Water proton spin saturation affects measured protein backbone ¹⁵N spin relaxation rates

Abstract

Protein backbone ¹⁵N NMR spin relaxation rates are useful in characterizing the protein dynamics and structures. To observe the protein nuclear-spin resonances a pulse sequence has to include a water suppression scheme. There are two commonly employed methods, saturating or dephasing the water spins with pulse field gradients and keeping them unperturbed with flip-back pulses. Here different water suppression methods were incorporated into pulse sequences to measure ¹⁵N longitudinal T₁ and transversal rotating-frame T₁ρ spin relaxation. Unexpectedly the ¹⁵N T₁ relaxation time constants varied significantly with the choice of water suppression method. For a 25-kDa Escherichia coli. glutamine binding protein (GlnBP) the T₁ values acquired with the pulse sequence containing a water dephasing gradient are on average 20% longer than the ones obtained using a pulse sequence containing the water flip-back pulse. In contrast the two T₁ρ data sets are correlated without an apparent offset. The average T₁ difference was reduced to 12% when the experimental recycle delay was doubled, while the average T₁ values from the flip-back measurements were nearly unchanged. Analysis of spectral signal to noise ratios (s/n) showed the apparent slower ¹⁵N relaxation obtained with the water dephasing experiment originated from the differences in ¹H_N recovery for each relaxation time point. This in turn offset signal reduction from ¹⁵N relaxation decay. The artifact becomes noticeable when the measured ¹⁵N relaxation time constant is comparable to recycle delay, e.g., the ¹⁵N T₁ of medium to large proteins. The ¹⁵N relaxation rates measured with either water suppression schemes yield reasonable fits to the structure. However, data from the saturated scheme results in significantly lower Model-Free order parameters (〈S²〉 = 0.81) than the non-saturated ones (〈S²〉 = 0.88), indicating such order parameters may be previously underestimated.

1. Introduction

Protein backbone ¹⁵N spin relaxation rates, the longitudinal (T₁) and the transversal (T₂) relaxation, are sensitive to the motion of protein N–H bond vectors. Accurately measured T₁ and T₂ are often used to derive both protein global diffusion parameters and local fast motions of the N–H bonds through numerical fitting, e.g. the Model-Free approach [1,2]. These parameters are useful in describing thermodynamic changes occurring in biomolecules as they function. In addition, since the ratio T₁/T₂ has been demonstrated to be dependent on the projection of the individual N–H bond to the protein rotational diffusion tensor frame [3], they can be applied as restraints to refine structures [4–6], as well as to relatively position domains within a protein complex [7–10]. The T₁/T₂ restraints are readily available and could be very helpful in characterizing larger systems. In contrast other NMR restraints are either requiring special treatment of the sample to measure, e.g. residual dipolar couplings (RDC) [11–13] and paramagnetic relaxation enhancement (PRE) [14], or not measurable, i.e. NOEs in deuterated system.

Similar to other solution NMR experiments accurate measurements of ¹⁵N T₁ and T₂ require satisfactory suppression of water ¹H signals. One approach is to dephase or saturate water ¹H spin coherence with the use of pulse field gradient or composite pulses [15–19]. Alternatively flip-back pulses, which keep water proton spins parallel to the B₀ field throughout the pulse sequence, became popular owing to the advantage in minimizing effects of radiation damping on cryogenic probes [20,21]. Here we have incorporated both water suppression schemes into 2D ¹H–¹⁵N pulse sequences for measuring ¹⁵N T₁ and T₁ρ. Significantly longer T₁ values were obtained when water proton spins were saturated by the gradient. Close analysis of relaxation curves and spectral signal to noise (s/n) ratio illustrated that the ¹⁵N relaxation difference originated from the varied ¹H_N recovery caused by water saturation. The conclusion was supported by the fact that doubling the recycle delay reduced the relaxation rate differences. Interestingly such ¹⁵N relaxation rate discrepancies did not produce any noticeable differences in their fit to the rotational diffusion tensor, however, the derived order parameters differed significantly. Thus saturating water proton spins causes artifacts in measuring ¹⁵N relaxation rates, which also affect the interpretation of fast motion dynamics.

