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., 2004b,a,2005a). As this model is linear, the assumption that k_AC = k_CA = 0 has been made. To simplify the optimisation space for the model, the assumption R_2A⁰ = R_2B⁰ = R_2C⁰ = 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_± = e^{a_±⋅τ_CPMG},

(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 R_2eff calculation and the m_j matrices. The rate is calculated as

R_2eff = - $\displaystyle {\frac{{1}}{{T}}}$ log $\displaystyle \left\{\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pma... ...p_{\textrm{A}}\\ p_{\textrm{B}}\\ p_{\textrm{C}}\end{pmatrix}\right] }\right.$ Re $\displaystyle \left[\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0&0\... ...pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\\ p_{\textrm{C}}\end{pmatrix}}\right.$ $\displaystyle {\frac{{0.5}}{{p_{\textrm{A}}}}}$ $\displaystyle \begin{pmatrix}1&0&0\end{pmatrix}$ ⋅ $\displaystyle \left(\vphantom{ \mathbf{AB} + \mathbf{CD} }\right.$ AB + CD $\displaystyle \left.\vphantom{ \mathbf{AB} + \mathbf{CD} }\right)$ ⋅ $\displaystyle \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\\ p_{\textrm{C}}\end{pmatrix}$ $\displaystyle \left.\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0&0\... ...pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\\ p_{\textrm{C}}\end{pmatrix}}\right]$ $\displaystyle \left.\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pmat... ..._{\textrm{A}}\\ p_{\textrm{B}}\\ p_{\textrm{C}}\end{pmatrix}\right] }\right\}$ .

(11.68)