On the use of dioxane as reference for determination of the hydrodynamic radius by NMR spectroscopy

Abstract

Measuring the compaction of a protein or complex is key to our understanding of the interactions within and between biomolecules. Experimentally, protein compaction is often probed either by estimating the radius of gyration (R_g) obtained from small-angle x-ray scattering (SAXS) experiments or the hydrodynamic radius (R_h) obtained, for example, by pulsed field gradient NMR (PFG NMR) spectroscopy. PFG NMR experiments generally report on the translational diffusion coefficient, which in turn can be used to estimate R_h using an internal standard to account for sample viscosity and uncertainty about the gradient strength. 1,4-Dioxane is one such commonly used internal standard, and the reference value of R_h is therefore important. We have revisited the basis for the commonly used reference value for the R_h of dioxane (2.12 Å) that is used to convert measured diffusion coefficients into a hydrodynamic radius. We followed the same approach that was used to establish the current reference value by measuring SAXS and PFG NMR data for a set of seven different proteins and using these as standards. Our analysis shows that the current R_h reference value for dioxane R_h is underestimated, and we instead suggest a new value of 2.27 ± 0.04 Å. Using this updated reference value results in a ∼7% increase in R_h values for proteins whose hydrodynamic radii have been measured by PFG NMR. These results are particularly important when the absolute value of R_h is of interest such as when determining or validating ensemble descriptions of intrinsically disordered proteins.

Significance

We have revisited the basis for the commonly used reference value for the R_h of dioxane (2.12 Å) that is used to convert measured diffusion coefficients into a hydrodynamic radius. Our analysis shows that the current R_h reference value for dioxane R_h is underestimated, and we instead suggest a new value of 2.27 ± 0.04 Å. Using this updated reference value results in a 7% increase in R_h values for proteins whose hydrodynamic radii have been measured by PFG NMR. We discuss the implications of these results including for how we calculate R_h for conformational ensembles of disordered proteins.

Introduction

Proteins are dynamic entities that exist in ensembles of states whose average properties vary depending on their sequence properties, structural context, environment, and posttranslational modifications. Folded proteins typically have a narrow distribution of conformations, whereas the structures of so-called intrinsically disordered proteins (IDPs) can vary substantially across the ensemble (1,2). When characterizing the structure and interactions of proteins it is often advantageous to be able to probe the global dimensions of the protein monomer or the size of any molecular assembly it is part of. Experimentally, this can, for example, be assessed via probing the radius of hydration, R_h, and the radius of gyration, R_g (3,4). R_h may be probed by methods such as pulsed field gradient NMR (PFG NMR) spectroscopy (5), fluorescence correlation spectroscopy (6), dynamic light scattering (7), or by size-exclusion chromatography (SEC) (8). R_g is most commonly probed by small-angle x-ray scattering (SAXS) (3,9). Many current estimates of R_h using PFG NMR rely on assumptions that for decades have mostly been left unexamined, prompting us to revisit one key step when determining the R_h of a protein by PFG NMR.

In PFG NMR, the three-dimensional (3D) location of the protein is “encoded” via a spatial field gradient which makes it possible to probe the translational diffusion coefficient (D_t) and in turn estimate the R_h. As PFG NMR reports on the conformationally averaged D_t, and thus average R_h, the technique is especially useful when studying IDPs and their interactions, where conformational ensemble compactness can provide structural information, and changes in R_h may be used to quantify interactions (10,11). In a standard PFG NMR one records a series of NMR spectra at varying gradient strengths, where the decay of the NMR peak intensities with increasing gradient strengths can be fitted to the Stejskal-Tanner equation (Eq. 1) (12,13,14), that relates D_t to the measured peak intensities:

$$I=I_0{e}^{{-g}^2{\gamma }^2{\delta }_{\mathrm{eff}}^2\left(\Delta -\frac{\delta }{3}\right)D_{\mathrm{t}}}$$ (1)

Here, g is the gradient strength, γ is the gyromagnetic ratio, δ_eff is the effective duration of the gradient according to the short gradient approximation, reducing the importance of the gradient pulse shape for the fit (15), and Δ is the diffusion time. Assuming that the resulting translational diffusion of the nuclei is equal to that of the parent molecule, and the molecule diffuses as a spherical entity, the R_h of the selected peak can then be derived from the Stokes-Einstein relation (Eq. 2).

$$R_h=\frac{k_{B}T}{6\pi \eta D_t}$$ (2)

where k_B is the Boltzmann constant, T the temperature, and η the solvent viscosity. For nonspherical molecules, Eq. 2 can also be used as an operational definition of R_h.

Complications may, however, arise when determining the R_h this way. First, estimating D_t on an absolute scale requires careful calibration of the field gradient. Second, the solvent viscosity is sensitive to the type of solvent, solute, and sample conditions. Thus, the buffer composition, temperature, and protein concentration should be carefully controlled. Furthermore, any added D₂O used to lock the NMR frequency must be corrected for as there is a slight difference in viscosity between H₂O and D₂O (11). As the solvent viscosity and field gradient may be difficult to measure and control precisely, deriving a reliable value of R_h of a protein from PFG NMR may be challenging. Instead, one often uses an internal reference compound that is added to the NMR sample. By knowing the R_h of the reference compound (16), the ratio of the D_t between the reference compound and the protein can be used to estimate the R_h of the protein according to Eq. 3.

$$R_{\mathrm{h},\mathrm{protein}}=\frac{D_{\mathrm{t},\mathrm{reference}}}{D_{\mathrm{t},\mathrm{protein}}}{R}_{\mathrm{h},\mathrm{reference}}$$ (3)

Using an internal reference removes the need for accurate calibration of the gradient and measurement of the viscosity. Often, the reference compound of choice is 1,4-dioxane (hereafter dioxane). Dioxane provides a single, easily discernible, ¹H NMR peak at approximately 3.75 ppm and has been found not to interact substantially with several proteins at experimentally used concentrations (17). While other reference compounds such as α- or β-cyclodextrin are sometimes suggested in place of dioxane (18), cyclodextrins have been shown to interact with proteins (19,20), and contribute with more signals in the NMR spectrum that may overlap with those of the protein.

