Accurate measurement of longitudinal paramagnetic relaxation enhancement through division after subtraction

Abstract

Paramagnetic relaxation enhancement (PRE) provides valuable distance constraints with respect to unpaired electron for the structural and dynamic characterization of biomolecules. While transverse PRE (Γ₂) has been widely used, its quantitative interpretation is often confounded by chemical exchange and other factors. In contrast, longitudinal PRE (Γ₁), though much smaller in magnitude than Γ₂, is a more faithful reporter of electron-nuclei distances. In this work, we introduce the division after subtraction (DAS) approach, which utilizes a synchronized sampling scheme to analytically cancel out common-mode artifacts. We validate this approach using site-directed covalent labeling and solvent PRE systems, and demonstrate improved accuracy compared to standard separate fitting routine. DAS is particularly effective for labile residues, where traditional nonlinear regression breaks down due to solvent-mediated effects. The enhanced precision with DAS in Γ₁ measurement allows for more rigorous investigation into biomolecular conformational landscape, thus providing a powerful complement to Γ₂-based PRE methodology.

Graphical abstract

1. Introduction

Paramagnetic relaxation enhancement (PRE) arises from dipole-dipole interactions between an unpaired electron and a nucleus, usually a proton, and accordingly, its magnitude is over 600 times greater than that of nuclear Overhauser effect (NOE), the dipole-dipole interactions between protons [[1], [2], [3]]. Thus, PRE can afford distance restraints up to ∼40 Å [[3], [4], [5]]. Furthermore, PRE is uniquely suited for elucidating the conformational fluctuation of biomacromolecules, providing that the electron-nucleus is shorter in the lowly populated species [3,[5], [6], [7], [8], [9], [10], [11], [12]]. PRE has been used to uncover and characterize alternative conformational states and provide valuable insights not accessible to other NMR observables [[13], [14], [15], [16]].

A paramagnetic probe can be covalently introduced to an otherwise diamagnetic protein system. Common strategies involve chemical reaction with site-specifically engineered cysteine or unnatural amino acid [4,17], or incorporation of metal-binding motif [[18], [19], [20]]. Probes such as nitroxide spin radicals [21,22], and Mn²⁺ or Gd³⁺-chelating complexes (which possess symmetric d or f electron orbitals) [3,23] are frequently used. These probes induce significant relaxation enhancement to the nuclei, i.e., Γ₁ and Γ₂ for longitudinal and transverse rates [1,2,24], respectively, without concurrent pseudo-contact shift (PCS) effect, which complicates the assessment of electron-nucleus distance [23]. While conventional PRE relies on covalently attached tags, solvent PRE (sPRE) with the use of paramagnetic co-solutes has gained traction in recent years [25,26]. The sPRE approach is experimentally advantageous as it obviates the need for site-specific conjugation [7,27,28]. Furthermore, by employing differentially charged co-solutes, sPRE has been used to map electrostatic potential at protein surface [[29], [30], [31]].

Compared to nitroxide spin radicals, probes with chelated Mn²⁺ or Gd³⁺ metal ions offer several distinct advantages. Though they require the additional step of metal coordination, the metal ion is located at the center of the probe. Previously, we have redesigned Gd³⁺ chelator, making the metal ion fully coordinated [32]. In comparison, the density of unpaired electron in a spin radical is localized at one vertex of a constrained five- or six-membered ring [22]. Furthermore, metal probes are usually more hydrophilic, thereby minimizing non-specific interactions with the protein. Critically, metal probes allow for the simultaneous measurement of transverse (Γ₂) and longitudinal (Γ₁) PRE [3,24]. This is usually not feasible with spin radical co-solutes, as the electron relaxation time (τ_e) of nitroxide radicals is extremely long (>100 ns), resulting in a negligible Γ₁ contribution [33].

The simplest approach for Γ₂ measurement is the "single-point" method, which estimates relaxation by comparing peak intensities between paramagnetic and diamagnetic HSQC spectra [3,4], under the assumption of a fixed transverse relaxation delay during the two INEPT transfers (∼10 ms). However, this approach is semi-quantitative at most. As paramagnetic probe alters longitudinal relaxation, Γ₂ value is contaminated by the Γ₁ effect. Accurate quantification, therefore, requires a "two-point" approach, as the pioneering work by Iwahara, Tang, and Clore demonstrated that implementing carefully selected relaxation delays can effectively suppress systematic errors arising from diamagnetic contamination and J coupling effects [24]. Iwahara et al. also established that a normalization strategy—utilizing identical relaxation intervals between synchronized paramagnetic and diamagnetic experiments—could successfully eliminate homonuclear ¹H–¹H scalar coupling artifacts, provided that the sampling points are away from multiples of 1/J_HN-HA [3]. More recently, Toyama, Rangadurai, and Kay developed optimized pulse sequences to further overcome the challenges of homonuclear scalar coupling modulations during backbone Hα transverse relaxation measurements. Their approach utilizes a spin-lock relaxation element—supplemented by an alternative band-selective adiabatic decoupling scheme—to suppress homonuclear Hα-Hβ scalar coupling evolution while simultaneously minimizing water radiation damping artifacts [34]. These key developments set the foundation for accurate measurement and processing of longitudinal PRE Γ₁ rates.

Γ₁ is typically smaller than Γ₂, yet its insensitivity to chemical exchange makes it a more faithful reporter of the paramagnetic effect. Since longitudinal relaxation does not involve phase coherence or chemical shift evolution, it is decoupled from these transverse dephasing artifacts. This advantage is particularly pronounced for metal ions characterized by rapid electron relaxation—such as Ni²⁺ and Fe³⁺—where Γ₁ can be substantial even in cases where Γ₂ is negligible [33,35,36]. Nevertheless, accurate quantification of Γ₁ is demanding, as fitting of longitudinal decay curves requires optimization of additional parameters compared to Γ₂ measurement. In this work, we introduce the "division after subtraction" (DAS) approach for highly accurate measurement of Γ₁ values. By employing a synchronized sampling scheme for both paramagnetic and diamagnetic samples during their recovery phases, our approach effectively cancels out NOE and solvent exchange contributions while suppressing artifacts arising from incomplete saturation.

2. Results and discussion

2.1. Derivatization of the DAS measurement approach

For the measurement of ¹H longitudinal relaxation rates, saturation recovery offers clear advantages over inversion recovery [37,38]. While inversion recovery is a standard approach, radiation damping makes it difficult to achieve complete signal inversion. In contrast, saturation recovery eliminates the need to flip the spin, while reducing the total experimental time by about half. In addition, saturation recovery is compatible with standard two-dimensional heteronuclear correlation pulse sequences by appending a gradient purging and relaxation delay block before the magnetization transfer.

Longitudinal decay during saturation recovery is described by:

$$I=I_{\text{eq}}-I_{\text{eq}}\times {\mathrm{e}}^{-R_{1}t}$$ (1)

Yet, factors such as pulse imperfections, field inhomogeneity, and incomplete saturation can cause the initial magnetization to deviate from zero. To account for such deviation and a potential timing offset, the recovery equation is more accurately expressed with a modified pre-factor, $I_1^{\prime }$:

$$I=I_{\text{eq}}+I_1^{\prime }\times {\mathrm{e}}^{-R_{1}t}$$ (2)

where $I_1^{\prime }=\left(I_0-I_{\text{eq}}\right){\mathrm{e}}^{R_1{t}_0}$. Here, $I_0$ represents the initial residual longitudinal magnetization resulting from incomplete or imperfect saturation. It should be noted that $I_0$ is treated as a signed variable that can assume either positive or negative values depending on the specific experimental conditions. Because the DAS methodology cancels out the pre-factor terms through synchronized subtraction and normalization, it remains mathematically valid across both regimes. Thus, the signal intensities for diamagnetic (dia) and paramagnetic (para) samples during respective saturation recoveries are defined as:

$$I^{\text{para}}=I_{\text{eq},\text{para}}+I_{1,\text{para}}^{\prime }\times {\mathrm{e}}^{-\left(R_1+{\Gamma }_1\right)t}$$ (3)

$$I^{\text{dia}}=I_{\text{eq},\text{dia}}+I_{1,\text{dia}}^{\prime }\times {\mathrm{e}}^{-R_{1}t}$$ (4)

Previously, PRE Γ₁ determination involves fitting of these two curves separately to determine R₁ + Γ₁ and R₁, then subtracting the two values [38]. However, because resonances in the paramagnetic sample usually relax faster, they may reach a different steady-state equilibrium magnetization, and consequently exhibit different pre-factors, especially when provided with a finite inter-scan recycle delay D₁. This discrepancy in residual longitudinal magnetization introduces systematic errors in the Γ₁ value.

The DAS method overcomes these limitations by comparing the differences between synchronized sampling points to cancel out the unknown constants I_eq and I₁′. Here, we define a difference signal between two points separated by an interval τ (Fig. 1A), which yields the following equation for the diamagnetic sample:

$$\Delta I_t^{\text{dia}}=\left[I_{\text{eq},\text{dia}}+I_{1,\text{dia}}^{\prime }\times {\mathrm{e}}^{-R_1\left(t+\tau \right)}\right]-\left[I_{\text{eq},\text{dia}}+I_{1,\text{dia}}^{\prime }\times {\mathrm{e}}^{-R_{1}t}\right]=I_{1,\text{dia}}^{\prime }\times \left[{\mathrm{e}}^{-R_{1}t}{\mathrm{e}}^{-R_1\tau }-{\mathrm{e}}^{-R_{1}t}\right]=I_{1,\text{dia}}^{\prime }{\times \mathrm{e}}^{-R_{1}t}\times \left[{\mathrm{e}}^{-R_1\tau }-1\right]$$ (5)

Fig. 1.

Schematic representation of the division after subtraction (DAS) sampling schemes. The scheme utilizes sampling points to define two difference signals, ΔI_t and ΔI_t+Δt, each characterized by a constant interval τ and separated by a relaxation delay Δt. This differencing step eliminates the equilibrium magnetization term (ΔI_eq) and compensates for initial saturation imperfections, with Δt < τ in (A) and Δt > τ in (B). The uniform sampling grid is illustrated in (C), where the relaxation increment is set equal to the differencing interval(Δt = τ). This equidistant sampling scheme maximizes experimental efficiency by utilizing overlapping data points while ensuring rigorous suppression of systematic artifacts.

We further normalize the difference signal when acquiring after a relaxation time Δt, which can be either smaller or larger than τ (Fig. 1A, Β):

$${{\rm X}}_{\Delta t}^{\text{dia},\text{norm}}=\frac{\Delta I_{t+\Delta t}^{\text{dia}}}{\Delta I_t^{\text{dia}}}=\frac{I_{1,\text{dia}}^{\prime }·e^{-R_1\left(t+\Delta t\right)}\left({\mathrm{e}}^{-R_1\tau }-1\right)}{I_{1,\text{dia}}^{\prime }·e^{-R_{1}t}\left({\mathrm{e}}^{-R_1\tau }-1\right)}={\mathrm{e}}^{-R_1\Delta t}$$ (6)

Similarly, for the paramagnetic sample:

$$X_{\Delta t}^{\text{para},\text{norm}}={\mathrm{e}}^{-\left(R_1+{\Gamma }_1\right)\Delta t}$$ (7)

