A device for the measurement of residual chemical shift anisotropy and residual dipolar coupling in soluble and membrane-associated proteins

Abstract

Residual dipolar coupling (RDC) and residual chemical shift anisotropy (RCSA) report on orientational properties of a dipolar bond vector and a chemical shift anisotropy principal axis system, respectively. They can be highly complementary in the analysis of backbone structure and dynamics in proteins as RCSAs generally include a report on vectors out of a peptide plane while RDCs usually report on in-plane vectors. Both RDC and RCSA average to zero in isotropic solutions and require partial orientation in a magnetic field to become observable. While the alignment and measurement of RDC has become routine, that of RCSA is less common. This is partly due to difficulties in providing a suitable isotopic reference spectrum for the measurement of the small chemical shift offsets coming from RCSA. Here we introduce a device (modified NMR tube) specifically designed for accurate measurement of reference and aligned spectra for RCSA measurements, but with a capacity for RDC measurements as well. Applications to both soluble and membrane anchored proteins are illustrated.

Appendix

The RCSA span is an important consideration in terms of its practical application. Generally a larger span allows more room for measurement errors and therefore is preferable. Here we derive the relationship between the upper and lower bounds of RCSAs and the order tensor. We use the symbol $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta }$ for the chemical shift tensor as suggested by Mason (1993).

RCSA, in ppm, is described by the following equation in an arbitrary reference frame:

$$ {\text{RCSA}} = \left\langle {{\text{B}}^{\text{T}} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } \cdot {\text{B}}} \right\rangle - \delta_{\text{iso}} $$

(A1)

where B is a column vector representing the direction of the B₀ field in the reference frame and B^T is the transposed row vector. The chemical shift (CS) tensor $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta }$ is also expressed in the same reference frame. The isotropic chemical shift δ_iso is a constant that equals the average of the 3 eigen values of the CS tensor, (δ₁₁ + δ₂₂ + δ₃₃)/3. The bracket represents averaging by molecular reorientation. Written in the principal axis frame (PAF) of the CS tensor, (A1) becomes:

$$ {\text{RCSA}} = \delta_{11} \left\langle {\cos^{2} \alpha } \right\rangle + \delta_{22} \left\langle {\cos^{2} \beta } \right\rangle + \delta_{33} \left\langle {\cos^{2} \gamma } \right\rangle - \delta_{\text{iso}} $$

(A2)

where α, β, and γ are the angles between B₀ and the x, y, and z axis of the CS PAF. By convention, the eigen values of the CS tensor are ordered so that δ₁₁ ≥ δ₂₂ ≥ δ₃₃ (This originates from the ordering convention for chemical shielding tensor that σ₃₃ ≥ σ₂₂ ≥ σ₁₁). Notice that in (A2), 〈cos²θ〉 (θ = α, β, or γ) can be regarded as weights for the eigen values, based on the fact that 0 ≤ 〈cos²θ〉 ≤ 1 and 〈cos²α〉 + 〈cos²β〉 + 〈cos²γ〉 = 1. Therefore, the upper bound of RCSA occurs when the largest eigen value δ₁₁ gets the largest weight and the smallest eigen value δ₃₃ gets the smallest weight. Similarly, the lower bound of RCSA occurs when the largest eigen value gets the smallest weight and the smallest eigen value gets the largest weight.

Next we examine the largest and smallest possible weights in terms of their relationships to the Saupe order matrix. The order matrix element in an arbitrary molecular frame is given by:

$$ {\text{S}}_{\text{ij}} = \frac{1}{2}\left( {3\left\langle {\cos \theta_{\text{i}} \cos \theta_{\text{j}} } \right\rangle - \Updelta_{\text{ij}} } \right) $$

(A3)

where θ_i is the angle between the B_o field and the i-th axis of the molecular frame, and Δ_ij is the delta function. If the molecular frame is chosen to be a CS PAF, then obviously the weights in (A2), namely 〈cos²α〉, 〈cos²β〉, and 〈cos²γ〉, are directly related to the diagonal elements (i = j) of the order matrix in the CS PAF. Clearly, the largest and smallest weights are associated with the largest and smallest diagonal elements, respectively. Note that for the diagonal elements, comparison is by the numerical value but not by the absolute value, i.e. a negative value of a high magnitude is smaller than a positive number of a low magnitude. It is easy to show that the diagonal elements of the order matrix in the CS PAF (or any other frame) can be expressed as a weighted average of the diagonal elements in the order tensor’s PAF:

$$ {\text{S}}_{\text{ii}} = {\text{U}}_{\text{xi}}^{ 2} {\text{S}}_{\text{xx}}^{\prime } + {\text{U}}_{\text{yi}}^{ 2} {\text{S}}_{\text{yy}}^{\prime } + {\text{U}}_{\text{zi}}^{ 2} {\text{S}}_{\text{zz}}^{\prime } \;({\text{i}} = {\text{x,y,z}}) $$