Using dioxane as a viscosity reference requires that its R_h is known and that the R_h value is insensitive to environmental changes. Early use of dioxane as a reference in PFG NMR established the R_h of the molecule to be 2.12 Å, and this value has since been used as a reference when using PFG NMR to determine the R_h of proteins (21). The reference value for dioxane was determined as described above but using a protein molecule as reference. Specifically, Wilkins et al. performed PFG NMR experiments on a solution of dioxane and hen egg white lysozyme (HEWL), where instead of an unknown R_h of the protein in Eq. 3, the R_h of dioxane was unknown and the R_h of the HEWL set to 19.8 Å. This R_h value originated from an earlier study where batch SAXS experiments on HEWL provided an experimental R_g for natively folded HEWL of 15.3 ± 0.2 Å (22). By assuming the ratio between R_g and R_h, ρ, for a globular protein such as HEWL to be that of a solid sphere (23), the R_h of HEWL was obtained with the R_g from SAXS using Eq. 4:

$$\rho =\frac{R_g}{R_h}={\left(3/5\right)}^{1/2}$$ (4)

Using this approach and experimental SAXS data of HEWL, the authors estimated the R_h of natively folded HEWL to be 19.8 Å (21). The ratio of the translational diffusion coefficients between HEWL and dioxane was estimated from the PFG NMR data to be 9.33 (24), leading to an R_h of dioxane of 2.12 Å.

Recently, a community study addressed the reproducibility of SAXS experiments across different instruments and proteins (25). It was found that SAXS experiments and analyses were generally reproducible and consistent. However, batch SAXS experiments on HEWL specifically showed significant variability between experiments. Considering that the original estimate of the R_h of dioxane was based on a batch SAXS measurement of HEWL, any uncertainty in the R_g of HEWL would result in uncertainty in the dioxane R_h of 2.12 Å and hence impact R_h measurements of proteins using PFG NMR and dioxane referencing. Further, the value depends on the ratio ρ between R_g and R_h, and while this value is expected to be close to 0.77, variations in ρ across different proteins would also lead to uncertainty.

While R_g and R_h, report on comparable physical properties, there are differences between the two including differences in sample requirements, and dependency of shape. Therefore, it is useful to be able to determine the absolute value of both. With the abovementioned possibilities for uncertainty on the R_h of dioxane in mind, we decided to revisit the foundation of this reference value by exploring a larger set of proteins, using the same approach as used originally (21). We estimated the average and uncertainty of the R_h of dioxane using a set of seven folded proteins for which we measured both translational diffusion coefficients by PFG NMR and determined the R_g by batch SAXS measurements. By considering the quality of the recorded data, we find that the established R_h value of dioxane is slightly underestimated. Our data suggest that it should be increased by approximately 7% compared with the previous value, and we propose an updated standard R_h value of 2.27 ± 0.04 Å, which would also result in a 7% increase in derived protein R_h using Eq. 4.

Materials and methods

Protein purifications

Protein samples used in this work were either prepared from bought lyophilized powder stocks or from frozen, prepurified stocks. Proteins from purchased powder stocks were equine myoglobin (Sigma-Aldrich), bovine ribonuclease A (RNaseA) (GE Healthcare), and HEWL (Sigma-Aldrich). Prepurified proteins include S100A13, prolactin, a Tyr73Phe variant of acyl-coenzyme A binding protein (ACBP_Y73F) (26), and 14-3-3 ζ. Purification of prolactin was performed as described in (27). Purification of ACBP_Y73F was performed as described in (28).

For S100A13, E. coli cells (BL21 DE3) (Biolabs) were transformed with a pET-24a plasmid coding for His₆-SUMO S100A13 (UniProt Q99584) and grown in high salt LB-broth medium (Sigma-Aldrich). Cells were grown until an OD₆₀₀ of 0.6–0.8 and expression induced with 0.1 mM isopropyl-β-D-1-thiogalactopyranoside (IPTG). After 4 h growth at 37°C, cells were harvested by centrifugation at 5000 × g for 15 min and stored at −20°C. Cells were lysed in 50 mM Tris (pH 8.0), 150 mM NaCl, 2 mM CaCl₂ through a French press cell disrupter at 25,000 psi (Constant Systems) followed by centrifugation at 20,000 × g where the clear lysate was subsequently loaded onto a 5 mL Ni-NTA Sepharose column (GE Healthcare) equilibrated with 50 mM Tris (pH 8.0), 150 mM NaCl, 2 mM CaCl₂. His₆-SUMO S100A13 was eluted with 50 mM Tris (pH 8.0), 150 mM NaCl, 500 mM Imidazole followed by overnight dialysis where 0.1 mg His-tagged ubiquitin-like protein protease 1 (ULP1) and 1 mM DTT was added to cleave the SUMO-tag. The sample was purified further using a reverse Ni-NTA step removing noncleaved protein, ULP1 and SUMO followed by removal of DNA on a 1 mL Heparin column with 50 mM Tris (pH 7.4), 2 mM CaCl₂ and 50 mM Tris (pH 7.4), 1 M NaCl, 2 mM CaCl₂. For a final step, the sample was run on a Superdex 75 10/300 (GE Healthcare) in 50 mM Tris (pH 7.4), 150 mM NaCl, fractions concentrated and stored at −20°C.