Finally, by dividing the normalized paramagnetic term by the diamagnetic term, the intrinsic longitudinal relaxation rate (R₁) is canceled, leaving only the paramagnetic contribution:

$$F\left(\Delta t;\Gamma \right)=\frac{X_{\Delta t}^{para,norm}}{X_{\Delta t}^{dia,norm}}=\frac{e^{-\left(R_1+{\Gamma }_1\right)\Delta t}}{e^{-R_1\Delta t}}=e^{-{\Gamma }_1\Delta t}$$ (8)

This approach provides a robust measurement of Γ₁ value that is mathematically immune to initial saturation imperfections and variations in equilibrium magnetization between samples. The expression for Γ₁ can further be written in terms of the raw intensity differences (ΔΙ) at two distinct pairs of time points with a time interval τ:

$${\Gamma }_1=\frac{\ln \left[\frac{\left(I_{1+\tau }^{\text{para}}-I_1^{\text{para}}\right)}{\left(I_{2+\tau }^{\text{para}}-I_2^{\text{para}}\right)}\times \frac{\left(I_{2+\tau }^{\text{dia}}-I_2^{\text{dia}}\right)}{\left(I_{1+\tau }^{\text{dia}}-I_1^{\text{dia}}\right)}\right]}{\Delta t}=\frac{1}{\Delta t}\left[\ln \left({\Delta I}_1^{\mathrm{p}}\right)-\ln \left({\Delta I}_2^{\mathrm{p}}\right)+\ln \left({\Delta I}_2^{\mathrm{d}}\right)-\ln \left({\Delta I}_1^{\mathrm{d}}\right)\right]$$ (9)

Accordingly, the variance of Γ₁ is determined by the sum of the squared partial derivatives with respect to each intensity measurement, multiplied by their respective variances:

$$\sigma \left({\Gamma }_1\right)=\frac{1}{\Delta t}\sqrt{\frac{{\sigma }_{I_{1+\tau }^{\text{para}}}^2+{\sigma }_{I_1^{\text{para}}}^2}{{\left({\Delta I}_1^{\mathrm{p}}\right)}^2}+\frac{{\sigma }_{I_{2+\tau }^{\text{para}}}^2+{\sigma }_{I_2^{\text{para}}}^2}{{\left({\Delta I}_2^{\mathrm{p}}\right)}^2}+\frac{{\sigma }_{I_{2+\tau }^{\text{dia}}}^2+{\sigma }_{I_2^{\text{dia}}}^2}{{\left({\Delta I}_2^{\mathrm{d}}\right)}^2}+\frac{{\sigma }_{I_{1+\tau }^{\text{dia}}}^2+{\sigma }_{I_1^{\text{dia}}}^2}{{\left({\Delta I}_1^{\mathrm{d}}\right)}^2}}$$ (10)

If we define σ^para and σ^dia as the respective noise levels for the paramagnetic and diamagnetic spectra, the uncertainty in Γ₁ measurement is expressed as:

$$\sigma \left({\Gamma }_1\right)=\frac{\sqrt{2}}{\Delta t}\sqrt{\left[\frac{1}{{\left({\Delta I}_1^{\mathrm{p}}\right)}^2}+\frac{1}{{\left({\Delta I}_2^{\mathrm{p}}\right)}^2}\right]{\left({\sigma }^{\text{para}}\right)}^2+\left[\frac{1}{{\left({\Delta I}_2^{\mathrm{d}}\right)}^2}+\frac{1}{{\left({\Delta I}_1^{\mathrm{d}}\right)}^2}\right]{\left({\sigma }^{\text{dia}}\right)}^2}$$ (11)

Unlike the fitting of longitudinal relaxation curves, the error in the DAS approach can be directly calculated from the data. Moreover, the 1/Δt pre-factor and ΔI denominators provide a clear mathematical guide for selecting optimal values for Δt and τ. Choosing a τ that is too short would result in a small ΔI, which amplifies the error. Conversely, a very long τ makes the signal to decay toward the equilibrium magnetization I_eq, which also reduces the magnitude of the difference and the overall signal-to-noise ratio. In practice, a τ value approximately equal to 0.5 times T₁ provides a robust difference signal while maintaining sufficient sensitivity. Concurrently, if Δt is excessively long, the signal in the second window (ΔI₂) would have largely decayed, leading to a sharp increase in uncertainty. Empirically, Δt should be chosen such that the signal has not decayed below 30% of the first difference intensity (ΔI₁) for either paramagnetic or diamagnetic sample.

By employing a uniform sampling grid where the relaxation increment is explicitly set equal to the differencing interval (Δt = τ), the acquisition time can be minimized (Fig. 1C). This equidistant sampling strategy enables the reuse of overlapping data points, and as a result, this 3-point measurement configuration maximizes experimental efficiency without compromising the mathematical elimination of systematic artifacts or incomplete saturation offsets, offering an accelerated acquisition approach for time-sensitive or unstable biomolecular samples. It should also be noted that although DAS approach is robust against very short recycle delays, longer D₁ values, e.g., 1.5 to 2 times of the longitudinal relaxation time of the diamagnetic sample can improve the overall signal-to-noise ratio.

2.2. NOE contributions are minimized with the DAS scheme

Transverse relaxation enhancement Γ₂ has contributions from alternative conformational states [3,5,8,22]. In contrast, longitudinal relaxation enhancement (Γ₁) is largely free of chemical exchange contributions, and therefore a faithful reporter of the electron-nucleus distance. However, Γ₁ measurements can be affected by cross-relaxation (NOE) between the spins within the protein and with their environment.

