Practical Aspects of ¹H Transverse Paramagnetic Relaxation Enhancement Measurements on Macromolecules

Abstract

The use of ¹H transverse paramagnetic relaxation enhancement (PRE) has seen a resurgence in recent years as method for providing long-range distance information for structural studies and as a probe of large amplitude motions and lowly populated transient intermediates in macromolecular association. In this paper we discuss various practical aspects pertaining to accurate measurement of PRE ¹H transverse relaxation rates (Γ₂). We first show that accurate Γ₂ rates can be obtained from a two time-point measurement without requiring any fitting procedures or complicated error estimations, and no additional accuracy is achieved from multiple time-point measurements recorded in the same experiment time. Optimal setting of the two time-points that minimize experimental errors is also discussed. Next we show that the simplistic single time-point measurement that has been commonly used in the literature, can substantially underestimate the true value of Γ₂, unless a relatively long repetition delay is employed. We then examine the field dependence of Γ₂, and show that Γ₂ exhibits only a very weak field dependence at high magnetic fields typically employed in macromolecular studies. The theoretical basis for this observation is discussed. Finally, we investigate the impact of contamination of the paramagnetic sample by trace amounts (≤5%) of the corresponding diamagnetic species on the accuracy of Γ₂ measurements. Errors in Γ₂ introduced by such diamagnetic contamination are potentially sizeable, but can be significantly reduced by using a relatively short time interval for the two time-point Γ₂ measurement.

1. Introduction

The history of paramagnetic relaxation enhancement (PRE) dates back to that of the nuclear Overhauser effect (NOE). Indeed, in Solomon's classic 1955 paper, the relevant equations for both the NOE and PRE were presented (1). Despite the long history of the PRE and its potential to provide unique long-range distance information in the 15-35 Å range, until relatively recently, applications of the PRE to studies of biological macromolecules that do not possess an intrinsic paramagnetic group, have been rather limited. Recent technical advances that permit easy and reliable conjugation of a paramagnetic group to a specific site have changed this situation, and PRE studies on macromolecules, to which an extrinsic paramagnetic group has been attached, are becoming increasingly popular.

The PRE arising from unpaired electrons with an isotropic g-tensor (such as a nitroxide spin-label or EDTA-Mn²⁺) has proved to be a versatile tool. PRE analysis for such systems is simple, since there are no pseudo-contact shifts, and Curie-spin relaxation that could potentially exhibit significant cross-correlation with other relaxation mechanisms (2, 3), is negligible. Using this type of PRE data, macromolecular structures have been characterized for soluble proteins (4-10), protein-protein complexes (11-14), protein-oligosaccharide complexes (15,16), protein-nucleic acid complexes (17-21), and membrane proteins (22,23). The PRE can also provide information relating to large-scale dynamics that accompany changes of paramagnetic center - ¹H distances, for example in non-specific protein-DNA interactions (24,25) and inter-domain motions (26). A recent major advance in the field is the finding that in the fast exchange regime the intermolecular PRE can provide a powerful probe to detect and characterize transient, lowly populated intermediates in macromolecular binding events, thereby providing structural information on encounter complexes that cannot be obtained by any other biophysical technique (27,28).

For such investigations, the PRE for ¹H-transverse magnetization (referred to here as ¹H-Γ₂) is often used. The large magnitude of ¹H-Γ₂ makes it a highly sensitive probe. This is due to the large nuclear gyromagnetic ratio of the proton and the primary dependence of ¹H-Γ₂ on the spectral density function at zero-frequency. In addition, ¹H-Γ₂ is much less susceptible to internal motions and cross relaxation than the longitudinal ¹H-PRE (referred to as ¹H-Γ₁) which is highly sensitive to these two factors (20). For quantitative analysis, it is very important to accurately and precisely measure ¹H-Γ₂ rates and to understand potential pitfalls in the measurements. In this paper, we describe practical aspects of ¹H-Γ₂ measurements for amide protons of proteins.

2. Materials and Methods

2.1 NMR samples

Experimental PRE data were acquired on two samples: a 20 kDa complex between ²H/¹⁵N-labeled SRY and a 14-bp DNA duplex with dT-EDTA located at the third base-pair (site c displayed in Fig. 2 of Ref. 20); and ¹⁵N-labeled HPr(E32C) with EDTA-Mn²⁺ conjugated at Cys32.

Fig. 2.

Theoretical relationship between the delay difference ΔT and the error σ(Γ₂) in the value of Γ₂ obtained from a two time-point measurement (first time point, T = 0; second time point T = ΔT). The error function is given by Eq. 6. The results are shown for I_dia(0)/σ(I_dia) = 200.0, where I_dia(0) is the signal height at T = 0 for the diamagnetic sample, and σ(I_dia) is the noise standard deviation. The relationship between I_dia(0) and I_para(0) was assumed to be given by Eq. 7.

Expression and purification of SRY was carried out as described by Murphy et al. (29). Samples of the complexes chelating Mn²⁺ or Ca²⁺ were prepared as described (19,20). Data were recorded on 0.3 mM complex dissolved in 10 mM Tris•HCl (pH 6.8), 20 mM NaCl and 7% D₂O.