2. Results

2.1. Pulse sequences

The 2D ¹H–¹⁵N ST2-PT TROSY pulse sequence [22–24] was modified to include relaxation delays (Fig. 1). Shown in Fig. 1a is the pulse sequence for measuring longitudinal T₁ relaxation. Pulses between points b and c in Fig. 1a are replaced with pulses in Fig. 1b to measure the transverse rotating frame relaxation T₁ρ. At point a of Fig. 1a the two-spin order H_zN_z is established and water proton spins are on the transverse plane. For the water flip-back version of the measurement the soft 1.2-ms long ¹H pulse, the open rectangle in Fig. 1a, was applied to flip water proton spins to +z. The in-phase N_x component was established at point b for the relaxation delays. Proton water-gate pulses, which flip non-water proton spins 180°, were applied periodically during T₁ and T₁ρ relaxation delays to cancel cross-correlations [25]. At point c gradient pulse G₅ is a z-filter to clean any transverse magnetizations. Shaped and water-gate proton pulses were implemented during the TROSY detection to ensure water spins stay on +z (Fig. 1a).

Fig. 1.

The 2D ¹H–¹⁵N pulse sequences for measuring ¹⁵N spin relaxation time constants T₁ (a) and T₁ρ (b) under TROSY detection. The pulses in panel 1a between points b and c are replaced with pulses in panel 1b to establish the pulse sequence for T₁ρ measurement. The ¹H and ¹⁵N carrier frequency was set to 4.663 ppm and 118.1 ppm, respectively. Narrow and wide filled rectangles correspond to hard pulses with flip angles of 90° and 180°, respectively. The open rectangle at point a is a 1.2-ms long 90° square pulse, turned on and off for water flip-back and dephase measurements, respectively. Other low-power filled squares on ¹H channel are 90° square pulses with a width of 1.0 ms. Filled bells are 90° sinc-shaped water-selective pulses with a duration of 1.9 ms. In panel 1b the spin lock (SL) and the following 2-ms purge pulses on ¹⁵N channel had a γB₁ field strength of 2000 Hz and were applied with phase φ₇. Pulses are x-phase by default. Phase cycles are listed as follows, φ₁ = 4(y), 4(−y); φ₂ = 8(x), 8(−x); φ₃ = 16(y), 16(−y); φ₄ = y, −y, x, −x; φ₅ = y; φ₆ = −y; φ₇ = 32(x), 32(−x); φ_rec = y, −y, x, −x, 2(−y, y, −x, x), y, −y, x, −x. Delay durations are listed as follows, τ₁ = 2.3 ms, τ₂ = 2.7 ms, τ₃ = 2 ms, τ₄ = 15 ms, τ₅ = 0.4 ms, τ₆ = 0.08 ms, ΔT₁ = 4.0 ms. All gradient pulses are along z-axis and G₆, G₇, and G₁₀ are in rectangular shape while the rest are sine-shaped. The duration, sign and strength for gradient pulses are as follows, G₁ = 3 ms, 15 G/cm; G₂ = 1 ms, −7.5 G/cm; G₃ = 2 ms, −25 G/cm; G₄ = 1 ms, 25 G/cm; G₅ = 2 ms, 25 G/cm; G₆ = 200 μs, −15 G/cm; G₇ = 200 μs, 15 G/cm; G₈ = 0.25 ms, −9 G/cm; G₉ = 1 ms, 15 G/cm; G₁₀ = 40.5 μs, 15 G/cm. Quadrature detection on the ¹⁵N dimension was achieved via Echo-Antiecho method [49–51] such that the second fid for each increment of t₁ was collected with signs of gradient pulses G₆ and G₇ being switched, and pulse phases of φ₄ = y, −y, −x, x, φ₅ = −y, and φ₆ = y being applied. For T₁ measurement the relaxation delay equals ΔT₁ × 4 × n where the loop number n was varied to yield different relaxation delays. For T₁ρ measurement the relaxation delay equals total SL-duration, ΔT₁ρ × 8, and ΔT₁ρ was varied. For T₁ measurement gradient pulse G₄ is a z-filter to clean any magnetization other than N_z. For T₁ρ measurement the ¹⁵N spin lock pulse was extended for 2 ms after the relaxation blocks to completely remove any component not in the direction of the spin lock (Fig. 1b); the ¹⁵N spin was then restored to the z direction by a 90°(y) pulse.