(A4)

where ${\text{S}}_{\text{xx}}^{\prime }$ is the diagonal element of the order matrix in its PAF (principal value), and U_ij is an element of the rotation matrix that relates the order tensor PAF and the CS PAF. The orthogonality of the axis systems related through the rotation matrix U requires that ${\text{U}}_{\text{xi}}^{2} + {\text{U}}_{\text{yi}}^{2} + {\text{U}}_{\text{zi}}^{2} = 1$. Therefore, for a diagonal element S_ii in an arbitrary frame, the relation holds that ${\text{S}}_{ \min }^{\prime }$ ≤ S_ii ≤ ${\text{S}}_{ \max }^{\prime }$, where ${\text{S}}_{ \max }^{\prime }$ and ${\text{S}}_{ \min }^{\prime }$ stand for the maximal and minimal principal values. According to (A3), the following relation is derived: (2${\text{S}}_{ \min }^{\prime }$ + 1)/3 ≤ 〈cos²θ〉 ≤ (2${\text{S}}_{ \max }^{\prime }$ + 1)/3, where θ = α, β, or γ.

The order matrix is traceless, i.e., S_xx + S_yy + S_zz = 0. By convention, the principal values are ordered so that |S_zz| ≥ |S_yy| ≥ |S_xx|. Therefore if ${\text{S}}_{\text{zz}}^{\prime }$ ≥ 0, ${\text{S}}_{ \max }^{\prime }$ = ${\text{S}}_{\text{zz}}^{\prime }$ and ${\text{S}}_{ \min }^{\prime }$ = ${\text{S}}_{\text{yy}}^{\prime }$; otherwise, ${\text{S}}_{ \max }^{\prime }$ = ${\text{S}}_{\text{yy}}^{\prime }$ and ${\text{S}}_{ \min }^{\prime }$ = ${\text{S}}_{\text{zz}}^{\prime }$. Going back to (A1), after some straightforward reorganization, gives the following result for ${\text{S}}_{\text{zz}}^{\prime }$ ≥ 0:

$$ \left\{ \begin{gathered} {\text{RCSA}}_{ \max } = \frac{2}{3}\left( {\delta_{11} {\text{S}}_{\text{zz}}^{\prime } + \delta_{22} {\text{S}}_{\text{xx}}^{\prime } + \delta_{33} {\text{S}}_{\text{yy}}^{\prime } } \right) \hfill \\ {\text{RCSA}}_{ \min } = \frac{2}{3}\left( {\delta_{11} {\text{S}}_{\text{yy}}^{\prime } + \delta_{22} {\text{S}}_{\text{xx}}^{\prime } + \delta_{33} {\text{S}}_{\text{zz}}^{\prime } } \right) \hfill \\ \end{gathered} \right. $$

(A5)

The upper bound occurs when the (x, y, z) axes of the CS tensor are collinear with the (z, x, y), (z, −x, −y), (−z, x, −y), or (−z, −x, y) axes of the order tensor, respectively, and the lower bound occurs when the (x, y, z) axes of the CS tensor are collinear with the (y, x, −z), (y, −x, z), (−y, x, z) or (−y, −x, −z) axes of the order tensor, respectively. If ${\text{S}}_{\text{zz}}^{\prime }$ < 0, the relationships in (A5) for RCSA_max and RCSA_max are simply swapped, and so are relationships between the CS tensor and the order tensor. Clearly, independent of the sign of ${\text{S}}_{\text{zz}}^{\prime }$, the RCSA span is given by:

$$ \left| {{\text{RCSA}}_{ \max } - {\text{RCSA}}_{ \min } } \right| = \frac{2}{3}\left( {\delta_{11} - \delta_{33} } \right)\,\left| {{\text{S}}_{\text{zz}}^{\prime } - {\text{S}}_{\text{yy}}^{\prime } } \right| $$

(A6)

Or using the asymmetry parameter, η, of the order tensor defined as η = (${\text{S}}_{\text{xx}}^{\prime }$−${\text{S}}_{\text{yy}}^{\prime }$)/${\text{S}}_{\text{zz}}^{\prime }$, (A6) is re-written as:

$$ \left| {{\text{RCSA}}_{ \max } - {\text{RCSA}}_{ \min } } \right| = \left( {\delta_{11} - \delta_{33} } \right)\left| {{\text{S}}_{\text{zz}}^{\prime } } \right|\left( {1 + {\frac{\eta }{3}}} \right) $$

(A7)

This result suggests that the RCSA span is basically the full chemical shift span scaled down by the order tensor. The RDC span is well known:

$$ \left| {{\text{RDC}}_{ \max } - {\text{RDC}}_{ \min } } \right| = \left| {{\text{DC}}_{ \max } } \right|\left| {{\text{S}}_{\text{zz}}^{\prime } } \right|\left( {1 + {\frac{\eta }{3}}} \right)\frac{3}{2} $$

(A8)

Therefore the ratio between RDC and RCSA spans is given by:

$$ {\text{k}} = \frac{3}{2}{\frac{{\left| {{\text{DC}}_{ \max } } \right|}}{{\nu_{0} \left( {\delta_{11} - \delta_{33} } \right)}}} $$

(A9)

where ν₀ is the Larmor frequency of the atom in MHz. This ratio depends only on the relative sizes of the spin interactions, but not on molecular alignments. Typical ratios for NH RDC over RCSAs of N, H, and C’ atoms on the peptide bond plane are 2.0, 3.0, and 0.9 on a 900 MHz spectrometer.