J-Spectroscopy in the presence of residual dipolar couplings: determination of one-bond coupling constants and scalable resolution

Abstract

The access to weak alignment media has fuelled the development of methods for efficiently and accurately measuring residual dipolar couplings (RDCs) in NMR-spectroscopy. Among the wealth of approaches for determining one-bond scalar and RDC constants only J-modulated and J-evolved techniques retain maximum resolution in the presence of differential relaxation. In this article, a number of J-evolved experiments are examined with respect to the achievable minimum linewidth in the J-dimension, using the peptide PA₄ and the 80-amino-acid-protein Saposin C as model systems. With the JE-N-BIRD^d,X-HSQC experiment, the average full-width at half height could be reduced to approximately 5 Hz for the protein, which allows the additional resolution of otherwise unresolved peaks by the active (J+D)-coupling. Since RDCs generally can be scaled by the choice of alignment medium and alignment strength, the technique introduced here provides an effective resort in cases when chemical shift differences alone are insufficient for discriminating signals. In favorable cases even secondary structure elements can be distinguished.

Appendix

Neglecting strong coupling artifacts, the evolution of transverse operators during a BIRD^d,X element applied to a three spin system consisting of one heteronucleus X, one directly coupled proton spin I ^d, and one remotely coupled proton spin I ^r with the corresponding couplings J _d,X  > J _d,r  ≈ J _r,X , results in the following transfers:

$$ \begin{aligned} I_x^d &\longrightarrow I_x^d\cos2\pi\delta^1J_{d,X}\cos2\pi\delta^n J_{d,r} +2I_z^dX_z\sin2\pi\delta^1J_{d,X}\cos2\pi\delta^n J_{d,r}\\ &\,-2I_z^dI_y^r\cos2\pi\delta^1J_{d,X}\sin2\pi\delta^n J_{d,r} +4I_x^dI_y^rX_z\sin2\pi\delta^1J_{d,X}\sin2\pi\delta^n J_{d,r}\\ I_y^d&\longrightarrow I_y^d\\ 2I_x^dX_z&\longrightarrow -2I_x^dX_z\cos2\pi\delta^1J_{d,X}\cos2\pi\delta^n J_{d,r} -I_z^d\sin2\pi\delta^1J_{d,X}\cos2\pi\delta^n J_{d,r}\\ &\,+4I_z^dI_y^rX_z\cos2\pi\delta^1J_{d,X}\sin2\pi\delta^n J_{d,r} -2I_x^dI_y^r\sin2\pi\delta^1J_{d,X}\sin2\pi\delta^n J_{d,r}\\ 2I_y^d X_z &\longrightarrow -2I_y^dX_z\\ X_x&\longrightarrow X_x\cos2\pi\delta^1J_{d,X}\cos2\pi\delta^1 J_{r,X} -2I_y^dX_y\sin2\pi\delta^1J_{d,X}\cos2\pi\delta^1 J_{r,X}\\ &\,-2I_y^rX_y\cos2\pi\delta^1J_{d,X}\sin2\pi\delta^1 J_{r,X} -4I_y^dI_y^rX_x\sin2\pi\delta^1J_{d,X}\sin2\pi\delta^1 J_{r,X}\\ X_y&\longrightarrow -X_y\cos2\pi\delta^1J_{d,X}\cos2\pi\delta^1 J_{r,X} -2I_y^dX_x\sin2\pi\delta^1J_{d,X}\cos2\pi\delta^1 J_{r,X}\\ &\,-2I_y^rX_x\cos2\pi\delta^1J_{d,X}\sin2\pi\delta^1 J_{r,X} +4I_y^dI_y^rX_y\sin2\pi\delta^1J_{d,X}\sin2\pi\delta^1 J_{r,X}\\ 2I_z^dX_x&\longrightarrow 2I_z^dX_x\cos2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r} -4I_z^dI_y^rX_y\sin2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r}\\ &\,+4I_x^dI_y^rX_x\cos2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r} -2I_x^dX_y\sin2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r}\\ 2I_z^dX_y&\longrightarrow -2I_z^dX_y\cos2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r} -4I_z^dI_y^rX_x\sin2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r}\\ &\,-4I_x^dI_y^rX_y\cos2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r} -2I_x^dX_x\sin2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r}\\ \end{aligned} $$