The 14-3-3ζ protein was expressed from a modified pET-24a vector designed to encode an N-terminal His₆-SUMO tag, which was to be cleaved using ULP1. The plasmid was transformed into NiCo21(DE3) competent E. coli cells (New England Biolabs) grown in LB medium containing 50 μg/mL kanamycin and the fusion protein expression was induced with 0.5 mM IPTG for 4–5 h before harvesting cells by centrifugation at 5000 × g, 15 min, 4°C. The pellet was lysed in lysis/equilibration buffer (20 mM Bis-Tris (pH 7.2), 10 mM Imidazole, 150 mM NaCl, 5 mM β-mercaptoethanol (bME)) using a French pressure cell disrupter (25 kpsi; Constant Systems, Daventry, UK), and the lysate was cleared by centrifugation at 20,000 × g for 45 min at 4°C. The His₆-SUMO-14-3-3ζ fusion protein was purified by immobilized metal affinity chromatography (IMAC) using Ni Sepharose 6 Fast Flow resin (5 mL; GE Healthcare) with standard IMAC procedures of sample application, high salt (1 M NaCl) washing step and imidazole elution. The eluted sample was dialyzed toward 2 L of buffer A (20 mM Bis-Tris (pH 6.5), 5 mM bME) before the sample was applied to a 1 mL HiTrap Heparin HP column (Cytiva). The column was washed with 15 mL of buffer A before the fusion protein was eluted with a linear two-step gradient of 0–30% over 3 mL and 30–100% over 20 mL of buffer B (20 mM Bis-Tris (pH 6.5), 1 M NaCl, 5 mM bME). The His₆-SUMO tag was cleaved off by supplementing the sample with 0.1 mg ULP1 and 2 mM DTT for at least 3 h. This sample was reapplied to the IMAC column to remove the His₆-SUMO-tag and ULP1, and the flow through containing pure 14-3-3ζ was collected.

NMR and SAXS sample preparations

To prepare samples from purified proteins or lyophilized protein stocks, each protein was applied to a SEC column (Superdex 75 10/300 GL, Cytiva) in a 20 mM sodium phosphate buffer (pH 7.4), 150 mM NaCl mounted on an Äkta Purifier system. Each collected sample was then analyzed by SDS-PAGE to verify the purity of the protein. The samples for PFG NMR experiment were prepared with the following proteins and concentrations: HEWL 200 μM, RNaseA 300 μM, myoglobin 400 μM, S100A13 150 μM, ACBP_Y73F 200 μM, prolactin 300 μM, and 14-3-3150 μM. DSS was added to a final concentration of 25 μM, D₂O to a final concentration of 10% (v/v), and dioxane to a final concentration between 0.04 and 0.06% (v/v). Sample volumes were either 100, 350, or 500 μL, depending on the use of 3 mm Shigemi, 5 mm Shigemi, or 5 mm glass single-use NMR tubes (Bruker), respectively. Samples for SAXS experiments were prepared by concentrating the proteins after SEC to multiple samples of 1, 2, and 3 mg/mL in the 20 mM sodium phosphate buffer also used in SEC. A buffer solution was also prepared for recording of buffer scattering and subtraction of the background. An overview of sample details is provided in Table S1.

PFG NMR spectroscopy

PFG NMR experiments were performed on a Bruker Avance III HD 600 MHz spectrometer equipped with a Bruker proton-optimized quadruple resonance NMR “inverse” QCI cryoprobe. The diffusion coefficients were determined from a series of ¹H-1D NMR spectra using the bipolar pulse pair with longitudinal eddy-current delay (16) with a smoothed squared gradient (“Difftrap” in the topspin library). Each PFG NMR experiment was preceded by a 1D ¹H-spectrum used for referencing the spectra to the DSS peak at 0 ppm. All sets of PFG and 1D NMR spectra for each protein were recorded at 20°C, except for spectra of myoglobin, which were recorded at 18°C. We used presaturation as water suppression during the recovery delay and big delta. Baseline correction was done in topspin applying a solvent filter first (qfil in topspin, with a filter width of 0.5 ppm) and then a subsequent first-order polynomial. To obtain the least bias in peak selection, the Bruker Dynamics software was set to choose the area under the curve providing the least variance between peaks. In this way, baseline shifts and the DSS doublets were also best avoided. Translational diffusion coefficients of proteins were determined by fitting peak intensity decays in the methyl and methylene region (2.5–0.5 ppm) to the Stejskal-Tanner equation (12,16). The dioxane translational diffusion coefficients were fitted to the intensity decay of the dioxane peak. More specifically, the following protein and dioxane peaks were analyzed; ACBP: 0.878, 3.773; RNAseA: 0.718, 3.751; HEWL: 0.950, 3.755; myoglobin: 1.366, 3.752; S100A13: 0.875, 3.750; prolactin: 0.967, 3.782; 14-3-3: 0.848, 3.751, where the two numbers refer to the chemical shifts (in ppm) for the protein and dioxane peak, respectively. Integration of selected peaks was performed by Bruker Dynamics Center v.2.5.6. For every PFG NMR experiment, 64 scans were recorded with a gradient strength interval from 2 to 98% (γ = 26,752 rad s⁻¹ Gauss⁻¹) with a diffusion time (Δ) of 200 ms and a gradient length (δ) of 2 ms. For our purpose, the exact value of the gradient strength is not critical as we only look at relative diffusion coefficients. To access if the results of the long diffusion times were affected by convection, the same type of data set was recorded on an identical sample of HEWL in three differently sized NMR tubes of 5 mm, 5 mm Shigemi and in a 3 mm tube and the results compared, mapping also the variance of R_h originating from different peaks. Baseline correction and assessment of the Stejskal-Tanner fitting intervals were performed in Bruker Topspin 3.6.2 and Dynamics Center v.2.5.6, while final fitting of the translation diffusion was performed in GraphPad Prism 8.2.1. In addition to error estimates from fits we also estimated an error using the technical replicate measurements of HEWL and RNaseA (Table S2).