To quantify the impact of NOE on longitudinal recovery, we model the system using a modified Solomon equation. In its traditional formulation, the recovery of longitudinal relaxation vectors is driven by a constant offset toward thermal equilibrium (H_eq), creating a system of non-homogeneous differential equations. To enable direct numerical evaluation via a single matrix exponential operation, we map these non-homogeneous boundary conditions directly into a time-independent linear matrix form. By introducing a time-invariant dummy variable (E = 1) as a third element in the state vector, the equilibrium driving forces can be integrated as constant elements (θ^G and θ^SS) within the relaxation matrix:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}{H}_{\mathrm{z}}^{\mathrm{G}}\\ H_{\mathrm{z}}^{\text{SS}}\\ E\end{array}\right]=-\left[\begin{array}{ccc}{R}_z^{\mathrm{G}}& \sigma /N& {\theta }^{\mathrm{G}}\\ \sigma & R_{\mathrm{z}}^{\mathrm{S}}& {\theta }^{\text{SS}}\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{H}_{\mathrm{z}}^{\mathrm{G}}\\ H_{\mathrm{z}}^{\text{SS}}\\ E\end{array}\right]$$ (12)

where R_z^G and R_z^S represent the apparent longitudinal relaxation rates of the target spins (H_z^G) and their surrounding environment (H_z^SS), respectively, while $\sigma$ accounts for the cross-relaxation rate between them. This algebraic modification effectively transforms the inhomogeneous system into a homogeneous matrix framework, streamlining the simulation of multi-exponential recovery behaviors.

Our simulation results clearly demonstrate the advantage of the DAS scheme in mitigating NOE-induced systematic errors. While separate fitting exhibits systematic errors exceeding 60% as the NOE increases, the DAS scheme has a vastly expanded low-error region (Fig. 2). This indicates that DAS effectively treats internal NOE contributions as common-mode factors that are mathematically canceled during the normalization and division steps.

Fig. 2.

Comparison of Γ₁ accuracy between separate fitting and the new DAS scheme under varying NOE regimes. The heatmaps illustrate the percentage error in Γ₁ as a function of the NOE cross-relaxation rate σ (negative NOEs within the protein). (Left) Separate fitting of the longitudinal relaxation following saturation recovery shows high sensitivity to NOE, with relative errors as high as 60%. (Right) The DAS method exhibits a much larger low-error region, maintaining high precision even in the presence of strong NOE contributions. Errors in the DAS scheme are calculated analytically from raw intensities rather than fitting residuals.

The utility of DAS further extends to the suppression of NOE and exchange contributions originating from the solvent. Our analysis indicates that the error profile is highly sensitive to the magnetization state of the surrounding solvent during the recovery period (Fig. 3). By preserving the solvent magnetization—for instance, by implementing a selective water flip-back pulse before the saturation block—the influence on the recovery of $H_{\mathrm{z}}^{\mathrm{G}}$ by the solvent magnetization and the errors associated with Γ₁ measurements can be further minimized.

Fig. 3.

Impact of solvent magnetization on Γ₁ measurement accuracy. Simulated error heatmaps comparing separate fitting (left columns) and DAS (right columns) under two solvent conditions: "Saturated Surroundings" (top row), where the solvent is saturated during the recovery block, and "Retained Surroundings" (bottom row), where solvent magnetization is preserved via a selective water flip-back pulse. The DAS method combined with retained surroundings allows for more accurate measurement of Γ₁ values, effectively suppressing both intramolecular and solvent-mediated artifacts.

2.3. Experimental demonstration of improved measurement accuracy with the DAS scheme

We first evaluated the performance of the DAS approach with a covalently conjugated paramagnetic probe, for characterizing the transient interaction between the UIM1 domain of human Rpn10 and ubiquitin (Ub). Rpn10, a key proteasomal subunit, utilizes its two ubiquitin-interacting motifs (UIM1 and UIM2) to recruit ubiquitinated substrate proteins for degradation [39]. The EDTA-Mn²⁺ was site-specifically introduced to residue E232C of UIM1. Upon addition of ¹⁵N-labeled Ub, longitudinal PRE (Γ₁) rates were determined using both the traditional separate fitting and the DAS scheme. Interestingly, the Γ₁ values obtained via DAS were systematically larger for most structured residues compared to those from separate fitting (Fig. 4A). This small but consistent difference is likely caused by incomplete saturation at the start of the experiment with I₀ < 0. While traditional separate fitting is sensitive to baseline errors, the subtraction in the DAS scheme cancels them out. Quantitatively, this baseline shift is quite small across the rigid core residues, averaging 0.08 ± 0.06 s⁻¹, and therefore, saturation imperfections introduce rather minor errors for well-structured regions under routine spectrometer conditions. However, the error-suppression with the DAS framework becomes indispensable for highly flexible and labile residues.

Fig. 4.

Comparative performance of separate fitting versus the DAS framework for protein system with a covalently conjugated paramagnetic probe. (A) Γ₁ measurements characterizing the transient interaction between ubiquitin (Ub) and the UIM1 domain of human Rpn10. Error bars for the conventional multi-point separate exponential fitting (SEP, black circles) were estimated using 1000 Monte Carlo iterations. The SEP routine yields an unphysical negative Γ₁ rate for residue T9, indicated by an asterisk (∗). (B) The ratio of Γ₁ measurements derived from the two processing routines. Error bars represent values calculated via standard error propagation. The DAS results reside systematically above unity across the structured regions. (C) Goodness-of-fit assessment for the SEP routine, represented by the Coefficient of Determination (R², COD) for both the diamagnetic (black) and paramagnetic (red) longitudinal recovery curves. Drops in R² signify deviations from ideal single-exponential relaxation behavior. (D, E) Experimental longitudinal recovery curves and corresponding mono-exponential SEP fits for residues T9 (D) and G76 (E) under diamagnetic (black) and paramagnetic (red) conditions.