Expression and purification of HPr (E32C) and the conjugation reaction with N-[S-(2-pyridylthio)cysteaminyl]ethylene-diamine-N,N,N′N′-teatracetic acid (Toronto Research Chemicals) were carried out as described (28). The final conjugated HPr(E32C)-EDTA-Mn²⁺ or -EDTA-Ca²⁺ samples were further purified by Mono-Q anion-exchange chromatography. NMR data on the paramagnetic (Mn²⁺) and diamagnetic (Ca²⁺) states were recorded using 0.3 mM protein dissolved in 10 mM Tris•HCl (pH 7.4) and 5% D₂O.

The chemical shifts for the Mn²⁺- and Ca²⁺-chelated states of the SRY/DNA complex are identical, indicating that pseudo-contact shifts are negligible and, therefore, the electronic g-tensor for the unpaired electrons of Mn²⁺ is indeed isotropic. An isotropic g-tensor for the unpaired electrons of Mn²⁺ in EDTA-Mn²⁺ conjugated to either DNA or protein was also found in previous studies, including systems ranging from 20 to 50 kDa (10,19,23-28).

2.2 NMR measurements

Measurements of PRE ¹H_N-Γ₂ rates were carried out using Bruker DMX-500, DRX-600 and DRX-800 spectrometers equipped with z-gradient triple resonance cryogenic probes. Identical T₂ experiments were performed on two samples, one with Mn²⁺(paramagnetic) conjugated to EDTA and the other with Ca²⁺(diamagnetic). The ¹H-Γ₂ rates are given by the difference in ¹H-R₂ rates between the paramagnetic and diamagnetic samples. For PRE applications, we found that conventional NMR tubes were better in terms of shimming than Shigemi micro-tubes, presumably because the magnetic susceptibility of the paramagnetic sample is quite different from that for a micro-cell matched for H₂O-based diamagnetic samples. The pulse sequence employed is given in Fig. 1. ¹H-T₂ measurements with multiple time-points were carried out in an interleaved manner. The recorded data were processed with the NMRPipe software (30). Shifted cosine bell window-functions were applied to both the ¹H and ¹⁵N time-domains, followed by zero-filling and Fourier transformation. (Note, we found that the type of window function employed does not affect the R₂ values obtained from multiple-time-point measurements). For the single-time-point approach, which requires Lorentzian line-shapes in the ¹H dimension, an exponential window-function was applied instead. Peak heights were quantified with either the NMRView (31) or NMRDraw (30) software. PRE ¹H_N-Γ₂ rates were determined from the peak heights for diamagnetic and paramagnetic samples (I_dia and I_para, respectively) as a function of the delay T (see Fig. 1) given by:

$$I_{\mathit{dia}}\left(T\right)=I_{\mathit{dia}}\left(0\right)\exp (-R_{2,\mathit{dia}}T)$$ [1]

$$I_{\mathit{para}}\left(T\right)=I_{\mathit{para}}\left(0\right)\exp \{-(R_{2,\mathit{dia}}+{\Gamma }_2)T\}$$ [2]

For two-time-point measurements, errors in Γ₂ were estimated as described in the Results and Discussion. Otherwise, errors were estimated using a Monte-Carlo approach.

Fig. 1.

Pulse sequence for ¹H_N-Γ₂ measurements. The delay T is changed for the relaxation measurement. Thin and bold bars indicate rectangular 90° and 180° pulses, respectively. Phases are along x unless indicated otherwise. Short bold bars represent soft rectangular 90° pulses (1.4 ms) selective for the ¹H₂O resonance. A half-bell shape for ¹H represents a half-Gaussian 90° pulse selective for water (2.0 ms). Delays are as follows: τ_a = 2.7 ms; τ_b = 2.25 ms; δ = (length of ¹³C WURST pulse). Phase cycling: ϕ₁=(y, y, −y, −y); ϕ₂ = (x,−x); ϕ₃ = (x,x,−x,−x,y,y,−y,−y); receiver=(x, −x, −x, x, −x, x, x, −x). The receiver phase and ϕ₂ were incremented for States-TPPI quadrature detection in the t₁ domain. Field gradients are optimized to minimize the solvent signal. Although ³J_HN-Hα is active for non-deuterated proteins during the period T, the resulting modulation is cancelled out when Γ₂ is calculated as described in the main text.

3. Results and Discussion

3.1 Experiment for Γ₂ measurements

At high magnetic fields (¹H frequency ≥ 500 MHz), the PRE rate, Γ₂, arising from the dipole-dipole interaction between a nucleus and unpaired electrons with an isotropic g-tensor is given by (1,32):

$${\Gamma }_2=\frac{1}{15}{\left(\frac{{\mu }_0}{4\pi }\right)}^2{\gamma }_I^2{g}^2{\mu }_B^2\mathrm{S}(\mathrm{S}+1)r^{-6}\left\{4{\tau }_c+\frac{3{\tau }_c}{1+{\left({\omega }_H{\tau }_c\right)}^2}\right\}$$ [3]

where r is the distance between the paramagnetic center and the observed nucleus; μ₀, the permeability of vacuum; γ_I, the nuclear gyromagnetic ratio; g, the electron g-factor; μ_B, the electron Bohr magneton; S, the electron spin quantum number; τ_c, the PRE correlation time defined as ${\tau }_c^{-1}={\tau }_r^{-1}+{\tau }_s^{-1}$; τ_r, the rotational correlation time; τ_s, the electron relaxation time; and ω_H/2π, nuclear Larmor frequency . Although Eq. 3 is valid only for a system where the paramagnetic group is fixed within the molecular frame, the equation can be readily modified to deal with a system where the paramagnetic center is mobile as described in our previous paper (20).