The dephase version of the pulse sequences stay nearly identical to the flip-back ones described above except for replacing the soft 1.2-ms long ¹H pulse (the open rectangle in Fig. 1a) with a 1.2-ms delay. Such modification allows water proton spin coherence to be destroyed by the gradient pulse G₃. Other than the difference of this soft ¹H pulse, the rest of the pulse sequence remains identical between the two water suppression schemes so that any imperfection in canceling ¹H–¹⁵N cross-correlation would affect the two measurements equally.

2.2. Measured T₁ and T₁ρ

The protein used to test the pulse sequences is Escherichia coli. Glutamine Binding Protein (GlnBP), which has a total of 226 residues and is highly anisotropic with a moment inertia ratio of 2. Chemical shift assignments of a total of 211 out of 219 non-proline ¹H–¹⁵N resonances [26] were confirmed using conventional 3D experiments. Amide proton of residues undergoing exchange, A1, D2, G21, D22, D73, N97, N99, and G171, could not be observed. Overlapping residues E74, F136, L146, L196, K199, and K205 were further excluded, and 205 resonances were left for analysis. The NMR sample of 250 μM ²H/¹⁵N labeled GlnBP was subject to relaxation measurements under either of the two water suppression schemes (Fig. 1) at 34 °C and a magnetic field of 14.1 T.

For T₁ measurements the average curve fitting errors for the exponential decay function were 1.3% and 1.8% for dephase and flip-back data sets, respectively. Our experimental reproducibility error was 2.4%. The correlation plots of T₁ from two different water suppression schemes (Fig. 2a) indicated significant differences between the two data sets with a large r.m.s.d. value of 181 ms between the two, or about 21% of the average T₁ of the flip-back data set, which was 869.4 ms. Residues with side-chain being exposed [27] were indicated (Fig. 2a) and there was no correlation between surface exposure and T₁ differences. The dephase pulse sequence which saturated water magnetizations resulted in a significantly slower T₁ relaxation.

Fig. 2.

The correlation between the two relaxation measurements using the dephase and the flip-back schemes of pulse sequences for T₁ (a) and T₁ρ (b). Surface exposed residues are shown in filled circles.

For T₁ρ the average curve fitting errors were 1.6% and 1.5% for dephase and flip-back data sets, respectively, and the r.m.s.d. between the two was 2.46 ms, about 3.4% of the average T₁ρ of the flip-back data set, which was comparable to the reproducibility error of 3.0%. In fact the T₁ρ correlation plot in Fig. 2b shows a much better agreement between the two measurements under different water suppression schemes than T₁s in Fig. 2a. Similarly surface exposed residues do not differentiate from others.

2.3. ¹⁵N relaxation rate differences and spectra s/n

Resonance intensities of ¹⁵N T₁ and T₁ρ relaxation data points for a buried residue A96 were plotted (Fig. 3). Overall the peak intensities for dephase measurements are significantly lower than flip-back ones due to saturation on water ¹H and protein ¹H_N spins. For all relaxation time points in T₁ρ experiments there is a relatively constant 50% reduction of peak intensities in the dephase measurement (Fig. 3b). And the fit of T₁ρ relaxation time constants from the two measurements are within 1.1%. However, for T₁ experiments such peak intensity reduction is not uniform for all time points (Fig. 3a), and the fit T₁ relaxation time constants differs by 30%. The varied peak intensity ratio of T₁ points indicated the initial ¹H_N magnetization population was not uniform within the water saturation experiment. The longer the T₁ relaxation delay time, the smaller the peak intensity difference between the two measurements (Fig. 3a). Earlier Grzesiek and Bax [28] pointed out a 10–20% increase of s/n of a protein HSQC spectrum was observed when water flip-back pulses were applied for water suppression instead of saturation using a gradient or rf pulses. And such s/n increase was not limited to solvent exposed amide protons because spin diffusion was quite efficient in transferring magnetizations within a protein [28,29]. Therefore it is plausible that saturation of water magnetization will result in variation of the initial ¹H_N magnetization that in turn will be modulated by the relaxation delay and can affect the measured resonance intensity decay.

