The NS MMQ 3-site linear model

This is the numerical model for 3-site exchange for proton-heteronuclear SQ, ZQ, DQ and MQ CPMG data, as derived in (Korzhnev et al., 2005a,2004b,a). As this model is linear, the assumption that kAC = kCA = 0 has been made. To simplify the optimisation space for the model, the assumption R2A0 = R2B0 = R2C0 = R20 has also been made.

The SQ, ZD and DQ equations

The basic evolution matrices for single, zero and double quantum CPMG-type data for this model are

A± = ea±τCPMG, (11.66)


a± = $\displaystyle \begin{pmatrix}-\textrm{k}_{\textrm{AB}}& \textrm{k}_{\textrm{BA}...
...C}}& -\textrm{k}_{\textrm{CB}}\pm\imath\Delta\omega_{\textrm{AC}} \end{pmatrix}$    
           - $\displaystyle \begin{pmatrix}\mathrm{R}_{\mathrm{2A}}^0& 0 & 0 \\ 0 & \mathrm{R}_{\mathrm{2B}}^0& 0 \\ 0 & 0 & \mathrm{R}_{\mathrm{2C}}^0 \end{pmatrix}$. (11.67)

The MQ equations

The formulae for multiple quantum CPMG-type data are the same as for the `NS MMQ 2-site' model except for the R2eff calculation and the mj matrices. The rate is calculated as

R2eff = - $\displaystyle {\frac{{1}}{{T}}}$log$\displaystyle \left\{\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pma...
..._{\textrm{A}}\\  p_{\textrm{B}}\\  p_{\textrm{C}}\end{pmatrix} \right] }\right.$Re$\displaystyle \left[\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0&0\...
...matrix}p_{\textrm{A}}\\  p_{\textrm{B}}\\  p_{\textrm{C}}\end{pmatrix} }\right.$$\displaystyle {\frac{{0.5}}{{p_{\textrm{A}}}}}$$\displaystyle \begin{pmatrix}1&0&0\end{pmatrix}$$\displaystyle \left(\vphantom{ \mathbf{AB} + \mathbf{CD} }\right.$AB + CD$\displaystyle \left.\vphantom{ \mathbf{AB} + \mathbf{CD} }\right)$$\displaystyle \begin{pmatrix}p_{\textrm{A}}\\  p_{\textrm{B}}\\  p_{\textrm{C}}\end{pmatrix}$$\displaystyle \left.\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0&0\...
...matrix}p_{\textrm{A}}\\  p_{\textrm{B}}\\  p_{\textrm{C}}\end{pmatrix} }\right]$$\displaystyle \left.\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pmat...
...{\textrm{A}}\\  p_{\textrm{B}}\\  p_{\textrm{C}}\end{pmatrix} \right] }\right\}$. (11.68)

The mj matrices are

\begin{subequations}\begin{align}\mathbf{m_1} &= \begin{pmatrix}-\textrm{k}_{\te...
... 0 & 0 & \mathrm{R}_{\mathrm{2C}}^0 \end{pmatrix}. \end{align}\end{subequations}

For the model, the assumption R2A0 = R2B0 = R2C0 = R20 is made.

More information about the NS MMQ 3-site linear model is available from:

The relax user manual (PDF), created 2016-10-28.