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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 [[Relaxation dispersion citation for relax|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 $\mathrm{R{:R1rho}_{1\rho}$ 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}
\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}
</math>
\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}
</math>
The relaxation evolution matrix is defined as
<math>
\begin{equation}
R = \begin{pmatrix}
-\mathrm{R}_{1\rho}prime{´}-\kAB textrm{k}_\textrm{AB} & -\delta_A & 0 & \kBA textrm{k}_\textrm{BA} & 0 & 0 \\ \delta_A & -\mathrm{R}_{1\rho}prime{´}-\kAB textrm{k}_\textrm{AB} & -\omega_1 & 0 & \kBA textrm{k}_\textrm{BA} & 0 \\ 0 & \omega_1 & -\Ronemathrm{R}_1-\kAB textrm{k}_\textrm{AB} & 0 & 0 & \kBA textrm{k}_\textrm{BA} \\ \kAB textrm{k}_\textrm{AB} & 0 & 0 & -\mathrm{R}_{1\rho}prime{´}-\kBA textrm{k}_\textrm{BA} & -\delta_B & 0 \\ 0 & \kAB textrm{k}_\textrm{AB} & 0 & \delta_B & -\mathrm{R}_{1\rho}prime{´}-\kBA textrm{k}_\textrm{BA} & -\omega_1 \\ 0 & 0 & \kAB textrm{k}_\textrm{AB} & 0 & \omega_1 & -\Ronemathrm{R}_1-\kBA textrm{k}_\textrm{BA} \\
\end{pmatrix},
\end{equation}
</math>
=== Essentials ===
It is essential to read in {{:R1}} values before starting a calculation:
<source lang="python">
relax_data.read(ri_id='R1', ri_type='R1', frq=cdp.spectrometer_frq_list[0], file='R1_values.txt', mol_name_col=1, res_num_col=2, res_name_col=3, spin_num_col=4, spin_name_col=5, data_col=6, error_col=7)
</source>
Where the data could be stored like
<source lang="text">
# mol_name res_num res_name spin_num spin_name value error
None 13 L None N 1.323940 0.146870
None 15 R None N 1.344280 0.140560
None 16 T None N 1.715140 0.136510
</source>
== Parameters ==
The NS R1rho 2-site model has the parameters {$R_{1\rho{:R1rhoprime}}'$, $...$, $p_A${{:pA}}, $\Delta\omega${{:Deltaomega}}, $k_{ex{:kex}}$}.
== 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{{#lst: an application to a Fyn SH3 domain. ''J. Am. Chem. Soc.'', '''127'''(2), 713-721. ([http://dx.doi.org/10.1021/ja0446855 10.1021/ja0446855]).Citations|Korzhnev05b}}
== Links ==
The [[Relaxation dispersion citation for relax|implementation of the NS R1rho 2-site model in relax ]] can be seen in the:* [http://www.nmr-relax.com/manual/NS_2_site_R1_modelThe_NS_2_site_R1_rho_model.html relax manual],
* [http://www.nmr-relax.com/api/3.1/lib.dispersion.ns_r1rho_2site-module.html API documentation],
* [http://www.nmr-relax.com/analyses/relaxation_dispersion.html#NS_R1rho_2-site relaxation dispersion page of the relax website].
== See also ==
[[Category:Relaxation_dispersionModels]][[Category:Dispersion models]][[Category:Relaxation dispersion analysis]]