Measurement of ¹H^α transverse relaxation rates in proteins: application to solvent PREs

Abstract

It has recently been demonstrated that accurate near surface electrostatic potentials can be calculated for proteins from solvent paramagnetic relaxation enhancements (PREs) of amide protons measured using spin labels of similar structures but different charges (Yu et al. in Proc Natl Acad Sci 118(25):e2104020118, 2021). Here we develop methodology for extending such measurements to intrinsically disordered proteins at neutral pH where amide spectra are of very poor quality. Under these conditions it is shown that accurate PRE values can be measured using the haCONHA experiment that has been modified for recording ¹H^α transverse relaxation rates. The optimal pulse scheme includes a spin-lock relaxation element for suppression of homonuclear scalar coupled evolution for all ¹H^α protons, except those derived from Ser and Thr residues, and minimizes the radiation damping field from water magnetization that would otherwise increase measured relaxation rates. The robustness of the experiment is verified by developing a second approach using a band selective adiabatic decoupling scheme for suppression of scalar coupling modulations during ¹H^α relaxation and showing that the measured PRE values from the two methods are in excellent agreement. The near surface electrostatic potential of a 103-residue construct comprising the C-terminal intrinsically disordered region of the RNA-binding protein CAPRIN1 is obtained at pH 5.5 using both ¹H^N and ¹H^α-based relaxation rates, and at pH 7.4 where only ¹H^α rates can be quantified, with very good agreement between potentials obtained under all experimental conditions.

Introduction

NMR spectroscopy is an extremely powerful technique for quantifying site-specific molecular dynamics (Mittermaier and Kay 2006; Palmer 2014; Anthis and Clore 2015). Most frequently this is accomplished through the measurement of heteronuclear (¹⁵N, ¹³C, ²H, ³¹P, ¹⁹F) spin relaxation rates that can then be recast in terms of motional parameters in the context of a preferred model of dynamics (Lipari and Szabo 1982a, b). In this regard, the use of heteronuclear spins as probes of motion, as opposed to measurements involving ¹H spins, offers several advantages. Importantly, it is often the case that the relaxation of heteronuclei can be quantitatively analyzed in terms of a small number of well-defined interactions, greatly simplifying data analysis. An example is the popular series of ¹⁵N R₁, R₂, and heteronuclear NOE experiments where a relatively simple two-spin ¹⁵N–¹H^N spin system is sufficient to describe the experiments (Kay et al. 1989). The situation is more complex for ¹³C, as coupled relaxation between spin interactions, such as ¹³C–¹H dipolar pairs in methylene and methyl groups (Vold and Vold 1976; Werbelow and Grant 1977), complicates the analysis (Kay and Torchia 1991), as does scalar coupling and relaxation between proximal ¹³C spins in uniformly ¹³C labeled samples (Yamazaki et al. 1994). The development of labeling schemes involving the placement of isolated ¹³C spins in the system of interest (Goto et al. 1999; Kainosho et al. 2006; Teilum et al. 2006; Lundström et al. 2007; Kasinath et al. 2013), in concert with the substitution of protons with deuterons (Ishima et al. 1999; Tugarinov and Kay 2005), can simplify the spin system so that it is well approximated as a two-spin ¹³C–¹H pair. In some cases experiments are relatively benign to the effects of scalar couplings that manifest in uniformly ¹³C labeled molecules, such as carbon-13 CEST (Vallurupalli et al. 2013), for example, while special pulse schemes have been developed for mitigating the effects of the homonuclear ¹³C couplings in ¹³C relaxation measurements (Yamazaki et al. 1994). Other applications exploit the inherent complexity that is introduced by the multiplicity of interactions within methylene and methyl groups to obtain additional insights into dynamics using experiments that rely on cross-correlated relaxation (Sun et al. 2011; Tugarinov and Clore 2021). Measuring ²H spin relaxation (Muhandiram et al. 1995) is advantageous in that the decay is dominated by the quadrupolar interaction, and the obtained rates can be cross-validated by recording as many as five independent decay times for each deuteron (Millet et al. 2002). A limitation is that the experiments are less sensitive than those quantifying ¹⁵N and, often, ¹³C relaxation, so that applications are largely focused on methyl groups (Kay et al. 1998).

Far fewer biomolecular applications involving ¹H relaxation have appeared in the literature, reflecting the fact that ¹H spins are most often both dipolar and scalar coupled to neighboring protons in fully protonated molecules, leading to the facile transfer of magnetization between protons and, therefore, contaminating measured relaxation rates. Labeling strategies involving partial deuteration are helpful in this regard, producing isolated ¹H spins at significant numbers of backbone and sidechain positions (Lundström et al. 2009; Hansen et al. 2012). However, it remains of interest to establish robust methods for measurement of relaxation rates in fully protonated proteins, in particular focusing on backbone ¹H^α spins, which is the subject matter of this report. Our interest in the measurement of ¹H^α relaxation rates concerns quantification of solvent paramagnetic relaxation enhancements (PREs) in an attempt to map near surface electrostatic potentials (Yu et al. 2021) in intrinsically disordered proteins (IDPs). As PRE effects scale with the square of the gyromagnetic ratios of the probe spins (Abragam 1961), there are clear advantages to proton-based experiments. The obvious choice is to record ¹⁵N–¹H^N HSQC spectra that measure ¹H^N rates, as these can be faithfully obtained via simple spin-echo schemes in which evolution from ¹H^N–¹H scalar couplings is refocused by the application of an ¹H^N-selective pulse in the center of a relaxation period (Donaldson et al. 2001). Alternatively, the effects of ¹H^N–¹H J-modulation can be “removed” during analysis of spectra recorded with non-selective ¹H^N chemical shift refocusing pulses by using identical relaxation times for both paramagnetic and diamagnetic samples and fitting paramagnetic relaxation rates directly from intensity ratios of corresponding peaks in the resulting pairs of spectra (Iwahara et al. 2004, 2007). Yet in some applications, especially those involving IDPs or intrinsically disordered regions (IDRs) in otherwise folded molecules that must be performed at neutral pH, amide spectra are severely compromised due to rapid hydrogen exchange. In these cases an approach that circumvents both the recording of ¹H^N chemical shifts and measurement of ¹H^N relaxation rates, that are likely to be contaminated by exchange with water, complicating extraction of robust exchange rates, would be preferred. Herein we develop a pseudo-4D experiment for measurement of ¹H^α relaxation rates based on the haCONHA pulse scheme that records (¹³CO_i,¹⁵N_i+1,¹H^α_i) correlations, where the chemical shifts of ¹³CO and ¹H^α spins of residue i are correlated with the ¹⁵N spin of the subsequent residue, i + 1 (Mäntylahti et al. 2011; Wong et al. 2020a). Important considerations for the design and optimization of the pulse scheme are described, along with applications to the C-terminal region of the RNA binding protein CAPRIN1 (Kedersha et al. 2016; Nakayama et al. 2017), so as to establish the robustness of the approach and its utility in studies of IDPs at neutral pH values and higher.

Material and methods

Sample preparation

The C-terminal region of CAPRIN1 (residues 607–709, Uniprot: Q14444) was expressed and purified as described previously (Kim et al. 2019; Wong et al. 2020a). As reported in our previous study (Wong et al. 2020a), residues N623-G624 and N630-G631 slowly form isoaspartate (IsoAsp)-Gly peptide linkages over time. As the formation of IsoAsp can alter the charge distribution of the CAPRIN1 molecule, N623T and N630T double mutations were introduced; the double Thr mutant was used in all of the experiments (and is referred to as CAPRIN1 in the discussion which follows). These mutations were introduced by using Quikchange site-directed mutagenesis (Agilent). Uniformly ¹³C, ¹⁵N-labeled CAPRIN1 was produced by bacterial growth, with expression using minimal media supplemented with [U-¹³C]-glucose and ¹⁵NH₄Cl as the sole carbon and nitrogen sources, respectively. The NMR samples were comprised of 280–300 μM U-¹³C,¹⁵N CAPRIN1, 25 mM MES-NaOH (pH 5.5) or 25 mM HEPES–NaOH (pH 7.4), and 3% D₂O. For solvent PRE measurements, 3-carboxy-PROXYL (Sigma-Aldrich) or 3-carbamoyl-PROXYL (Sigma-Aldrich) was added to a final concentration of 5 mM from a ~ 100 mM stock solution. The concentration of the paramagnetic co-solutes in the stock solution was measured by ¹H 1D NMR after reducing the spin-label, using a procedure established by Iwahara and co-workers (Yu et al. 2021).

