Measuring radiofrequency fields in NMR spectroscopy using offset-dependent nutation profiles

Abstract

The application of NMR spectroscopy for studying molecular and reaction dynamics relies crucially on the measurement of the magnitude of radiofrequency (RF) fields that are used to nutate or lock the nuclear magnetization. Here, we report a method for measuring RF field amplitudes that leverages the intrinsic modulations observed in offset-dependent NMR nutation profiles of small molecules. Such nutation profiles are exquisitely sensitive to the magnitude of the RF field, and B₁ values ranging from 1-2000 Hz, as well the inhomogeneity in B₁ distributions, can be determined with high accuracy and precision using this approach. In order to measure B₁ fields associated with NMR experiments carried out on protein or nucleic acids, where these modulations are obscured by the large transverse relaxation rate constants of the analyte, our approach can be used in conjunction with a suitable external small molecule standard, expanding the scope of the method for large biomolecules.

1. Introduction

The NMR toolbox for characterizing molecular dynamics has diversified considerably in the last two decades with the introduction of methods such as chemical exchange saturation transfer (CEST) and relaxation dispersion (RD) (1–12). This has led to the detection and structural characterization of a number of sparsely populated biomolecular conformations implicated in enzyme catalysis (13, 14), molecular recognition (15–18) and protein folding (19, 20), as well as in aggregation (21, 22) and disease (23–25). While R_1ρ RD experiments have identified transient Hoogsteen base pairs in duplex DNA (26, 27), mechanisms underlying base misincorporation during DNA replication (14, 28) and invisible excited states of HIV-1 TAR RNA (29), CEST has been employed to visualize higher energy functional conformations of Abl kinase (25), superoxide dismutase (23, 24) and the fluoride riboswitch (17), as well as to define the conformational selection mechanism underlying Hsp70 chaperone-substrate interactions (30). Moreover, the populations and lifetimes of transiently populated reaction intermediates in organic and metallorganic chemistry have recently been estimated with the CEST approach (31–35).

In both CEST and R_1ρ RD experiments, an accurate measurement of the radiofrequency (RF) field strength is essential for quantifying thermodynamic, kinetic and structural parameters of the conformational exchange event (11, 36, 37). Over the years, a number of methods have been developed for determining the amplitude of the B1 field, beginning with the sideband strategy outlined by Bloch (38) and demonstrated by Anderson in 1956 (39). With the advent of pulsed NMR, nutation resulting from on-resonance irradiation was proposed as an efficient method for calibrating the applied RF field (40, 41). In heteronuclear NMR involving ¹⁵N or ¹³C isotope-labeled samples, the magnitude B₁ field can be measured from the residual splitting observed in a multiplet pattern when the decoupling of the scalar-coupled nucleus is applied off-resonance (36, 42–44). While this method is ideal for B1 amplitudes comparable to or larger than the heteronuclear coupling constant (90 - 150 Hz), much smaller B₁ fields are used routinely used in CEST experiments.

The current method for measuring weak B₁ fields proposed by Guenneugues et al. (45) is a variation of the nutation experiment (41, 46) where the RF field of desired strength is applied on z-magnetization for a systematically incremented time duration. Transverse components of magnetization are subsequently dephased with a gradient and the z-component is quantified through a read-pulse. Fourier transformation of this time-dependent signal provides both the amplitude and the probability distribution of the B₁ field across the sample. While this approach has been successful over a broad range of B₁ field strengths, it is challenging to use when measuring small RF fields of the order of 1 - 10 Hz. This is because of the need to place the RF transmitter on-resonance to within a value much smaller than the magnitude of the B₁ field, or alternatively to treat chemical shift offset as a fitting parameter while modeling intensities in the nutation spectrum.

In this manuscript, we report a method for measuring radiofrequency (RF) fields that makes use of modulations observed in the constant-time offset-dependent nutation profile of z-magnetization (CONDENZ) under the influence of RF radiation. This method enables the precise and accurate calibration of RF fields and is particularly useful for weak RF fields of the order of 1-10 Hz employed in CEST experiments. In addition, the CONDENZ approach provides a robust estimate of the inhomogeneity in the B₁ field that is required for the quantitative analysis of CEST and R1p RD profiles.

