is not the case for most conventional spin labels (e.g. MTSL). For state A, with the nucleus far from the unpaired electron, Γ_2,A = 0·12 s⁻¹ (effectively zero). However, in state B, where they are in much closer proximity, Γ_2,B = 1900 s⁻¹.

If the protein in this example is in fast exchange between states A and B (k _ex ≫ Γ_2,A − Γ_2,B = 1900 s⁻¹; τ _ex ≪ 500 μs), the apparent PRE would be the population-weighted average of Γ_2,A and Γ_2,B (Eq. (56)). Thus, even if state A comprises 99% of the total population, and state B only comprises 1% of the total population, Γ₂^app = 0·99 × 0·12 s⁻¹ + 0·01 × 1900 s⁻¹ ≈ 19 s⁻¹. Thus, even though state B comprises only 1% of the total population, Γ₂^app arises almost exclusively from the minor state.

Two conditions are necessary for detection of a minor state by PRE: (1) a shorter distance between probe and reporter in the minor state, and (2) fast exchange between the states. Having a shorter distance to measure in the minor state is a necessary requirement for PRE measurements on minor states, and detecting such a state is most straightforward for the transient association between two molecules or for a minor state that involves forming a more compact structure than in the major state (e.g. CaM; Anthis et al. Reference Anthis, Doucleff and Clore2011). However, other states can also be measured, as long as a paramagnetic probe position can be found where at least some measurable nuclei are in closer proximity to the probe in the minor state than in the major state.

For quantitative PRE measurements (i.e. for the overall measured PRE profile to be exactly the population-weighted average of the PRE profiles from the individual states), the rate of exchange (k _ex = k _AB + k _BA) must be fast on the PRE timescale, meaning that k _ex ≫ Γ_2,B − Γ_2,A. Otherwise, the apparent PRE becomes highly dependent on k _ex, as seen in Fig. 2. That being said, under conditions where k _ex ~ Γ_2,B − Γ_2,A, the Γ₂^obsrate is only decreased by a factor of about 2 relative to the limiting value in the fast exchange limit (Fig. 2c); given the 〈r ⁻⁶〉 dependence of the PRE, the assumption of fast exchange in the intermediate PRE regime would only result in a ~10% overestimation in the corresponding distance. As the exchange rate decreases, the apparent PRE decreases, eventually becoming equal to that of state A, providing no information on the minor state. If exchange is not fast on the PRE timescale, the apparent PRE also becomes dependent on the chemical shift difference between states A and B (Ω_A − Ω_B), decreasing with decreasing k _ex or increasing Ω_A − Ω_B (unless k _ex ≫ |Ω_A − Ω_B|) (Clore & Iwahara, Reference Clore and Iwahara2009). However, under these conditions, the peak will also be subject to additional line broadening, making measurements difficult or impossible as |Ω_A − Ω_B| approaches k _ex. If k _ex ≪ |Ω_A − Ω_B|, then the two states will display two separate peaks; the minor state peak will likely be invisible, and the major state peak will not contain PRE information on the minor state (Clore & Iwahara, Reference Clore and Iwahara2009).

3.2.1 Producing paramagnetic samples

Broadly speaking, two types of paramagnetic labels are available for measuring PREs: those containing a nitroxide group and those that act as a ligand for a paramagnetic cation (e.g. Mn²⁺, Cu²⁺, and Gd³⁺). If a protein has an intrinsic cation-binding site, a paramagnetic cation can often be introduced in place of the natural metal ligand. For proteins that are not native metalloproteins, short metal-binding sequences can be genetically encoded. Examples of such sequences include the N-terminal ATCUN motif (XXH), which binds paramagnetic Cu²⁺ (Donaldson et al. Reference Donaldson, Skrynnikov, Choy, Muhandiram, Sarkar, Forman-Kay and Kay2001), the sequence YVDTNNDGAYEGDEL, which specifically binds lanthanides (Wohnert et al. Reference Wohnert, Franz, Nitz, Imperiali and Schwalbe2003), and the N-terminal sequence HHP, which forms a dimer with two peptides bound to paramagnetic Ni²⁺ (Jensen et al. Reference Jensen, Lauritzen, Dahl, Pedersen and Led2004). A paramagnetic Cu²⁺ ion can also be introduced into a protein by making a double histidine mutation at i and i + 4 positions in an α-helix and then adding the chelating agent nitrilotriacetic acid in a 1:1 ratio with Cu²⁺ (which prevents Cu²⁺-mediated dimerization) (Liu et al. Reference Liu, Gong, Guo, Zhang and Tang2014).

The most common method for attaching a paramagnetic probe to a protein of interest is site-directed spin labeling (Hubbell et al. Reference Hubbell, Cafiso and Altenbach2000; Todd et al. Reference Todd, Cong, Levinthal, Levinthal and Hubbell1989), where a paramagnetic tag is conjugated to an exposed cysteine residue introduced by site-directed mutagenesis (Fig. 6). Conjugation by disulfide chemistry (Battiste & Wagner, Reference Battiste and Wagner2000; Dvoretsky et al. Reference Dvoretsky, Gaponenko and Rosevear2002; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000) offers high specificity for cysteine and is straightforward to perform using a spin label molecule with a methanethiosulfonate or pyridylthiol functional group (Fig. 6a). Disulfide linkages are somewhat unstable, and over time some of the paramagnetically tagged protein is replaced by disulfide cross-linked protein dimers, although over a short time period (up to ~1 month, often longer), the disulfide-conjugated paramagnetic protein sample is pure and stable enough to perform PRE measurements. Paramagnetic probes can also be conjugated using iodo- or bromoacetamide derivatives, which irreversibly form a C–S bond with cysteine (Gillespie & Shortle, Reference Gillespie and Shortle1997b; Ogawa & McConnel, Reference Ogawa and Mcconnel1967) (Fig. 6b); however, although they are advantageous for their exceptional stability, they exhibit lower specificity (also reacting at a slower rate with histidine, methionine, and lysine). Maleimide derivatives, however, are not recommended due to their instability (Kosen, Reference Kosen1989) (Fig. 6c). As these tags conjugate to exposed cysteine residues, any natural exposed cysteine residues will need to be mutated to serine or alanine. Even buried cysteines can become transiently exposed by the protein's intrinsic motions, so one will need to ensure that they do not react with the tag if they are to remain unmutated.

Fig. 6.

Site-directed spin labeling. Shown are four different chemistries for conjugating a spin label to a cysteine residue (a–c) or to an unnatural amino acid (d). All four examples are nitroxide spin labels (with the unpaired electron denoted with a dot), but the same chemistry can be used for spin labels that chelate a paramagnetic metal ion. The C_α of the peptide backbone is labeled. (a) Reaction of a methanethiosulfonate with cysteine to form a disulfide bond. The compound (1-oxyl-2,2,5,5-tetramethyl-∆3-pyrroline-3-methyl) methanethiosulfonate (MTSL; left) reacts with cysteine to form the R1 side chain (right) (Battiste & Wagner, Reference Battiste and Wagner2000; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000). (b) Reaction of an iodoacetamide with cysteine to form an S–C bond. Shown on the left is 3-(2-iodoacetamido)-2,2,5,5-tetramethyl-1-pyrolidinyloxy (iodoacetamido-PROXYL) (Gillespie & Shortle, Reference Gillespie and Shortle1997a). (c) Reaction of a maleimide with cysteine to form an S–C bond. Shown on the left is N-(1-oxyl-2,2,6,6-tetramethyl-4-piperidinyl) maleimide (maleimide-TEMPO) (Griffith & McConnell, Reference Griffith and Mcconnell1966; Tang et al. Reference Tang, Ghirlando and Clore2008a). (d) Reaction of 3-aminooxymethyl-2,2,5,5-tetramethyl-2,5-dihydro-1H-pyrrol-1-yloxyl radical (HO-4120) with the unnatural amino acid p-acetyl-l-phenylalanine to form the side chain K1 (Fleissner et al. Reference Fleissner, Brustad, Kalai, Altenbach, Cascio, Peters, Hideg, Peuker, Schultz and Hubbell2009).

An alternative method, which is especially useful when the removal of natural cysteines is difficult or impossible, involves the incorporation of an unnatural amino acid using an orthogonal-tRNA/aminoacyl-tRNA synthetase pair (Wang et al. Reference Wang, Brock, Herberich and Schultz2001). The spin label site is generated by introducing an amber (TAG) codon into the DNA sequence of the protein of interest by site-directed mutagenesis. The protein is then coexpressed with the unnatural tRNA/aminoacyl-tRNA synthetase pair. This method can be used to produce a protein containing the unnatural amino acid p-acetyl-l-phenylalanine, which can be conjugated to HO-4120, a hydroxylamine-functionalized nitroxide spin-label (Fleissner et al. Reference Fleissner, Brustad, Kalai, Altenbach, Cascio, Peters, Hideg, Peuker, Schultz and Hubbell2009) (Fig. 6d). This method has been used to produce spin-labeled proteins at high yields, even in perdeuterated media, where it has been demonstrated that the key for sufficient yields is to induce expression of the tRNA/aminoacyl-tRNA synthetase pair while still in rich, protonated media before changing to deuterated media for expression of the target protein (Venditti et al. Reference Venditti, Fawzi and Clore2012). For shorter sequences (up to ~50 residues), a spin-labeled amino acid can be directly incorporated through solid-phase peptide synthesis (Hanson et al. Reference Hanson, Millhauser, Formaggio, Crisma and Toniolo1996; Lindfors et al. Reference Lindfors, De Koning, Drijfhout, Venezia and Ubbink2008; Milov et al. Reference Milov, Tsvetkov, Gorbunova, Mustaeva, Ovchinnikova and Raap2002; Monaco et al. Reference Monaco, Formaggio, Crisma, Toniolo, Hanson, Millhauser, George, Deschamps and Flippen-Anderson1999a, Reference Monaco, Formaggio, Crisma, Toniolo, Hanson and Millhauserb; Shenkarev et al. Reference Shenkarev, Paramonov, Balashova, Yakimenko, Baru, Mustaeva, Raap, Ovchinnikova and Arseniev2004).

Some spin label advances have been summarized in recent reviews (Hubbell et al. Reference Hubbell, Lopez, Altenbach and Yang2013; Jeschke, Reference Jeschke2013; Keizers & Ubbink, Reference Keizers and Ubbink2011), and Fig. 7 shows some nitroxide and metal-chelating paramagnetic tags. The paramagnetic nitroxide side chain R1, which is formed by disulfide conjugation of the spin label MTSL to cysteine (Fig. 7a; Battiste & Wagner, Reference Battiste and Wagner2000; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000), is commonly used in EPR studies and has been used to measure PREs in several studies (Anthis & Clore, Reference Anthis and Clore2013; Anthis et al. Reference Anthis, Doucleff and Clore2011; Bermejo et al. Reference Bermejo, Strub, Ho and Tjandra2009; Fawzi et al. Reference Fawzi, Fleissner, Anthis, Kalai, Hideg, Hubbell and Clore2011a; Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006). For studies involving R1, an acylated diamagnetic control is available (Fig. 7b). Alternatively, for PRE measurements, an EDTA-based tag (Fig. 7e), which can chelate a paramagnetic ion (e.g. Mn²⁺) or a diamagnetic control (e.g. Ca²⁺), can be used (Dvoretsky et al. Reference Dvoretsky, Gaponenko and Rosevear2002; Ebright et al. Reference Ebright, Chen, Pendergrast and Ebright1992; Ermacora et al. Reference Ermacora, Delfino, Cuenoud, Schepartz and Fox1992). Such tags have the advantage of giving larger PRE values due to larger electron spin number. However, these metal-bound tags are problematic for PCS studies because their flexibility greatly complicates analysis of PCSs, which are orientation dependent. Additionally, metal-bound EDTA forms two stereoisomers, each with a slightly different χ-tensor, causing the appearance of multiple peaks due to slow metal/tag exchange (Ikegami et al. Reference Ikegami, Verdier, Sakhaii, Grimme, Pescatore, Saxena, Fiebig and Griesinger2004; Pintacuda et al. Reference Pintacuda, Moshref, Leonchiks, Sharipo and Otting2004b; Vlasie et al. Reference Vlasie, Comuzzi, Van Den Nieuwendijk, Prudencio, Overhand and Ubbink2007). Metal-binding tags have also been developed with multiple functionalities, for example 4^′-mercapto-2,2^′:6^′,2″-terpyridine-6,6″-dicarboxylic acid (4MTDA), which binds to lanthanides and can be used for the measurement of PRE and PCSs, but also fluoresces in the UV–vis range (Huang et al. Reference Huang, Pei, Zuo, Chen, Yang and Su2013).