NMR measurements

All NMR measurements were performed at 23.5 Tesla (1 GHz ¹H frequency) on a Bruker Avance Neo spectrometer or at 14.0 Tesla (600 MHz ¹H frequency) on a Bruker Avance III HD spectrometer, equipped with cryogenically cooled x, y, z pulsed-field gradient triple-resonance probes. All spectra were processed and analyzed using the NMRPipe suite of programs (Delaglio et al. 1995) and visualized using the Python package nmrglue (Helmus and Jaroniec 2013). Peak intensities were extracted either by using the Peakipy software package (https://github.com/j-brady/peakipy) for 2D datasets, or by analyzing the time-domains of pseudo-4D datasets (haCONHA of Fig. 1A.1 or A.3), as described previously (Long et al. 2015; Wong et al. 2020b). In time-doming fitting, the reference 3D spectrum recorded with the first relaxation delay was reconstructed using SMILE (Ying et al. 2017), and the peak list required for the time-domain fitting was obtained by analyzing the processed data.

Fig. 1

The haCONHA pulse sequence for measuring ¹H^α R₂ rates by observing (¹³CO_i,¹⁵N_i+1,¹H^α_i) correlations. Many of the details of the pulse scheme are as described previously (Wong et al. 2020a); however, for completeness they are repeated here. All 90° (180°) rectangular pulses are denoted by narrow (wide) bars and applied along the x-axis unless otherwise indicated. The ¹H carrier is on resonance with the water line (~ 4.7 ppm), the ¹³C carrier is at 176 ppm between points b and c and otherwise at 58 ppm, and the ¹⁵N carrier is at 119 ppm (but see below for specific details about each of the schemes in panel A). ¹H WALTZ-16 decoupling is applied with a field of ~ 6.25 kHz. ¹³C^α and ¹³CO 90° and 180° rectangular pulses are applied with fields of ΔΩ/√15 and ΔΩ/√3, respectively, where ΔΩ = 118 ppm, ensuring minimal excitation of ¹³CO spins when ¹³C^α pulses are applied and vice versa (Kay et al. 1990). ¹³C^α WALTZ-16 decoupling during t₃ acquisition uses a field of ~ 2 kHz (600 MHz spectrometer). Inset A shows three approaches for measuring ¹H^α R₂ rates, including (i) application of a ¹H spin lock (A.1), (ii) application of ¹H^β, ¹H^N adiabatic decoupling (A.2), and a spin-echo scheme (A.3). In scheme A.1 the ¹H and ¹³C carriers are placed at 4.35 ppm and 50 ppm, respectively, with the placement of the ¹³C carrier so as to reduce off-resonance effects for ¹³C^α of Gly. The ¹H spin-lock is achieved with a 1 kHz CW field along x, at the center of which a high power 180_y pulse is applied. Prior to the spin-lock, ¹H spins are aligned along their effective fields via a pulse/delay scheme, described previously (Hansen and Kay 2007), where χ and ζ are set to 1/ω_SL − (4/π)pw and (2/π)pw, respectively, where ω_SL is the RF field strength for the ¹H spin-lock and pw is the ¹H high power 90° pulse width. The relaxation delay, T_relax, was varied from 0 to 30 ms. In scheme A.2, selective ¹H decoupling is achieved using a constant adiabaticity WURST decoupling element (Kupce and Wagner 1996) swept from 1.1  to 3.3 ppm (8 ms WURST (Kupce and Freeman 1995) pulse width) centered at 2.2 pm, a typical ¹H^β shift value, along with a second field swept from 7.4 to 8.6 ppm centered on the amide ¹H^N protons. T_relax was varied from 0 to 40 ms in 8 ms spacing intervals so as to be synchronous with the ¹H decoupling sequence. The delays are: τ₁ = 1.7 ms, τ₃ = 4.5 ms, τ₄ = 15 ms, with τ_e sufficiently long to accommodate gradient g13. The delay τ₂ is set to 2.3 ms, a compromise so as to obtain cross peaks from all residues including Gly. ¹³C^β decoupling is achieved using a constant adiabaticity WURST decoupling element swept from 41 to 15 ppm (5 ms WURST pulse width), along with a second field swept from 68 to 72 ppm. ¹³CO chemical shift evolution during t₁ is acquired in a semi-constant time mode (Grzesiek et al. 1993; Logan et al. 1993) as depicted in B. The phase cycle used is: φ₁ = 2(x),2(− x); φ₂ = y + 48.5° (600 MHz); φ₃ = x, − x; and φ_rec = x,2(− x),x. The phase change applied to φ₂ corrects for the Bloch-Siegert shift caused by application of the uncompensated ¹³C^α pulse during the t₁ period. Quadrature detection in t₁ and t₂ is achieved by STATES-TPPI (Marion et al. 1989) of φ₁ and φ₃, respectively. Gradients are applied with the following durations (ms) and strengths (in % maximum): g1: (0.5, 24%), g2: (1.0, 24%), g3: (0.256, 15%), g4: (0.5, 52.8%), g5: (1.0, 40%), g6: (1.25, 80%), g7: (1.5, 80%), g8: (0.9, 50%), g9: (1.0, 15%), g10: (0.512, 90%), g11: (0.4, 40%), g12: (0.3, 15%), g13: (0.256, 90.3%)

¹H^α R₂ or ¹H^α–¹³C^α longitudinal order relaxation measurements were recorded with pulse schemes that are based on the haCONHA experiment (Wong et al. 2020a) (see Fig. 1), and were performed in a pseudo-4D manner where the indirect ¹³CO and ¹⁵N dimensions were non-uniformly sampled using a Poisson-gap sampling schedule (Hyberts et al. 2010) (¹H^α R₂, measured using the schemes of Fig. 1A.1 or A.3) or by measuring 2D ¹³CO–¹H^α or ¹⁵N–¹H^α planes (¹H^α R₂, measured by the adiabatic scheme of Fig. 1A.2; $2{I}_{z}^{\alpha }{C}_{z}^{\alpha }$ longitudinal order relaxation using the scheme shown schematically in Fig. 3B that replaces A in Fig. 1). Measurements were performed at 600 MHz and 25 °C with relaxation delays set to 0, 4, 8, 12, 16, 20, 25, and 30 ms for the scheme of Fig. 1A.1 and A.3, or 0–40 ms, in 8 ms steps, for the scheme of Fig. 1A.2. Longitudinal order decay rates were quantified with delays of 0, 4, 8, 12, 16, 20, 25, and 30 ms.

¹H^N R₂ relaxation measurements (pH 5.5 sample) were performed using a transverse relaxation-optimized spectroscopy (TROSY) scheme (Pervushin et al. 1997), with a ¹H spin-echo variable delay interval inserted immediately prior to direct detection. A selective REBURP pulse (Geen and Freeman 1991) (length of 1800 μs and centered at 7.7 ppm, 1 GHz) during the ¹H spin-echo period refocuses homonuclear J-evolution of ¹H^N spins. The measurements were performed at 1 GHz and 25 °C, with relaxation delays of 2, 4, 6, 8, 12, 16, 22, and 30 ms.