2. Materials and Methods

2.1. Sample preparation

An NMR sample of 100 mM unlabeled sucrose was prepared in 25 mM tris buffer at pH 7 with 90% D₂O/10% H₂O, 0.03 % NaN₃ and 1 mM DSS. This sucrose sample was used for all data collection for the ¹³C CONDENZ profiles shown in Figure 1 and Figure S1. ¹³C RF field strengths were also measured on a sample containing a mixture of 100 mM unlabeled sucrose, 1 mM methyl-¹³C α-ketobutyric acid and 50 mM benzaldehyde prepared in 25 mM tris buffer at pH 7 with 90% D₂O/10% H₂O, 0.03 % NaN₃ and 1 mM DSS.

Figure 1.

A) ¹³C CONDENZ profiles for RF amplitude settings of 1 Hz (cyan, t_nut = 400 ms), 10 Hz (red, t_nut = 400 ms), 100 Hz (green, t_nut = 50 ms) and 1000 Hz (orange, t_nut = 5 ms). Each profile is plotted as the intensity (I) of the sucrose anomeric ¹H peak, normalized to the intensity in a reference spectrum acquired without t_nut (I₀), as a function of the ¹³C offset at which the B₁ field is applied. Solid lines are fits of the data (circles) to the Bloch equations (Eq. 9). The B₁ estimate from the fit is indicated at the top of the plot along with the error obtained through a bootstrap routine. The ${\chi }_{red}^2$ surface for the fit, plotted as the increase in ${\chi }_{red}^2$ from the best fit value, (B) evaluated by keeping the B₁ field fixed at various values during the fitting routine, as well as the bootstrap distribution for each fit (C) are shown below each CONDENZ profile.

RF fields on the ¹⁵N nucleus were measured on two different NMR samples. The first one contained 1 mM ¹⁵N^ε-labeled tryptophan and 0.8 mM U-¹⁵N ubiquitin, prepared in 25 mM sodium phosphate buffer at pH 7.0 with 25 mM NaCl, 1 mM EDTA, 0.03% NaN₃ and 2.5 % d₆-DMSO. The second one was used as an external standard for ¹⁵N B₁ calibration and contained 1 mM ¹⁵N^ε-labelled tryptophan prepared in the same buffer as the ¹⁵N^ε-labeled tryptophan/U-¹⁵N ubiquitin sample mentioned above. The aprotic solvent d₆-DMSO served as the lock solvent in both samples in order to eliminate artifacts from H/D solvent exchange (47).

U-¹⁵N ubiquitin was overexpressed as a construct without any purification tag in Escherichia coli (E.coli) BL21(DE3) cells grown in 2xM9 media (48) with 1 g/L of ¹⁵NH₄Cl as the sole nitrogen source. Cell pellets were lysed by sonication and purified as described earlier (49). Briefly, the pH of the clarified cell lysate was adjusted to 4.5 by drop-wise addition of acetic acid to precipitate many of the endogenous E.coli proteins. The mixture was centrifuged at 15000 rpm for 1 hour and the supernatant was dialysed against 50 mM acetic acid/sodium acetate buffer at pH 4.5. Ubiquitin was then loaded on an SP-sepharose cation exchange column and eluted with a 0 - 500 mM NaCl gradient in the same buffer. Ubiquitin eluted at an NaCl concentration of 165 mM. Fractions containing ubiquitin were pooled, concentrated and loaded on a 16 x 60 Superdex 75 size exclusion chromatographic column for subsequent purification. Pure fractions were pooled, concentrated, aliquoted, flash frozen and stored at -80 °C.

2.2. NMR spectroscopy

All NMR experiments were performed at 25 °C on a 700 MHz Bruker spectrometer equipped with a room temperature triple resonance single-axis gradient TXI probe.

2.3. CONDENZ measurements

CONDENZ data were acquired in pseudo-2D mode using the pulse sequence shown in Figure 2. In each dataset, the RF field is positioned at a specific offset value and a 1D spectrum is acquired that typically contains only one peak at the chemical shift of proton which is scalar coupled to the target ¹³C (¹⁵N) nucleus. The offset position of the RF field is then swept through the ¹³C (¹⁵N) chemical shift of the peak of interest, generating one 1D spectrum corresponding to each offset. Every pseudo-2D CONDENZ dataset also contains a reference 1D spectrum acquired with the same pulse sequence, but with t_nut set to 0. The sweep range and the offset step-size depend on the magnitude of the applied RF field and are tabulated in Table S1. For example, for a 10 Hz B₁ field, a pseudo-2D dataset containing 92 1D spectra was generated by sweeping -90 and 90 Hz using a uniform spacing between successive offsets of 2 Hz. Signal averaging over 16 transients for each 1D spectrum, corresponding to an average acquisition time of ~ 40 min per B₁ field, usually gave sufficient signal-to-noise (SNR) for quantitative analysis. The relaxation delay used in all CONDENZ measurements was 1.5 s.