Fig. 7.

Spin label side chains for measuring PRE. Shown are three nitroxide spin label side chains (a, c, d), a diamagnetic control (b), and two side chains for chelating paramagnetic metal ions (e, f). The unpaired electron in each nitroxide side is denoted with a dot, and metal chelating sites are denoted with stars. The C_α of the peptide backbone is labeled. (a) The side chain R1, introduced by the reaction of (1-oxyl-2,2,5,5-tetramethyl-∆3-pyrroline-3-methyl) methanethiosulfonate (MTSL) with a cysteine side chain (Battiste & Wagner, Reference Battiste and Wagner2000; Gaponenko et al. Reference Gaponenko, Howarth, Columbus, Gasmi-Seabrook, Yuan, Hubbell and Rosevear2000). (b) An acylated derivative of R1, which can be used as a diamagnetic control for R1/MTSL PRE studies, introduced by the reaction of (1-acetoxy-2,2,5,5-tetramethyl-δ-3-pyrroline-3-methyl) methanethiosulfonate with a cysteine side chain (Altenbach et al. Reference Altenbach, Oh, Trabanino, Hideg and Hubbell2001). (c) The side- chain R1p, a derivative of R1 that adopts a narrower range of conformations and simplifies analysis, introduced by the reaction of 3-methanesulfonilthiomethyl-4-(pyridin-3-yl)-2,2,5,5-tetramethyl-2,5-dihydro-1H-pyrrol-1-yloxyl radical (HO-3606) with a cysteine side chain (Fawzi et al. Reference Fawzi, Fleissner, Anthis, Kalai, Hideg, Hubbell and Clore2011a). (d) The side chain RX, which also is more rigid than R1, introduced by the reaction of 3,4-bis-(methanethiosulfonylmethyl)-2,2,5,5-tetramethyl-2,5-dihydro-1H-pyrrol-1-yloxy radical (HO-1944) with two cysteine side chains located within close proximity (specifically at i and i + 3 or i + 4 in a helix or at i and i + 2 in a β-strand) (Fleissner et al. Reference Fleissner, Bridges, Brooks, Cascio, Kalai, Hideg and Hubbell2011). (e) A side chain for chelating paramagnetic metal ions (e.g. Mn²⁺) or diamagnetic controls (e.g. Ca²⁺), introduced by the reaction of S-(2-pyridylthio)cysteaminyl-EDTA with a cysteine side chain (Dvoretsky et al. Reference Dvoretsky, Gaponenko and Rosevear2002; Ebright et al. Reference Ebright, Chen, Pendergrast and Ebright1992; Ermacora et al. Reference Ermacora, Delfino, Cuenoud, Schepartz and Fox1992). (f) The CLaNP (caged lanthanide NMR probe) tags chelate lanthanide ions and offer a more rigid paramagnetic center, suitable for both PRE and PCS measurements. Shown here is the side chain introduced by conjugating 1,4,7,10-tetraazacyclododecane-1,7-[di-(N-oxido-pyridine-2-yl)methyl]-4,10-bis(2-(acetylamino)ethylmethanesulfonothioate) (CLaNP-5) to two cysteine side chains located in close proximity to one another (Keizers et al. Reference Keizers, Desreux, Overhand and Ubbink2007, Reference Keizers, Saragliadis, Hiruma, Overhand and Ubbink2008).

A cysteine-conjugated MTSL tag (i.e. residue R1; Fig. 7a), has five torsional angles about which rotation is possible. This allows the tag to sample a large range of conformational space and makes it highly dynamic (Bridges et al. Reference Bridges, Hideg and Hubbell2010; Chou et al. Reference Chou, Case and Bax2003) (overall correlation time of the order of ~0·7 ns (Oganesyan, Reference Oganesyan2011)). However, the use of a shorter spin label increases the likelihood of undesired perturbations to the structure or dynamics of the molecule under study (Jeschke, Reference Jeschke2013). Thus, whereas the dynamics of the spin label have complicated analysis and impeded progress in the past, newer computational methods described below have made it possible to take these dynamics into account, making it preferable to use the more flexible tag that is less likely to disrupt the protein under study. NMR studies attempting to quantitatively interpret the PREs arising from R1 (Bermejo et al. Reference Bermejo, Strub, Ho and Tjandra2009), like any other label with internal motions, must take into account the ensemble of nitroxide positions, their interconversion dynamics, and the order parameters for the interspin (nitroxide–nucleus) vectors (Clore & Iwahara, Reference Clore and Iwahara2009; Iwahara et al. Reference Iwahara, Schwieters and Clore2004b).

Various analogs of R1 (Columbus et al. Reference Columbus, Kalai, Jeko, Hideg and Hubbell2001) have shown reduced flexibility, thus potentially simplifying analysis. A series of R1 analogs aromatically substituted at position 4 of the paramagnetic pyrroline ring exhibit restricted motion, and the 4-pyridyl analog of R1 (R1p; Fig. 7c) shows particular promise. In one study, multiple tag conformations were required to fit PRE data from R1, but a single conformation could be used for the R1p tag (Fawzi et al. Reference Fawzi, Fleissner, Anthis, Kalai, Hideg, Hubbell and Clore2011a). A different nitroxide side chain (V1), formed by the reaction between a free cysteine and bis(2,2,5,5-tetramethyl-3-imidazoline-1-oxyl-4-il)-disulfide (IDSL), also displays more constrained geometry than R1 (Toledo Warshaviak et al. Reference Toledo Warshaviak, Khramtsov, Cascio, Altenbach and Hubbell2013). If two cysteines are available in close proximity (at i and i + 3 or i + 4 in a helix or at i and i + 2 in a β-strand), the side chain RX (Fig. 7d) can be introduced, forming two disulfide linkages and maintaining a more rigid conformation than R1 (Fleissner et al. Reference Fleissner, Bridges, Brooks, Cascio, Kalai, Hideg and Hubbell2011). An alternative strategy for a more rigid tag is the incorporation of the unnatural amino acid TOAC into the peptide of interest for PRE studies (Lindfors et al. Reference Lindfors, Venkata, Drijfhout and Ubbink2011). TOAC contains a nitroxide radical that is contained within a single piperidine ring that includes the backbone C_α. Its downsides are that it can only be incorporated into synthetic peptides (not recombinant proteins), and it can still adopt multiple conformations (Crisma et al. Reference Crisma, Deschamps, George, Flippen-Anderson, Kaptein, Broxterman, Moretto, Oancea, Jost, Formaggio and Toniolo2005; Flippen-Anderson et al. Reference Flippen-Anderson, George, Valle, Valente, Bianco, Formaggio, Crisma and Toniolo1996).

More rigid tags have also been designed to introduce paramagnetic lanthanides into proteins via chelating groups. These are particularly valuable for PCS studies, where rigid electron–nucleus vectors are needed due to the orientational dependence of the PCS. A series of CLaNP (caged lanthanide NMR probe) tags have been developed, which form a much more rigid structure. After chelating a lanthanide ion, the CLaNP tag can then be conjugated to a protein via two cysteine residues in close proximity on the surface of the protein. One of these, CLaNP-5 (Keizers et al. Reference Keizers, Desreux, Overhand and Ubbink2007, Reference Keizers, Saragliadis, Hiruma, Overhand and Ubbink2008) is shown in Fig. 7f. CLaNP-7, has a pH-dependent χ-tensor and has a lower charge (+1 versus +3 for CLaNP-5), lessening the potential for electrostatic interactions with negatively charged molecules (Liu et al. Reference Liu, Keizers, Hass, Blok, Timmer, Sarris, Overhand and Ubbink2012). Another tag, 4-mercaptomethyl-dipicolinic acid, adopts a rigid conformation after conjugation to a surface cysteine located near a carboxyl-bearing side chain, followed by addition of a lanthanide ion (Shen et al. Reference Shen, Vernon, Baker and Bax2009). Another strategy for introducing a paramagnetic lanthanide uses stereospecifically methylated DOTA (DOTA-M8), which is conjugated to the protein via a single surface cysteine, and adopts a more rigid conformation due to hydrophobic interactions on the protein surface (Haussinger et al. Reference Haussinger, Huang and Grzesiek2009).

It is essential to ensure that the introduction of the mutation and the introduction of the tag do not affect the structure or function of the protein. This can preferably be done through a functional assay if one is available. Biophysical measures (e.g. CD, melting temperature) can be used to ensure that there are no changes to protein structure or stability. If the protein is not expressed or becomes insoluble, this likely indicates that the mutation has disrupted the structure of the protein and made it unstable. If it is possible to acquire an NMR spectrum (e.g. ¹H–¹⁵N HSQC) of the mutant protein and compare it to the spectrum of the wild-type protein, there should not be any broadened or disappearing peaks, and only peaks near the site of the mutation should exhibit changes in chemical shift. One can then check a spectrum of the protein conjugated to a diamagnetic tag. These results should also be the same. Some very localized disruption of the protein structure may be inevitable and they can be considered acceptable if they are restricted to adjacent residues. When PRE data are acquired, the R ₂ rate of the diamagnetic control should not vary from that of the wild type. Enhancements to the diamagnetic R ₂ can indicate some disruption of local structure. Any significant disruption needs to be taken into account and may indicate that a particular tag position is not appropriate. Finding the best sites for a paramagnetic tag often involves trial and error, as tags at various different positions may cause disruptions to protein structure.

3.2.2 Measuring PRE data

The transverse PRE rate (Γ₂) of a nucleus is given by the difference in its transverse relaxation rate between the paramagnetic (R _2,para) and diamagnetic (R _2,dia) states of the molecule:

(58)$${\rm \Gamma} _2 = R_{2,{\rm para}} - R_{2,{\rm dia}} $$

The PRE is determined by measuring R ₂ values for a paramagnetic sample and for a diamagnetic sample and then subtracting the diamagnetic R ₂ values from the paramagnetic R ₂ values. The most appropriate diamagnetic sample will depend on the nature of the paramagnetic sample. If the PREs arise from a paramagnetic ion, an appropriate diamagnetic control would be a similar sample with the paramagnetic ion replaced with a diamagnetic ion of similar properties. For a metalloprotein, this might be the natural metal ligand. Or, for a chelating extrinsic tag for example, one could use Mn²⁺ for the paramagnetic sample and Ca²⁺ for the diamagnetic sample. For a protein tagged with a nitroxide label, a good diamagnetic sample is a protein tagged with a similar diamagnetic label. For MTSL, one could use an acylated analogous tag ((1-acetoxy-2,2,5,5-tetramethyl-δ-3-pyrroline-3-methyl) methanethiosulfonate) for the diamagnetic control. An alternative is to add ascorbic acid directly to the paramagnetic sample, which reversibly reduces nitroxides, forming a diamagnetic product. This method has the advantage of only requiring one sample, but has the downside of requiring a large excess of ascorbic acid, which must remain present during acquisition (Venditti et al. Reference Venditti, Fawzi and Clore2011). These methods are generally preferable, however, to using a wild-type or untagged protein as the diamagnetic control, as the tag can have subtle local effects that will not be taken into account in the wild-type protein.