Fitting of ¹H^α and ¹H^N relaxation rates

¹H^α PREs were quantified by fitting intensity ratios of corresponding peaks in the “paramagnetic” (with 5 mM 3-carboxy-PROXYL or 5 mM 3-carbamoyl-PROXYL, denoted by − or N, respectively) and “diamagnetic” (no PRE co-solute molecules) experiments to the single exponential decay function,

$$\frac{{I}^{para.,i}({T}_{relax})}{{I}^{dia.}({T}_{relax})}=exp(-{\Gamma }_{2,i}{T}_{relax})$$

(1)

where ${I}^{para.,i}({T}_{relax})$ and ${I}^{dia.}\left({T}_{relax}\right)$ are signal intensities at time T_relax for peaks in the paramagnetic and diamagnetic samples, and Γ_2,i is the PRE contribution to the ¹H^α R₂ rate (i $\in$ {− , N}). As described by Iwahara, Clore and co-workers (Iwahara et al. 2004, 2007; Yu et al. 2022), by taking the ratio of intensities it is possible to divide out contributions from ¹H–¹H J-modulations that would otherwise contaminate the relaxation rates. Nevertheless, scalar-coupled evolution does attenuate the signals so that it is highly desirable to suppress the modulations in the first place, and this is possible for ¹H^α spins from all residues with the exception of Ser and Thr, as described in detail below. We have also observed that for a number of non-Ser/Thr residues there is a slight deviation from single exponential decay, presumably because of modulation from couplings that are not completely suppressed by the 1 kHz spin-lock field of Fig. 1A.1 that selectively locks ¹H^α magnetization. Use of Eq. (1) is beneficial for these cases as well.

¹H^N PREs are quantified by fitting the decay of signals to a single exponential function to obtain ${R}_{2}^{para.,i}$ and ${R}_{2}^{dia.}$ rates, from which the PRE contribution is calculated as ${\Gamma }_{2,i}={R}_{2}^{para.,i}-{R}_{2}^{dia.}$. Fits made use of in-house written programs (Python 3.7), exploiting the Levenberg–Marquardt algorithm of the Lmfit python software package (https://lmfit.github.io/lmfit-py/).

Calculations of near-surface electrostatic potentials

Near-surface electrostatic potentials were calculated from the PRE rates obtained with 3-carboxy-PROXYL (${\Gamma }_{2,-}$) and 3-carbamoyl-PROXYL (${\Gamma }_{2,N}$) derivatives using the following equation, as described previously (Yu et al. 2021),

$${\phi }_{ENS}=-\frac{{k}_{B}T}{e}\mathrm{ln}\left(\frac{{\Gamma }_{2,N}}{{\Gamma }_{2,-}}\right)$$

(2)

where k_B is Boltzmann’s constant (8.62 × 10^–5 eV/K), T is temperature (298.15 K), and e is the charge of an electron. Note that the denominator was set to 1e as the difference in charge between 3-carboxy-PROXYL and 3-carbamoyl-PROXYL is 1. In the calculations, residues with ${\Gamma }_{2,-}$ or ${\Gamma }_{2,N}$ larger than 0.5 s⁻¹ were used.

Simulations of ¹H^α scalar coupled evolution with and without a spin-locking field

The scalar-coupled evolution of ¹H^α magnetization was simulated by calculating the time-evolution of the density matrix (Sørensen et al. 1984). The operative Hamiltonian (${\widehat{\mathcal{H}}}_{0}$) is composed of chemical shift (${\widehat{\mathcal{H}}}_{CS}$), scalar coupling (${\widehat{\mathcal{H}}}_{J}$), and spin-lock field (${\widehat{\mathcal{H}}}_{SL}$) terms, as follows,

$${\widehat{\mathcal{H}}}_{0}={\widehat{\mathcal{H}}}_{CS}+{\widehat{\mathcal{H}}}_{J}+{\widehat{\mathcal{H}}}_{SL}=\sum_{i}{{\Omega} }_{i}{I}_{z}^{i}+\sum_{i\ne j}2\pi {J}_{ij}{{\varvec{I}}}^{i}\cdot {{\varvec{I}}}^{j}+\sum_{i}{\omega }_{1}{I}_{x}^{i}$$

(3)

where Ω_i is the offset frequency (rad/sec) of proton spin i from the radio frequency carrier, I_x and I_z denote the x and z components of ¹H spin angular momentum, respectively, J_ij is the homonuclear scalar coupling constant between spins i j, ω₁ is the ¹H spin-lock field strength in rad/sec (1000 × 2π rad/sec was used in experiments), and it is understood that only one of the terms of the form 2πJ_αβ1I^α·I^β1 or 2πJ_β1αI^β1·I^α is included, for example. The number of spins, and the chemical shifts and J coupling constants used in each simulation are indicated in the schematics of Fig. 2. In the calculations, only 2-bond or 3-bond homonuclear J couplings were considered and heteronuclear couplings were not included. In the case of a “typical amino-acid”, such as shown in Fig. 2A (top left), for example, a set of 4 spins i,j $\in$ {α, β¹, β², H^N} was considered. Relaxation was not included in the simulations. The time evolution of the density matrix, evolving with the scheme of Fig. 2A (top right), was calculated by numerically solving the Liouville von-Neumann equation,

Fig. 2

Simulating the evolution of ¹H^α magnetization due to ¹H^α-¹H scalar couplings. A Evolution of ¹H^α magnetization in a spin system typical for amino acids, where ¹H^N and two ¹H^β protons are scalar-coupled to ¹H^α. (top left) The chemical shifts of each spin and the J-coupling constants used are shown. (top right) Pulse scheme elements, similar to those used in experiments of Fig. 1A.1 (orange; ¹H^α magnetization, $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$, is locked along its effective field at the start of the spin-lock, as is done experimentally) and in 1A.3 (navy, starting from ${I}_{x}^{\alpha }$). (bottom) Plots of the trajectories of ¹H^α x-magnetization calculated with different ¹H^β chemical shifts (left: 2 ppm, center: 3 ppm, and right: 3.5 ppm), in the presence (orange) and absence (navy) of a 1 kHz spin-lock field centered at 4.35 ppm. B Evolution of ¹H^α magnetization in methionine (left), serine (center), and threonine (right) spin systems. (top) The chemical shifts of each spin and the J-coupling constants are shown. (bottom) Trajectories of calculated ¹H^α x-magnetization with (orange) and without (navy) a 1 kHz spin-lock field centered at 4.35 ppm. The chemical shifts of each ¹H spin were taken from a tabulation of random coil values (Wishart et al. 1995) and the J-coupling constants were set to those measured in unfolded proteins (Hähnke et al. 2010) or those typically observed in folded proteins

$$\begin{array}{cc}& \sigma \left({T}_{relax}\right)=U\sigma \left(0\right){U}^{-1}\\ & U=\mathrm{exp}\left(-i{\widehat{\mathcal{H}}}_{0}\frac{{T}_{relax}}{2}\right)\mathrm{exp}(-i\pi \!\sum_{i}{I}_{y}^{i})\mathrm{exp}\left(-i{\widehat{\mathcal{H}}}_{0}\frac{{T}_{relax}}{2}\right)\\ & {U}^{-1}=\mathrm{exp}\left(i{\widehat{\mathcal{H}}}_{0}\frac{{T}_{relax}}{2}\right)\mathrm{exp}(i\pi\!\sum_{i}{I}_{y}^{i})\mathrm{exp}\left(i{\widehat{\mathcal{H}}}_{0}\frac{{T}_{relax}}{2}\right)\end{array}$$

(4)

with σ(0) = ${I}_{x}^{\alpha }$ (in-phase ¹H^α magnetization). Terms from other spins were set to zero initially. Note that, experimentally, the proton magnetization of interest during the relaxation period is antiphase with respect to the attached ¹³C spin ($2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$, see Fig. 1) and other transverse magnetization components coupled to ¹³C^α (such as $2{I}_{x}^{\beta }{C}_{z}^{\alpha }$) are not present at the beginning of this period. Thus, when considering a homonuclear spin system exclusively, the analogous situation is one where initial magnetization components with the exception of ${I}_{x}^{\alpha }$ are set to 0. The expectation value of the x-transverse magnetization at time T_relax (M(T_relax)) was calculated by taking the trace of the product of σ(T_relax) and ${I}_{x}^{\alpha }$. M(T_relax) profiles for 0 ≤ T_relax ≤ 30 ms, with a time step of 1 ms, were calculated from

$${M\left( {T_{{relax}} } \right) = \frac{{{\text{tr}}\left\{ {\sigma \left( {T_{{relax}} } \right) \cdot I_{x}^{\alpha } } \right\}}}{{{\text{tr}}\left\{ {\left( {I_{x}^{\alpha } } \right)^{\dag } \cdot I_{x}^{\alpha } } \right\}}}}$$

(5)

and plotted in Fig. 2A, B.