Figure 2.

The selective ¹³C 1D-based pulse sequence used for acquiring ¹³C CONDENZ profiles. Hard 90° and 180° pulses on both ¹H and ¹³C channels are depicted as black narrow and wide rectangles respectively and applied at the maximum available power. The ¹H transmitter is positioned on-resonance to the target proton signal in-between points a and b, and on-resonance to water through the rest of the pulse sequence. The ¹³C transmitter is placed on-resonance to the target carbon throughout the pulse sequence except during the nutation period, when it is varied systematically to generate ¹³C offset-dependent intensities. In this pulse sequence, ¹H magnetization is first destroyed by a ${90}_x^{\circ }-G_z-{90}_y^{\circ }-G_z$ module and ¹H z-magnetization is allowed to recover during the subsequent 1.5 s relaxation delay (d1). This module ensures that the same magnitude of ¹H polarization is available at the beginning of each slice of a CONDENZ dataset. Following the d1 period, equilibrium ¹³C polarization is dephased with a gradient and ¹H magnetization is transferred to ¹³C through a selective cross-polarization (SCP) element (51). SCP is achieved by application of matched RF fields with an amplitude of 130 Hz on the ¹H and ¹³C channels for a period of 6.3 ms for AX and 4.6 ms for AX₃ spin systems. After magnetization transfer, ¹³C coherence is flipped onto the z-axis and residual transverse ¹H and ¹³C components are destroyed with a gradient pulse. Subsequently, ¹³C transverse magnetization is created with a ${90}_{-y}^{\circ }$ pulse and passed through a filter for eliminating residual magnetization originating from resonances having similar ¹³C chemical shifts (within ~ 1 ppm). In this filter, transverse magnetization is maintained for a period $\zeta =\frac{1}{4\delta }$, where δ is the difference in chemical shifts (Hz) between the target ¹³C chemical shift and the second nucleus in its vicinity (53). In the ζ period, coherence from the on-resonance target ¹³C does not precess in the rotating frame and remains oriented along the -x axis, while the coherence from the second nucleus precesses by 90° to align along the y or the -y axis (depending on the sign of its ¹³C offset in the rotating frame). The application of a second ${90}_y^{\circ }$ pulse places the target polarization along the z-axis but does not affect the coherence of the second nucleus, which is then dephased by a gradient before the t_nut delay. ¹H decoupling during both the ζ and t_nut periods is implemented using a 4 kHz DIPSI-2 composite decoupling scheme (52). Following B1 irradiation along the x-axis during t_nut, residual transverse ¹³C magnetization is again eliminated with a gradient pulse and the z-component is transferred to ¹H through an SCP element. Water suppression is implemented using 1.5 ms rectangular water-selective pulses which are shown as open curves. ¹³C decoupling during acquisition is carried out using a WALTZ-16 train (54). A reference spectrum is acquired using the same sequence but lacking the t_nut nutation period. In order to ensure that heating from the DIPSI-2 ¹H decoupling is the same in the reference spectrum as well as in spectra acquired with different values of t_nut, a heat compensation element is inserted in the pulse sequence immediately after completion of data acquisition. During this heat compensation element, the 4 kHz DIPSI-2 ¹H decoupling field is turned on for a period of t_nut,_max - t_nut, where t_nut,_max is the maximum nutation time. The complete phase cycling for this sequence is: ϕ₁= {y,y,y,y,y,y,y,y,-y,-y,-y,-y,-y,-y,-y,-y}, ϕ₂ = {-x,x}, ϕ₃ = {x,x,x,x,-x,-x,-x,-x}, ϕ₄ = {x,x,-x,-x} and ϕ_R = {x,-x,-x,x,-x,x,x,-x,-x,x,x,-x,x,-x,-x,x}, but an 8-step phase cycle is sufficient. Gradient strengths are applied in the smooth square shape with the following strengths (% of maximum of ~ 50 G/cm) and durations: G₀ (11 %, 500 μs), G₁ (17 %, 500 μs), G₂ (83 %, 500 μs), G₃ (93 %, 500 μs), G₄ (71 %, 500 μs), G₅ (66 %, 300 μs). ¹⁵N CONDENZ data are acquired with the same pulse sequence but with ¹³C pulses replaced by ¹⁵N ones. The B₁ field strength and duration for SCP in this case are 90 Hz and 11 ms respectively.