Commonly, one measures the PRE of the backbone amide protons (¹H_N) using a 2D ¹H–¹⁵N HSQC-based pulse sequence (Iwahara et al. Reference Iwahara, Tang and Clore2007). Two examples are shown in Fig. 8. In the first example (Fig. 8a), the period for the ¹H transverse relaxation measurement (T) is incorporated in the first INEPT scheme. Identical experiments are performed for the paramagnetic and diamagnetic samples to obtain Γ₂. This pulse sequence can readily be modified to a transverse relaxation optimized (TROSY) version, although Boltzmann ¹⁵N-magnetization should be destroyed at the beginning the pulse sequence so that all observables are modulated solely by ¹H transverse relaxation during the time interval T (Clore & Iwahara, Reference Clore and Iwahara2009). If a TROSY pulse sequence is being used, it is advantageous to measure the slower TROSY R ₂ rate for each sample by placing the delay T at the end of the sequence (Fig. 8b). Additionally, 3D experiments can also be employed to overcome peak overlap. HNCO-based experiments measuring ¹H_N–Γ₂ have been developed for this purpose (Anthis et al. Reference Anthis, Doucleff and Clore2011; Hu et al. Reference Hu, Vogeli and Clore2007).

Fig. 8.

Pulse sequences for 2D ¹H_N–Γ₂ PRE experiments. (a) HSQC-based pulse sequence for measuring ¹H_N–Γ₂. Full details are given in Iwahara et al. (Iwahara et al. Reference Iwahara, Tang and Clore2007). The delay T is incremented to measure ¹H_N transverse relaxation. Two 180° ¹⁵N pulses are applied to eliminate longitudinal cross-correlated relaxation interference between the ¹⁵N–¹H dipolar interaction and ¹⁵N chemical shift anisotropy. Although ³J _HN−Hα is active for non-deuterated proteins during the period T, the resulting modulation is cancelled out when Γ₂ is measured by the two-time-point method, using an equal delay T in both the paramagnetic experiment and the diamagnetic control. Alternatively, ³J _HN−Hα can be directly cancelled by replacing the 180° ¹H pulse separating the two halves of delay T (arrow) with an IBURP2 pulse that selectively excites only the amide region of the ¹H spectrum. (b) TROSY pulse sequence for measuring ¹H_N–Γ₂. Note that in this pulse sequence the delay T (located at the end of the pulse sequence) is incremented to measure the slower-relaxing TROSY component of the ¹H_N transverse relaxation rate (although ¹H_N–Γ₂ is the same as for R ₂ measurements). For proteins that are not deuterated, ³J _HN−Hα can be cancelled out either by employing the two-time-point method or by replacing the final 180° ¹H pulse and adjacent soft water flip-back pulses separating the two halves of delay T (arrow) with an IBURP2 pulse that selectively excites only the amide region of the ¹H spectrum. Unless otherwise indicated, all pulses have phase x; φ ₁ = y, −y, −x, x; φ _rec = y, −y, x, −x. Quadrature detection in the indirect ¹⁵N dimension is achieved using the Rance-Kay echo/anti-echo scheme (Kay et al. Reference Kay, Keifer and Saarinen1992) by inverting gradients G3 and −G3 and using phase φ ₁ = y, −y, x, −x for the second free induction decay (FID) of each quadrature pair.

The ¹H–Γ₂ PRE can also be measured for protons directly bonded to ¹³C (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b), and the measurement of methyl ¹H–Γ₂ can be particularly useful for larger proteins (Fig. 8). (Note that although R ₂ for methyl CH₃ protons is biexponential due to cross-correlation (Ollerenshaw et al. Reference Ollerenshaw, Lidar and Kay2003), the two-time point procedure described below still gives reasonably accurate Γ₂ values as long as the PRE affects both components of the biexponential decay equally (Clore & Iwahara, Reference Clore and Iwahara2009)). A scheme has been developed to automatically assign the methyl resonances of a protein given only the measured PRE values and the 3D atomic coordinates of the molecule (Venditti et al. Reference Venditti, Fawzi and Clore2011).

PRE rates are measured by recording spectra with different time delays (T) in an interleaved manner. R ₂ rates are traditionally measured using multiple time points (~8), from 0 × to ~1·3 × T ₂. For a paramagnetic experiment, however, more time points may be required to cover the slow and fast relaxing peaks. For each peak, a plot of its intensity versus time is fit to a single-exponential decay to determine R ₂. Errors can be determined by Monte Carlo analysis. Peaks with a very high PRE will have very low intensity or will have completely disappeared, and it will not be possible to accurately measure their R ₂ values.

For measuring PRE rates, however, one can also employ a two-time-point measurement. The merits of such a strategy have been discussed at length previously (Clore & Iwahara, Reference Clore and Iwahara2009; Iwahara et al. Reference Iwahara, Tang and Clore2007), and the use of two time points can offer several-fold time savings over multiple time points. Two-time-point measurements are particularly suitable for PRE measurements because, by subtracting the diamagnetic R ₂ from the paramagnetic R ₂, relaxation mechanisms common to both states are cancelled out, leaving only the PRE. It is important to note that the same delay (ΔT) must be used for both the diamagnetic and paramagnetic samples to cancel artifacts present in both experiments. (In the case of paramagnetic systems with an anisotropic g-tensor, however, exchange contributions can be significantly different between the paramagnetic and diamagnetic states; Clore & Iwahara, Reference Clore and Iwahara2009).

In addition to the time savings of reduced data acquisition times, the two-time-point measurement offers the added advantage of allowing one to calculate the PRE rate directly from the intensity of a peak in just four spectra without requiring any curve fitting:

(59)$${\rm \Gamma} _2 = \displaystyle{1 \over {T_b - T_a}} \ln \displaystyle{{I_{{\rm dia}} (T_b )I_{{\rm para}} (T_a )} \over {I_{{\rm dia}} (T_a )I_{{\rm para}} (T_b )}},$$

where I _dia(T) and I _para(T) are the peak intensities for the diamagnetic and paramagnetic states, respectively, at time T. The total transverse relaxation time, ΔT, is given by T _b − T _a. The error in PRE can be propagated from the spectral noise using:

(60)$$\sigma {\rm (}{\rm \Gamma} _2 {\rm )} = \displaystyle{1 \over {T_b - T_a}} \sqrt {\left( {\displaystyle{{\sigma _{{\rm dia}}} \over {I_{{\rm dia}} (T_a )}}} \right)^2 + \left( {\displaystyle{{\sigma _{{\rm dia}}} \over {I_{{\rm dia}} (T_b )}}} \right)^2 + \left( {\displaystyle{{\sigma _{{\rm para}}} \over {I_{{\rm para}} (T_a )}}} \right)^2 + \left( {\displaystyle{{\sigma _{{\rm para}}} \over {I_{{\rm para}} (T_b )}}} \right)^2}, $$

where σ _dia and σ _para are the standard deviations of the noise in the spectra of the diamagnetic and paramagnetic states, respectively. When using the two-time-point method, the relative error in the recorded PRE value can be minimized by carefully choosing the ΔT value between the two time points. The exact choice of time points will depend on the intrinsic R ₂ of the molecule and the anticipated PRE values. The value of ΔT that gives the minimum percent error in a two-time-point measurement is given by:

(61)$${\rm \Delta} T \approx \displaystyle{{1 \!\cdot \!15} \over {R_2 + {\rm \Gamma} _2}}, $$

where the exact value of the numerator in Eq. (61) depends on the specific values of R ₂ and Γ₂ (ranging from ~1·1 to ~1·2, but 1·15 is a useful rule of thumb). The Γ₂ value will vary significantly between different ¹H_N nuclei (as will R ₂ to a lesser extent), so one must choose the PRE value that one will want to measure with the greatest precision, giving larger relative errors for values of R ₂ + Γ₂ higher and lower than the chosen value. For a protein with average ¹H_N–R₂ values of 20 s⁻¹, to optimally measure a PRE of 40 s⁻¹ (in the middle of the generally useful range of 0 to ~80 s⁻¹), the optimal value of ΔT would be ~ 1·15/(20 s⁻¹ + 40 s⁻¹) = 19 ms.

A limitation of the two-time-point method is that the results can be heavily influenced by the presence of diamagnetic contamination (e.g. incomplete tagging or dimerization in aged samples). This effect increases for higher amounts of contamination and for larger PRE values, and under extreme cases of diamagnetic contamination causes an apparent PRE much smaller than the true PRE. If one is not careful, peaks close to the noise in cases with large PREs can also suffer from this problem. If diamagnetic contamination is greater than ~3%, smaller values of ΔT are required for accurate determination of the PRE than predicted by Eq. (61). Diamagnetic contamination can be caused by a variety of factors, including: (a) incomplete conjugation of the extrinsic paramagnetic group and insufficient purification of the conjugated species; (b) the presence of diamagnetic impurities in the paramagnetic stock solution (e.g. trace amounts of diamagnetic metals such as Zn²⁺ and Ca²⁺ in stock solutions of Mn²⁺ or the reduced species of a nitroxide spin label); (c) chemical instability of the conjugated states, especially for disulfide linkages. Thus, it is important to determine the extent of such contamination and what effect it may have on the measured ¹H–Γ₂ values (Clore & Iwahara, Reference Clore and Iwahara2009; Iwahara et al. Reference Iwahara, Tang and Clore2007). On the other hand, intermolecular PRE measurements are not as drastically affected by diamagnetic contamination. As long as the intermolecular association process occurs on a fast timescale, the apparent PRE is simply the true PRE scaled by a factor of (1 − p _dia) (Clore & Iwahara, Reference Clore and Iwahara2009).

For a sample with no diamagnetic contamination and an infinite S/N ratio, a plot of apparent PRE versus true PRE is a straight line with a slope of 1. In the presence of diamagnetic contamination or significant noise, such a plot still gives a slope of ~1 for low PRE values, but as the true PRE increases, the slope of the line decreases, eventually becoming negative, yielding apparent PRE values of ~0 for very large true PREs. One method to check for whether a PRE value is inaccurate due to diamagnetic contamination or another problem (e.g. a low S/N ratio) is to plot the PRE versus log(I _dia/I _para). Most peaks should follow a straight line with a positive slope, but outliers far below this line deserve further scrutiny.

Some studies have taken an approach that we will term a ‘single-time-point’ method, analyzing the ratio of I _para to I _dia, sometimes inappropriately referring to this ratio as the PRE. Because such a ratio depends on ¹H–Γ₂ and ¹H–Γ₁, it is influenced by the delays for coherence transfers, the delays between experiments, and the type of data processing (Clore & Iwahara, Reference Clore and Iwahara2009). Thus, it alone is not physically meaningful, and it is more suitable for qualitative analysis, although quantitative analysis can be performed with certain caveats. Under conditions of full magnetization recovery between experiments, the ratio of peak intensities in the diamagnetic and paramagnetic spectrum are given by:

(62)$$I_{{\rm para}} (0) = I_{{\rm dia}} (0)\displaystyle{{R_{2,{\rm dia}}} \over {R_{2,{\rm dia}} + {\rm \Gamma} _2}} \exp ( - {\rm \Gamma} _2 \tau ),$$

where τ is the overall ¹H transverse period for the given pulse sequence (e.g. ~10 ms). In addition to requiring a long inter-scan delay (T _rep ≫ ¹H–T ₁), such an approach requires identical experimental conditions (sample concentration, number of scans, etc.) for the diamagnetic and paramagnetic samples as well as a Lorentzian line shape in the ¹H dimension (Iwahara et al. Reference Iwahara, Tang and Clore2007). If the ¹H–R ₂ relaxation rates for the diamagnetic sample are not already known, they will need to be measured separately. Additionally, if an inter-scan delay long enough for full-spin relaxation is not used, one must also take into account the faster longitudinal relaxation in the paramagnetic sample due to the longitudinal PRE (¹H–Γ₁):

(63)$$I_{{\rm para}} (0) = I_{{\rm dia}} (0)\displaystyle{{R_{2,{\rm dia}}} \over {R_{2,{\rm dia}} + {\rm \Gamma} _2}} \displaystyle{{1 - \exp \left\{ { - (R_{1,{\rm dia}} + {\rm \Gamma} _1 )T_{{\rm rep}}} \right\}} \over {1 - \exp ( - R_{1,{\rm dia}} T_{{\rm rep}} )}}\exp ( - {\rm \Gamma} _2 \tau ).$$

In the absence of internal dynamics, ¹H–Γ₁ can be calculated directly from ¹H–Γ₂, using a formula based on Eqs. (44), (46), and (47):

(64)$${\rm \Gamma} _1 = {\rm \Gamma} _2 \left( {\displaystyle{2 \over 3}\omega _{\rm I}^2 \tau _{\rm c}^2 + \displaystyle{7 \over 6}} \right)^{ - 1}, $$

where τ _c is in units of seconds and ω _I is the ¹H frequency in radians·s⁻¹. To avoid having to use Eq. (64), one must use a long repetition delay (>4 × T _1,dia), especially for large and/or deuterated molecules (Iwahara et al. Reference Iwahara, Tang and Clore2007). Thus, under these circumstances, performing a single-time-point measurement offers no appreciable time savings over directly measuring the diamagnetic and paramagnetic transverse relaxation rates using multiple time points. Additionally, the requirement of a Lorentzian line shape for the single-time-point experiment limits data processing to the use of either an exponential window function or no window function at all for the direct ¹H dimension. The use of broad Lorentzian lines can lead to increased peak overlap and can be especially problematic for larger proteins. This restriction does not apply to multiple-time-point experiments.