Simulating the effects of cross-relaxation

As described in the text, a spin-lock field has been used to minimize evolution of magnetization due to homonuclear scalar couplings. We wondered whether dipolar cross-relaxation between neighboring spins would become an issue (ROE effect) under these conditions, leading to non-exponential decay of ¹H^α magnetization and to mixing of PREs from proximal ¹H spins. We have, therefore, simulated an I-C-M three spin system, where I, C, and M are ¹H^α, ¹³C^α, and ¹H^β spins, respectively, considering relaxation and scalar coupled evolution. In principle, a complete description of this spin system requires a basis of 64 elements. However, as C_z is the only ¹³C operator considered (¹³C pulses are not applied), a reduced basis set suffices, composed of 30 elements (excluding the identity operator). The normalized basis set can be expressed by using a column vector with 30 Cartesian product operators (Ernst et al. 1987; Allard et al. 1998),

$$\sigma = [I_{x} ,I_{y} ,I_{z} ,M_{x} ,M_{y} ,M_{z} ,2I_{x} M_{z} ,2I_{y} M_{z} ,2I_{z} M_{x} , 2I_{z} M_{y} ,2I_{x} M_{x} ,2I_{x} M_{y} ,2I_{y} M_{x} ,2I_{y} M_{y} ,2I_{z} M_{z} , 2I_{x} C_{z} ,2I_{y} C_{z} ,2I_{z} C_{z} , \cdots ,4I_{x} M_{z} C_{z} , \cdots ,4I_{z} M_{z} C_{z} ] ^{+}$$

(6)

where ⁺ denotes the transpose operation.

The operative Hamiltonian in this case is,

$$\widehat{\mathcal{H}}={{\Omega} }_{I}{I}_{z}+{{\Omega} }_{M}{M}_{z}+2\uppi {J}_{IM}{\varvec{I}}\cdot {\varvec{M}}+2\pi {J}_{IC}{I}_{z}{C}_{z}+{\omega }_{1}({I}_{x}+{M}_{x})$$

(7)

where Ω_I and Ω_M are the offsets of ¹H spins I and M from the ¹H carrier, J_IM and J_IC are I-M and I-C homo- and hetero-nuclear scalar coupling constants, respectively, and ω₁ (1000 × 2π rad/sec) is the strength of an applied field on spins I and M. In all simulations, J_IM = 7 Hz, J_IC = 140 Hz and a static magnetic field of 14.0 Tesla (600 MHz ¹H resonance frequency) was used.

The evolution of the density matrix, σ, during the scheme of Fig. 2A (top right) was calculated by numerically solving

$$\sigma \left({T}_{relax}\right)=\mathrm{exp}\{-\left(i\widehat{\widehat{\mathcal{L}}}+\widehat{\widehat{\mathcal{R}}}\right)\frac{{T}_{relax}}{2}\}{\widehat{\widehat{U}}}^{y,\pi }\mathrm{exp}\{-\left(i\widehat{\widehat{\mathcal{L}}}+\widehat{\widehat{\mathcal{R}}}\right)\frac{{T}_{relax}}{2}\}\sigma \left(0\right)$$

(8)

In Eq. (8) $\widehat{\widehat{\mathcal{L}}}$ is a 30 × 30 Liouvillian matrix that includes J and chemical shift evolution, $\widehat{\widehat{\mathcal{R}}}$ is a relaxation matrix, including auto- and cross-relaxation terms, and ${\widehat{\widehat{U}}}^{y,\pi }$ is a rotation matrix that “applies” a ¹H 180° pulse along y in the center of the spin lock element. Each component of $\widehat{\widehat{\mathcal{L}}}$ and ${\widehat{\widehat{U}}}^{y,\pi }$ was calculated as

$${\widehat{\widehat{\mathcal{L}}}}_{rs}=\frac{\langle r|[\widehat{\mathcal{H}},s]\rangle }{\langle r|r\rangle }$$

(9)

$${\widehat{\widehat{U}}}_{rs}^{y,\pi }=\frac{\langle r|[\exp({-i\pi({I}_{y}+{M}_{y})})s\exp({i\pi({I}_{y}+{M}_{y})})]\rangle }{\langle r|r\rangle }$$

(10)

where |r > and |s > are density elements listed in Eq. (6).

The relaxation matrix, $\widehat{\widehat{\mathcal{R}}}$, includes auto-relaxation terms for each operator and cross-relaxation terms coupling I_x ↔ M_x, I_y ↔ M_y, I_z ↔ M_z, 2I_xM_z ↔ 2I_zM_x, 2I_yM_z ↔ 2I_zM_y, 2I_xM_y ↔ 2I_yM_x, 2I_xC_z ↔ 2M_xC_z, 2I_yC_z ↔ 2M_yC_z, 2I_zC_z ↔ 2M_zC_z, 4I_xM_zC_z ↔ 4I_zM_xC_z, 4I_yM_zC_z ↔ 4I_zM_yC_z, and 4I_xM_yC_z ↔ 4I_yM_xC_z. Auto-relaxation was calculated by including ¹H-¹H dipolar interactions between each of spins I and M and a pair of external proton spins (one unique external ¹H for each of I and M; r_HH,ext = 2.1 Å for both I and M spins), ¹H-¹H dipolar interactions between spins I and M (r_HH,IM = 1.9 Å), and ¹H-¹³C dipolar interactions between spins I and M and their directly bonded ¹³C nuclei (r_CH = 1.1 Å). Only terms proportional to the spectral density evaluated at zero frequency are included in our analysis. Although a separate carbon spin one-bond coupled to spin M is not explicitly included in the spin system under consideration (Eq. (6)), we have, nevertheless, included a C-M dipolar interaction that would normally be present in the U-¹³C,¹⁵N proteins that are studied experimentally. The relaxation matrix can, thus, be defined as follows (Allard et al. 1997),

(11)