In practice, Γ₂ is measured as a difference in transverse relaxation rates between the paramagnetic (R_2,para) and diamagnetic (R_2,dia) states:

$${\Gamma }_2=R_{2,\mathit{para}}-R_{2,\mathit{dia}}$$ [4]

This subtraction cancels out all relaxation mechanisms common to both states, including exchange contributions to the transverse relaxation rate R₂, such that the only remaining relaxation mechanism arises from electron-nucleus interactions. Fig. 1 shows the pulse sequence we have been using for ¹H-Γ₂ measurements (19,23,24,25,27), which is essentially the same as that proposed by Kay and co-workers (8). The period for the ¹H transverse relaxation measurement is incorporated in the first INEPT scheme. During the ¹H transverse period of T + 2τ_a, ¹⁵N 180 pulses swap the slow and fast relaxing components (represented by H_y - 2H_yN_z and H_y + 2H_yN_z, respectively) and the contributions from the two components are identical, making the overall decay a single-exponential process with an average relaxation rate. The observed relaxation rates correspond to 1/T₂ for ¹H in-phase terms, since the transverse relaxation rates for H_y and 2H_xN_z terms are expected to be virtually identical for macromolecules. (Note that T₁ relaxation of N_z is much slower than ¹H T₂). Identical experiments are performed for the paramagnetic and diamagnetic samples to obtain Γ₂.

3.2 Two-time-point experiment for measurement of Γ₂ rates

A two-time-point measurement provides a simple means of obtaining Γ₂ rates and their corresponding errors without making use of any fitting procedures. In this approach, ¹H-Γ₂ rates are determined from two time points (T = 0 and ΔT) for transverse relaxation using Eqs. 1 and 2 as follows:

$${\Gamma }_2=R_{2,\mathit{para}}-R_{2,\mathit{dia}}=\frac{1}{T_b-T_a}\ln \frac{I_{\mathit{dia}}\left(T_b\right)I_{\mathit{para}}\left(T_a\right)}{I_{\mathit{dia}}\left(T_a\right)I_{\mathit{para}}\left(T_b\right)}$$ [5]

where I_dia and I_para are the peak intensities for the diamagnetic and paramagnetic states, respectively. It should be noted that effects of homonuclear ³J_HNHα-modulation during T are cancelled out by using identical times and taking ratios for the two states. The errors in Γ₂ can be propagated from Eq. 5 and are given by:

$$\sigma \left({\Gamma }_2\right)=\frac{1}{T_b-T_a}\sqrt{{\left\{\frac{{\sigma }_{\mathit{dia}}}{I_{\mathit{dia}}\left(T_a\right)}\right\}}^2+{\left\{\frac{{\sigma }_{\mathit{dia}}}{I_{\mathit{dia}}\left(T_b\right)}\right\}}^2+{\left\{\frac{{\sigma }_{\mathit{para}}}{I_{\mathit{para}}\left(T_a\right)}\right\}}^2+{\left\{\frac{{\sigma }_{\mathit{para}}}{I_{\mathit{para}}\left(T_b\right)}\right\}}^2}$$ [6]

where σ_dia and σ_para are the standard deviations of the noise in the spectra recorded for the diamagnetic and paramagnetic states, respectively.

The two time-points should be chosen to minimize the errors in the Γ₂ rates. Fig. 2 illustrates the theoretical relationship between ΔT and the error in Γ₂, obtained using Eqs. 1, 2 and 6 together with the following equation:

$$I_{\mathit{para}}\left(0\right)=I_{\mathit{dia}}\left(0\right)\frac{R_{2,\mathit{dia}}}{R_{2,\mathit{dia}}+{\Gamma }_2}\exp (-{\Gamma }_2\tau )$$ [7]

where τ represents the overall ¹H transverse period for the coherence transfers (9.9 ms). Eq. 7 assumes a Lorentzian line-shape in the ¹H-dimension, equal recovery levels during the repetition delay, and the same concentration and number of scans for both the diamagnetic and paramagnetic samples. Under these conditions, the error σ(Γ₂) is minimal when T_a = 0 and T_b ∼ 1.15/(R_2,dia+Γ₂). As can be seen from Fig. 2, the larger the value of Γ₂, the narrower the optimal range of T. Therefore, the second time-point should be set to be optimal for a relatively large Γ₂ expected for the system under study. For example, if the range of expected Γ₂ rates is 0 to 75 s⁻¹, a second time point at ∼1.15/(R_2,dia+50) sec. represents a reasonable choice. (However, if diamagnetic contamination in the paramagnetic sample is greater than ∼3%, smaller values are required for accurate measurement of intramolecular Γ₂ rates; see below).