When conducting PRE experiments, the use of concentrations of paramagnetic protein that are too high can lead to PREs from random collisions between molecules, similar to solvent PREs. Relatively low concentrations (~0·3 mM) of sample can be used to avoid this undesirable effect, which can be detected by performing experiments with different concentrations, as the solvent PRE is directly proportional to the concentration of the paramagnetic species. For intramolecular PRE measurements, one can perform a control intermolecular experiment, where instead of having the paramagnetic spin label attached to the isotopically labeled species, one can make a sample with a mix of isotopically labeled (but not spin labeled) protein and paramagnetically tagged (but not isotopically labeled) protein. Any PRE measured in such an experiment would be from intermolecular effects.

Two-time-point PRE measurements can be performed within 1–2 days for a well-behaved small-medium-sized protein. For example, after 20 h each of measurement time for the paramagnetic and diamagnetic control, samples of 0·3 mM perdeuterated CaM–4Ca²⁺ measured using a two-time-point (20 ms delay) TROSY 2D ¹⁵N–¹H_N experiment at 600 MHz on a cryogenically cooled probe gave errors in Γ₂ (calculated from S/N using Eq. (60)) of 1–2% for Γ₂ ≈ 20 s⁻¹ and 2–10% for Γ₂ ≈ 50 s⁻¹ (Anthis et al. Reference Anthis, Doucleff and Clore2011).

3.3.1 Fitting PRE data

Once PRE data are acquired, further analysis is needed to gain insight into macromolecular structure; this often involves back-calculating PRE values from atomic coordinates (either to test the agreement between experimental PRE values and a known structure or as part of a routine to calculate a new structure or ensemble from PRE data). Although it is tempting to interpret PREs directly as distances, this approach is problematic due to the flexibility of most common PRE spin labels (and the inherent flexibility of the macromolecules under study). As the PRE is a population-weighted 〈r ⁻⁶〉 average of all states in exchange on a fast timescale, the PRE generally does not correspond to a single distance, and the contribution of a particular state can be very different than its fractional population (hence the utility of PRE for detecting minor states). Historically, this has made rigorous treatment of PRE data complicated, but computational advances have made such studies feasible.

One solution has been the incorporation of an ensemble average approach (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b), which has been implemented in the macromolecular structure determination software package Xplor-NIH (Schwieters et al. Reference Schwieters, Kuszewski, Tjandra and Clore2003, Reference Schwieters, Kuszewski and Clore2006). Here, the paramagnetic group is represented by an ensemble of multiple conformers, and PRE values are the sum of the PREs for the individual conformers. Thus, using the ‘prePot’ pseudoenergy term in Xplor-NIH, one can refine directly against the experimental PRE values. Scripts and detailed instructions for these procedures are available online at http://spin.niddk.nih.gov/clore.

The first step in back-calculating PRE data in Xplor-NIH is the preparation of a suitable .pdb file, beginning with the in silico attachment of a spin label. The process is streamlined for common MTSL and EDTA tags. To prepare the .pdb file, one first changes the name of the residue where the tag is to be attached to ‘CYSP’ (for MTSL) or ‘CED’ (for EDTA). A script can then be run that fills in the missing atoms and regularizes their geometry (using the regularize module in Xplor-NIH). If the tag is not already in the Xplor-NIH library, one must generate parameter files and then attach the tag to the structure in silico. In order to use the SBMF PRE method, multiple conformations (at least 3) of the tag need to be generated.

Tag conformations are then refined by minimizing the target function for the PRE restraints, E _PRE, defined as:

(65)$$E_{{\rm PRE}} = k_{{\rm PRE}} \sum\limits_i {w_i \left( {{\rm \Gamma} _2^{{\rm obs}} (i) - {\rm \Gamma} _2^{{\rm calc}} (i)} \right)^2}, $$

where k _PRE is a force constant, w _i is a weighting factor, Γ₂^obs(i) is the observed PRE, and Γ₂^calc(i) is the back-calculated PRE. The paramagnetic group is represented by an ensemble of N different conformers, while the remainder of the molecule is represented by a single structure. The individual members of the ensemble are treated independently and are allowed to overlap by excluding their interactions from the non-bonded energy term. This approach provides equal weights of 1/N to each member of the ensemble, but such an approach can account for uneven distributions if a larger N (>3) is used, since members of the ensemble can overlap in the simulation. The ensembleSimulation (Clore & Schwieters, Reference Clore and Schwieters2004b) module in Xplor-NIH (Schwieters et al. Reference Schwieters, Kuszewski, Tjandra and Clore2003, Reference Schwieters, Kuszewski and Clore2006) can also be used to explicitly specify or optimize different weights for each member of the ensemble (Deshmukh et al. Reference Deshmukh, Schwieters, Grishaev, Ghirlando, Baber and Clore2013).

To calculate the PRE value, the ensemble-averaged distance 〈r ⁻⁶〉 for the distance between the observed nucleus (n) and the paramagnetic center (e) must be calculated as follows (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b):

(66)$$\left\langle {r^{ - 6}} \right\rangle = \displaystyle{1 \over {Nn_{\rm p}}} \sum\limits_e^N {\sum\limits_n^{n_{\rm p}} {r_{en}^{ - 6}}}, $$

where N is the number of ensembles used to represent the paramagnetic group and n _p is the number of equivalent protons (e.g. 1 for NH or CH, 2 for NH₂ or CH₂, and 3 for CH₃). Γ₂ can then be directly calculated from the SB (Eq. (44)) or SBMF (Eq. (49)) equations; the SBMF equation is more accurate but requires additional computational time). Use of the SBMF equation requires the determination of an order parameter (S²) for each electron–nucleus vector, as defined by Eq. (51). In Xplor-NIH, order parameters are calculated on the fly from the ensemble of paramagnetic conformers, operating under the assumption that the fluctuations in the position of the measured nucleus are small compared with those of the paramagnetic center. Order parameters can be calculated using Eq. (52) together with the following equations (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b):

(67)$${\bi S}_{{\rm PRE,angular}}^2 = \displaystyle{1 \over {N^2 n_{\rm p}^2}} \sum\limits_{e,d}^N {\sum\limits_{n.m}^{n_{\rm p}} {\left\{ {\displaystyle{3 \over 2}\left( {\displaystyle{{\vec r_{en} \bullet \vec r_{dm}} \over {r_{en} r_{dm}}}} \right)^2 - \displaystyle{1 \over 2}} \right\}}}, $$

(68)$${\bi S}_{{\rm PRE,radial}}^2 = \displaystyle{{\left( {\sum\limits_e^N {\sum\limits_n^{n_{\rm p}} {r_{en}^{ - 3}}}} \right)^2} \over {Nn_{\rm p} \sum\limits_e^N {\sum\limits_n^{n_{\rm p}} {r_{en}^{ - 6}}}}}, $$

where r _en and r _dm are vectors between the paramagnetic center (e, d) and the observed nucleus (n, m) in two different conformers. The values calculated in Eqs. (67) and (68) are the sum of contributions for each pair of conformers in the ensemble.

To fit the PRE values to the structure of a protein, a simulated annealing routine is performed with the protein molecule held rigid and the tags allowed to move to fit the data. Adjacent side chains can also be allowed to move to resolve steric overlap with the tag. At each step, Xplor-NIH will recalculate the order parameter for each PRE vector from the position of the different members of the tag ensemble. Fitting also requires the specification of correlation times, which can be explicitly provided or fit along with the tag positions in Xplor-NIH. If using the SB mode (i.e. Eq. (44)), only a single overall correlation time (τ _c) is required, which depends on both the macromolecule rotational correlation time (τ _r) and the electron spin relaxation time (τ _s) (Eq. (45)). If τ _s ≫ τ _r (e.g. nitroxide spin labels (Bernini et al. Reference Bernini, Venditti, Spiga and Niccolai2009)), then the value of τ _s can be ignored, and τ _c is just simply τ _r. However, if τ _s is of the order of, or shorter than, τ _r (e.g. paramagnetic metal ions), then the value of τ _s must be taken into account. Under rigid SB assumptions, τ _c can be calculated directly from the ratio of transverse and longitudinal PRE values (Iwahara et al. Reference Iwahara, Anderson, Murphy and Clore2003):

(69)$$\tau _{\rm c} = \displaystyle{1 \over {\omega _{\rm I}}} \sqrt {\displaystyle{3 \over 2}\displaystyle{{{\rm \Gamma} _2} \over {{\rm \Gamma} _1}} - \displaystyle{7 \over 6}}, $$

where τ _c is in units of seconds and ω _I is the ¹H frequency in rad·s⁻¹. Equation (69) is only formally true for the SB approximation; however, even under SBMF conditions it offers a useful estimate of τ _c (yielding maximum errors in the effective Γ₂^−1/6-calculated distance r of 4% for an order parameter of S² = 0·5, with smaller errors for larger order parameters) (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). To avoid these errors, however, and in the absence of additional data, τ _c can instead be optimized within a pre-defined range (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). If one is using the SBMF mode (i.e. Eq. (49)), which is more appropriate unless a rigid paramagnetic center is used, one must also take into account (in addition to τ _c) the total correlation time τ _t that incorporates τ _r, τ _s, and the internal correlation time for the PRE vector (τ _i) (Eq. (G)). If τ _r ≫ τ _s and τ _r ≫ τ _i, one can use the approximation τ _t ≈ 0, which assumes rapid motion of the PRE spin label. This in effect makes the order parameter S² a scaling factor for the back-calculation of PRE data. This approximation does not hold true for shorter values of τ _s. The prePot potential in Xplor-NIH also offers the option of optimizing τ _c and/or τ _t (or τ _i) as part of the simulated annealing routine. Doing so then allows calculations to be performed with no prior knowledge of correlation times.

With these factors in place, Xplor-NIH can perform a simulated annealing calculation, refining directly against the experimental PREs, minimizing a target function including a pseudoenergy term calculated from the difference between back-calculated and experimental PRE values as well as terms to maintain proper geometry and avoid steric clashes. The goodness of fit of the PREs can be assessed by the Q-factor, which is analogous to the crystallographic R-factor and decreases with improved agreement between the observed (Γ₂^obs) and back-calculated (Γ₂^calc) rates (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b):

(70)$$Q = \sqrt {\displaystyle{{\sum\limits_i {\left( {{\rm \Gamma} _2^{{\rm obs}} (i) - {\rm \Gamma} _2^{{\rm calc}} (i)} \right)^2}} \over {\sum\nolimits_i {{\rm \Gamma} _2^{{\rm obs}} (i)^2}}}}. $$

Alternatively, one can also use a reduced χ ² statistic:

(71)$$\chi ^2 = \displaystyle{1 \over M}\sum\limits_N {\left( {\displaystyle{{{\rm \Gamma} _2^{{\rm obs}} - {\rm \Gamma} _2^{{\rm calc}}} \over {\sigma \left( {{\rm \Gamma} _2^{{\rm obs}}} \right)}}} \right)} ^2, $$

where the degrees of freedom (M) are given by:

(72)$$M = N - 3C,$$

where N is the number of observed PREs and C the number of spin label conformers in the fitting ensemble (C is multiplied by three for the three translational degrees of freedom involved in fitting each nitroxide position). Other measures of the goodness of fit include the root mean-square-deviation (RMSD) between the observed and back-calculated PREs and the correlation coefficient from a linear regression fit of a scatter plot of the back-calculated versus observed PREs.