Fig. 3.

The T₁ (a) and T₁ρ (b) relaxations curves for a buried residue A96 with absolute intensities being plotted. Data points from the dephase and the flip-back measurements are shown in squares and circles, respectively. The fit relaxation time constants in (a) are 1019 ms and 783.0 ms for the dephase and the flip-back measurements, respectively. The fit relaxation times constants in (b) are 71.94 ms and 71.15 ms for the dephase and the flip-back measurements, respectively.

To further confirm the cause-relationship between the initial ¹H_N intensity and ¹⁵N T₁ relaxation rate differences, we plotted the s/n differences of the first time-point spectra of dephase and flip-back T₁ experiments against the measured ¹⁵N T₁ differences between the two (Fig. 4). A strong correlation was observed and this confirmed the offset of ¹⁵N T₁ in the two measurements was an artifact due to the difference in the recovery of ¹H_N magnetizations that was influenced by water saturation. The longer the relaxation delay, the closer the ¹H_N magnetizations return to its equilibrium, thus resulting in higher resonance intensity. This will cause an apparent slower decay of the resonance intensity as a function of ¹⁵N T₁ relaxation delay.

Fig. 4.

The correlation between the normalized differences of s/n and T₁ between dephase and flip-back measurements. The spectra s/n were taken from the first time-point in each measurement.

2.4. ¹⁵N relaxation rate differences and recycle delay

A simple check to the ¹H_N recovery problem in ¹⁵N T₁ relaxation experiments is to increase the recycle delay so that the recovery difference can be reduced. We doubled the recycle delay from 1.5 s to 3.0 s and repeated the same ¹⁵N T₁ measurements using both dephase and flip-back pulse sequences. Indeed the average difference between the T₁ data sets was reduced to 12% from 21%. The two flip-back measurements with different recycle delays are correlated with an r.m.s.d. of 25.1 ms (Fig. 5a), 2.9% of the averaged T₁ value. The two dephase measurements have an r.m.s.d. of 72.6 ms, and the T₁ values with the 3-s of recycle delay are shorter (Fig. 5b). The dephase measurements do not provide reliable ¹⁵NT₁ values because they depend on recycle delays. In contrast the flip-back measurement is relatively insensitive to changes in the recycle delay and will yield T₁ relaxation times that are closer to the true values.

Fig. 5.

The T₁ correlation between the two sets of repeated experiments at varied recycle delay of 1.5 s and 3.0 s. Either the flip-back method (a) or the dephase (b) method was applied for water suppression.

2.5. Dynamics derived from T₁ and T₂

Since most published relaxation measurements were carried out using the dephase scheme, it would be interesting to test whether dynamics derived from those ¹⁵N relaxation rates were reasonable. We used the GlnBP data sets collected with pulse sequences having dephase or flip-back schemes and 1.5-s recycle delay. Both sets of measured relaxation time constants ratios (T₁/T₂) were fit against the crystal structure of GlnBP [30] to obtain the rotational diffusion tensor [3,31]. After the removal of residues undergoing large amplitude fast motion, K3, K4, A168, T223, E234 and K226, a total of 199 T₁/T₂ data points were selected for fitting. To avoid over fitting only 20% of data points were randomly chosen to fit the fully anisotropic rotational diffusion tensor. The randomization were repeated 1000 times to yield a distribution of individual tensor parameters, and such distributions were fit to a Gaussian function to yield the means and standard deviations in Table 1. The overall diffusion D_zz from the dephase data set is 8% slower than the value from the flip-back measurement, due to an apparent slower T₁ relaxation. The values of anisotropy (2D_zz/(D_xx + D_yy)) and asymmetry (D_yy/D_xx) from the two sets are close to each other, within a standard deviation, so are the Euler angles defining the tensor directions. The accuracy of tensor directions were checked by calculating the projection angle between the principal Z axes of each rotational diffusion tensor out of the T₁/T₂ fitting and the inertia tensor. The mean values are 8.3° for both data sets (Table 1). Thus both sets of measurements yielded reasonable tensor directions in comparison to a projection angle of 18° that was reported for a smaller anisotropic protein GB3 [32].