The most pronounced discrepancies between the two data-processing routines were localized to residues around T9 and G76 (Fig. 4B). Statistical evaluation revealed that these residues exhibited a breakdown from the single-exponential model during traditional separate fitting, characterized by marked dips in the Coefficient of Determination (R², COD) (Fig. 4C). Given that T9 resides in a solvent-accessible loop connecting the Ub β1 and β2 strands and G76 is a highly flexible C-terminal residue, these deviations are likely driven by localized cross-relaxation (σ) with solvent water.

Mechanistically, the directional nature of these artifacts—whether the traditional separate fitting (SEP) routine overestimates or underestimates the PRE rate—depends on whether the multi-exponential cross-relaxation distortion dominates the diamagnetic or the paramagnetic datapoints. Specifically, as shown in the experimental recovery trajectories (Fig. 4D), attempting to fit the diamagnetic curve of residue T9 using a standard mono-exponential regression resulted in a severely overshoot of the initial slope, thus inflating the extracted R_1,dia while underestimating the equilibrium plateau (I_eq). As the PRE Γ₁ rate is determined by subtraction R_1,dia from R_1,para, this artificially inflated R_1,dia results in a large underestimation of the final calculated value with the standard SEP routine. Conversely, for the dynamic C-terminal residue G76, the initial data points of the paramagnetic sample exhibit an accelerated recovery trajectory, causing the mono-exponential regression to overestimate the initial slope and overestimate the fitted R_1,para. Upon subtraction, this inflated R_1,para mathematically propagates as an exaggerated Γ₁value.

Fundamentally, cross-relaxation with solvent water distorts an ideal single-exponential recovery into a complex, multi-exponential process. Our numerical fittings indicate that when the cross-relaxation rate $\sigma$ matches local longitudinal relaxation rates (R₁ or R₁+Γ₁; Fig. 3), the underlying fast and slow components interfere constructively or destructively with standard non-linear regressions. In contrast, by implementing a synchronized time-point subtraction step, the DAS framework treats these non-exponential cross-relaxation components as common-mode terms, thus analytically cancelling them out prior to rate determination.

To further demonstrate the utility of the DAS framework, we applied it to solvent PRE (sPRE) measurements of Ub in the presence of the soluble paramagnetic co-solute Gd-DTPA-BMA (Fig. 5A). The most pronounced discrepancies between separate fitting and the DAS scheme were also observed for residues in the solvent-exposed loop connecting strands β1 and β2 and the highly flexible C-terminus (Fig. 5B). For these residues, the traditional multi-point SEP approach showed a substantial reduction in the coefficient of determination (R², COD) values (Fig. 5C). These localized dips in fit quality provide clear evidence that the underlying longitudinal recovery curves deviate from ideal single-exponential behavior due to non-negligible cross-relaxation (σ) with the bulk solvent water pool (Fig. 3). In contrast, by utilizing a synchronized time-point subtraction, the DAS approach analytically cancels out multi-exponential distortions as common-mode factors, and consequently yields a more accurate and consistent determination for the relaxation rates. This ensures that the extracted Γ₁ values simply reflect the outer-sphere paramagnetic enhancement uncontaminated by localized water-exchange dynamics, making DAS an invaluable tool for quantitative surface-mapping studies.

Fig. 5.

Comparative performance of separate fitting versus the DAS framework for solvent PRE (sPRE) measurement. (A) Concentration-normalized longitudinal PRE Γ₁ measurements for a 0.6 mM Ub in the presence of 2 mM paramagnetic co-solute Gd-DTPA-BMA, using traditional multi-point separate non-linear regression (SEP, black circles) and the synchronized 3-point DAS approach (red circles). (B) Absolute difference between the Γ₁ rates derived from the two processing routines. (C) Goodness-of-fit analysis for the separate longitudinal recovery curves, represented by the Coefficient of Determination (R², COD) for the diamagnetic (black) and paramagnetic (red) data arrays. The dips in R² are suggestive of multi-exponential solvent NOE perturbations.

3. Conclusion

In contrast to transverse paramagnetic relaxation enhancement (Γ₂), which is frequently convoluted by contributions from chemical exchange effects, the longitudinal component (Γ₁) is less susceptible to coherent dephasing artifacts. Furthermore, since Γ₁ values are inherently smaller in magnitude than Γ₂, the condition for the fast-exchange limit, i.e., k_ex ≫ ΔΓ [5], is much more accessible. Specifically, for a two-state conformational exchange model (A ↔ B), we define ΔΓ₁ = |Γ_1,A – Γ_1,B | and ΔΓ₂ = |Γ_2,A – Γ_2,B |, representing the differences in intrinsic PRE rates between the exchanging states. Transverse PRE values (Γ₂) are typically an order of magnitude larger than longitudinal PRE values (Γ₁), with Γ₂ easily reaching thousands of s⁻¹ for nuclei in close proximity to the paramagnetic center. Thus, for Γ₂, to achieve the true fast-exchange regime with the observed PRE rates equal to population-weighted average of the underlying states necessitates exceptionally rapid dynamics, approaching microsecond (μs) regime or faster. In this regard, Γ₁ measurements have a significant advantage in characterizing complex or heterogeneous conformational landscapes.

The DAS scheme presented in this study is conceptually analogous to intensity-ratio approach that has been established for Γ₂ measurement [22,24,34]. By normalizing the difference in signal intensities, the DAS scheme effectively cancels out systematic errors that impact both diamagnetic and paramagnetic samples. This error-suppression capability is critical for Γ₁ measurements, which often require extended experimental acquisition times where spectrometer drift or pulse instabilities may compromise data integrity. Our methodology can be readily incorporated into standard NMR pulse sequences without significant modification. Notably, existing ¹H relaxation datasets with equidistant time points may be reanalyzed using the DAS scheme to recover more accurate measurement for residues previously deemed erroneous due to fitting uncertainties.