The representation of the paramagnetic label as an ensemble of multiple conformers is essential for fitting PRE data arising from flexible labels (e.g. MTSL). The use of a three-conformer ensemble improved the fit of PRE data from SRY/DNA complexes incorporating dT-EDTA-Mn²⁺ without overfitting the data – as indicated by a reduction in the free Q-factor calculated from data excluded from structure calculations and by the improvement of the fit of RDCs to the calculated coordinates (Iwahara et al. Reference Iwahara, Schwieters and Clore2004b). More rigid spin labels, however, do not require an ensemble representation. For example, it was demonstrated that, whereas multiple tag conformations were required to fit PRE data in CaM conjugated to MTSL, a single conformation was sufficient when CaM was conjugated to a pyridyl analog of MTSL (Fawzi et al. Reference Fawzi, Fleissner, Anthis, Kalai, Hideg, Hubbell and Clore2011a).

A scatter plot of the back-calculated PREs versus the experimental PREs is revealing of how good the fit is. However, for understanding the PREs in terms of structural features, a plot of PRE versus residue number (plotting both the back-calculated PREs and the experimental PREs) can be more informative. The key feature that indicates that a minor state might be present is the measurement of significantly higher PRE values than those back-calculated from the structure. However, before attempting to identify and characterize minor populations, one must first ensure that the experimental PRE data cannot be reasonably explained by a single structure. If these efforts fail to sufficiently explain the measured PREs, one can then attempt to calculate an ensemble of new structures to explain the data.

3.3.2 Ensemble simulated annealing calculations

If the PRE data collected on a system cannot be explained by known structures, the calculation of new structures or ensembles of structures may be necessary to satisfy the experimental data. The calculation of a molecular ensemble from average (in this case PRE) data is an inverse problem that is ill-posed and underdetermined for large ensembles and highly flexible systems. Thus, various approaches have been proposed (see below). Here, we describe the calculation of minor state structures using ensemble rigid body simulated annealing calculations using the ensembleSimulation module of Xplor-NIH (Clore & Schwieters, Reference Clore and Schwieters2004b).

In performing a rigid body refinement, the internal structure of all or part of a macromolecule is constrained to behave as a rigid body that is only able to translate and rotate as a cohesive unit (Clore, Reference Clore2000; Clore & Bewley, Reference Clore and Bewley2002). For example, CaM comprises two domains connected by a flexible linker; thus in PRE-driven ensemble simulated annealing calculations, each domain was treated as a rigid body, and residues in the linker were given full torsional degrees of freedom (Anthis et al. Reference Anthis, Doucleff and Clore2011). Side-chain residues at interfaces can also be allowed to move to prevent steric overlap. Rigid body refinement is appropriate only when the structure of the domains held rigid are not expected to change appreciably from the starting structure, a condition that can be validated by checking the agreement between experimentally measured RDCs and those back-calculated from the starting structure. The need for rigid body refinement arises from the sparse and long-range nature of the PRE, compared with conventional (NOE-based) NMR structural restraints. If sufficient additional data are available, full atomic structure calculations can be performed, although they will be computationally more expensive and the results may be ill defined compared with the results of a rigid-body refinement.

Additional points to consider are how many individual sets of PRE data (each from a molecule with a single paramagnetic tag) need to be included in the structure calculation and where the paramagnetic tags should be placed. Unless detailed information about the minor state(s) is known beforehand, the goal will be to have enough paramagnetic tags so that each position on the surface of the molecule would be within a distance that would generate a measurable PRE from at least one possible minor state conformation. Because some tag positions may disrupt the structure of the protein or have other undesirable effects, it is often necessary to test many more potential tag positions than might ultimately be required. In the case of CaM, which comprises two 8-kDa globular domains, we found that reproducible results required at least one tag on each of its two domains. Regardless of how many data sets are ultimately used, they can all be included in a single simulated annealing calculation.

Before calculating new ensemble structures to fit PRE data, the conformations of the spin labels must first be set using the fitting procedure described in Section 3.3.1, incorporating only the intramolecular or intradomain PRE data (i.e. data that comes primarily from the major state). These tag positions can then be maintained in the same conformations for the subsequent rigid body fitting. When performing simulated annealing calculations involving ensembles of minor states, one must determine an appropriate number of minor states and an appropriate population to fit the experimental data without overfitting. This can be accomplished by performing a grid search, i.e. performing multiple calculations while independently varying the number of minor states and the population attributed to each minor state. Increasing the number of minor states or the population attributed to each state should improve the fit (i.e. reduce the Q-factor) up to a certain threshold. The number N of minor states above which the Q-factor exhibits only marginal reduction is a good working value to use for calculations, as larger values of N will offer little improvement to the fit and will require proportionally more computational time. The minimum population and value of N needed to fit the data can be more rigorously determined by also calculating Q _free (calculated the same as the Q-factor (Eq. (70)), but using a small percentage (e.g. 5%) excluded from the data set used for refinement). The value of N needed to fit the data varies from system to system, and a value under 10 is generally sufficient, even for fairly heterogeneous minor states. Likewise, the population above which the Q-factor exhibits only marginal reduction offers a good estimate of the true minor state population. For example, N = 8 was sufficient for CaM, in the presence or absence of calcium. A minor state of 10% was necessary to fit the calcium-loaded CaM data, but only 5% was needed for apo CaM (Anthis et al. Reference Anthis, Doucleff and Clore2011). New updates to Xplor-NIH allow users to directly fit the populations of ensemble members (Deshmukh et al. Reference Deshmukh, Schwieters, Grishaev, Ghirlando, Baber and Clore2013); however, this approach is applicable primarily to ensembles of more highly populated states because the populations of individual members of a minor state ensemble derived from PRE data can be ill-defined due to the dependence of the PRE on both the distance from the paramagnetic center and the population of each ensemble member.

Ensemble calculations can be computationally expensive, although the time required can be minimized through parallelization. The ensembleSimulation module in Xplor-NIH offers two types of parallelization (Clore & Schwieters, Reference Clore and Schwieters2004b), which can be used simultaneously. As with conventional structural determination in Xplor-NIH, individual ensembles can be calculated independently on different processors, as individual structures can. An additional level of parallelization can be added when using ensembleSimulation, as each member of the ensemble can also be calculated semi-independently on different processors.

The results of an ensemble simulated annealing calculation can be difficult to visualize, given that each ensemble could have tens of members, and the calculation might be repeated 100 or more times. Trying to visualize all individual ensemble members, especially at atomic resolution, can be both overwhelming and uninformative. Other visualization schemes, such as showing just the center of mass for each ensemble member, involve losing most detailed structural information. Reweighted atomic probability density maps, however, allow the consolidation of the full ensemble into a single representation, preserving many structural features and highlighting the major contributing members to the ensemble – a feature that may be lost in other representations (Schwieters & Clore, Reference Schwieters and Clore2002).

For visualizing NMR ensembles, instead of using an evenly weighted map, it is advantageous to calculate a probability density map that has been reweighted so that the maximum contribution of each atom is equal, allowing visualization of even the highly flexible regions of the molecule (Schwieters & Clore, Reference Schwieters and Clore2002). This is accomplished by defining the density at each point q as (Schwieters & Clore, Reference Schwieters and Clore2002):

(73)$$\rho _{{\rm reweight}} ({\bf q}{\rm )}{\bf =} \displaystyle{1 \over {N_s}} \sum\limits_{l = 1}^{N_{\rm a}} {w_l \sum\limits_{k = 1}^{N_{\rm s}} {\rho _a ({\bf q,q}_{kl} )}}, $$

where N _s is the number of member structures in the ensemble, N _a the number of atoms in each structure, q_kl the position of atom l in each ensemble member, and w _l a weighting factor. ρ _a(q, q_lk) is a quartic function, ranging from 0 to 1, that represents the electron density for each atom (Schwieters & Clore, Reference Schwieters and Clore2002):

(74)$$\rho _a ({\bf q,q}_{kl} ) = \left\{ {\matrix{ {\displaystyle{{105} \over {32\pi a^7}} \left( {d({\bf q,q}_{kl} )^2 - a^2} \right)} & {d({\bf q,q}_{kl} )\;{\rm \lt} \;a,} \cr 0 & {d({\bf q,q}_{kl} ) \ge a,} \cr}} \right.$$

where d(q, q_lk) is the distance between point q and the center of the atom q_kl, and a (set to 1 Å in Xplor-NIH) is defined so that ρ _a(q, q_lk) = 0·5 at d(q, q_lk) = a × (1 − 1/√2)^1/2. The weighting factor for each atom is defined as (Schwieters & Clore, Reference Schwieters and Clore2002):

(75)$$\displaystyle{1 \over {w_l}} = \max \sum\limits_{k = l}^{N_{\rm s}} {\rho _a ({\bf q,q}_{kl} )}. $$

These maps can be generated in Xplor-NIH and then visualized in a molecular graphics program such as VMD-Xplor (Schwieters & Clore, Reference Schwieters and Clore2001) or PyMOL (http://www.pymol.org). Showing this map at different contour levels gives a good indication of where the minor states are located, rather than looking at a large jumble of atoms, bonds, or ribbons. For the clearest results, it is often advantageous to generate multiple different maps in order to visualize the ensembles from multiple perspectives. For example, first, all structures could be aligned to rigid body A, and density maps could be calculated and visualized for rigid body B. Then, all structures could be aligned to rigid body B, and density maps could be calculated and visualized for rigid body A. The calculation and visualization of minor state ensembles allow structural insight into these important, transient species that have previously eluded structural characterization. The extent of insight possible will depend upon the heterogeneity of these ensembles, but if the structures are consistent from one ensemble to the next, it will be possible to analyze specific molecular contacts with a high degree of certainty. For larger or more heterogeneous ensembles, one might be restricted to identifying binding interfaces or other large-scale structural features. Regardless, the insights gained through these calculations shed substantial light on these elusive states and can provide a stepping stone for further biochemical study and manipulation of these minor states.

The PRE-driven ensemble calculation approach has been used to visualize transient interactions between the N-terminal domain of enzyme I (EIN) and the phosphocarrier protein HPr (Tang et al. Reference Tang, Iwahara and Clore2006, Reference Tang, Ghirlando and Clore2008a), between cytochrome c and cytochrome c peroxidase (Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006, Reference Volkov, Ubbink and Van Nuland2010), between the N- and C-terminal domains of apo MBP (Tang et al. Reference Tang, Schwieters and Clore2007), between subunits of the HIV-1 protease precursor protein (Tang et al. Reference Tang, Ghirlando and Clore2008a), and between the N- and C-terminal domains of the calcium-sensing protein CaM (Anthis et al. Reference Anthis, Doucleff and Clore2011). Because these interdomain or intermolecular contacts only exist in a minor proportion (<10%) of the total population, they are invisible to conventional structural methods and could only be visualized by long-range paramagnetic NMR measurements. Thus, PRE can unlock fundamental insights into macromolecular-binding events, and many of these studies highlight the subtle interplay between induced fit and conformational selection in these binding events (Clore, Reference Clore2013).