To obtain reasonably precise data with errors smaller than ∼10% for Γ₂ rates of ∼50 s⁻¹, the number of accumulated scans per FID should typically be set to at least 32 for measurements using a cryogenic probe on a ∼0.3 mM paramagnetic sample of a non-deuterated ∼20 kDa protein or protein complex. Although the measurements could be somewhat time-consuming without a cryogenic probe, the sample concentration cannot be greatly increased beyond ∼0.5 mM, since undesired additional PRE effects arising from transient, random collisions (“solvent PRE” effect) may become significant at a higher concentrations (8).

3.3 Impact of increasing the number of time-points on the accuracy of Γ₂ rates

Since one might expect that a large number of time points may yield superior quality Γ₂ data, we investigated the impact of increasing the number of acquired time-points on the accuracy and precision of Γ₂ measurements using a Monte-Carlo approach. 100 sets of synthetic data with random Gaussian noise were generated using Eqs. 1, 2 and 7. Non-linear least-squares fitting was then carried out for each dataset to obtain the standard deviation of the apparent Γ₂ values. The results for 2, 4, 8 and 16 time-points were compared.

First, we simulated the case where the same number of accumulated scans is employed per time-point (Table I). Under these circumstances, the total length of data collection is proportional to the number of time-points. For each simulation, the noise standard deviation σ_I was set to 0.5% of I_dia(0) in Eq. 1. The relaxation rate for the diamagnetic state was set to 50 s⁻¹ and four values of the true Γ₂ (10, 25, 50, and 75 s⁻¹) were examined. As Table I shows, increasing the number of time-points without sacrificing the number of accumulated scans does indeed improve the precision in the Γ₂ measurement. (Note in these calculations with synthetic data, precision and accuracy are equivalent).

Table I.

Impact of increasing time-points without reducing scan-accumulations on the precision of of Γ₂ measurements.^a

			R2,dia = 50 s−1
n	Idia(0)/σI	ΔT	Γ2 = 10 s−1 σ(Γ2)	Γ2 = 25 s−1 σ(Γ2)	Γ2 = 50 s−1 σ(Γ2)	Γ2 = 75 s−1 σ(Γ2)
2	200	0.012	1.54	2.35	4.99	10.60
4	200	0.004	1.33	1.93	3.68	6.87
8	200	0.002	0.99	1.43	2.73	5.13
16	200	0.001	0.71	1.02	1.96	3.72

^aCalculations were carried out using a Monte-Carlo approach with 100 synthetic datasets as described in the main text. Definitions of symbols are as follows: n, number of time-points; I_dia(0), peak intensity at T=0 for the diamagnetic state; σ_I , noise standard deviation; ΔT, interval for linear sampling of T between 0 and (n-1) ΔT; σ(Γ₂), standard deviation of calculated Γ₂. The deviation of the calculated Γ₂ from the true Γ₂ was found to be almost same as σ(Γ₂). Thus, for these calculations, precision and accuracy are essentially equivalent.

In practice, however, spectrometer access may be limited, in which case increasing the number of time points would entail a proportionate decrease in the number of accumulated scans per time point to keep the total measurement time the same. Interesting questions arise in this regard. Should one use many time-points sacrificing signal-to-noise (S/N) ratios for individual spectra? Or should one acquire high S/N ratio data with fewer time points? To examine this issue, we carried out a second set of Monte-Carlo simulations in which the number of accumulated scans per time point was assumed to be inversely proportional to the number of time-points (n), making the overall experiment length constant and I_dia(0)/σ_I inversely proportional to √n . Table II shows the results of simulations using different I_dia(0)/σ_I ratios but in other respects identical to the simulations reported in Table I. Under these conditions, the simulations reveal that increasing the number of time-points at a cost of a lower S/N ratio results in no gain in Γ₂ precision. Therefore, a two-time-point Γ₂ measurement with high S/N ratio represents a good choice in practice, since it requires neither fitting procedures nor complicated error estimations. This conclusion is confirmed experimentally by the excellent correlation between the measured Γ₂ rates for the SRY-DNA complex using 2 and 8 time-points (Fig. 3).

Table II.

Impact of increasing time-points with reduced scan accumulations, keeping the overall measurement time constant, on the precision of Γ2 measurements.^a

			R2,dia = 50 s−1
n	Idia(0)/σI	ΔT	Γ2 = 10 s−1 σ(Γ2)	Γ2 = 25 s−1 σ(Γ2)	Γ2 = 50 s−1 σ(Γ2)	Γ2 = 75 s−1 σ(Γ2)
2	200	0.012	1.54	2.35	4.99	10.60
4	141	0.004	1.89	2.74	5.23	9.78
8	100	0.002	1.98	2.86	5.48	10.33
16	71	0.001	1.99	2.86	5.53	10.54

^aCalculations were carried out in the same manner as for those in Table I, except that different values of I_dia(0)/σ_I were employed to simulate a constant overall measurement time in all cases. For example, to acquire 8 points for T, the number of scans should be a fourth of that for a 2 time-point measurement, resulting in a reduction of 2 in S/N ratio. Definitions of symbols are as given in Table I.

Fig. 3.

Correlation between experimentally determined ¹H_N-Γ₂ rates derived from two (ΔT = 14 ms) and 8 time-point (ΔT = 2.8 ms) measurements for the ²H/¹⁵N-labeled SRY/DNA-EDTA-Mn²⁺ complex (0.3 mM). 64 scans per FID were acquired for the 2-time-point measurement and 16 scans per FID for the 8-time-point measurement, resulting in the same total measurement time (∼21 hours). Identical measurements were carried out for the paramagnetic (Mn²⁺) and diamagnetic (Ca²⁺) states. The data were measured at a ¹H-frequency of 500 MHz.

3.4 Inaccuracies of the single-time-point approach for the measurement of Γ₂ rates

Many studies in the literature have made use of a very simplistic approach in which only two regular HSQC spectra, one of the diamagnetic state and the other of the paramagnetic state at the same concentration, are employed to determine ¹H_N-Γ₂ rates using Eq. 7. We refer to this method as the single-time-point approach. Eq.7, however, only holds true if the repetition delay is sufficiently long to ensure that magnetization recovery levels are identical for the diamagnetic and paramagnetic states. However, for conventional HSQC experiments on biological macromolecules, the repetition delay between the scans is generally set to 0.8-2.0 s, which is not sufficient for 100% recovery of ¹H longitudinal magnetizations. Indeed, for a ²H-labeled protein, the amide ¹H T₁ relaxation time at regular magnetic fields (¹H frequency of 500-800 MHz) is very long and a repetition delay longer than ∼20 s would be required to achieve 100% recovery for all amide ¹H nuclei (33). With a shorter repetition delay, T_rep, the recovery levels for the paramagnetic sample are always higher than those for the corresponding diamagnetic sample owing to the PRE on longitudinal relaxation rates (Γ₁). When this effect is taken into account, Eq. 7 is modified to:

$$I_{\mathit{para}}\left(0\right)=I_{\mathit{dia}}\left(0\right)\frac{1-\exp \{-(R_{1,\mathit{dia}}+{\Gamma }_1)T_{\mathit{rep}}\}}{1-\exp (-R_{1,\mathit{dia}}{T}_{\mathit{rep}})}\frac{R_{2,\mathit{dia}}}{R_{2,\mathit{dia}}+{\Gamma }_2}\exp (-{\Gamma }_2\tau )$$ [8]

From the Solomon-Bloembergen equations, Γ₁ can be calculated from Γ₂ as follows:

$${\Gamma }_1={\Gamma }_2{\left(\frac{2}{3}{\omega }_H^2{\tau }_c^2+\frac{7}{6}\right)}^{-1}$$ [9]

where ω_H/2π is the ¹H frequency. Fig. 4 shows the theoretical relationship between the true Γ₂ and the apparent Γ₂ derived from the single-time-point approach $\left({\Gamma }_{2,\mathit{SP}}^{\mathit{app}}\right)$ using Eqs. 8 and 9. Depending on the PRE correlation time τ_c, the value of ${\Gamma }_{2,\mathit{SP}}^{\mathit{app}}$ can be significantly smaller than the true Γ₂ value, especially for deuterated molecules. In the case of the PRE arising from Mn²⁺ (where τ_s is much smaller than that for a nitroxide spin label), the single time-point approach only yields accurate Γ₂ values when the repetition delay exceeds 4/R_1,dia. The requirement of a long repetition delay makes the single-time-point approach less attractive, because multiple time-points can be acquired with a shorter delay and therefore a comparable overall experiment time. Another practical problem of the single time-point approach is that either an exponential window function or no window function at all must be employed for the ¹H dimension since Eq. 7 assumes a Lorentzian line-shape. In practice, this type of data processing is suboptimal for relatively large systems because Lorentzian shapes are broad and hence significantly decrease the number of analyzable peaks owing to partial overlaps. In addition, the single time-point approach requires the use of an appropriate scaling factor to account for slight differences in sample concentrations while multiple time-point methods do not. For a small protein, enhanced relaxation rates and decreased peak intensities are observed for virtually all residues in the presence of a paramagnetic probe, making it difficult to scale the two spectra reliably. For these reasons, it is our view that the single time-point approach should be restricted to qualitative use, especially for deuterated proteins.

Fig. 4.

Theoretical relationship between the true Γ₂ and the apparent Γ₂ derived from the single-time-point approach $\left({\Gamma }_{2,\mathit{SP}}^{\mathit{app}}\right)$ at ¹H-frequency of 600 MHz, illustrating the effect of neglecting different recovery levels for the paramagnetic and diamagnetic states on the accuracy of ¹H-Γ₂ rates derived using the single time-point approach. Two different values for the PRE correlation time τ_c were considered: τ_c = 4 ns (panels a and b) and τ_c = 10 ns (panels c and d). Those two τ_c values correspond to a ∼20-kDa molecule conjugated to EDTA-Mn²⁺ and a nitroxide spin label, respectively. Results for a non-deuterated protein (T_2,dia = 20 ms and T_1,dia = 0.8-1.2 s; panels a and c) and the corresponding deuterated protein (T_2,dia = 40 ms and T_1,dia = 2.4 – 4.8 s; panels b and d) are displayed separately. The repetition delays between scans (T_rep) in the HSQC experiments (τ in Eq. 7 is 9.2 ms) were assumed to be 1.0 s for the non-deuterated sample and 2.0 s for the deuterated one.