The auto-relaxation rates (R₁₁–R₁₅₁₅), anti-phase and multiple-quantum cross-relaxation terms (R₇₉, R₉₇, R₈₁₀, R₁₀₈, R₁₂₁₃ and R₁₃₁₂) are given by Desvaux et al. (1994) and Allard et al. (1997)

$$\begin{array}{cc}& \\ & \\ & {R}_{11}={R}_{22}={R}_{44}={R}_{55}=\frac{{d}_{HH}^{ext}}{4}+\frac{{d}_{HH}^{IM}}{4}+\frac{{d}_{HC}}{5}\\ & {R}_{33}={R}_{66}=\frac{{d}_{HH}^{ext}}{10}+\frac{{d}_{HH}^{IM}}{10}\\ & {R}_{77}={R}_{88}={R}_{99}={R}_{1010}=\frac{7{d}_{HH}^{ext}}{20}+\frac{{d}_{HH}^{IM}}{4}+\frac{{d}_{HC}}{5}\\ & {R}_{79}={R}_{97}={R}_{810}={R}_{108}=\frac{{d}_{HH}^{IM}}{5}\\ & {R}_{1111}={R}_{1414}=\frac{{d}_{HH}^{ext}}{2}+\frac{2{d}_{HC}}{5}\\ & {R}_{1212}={R}_{1313}=\frac{{d}_{HH}^{ext}}{2}+\frac{{d}_{HH}^{IM}}{10}+\frac{2{d}_{HC}}{5}\\ & {R}_{1213}={R}_{1312}=-\frac{{d}_{HH}^{IM}}{10}\\ & {R}_{1515}=\frac{{d}_{HH}^{ext}}{5}\\ & {d}_{HH}^{ext}={\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}\frac{{\mathrm{\hslash }}^{2}{\gamma }_{H}^{4}{\tau }_{c}}{{r}_{HH,ext}^{6}}\\ & {d}_{HH}^{IM}={\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}\frac{{\mathrm{\hslash }}^{2}{\gamma }_{H}^{4}{\tau }_{c}}{{r}_{HH,IM}^{6}}\\ & {d}_{HC}={\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}\frac{{\mathrm{\hslash }}^{2}{\gamma }_{H}^{2}{\gamma }_{C}^{2}{\tau }_{c}}{{r}_{HC}^{6}}\end{array}$$

(12)

and the transverse and longitudinal cross-relaxation rates, σ_ROE and σ_NOE, are defined as

$${\sigma }_{ROE}=\frac{1}{5}{\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}\frac{{\hslash}^{2}{\gamma }_{H}^{4}{\tau }_{c}}{{r}_{HH,IM}^{6}}{\sigma }_{NOE}=-\frac{1}{10}{\left(\frac{{\mu }_{0}}{4\pi }\right)}^{2}\frac{{\hslash}^{2}{\gamma }_{H}^{4}{\tau }_{c}}{{r}_{HH,IM}^{6}}$$

(13)

where μ₀ denotes the vacuum permeability, γ_H and γ_C are the gyromagnetic ratios of ¹H and ¹³C spins, respectively, ℏ is Planck’s constant devided by 2π, and τ_C is the correlation time (Abragam 1961; Cavanagh et al. 2007). Relaxation of longitudinal magnetization to its thermal equilibrium is not included in the calculation, as the evolution of anti-phase magnetization is considered (see below).

In our experimental scheme (Fig. 1A.1) antiphase ¹H^α magnetization ($2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$) is spin-locked along its effective field using a previously described alignment element (Hansen and Kay 2007). Thus, in our simulations the initial value of the density matrix is given by

$$\sigma \left(0\right)=\mathrm{sin}{\theta }_{I}2{I}_{x}{C}_{z}+\mathrm{cos}{\theta }_{I}2{I}_{z}{C}_{z}=2{I}_{z}^{^{\prime}}{C}_{z}$$

(14)

where ${I}_{z}^{^{\prime}}$ is the aligned magnetization in the tilted frame and θ_I is the angle between the z-axis of the tilted frame and the axis parallel to the static magnetic field (tanθ = ω₁/Ω_I). The expectation value of the spin-locked magnetization at time T_relax (M(T_relax)) was obtained by solving Eq. (8) and then extracting the $2{I}_{z}^{^{\prime}}{C}_{z}$ element as,

$$M\left({T}_{relax}\right)=\frac{\langle 2{I}_{z}^{^{\prime}}{C}_{z}|\sigma ({T}_{relax})\rangle }{\langle 2{I}_{z}^{^{\prime}}{C}_{z}|2{I}_{z}^{^{\prime}}{C}_{z}\rangle }$$

(15)

In our simulations a value of τ_C = 2 × 10^–9 s was assumed, consistent with previous relaxation measurements (Kim et al. 2021), so that the corresponding cross-relaxation rates are σ_ROE = 4.84 s⁻¹ and σ_NOE =  − 2.42 s⁻¹. In all simulations, the chemical shift of spin I and the carrier position were fixed to 4.4 and 4.35 ppm, respectively (Ω_I = (4.4 − 4.35) × 600 × 2π rad/sec), and the chemical shift of spin M was varied from 2 to 4 ppm (Ω_M = (2 to 4 − 4.35) × 600 × 2π rad/sec). The trajectory of $2{I}_{z}^{^{\prime}}{C}_{z}$ was calculated from 0 to 120 ms (fourfold longer than experimental relaxation times, Fig. 3A) with a time step of 1 ms. Similar simulations were performed in the absence of cross-relaxation by setting σ_ROE and σ_NOE to 0 in the relaxation matrix of Eq. (11).

Fig. 3

Cross-relaxation has a minimal effect on the evolution of $2{I{^{\prime}}}_{z}^{\alpha }{C}_{z}^{\alpha }$ magnetization. A (top) Schematic of the 3-spin {I, C, M}-spin system used in the simulations; a pair of external protons are included, in addition, contributing only to the auto-relaxation of proton spins I and M. (bottom) The evolution of anti-phase ¹H^α magnetization ($2{I{^{\prime}}}_{z}^{\alpha }{C}_{z}^{\alpha }$) spin-locked along its effective field (Eq. (14)) in the presence (navy, line) and absence (pink, dotted line) of cross-relaxation. The chemical shift of spin M (¹H^β mimic) was set to 2 (left), 3 (center), and 4 (right) ppm, respectively. A 1 kHz spin-lock field is applied, centered at 4.35 ppm. Note that the distinctly non-exponential profile for the case where the ¹H^β resonance frequency is 4 ppm is due to homonuclear J-evolution that is not suppressed using the ¹H spin-lock field. B Relaxation of longitudinal order ($2{I}_{z}^{\alpha }{C}_{z}^{\alpha }$) was quantified using the scheme (left) that replaces A in Fig. 1. The delay τ₁ and gradient strengths of g3 and g5 are indicated in the Fig. 1 legend. Water magnetization is either initially in the transverse plane (Scheme 1) or dephased at the start of the relaxation interval (Scheme 2) during T_relax. The relaxation rates of $2{I}_{z}^{\alpha }{C}_{z}^{\alpha }$ two-spin order (CAPRIN1, 25 °C, 600 MHz) are shown as a function of ¹H^α chemical shift (right)

Results and discussion

Description of pulse scheme for the measurement of ¹H^α transverse relaxation rates in IDPs

The original experiments to quantify near surface electrostatic potentials of proteins, developed by Iwahara and co-workers (Yu et al. 2021), based in part on work from the Clore group (Okuno et al. 2020), focused on the measurement of amide ¹H^N transverse relaxation rates in the presence and absence of variously charged spin labels. The measurement of ¹H^N as opposed to other proton relaxation rates has the obvious advantage in that there is only a single homonuclear scalar coupling to consider, involving ¹H^α spins, and evolution from the ³J_HN-Hα coupling can be refocused by the application of a ¹H^N-selective 180° pulse in the center of the ¹H evolution period that is required to quantify transverse relaxation (Donaldson et al. 2001). Unfortunately, however, studies of IDPs at physiological pH values cannot be performed using amide correlation spectra as the rapid exchange of amide protons with water deteriorates the quality of the resulting spectra. Moreover, in such cases the relaxation of ¹H^N spins is contaminated by exchange with water, with effective rates that are often non-exponential. These rates, further, can vary significantly depending on where in the pulse scheme they are interrogated (Ishima et al. 1998; Yuwen et al. 2016). By contrast, relaxation rates of ¹H^α protons are not sensitive to pH (exchange with water) and ¹H^α-detect experiments remain of high quality even when amide protons exchange rapidly with solvent. As the resolution of ¹H^α-¹³C^α correlations in 2D heteronuclear spectra of IDPs is poor, we prefer to measure relaxation data using 3D haCONHA-type experiments in which correlations of the form (ω_CO, ω_N, ω_Hα) are recorded, exploiting the resolution in ¹³CO and ¹⁵N dimensions (Mäntylahti et al. 2011; Wong et al. 2020a). A pseudo-4D dataset is, thus, obtained, in which each 3D spectrum corresponds to a single time point that is used to quantify transverse relaxation.