3.4.1 Maximum occurrence (MO), ensemble selection, and MD

In the previous sections, we discussed analyzing PRE data by fitting to known structures using simulated annealing calculations and characterizing minor states by using PRE-driven ensemble simulated annealing calculations. Various other methods are also available for calculating representative structural ensembles of a dynamic system (based on PRE data or on a variety of other NMR and non-NMR [e.g. SAXS] measurements), and these can broadly be placed in two categories (Fisher & Stultz, Reference Fisher and Stultz2011). The first category is the calculation of ensembles via simulated annealing calculations (Clore & Gronenborn, Reference Clore and Gronenborn1991; Clore & Schwieters, Reference Clore and Schwieters2004a, Reference Clore and Schwietersb, Reference Clore and Schwieters2006), as described above. Another broad class of methods involves first the generation of a large number of structures and then selection amongst these of the most appropriate ensemble to fit the experimental data. Examples include ENSEMBLE (Choy & Forman-Kay, Reference Choy and Forman-Kay2001), the ensemble optimization method (EOM) (Bernado et al. Reference Bernado, Mylonas, Petoukhov, Blackledge and Svergun2007), ASTEROIDS (Nodet et al. Reference Nodet, Salmon, Ozenne, Meier, Jensen and Blackledge2009), minimum ensemble selection (MES) (Pelikan et al. Reference Pelikan, Hura and Hammel2009), MO (Bertini et al. Reference Bertini, Giachetti, Luchinat, Parigi, Petoukhov, Pierattelli, Ravera and Svergun2010), and sparse ensemble selection (SES) (Berlin et al. Reference Berlin, Castaneda, Schneidman-Duhovny, Sali, Nava-Tudela and Fushman2013).

Some of these methods use molecular dynamics (MD) simulations to generate the initial set of individual conformers from which the final ensemble is selected. However, MD can be incorporated into the analysis of paramagnetic data in various ways. The simulated annealing approach discussed previously involves MD calculations with a simplified force field. Paramagnetic data and other NMR observables can also be incorporated directly into equilibrium MD simulations that also include explicit solvent and electrostatic force fields. Additionally, MD can be used to generate ensembles that can be analyzed for the fit of the experimental data. This approach was used to characterize CaM bound to the IQ-recognition motif from the voltage-gated calcium channel Ca_v1·2; the CaM/IQ complex had been crystallized in three different conformations, but in order to fit the PCS and paramagnetic RDC data on this complex in solution, all three crystal conformations plus four additional conformations calculated by MD had to be included (Russo et al. Reference Russo, Maestre-Martinez, Wolff, Becker and Griesinger2013).

Many different ensemble selection methods have been used for dynamic systems, but one that has been used extensively with paramagnetic data is the MO or maximum allowed probability (MAP) method described by Bertini and colleagues (Bertini et al. Reference Bertini, Gupta, Luchinat, Parigi, Peana, Sgheri and Yuan2007, Reference Bertini, Giachetti, Luchinat, Parigi, Petoukhov, Pierattelli, Ravera and Svergun2010, Reference Bertini, Ferella, Luchinat, Parigi, Petoukhov, Ravera, Rosato and Svergun2012a; Fragai et al. Reference Fragai, Luchinat, Parigi and Ravera2013) based on prior theoretical work (Gardner et al. Reference Gardner, Longinetti and Sgheri2005; Longinetti et al. Reference Longinetti, Luchinat, Parigi and Sgheri2006). This method has been used for paramagnetic PCS and RDC data (in combination with SAXS data) (Bertini et al. Reference Bertini, Giachetti, Luchinat, Parigi, Petoukhov, Pierattelli, Ravera and Svergun2010) as well as PRE data (Bertini et al. Reference Bertini, Luchinat, Nagulapalli, Parigi and Ravera2012b). A similar approach for PRE data has also been taken by Volkov et al. (Reference Volkov, Ubbink and Van Nuland2010). The MO approach involves first generating individual structures, as with other ensemble selection methods, and then determining a maximum population (‘occurrence’) that can be ascribed to each structure without violating the data. The result is a statistically rigorous view of the minor conformations observed by paramagnetic data, although without producing a tangible ensemble that can be used for further analysis.

The MO approach first requires generating a pool of random structures that comprehensively sample the rotational and translational conformational space accessible to the system. For a protein with two domains connected by a flexible linker (e.g. CaM), ~50 000 individual structures were required (Bertini et al. Reference Bertini, Giachetti, Luchinat, Parigi, Petoukhov, Pierattelli, Ravera and Svergun2010). For the transient interaction between two globular proteins (e.g. cytochrome c and cytochrome c peroxidase) ~1 000 000 individual structures were required to sample the interaction surface in 1-Å increments (Volkov et al. Reference Volkov, Ubbink and Van Nuland2010). A random structure generation protocol can be used to generate individual structures that fulfill basic criteria, such as lack of steric clashes between proteins or domains (Grishaev et al. Reference Grishaev, Anthis and Clore2012).

The MO procedure (Bertini et al. Reference Bertini, Ferella, Luchinat, Parigi, Petoukhov, Ravera, Rosato and Svergun2012a) is then performed for some subset of these (thousands to millions) individual structures. First, a structure is randomly selected and given a population. Then, an ensemble of a pre-determined size is selected to best fit the experimental data. A target function (e.g. Eq. (65)) is calculated, and this procedure is repeated for a range of different populations of the selected structure. The MO of this structure is defined as the maximum population that does not exceed the pre-set threshold value for the target function. The entire procedure is then repeated for a certain subset of the random structures available. The MO of each structure corresponds to an upper limit of the structure's population, but as the quantity of experimental data increases, the MO for each conformation should converge to its true population (Fragai et al. Reference Fragai, Luchinat, Parigi and Ravera2013). The different types of data that have been used in MO studies are highly complementary, being sensitive to either distance from a paramagnetic center (PRE), relative orientation (RDC), both of these (PCS), or overall molecular shape (SAXS). Additionally, the incorporation of multiple paramagnetic tag positions or the use of multiple paramagnetic metals with different g-tensors can also help to further refine the MO values (Fragai et al. Reference Fragai, Luchinat, Parigi and Ravera2013).

The MO method can be computationally expensive, although it lends itself well to parallelization. For this reason, the MaxOcc web portal (http://py-enmr.cerm.unifi.it/access/index/maxocc) has been set up to allow MO calculations on the e-NMR grid infrastructure (Bertini et al. Reference Bertini, Ferella, Luchinat, Parigi, Petoukhov, Ravera, Rosato and Svergun2012a). Another difficulty of MO analysis is finding a reasonable way to visualize the data. To simplify the presentation, the center of mass of each protein or domain can be represented as a dot, although this removes any orientational information. Instead of a dot, a set of axes can be used to represent the orientation of the molecule or domain, although information on interfaces and molecular contacts are still lacking. The MO can be indicated on such a representation through color coding. Also, unlike the ensemble simulated annealing approach, the MO approach does not generate concrete structural ensembles that can be used for further analysis.

The MO approach has been used extensively to characterize CaM (Bertini et al. Reference Bertini, Giachetti, Luchinat, Parigi, Petoukhov, Pierattelli, Ravera and Svergun2010, Reference Bertini, Luchinat, Nagulapalli, Parigi and Ravera2012b; Dasgupta et al. Reference Dasgupta, Hu, Keizers, Liu, Luchinat, Nagulapalli, Overhand, Parigi, Sgheri and Ubbink2011; Luchinat et al. Reference Luchinat, Nagulapalli, Parigi and Sgheri2012). These studies have incorporated PCS, RDC, PRE, and/or SAXS data from CaM, primarily using an N60D mutant that selectively binds lanthanide ions in one of the two N-terminal Ca²⁺-binding sites of CaM. These studies indicate that the two domains of CaM are capable of sampling a wide range of conformational space with respect to one another. Another study demonstrated that even when bound to a peptide from myelin basic protein, CaM was still highly dynamic (Nagulapalli et al. Reference Nagulapalli, Parigi, Yuan, Gsponer, Deraos, Bamm, Harauz, Matsoukas, De Planque, Gerothanassis, Babu, Luchinat and Tzakos2012). This flexible complex differed from the more rigid complex that Ca²⁺-loaded CaM forms with many peptide targets, but similar to that formed between apo CaM and a peptide from myosin light-chain kinase (Anthis et al. Reference Anthis, Doucleff and Clore2011). The MO approach was also used with SAXS, RDC, and PCS restraints to characterize matrix metalloproteinase-1 (MMP-1), revealing that MMP-1 adopts a more open structure in solution than that observed by X-ray crystallography (Cerofolini et al. Reference Cerofolini, Fields, Fragai, Geraldes, Luchinat, Parigi, Ravera, Svergun and Teixeira2013). A similar approach was also used to map the encounter complex between cytochrome c and cytochrome c peroxidase, although PRE-driven ensemble simulated annealing calculations were used to further refine the model of the encounter complex (Volkov et al. Reference Volkov, Ubbink and Van Nuland2010), demonstrating that the encounter complex adopts conformations nearby in space to the major species observed by X-ray crystallography (Volkov et al. Reference Volkov, Worrall, Holtzmann and Ubbink2006).

3.4.2 PRE studies on intrinsically disordered systems

PRE has proven to be a powerful method for studying unfolded and intrinsically disordered proteins, which, due to their dynamic nature, often defy characterization by other methods (Clore & Iwahara, Reference Clore and Iwahara2009; Clore et al. Reference Clore, Tang and Iwahara2007). Many studies on denatured systems have revealed evidence of residual structure in the unfolded state; examples include denatured staphylococcal nuclease (Gillespie & Shortle, Reference Gillespie and Shortle1997a, Reference Gillespie and Shortleb), the unfolded form of the N-terminal SH3 domain of drk (in slow-exchange equilibrium with the folded form) (Choy & Forman-Kay, Reference Choy and Forman-Kay2001; Marsh & Forman-Kay, Reference Marsh and Forman-Kay2009; Marsh et al. Reference Marsh, Neale, Jack, Choy, Lee, Crowhurst and Forman-Kay2007), acid-denatured apomyoglobin (Felitsky et al. Reference Felitsky, Lietzow, Dyson and Wright2008; Lietzow et al. Reference Lietzow, Jamin, Dyson and Wright2002; Silvestre-Ryan et al. Reference Silvestre-Ryan, Bertoncini, Fenwick, Esteban-Martin and Salvatella2013), denatured acyl coenzyme A-binding protein (Kristjansdottir et al. Reference Kristjansdottir, Lindorff-Larsen, Fieber, Dobson, Vendruscolo and Poulsen2005; Ozenne et al. Reference Ozenne, Noel, Heidarsson, Brander, Poulsen, Jensen, Kragelund, Blackledge and Danielsson2014), and the γ-subunit of cGMP phosphodiesterase (Song et al. Reference Song, Guo, Muradov, Artemyev, Ruoho and Markley2008). (However, PRE data on other systems have indicated that native-like populations constitute a very small or non-existent population in the disordered state (Cliff et al. Reference Cliff, Craven, Marston, Hounslow, Clarke and Waltho2009; Vise et al. Reference Vise, Baral, Stancik, Lowry and Daughdrill2007)). Studies on the intrinsically unstructured and aggregation-prone protein α-synuclein have revealed a more compact average hydrodynamic radius than expected for a truly random coil peptide as well as transient long-range interactions that appear to inhibit oligomerization and aggregation (Bertoncini et al. Reference Bertoncini, Jung, Fernandez, Hoyer, Griesinger, Jovin and Zweckstetter2005; Dedmon et al. Reference Dedmon, Lindorff-Larsen, Christodoulou, Vendruscolo and Dobson2005). PRE can also be used to identify interactions that could be productive for aggregation, as observed for the protein tau (Peterson et al. Reference Peterson, Zhou, Dahlquist and Lew2008).

Since disordered systems cannot be described by a single structure, ensemble approaches are usually appropriate (Fisher & Stultz, Reference Fisher and Stultz2011), including some of the methods described in Sections 3.3.2 and 3.4.1), although a very large and heterogeneous ensemble may be required to describe an unstructured system. Other alternative treatments of the data are also possible. For example, it has been demonstrated that the calculation of a single average structure can provide high-resolution information about residual molecular contacts in acid-unfolded apomyoglobin (Silvestre-Ryan et al. Reference Silvestre-Ryan, Bertoncini, Fenwick, Esteban-Martin and Salvatella2013), although it should be recognized that this does not provide an accurate representation of the reality in solution.

