Quantitative Interpretation of Solvent Paramagnetic Relaxation for Probing Protein-Cosolute Interactions

Abstract

Protein-small cosolute molecule interactions are ubiquitous and known to modulate the solubility, stability, and function of many proteins. Characterization of such transient weak interactions at atomic resolution remains challenging. In this work, we develop a simple and practical NMR method for extracting both energetic and dynamic information on protein-cosolute interactions from solvent paramagnetic relaxation enhancement (sPRE) measurements. Our procedure is based on an approximate (non-Lorentzian) spectral density that behaves exactly at both high and low frequencies. This spectral density contains two parameters, one global related to the translational diffusion coefficient of the paramagnetic cosolute, and the other residue specific. These parameters can be readily determined from sPRE data, and then used to calculate analytically a concentration normalized equilibrium average of the interspin distance, ⟨r^-6⟩_norm, and an effective correlation time, τ_C, that provide measures of the energetics and dynamics of the interaction at atomic resolution. We compare our approach with existing ones, and demonstrate the utility of our method using experimental ¹H longitudinal and transverse sPRE data recorded on the protein ubiquitin in the presence of two different nitroxide radical cosolutes, at multiple static magnetic fields. The approach for analyzing sPRE data outlined here provides a powerful tool for deepening our understanding of extremely weak protein-cosolute interactions.

Figure 1.

Definition of the interspin vector $\overrightarrow{r}$.

Figure 2.

Validation of the approximate spectral density function, eq 15, for hard-sphere protein–cosolute interactions. (A) and (B) comparison of the exact force-free hard-sphere spectral densities (continuous gray lines) calculated using eq 20 with those calculated using either our approximate (eq 15, panel A) or single Lorentzian (eq 9, panel B) spectral density functions (shown as dashed red lines) at several distances of the ¹H spin to the surface of an idealized protein represented by a sphere. The parameters used to generate the exact spectral densities are ${\tau }_{\text{P}}=3.8\text{ns}$ and ${\tau }_{\text{S}}=39\text{ps}$ (corresponding to spheres of radii ∼17 Å and ∼3.5 Å, respectively). The ¹H spin is located within the protein sphere at distances of 1, 2, 3, 4, and 5 Å from the protein surface. The electron spin is located at 1 Å from the surface of the cosolute. The approximate spectral densities obtained using eq 15 superimpose almost exactly on the exact spectral densities calculated using eq 20. The blue circles represent the points from the exact spectral densities used to fit the approximate spectral density functions. (C) and (D) results of complete cross-validation fitting the Γ₁ data at 2 fields using eqs 15 and 9, respectively, and comparing the predicted and exact Γ₁ values at the third field (free data set). Q_free is a measure of agreement between predicted ${\mathrm{\Gamma }}_1^{\text{calc,free}}(i)$ and experimental ${\mathrm{\Gamma }}_1^{\text{obs},\text{free}}(i)$ values for the free data set (i.e., not used in fitting) and: $${\left\{\sum _i{\left[{\mathrm{\Gamma }}_1^{\text{obs},\text{free}}(i)-{\mathrm{\Gamma }}_1^{\text{calc},\text{free}}(i)\right]}^2/\sum _i{\mathrm{\Gamma }}_1^{\text{obs},\text{free}}{(i)}^2\right\}}^{1/2}$$ Where the summation with index i runs over all data points i at a given field. (E) and (F) ratios of approximate to exact ${\langle r^{-6}\rangle }_{\text{norm}}$ values obtained using eqs 15 and 9, respectively, for 2- and 3-field fits. (G) and (H) correlation plots between exact and approximate ${\langle r^{-6}\rangle }_{\text{norm}}$ values obtained using eqs 15 and 9, respectively, for a range of parameters: ${\tau }_{\text{P}}=2,5,10,$ and $15\text{ns};{\tau }_{\text{S}}=5,25,50,100,$ and $250\text{ps};$ and the ¹H spin located at 2 (red), 4 (green), and 6 (blue) Å from the protein surface. The electron spin is located at the center of the cosolute sphere.

Figure 3.