2.4. Analysis of CONDENZ datasets

Pseudo 2D CONDENZ data were processed and analysed using the Bruker Topspin version 4.0.7 software package. The 2D datasest was first split into individual 1D spectra using the command 'splitser', following which the peaks in 1D spectra acquired at different offsets were picked using the automated Topspin routine 'pps'. The intensities in the offset-dependent 1D spectra (I) were plotted as a ratio against the corresponding intensity in the reference spectrum (t_nut = 0, I₀) as a function of the offset at which the RF field is applied to generate CONDENZ profiles. Offsets were measured as differences from the on-resonance chemical shift of the target ¹³C nucleus, which was set to 0 Hz.

The errors in the intensity values were obtained using the 'SINO' routine. First, the regions corresponding to the signal and to noise in each individual 1D spectrum are selected. SINO is then used to determine the SNR for each spectrum in the CONDENZ dataset. The noise, which is used as the error estimate in intensity, is then evaluated from the offset-specific intensity and SNR values. Typically, 0.05 ppm surrounding the peak of interest and 0.5 ppm in the region 0.5 - 1.0 ppm are chosen as the signal and noise regions in SINO. For offset values near resonance, the intensity is close to 0. The error value assigned in such cases is the average of the errors estimated from spectra belonging to the same pseudo-2D dataset where intensity is sufficiently large to quantify the error.

3. Results and Discussion

3.1. Intensity modulations observed in offset-dependent nutation profiles

Figure 1A and Figure S1 show CONDENZ nutation profiles of the anomeric ¹³C magnetization of the glucose ring in sucrose (referred to herewith as the sucrose anomeric carbon) for B₁ fields ranging from 1 - 2000 Hz. Nutation data were acquired using a selective 1D-based ¹³C-CEST pulse sequence shown in Figure 2. In this sequence, ¹³C z-magnetization of the anomeric sucrose carbon (C_z) is created from the single-bonded anomeric ¹H magnetization by transfer via a selective J cross-polarization module (50, 51) (Fig. 2). C_z is then subjected to a B₁ field applied at a specific ¹³C offset for a constant nutation time t_nut, during which scalar coupled protons are decoupled via a DIPSI-2 (52) composite pulse decoupling train. The ¹³C z-magnetization at the end of t_nut is then transferred back to ¹H using the same selective Hartmann-Hahn transfer for detection. A reference spectrum is acquired for each value of the B₁ field using the same pulse sequence but where the nutation period is absent. CONDENZ profiles in Figure 1A graph the intensity of the sucrose anomeric ¹H peak (I) as a ratio against the intensity of the target peak in the reference spectrum (I₀) as a function of the ¹³C offset at which the B₁ field is applied.

3.2. Theoretical basis for modulations seen in CONDENZ profiles

The presence of offset-dependent modulations observed in CONDENZ profiles can be explained through an analysis of the Bloch equations. For a single-spin-1/2 system, the Bloch equations in the rotating frame for the three components of magnetization (M_x, M_y and M_z) in the presence of a RF field of amplitude ω₁ (rad/s) applied along the x-axis are given by (55):

$$\frac{d\overrightarrow{M}}{dt}=R\overrightarrow{M}$$ (1)

where $\overrightarrow{M}={\left[E{M}_x{M}_y{M}_z\right]}^T$ (T being the transpose operation), Δ is the chemical shift offset (rad/s) and

$$R=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& -R_2& -\Delta & 0\\ 0& \Delta & -R_2& -{\omega }_1\\ R_1{M}_0& 0& {\omega }_1& -R_1\end{array}\right].$$ (2)

The formal solution to this set of coupled differential equations is:

$$\overrightarrow{M}\left(t\right)=e^{Rt}\overrightarrow{M}\left(0\right).$$ (3)

For small molecules where relaxation rates are slow compared to Δ or ω₁, we can neglect relaxation and the simplified equations of motion in the rotating frame become:

$$\begin{array}{l}\frac{d{M}_x\left(t\right)}{dt}=-\Delta M_y\left(t\right)\\ \frac{d{M}_y\left(t\right)}{dt}=\Delta M_x\left(t\right)-{\omega }_1{M}_z\left(t\right)\\ \frac{d{M}_z\left(t\right)}{dt}={\omega }_1{M}_y\left(t\right)\end{array}$$ (4)