3.5 Field-dependence of ¹H-Γ₂

In this section, we describe the field-dependence of ¹H-Γ₂. Since Γ₂ for a macromolecule is dominated by the value of the spectral density function at zero frequency, ¹H-Γ₂ rates measured at high magnetic field (B₀ > 10 Tesla) should be dependent on B₀ if (a) the PRE correlation time τ_c is field-dependent, or (b) the contribution from Curie-spin relaxation (whose rate is proportional to B₀²) is non-negligible. According to Gueron (34), Γ₂ due to the Curie-spin relaxation mechanism is given by:

$${\Gamma }_{2,\mathrm{Curie}-\mathrm{spin}}=\frac{1}{5}{\left(\frac{{\mu }_0}{4\pi }\right)}^2\frac{{\omega }_H^2{g}^4{\mu }_B^4{S}^2{(S+1)}^2}{{\left(3kT\right)}^2{r}^6}\left(4{\tau }_r+\frac{3{\tau }_r}{1+{\left({\omega }_H{\tau }_r\right)}^2}-4{\tau }_c-\frac{3{\tau }_c}{1+{\left({\omega }_H{\tau }_c\right)}^2}\right)$$ [10]

where k is the Boltzman constant, and T the temperature in Kelvin (Sometimes the last two negative terms in Eq. 10 are neglected, but this is only valid when τ_s << τ_r). In the case of SRY/DNA-EDTA-Mn²⁺ complex, the rotational correlation time τ_r is 8.5 ns from the ¹⁵N relaxation data, and the PRE correlation time τ_c, derived from the PRE Γ₁ and Γ₂ data, is 4.5 ns (20). Thus, the electron relaxation time τ_s is calculated to be 9.6 ns (20). For this system, Eqs. 3 and 10 predict that the contribution of the Curie-spin relaxation mechanism to the overall Γ₂ at 308 K is only 2%, even at a ¹H-frequency of 800 MHz. The relative contribution of the Curie-spin relaxation depends on the size of the system. For a very large system with τ_r larger than 50 ns (corresponding to a molecular weight in excess of 100 kDa), the contribution could be larger than 20% at 800 MHz. In the case of a nitroxide spin label, the electron spin relaxation time τ_s (>10⁻⁷ s) is much longer than τ_r (35,36) and, therefore, the PRE correlation time τ_c, defined as (τ_r⁻¹+τ_s⁻¹)⁻¹, is virtually identical to τ_r, resulting in a field-independent ¹H-Γ₂. For a macromolecular system with conjugated EDTA-Mn²⁺, on the other hand, τ_s is comparable to τ_r (10,19,20) and consequently ¹H-Γ₂ could be field-dependent because of the field-dependence of τ_s.

To examine the field-dependence of ¹H-Γ₂ arising from Mn²⁺, we measured ¹H_N-Γ₂ rates for the ¹³C/¹⁵N-labeled SRY/DNA-EDTA-Mn²⁺ complex at ¹H-frequencies of 500, 600 and 800 MHz. Fig. 5 shows the correlations between the Γ₂ rates measured at the different fields. Linear regressions indicate that the ¹H-Γ₂ rates at 600 MHz and 800 MHz are higher than the corresponding rates at 500 MHz by factors of 1.02 and 1.08, respectively. (Note that these numbers are not precise because of experimental errors). The field dependence of ¹H-Γ₂ arising from Mn²⁺ is very weak in this magnetic field range (11.7-18.8 Tesla). This finding leads one to two conclusions. First, the contribution from Curie-spin relaxation is indeed negligible, which is consistent with the considerations described above. Second, the PRE correlation time τ_c is almost field-independent. Considering that the experimentally determined τ_c is smaller than the rotational correlation time τ_r, this suggests a very weak field-dependence for the electron relaxation time τ_s of Mn²⁺ at high magnetic field (>10 Tesla) in contrast to the strong field dependence observed at low (< 3 Tesla) magnetic field (37).

Fig. 5.

Experimental field-dependence of ¹H_N-Γ₂ measured for the ²H/¹⁵N-labeled SRY/DNA-EDTA-Mn²⁺ complex (0.3 mM) at 500 MHz, 600 MHz, 800 MHz. (a) Correlation between ¹H_N-Γ₂ rates determined at 500 MHz and 600 MHz. (b) Correlation between ¹H_N-Γ₂ rates determined at 500 MHz and 800 MHz. The dotted lines are diagonals. Red solid lines represent linear regressions. Slopes for a and b are 1.024 and 1.078, respectively. 64, 40, and 32 scans per FID were acquired for the data recorded at 500 MHz, 600 MHz and 800 MHz, respectively.

Until recently, it was thought that the electron relaxation time τ_s for Mn²⁺ is dominated by collisional modulation of the zero-field-splitting (ZFS) tensor, arising from distortion of a metal coordination system by collision with solvent molecules (37,38). In this instance, τ_s should be proportional to B₀² at high magnetic fields that satisfy the condition ω_sτ_ν >> 1, where ω_s/2π is the electron Larmor frequency and τ_ν the correlation time for collisional modulation of the ZFS tensor (usually on the order of 10⁻¹¹ s). If this were the case, the ratio of the 800 MHz to 500 MHz ¹H-Γ₂ rates should be 1.52 for the SRY/DNA-EDTA-Mn²⁺ complex whose rotational correlation time τ_c is 8.5 ns. Hence, this mechanism clearly does not explain our experimental observations.