Experimental backbone amide proton sPRE measured for 0.5 mM ²H/¹⁵N-labeled ubiquitin in the presence of 25 mM (A) 3-carboxy-PROXYL or (B) 3-carbamoyl-PROXYL paramagnetic cosolutes. The chemical structures of the paramagnetic cosolute are shown at the top. The middle and bottom panels display the experimental ${}^1{\text{H}}_{\text{N}}-{\mathrm{\Gamma }}_2$ and ${}^1{\text{H}}_{\text{N}}-{\mathrm{\Gamma }}_1$ profiles, respectively, at 500 (blue), 800 (green), and 900 (red) MHz spectrometer fields. (C) and (D) complete cross-validation showing the correlation between experimental and predicted ${}^1{\text{H}}_{\text{N}}-{\mathrm{\Gamma }}_1$ values at 500, 800, and 900 MHz (free data sets) for 3-carboxyl-PROXYL and 3-carbamoyl-PROXYL, respectively, when eq 15 is used to fit the working data sets at the other two fields (800 and 900 MHz, 500 and 900 MHz, and 800 and 900 MHz, respectively).

Figure 4.

Experimentally derived ${\langle r^{-6}\rangle }_{\text{norm}}$ and ${\tau }_{\text{C}}$ values obtained from backbone amide proton sPRE measurements at multiple magnetic fields (500, 800, and 900 MHz) on 0.5 mM ²H/¹⁵N-labeled ubiquitin in the presence of 25 mM 3-carboxy-PROXYL and 3-carbamoyl-PROXYL. ${\langle r^{-6}\rangle }_{\text{norm}}$ and ${\tau }_{\text{C}}$ profiles as a function of residue obtained with (A) 3-carboxy-PROXYL and (B) 3-carbamoyl-PROXYL. For comparison, the ${\langle r^{-6}\rangle }_{\text{norm}}^{\text{exc}}$ profiles calculated directly from the coordinates of ubiquitin (PDB 1D3Z), taking into account only the excluded volume with no intermolecular forces between protein and cosolute, are displayed as continuous red lines. The ${\langle r^{-6}\rangle }_{\text{norm}}^{\text{exc}}$ values were calculated for the 10 different structures in the 1D3Z PDB file to account for different surface side-chain conformers, and the error bars (red vertical lines) represent the standard deviations among these 10 structures. Large standard deviations in ${\langle r^{-6}\rangle }_{\text{norm}}^{\text{exc}}$ are only observed at the C-terminus, largely due to variations in the conformation of the backbone associated with a flexible C-terminal tail.

Figure 5.

Mapping sites of preferential protein-cosolute interactions on the surface of ubiquitin. $\left[{\langle r^{-6}\rangle }_{\text{norm}}-{\langle r^{-6}\rangle }_{\text{norm}}^{\text{exc}}\right]$ profiles obtained from backbone amide proton sPRE measurements on ²H/¹⁵N labeled ubiquitin in the presence of 25 mM (A) 3-carboxy-PROXYL and (B) 3-carbamoylPROXYL radicals. The dashed line is drawn at 3 × 10²⁸ m⁻³ and the points corresponding to residues showing significant preferential interactions are colored in purple. Mapping of $\left[{\langle r^{-6}\rangle }_{\text{norm}}-{\langle r^{-6}\rangle }_{\text{norm}}^{\text{exc}}\right]$ on the surface of ubiquitin color graded from purple (∼3 × 10²⁹ m⁻³) to white for (C) 3-carboxy-PROXYL and (D) 3-carbamoyl-PROXYL. Two views related by a 180° rotation are shown; the top panels display the molecular surface, and the bottom panels, the corresponding ribbons diagrams with the side chains of residues showing significant preferential interactions in purple. For comparison, the (E) hydrophobicity scale (color coded from green to white) and (F) electrostatic potential (graded from blue, positive; white, neutral; and red, negative) are displayed on the molecular surface of ubiquitin. Hydrophobicity and electrostatic potential were calculated using the PyMol Molecular Graphics System (version 2.0 Schrödinger, LLC).