Assuming initial conditions of:

$$\begin{array}{c}{M}_x\left(0\right)=M_y\left(0\right),\hfill \\ M_z\left(0\right)=M_0\hfill \end{array},$$ (5)

the analytical solution for M_z (t_nut) is given by:

$$\frac{M_z\left(t_{nut}\right)}{M_0}=1-\left[\frac{{\omega }_1^2{t}_{nut}}{2}{\left(\frac{\sin \varphi }{\varphi }\right)}^2\right],$$ (6)

where

$$\varphi =\frac{\sqrt{{\Delta }^2+{\omega }_1^2}{t}_{nut}}{2}$$ (7)

The trajectory of z-magnetization as a function of the offset Δ is thus a squared sinc function, matching experimental observations shown in Figure 1A. The argument of the sinc function depends on the variable chemical shift offset Δ, as well as the values of the nutation time and the amplitude of the applied radiofrequency field, which are held constant while acquiring nutation profiles.

3.3. CONDENZ profiles are exceptionally sensitive to the B₁ field strength

Interestingly, simulations of the Bloch equations (Fig. 3) reveal that these nutation profiles are exquisitely sensitive to the magnitude of the RF field; for example, profiles simulated with B₁ values of 3 and 3.2 Hz are visibly different. Therefore, we reasoned that experimentally acquired nutation profiles should enable us to measure the corresponding B₁ field with high precision.

Figure 3.

CONDENZ profiles simulated for a single ¹³C spin for pairs of B₁ fields (3 Hz, 3.2 Hz, left), (10 Hz, 10.5 Hz, middle) and (25 Hz, 26 Hz, right) demonstrating the high sensitivity of these profiles to small changes in B₁ field amplitude. Datasets were generated using the following parameters: t_nut = 0.4 s, R₁ = 1.7 s^-1, R₂ = 2.0 s^-1 and I_B1 = 5 %.

Accordingly, we modeled nutation profiles shown in Figure 1A using the Bloch equations, assuming the anomeric ¹³C to be a single-spin-1/2 system. Longitudinal (R₁) and transverse (R₂) relaxation rates of the anomeric carbon, as well as its resonance offset and the B₁ field were treated as fitting parameters. The RF amplitude was assumed to follow a Gaussian distribution with its standard deviation defined as the B₁ inhomogeneity (I_B1) (56). N (typically, N = 11) samples of the B₁ field (B_1s) were drawn from this distribution within the range [−(B₁ + 2I_B1),(B₁ + 2I_B1)] and the probabilities of their occurrence were determined from the equation for the Gaussian probability distribution function as:

$$P\left(B_{1s}\right)=\frac{1}{\sqrt{2\pi I_{B1}^2}}\exp \left(-\frac{{\left(B_{1s}-B_1\right)}^2}{2I_{B1}^2}\right)$$ (8)

The formal solution of the Bloch equations depends on the particular value of B_1s, so that the overall magnetization at the end of the time-evolution is a weighted sum of the individual evolutions, with the weights given by P(B_1s) as:

$$\overrightarrow{M}\left(t\right)={\displaystyle \sum _{N}P\left(B_{1S}\right)M_S\left(t\right)={\displaystyle \sum _{N}P\left(B_{1S}\right)e^{R\left(B_{1S}\right)t}\overrightarrow{M}\left(0\right)}}$$ (9)

where the evolution matrix R now is a function of the particular sampled value of B_1S.

Nutation profiles can be fit very well using the single-spin-1/2 Bloch equations described by Eq. 9 to recover the magnitude of the RF field (Fig. 1A, Fig. S1). ${\chi }_{red}^2$ surfaces as a function of B₁ are very steep (Fig. 1B, Fig. S1), demonstrating the robustness of the B₁ values obtained from data fitting. In order to determine the precision of the measured B₁ values, we estimated errors using a bootstrap algorithm (57), where 1000 bootstrapped datasets for each B₁ field were constructed from the nutation profile by random sampling with replacement and fit to the Bloch equations. The resulting bootstrapped B₁ distributions (Fig. 1C, Fig. S1, Table 1) are narrow with standard deviations ranging from 0.02 - 1% of the measured B₁ value, clearly showing that highly precise B₁ measurements can be made from the modeling of offset-dependent nutation curves.

Table 1.