SAXS experiments

SAXS experiments were performed at the CPHSAXS Facility, University of Copenhagen, on a Xenocs BioXolver L with a wavelength of λ = 1.34 Å. All scattering curves were recorded at 20°C. Primary data reduction was made in BIOXTAS RAW. For each protein, the largest sample concentration possible yielding a high signal/noise ratio while avoiding aggregation was chosen by inspecting the low-q region of the scattering curves in a Guinier plot of the sample data. Each scattering curve consisted of an accumulation of 10 measurements on each sample. Subsequent data analysis was performed in the ATSAS 3.0.5 Primus suite (29), with the “merge” function used on multiple scattering curves for each sample protein. Merged scattering curves of n SAXS experiments for each protein sample were then used for Guinier derivation of the protein R_g. The Primus “AutoRg” function was used to determine q-range for R_g derivation. Pair distance distribution plots of the consensus curves were also calculated in the ATSAS 3.0.5 Primus suite with the D_max value being set based on a qualitative assessment of the p(r)-function reaching 0. In addition to error estimates from fits we also estimated an error using the technical replicate measurements of HEWL and RNaseA (Table S3).

Calculating the radius of hydration and -gyration from atomic coordinates

The (anhydrous) R_g from the atomic coordinates of proteins was calculated as the mass-weighted average distance of each atom from the protein’s center of mass. For a better comparison to R_g values obtained from experimental SAXS data, we also used the WAXSiS webserver (30,31) to explicitly include a water layer around the protein, calculate the SAXS profile of the protein and envelope and fit an R_g from this with the Guinier approach. The R_h was calculated with the HullRadSAS software (32). For PDB entries that are NMR ensembles, the R_g and R_h were calculated on all conformers of the ensembles and then averaged as $⟨R_g⟩=n^{-1}\sum _i^n{R}_{g,i}$ and $⟨R_h⟩={\left(n^{-1}\sum _i^n{R}_{h,i}^{-1}\right)}^{-1}$. Missing C-terminal residues in the 14-3-3 ζ structure were added with MODELLER (33,34) before calculating R_h and R_g.

R_h determination of dioxane

To determine the R_h of dioxane, the ratio of D_t between the sample protein and dioxane was used in Eq. 3 alongside the R_h values for the protein estimated from experimental SAXS R_g values and Eq. 4. To ensure an accurate estimate of the final R_h value of dioxane, the data quality was factored in by weighing the data by its quality using a χ² approach. The χ² value was calculated by first calculating the observed ratios of diffusion coefficients from PFG NMR, and then by subtracting the expected ratios of the diffusion coefficients from SAXS R_g-derived R_h values and an assumed dioxane R_h value spanning an interval of 2.0 to 2.5 Å. The deviation in the diffusion ratio from the observed NMR-data and SAXS-derived expected data was then divided by the sum of the squared experimental standard error of fits from both PFG NMR and SAXS, leaving a χ² value representing the data quality and fit to different possible R_h values for dioxane spanning 2.0 to 2.5 Å. We thus calculated χ² using the following equation for different estimates of the R_h value of dioxane.

$${\chi }^2=\frac{{\left(D_{\mathrm{ratio},\mathrm{NMR}}-D_{\mathrm{ratio},\mathrm{expected}}\right)}^2}{\left(S{E}_{\mathrm{fit},\mathrm{NMR}}^2+S{E}_{\mathrm{fit},\mathrm{SAXS}}^2\right)}$$ (5)

where $D_{\mathrm{ratio},\mathrm{NMR}}=\frac{D_{\mathrm{t},1,4-\mathrm{dioxane}}}{D_{\mathrm{t},\mathrm{protein}}}$, and $D_{\mathrm{ratio},\mathrm{expected}}=\frac{R_{\mathrm{h},\mathrm{SAXS}}}{R_{\mathrm{h},1,4-\mathrm{dioxane}}}$ .