The improved precision and experimental efficiency of the DAS method will facilitate the broader application of PRE experiments to challenging biological systems. This includes, but is not limited to, mapping conformational exchange processes across multiple timescales, and characterizing intrinsically disordered proteins (IDPs) and regions (IDRs), where subtle structural preferences are masked by high flexibility as well as solvent exchange.

4. Materials and methods

4.1. Protein expression and purification

Ubiquitin plasmid (pET-28a vector, kanamycin resistance) was synthesized by Tsingke Biotechnology. For protein expression and purification, the plasmid was transformed into Lemo-21 E. coli competent cells. Bacteria were cultured in M9 minimal medium containing ¹⁵NH₄Cl as the sole nitrogen source. Upon reaching OD600 = 0.6–1.0, protein expression was induced by adding 1 mM IPTG and incubating at 37 °C 220 rpm for another 5 h. Cells were harvested by centrifugation at 4 °C 5000 rpm for 20 min. The cell pellet was resuspended in 40 mL of SPA buffer (20 mM sodium acetate, pH 4.8 at room temperature) and lysed. The lysate was heated at 65 °C for 25 min to precipitate other proteins, followed by centrifugation at 4 °C 16 000 rpm for 30 min. The supernatant was collected and loaded onto SP cation exchange column (Cytiva) with SPA buffer, and elute with 60 mL SPB buffer (20 mM sodium acetate, 0.5 M NaCl, pH 4.8 at room temperature). The sample was further purified by size exclusion chromatography on a Superdex 75 120 mL column (Cytiva) with SEC buffer (22.2 mM sodium acetate, 111 mM NaCl, pH 6.0 at room temperature). Fractions were identified by SDS-PAGE, and the purified protein was flash-frozen in liquid nitrogen and stored at −80 °C.

The gene encoding of UIM1 from PSMD4_HUMAN (residues 195–252; UniProt P55036-1) was codon-optimized for E. coli expression and synthesized by Shenggong Biotechnology. The construct included an N-terminal 6 × His tag and was cloned into a pET-28a vector. To introduce paramagnetic probe, E232C mutations were introduced. The plasmid was transformed into Lemo21(DE3) cells and plated on LB agar containing kanamycin (50 μg/mL). A single colony was picked to inoculate a 20 mL overnight culture in LB with kanamycin at 37 °C, 225 rpm. This culture was then scaled to 1 L of fresh LB-kanamycin and grown at 37 °C to an OD 600 of ∼0.8. Expression was induced with 0.2 mM IPTG at 20 °C for 20 h. Cells were harvested and lysed in PBS buffer with 10 mM imidazole at pH 7.4, and the supernatant was applied to a HisTrap column (Cytiva), eluted by PBS with 500 mM imidazole. Final purification was performed using a Superdex 75 pg size-exclusion column (Cytiva) in PBS buffer. Protein purity was assessed by 20% SDS–PAGE after each purification step.

4.2. Synthesis of paramagnetic co-solute

Using a synthesis route similar to previously described [32], the synthesis of DTPA-BAE starts from 30 mL of DMF (N,N-dimethylformamide), 1.07 g of DTPAA (Diethylenetriamine-pentaacetic acid anhydride), and 2.1 mL of triethylamine were added sequentially to a 100 mL round-bottom flask. The mixture was stirred for 30 min, during which the solid gradually dissolved, yielding a yellow solution. Then, 438 μL of 1,8-diamino-3,6-dioxaoctane was added dropwise, resulting in the immediate formation of a large amount of white precipitate. The reaction was stirred at room temperature for 1 day, producing a pale-yellow solution with a light orange solid at the bottom. The solvent was removed by rotary evaporation, affording a light orange solid.

The light orange solid was dissolved in 30 mL water to yield a yellow solution. After adding 1 mL of 5 M hydrochloric acid to adjust the pH to ∼2, 0.76 g of Gd₂O₃ was introduced. The mixture was heated under stirring at 95 °C for 21 h, resulting in a light-yellow solution. The reaction mixture was cooled to room temperature, filtered through a 0.45 μm organic filter, and rotary evaporated to produce a beige solid. The solid was sequentially washed with 100 mL ethanol and 50 mL acetone using F-type sand-core funnel, followed by dissolution in 25 mL water to obtain a light-yellow solution. Filtration through the same funnel was conducted via three columns: 6 mL Na-type SCX cation exchange column, 6 mL Cl-type MAX anion exchange column, and 6 mL Na-type SCX cation exchange column, yielding a colorless effluent. Solvent removal by rotary evaporation produced 0.78 g of white powder. The successful synthesis of DTPA-BAE-Gd³⁺ was confirmed by electrospray mass spectrometry and element analysis.

The light orange solid obtained from the previous step was dissolved in 30 mL of ultrapure water, resulting in a yellow solution. The pH was adjusted to approximately 2 by adding 1 mL of 5 M hydrochloric acid, followed by the addition of 0.76 g of Gd₂O₃. The mixture was heated to 95 °C and stirred for 21 h, during which the solid gradually dissolved, yielding a light-yellow solution. After the reaction mixture had cooled to room temperature, it was filtered through a 0.45 μm organic membrane filter. The solvent was removed by rotary evaporation, affording an off-white solid. The solid was washed multiple times on a Synthware F-type sintered glass funnel, first with 100 mL of ethanol and then with 50 mL of acetone. The washed solid was dissolved in 25 mL of ultrapure water, and the solution was filtered through a Synthware F-type sintered glass funnel, and the filtrate was sequentially passed through a 6 mL Na-type SCX cation exchange column, a 6 mL Cl-type MAX anion exchange column, and another 6 mL Na-type SCX cation exchange column, yielding a colorless effluent. The solvent was removed by rotary evaporation, affording 0.78 g of a white powder, corresponding to a ∼28% yield. ESI-MS analysis: calcd. for C₂₀H₃₂GdN₅O₁₀Na⁺, m/z = 683.1283; found, m/z = 683.1292. EA analysis: calcd. C, 36.41; H, 4.89; N, 10.62; found C, 25.90; H, 4.24; N, 7.16. Accordingly, the purity of Gd-DTPA-BAE in the product was approximately 25.90/36.41 × 100% = 71.13%, with the remainder attributed to NaCl introduced during the ion-exchange column process.