Figure 1 highlights the pulse scheme that we prefer (Scheme A.1), along with a sequence that is used to cross-validate the results (Scheme A.2). For completeness, we also show a simple spin-echo variant, similar to a recently described experiment by Yu et al. (2022), to illustrate some of the challenges with recording ¹H^α relaxation rates that must be overcome in the design of a robust pulse scheme. The initial element of the pulse sequence (A, in Fig. 1) is of interest to the relaxation experiment described here and in what follows we provide a brief overview of the magnetization pathway during this interval. Focusing on A.1, after the creation of longitudinal order ($2{I}_{z}^{\alpha }{C}_{z}^{\alpha }$) by the first INEPT element, where I^j and C^j are the proton and carbon spins that are one-bond coupled and j = α denotes either the ¹H^α or ¹³C^α spin, the water magnetization is dephased by application of a pulsed field gradient to minimize the radiation damping field (gradient g4); failure to do so can lead to apparent ¹H^α PREs that are significantly elevated, by 1.5 to 2-fold, in applications involving CAPRIN1 (see below). Subsequently, the ¹H^α spins are locked along their respective effective fields in a manner that is efficiently achieved for the narrow ¹H^α chemical shift range for IDPs (~ 4–4.8 ppm for CAPRIN1) using a 1 kHz ¹H continuous-wave (cw) field applied in the center of the ¹H^α spectrum (along the x-axis; a ¹H 180_y pulse is included in the center of the cw element), and the magnetization subsequently restored to the z-axis prior to magnetization transfers to ¹³CO(t₁) and ¹⁵N(t₂) that are identical to those in a regular haCONHA experiment (Wong et al. 2020a).

Although the haCONHA approach circumvents issues with hydrogen exchange, other problems are introduced when using aliphatic proton spins, such as ¹H^α, and Scheme A.1 of Fig. 1 is our best attempt to minimize these. For example, ¹H^α protons are three-bond scalar coupled to ¹H^N and ¹H^β spins and evolution of ¹H^α transverse magnetization from ³J_Hα-Hβ couplings is not as readily refocused as for ³J_HN-Hα couplings in the context of ¹H^N-based measurements, for example. In a simple spin-echo scheme that might be considered for measurement of transverse relaxation rates (Fig. 1, Scheme A.3, starting from ${I}_{x/y}^{\alpha }$) this evolution would proceed for the complete T_relax period, modulating the signal, and complicating extraction of accurate transverse relaxation rates. To illustrate this, as well as our solution to the problem (Scheme A.1, starting from $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$), in more detail, we consider the “generic” spin system shown in Fig. 2A, top left, and simulate the evolution of transverse ¹H^α x-magnetization during the spin-echo element shown in Fig. 2A, top right. We consider the case where ¹H^α spins are locked along their respective effective fields or when the spin lock field is removed. In the simulations shown in Fig. 2A (bottom) ω_Hα = 4.4 ppm and ω_Hβ is varied from 2 to 3.5 ppm with the orange (navy) profiles obtained with (without) the 1 kHz spin-lock field centered at 4.35 ppm. Notably, for ω_Hβ = 2 ppm (bottom left) or 3.0 ppm (bottom center) the orange profiles are flat, as if the scalar couplings involving ¹H^α were “turned off” (³J_Hα,NH = 7, ³J_Hα,Hβ1 = 5.5, and ³J_Hα,Hβ2 = 7.5 Hz are used in the simulation). In contrast, if the spin-lock field is omitted the navy curves result, clearly showing modulation from ¹H^α–¹H^β and ¹H^α–¹H^HN scalar couplings. When the chemical shifts of the ¹H^β protons are increased to 3.5 ppm (bottom right), closer to the position of the spin lock, a slight amount of modulation (approximately 1–2%—between 1 and 0.98) is obtained for the spin lock case.

It is noteworthy that in IDPs all ¹H^β protons resonate upfield of 3.5 ppm with the exception of those from Ser and Thr (Wishart et al. 1995), so that flat profiles (i.e., unmodulated by homonuclear scalar couplings) would be expected for all non-Ser/Thr ¹H^α protons using a spin lock scheme to measure relaxation. As might be expected, the presence of more spins on the side-chain does not affect the trajectory of x-magnetization, as shown in the simulation for a Met spin system containing 6 spins, where random coil chemical shifts are taken from Wishart et al. (1995) (Fig. 2B, left). The proximity of ¹H^α and ¹H^β chemical shifts in Ser (ω_Hα = 4.47 ppm, ω_Hβ = {3.87, 3.89} ppm) leads to significant modulation that cannot be suppressed by the ¹H spin-lock field (Fig. 2B, center), with a similar situation occurring for Thr (Fig. 2B, right). For both of these residues homonuclear J-modulation simply decreases magnetization intensity; there is no net transfer of observable magnetization between spins, as there would be in a Hartmann-Hahn scheme where in-phase ¹H^α/β… magnetization is initially created. This is because the initial magnetization for each proton is anti-phase with respect to its attached carbon and ¹³C–¹H scalar coupled evolution is largely suppressed by the ¹H cw field. Thus, while the transfer, $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }\stackrel{{J}_{H\alpha H\beta }}{\to }2{I}_{x}^{\beta }{C}_{z}^{\alpha }$, does occur, the transfer, $2{I}_{x}^{\beta }{C}_{z}^{\beta }\stackrel{{J}_{H\alpha H\beta }}{\to }2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$, does not, with only antiphase magnetization of the form $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$ ultimately detected. This ensures that the measured ¹H^α relaxation rates are not ‘contaminated’ by contributions from relaxation of other scalar coupled protons in the case of Ser and Thr. For ¹H^α spins from these residues, however, the scalar coupled evolution, $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }\stackrel{{J}_{H\alpha H\beta }}{\to }2{I}_{x}^{\beta }{C}_{z}^{\alpha }$, prohibits extraction of accurate relaxation rates from exponential fits of the “decay” curves. In contrast, as J-coupled modulation of non-Ser/Thr ¹H^α protons does not occur, there is no “leakage” of magnetization from $2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$ to $2{I}_{x}^{\beta }{C}_{z}^{\alpha }$ in these cases, and exponential decays are expected. Finally, for Gly, the ¹H^α spins become very strongly coupled in the presence of the cw field (i.e., essentially equivalent), and the sum of ¹H^α magnetization does not evolve under scalar coupling in this limiting case since $\left[{I}_{x}^{\alpha 1}+{I}_{x}^{\alpha 2},{{\varvec{I}}}^{\alpha 1}\cdot {{\varvec{I}}}^{\alpha 2}\right]=0$, where [] denotes the commutator operation. Thus, flat profiles are observed for the ¹H^α spins of Gly in simulations even when scalar coupled evolution is considered. Of course, even in the case where the two ¹H^α Gly spins are resolved (typically with chemical shift differences of 0.05–0.1 ppm), the observed relaxation rates would represent the average of the values from the two H^α positions, which are expected to be very similar, as magnetization is very efficiently transferred between the ¹H^α spins during the application of the spin-lock. It is worth noting that complications from J-modulation can be prevalent in non-¹H homonuclear spin systems and similar strategies involving band-selective locking schemes have been used previously to measure ¹³C^α relaxation rates in uniformly ¹³C-labeled proteins (Yamazaki et al. 1994).

Figure 2 illustrates the importance of the ¹H cw spin-lock field in the suppression of homonuclear J-modulation of magnetization for the majority of the spin systems in IDPs. Spin-locking of magnetization can, however, potentially lead to non-exponential relaxation from magnetization transfer mediated by dipolar cross-relaxation (spin diffusion). As described above, in the context of magnetization transfer through scalar couplings, the dipolar transfer can also be minimized by recording relaxation rates of ¹H^α magnetization that is anti-phase with respect to the one-bond coupled ¹³C (Sekhar et al. 2016). Thus, in the case of a pair of dipolar coupled ¹H spins, I and M, relaxation proceeds as

$$\frac{d2{I}_{z}^{^{\prime}}{C}_{z}^{I}}{dt}=-{\rho }_{{I}^{^{\prime}}}2{I}_{z}^{^{\prime}}{C}_{z}^{I}-{\sigma }_{{I}^{^{\prime}}{M}^{^{\prime}}}2{M}_{z}^{^{\prime}}{C}_{z}^{I}$$

(16)

where ρ_I' and σ_I'M' are auto- and cross-relaxation rates of the aligned magnetization in the tilted spin-lock frame that is germane for spin-locked magnetization considered in our experiments (the primes in ${I}_{z}^{^{\prime}}$ and ${M}_{z}^{^{\prime}}$ denote ‘tilted’ magnetization). As hetero-spins M and C^I are not scalar coupled (C^I is one bond coupled to proton spin I) there is no longitudinal order of the form $2{M}_{z}^{^{\prime}}{C}_{z}^{I}$ initially so that the decay of the magnetization of interest is essentially single exponential, with magnetization transfer between $2{M}_{z}^{^{\prime}}{C}_{z}^{I}$ and $2{I}_{z}^{^{\prime}}{C}_{z}^{I}$ effectively suppressed over the relatively short range of relaxation times considered (T_{relax_max} = 30 ms in our experiments). This is in contrast to what would be expected if the relaxation of in-phase ¹H magnetization was quantified. Figure 3A illustrates the evolution of anti-phase I spin magnetization during the relaxation element for a 3-spin {I, C, M}-spin system, with the chemical shift of the M spin varied between 2 and 4 ppm; cross-relaxation introduces a negligible effect, using an effective I–M distance of 1.9 Å for relaxation times extending to 120 ms and for a rotational correlation time of 2 ns, appropriate for the experimental system considered here (Kim et al. 2021). Note that the evolution of magnetization is decidedly non-exponential when ω_Hβ = 4 ppm, due to I–M scalar coupled evolution.

The importance of “water” management, even in ¹H^α-based experiments is illustrated in Fig. 3B. Here, by means of example, we consider the relaxation of longitudinal order, $2{I}_{z}^{\alpha }{C}_{z}^{\alpha }$, during an interval, T_relax, where a gradient that dephases water is applied either at the start or the end of the relaxation period. If the water transverse magnetization is not dephased initially, its precession induces an oscillating current in the receiver coil, and hence, a magnetic field, oscillating at the frequency of precession. This induced field rotates water magnetization and other spins resonating close to the water line back to their equilibrium (Krishnan and Murali 2013). Thus, the effect of the induced magnetic field would be expected to be more pronounced for ¹H^α spins whose chemical shifts are closer to the water line. With this in mind, the measured longitudinal order relaxation rate for each ¹H^α–¹³C^α spin pair in CAPRIN1 (described below) is plotted as a function of ¹H^α chemical shift, showing a clear elevation in rates for spins resonating near the water line when water is not dephased. As the water is initially in the transverse plane in this experiment, as it would be at point a in Schemes A.1–A.3 of Fig. 1 the radiation-damping field from bulk water can be considerable, unless water magnetization is initially dephased (Saturation of the water line significantly attenuates sensitivity of the experiment and is not a good option). Similar experiments, starting from anti-phase ¹H^α magnetization ($2{I}_{x}^{\alpha }{C}_{z}^{\alpha }$), show a 1.5- to 2-fold increase in measured ¹H^α PRE rates, in the absence of dephasing. Water magnetization is therefore dephased in schemes A.1 and A.2 immediately after the initial INEPT transfer and prior to the T_relax period.

Experimental validation

The RNA binding protein CAPRIN1 has been shown to play an essential role in the formation of neuronal and stress granules in cells (Kedersha et al. 2016; Nakayama et al. 2017), and the C-terminal low complexity disordered region comprising residues 607–709, and referred to in what follows as CAPRIN1, phase separates in vitro (Kim et al. 2019). Because of the small size of CAPRIN1 it has been used as a model system in our laboratory, both for the development of NMR methodology for characterizing IDPs in condensates, and, importantly, to understand the interactions that give rise to phase separation in the first place (Kim et al. 2021). CAPRIN1 has a pI of 11.5 and a charge of + 13 under the conditions of our experiments and the resulting unfavorable electrostatic interactions between proximal molecules must be screened before phase separation can occur. This can be achieved typically by the addition of negatively charged molecules or by adding salt (Kim et al. 2019; Wong et al. 2020a). Here we have used low salt buffers (25 mM MES-NaOH (pH 5.5) or 25 mM HEPES–NaOH (pH 7.4)) to ensure that the protein solutions studied are fully mixed (i.e., not phase separated), and, therefore, CAPRIN1 is expected to have a positive electrostatic potential, as established below. Figure 4A, top, shows the ¹³CO–¹⁵N projection of a 3D haCONHA dataset recorded with the pulse sequence of Fig. 1 (Scheme A.1), T_relax = 0 ms, along with the magnetization transfer pathway that gives rise to the spectrum (bottom). Three peaks are highlighted, along with the residues from where the correlations originate; analysis of these peaks in a series of 3D datasets recorded as a function of T_relax, generates the decay curves in Fig. 4B. As our goal is to calculate the near surface electrostatic potential (Yu et al. 2021) of CAPRIN1, we have measured ¹H^α transverse relaxation rates in the presence or absence (Diamagnetic, purple) of 5 mM negative (Carboxy-PROXYL, red) or neutral (Carbamoyl-PROXYL, grey) solvent spin labels. A comparison of intensity profiles using Schemes A.3 and A.1 (titled Scheme A.3 and Scheme A.1 in the figure) clearly shows the effects of scalar coupling on the evolution of ¹H^α magnetization when recording data with the spin-echo scheme (Fig. 1A.3) relative to the spin-lock element of Fig. 1A.1. Notably, J-modulation gives rise to decidedly non-exponential decays of ¹H^α magnetization (left column, Scheme A.3), as is particularly apparent in the profile of G609, where the magnetization becomes negative for T_relax values in excess of approximately 20 ms. The effective intensity decays are slower when using the spin-lock, including for T705, despite the fact that scalar coupling effects are not completely eliminated for ¹H^α spins of this residue when magnetization is locked (see above).

Fig. 4

Suppression of J-modulation via spin locking of ¹H^α magnetization. A ¹³CO-¹⁵N projection of a 3D haCONHA dataset recorded with the pulse sequence of Fig. 1A.1, T_relax = 0 ms. Several peaks are highlighted from which the ¹H^α relaxation profiles shown are derived. The magnetization transfer pathway is indicated at the bottom. B Decay curves of selected residues measured in the presence or absence (Diamagnetic, purple) of 5 mM negative (carboxy-PROXYL, red) or neutral (carbamoyl-PROXYL, grey) solvent spin labels. Solid lines connect the experimental points in the panels titled Scheme A.3 and Scheme A.1. Ratio of corresponding peak intensities in spectra recorded with the sequences of either Fig. 1A.3 (left) or Fig. 1A.1 (right) and either with, ${I}^{para.i}({T}_{relax})$, or without, ${I}^{dia.}({T}_{relax})$, solvent spin-labels, along with exponential fits of the data and extracted PRE rates, are shown. All measurements were performed on a 300 μM U-¹³C, ¹⁵N CAPRIN1 sample at pH 5.5, 25 °C and 14.0 Tesla