PRE data can be used to characterize flexible parts of otherwise more ordered systems. For example, recently the pH gating ability of loop 6 of the β-barrel outer membrane protein G was studied by PRE (Zhuang et al. Reference Zhuang, Chisholm, Chen and Tamm2013). PRE data have been used for studying systems involving structured domains linked by a flexible linker, including CaM (Anthis et al. Reference Anthis, Doucleff and Clore2011). The nature of the linker in this case was explored in further detail, as the extent of association between the two domains of CaM was determined by interdomain PRE measurements for linkers of varying length, and the results were fit to a simple random-coil model of the linker describing the effective concentration of the ends of the linker as a function of linker length (Anthis & Clore, Reference Anthis and Clore2013).

3.4.3 Solvent PRE

In addition to the measurement of PREs arising from a spin label or paramagnetic ion bound directly to a macromolecule (intramolecular PRE) or from a tag or ion bound to another macromolecule (intermolecular PRE), PREs arising from small co-solute molecules (solvent PRE) can also be measured. Such co-solute molecules include molecular oxygen (O₂ (Hernandez et al. Reference Hernandez, Teng, Bryant and Lemaster2002; Sakakura et al. Reference Sakakura, Noba, Luchette, Shimada and Prosser2005)), nitroxides (e.g. TEMPOL (Fesik et al. Reference Fesik, Gemmecker, Olejniczak and Petros1991; Petros et al. Reference Petros, Mueller and Kopple1990)) and chelated metal ions (e.g. Gd-diethylenetriamine pentacetic acid-bismethylamide [Gd-DTPA-BMA or ‘Omniscan’] (Iwahara et al. Reference Iwahara, Zweckstetter and Clore2006; Pintacuda & Otting, Reference Pintacuda and Otting2002)). These particular co-solutes are neutrally charged, making them particularly appropriate as probes for solvent accessibility, although care must be taken with TEMPOL, which can preferentially interact with hydrophobic surfaces. Solvent PRE measurements can be used as structural restraints as a measure of solvent accessibility or proximity to the surface of a macromolecule, and they can also be used to detect intermolecular interactions (i.e. decreased PRE values near the interaction surface) (Iwahara et al. Reference Iwahara, Zweckstetter and Clore2006; Sakakura et al. Reference Sakakura, Noba, Luchette, Shimada and Prosser2005).

Analyzing solvent PRE data is complicated by the need to take into account the diffusion rates and correlation times of many co-solute molecules in addition to those of the macromolecule. Different theoretical frameworks have been proposed (Bernini et al. Reference Bernini, Venditti, Spiga and Niccolai2009; Clore & Iwahara, Reference Clore and Iwahara2009), including an inner-sphere model and an outer sphere model. In the inner-sphere model, the solvent PRE is assumed to arise from interactions between the paramagnetic protein and the macromolecular surface (Bloembergen, Reference Bloembergen1957; Solomon, Reference Solomon1955; Solomon & Bloembergen, Reference Solomon and Bloembergen1956), whereas in the outer-sphere model, the solvent PRE is assumed to arise from paramagnetic centers that behave as diffusive solvent molecules (Freed, Reference Freed1978; Hwang & Freed, Reference Hwang and Freed1975). Hwang and Freed have developed a theoretical framework to describe solvent PRE values by such an outer-sphere model, although due to its complexity it is difficult to apply to macromolecular structures.

Otting and LeMaster independently proposed an empirical method for back-calculating solvent PRE values from macromolecular structures (Hernandez et al. Reference Hernandez, Teng, Bryant and Lemaster2002; Pintacuda & Otting, Reference Pintacuda and Otting2002). The macromolecule is imbedded in a tightly spaced cubic lattice in silico, and the solvent PRE for a given nucleus in the macromolecule is the sum of the r ⁻⁶ distances to each of the lattice points (excluding those that overlap with the macromolecule), weighted by a single empirical factor, a, which takes into account the concentration of the paramagnetic co-solute as well as its dynamics. This approach has been applied to solvent PRE data from O₂ on ¹H nuclei in rubredoxin (Hernandez et al. Reference Hernandez, Teng, Bryant and Lemaster2002), from Gd–DTPA–BMA on ¹H nuclei in ubiquitin (Pintacuda & Otting, Reference Pintacuda and Otting2002), and from Gd–DTPA–BMA on ¹H nuclei in specific and non-specific protein/DNA complexes (Iwahara et al. Reference Iwahara, Zweckstetter and Clore2006).

Some of these same solvent PRE co-solvents can be used in imaging studies as contrast agents. Conventionally, a non-toxic paramagnetic molecule (generally chelated Gd³⁺) is injected, and contrast is provided by the enhancement to longitudinal relaxation, measured as an increase in signal intensity where greater concentrations of the contrast agent are present (Caravan et al. Reference Caravan, Ellison, Mcmurry and Lauffer1999). Contrast agents have also been developed that take advantage of CEST and, more recently, paramagnetic contrast agents have been developed for paraCEST (chelated lanthanide ions, especially Eu³⁺) to generate even more dramatic chemical shift differences by hyperfine shifts for CEST imaging (Hancu et al. Reference Hancu, Dixon, Woods, Vinogradov, Sherry and Lenkinski2010; Vinogradov et al. Reference Vinogradov, Sherry and Lenkinski2013).

4.1.1 Lifetime line broadening

Here, we consider a process, such as protein aggregation or the interaction of a protein with a larger macromolecular complex, where the system is in exchange between two states:

(76)$${\rm A}\; {\mathop{\rightleftharpoons}_{k_{\rm off}}^{k_{\rm on}^{\rm app}}} \; {\rm B,}$$

where state A is the visible species and state B is the invisible dark state. Exchange from states A to B is described by the pseudo-first-order rate constant k _on^app, and exchange from states B to A is described by the first-order rate constant k _off. Thus, at equilibrium:

(77)$$k_{{\rm on}}^{{\rm app}} [{\rm A}] = k_{{\rm off}} [{\rm B}].$$

The rate constant k _on^app is a pseudo-first-order rate constant because although it describes a reaction that is first-order with respect to A, the actual reaction could be of higher order.

In such an exchanging system, the observed transverse relaxation rate (R ₂^obs) of state A can be enhanced by broadening due to differences in chemical shift between states A and B (R _ex) and by broadening due to differences in transverse relaxation rate between the two states (lifetime line broadening, R _llb). Overall, the enhancement (ΔR ₂^obs) to the transverse relaxation rate of state A is given by:

(78)$$R_2^{{\rm obs}} = R_{2,{\rm A}}^0 + {\rm \Delta} R_2^{{\rm obs}}, $$

where R _2,A⁰ is the intrinsic relaxation of state A in the absence of exchange. The relative contributions of R _ex and R _llb to ΔR ₂^obs depend on the exchange rate and on the magnitude of chemical shift difference (Δω = |ω _A − ω _B|) and transverse relaxation rate difference (ΔR ₂⁰ = R _2,B⁰ − R _2,A⁰) between the states (Fig. 5).

If the rate of transverse relaxation in state B (the ‘dark’ state; R _2,B⁰ = R ₂^dark) is very rapid (due a large size and therefore slow tumbling), then any transverse magnetization that is transferred from A to B will effectively be destroyed. If the exchange is slow on the transverse relaxation timescale (k _ex ≪ ΔR ₂⁰), lifetime line broadening will be equal to the apparent first-order rate constant (R _llb = k _on^app). If, however, exchange is fast on the transverse relaxation timescale (k _ex ≫ ΔR ₂⁰), R ₂^obs will be a population-weighted average of the two relaxation rates:

(79)$$R_2^{{\rm obs}} = p_{\rm A} R_{2,{\rm A}}^0 + p_{\rm B} R_{2,{\rm B}}^0 $$

(providing k _ex ≪ 1/τ _r, where τ _r is the rotational correlation time) and R _llb is given by:

(80)$$R_{{\rm llb}} = R_2^{{\rm obs}} - R_{2,{\rm A}}^{\rm 0} = p_{\rm B} {\rm \Delta} R_2^0. $$

For cases where the dark state B is very large and has a very rapid rate of transverse relaxation (e.g. large macromolecular assemblies and aggregated species), lifetime line broadening can only be measured in the slow exchange regime, unless p _B is very small, because as R ₂^obs approaches values of ~100 s⁻¹ or larger, it becomes difficult or impossible to measure. For smaller assemblies with more moderate R _2,B⁰, lifetime line broadening can be measured in the intermediate and fast exchange regimes as well.

If states A and B differ both in chemical shift and transverse relaxation rate, the manner by which R _ex and R _llb add up to ΔR ₂^obs depends on the value of the exchange rate (Fig. 5). For all exchange rates, the ΔR ₂^obs will be at least the magnitude of the greater of R _ex and R _llb, and ΔR ₂^obs will be at most the sum of the two. In the slow exchange regime (k _ex ≪ ΔR ₂⁰ and k _ex ≪ Δω), ΔR ₂^obs = R _ex = R _llb = k _on^app. In the fast exchange regime (k _ex ≫ ΔR ₂⁰ and k _ex ≫ Δω), ΔR ₂^obs = R _ex + R _llb (and in the very fast regime, ΔR ₂^obs = R _llb, as R _ex approaches zero). In the intermediate exchange regime, ΔR ₂^obs will occupy a value intermediate between these two extremes (Fig. 5).

Values of ΔR ₂^obs, R _ex, and R _llb can be calculated by solving the McConnell equations (Eqs. (16)–(20)) and fitting the resulting time series to a single-exponential decay to derive a single apparent transverse relaxation rate (R ₂^obs). Subtracting R _2,A⁰ from R ₂^obs yields ΔR ₂^obs. The same procedure can be used to calculate R _ex and R _llb by setting ΔR ₂⁰ = 0 (to calculate R _ex) and by setting Δω = 0 (to calculate R _llb). Because R _llb does not depend on chemical shift differences, it can be calculated by a simplified procedure only requiring the McConnell equations to be solved once. If all chemical shifts are set to zero (or if the −iΩ matrix in Eq. (16) is omitted), R _llb can be calculated by the following expression:

(81)$$R_{{\rm llb}} = - \displaystyle{1 \over {T_R}} \ln \left( {\displaystyle{{M_{x,{\rm A}} (T_{\rm R} )} \over {M_{x,{\rm A}} (0)}}} \right) - R_{2,{\rm A}}^0, $$

where M _x,A(0) is the initial value of x transverse magnetization on state A and M _x,A(T _R) is the magnetization at time T _R (the specific value of T _R is unimportant, but it can be set to an experimentally realistic value such as 0·1 s).

4.1.2 DEST

If a weak CW RF field is applied to a rapidly tumbling macromolecule, NMR signals at the same frequency as the CW RF radiation will be saturated, signals sufficiently far off resonance will be unaffected, and those near the RF frequency will be partially saturated. The width of this saturation profile increases with the transverse relaxation rate (R ₂) of the nucleus and the strength of the CW RF field. For proteins or peptides that tumble rapidly and exhibit slow R ₂ rates – and are thus visible by NMR – this profile is narrow. Large macromolecular complexes and assemblies (≥1 MDa), on the other hand, tumble slowly and exhibit large R ₂ rates that make direct observation by solution NMR difficult or impossible due to the dramatically broadened linewidths; these large systems exhibit much broader saturation profiles. However, if such a dark state is in exchange with a visible species, it may be possible to indirectly study it via DEST, even if the dark state is also sparsely populated and/or has an amorphous or heterogeneous structure.