Table 1.

The rotational diffusion tensor derived from ¹⁵N T₁/T₂.

Measurementa	Dzz (107s−1)	2Dzz/(Dyy + Dxx)	Dyy/Dxx	α (°)	β (°)	γ (°)	χ2	Projection angle (°)b
Dephase	1.74 ± 0.04	1.44 ± 0.05	1.07 ± 0.04	96.4 ± 4.5	122 ± 4	33.8 ± 29.8	11.3	8.28 ± 3.26
Flip-back	1.89 ± 0.06	1.41 ± 0.07	1.09 ± 0.07	98.0 ± 6.9	124 ± 5	36.9 ± 34.6	16.2	8.33 ± 4.05

^aThe standard deviations in diffusion tensor parameters and projection angles were derived from a Gaussian fitting on a total of 1000 individual T₁/T₂ fitting results using randomly chosen 20% of the 199 data points.

^bThe angle between the long axes (Z) of the fit diffusion tensor and the inertia tensor.

Further minimizations on T₁ and T₂ with the obtained diffusion tensor parameters resulted in local Model-Free order parameters S². Shown in Fig. 6 is the correlation between derived S² from the two measurements with the surface exposed residues indicated. The dephase data set yielded lower order parameter with an average value 〈S²〉 of 0.812, and the 〈S²〉 for the flip-back data set was 0.879. Such order parameter difference is significant.

Fig. 6.

The correlation between the dephase data and the flip-back data derived Model-Free order parameters S². Surface exposed residues are shown in filled circles.

3. Summary and discussion

We measured ¹⁵N spin relaxation time constants T₁ and T₁ρ with the two different choices of water suppression methods. Measurements using the flip-back scheme keep water proton spins on +z axis while those employing dephasing gradient destroy water spin coherence on the transverse plane. The dephase experiment saturates water proton magnetizations while the flip-back experiment does minimal saturation. Pulse sequences were tested on both a medium sized protein GlnBP and a small protein Ubiquitin (Fig. S1). For both proteins the measured T₁ relaxation time constants were significantly longer when the spectra were collected with the dephase pulse sequence. And the longer the T₁ values, the larger the difference that was observed between the dephase and the flip-back measurements. Similar T₁ differences were also found when HSQC detection scheme was utilized instead of the TROSY method (data not shown). This discrepancy was also observed when the experiments were carried out using a room-temperature probe instead of a cryogenic probe. This phenomenon does not seem to be sensitive to variation in the temperature, or whether the protein was deuterated or protonated. The difference for T₁ρ is less prominent.

The apparent slower ¹⁵N T₁ relaxation obtained from the dephase measurement is due to the difference in ¹H_N recovery. The variation of T₁ relaxation delay results in the different initial ¹H_N magnetization, which adds to the effect of ¹⁵N relaxation in the final signal intensity. The longer the relaxation delay, the closer the ¹H_N magnetization returns to its equilibrium and the more signal will be observed. This counteracts the signal decay due to ¹⁵N relaxation being measured, thus resulting in an apparent longer relaxation time constant. This artifact is only noticeable when the measured ¹⁵N relaxation time constant is comparable to or longer than the recycle delay (1–2 s), e.g. ¹⁵N T₁ of medium to large proteins. The effect is less or unnoticeable if ¹⁵N T₁ of small proteins (<500 ms) or T₂ are measured. Thus the flip-back method for water suppression should be preferable, which cause minimal water saturation [20,21]. The observations that water saturation could reduce the NMR spectrum s/n were made previously [28,29]. Furthermore applications of selective excitation pulses for fast spectrum acquisition were based on the similar principle of minimal water saturation [33,34]. Here we showed that this same effect could also influence measured heteronuclear relaxation times by modulating the initial proton magnetization.