A comparison of ¹³C RF field amplitudes measured on a sucrose sample using the on-resonance nutation experiment (B_{1 nutation}) and CONDENZ (B_{1 CONDENZ}). B_{1 nutation} values are reported to the same number of decimal places as the B_{1 CONDENZ} values to facilitate comparison. B₁ inhomogeneities extracted from the CONDENZ profiles at each B1 field are listed in column 4. (ND: not determined)

B1 setting (Hz)	B1 CONDENZ (Hz)	IB1 (%)	B1 nutation (Hz)
1	0.98 ± 0.01	ND	0.91
2	2.00 ± 0.01	ND	2.04
3	2.95 ± 0.01	ND	3.03
5	4.95 ± 0.04	ND	5.03
7.5	7.47 ± 0.02	5.3 ± 0.5	7.50
10	9.94 ± 0.01	5.0 ± 0.2	9.98
15	14.98 ± 0.02	4.6 ± 0.3	14.99
20	19.79 ± 0.03	4.2 ± 0.2	20.07
25	24.99 ± 0.03	4.1 ± 0.2	24.98
30	29.67 ± 0.04	4.6 ± 0.3	30.04
100	97.20 ± 0.04	4.65 ±0.04	96.30
500	486.3 ± 0.1	5.42 ± 0.05	485.9
750	731.6 ± 0.2	5.21 ± 0.04	731.1
1000	973.4 ± 0.4	4.72 ± 0.03	975.6
1500	1456.9 ± 0.3	5.16 ± 0.04	1465.0
2000	1943 ± 3	6.0 ± 0.1	1965

3.4. Evaluating the accuracy of B₁ amplitudes extracted from CONDENZ profiles

As the next step, we evaluated the data for the presence of systematic errors that could bias the B₁ estimates. First, we experimentally measured the magnitude of B₁ using a second method proposed by Guenneugues et al (45, 58), in which magnetization is nutated by an external B₁ field applied on-resonance to the peak of interest for a variable nutation time. The time-dependent signal intensity is then Fourier transformed to determine the B₁ field. B₁ values obtained from CONDENZ profiles and the on-resonance nutation method agree very well with an R² value of 0.99 (Fig. 4, Table 1), demonstrating the reliability of the RF field measurements that can be made with the CONDENZ method.

Figure 4.

A comparison of the B₁ field amplitude obtained by modeling CONDENZ profiles (y-axis) with the values determined from an on-resonance nutation experiment (x-axis) for weak (1-35 Hz, panel A) and strong (100 - 2000 Hz, panel B) B₁ fields. The solid line in both panels is a y=x function. B₁ data points obtained by fitting CONDENZ profiles shown in Figure 1 are indicated with the same colour scheme as in Figure 1. Error bars are generally smaller than the size of the data points and not visible in the plot.

Second, we addressed the possibility of systematic deviations originating from the use of single-spin-1/2 Bloch equations for modeling nutation profiles. While the anomeric proton in sucrose is decoupled from the covalently bonded carbon (¹J_CH = 169 Hz) during t_nut through the use of a 4 kHz DIPSI-2 composite pulse decoupling scheme, the anomeric carbon is also scalar coupled to the proton on the neighbouring carbon that is off-resonance to the DIPSI-2 field by 1.86 ppm, through a two-bond ²J_CH of 7 Hz (59). These two-bond and three-bond couplings could impact the fit B₁ values especially at small amplitudes of the RF field. In order to probe this possibility, we first simulated nutation profiles at various B₁ fields using product operators for an AMX spin system, where A and M are the anomeric carbon and proton (¹J_CH = 169 Hz) that are on-resonance to the B1 and decoupling fields respectively, and X is the neighbouring proton that is off-resonance to the decoupling field and scalar-coupled to the anomeric carbon with a ²J_CH of 7 Hz and to the anomeric proton with a ³J_HH of 4 Hz. The simulated profiles were then fit with the single-spin-1/2 equations above to extract the amplitude of the B₁ field (Fig. S2). The input and fit B₁ values correlate excellently (R² = 0.99) and agree to within 0.5 % over a range of B₁ values from 0.5 - 10 Hz (Fig. S2, Table S2), confirming that the use of the single-spin-1/2 equations for modeling the nutation profiles does not systematically distort the measured B₁ values.

3.5. CONDENZ profiles provide robust estimates of the RF inhomogeneity

Having established the utility of our method for accurately and precisely measuring RF fields, we next asked whether the offset-dependent modulations observed in our experiments are sensitive to B₁ inhomogeneity. In order to address this question, we constructed ${\chi }_{red}^2$ curves that reflect the robustness of the parameter estimates of I_B1. These curves show pronounced minima for B₁ > 5 Hz (Fig. 5A, Fig. S1), confirming that I_B1 can also be measured reliably with our method. The errors in I_B1 range from ~ 9 % at 7.5 Hz to < 1% at 1 kHz (Table 1), showing that the estimates are more precise at higher RF field strengths. For B₁ ≤ 5 Hz, ${\chi }_{red}^2$ curves are often flat on the lower limit of I_B1, suggesting that only an upper estimate can be extracted reliably (Fig. S1). We verified these conclusions by fitting data simulated using the procedure detailed above to Eq. 9. I_B1 values recovered from the fit match very well with the input values to within 0.5 % for B₁ > 5 Hz (Table S2), while there is a systematic difference of the order of 5 % between input and fit values for B₁ ≤ 5 Hz.