For a visible species in exchange with an invisible species, the overall R ₂ rate for a nucleus signal will have contributions from both states. For the type of very large species that one might study by DEST (e.g. MW > 1 MDa), the transverse relaxation rate can become incredibly fast (e.g. >1000 s⁻¹). Thus, transverse magnetization in the invisible species will quickly disappear, whereas more slowly relaxing longitudinal magnetization will remain intact. Thus, by just partially tilting the bulk magnetization of the sample toward the transverse plane (i.e. only partially saturating the magnetization) by the application of an off-resonance CW RF field, the overall signal will be attenuated but not destroyed. Under these conditions, nuclei in the invisible state will relax more quickly than nuclei in the visible state, but their magnetization will not disappear completely. Thus, due to the large difference in R ₂ between the visible and invisible, the otherwise invisible large species can be detected by DEST (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2012).

The data from a DEST experiment can be modeled using a homogeneous form of the McConnell equations describing a system in equilibrium (Helgstrand et al. Reference Helgstrand, Hard and Allard2000). If the spins can be treated as isolated spins free of spin diffusion (e.g. ¹⁵N), the time dependence of the magnetization in this two-site exchanging system is given by (Helgstrand et al. Reference Helgstrand, Hard and Allard2000):

(82)$$\displaystyle{{\rm d} \over {{\rm d}t}}{\bf M =} -\left( {{\bf E + R + K + \Omega + S}} \right){\bf M}{\rm,} $$

where M is the vector describing the transverse magnetization of each state, R is the matrix describing the transverse relaxation rates of each state, K is the matrix describing the kinetic rates of exchange between each state, Ω is the matrix describing the chemical shift offset of the saturation pulse relative to the chemical shift of each state, S is the matrix describing the field strength of the saturation pulse, and E is the matrix describing the equilibrium magnetization. The exact equations will be dictated by the kinetic model describing the system, and here we consider a system in exchange between two states (Eq. (76)), where state A is the visible species and state B is the invisible dark state:

(83)$${\bf M =} \left[ {\matrix{ {E/2} \cr {M_{x,{\rm A}}} \cr {M_{y,{\rm A}}} \cr {M_{z,{\rm A}}} \cr {M_{x,{\rm B}}} \cr {M_{y,{\rm B}}} \cr {M_{z,{\rm B}}} \cr}} \right],$$

(84)$$\eqalign{&{\bf E} + {\bf R} + {\bf K} + {\rm \Omega} + {\bf S} \cr &\quad = \left[ {\matrix{ 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr 0 & {R_{2,{\rm A}}^0 + k_{{\rm on}}^{{\rm app}}} & {{\rm \Omega} _{\rm A}} & {-\omega _y} & {-k_{off}} & 0 & 0 \cr 0 & {-{\rm \Omega} _{\rm A}} & {R_{2,{\rm A}}^0 + k_{{\rm on}}^{{\rm app}}} & {\omega _x} & 0 & {-k_{off}} & 0 \cr {-2R_{1,{\rm A}} M_{{\rm eq,A}}} & {\omega _y} & {-\omega _x} & {R_{1,{\rm A}} + k_{{\rm on}}^{{\rm app}}} & 0 & 0 & {-k_{off}} \cr 0 & {-k_{{\rm on}}^{{\rm app}}} & 0 & 0 & {R_{2,{\rm B}}^0 + k_{off}} & {{\rm \Omega} _{\rm B}} & {-\omega _y} \cr 0 & 0 & {-k_{{\rm on}}^{{\rm app}}} & 0 & {-{\rm \Omega} _{\rm B}} & {R_{2,{\rm B}}^0 + k_{off}} & {\omega _x} \cr {-2R_{1,{\rm B}} M_{{\rm eq,B}}} & 0 & 0 & {-k_{{\rm on}}^{{\rm app}}} & {\omega _y} & {-\omega _x} & {R_{1,{\rm B}} + k_{off}} \cr}} \right],}$$

where E is unity, M is the magnetization of the A or B spin along each axis, and M _eq is the equilibrium magnetization of each state. The longitudinal relaxation rates are given by R ₁, and the intrinsic transverse relaxation rates (in the absence of exchange) are given by R ₂⁰. ω is the strength of the CW RF B ₁ saturation field applied to each axis (ω [rad/s] = 2π × ν [Hz]), and Ω_A and Ω_B are the frequency offsets between the resonance frequency of the nucleus and the carrier frequency (in rad s⁻¹). Equations describing three-state exchange (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b) and two-state-exchange involving a multi-spin system (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b) have been described elsewhere.

A DEST (or CEST) profile can be calculated by solving Eq. (82) for a series of CW RF frequency offsets. For each offset, the signal intensity (I _sat) is given by solving these equations for the time that the CW RF (ω _x) field is applied. The DEST profile is plotted as I _sat/I _refversus carrier offset, where, I _ref is the intensity in the absence of the saturation CW RF field, calculated as for I but with ω _x = 0. Calculations are performed by starting with magnetization on the z-axis (M _z,A[t = 0] = 1 × p _A; all other components equal zero, including M _z,B[t = 0], since the dark state magnetization would not survive the INEPT transfer if R _2,B⁰ is very large). Longitudinal magnetization decays to the equilibrium values of M _eq,A and M _eq,B. For ¹⁵N DEST, a value of M _eq,A = M _z,A[t = 0]/10 can be used, because γ _H/γ _N ≈ 10. A calculation is performed with magnetization beginning on +z, then a calculation is performed with magnetization beginning on −z (i.e. M _z,A[t = 0] = 1 × p _A, then M _z,A[t = 0] = −1 × p _A). The values from the second iteration (beginning on −z) are subtracted from the first (beginning on +z) because this difference yields a single-exponential curve that decays to zero in the absence of a saturation field). To properly simulate experimental data, the inhomogeneity of the B ₁ saturation field should also be taken into account, which can be accomplished by repeating the calculations with a series of slightly different values of ω _x (e.g. a Gaussian distribution about the desired value of ω _x if the standard deviation of the field strength is known (Vallurupalli et al. Reference Vallurupalli, Bouvignies and Kay2012) or just ~10 fields strengths evenly spaced over a range of ω _x ± 10% (Fawzi et al. Reference Fawzi, Libich, Ying, Tugarinov and Clore2014)).

A DEST (or CEST) profile can also be approximated by treating it as an off-resonance R _1ρ experiment (Palmer, Reference Palmer2014):

(85)$$I_{{\rm sat}} /I_{{\rm ref}} = \cos ^2 \theta \exp \,(-T_{{\rm sat}} R_{1\rho} ),$$

where T _sat is the total saturation time, the angle θ is given by Eq. (34), and R _1ρ is given by Eq. (40) (as long as p _A ≫ p _B) for a DEST experiment (where R _2,A⁰ and R _2,B⁰ differ) or by simpler expressions (e.g. Eq. (39)) for CEST (as long as R _2,A⁰ = R _2,B⁰). Palmer has demonstrated good agreement between this analytical approximation, especially when Eq. (40) is used, compared to numerical solutions of the McConnell equations (Palmer, Reference Palmer2014).

4.2.1 Measuring lifetime line broadening

Measuring lifetime line broadening requires being able to measure the R ₂ values for the major state alone and in the presence of the dark state. To characterize the exchange of Aβ monomers with protofibrils, this was accomplished by acquiring data at both low concentrations of Aβ (no dark state present) and high concentrations (dark protofibrils present; Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2010b, Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b, Reference Fawzi, Libich, Ying, Tugarinov and Clore2014). To characterize the interaction of Aβ with GroEL, data were measured in the absence and presence of GroEL (Libich et al. Reference Libich, Fawzi, Ying and Clore2013). Transverse relaxation can be measured using conventional pulse sequences (Cavanagh et al. Reference Cavanagh, Fairbrother, Palmer, Skelton and Rance2007) under conditions that eliminate any R _ex contributions, using either a CPMG experiment with a large value of ν _CP or a R _1ρ experiment with high spin-lock field strength. Under these conditions:

(86)$$R_{{\rm llb}} = {\rm \Delta} R_2^{{\rm obs}} = R_2^{{\rm vis + dark}} -R_2^{{\rm vis}}, $$

where R ₂^vis is R ₂^obs in the absence of the dark state and R ₂^vis+dark is in the presence of the dark state. R ₂ is measured by varying, in an interleaved manner, the length of a delay during which the magnetization of the observed nucleus is on the xy-plane (generally at least ~8 different times, ranging from 0 × to ~1·3 × T ₂), and fitting the intensity of each peak to an exponential decay function. Because it can be more straightforward to measure R _llb, it is usually advisable to attempt to measure R _llb before attempting to make more complex DEST measurements. Detection of lifetime broadening of R _llb ≥ 1 s⁻¹ is a strong indicator that DEST measurements will also be successful.

4.2.2 Acquiring DEST data

Most DEST experiments to date have been performed on ¹⁵N, which is advantageous because it provides one probe per residue, and it is not subject to spin diffusion – which complicates analysis of ¹H DEST measurements – or spin–spin coupling – which complicates ¹³C measurements in universally labeled samples. The setup and analysis of a DEST experiment, which has recently been described in greater detail (Fawzi et al. Reference Fawzi, Ying, Torchia and Clore2012), will be summarized here.

¹⁵N DEST can be measured with a 2D HSQC-based experiment using the pulse sequence in Fig. 10. Pulse sequences for measuring DEST are available online at http://spin.niddk.nih.gov/clore/Software/software.html. A series of interleaved 2D experiments are performed, varying the frequency of the ¹⁵N CW saturation field. The optimal duration of the CW period is T ₁ = 1/R ₁. Experiments are performed at multiple saturation field strengths (e.g. 250 and 500 Hz) to assist in optimizing parameter values and discriminating between models. A reference experiment also needs to be performed with the saturation field turned off (i.e. strength of 0 Hz). The intensity of each peak is measured for each of these 2D experiments and plotted against the frequency offset of the CW saturation radiation. The spectrometer time required to acquire DEST data varies, but about 1 week was required to acquire the complete set of DEST and R ₂ data on ~150 μM ¹⁵N-labeled Aβ using a cryogenically cooled probe (including the measurement of DEST at two different saturation field strengths; Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b).

Fig. 10.

Pulse sequence for the 2D ¹⁵N–DEST experiment. Full details are given in Fawzi et al. (Fawzi et al. Reference Fawzi, Ying, Ghirlando, Torchia and Clore2011b). During saturation experiments, the ¹⁵N CW saturation field is applied at a range of offset frequencies and at two different RF field strengths. During this period, a 180° ¹H pulse train is applied (with 100 ms spacing between pulses) in order to eliminate longitudinal cross-correlated relaxation interference between the ¹⁵N–¹H dipolar interaction and the ¹⁵N chemical shift anisotropy. Thus, for a saturation time of 0·9 s, for example, the saturation block would be repeated a total of n = 9 times. The optimal time for application of ¹⁵N saturation is the ¹⁵N–T ₁ longitudinal relaxation time. In the reference experiment, the ¹⁵N saturation pulse is turned off (but the 180° ¹H pulse train is still applied).

In order for DEST to be measured, the visible species, which gives rise to a 2D ¹H–¹⁵N spectrum with well-resolved peaks and high S/N ratio, must be in exchange with a high-molecular-weight species (i.e. the ‘dark state’). Additionally, both k _on^app and the rate constant for the transition of the dark state to the light state (k _off) must be of the order of, or faster than, the R ₁ rate of the nucleus undergoing saturation (e.g. ≥1 s⁻¹ for backbone ¹⁵N). The lower limit on k _off arises because longer residency periods in the dark state would cause too great a loss of signal before a significant number of transitions from the dark to visible state could be measured. The reason for the lower limit on k _on^app is that, due to the extremely fast R ₂ relaxation rate in the dark state, the creation of ¹⁵N_z magnetization must take place in the light state due to the inefficiency of the INEPT magnetization transfer in the dark state. Thus, it is also necessary to have a significant number of transitions from the light to the dark state within a time of T ₁ = 1/R ₁. Under conditions where R ₂^dark ≫ k _off, there is also an upper limit of k _on^app ≤ 100 s⁻¹ as larger values will cause too great a loss of NMR signal and make measurement of DEST or R ₂ difficult or impossible. Additionally, a minimum population of the dark state (~0·5% or higher) is required; leading to a rough lower limit of k _on^app/k _off > 0·005.