Results and discussion

Proteins and experimental measurements

To establish a foundation for determining the R_h of dioxane we chose seven globular proteins, varying in size between 86 and 245 residues, and recorded PFG NMR and batch SAXS experiments for each protein at different concentrations at a temperature of 20°C. The seven proteins were chosen based on availability, ease of handling (e.g., solubility and stability), available 3D structures and size. Their known structures show that they have roughly spherical shapes (see also further below), thus supporting the estimation of R_h from R_g using Eq. 4. The seven proteins were ACBP_Y73F, RNaseA, HEWL, myoglobin, S100A13 (dimer), prolactin, and 14-3-3 ζ (dimer) (Fig. 1; Table 1). All proteins were checked for purity and homogeneity by SEC and SDS-PAGE (Fig. S1) before NMR and SAXS data acquisition.

Figure 1.

Surface contour representations of the seven proteins. Proteins are arranged in order of molecular weight and shown on comparable scales. The structures used are from PDB: 1NTI (28), 2AAS (35), 1E8L (36), 5ZZE (37), 1YUU (38), 1RW5 (27), 2O02 (39). When more than one model was present in the PDB entry we show model 1. S110A13 and 14-3-3 are shown as dimers.

Table 1.

Overview of protein properties

Protein	ACBPY73F	RNASEA	HEWL	Myoglobin	S100A13	Prolactin	14-3-3ζ
Organism	B. taurus	B. taurus	G. gallus	E. caballus	H. sapiens	H. sapiens	H. sapiens
Source	E. coli expression	GE Healthcare	Sigma-Aldrich	Sigma-Aldrich	E. coli expression	E. coli expression	E. coli expression
UniProt ID (sequence)	P07108 (2–87, Y73F)	P61823 (27–150)	P00698 (19–147)	P68082 (2–154)	Q99584 (1–98)	P01236 (29–227)	P63104 (1–245)
PDB	1NTI	2AAS	1E8L	5ZZE	1YUU	1RW5	2O02a
MW	9.9 kDa	13.7 kDa	14.3 kDa	17.1 kDa	22.9 kDa dimer	22.9 kDa	54 kDa dimer

^aLacks 15 C-terminal residues compared with the protein used in these experiments.

We first probed the diffusion coefficients of each protein using PFG NMR experiments with the addition of dioxane as an internal standard. After picking peaks corresponding to either dioxane or protein, we fitted the intensity decays as a function of the gradient strength to the Stejskal-Tanner equation (Eq. 1) to estimate the values of D_t for the proteins and dioxane (Figs. 2 and S3; Table 2). Note that while the effective pulse duration (δ_eff) will depend on the shape of the pulse, we analyze ratios of diffusion coefficients from the same experiment (protein and dioxane); this also means that the reported diffusion coefficients will have absorbed effects of the pulse shape. To assess the potential influence of convection at long diffusion times, as well as assessing whether different peaks yielded the same diffusion coefficients, we analyzed three identical HEWL samples in different NMR tubes and with six separate peaks being fitted (Fig. S2). We find that the results depend only little across these variations.

Figure 2.

Peak intensity decays as a function of gradient strength in PFG NMR experiments. Intensity decays for seven samples containing both dioxane (top) and protein (bottom). The panels are labeled and shown in the following order: ACBP_Y73F, RNaseA, HEWL, myoglobin, S100A13, prolactin, 14-3-3ζ. Differences in peak intensity of dioxane between samples are due to variations in concentration and differences in automatic selection of integration intervals performed in Bruker Dynamics Center. Fig. S3 shows the same figure with the inclusion of outliers and residuals for the fits.

Table 2.

Diffusion coefficients of dioxane and seven proteins and their ratios based on single measurements

	ACBP	RNaseA	HEWL	Myoglobin	S100A13	Prolactin	14-3-3
Dioxane $D_{\mathrm{t}}$$\left(m^2/s\right)$	$1.057\times {10}^{-9}\pm 6\times {10}^{-12}$	$1.062\times {10}^{-9}\pm 7\times {10}^{-13}$	$1.091\times {10}^{-9}\pm 3\times {10}^{-12}$	$9.919\times {10}^{-10}\pm 8\times {10}^{-13}$	$1.074\times {10}^{-9}\pm 8\times {10}^{-13}$	$1.060\times {10}^{-9}\pm 3\times {10}^{-12}$	$1.043\times {10}^{-9}\pm 7\times {10}^{-13}$
Protein $D_{\mathrm{t}}$$\left(m^2/s\right)$	$1.373\times {10}^{-10}\pm 1\times {10}^{-12}$	$1.197\times {10}^{-10}\pm 6\times {10}^{-13}$	$1.217\times {10}^{-10}\pm 4\times {10}^{-13}$	$1.069\times {10}^{-10}\pm 9\times {10}^{-13}$	$9.837\times {10}^{-11}\pm 8\times {10}^{-13}$	$9.744\times {10}^{-11}\pm 5\times {10}^{-13}$	$6.223\times {10}^{-11}\pm 4\times {10}^{-13}$
$D_{\mathrm{t}}$ ratio	$7.69\pm 0.09$	$8.87\pm 0.05$	$8.96\pm 0.04$	$9.28\pm 0.08$	$10.92\pm 0.09$	$10.88\pm 0.06$	$16.8\pm 0.1$

While the dioxane data fit well to the Stejskal-Tanner equation, the protein data showed greater variation. As expected, the ratio of the diffusion coefficients between protein and dioxane increased with the molecular weight of the protein. Differences in total intensity of dioxane was observed, in part due to using slightly different dioxane concentration in some of the samples (0.06% in S100A13 and 14-3-3, 0.04% in ACBP, HEWL, RNaseA, prolactin, and myoglobin). We also observed some variation in dioxane intensity in experiments on triplicates of identical samples (Fig. S4), likely due to the automatic selection of integration area by the Dynamics Center software.

