# 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 pA > 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 R value for state A magnetisation is defined as

$$$\mathrm{R}_{1\rho} = - \frac{1}{T_\textrm{relax}} \cdot \ln \left( M_0^T \cdot e^{R \cdot T_\textrm{relax}} \cdot M_0 \right),$$$

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

$$$R = \begin{pmatrix} -\mathrm{R}_{1\rho}{´}-\textrm{k}_\textrm{AB} & -\delta_A & 0 & \textrm{k}_\textrm{BA} & 0 & 0 \\ \delta_A & -\mathrm{R}_{1\rho}{´}-\textrm{k}_\textrm{AB} & -\omega_1 & 0 & \textrm{k}_\textrm{BA} & 0 \\ 0 & \omega_1 & -\mathrm{R}_1-\textrm{k}_\textrm{AB} & 0 & 0 & \textrm{k}_\textrm{BA} \\ \textrm{k}_\textrm{AB} & 0 & 0 & -\mathrm{R}_{1\rho}{´}-\textrm{k}_\textrm{BA} & -\delta_B & 0 \\ 0 & \textrm{k}_\textrm{AB} & 0 & \delta_B & -\mathrm{R}_{1\rho}{´}-\textrm{k}_\textrm{BA} & -\omega_1 \\ 0 & 0 & \textrm{k}_\textrm{AB} & 0 & \omega_1 & -\mathrm{R}_1-\textrm{k}_\textrm{BA} \\ \end{pmatrix},$$$

### Essentials

It is essential to read in R1 values before starting a calculation:

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)

Where the data could be stored like

# 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

## Parameters

The NS R1rho 2-site model has the parameters {R', ..., pA, Δω, 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: an application to a Fyn SH3 domain. J. Am. Chem. Soc., 127(2), 713-721. (DOI: 10.1021/ja0446855)