It is also interesting to notice the practical effects of different water treatments on the dynamics analysis. The overall rotational diffusion tensor directions from the two measurements in general agree with the inertia tensor. Therefore structure refinement using T₁/T₂ ratios should still be valid. The other aspect of dynamics analysis is the Model-Free order parameters S² describing amplitude of local motions. The data obtained using the flip-back scheme resulted in S² (〈S²〉 = 0.88) that are globally higher than the values calculated from the dephase data set, which averaged at 0.81. Even though the dephase S² values are closer to other published values being measured on other proteins at temperatures close to 34 °C [35–37], it is important to note that all of these published order parameters were derived from data acquired using the dephase scheme for water suppression. Thus the true range of Model-Free order parameters could be underestimated. The interpretation of fast-motion dynamics based on ¹⁵N spin relaxation depends on water suppression method and the real range of protein backbone N–H bond order parameters need further cross-validation.

4. Materials and methods

4.1. Isotope labeling and protein purification

The E. coli BL21(DE3) cells carrying the cDNA for its periplasmic protein GlnBP [38] constructed in a pET vector were cultured in 500 ml M9 minimal medium prepared in 99% D₂O, with ¹⁵NH₄Cl and glucose as the sole nitrogen and carbon sources, respectively. All isotopes were purchased from Cambridge Isotope Laboratories (Andover, MA). Cells were allowed to grow and express proteins at 37 °C for 2 days. Cell culture was then spun down and supernatant was saved. The cell pellet was further incubated at 25 °C for another 2 days in 100 ml of 20 mM Tris buffer pH 8.0 and 150 mM NaCl. This step allowed the complete secretion of GlnBP proteins from periplasmic space into the buffer. All buffer and the previously collected growth medium were concentrated to 100 ml and dialyzed against the same Tris buffer. The dialyzed sample with added glutamine to 1 mM was then loaded onto a 5-ml gravity column filled with Q-Sepharose resins. The flow-through, containing GlnBP in ligand-bound form, were collected, concentrated to 20 ml, and then dialyzed against 20 mM KPi buffer at pH 6. The dialyzed sample was then loaded onto a 5-ml gravity column filled with SP-Sepharose resins. Flow through was collected and concentrated to 5 ml and mixed with 25 ml of 6 M GuHCl. The mixture was incubated at 37 °C overnight to allow amide deuterons being exchanged to protons. Denatured GlnBP was refolded by a 10-fold dilution into 50 mM potassium phosphate buffer at pH 6.9. Further concentrating and buffer exchange were carried out as needed. The NMR sample for the study is composed of 250 μM ²H/¹⁵N labeled GlnBP in glutamine bound form in 50 mM potassium phosphate buffer at pH 6.9 with 0.5 mM EDTA, and 0.02% NaN₃.

4.2. T₁ and T₂ measurements

All experiments were performed on a Bruker Avance 600 MHz spectrometer equipped with cryogenic probe and Z-gradient coil. Temperature was set at 34 °C. Pulse sequences were shown in Fig. 1 with descriptions provided in the caption. The spectral widths were set to 7212 Hz and 2083 Hz for ¹H and ¹⁵N, respectively. The number of complex data points collected for ¹H and ¹⁵N dimensions were 1024 and 128, respectively. For T₁ measurements a series of seven 2D datasets at varied longitudinal relaxation delays from 80 to 1376 ms were collected in a randomized order. For T₁ρ measurements a series of six 2D datasets at varied transverse rotating-frame relaxation delays from 4 to 91 ms were collected in a randomized and interleaved manner. With recycle delay of 1.5 s the total acquisition time for T₁ and T₁ρ experiments were 37 and 48 h, respectively. NMRPipe [39] was used to process all spectra and relaxation time constants and fitting errors were extracted using minimization routine in Sparky (T.D. Goddard and D.G. Kneller, University of California, San Francisco).

$$1/T_2=(1/T_{1\rho }-{sin}^2\theta /T_1)/{cos}^2\theta$$ (1)

$$\theta ={tan}^{-1}({\mathrm{\Omega }}_{\text{N}}/\gamma B_1)$$ (2)

The T₂ values were calculated from T₁ρ, T₁, ¹⁵N chemical shift offset Ω_N, and spin lock field strength γB₁ using Eqs. (1) and (2). Ω_N was expressed in Hz and a uniform shift of −45 Hz (≈J_HN/2) was applied to remove TROSY frequency offset for ¹⁵N spins. The γB₁ value was fixed at experimental setup to 2000 Hz.