Notably, our estimate of the ratio of the diffusion coefficients between HEWL and dioxane in our experiments (8.96 ± 0.05) was ca. 4% lower than the value of 9.33 reported previously (21,24). This difference could possibly be explained by differences in pH or protein concentration between measurements, as the original diffusion coefficient ratio of 9.33 was measured at pH 2, and with a HEWL concentration of ca. 1 mM, whereas our experiments were recorded at neutral pH and with 200 μM of HEWL. To test whether the observed variation in diffusion ratio between our experiments and the original experiments could be explained by random error, we recorded technical triplicate measurements of HEWL and RNaseA samples prepared in the same way from the same protein stock (Fig. S4; Table 2). These results showed an approximately 1% variance in the ratio of the diffusion coefficients between technical replicates, smaller than the difference between 8.96 and 9.33.

Next, we recorded multiple SAXS curves for each protein and used the merge function in the ATSAS suite to derive a protein consensus curve (Fig. 3). We then performed a Guinier analysis of each consensus curve to estimate R_g, to help minimize effects from aggregation that could otherwise increase variance between measurements. The quality of the consensus scattering curves was evaluated by examining the pair distance distribution plots of each curve (Fig. S5).

Figure 3.

Consensus scattering curves for each of the seven proteins. Total scattering curves used in each consensus curve: ACBP (from two measurements), RNaseA (four measurements), HEWL (five measurements), myoglobin (six measurements), S100A13 (five measurements), prolactin (three measurements), and 14-3-3 (three measurements). Error bars represent the standard error. For each panel, the insert shows the low-q region on a logarithmic scale and the linear fit used for the Guinier analysis.

We estimated the R_g values of the proteins by Guinier analysis (Table 3) and from the pair distance distribution plots (Table S4); only values from Guinier analysis were used for estimating the dioxane R_h. As expected, the R_g values increased with molecular weight. The value of R_g that we estimate for HEWL using this Guinier analysis (15.16 ± 0.08 Å) is similar to previous measurements including the value originally used to calibrate the dioxane R_h (15.3 ± 0.2 Å; Chen et al. (22)), and to the mean batch-SAXS R_g (15.3 ± 0.8 Å; Trewhella et al. (25)).

Table 3.

Experimentally derived diffusion coefficient ratios from PFG NMR, experimentally derived R_g values by Guinier analysis from batch SAXS, estimated protein R_h values from the SAXS data and resulting estimated R_h values for dioxane

	ACBP	RNASEA	HEWL	Myoglobin	S100A13	Prolactin	14-3-3
$D_{\mathrm{t}}$ ratio	$7.69\pm 0.09$	$8.87\pm 0.05$	$8.96\pm 0.04$	$9.28\pm 0.08$	$10.92\pm 0.09$	$10.88\pm 0.06$	$16.8\pm 0.1$
Protein Rg (Å) (Guinier)	$14.8\pm 0.2$	$15.20\pm 0.08$	$15.16\pm 0.08$	$16.41\pm 0.04$	$19.8\pm 0.2$	$20.3\pm 0.2$	$27\pm 1$
Estimated protein Rh (Å)	$19.1\pm 0.2$	$19.6\pm 0.1$	$19.6\pm 0.1$	$21.19\pm 0.05$	$25.5\pm 0.3$	$26.2\pm 0.2$	$35\pm 1$
Derived dioxane Rh (Å)	$2.48\pm 0.06$	$2.21\pm 0.02$	$2.18\pm 0.02$	$2.28\pm 0.02$	$2.34\pm 0.05$	$2.41\pm 0.03$	$2.12\pm 0.09$

All reported errors are propagated standard errors of the original fits.

We then estimated the R_h-values for the seven proteins from their measured R_g values using Eq. 3 (Table 3). We used these values with the measured D_t ratios to estimate the R_h for dioxane (Eq. 4). This approach yielded an estimated dioxane R_h from each protein data set with values ranging from 2.12 to 2.48 Å (Table 3). The original estimate of R_h = 2.12 Å lies at the edge of this range and is lower than the value estimated for six of the seven proteins. Taking the average dioxane R_h from our data, factoring in the errors for each protein set, we arrive at a weighted average of 2.27 Å, with an error estimated by bootstrapping to be 0.02 Å.

Examining the relationship between R_g and R_h for proteins

It is possible that differences in the R_h values estimated for dioxane from the SAXS and NMR data across the seven proteins can in part be explained by deviation of the protein shapes from sphere of uniform mass distribution. As a consequence, the ratio between R_g and R_h may not be equal to ρ=(3/5)^1/2, as assumed in Eq. 4. To examine whether this assumption is reasonable for the seven proteins, we used both the crystallographic and solution structures of each of the seven proteins to calculate R_g and R_h, and compared their ratio to the assumed value of ρ=(3/5)^1/2 (Fig. 4) (31). Deviations from an ideal value of ρ would affect our analysis and the estimates of the R_h value for dioxane presented above. In line with the selection of the seven proteins to have roughly spherical shapes, we observe the calculated R_g/R_h values to be close to ρ=(3/5)^1/2, with a small underestimation on average (Fig. 4). As the R_g values used in this analysis were calculated only for the protein coordinates, we subsequently included the contribution of the hydration layer to the calculated R_g to investigate if the apparent underestimation of R_g/R_h could be accounted for by considering the hydration of the proteins (Fig. S6) (32). This, however, led to a large overestimation of R_g/R_h compared with ρ=(3/5)^1/2. We examined whether the deviation from to ρ=(3/5)^1/2 could be explained by deviations from a spherical shape by calculating the asphericity of the proteins (40); however, the asphericity is generally small and if anything, anticorrelated with the deviation (Fig. S5).