Recently, Miller and Sharp pointed out that other contributions to the electron relaxation time τ_s need to be taken into account and that τ_s for a Mn²⁺ system is actually given by (39):

$${\tau }_s^{-1}={\tau }_{s,\mathit{col}}^{-1}+{\tau }_{s,\mathit{rot}}^{-1}+{\tau }_{s,\mathit{vib}}^{-1}$$ [11]

The first term, ${\tau }_{s,\mathit{col}}^{-1}$, is due to collisional modulation of the ZFS tensor, which was discussed above. The second term, ${\tau }_{s,\mathit{rot}}^{-1}$, is the reorientational modulation of the ZFS tensor, but its contribution is negligible for a macromolecule. The third term, ${\tau }_{s,\mathit{vib}}^{-1}$, is for electron relaxation due to vibrational modulation of the ZFS tensor, which is essentially field-independent. The contribution of ${\tau }_{s,\mathit{vib}}^{-1}$ to the overall ${\tau }_s^{-1}$ could be dominant at high-magnetic field since the first term ${\tau }_{s,\mathit{col}}^{-1}$ is inverse-proportional to B₀². In fact, the magnitude of the ZFS tensor and the τ_ν value reported for EDTA-Mn²⁺ systems (40) predict a value 3×10⁶ s⁻¹ for ${\tau }_{s,\mathit{col}}^{-1}$ at 11.7 Tesla, which is significantly smaller than the experimentally measured value of 1.0×10⁹ s⁻¹ for ${\tau }_s^{-1}$. Judging from our ¹H-Γ₂ data, it is likely that the field-independent term ${\tau }_{s,\mathit{vib}}^{-1}$ is dominant in overall electron relaxation at high magnetic fields between 11.7 and 18.1 Tesla, whereas ${\tau }_{s,\mathit{col}}^{-1}$ is dominant at low magnetic fields.

3.6 Effect of diamagnetic contamination on ¹H-Γ₂

In practice, no matter how carefully a paramagnetic sample may be prepared, contamination by trace amounts (∼1-5%) of the corresponding diamagnetic species is almost impossible to avoid for the following reasons: (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⁺ for the EDTA-Mn²⁺ system; the reduced species for a nitroxide spin label); (c) chemical instability of the conjugated states (discussed in a later section). Thus, it is important to ascertain the impact of diamagnetic contamination on the measured ¹H-Γ₂ data.

Under conditions where the chemical shifts for the diamagnetic and paramagnetic states are identical and they do not exchange with each other, the signal intensity for the paramagnetic sample containing a trace amount of the diamagnetic species with population p_d is given by:

$$I\left(T\right)=(1-p_d)I_{\mathit{para}}\left(T\right)+p_d{I}_{\mathit{dia}}\left(T\right)$$ [12]

Using Eqs. 1, 2, 5, 7 and 12, we calculated the theoretical relationship between the true value of ¹H-Γ₂ and the apparent value that would be obtained from a two-time-point measurement for two values of ΔT and diamagnetic contaminations ranging from 0 to 5% (Fig. 6). The apparent value Γ₂ is always smaller than the true value. For the case with R_2,dia = 50 s⁻¹, Γ₂ = 60 s⁻¹ and p_d = 2%, the apparent value of Γ₂ from the two-time-point measurement with ΔT = 18 ms is 52 s⁻¹. For p_d = 5%, the apparent value of Γ₂ is reduced to 44 s⁻¹. Note that the percentage error in Γ₂ is much larger than p_d. This is due to the fact that the PRE significantly reduces the contribution of the first term in Eq. 12. The deviation from the true value of Γ₂ can be reduced by using a relatively small value of ΔT, thereby reducing the relative contribution of the second term. For example, the apparent values of Γ₂ obtained with ΔT = 6 ms are 55 s⁻¹ for p_d = 2% and 48 s⁻¹ for p_d = 5%.

Fig. 6.

Theoretical effect of contamination by an equivalent diamagnetic species in a paramagnetic sample on Γ₂ accuracy. The relationship between the true values of Γ₂ and the apparent values derived from two time-point measurements $\left({\Gamma }_{2,\mathit{TP}}^{\mathit{app}}\right)$ at different levels of diamagnetic contamination were computed using Eq. 1, 2, 5, 7 and 12 for ΔT values of (a) 6 ms and (b) 18 ms. The curves shown are for contaminant population p_d of 0, 1, 2, 3 and 5% as indicated in the panels.

Fig. 7a shows correlations between experimental ¹H_N-Γ₂ data measured with ΔT = 4 ms and 40 ms on ¹⁵N-HPr(E32C) conjugated with EDTA-Mn²⁺ at Cys32. As expected from the above considerations, the measured Γ₂ values derived from the experiment with ΔT = 40 ms are systematically smaller than those obtained with ΔT = 4 ms, indicating the presence of a diamagnetic impurity. From this correlation, p_d was estimated to be 3%. The same population for the diamagnetic impurity can also obtained from the intensities of residual peaks that should be completely broadened beyond detection, yet appear with low intensity because of the diamagnetic contamination (Fig. 7b).

Fig. 7.