If we assume that long-range heteronuclear and homonuclear couplings can be neglected during the BIRD^d,X element, the transfers can be simplified to

$$ \begin{array}{lll} I_x^d &\longrightarrow &I_x^d\cos2\pi\delta^1J_{d,X}+2I_z^dX_z\sin2\pi\delta^1J_{d,X}\\ I_y^d &\longrightarrow &I_y^d\\ 2I_x^dX_z& \longrightarrow &-2I_x^dX_z\cos2\pi\delta^1J_{d,X}-I_z^d\sin2\pi\delta^1J_{d,X}\\ 2I_y^dX_z &\longrightarrow &-2I_y^dX_z\end{array} $$

for proton transverse operators and

$$ \begin{array}{lll} X_x& \longrightarrow &X_x\cos2\pi\delta^1J_{d,X}-2I_y^dX_y\sin2\pi\delta^1J_{d,X}\\ X_y& \longrightarrow &-X_y\cos2\pi\delta^1J_{d,X}-2I_y^dX_x\sin2\pi\delta^1J_{d,X}\\ 2I_z^dX_x &\longrightarrow &2I_z^dX_x\\ 2I_z^dX_y &\longrightarrow &-2I_z^dX_y \end{array}$$

for the corresponding transverse operators of the heteronucleus. The magnetization of every other operator is scaled by ${\cos(2\pi\delta^{1}J_{d,X})},$ resulting in an average loss of ${0.5^\ast(1- \cos(2\pi\delta^{1}J_{d,X}))}$ due to incomplete transfer during the BIRD^d,X filter.

To examine the influence of homonuclear and long-range heteronuclear couplings, we assume a perfectly matched delay δ, leading to ${\hbox{cos}(2\pi\delta^{1}J_{d,X})=-1}$ and ${\sin(2\pi\delta^{1}J_{d,X})=0}.$ The transfers are then

$$ \begin{array}{lll} I_x^d &\longrightarrow &-I_x^d\cos2\pi\delta^nJ_{d,r}+2I_z^dI_y^r\sin2\pi\delta^nJ_{d,r}\\ I_y^d &\longrightarrow &I_y^d\\ 2I_x^dX_z &\longrightarrow &2I_x^dX_z\cos2\pi\delta^nJ_{d,r}-4I_z^dI_y^rX_z\sin2\pi\delta^nJ_{d,r}\\ 2I_y^dX_z &\longrightarrow &-2I_y^dX_z\end{array} $$

and

$$ \begin{array}{lll} X_x& \longrightarrow &-X_x\cos2\pi\delta^1J_{r,X}+2I_y^rX_y\sin2\pi\delta^1J_{r,X}\\ X_y& \longrightarrow &X_y\cos2\pi\delta^1J_{r,X}+2I_y^rX_x\sin2\pi\delta^1J_{r,X}\\ 2I_z^dX_x&\longrightarrow &2I_z^dX_x\cos2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r} -4I_z^dI_y^rX_y\sin2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r}\\ & &+4I_x^dI_y^rX_x\cos2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r} -2I_x^dX_y\sin2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r}\\ 2I_z^dX_y&\longrightarrow &-2I_z^dX_y\cos2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r} -4I_z^dI_y^rX_x\sin2\pi\delta^1J_{r,X}\cos2\pi\delta^n J_{d,r}\\ & &-4I_x^dI_y^rX_y\cos2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r} -2I_x^dX_x\sin2\pi\delta^1J_{r,X}\sin2\pi\delta^n J_{d,r}\end{array} $$

As with the direct one-bond coupling, every other transverse component is scaled by the homonuclear coupling ${\cos(2\pi \delta ^{1}J_{d,r})},$ causing an average loss of magnetization of 0.5*(1- ${\cos(2\pi \delta ^{1}J_{d,r}))}.$ Heteronuclear long-range couplings, instead, do not affect proton transverse operators, but reduce all transverse operators on the heteronucleus according to ${(1-\cos(2\pi \delta ^{1}J_{r,X}))}.$