Figure 4.

Ratios of R_g/R_h obtained from 3D structures. For each protein, three to five 3D structures were used, with the following PDB codes for each protein: HEWL (1e8l (36), 1dpx (41), 6abn (42), 5a3e (43)), RNaseA (2aas (35), 1fs3 (44), 1jvt (45), 4ooh, 1kf7 (46)), myoglobin (5zze (37), 1wla (47), 4dc8 (48), 5d5r (49), 5cn4 (49)), S100A13 (1yuu (38), 2h2k (50), 2egd (51)), ACBP (1nti (28), 1hb6 (52), 1hb8 (52), 2abd (53)), prolactin (1rw5 (27), 2q98 (54)), 14-3-3ζ (2o02 (39), 1qja (55), 1gjb (56)). AlphaFold structures (57,58) were also included for each protein. Individual ratios are shown as partially transparent markers. For each protein we calculate the mean ratio over different structures and the standard error of the mean (solid markers). Dashed blue line represents the calculated average value of the R_g/R_h, and the black horizontal line indicates ρ=(3/5)^1/2.

Factoring in the data quality to estimate the R_h of dioxane

As described above, we observed some variation in the estimated value of R_h for dioxane depending on which dataset we used. Rather than focusing on the sources of variation across the different estimates, we therefore performed a global analysis of all seven protein data sets to help minimize protein-specific effects. This was done by calculating a χ² value (Eq. 5) between the measured ratio of diffusion coefficients and the value expected depending on 1) the estimated R_h for each protein and 2) the R_h for dioxane (Eq. 3). With this approach, we can assess an interval of likely dioxane R_h values while taking into account the errors on the measured D_t ratios and the estimated value for the R_h for the seven proteins (from SAXS and ρ=(3/5)^1/2) (16).

The resulting plot of χ² vs. the R_h for dioxane (Fig. 5) shows a minimum around R_h of 2.27 Å and, as discussed above, suggests that the previously determined value of 2.12 Å is an underestimate (dashed line in Fig. 5). To estimate an error on the χ²-estimated dioxane R_h of 2.27 Å, we performed a leave-one-out analysis, which provided an error estimate of 0.02 Å, corresponding to the earlier weighted average dioxane R_h and bootstrap error estimation. We also estimated an error using the technical replicate measurements of HEWL and RNaseA in both PFG NMR and SAXS (Tables S2 and S3, respectively). From the technical replicates, we found a 1.4 and 1.1% error across samples measured by PFG NMR and SAXS Guinier analysis, respectively. Propagating these relative errors from the technical replicates to the χ²-estimated dioxane R_h, we determine the R_h of dioxane to be 2.27 ± 0.04 Å

Figure 5.

χ² value for the seven protein data sets calculated as described in Eq. 5 for different possible R_h values for dioxane. The dashed vertical line highlights the R_h value of dioxane of 2.12 Å determined in (21).

Impact of an increase in the size of the hydrodynamic radius of dioxane

With an added uncertainty to the estimated dioxane R_h, the above results suggest a ∼7% (2.27 Å/2.12 Å = 1.07) increase in R_h compared with the commonly used reference. This change in reference value can be accounted for when examining previously published PFG NMR data that have used dioxane as a reference by increasing the derived protein R_h by 7% as well. For example, we previously reported the R_h of prothymosin-α to be 28.9 ± 0.8 Å using PFG NMR and with dioxane as a reference with the R_h set as 2.12 Å (59). Using the updated reference value for dioxane and propagating the errors, the reestimated R_h would be 31 ± 1 Å. Often, R_h values from PFG NMR are used to track changes in protein dimensions following a selected perturbation. In these cases, the increase in dioxane R_h would not be an issue, as the relative changes are unaffected by the increase in the absolute R_h. An example of this would be to use PFG NMR to study the oligomeric state of a protein assembly from the observed R_h at different concentrations (60). On the other hand, the compaction of a protein is a sensitive reporter on intermolecular interactions and is hence useful for probing the accuracy of force fields for molecular simulations. In such cases it is important to know the absolute value of R_h.

Consequences for estimating the conformational ensembles of IDPs

One important consequence of the results presented here is that they affect our understanding of how to compare conformational ensembles with experimental measurements of R_h. We recently compiled a list of the values of R_h measured by PFG NMR for 11 IDPs, and used these together with SAXS experiments to evaluate different models to calculate R_h from conformational ensembles of IDPs (17). Our work led to the conclusion that, among the different approaches tested, the Kirkwood-Riseman equation (61) resulted in the best agreement between computational models and the R_h values measured by PFG NMR. We noted, however, how possible inaccuracies in the reference R_h of dioxane would affect our results, by changing the experimentally determined R_h values and leading to a different conclusion.