Figure 5.

${\chi }_{red}^2$ surfaces (A) and bootstrap (B) distributions for B₁ inhomogeneity (I_B1) evaluated by modeling CONDENZ profiles. The inhomogeneity is depicted as a percentage of the B₁ field. The values of B₁ and I_B1 obtained from the CONDENZ profiles are indicated at the top of the figure. B₁ measurements made from profiles shown in Figure 1 are coloured with the same scheme.

3.6. Estimating ¹⁵N RF amplitudes using CONDENZ data

Since R_1ρ and CEST data are acquired primarily on ¹³C and ¹⁵N nuclei (36), we next explored the possibility of measuring ¹⁵N RF fields using the CONDENZ approach. We chose ¹⁵N^ε-labeled Trp as a suitable small molecule because of a number of favorable properties such as easy availability, the slow solvent exchange rate of the indole ¹H^ε(60), a sharp indole ¹H^ε-¹⁵N^ε correlation, and a ¹⁵N chemical shift that falls within the resonance frequency range of typical protein and nucleic acid molecules. In order to eliminate potential interference from H/D exchange in the nutation profiles, we used 2.5 % DMSO-d₆ as the lock solvent (47). CONDENZ profiles of the Trp indole ¹⁵N^ε nucleus also show offset-dependent modulations in the presence of a B₁ field similar to those seen for the sucrose anomeric ¹³C (Fig. 6A, Fig. S3), confirming that these modulations are independent of the identity of the nucleus. Nutation profiles were modeled using the Bloch equations (Eq. 9) to extract values of B₁ and I_B1. Similar to ¹³C, the amplitude of B₁ fields applied on the ¹⁵N channel also can be measured accurately and precisely with steep ${\chi }_{red}^2$ and small errors (0.05 - 0.5 %) (Fig. S3, Table S3) for B₁ fields larger than 2 Hz, demonstrating the generality of the methodology. In contrast, there is a 6 % difference between the B₁ fields measured using the on-resonance and CONDENZ approaches for B1 ≤ 2 Hz, highlighting the difficulty of locating the exact ¹⁵N chemical shift of the Trp indole ¹⁵N^ε while acquiring on-resonance nutation data.

Figure 6.

A) ¹⁵N CONDENZ profiles for B₁ settings of 7.5 Hz (blue, left, t_nut = 200 ms) and 20 Hz (magenta, right, t_nut = 200 ms), plotted as the intensity ratio of the indole H^ε resonance in spectra acquired with (I) and without t_nut (I₀) against the ¹⁵N offset at which the B₁ field is applied. Solid lines are fits of the data to the Bloch equations (Eq. 9). The B₁ values obtained by modeling the CONDENZ profiles and the corresponding standard deviations recovered from a bootstrapping procedure are indicated at the top of the plot. Data were acquired on a sample containing both ¹⁵N^ε-Trp (internal standard) and U-¹⁵N ubiquitin. B) On-resonance nutation spectra acquired using the pulse sequence of Guenneugues et al. (45, 58) on a peak belonging to ¹⁵N-labeled ubiquitin for a B₁ setting of 7.5 Hz (blue, left) and 20 Hz (magenta, right). The same sample as for panel A was used in these experiments. C) Histogram depicting the comparison between B₁ field strengths measured on U-¹⁵N ubiquitin (yellow), an internal ¹⁵N^ε-Trp standard (green) and an external ¹⁵N^ε-Trp standard (pink) for B₁ amplitude settings ranging from 1 - 40 Hz, showing excellent agreement between the three values for all B₁ fields. Error bars are of the order of the line thickness and not readily visible.