4.3. Covalent conjugation of paramagnetic probe

For the standard PRE experiment, an EDTA-maleimide probe (Toronto Research Chemicals, cat# M138480) was utilized. The UIM1 domain of human Rpn10, harboring an E232C point mutation and concentrated to a final concentration of 70 μM, was treated with 1 mM tris(2-carboxyethyl)phosphine (TCEP). Conjugation was initiated by removing the TCEP by desalting and the addition of 1 mM EDTA-maleimide probe and 1.1 mM of MnCl₂, followed by incubation at 20 °C overnight. Unreacted probe and excess metal ions were subsequently removed with extensive dialysis. The final paramagnetic sample was exchanged into an NMR buffer consisting of 20 mM HEPES at pH 6.8.

4.4. NMR data acquisition and analysis

The PRE measurement for 500 μM EDTA-Mn²⁺ conjugated Rpn10 UIM1 with the addition of equimolar ¹⁵N-labeled Ub was conducted at 298 K on a Bruker Avance III 600 MHz spectrometer. Resonance assignments were confirmed via standard ¹H–¹⁵N HSQC experiments. Longitudinal relaxation was measured using a saturation recovery block with eight relaxation delays: 0.1, 0.3, 0.5, 0.9, 1.1, 1.3, 1.5, and 1.7 s. Spectra were acquired with 1024 × 128 complex points, utilizing 48 scans for the diamagnetic reference and 88 scans for the paramagnetic sample.

The sPRE measurement was performed at 298 K on a Bruker Avance III 700 MHz spectrometer equipped with a TCI cryoprobe. Longitudinal sPRE rates (R₁) were measured for 600 μM ¹⁵N-labeled Ub using a saturation recovery pulse sequence in the presence of 2 mM Gd-DTPA-BAE. Six relaxation delays were employed: 0.01, 0.1, 0.35, 0.6, 1.1, and 1.6 s. For both diamagnetic and paramagnetic samples, data were acquired with 2818 × 128 complex points and 64 scans per increment. To ensure high-quality data while optimizing instrument time, a minimal repetition delay was used for disk I/O, resulting in approximately 12 h per multi-point dataset; for the accelerated three-point equidistant DAS implementation, acquisition time was reduced to approximately 6 h.

All spectra were processed using the NMRPipe software package [40]. A cosine-bell window function was applied in both dimensions, and data were zero-filled to 2048 × 128 points prior to fast Fourier transformation. Peak intensities were extracted using the ipap.tcl script in the NMRPipe package [40]. For the conventional analysis approach, longitudinal relaxation rates (R₁) for diamagnetic and paramagnetic states were determined by fitting intensities to a mono-exponential longitudinal recovery model. The resulting Γ₁ values were calculated as the R₁ rate differences between diamagnetic and paramagnetic samples. Uncertainties in the fitted rates were estimated via 1000 Monte Carlo iterations.

For the DAS approach, intensities from synchronized time points were used to calculate difference signals (ΔI). In the three-point implementation (utilizing delays of 0.1, 0.6, and 1.1 s), the differences ΔI₁ = I_{0.6 s} – I_{0.1 s} and ΔI₂ = I_{1.1 s} – I_{0.6 s} were obtained. The Γ₁ value was then obtained by calculating the ratio of these normalized differences between the paramagnetic and diamagnetic experiments. In this equidistant scheme, Δt and τ are both 0.5 s. Precision and measurement uncertainties for the DAS-derived Γ₁ rates were calculated using the analytical error propagation formulas derived in Eq. (11). For multi-point datasets, all valid pairs, such as (0.1 s, 0.9 s) and (0.9 s, 1.7 s), suitable for DAS processing were utilized.

4.5. Numerical simulation of cross-relaxation contributions to systematic errors

Numerical simulations were conducted in Python to evaluate the impact of cross-relaxation (NOE) on the accuracy of Γ₁ measurements. The longitudinal relaxation of the spins of interest (H_z^G) and their surrounding environment (H_z^SS) was modeled using the matrix form of the Solomon equations [41,42]:

$$\left[\begin{array}{c}{H}_{\mathrm{z}}^{\mathrm{G}}\\ H_{\mathrm{z}}^{\text{SS}}\\ E\end{array}\right]=\exp \left(-T\left[\begin{array}{ccc}{R}_{\mathrm{z}}^{\mathrm{G}}& \sigma /N& {\theta }^{\mathrm{G}}\\ \sigma & R_{\mathrm{z}}^{\mathrm{S}}& {\theta }^{\text{SS}}\\ 0& 0& 0\end{array}\right]\right){\left[\begin{array}{c}{H}_{\mathrm{z}}^{\mathrm{G}}\\ H_{\mathrm{z}}^{\text{SS}}\\ E\end{array}\right]}_{t=t_0}$$ (13)

In this framework, E = 1 acts as a constant component representing the unit magnetization, while ${\theta }^{\mathrm{G}}=-R_{\mathrm{z}}^{\mathrm{G}}{H}_{\text{eq}}^{\mathrm{G}}-\frac{\sigma }{N}{H}_{\text{eq}}^{\text{SS}}$ and ${\theta }^{\text{SS}}=-R_z^{\mathrm{G}}{H}_{\text{eq}}^{\text{SS}}-\sigma H_{\text{eq}}^{\mathrm{G}}$ account for the recovery toward equilibrium. Here, σ represents the cross-relaxation rate between the target spins (H_z^G) and their surrounding environment (H_z^SS), where the equilibrium magnetization of the surrounding pool is scaled by the size factor N ($H_{\text{eq}}^{\text{SS}}=N\times H_{\text{eq}}^{\mathrm{G}}$).