Experimental manifestations of diamagnetic contamination. (a) Correlation between experimentally determined Γ₂ values measured for ¹⁵N-labeled HPr(E32C)-EDTA-Mn²⁺ using the two time-point approach with ΔT values of 4 and 40 ms (red solid circles with the error bars representing measurement precision). Also shown are the theoretical correlations between apparent Γ₂ values calculated for HPr (assuming a Lorenzian lineshape and a diamagnetic ¹H-R₂ rate of 25 s⁻¹) with ΔT values of 4 and 40 ms in the presence of 3% (blue line) and 5% (green line) diamagnetic species contamination. (b) Comparison of cross-peak intensities of Thr-34 in the diamagnetic (black line) and paramagnetic (red line) states. The right-hand panel is plotted a ten times the magnification level of the left-hand panel. The peak intensity of the paramagnetic state measured for HPr(E32C)-Cys-EDTA-Mn²⁺ is ∼3.5% of that in the diamagnetic control. As the amide proton of Thr-34 is in close proximity to the paramagnetic center (< 8 Å), the peak should be broadened beyond detection in the paramagnetic state in the complete absence of diamagnetic contamination.

In the case of intermolecular PRE measurements on a complex where dissociation and association processes are in fast exchange on the relaxation time scale, the observed Γ₂ is simply scaled down by 1-p_d and the effect of a diamagnetic impurity is much weaker than for the intramolecular case considered above.

3.7 Considerations of sample stability for ¹H-Γ₂ measurements

Since a relatively small percentage of diamagnetic contamination can affect ¹H-Γ₂, another concern is the stability of the paramagnetic sample itself. In the case of EDTA-conjugation to proteins, disulfide bond linkage to a Cys side-chain is commonly used (9,41-43). As the reaction is reversible, the conversion rate cannot be 100% and there is always a small percentage of unconverted diamagnetic species. Although the desired product after the conjugation reaction can be purified by ion-exchange chromatography (28) or organic mercury columns (5), undesired cross- linked protein dimer, which is diamagnetic, is gradually formed via disulfide exchange. We have found that for a 6-month old sample comprising a cysteaminyl-EDTA-Mn²⁺ conjugate of ¹⁵N-labeled HPr(E32C), the dimer population can be as high as 30%, as confirmed by electrophoresis and mass spectrometry (data not shown). Therefore, a freshly prepared sample should always be used for the measurement of PREs arising from Cys-EDTA-Mn²⁺. Alternatively, different chemistry can be used for conjugation of the paramagnetic group. For example, both maleimide (44) and iodide (5, 45) functional groups selectively and irreversibly react with a free sulfhydryl group under mild conditions, enabling the paramagnetic center to be linked to the protein via a stable C-S bond.

On the other hand, we found no evidence of this problem for EDTA-derivatized DNA, presumably because the EDTA group is conjugated through a stable linker without a disulfide bond (19,46). The SRY/DNA complex containing dT-EDTA-Mn²⁺ exhibited the same ¹H_N-Γ₂ values within experimental errors even 2 years after the first measurement (data not shown).

4. Concluding Remarks

In this paper, we have described various practical aspects of ¹H-Γ₂ measurements at high magnetic fields (¹H frequency ≥500 MHz) as applied to macromolecules (proteins, nucleic acids, and their complexes). We show that the two-time-point measurement approach provides a simple means to accurately measure ¹H-Γ₂ without requiring any fitting procedures or complicated error estimations. Optimal settings required to minimize the experimental errors are also discussed. Increasing time-points while sacrificing S/N ratio for individual spectra to complete the measurement within a given timeframe does not result in any improvement in the precision or accuracy of the Γ₂ rates. The single-time-point approach, on the other hand, which has been extensively used in the literature, yields inaccurate Γ₂ rates unless the magnetization recovery levels for the diamagnetic and paramagnetic samples are identical. The ¹H-Γ₂ rates arising from conjugated EDTA-Mn²⁺ are found to be virtually field-independent for a 20 kDa system at high magnetic field (B₀ > 11 Tesla). This is relevant when experiments are carried out in a research environment with multiple NMR spectrometers operating at different magnetic fields. Finally, PRE measurements involving conjugation of the paramagnetic center via a labile disulfide bond should be carried out immediately after sample preparation since trace amounts of the diamagnetic species can cause significant underestimation of Γ₂ rates.

The PRE arising from unpaired electrons with an isotropic g-tensor located on EDTA-Mn²⁺ or a nitroxide free radical conjugated via a flexible linker to the macromolecule of interest probably represents the only practical tool in paramagnetic NMR to characterize macromolecules that do not possess an intrinsic metal binding site or paramagnetic center. Although other observables, such as pseudo-contact shifts and cross-correlation involving the Curie-spin relaxation, arising from unpaired electrons with an anisotropic g-tensor have been shown to be useful for the investigation of metal-binding proteins (2,47-51), similar applications to macromolecules with an extrinsic paramagnetic group tend to problematic owing to various practical considerations (e.g. the presence of enantiomers in the EDTA coordination system and linker flexibility; cf. refs. 10 and 52). In conclusion, PRE measurements afford a wealth of highly valuable information related to both structure and dynamics of biological macromolecules (5-28), and the practical considerations described in this paper are important for optimal experimental design and data interpretation in such investigations.