3.7. Practical aspects of using CONDENZ profiles to determine B₁ field strength

There are a few practical considerations that govern the utilization of this methodology. First and foremost, the nutation parameters such as t_nut and the offset spacing (δΔ) must be adjusted so that the squared-sinc modulation is clearly observed in the CONDENZ profile. If t_nut or δΔ are too large, or if the signal-to-noise ratio (SNR) is too low so that clear modulations are not observed, the B₁ field cannot be extracted reliably. Parameters that provide tractable nutation profiles and accurate B1 fields in the range from 1 - 2000 Hz are listed in Table S1 and can be used as initial estimates, though we have observed that t_nut values may have to be slightly modified based on the R₂ of the observed nucleus; if the R₂ is higher than for the anomeric carbon of sucrose, smaller t_nut values can be employed to visualize the modulations clearly. Second, a vast number of CEST and R_1ρ experiments are carried out on large biomolecules such as proteins and nucleic acids, where R₂ is too large to observe squared-sinc modulations. In such cases, we asked whether small molecules can be used as internal or external standards for measuring the B₁ field. First, we doped a ¹⁵N-labeled ubiquitin sample with an internal ¹⁵N^ε-Trp standard and measured the RF field amplitude for B₁ values ranging from 1 - 50 Hz using the on-resonance nutation experiment for ubiquitin and the CONDENZ approach for Trp (Fig. 6B). The values agree to within 1 % and an R² of 0.99 (Table S4), demonstrating that both molecules experience the same B₁ field. Next, we made an external standard of ¹⁵N^ε-Trp in the same buffer and measured the B₁ field using CONDENZ. The magnitude of the RF field matches excellently with those estimated from the internal Trp standard as well as with ubiquitin (to within 1 %), provided the tuning, matching, pulse widths and power levels are left unchanged between the analyte and the external standard (Table S4). These results unequivocally show that the CONDENZ approach for determining the RF amplitude can be used in conjunction with an external small molecule standard and extends its utility for sensitive biomolecular samples that may be affected by the addition of small molecule standards. Third, R_1ρ and CEST experiments on biomolecules span a wide range of resonance offsets (28, 29, 56, 61–68). Specifically, experiments on ¹³C nuclei in proteins are acquired on moieties with chemical shifts from 5 ppm (¹³C^δ of Ile) to 175 ppm (carbonyl carbons) that corresponds to 34 kHz on an 800 MHz spectrometer. In order to determine whether the same external standard (eg. the anomeric ¹³C of sucrose resonating at 92 ppm) can be used to estimate the B₁ amplitude for such a wide range of ¹³C transmitter frequencies, we prepared a sample containing a mixture of benzaldehyde, sucrose and α-ketobutyric acid, and acquired CONDENZ profiles on the methyl-¹³C of ¹³CH₃ α-ketobutyric acid (ϖ = 6.5 ppm), the anomeric ¹³C of sucrose (ϖ = 92 ppm), an aromatic ¹³C of benzaldehyde (ϖ = 135.5 ppm) and the aldehydic ¹³C of benzaldehyde (ϖ = 196.5 ppm). The magnitude of the B₁ field determined with these four ¹³C nuclei spanning a range of ~ 200 ppm in chemical shift agree to within a maximum deviation of 1.2 % for B₁ fields of 5 Hz or larger, while a deviation of 12 % is observed for the 2 Hz case (Table S6). Though the deviations observed in the mixture sample are small, these experiments suggest that it is preferable to use a calibration standard whose resonance frequency matches well with the nucleus on which R_1ρ or CEST experiments are carried out, especially where small (1-3 Hz) B₁ fields are involved. Finally, a SNR of the reference standard of 200 is sufficient to guarantee good quality nutation profiles which can be analyzed to determine the amplitude of the RF field. For a 100 mM unlabeled sucrose sample (natural abundance ¹³C) or 1 mM ¹⁵N^ε-Trp, nutation data with such SNR can be obtained in 45 min on a 700 MHz spectrometer equipped with a room-temperature probe, underscoring the accessibility and ease of application of our method.

4. Conclusions

The CONDENZ approach is particularly useful for measuring weak B1 fields of the order of 1-10 Hz that are routinely employed in CEST experiments. In this B1 regime, the on-resonance nutation method is susceptible to interference from off-resonance effects, because locating the resonance frequency to within a few Hz is difficult for biomolecules with 10 Hz or larger linewidths. The CONDENZ method, however, is immune to off-resonance effects as the approach relies on measurements made by varying the chemical shift offset, the chemical shift offset is a fitting parameter and therefore does not have to be exactly identified. In addition, our experiments demonstrate that B1 calibration can be carried out with a small molecule external standard, eliminating the necessity for finding an isolated protein or nucleic acid resonance that does not undergo conformational exchange for this purpose. We anticipate such flexibility to be particularly useful for large biomolecules and intrinsically disordered proteins, whose 2D correlation spectra are characterized by severe peak overlap.