Numerical modeling was performed in a Python 3.12 environment using the SciPy and NumPy libraries for matrix operations. The simulation pipeline proceeds through the following steps.

• a)The spin environment size was set to 100. For the diamagnetic reference system, the apparent relaxation rates were defined as R_z^G = 1 – σ and R_z^SS = 1 – σ/N. For the paramagnetic system, the intrinsic longitudinal enhancement (Γ₁) was directly added to the diagonal target spin component (R_z^G). To account for the heterogeneous paramagnetic effect on the environment, the surrounding nuclei were modeled to experience a PRE magnitude equivalent to 80% of that of the measured nuclei.
• b)Two boundary cases for the initial magnetization state at t = t₀ were evaluated: (i) a fully saturated environment represented by the initial vector M_t=t0 = [0, 0, 1]^T, and (ii) a retained environment configuration represented by M_t=t0 = [0, N, 1]^T, which simulates selective water flip-back regimen.
• c)Longitudinal recovery curves were computed via the matrix exponential operator across a 20-point sampling grid spanning relaxation delays from 0.1 s to 2.0 s in 0.1 s steps. Apparent rates were then extracted using a multi-point non-linear regression for traditional separate fitting, and through synchronized 10-point pair calculations for the DAS framework.

4.6. Data processing and analysis

For the conventional separate fitting, longitudinal relaxation rates (R₁) were determined by fitting peak intensities (I) as a function of the relaxation delay (t) using a standard non-linear least-square regression algorithm implemented in Origin (OriginLab, Northampton, MA; verified against fitXY.tcl script in NMRPipe package [40] to ensure numerical consistency).

The intrinsic longitudinal relaxation rate for the paramagnetic sample (R_{1, para}) or the diamagnetic reference sample (R_{1, dia}) was extracted for each residue by fitting the recovery curve to the mono-exponential expression before subtraction. For sPRE measurement, the raw multi-point longitudinal PRE rate (Γ₁) was obtained and normalized against the concentration of the paramagnetic agent (c_probe), calculated as:

$${\Gamma }_{1,\text{SEP}}=\frac{R_{1,\text{para}}-R_{1,\text{dia}}}{c_{\text{probe}}}$$ (14)

To evaluate the reliability and single-exponentiality of the individual relaxation trajectories, the coefficient of determination (R², COD) was extracted from the non-linear regression matrix. The R² value represents the fraction of the total variance in the peak intensity data that is explained by the exponential recovery model, calculated as:

$$R^2=1-\frac{\sum _{\mathrm{i}}{\left(I_{\mathrm{i}}-I_{\text{fit},\mathrm{i}}\right)}^2}{\sum _{\mathrm{i}}{\left(I_{\mathrm{i}}-\overline{I}\right)}^2}$$ (15)

where I_i is the experimental peak intensity at delay point i, I_fit,i is the corresponding predicted value from the regression model, and $\overline{I}$ is the mean experimental intensity across the delays. Decreases in this metric below 1 were used as diagnostic indicators for residues experiencing experimental artifacts or severe cross-relaxation perturbations.

A delay list containing 20 points (ranging from 0.1 s to 2.0 s in 0.1 s increments) was generated. These 20 points were used for conventional mono-exponential fitting in the SEP method, while the 10 corresponding normalized ratios derived from synchronized pairs were used for the DAS methodology. Fitting errors for both methods were calculated by comparing the extracted rates against the true Γ₁ values and were plotted as heat maps.

CRediT authorship contribution statement

Lingshen Meng: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Formal analysis, Data curation, Conceptualization. Xinyuan Lu: Writing – review & editing, Writing – original draft, Validation, Resources, Formal analysis, Data curation. Chun Tang: Writing – review & editing, Writing – original draft, Supervision, Project administration, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Chun Tang is an editorial board member of Magnetic Resonance Letters but was not involved in the editorial review or the decision to publish this article.

Biographies

Lingsheng Meng received his BS degree from Peking University in 2020 and continued his graduate carreer at Peking University under the mentorship of Prof. Chun Tang. His research focuses on novel NMR method development, with special emphasis on dynamics and paramagnetism, as well as integrative structural biology, with applications to disease-related viral proteins.

Xinyuan Lu received his BS degree from Peking University in 2021 and continued his graduate career under the mentorship of Prof. Chun Tang. His research focuses on the development of novel solvent-based paramagnetic probes and photo-crosslinking reagents. A central theme of his work is investigating the conformational ensembles of intrinsically disordered proteins and their modulation by small-molecules.

Chun Tang received his BS degree from Zhejiang University in 1998 and PhD from the University of Maryland under the mentorship of Michael F. Summers. Following postdoctoral training with G. Marius Clore at the National Institutes of Health, he embarked his independent career as an Assistant Professor at the University of Missouri in 2008. He subsequently joined the Chinese Academy of Sciences under the Hundred Talents Program. Currently, he is a Boya Distinguished Professor at the College of Chemistry and Molecular Engineering, Peking University. His research focuses on elucidating the structural dynamics of biological macromolecules. To this end, his group develops advanced NMR methodologies integrated with computational simulations and artificial intelligence to better characterize complex biomolecular processes.