As we here have shown that the R_h of dioxane was previously underestimated, we reexamine the conclusion of our previous work considering the new reference value for the R_h of dioxane. We previously found that DSS1 was an outlier (17) and therefore here excluded it from our assessment of methods to calculate R_h. First, we rescaled the previously experimentally derived R_h values for the remaining ten IDPs by 7% (Table S5) and applied an uncertainty of 2.1% (average relative error associated with the R_h of the IDPs) to the corrected R_h values, as we previously proposed to use a uniform uncertainty in this type of analysis. We used the same SAXS-optimized conformational ensembles as in our previous work (produced with the CALVADOS model (62)). In our previous work we used the Kirkwood-Riseman equation (61), two empirical relationships relating R_g and R_h (63) and HullRad (64) to calculate R_h from the conformational ensembles generated by CALVADOS. We here replace the latter with the more recent HullRadSAS approach (32).

When we compare the R_h values calculated from the SAXS-refined ensembles to the revised experimental values, the results are less clear than when we compared with the R_h values based on the original reference value for dioxane. Specifically, we now find that the Kirkwood-Riseman systematically underestimates the R_h, whereas the other approaches overestimate R_h (Fig. 6). Of the four methods examined, we find that HullRadSAS agrees more closely with the data. This observation is due to the finding that HullRadSAS gives rise to very good agreement with the experimental values for four of the IDPs; for the remaining the result is more ambiguous as the experimental value lies in between the predictions from the different models (Fig. S8). Based on this result, we suggest the use of HullRadSAS when calculating the R_h of IDP conformations.

Figure 6.

Assessment of models to calculate R_h from structural ensembles of intrinsically disordered proteins using a previously described approach (17). We compare three approaches to calculate the R_h from conformational ensembles of ten IDPs (Table S5): the Nygaard equation (in blue) (63), the Kirkwood-Riseman equation (in orange) (65), and HullRadSAS (in green) (32). (a) Agreement between calculated and experimentally derived R_h values. (b) As the value for the R_h of dioxane varies, so does our assessment of the models to calculate R_h from conformational ensembles of IDPs: when using a value of 2.12 Å, the Kirkwood-Riseman equation leads to the best agreement with PFG NMR measurements. With the new proposed value of 2.27 ± 0.04 Å, HullRadSAS leads to the best agreement with PFG NMR measurements.

Conclusions

From the PFG NMR and SAXS data obtained for seven proteins, and subsequent weighted average and χ² estimations of the dioxane R_h, we find that the original 2.12 Å R_h is likely underestimated and suggest a value of 2.27 ± 0.04 Å to be used as the standard dioxane R_h when performing PFG NMR to determine the hydrodynamic radius of a protein. The error on this final value is determined as the propagated relative error from technical replications measured by both PFG NMR and SAXS on HEWL and RNaseA; however, this uncertainty will rarely be the limiting factor in the accuracy of derived protein R_h values. Previously published PFG NMR protein R_h values can easily be rereferenced to our suggested dioxane R_h by increasing the protein R_h with 7%, however, this is only needed if absolute R_h values are reported.

While our data suggest that the dioxane R_h is greater than 2.12 Å, one would ideally use an even greater protein dataset and/or other methods to analyze dioxane. Furthermore, the R_h of dioxane might be both pressure- and temperature dependent, and as such, the R_h of dioxane at different experimental conditions should also be examined further (66,67). It would also be useful to complement our studies with a more direct measurement of the hydrodynamic radius via calibration of the gradient strength and measurement of the solution viscosity.

In some analyses, such as for example, of titrations of ligand binding, it is often only the relative change in R_h that is important; in such cases variation in the reference value is not important. In other cases, one will compare absolute values of R_h. For example, since R_h depends on both inter- and intramolecular interactions, it may be used to assess conformational ensembles. We have shown that the revised reference value can be used to reassess methods for comparing conformational ensembles to experiments, and we envision that it will be particularly important when deriving conformational ensembles or benchmarking simulation methods and force fields.

Data and code availability

Experimental data, processed data, and scripts to reproduce the content of this work are available at: https://github.com/KULL-Centre/_2023_dioxane-tranchant/.

Acknowledgments

We thank Signe A. Sjørup for skilled technical assistance and Johan G. Olsen for valuable discussions on water layers in relation to SAXS measurements. We thank Andreas Prestel, manager of the cOpenNMR facility (www.bio.ku.dk/copennmr; grant no. NNF18OC0032996) for NMR assistance. We acknowledge access to the University of Copenhagen small-angle x-ray scattering facility, CPHSAXS, funded by the Novo Nordisk Foundation (drug.ku.dk/core-facilities/cphsaxs/; grant no. NNF19OC0055857), and thank Pernille Sønderby Tuelung for assistance. We acknowledge access to computational resources from the Biocomputing Core Facility at the Department of Biology, University of Copenhagen. This work was supported by grants from the Novo Nordisk Foundation to the Challenge centres PRISM (NNF18OC0033950 to K.L.-L.) and REPIN (NNF18OC0033926 to B.B.K.), the Lundbeck Foundation BRAINSTRUC initiative (R155-2015-2666 to B.K.K. and K.L.-L.), and the Danish Research Councils (9040-00164B to B.B.K.).

Author contributions

E.E.T. measured NMR and SAXS data. F.P. performed structure-based calculations. E.E.T. and F.P. performed statistical analyses. E.E.T., N.L.J., and C.T.B. purified proteins. E.E.T., F.P., B.B.K., and K.L.-L. analyzed data. B.B.K. and K.L.-L. designed the study. E.E.T., B.B.K., and K.L.-L. wrote the paper with input from all authors.

Declaration of interests

K.L.-L. holds stock options in and is a consultant for Peptone Ltd.

Editor: Scott Showalter.