4.3. T₁ and T₂ calculations and minimizations

Spin relaxation rates are described by sums of the spectral density functions J(ω) of the N–H bond. The mathematical descriptions of the heteronuclear spin relaxation were described quite sometime ago [40]. The theoretical expressions for amide ¹⁵N spin relaxation rates in the absence of conformational exchange contributions are given in Eqs. (3)–(6) [41], where ω_H and ω_N are Larmor frequencies of ¹H and ¹⁵N spins in rad/s, respectively, μ₀ (4π× 10⁻⁷ kg m s⁻² A⁻²) is the vacuum permeability, h (6.6262 × 10⁻³⁴ kg m² s⁻¹) is Planck’s constant, γ_H (2.6752 × 10⁸ rad kg⁻¹ s A) and γ_N (−2.7126 × 10⁷ rad kg⁻¹ s A) are gyromagnetic ratios of ¹H and ¹⁵N spins, respectively, r_HN (1.02 × 10⁻¹⁰ m) is the inter-nuclei distance between ¹H and ¹⁵N nuclei, and Δσ_N (172 × 10⁻⁶) is the chemical shift anisotropy of amide ¹⁵N nucleus [42,43], and generally the largest component of ¹⁵N chemical shift tensor is assumed to be parallel to the N–H bond [44,45].

$$1/T_1=q_{DD}[J({\omega }_{\text{N}}-{\omega }_{\text{H}})+3J({\omega }_{\text{N}})+6J({\omega }_{\text{N}}+{\omega }_{\text{H}})]+q_{\text{CSA}}J({\omega }_{\text{N}})$$ (3)

$$1/T_2=(1/2)q_{DD}[4J(0)+J({\omega }_{\text{N}}-{\omega }_{\text{H}})+3J({\omega }_{\text{N}})+6J({\omega }_{\text{H}})+6J({\omega }_{\text{N}}+{\omega }_{\text{H}})]+(1/6)q_{\text{CSA}}[4J(0)+3J({\omega }_{\text{N}})]$$ (4)

$$q_{DD}=(1/10){({\mu }_0/4\pi )}^2{(h/2\pi )}^2{\gamma }_{\text{N}}^2{\gamma }_{\text{H}}^2{r}_{\text{HN}}^{-6}$$ (5)

$$q_{\mathit{CSA}}=(2/15){\omega }_{\text{N}}^2\mathrm{\Delta }{\sigma }_{\text{N}}^2$$ (6)

The spectral density function J(ω), the only variable in calculating T₁ and T₂, can be obtained from the Fourier transform of the N–H bond rotational correlation function. The N–H bond motion can be decomposed as the sum of local and global motions within Model-Free approach [1,2]. The global rotational diffusion model can be isotropic, axial symmetric, and fully anisotropic [46,47]. In our analysis we adopted a fully anisotropic diffusion model. The spectral density function can be calculated according to the formula in Lee et al. [41]. The structure coordinates of GlnBP are from the X-ray crystal structure at 1.94 Å resolution (1WDN) [30]. Amide protons were added to the structure using XPLOR-NIH [48].

$${\chi }^2=(1/N)\sum _{i=1}^N{\left[\frac{{(T_{1i}/T_{2i})}_{\mathit{calc}}-{(T_{1i}/T_{2i})}_{\mathit{meas}}}{\delta {(T_{1i}/T_{2i})}_{\mathit{meas}}}\right]}^2$$ (7)

Numerical minimization on a total number (N) of available relaxation rate ratios T₁/T₂ was carried out to obtain the global rotational diffusion tensor according to Eq. (7), where the differences between the above calculated ratios and the measured ratios were minimized, and δ(T₁/T₂) is measurement error. We also carried out a robustness test on the resulting rotational diffusion tensor by repeating the minimization in which the input consists of randomly chosen 20% of the available data points, and yielded estimation for the mean and error range. The Simplex algothrim written in MATLAB 2007b (The Mathworks, MA) was employed for all data minimizations. Residue specific amide bond Model-Free order parameters S² were obtained by minimizing the χ² differences between calculated and measured T₁ and T₂ using the available global diffusion tensor parameters from the previous T₁/T₂ ratio fitting.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jmr.2011.09.042.