Recognizing the deleterious effects of homonuclear scalar couplings to the measurement of ¹H^α relaxation rates using spin-echo type experiments (Fig. 1A.3), Iwahara and co-workers determined PRE rates by simultaneous analysis of the ratios of signal intensities in paramagnetic and diamagnetic samples in spectra recorded with identical T_relax values (Iwahara et al. 2004; Clore and Iwahara 2009; Yu et al. 2022). In this way the scalar coupling terms cancel, and ratios are sensitive only to the PRE. However, the signal intensities themselves are reduced by the modulation making this approach more error prone than if coupled evolution was not present in the first place. For example, the large germinal ¹H^α coupling in Gly residues results in low peak intensities for T_relax values greater than approximately 15 ms (Fig. 4B); in our applications T_relax values between 16 and 25 ms had to be omitted when data were recorded using a spin-echo based sequence (Fig. 1A.3). For other residue-types (non-Gly residues), the reduction in intensities of resonances is less severe, on average a ratio of 0.46 ± 0.12 for T_relax = 30 ms is obtained, when comparing the schemes shown in Fig. 1A.1 and A.3. Also shown in Fig. 4B are exponential fits of intensity ratios, I^para.,i(T_relax)/I^dia.(T_relax), of cross-peaks from spectra recorded with the spin-echo and spin-lock schemes. Notably, while the PRE rates from Schemes A.1 and A.3 are somewhat different, the ratio of rates recorded with different combinations of solvent spin labels tends to be similar (with the exception of a number of Gly residues, for which the germinal coupling is particularly detrimental for the spin-echo scheme).

The simulations and experimental data presented in Figs. 2, 3, and 4 strongly suggest that the spin-lock scheme of Fig. 1A.1 suppresses J-modulation (except for ¹H^α from Ser and Thr) without introducing magnetization transfer via the ROE. A more rigorous evaluation of the robustness of the experiment can be made through comparison to an analogous yet distinct approach, illustrated in Fig. 1A.2 where the relaxation of transverse (not spin-locked) ¹H^α magnetization is measured. In this pulse scheme suppression of ¹H^α-¹H scalar coupled evolution is achieved for the majority of amino acids through the use of band-selective ¹H adiabatic decoupling (Kupce and Wagner 1996) that is carefully adjusted so as to minimally perturb the ¹H^α signals of interest, while decoupling ¹H^β and ¹H^N proton spins. Since adiabatic decoupling of ¹H^β is applied over a chemical shift range of ~ 2.2 ± 1.1 ppm, Ser, and Thr, whose ¹H^β chemical shifts do not fall within this region (and overlap with those of ¹H^α) are not effectively decoupled. In addition, the large germinal (two-bond) ¹H^α1–¹H^α2 coupling (~ −15 Hz) for Gly results in a severe modulation of the ¹H^α signals for this residue, as is observed in experiments recorded with Fig. 1A.3. Figure 5 compares carboxy-PROXYL PRE values obtained via the schemes of Fig. 1A.1 and A.2 omitting Gly residues, and the agreement is excellent (RMSD = 0.69 s⁻¹ for ${\Gamma }_{2,-}$, significantly better than when PRE rates are compared between spin-echo (Fig. 1A.3) and spin lock (Fig. 1A.1) schemes (RMSD = 2.21 s⁻¹). Thus, accurate PRE values are obtained by using the ¹H cw spin-lock field in Fig. 1A.1, without deleterious effects from residual J-evolution or cross-relaxation. As the ¹H^α signals from Gly residues are not modulated by geminal scalar couplings using the sequence of Fig. 1A.1, we prefer it, and in what follows all results were obtained using this experiment.

Fig. 5

Correlation plot of carboxy-PROXYL ¹H^α PRE rates using schemes A.1 and A.2 of Fig. 1. The carboxy-PROXYL ¹H^α PRE rates (Γ_2,− = paramagnetic − diamagnetic) measured using two schemes of Fig. 1 are plotted. The PRE rates plotted along the y-axis were measured using a ¹H adiabatic decoupling approach (A.2) and those plotted along the x-axis were quantified via a spin-lock element (A.1). PRE measurements were performed on a 300 μM U-¹³C, ¹⁵N CAPRIN1 sample at pH 5.5, 25 °C and 14.0 Tesla, with and without 5 mM 3-carboxy-PROXYL. R² is the squared Pearson correlation coefficient

Figure 6A shows ¹⁵N–¹H^N TROSY-HSQC spectra of CAPRIN1 recorded at pH 5.5 (left) and at pH 7.4 (center); all other experimental conditions are identical. The degradation of spectral quality with pH is obvious. However, the ¹³CO–¹⁵N projection of the 3D haCONHA dataset (non-TROSY acquisition in the ¹⁵N dimension) recorded on the pH 7.4 sample is of high quality (right), so that electrostatic potentials (ϕ_ENS) can be obtained at neutral pH values using relaxation rates measured with the sequence of Fig. 1A.1. A comparison of ϕ_ENS values measured at pH 5.5 using haCONHA and ¹⁵N-¹H^N TROSY pulse schemes is presented in Fig. 6B, C (left), with good agreement between the two methods. A strong correlation between ϕ_ENS values measured at pH 5.5 and pH 7.4 is also found (Fig. 6B, C, right), as expected, since CAPRIN1 does not contain His residues. Slightly higher errors in the haCONHA based ϕ_ENS measurements at pH 7.4 are noted compared to those at pH 5.5. This reflects the fact that solvent exchange is close to two orders of magnitude more rapid at the higher pH so that the amide protons are more effectively saturated through exchange with water (that is saturated using this pulse scheme). In turn, this leads to saturation transfer to the ¹H^α protons via spin-diffusion, decreasing the initial ¹H^α polarization and hence the resulting signal intensities in 3D datasets.

Fig. 6

Validation of the methodology. A ¹⁵N-¹H^N TROSY-HSQC spectra of CAPRIN1 recorded at pH 5.5 (left) and at pH 7.4 (center) with all other experimental conditions identical (25 °C, 23.5 T), along with a ¹³CO-¹⁵N projection of the 3D haCONHA dataset (right, pH 7.4, 14.0 T) B, C comparison of ϕ_ENS values measured at pH 5.5 using haCONHA and ¹⁵N-¹H^N TROSY pulse schemes (left) and between ϕ_ENS values measured at pH 5.5 and pH 7.4 using the haCONHA experiment (right)

Concluding remarks

Herein we have described a robust method for the measurement of backbone ¹H^α relaxation rates in IDPs, a first step for obtaining near surface electrostatic potentials of IDPs at neutral pH values. The experiment avoids ¹H^N magnetization, leading to high quality IDP spectra even when recorded at high pH where solvent exchange can often be a limiting factor. A number of issues associated with the measurement of ¹H^α relaxation in fully protonated protein systems are discussed and solutions presented so that robust rates can be obtained. Notably, the use of a band-selective spin-lock significantly suppresses homonuclear scalar coupling modulation for all ¹H^α protons, except those from Ser and Thr, improving the accuracy of measured PRE values, since signal decay is attenuated only by relaxation. The excellent agreement between PRE rates measured using ¹H spin-lock and ¹H adiabatic decoupling schemes, the close correlation between ϕ_ENS values measured on CAPRIN1 at pH 5.5 using ¹H^N- and ¹H^α-based experiments, where amide exchange is not limiting, and the good agreement for CAPRIN1 ϕ_ENS calculated from experiments on samples at pH 5.5 and 7.4 (where exchange is severe), provides strong confidence in the developed methodology. During the completion of this study we became aware of related work by Yu et al (2022) where ¹H^α transverse relaxation rates were used to establish the surface potential of ubiquitin using 2D (HCACO)NH-based experiments. This approach is most clearly appropriate for studies of folded proteins where solvent exchange is not limiting, although it seems likely that here, too, there would be considerable benefit with spin-locking of ¹H^α magnetization during the relaxation measurement. This work sets the stage for the measurement of electrostatic potentials in CAPRIN1 condensates, in order to establish the role of electrostatics in phase separation.