NS R1rho 2-site
The relaxation dispersion model for the numeric solution (NS) to the Bloch-McConnell equations for 2-site exchange for R1rho-type data. This model is labelled as NS R1rho 2-site in relax.
Equation
This is the numerical model for 2-site exchange using 3D magnetisation vectors. It is selected by setting the model to NS R1rho 2-site. The simple constraint $\textrm{pA} > \textrm{pB}$ is used to halve the optimisation space, as both sides of the limit are mirror image spaces.
For this model, the equations from Korzhnev05 have been used. The $\mathrm{R}_{1\rho}$ value for state A magnetisation is defined as \begin{equation}
\mathrm{R}_{1\rho} = - \frac{1}{T_\textrm{relax}} \cdot \ln \left( M_0^T \cdot e^{R \cdot T_\textrm{relax}} \cdot M_0 \right),
\end{equation}
For this model, the equations from \citet{Korzhnev05a} have been used. The $\mathrm{R}_{1\rho}$ value for state A magnetisation is defined as \begin{equation}
\mathrm{R}_{1\rho} = - \frac{1}{T_\textrm{relax}} \cdot \ln \left( M_0^T \cdot e^{R \cdot T_\textrm{relax}} \cdot M_0 \right),
\end{equation}
For this model, the equations from \citet{Korzhnev05a} have been used. The $\mathrm{R}_{1\rho}$ value for state A magnetisation is defined as \begin{equation}
\mathrm{R}_{1\rho} = - \frac{1}{T_\textrm{relax}} \cdot \ln \left( M_0^T \cdot e^{R \cdot T_\textrm{relax}} \cdot M_0 \right),
\end{equation}
where \begin{align}
M_0 &= \begin{pmatrix} \sin{\theta} \\ 0 \\ \cos{\theta} \\ 0 \\ 0 \\ 0 \end{pmatrix}, \\
\theta &= \arctan \left( \frac{\omega_1}{\Omega_\textrm{A}} \right).
\end{align}
The relaxation evolution matrix is defined as \begin{equation}
R = \begin{pmatrix}
-\mathrm{R}_{1\rho}prime-\kAB & -\delta_A & 0 & \kBA & 0 & 0 \\
\delta_A & -\mathrm{R}_{1\rho}prime-\kAB & -\omega_1 & 0 & \kBA & 0 \\
0 & \omega_1 & -\Rone-\kAB & 0 & 0 & \kBA \\
\kAB & 0 & 0 & -\mathrm{R}_{1\rho}prime-\kBA & -\delta_B & 0 \\
0 & \kAB & 0 & \delta_B & -\mathrm{R}_{1\rho}prime-\kBA & -\omega_1 \\
0 & 0 & \kAB & 0 & \omega_1 & -\Rone-\kBA \\
\end{pmatrix},
\end{equation}
Parameters
The NS R1rho 2-site model has the parameters {$R_{1\rho}'$, $...$, $p_A$, $\Delta\omega$, $k_{ex}$}.
Reference
The reference for the NS R1rho 2-site model is:
- Korzhnev, D. M., Orekhov, V. Y., and Kay, L. E. (2005). Off-resonance R(1rho) NMR studies of exchange dynamics in proteins with low spin-lock fields: an application to a Fyn SH3 domain. J. Am. Chem. Soc., 127(2), 713-721. (10.1021/ja0446855).
Links
The implementation of the NS R1rho 2-site model in relax can be seen in the: