Characterization of slow conformational dynamics in solids: dipolar CODEX

Abstract

A solid state NMR experiment is introduced for probing relatively slow conformational exchange, based on dephasing and refocusing dipolar couplings. The method is closely related to the previously described Centerband-Only Detection of Exchange or CODEX experiment. The use of dipolar couplings for this application is advantageous because their values are known a priori from molecular structures, and their orientations and reorientations relate in a simple way to molecular geometry and motion. Furthermore the use of dipolar couplings in conjunction with selective isotopic enrichment schemes is consistent with selection for unique sites in complex biopolymers. We used this experiment to probe the correlation time for the motion of ¹³C, ¹⁵N enriched urea molecules within their crystalline lattice.

Appendix

First, we will describe the recoupling of the heteronuclear dipolar coupling under MAS condition. The average Hamiltonian dominant in the evolution during an interval (t ₀, t) is

$$ \overline{H} (t_{0} ,t) = {\frac{{\phi (t_{0} ,t)}}{{t - t_{0} }}}2\vec{I}_{z} \vec{S}_{z} $$

(3)

where ϕ(t ₀, t) is the phase developed by the dipolar coupling during the interval,

$$ \phi (t_{0} ,t) = \int\limits_{{t_{0} }}^{t} {{\frac{1}{\sqrt 6 }}\Uplambda_{20}^{\text{Lab}} (\omega_{R} t){\text{d}}t} . $$

(4)

The laboratory frame tensor $\Uplambda_{20}^{\text{Lab}} (\omega_{R} t)$ can be obtained by transforming the dipolar coupling’s uniaxial tensor from the principal axis frame to the rotor frame through the Euler angles {α, β, γ}, and then to the laboratory frame, which causes a time-dependent due to magic angle spinning. For the REDOR pulse sequence element used during the dephasing and refocusing parts, π pulses invert the sign of the dipolar coupling Hamiltonian for every other half rotor period. Therefore, the phase accumulated in one rotor period (Tr) is

$$ \begin{aligned} \phi_{Tr} = & \int\limits_{{t_{0} }}^{{t_{0} + Tr/2}} {\Uplambda_{20}^{\text{Lab}} (\omega_{R} t){\text{d}}t} - \int\limits_{{t_{0} + Tr/2}}^{{t_{0} + Tr}} {\Uplambda_{20}^{\text{Lab}} (\omega_{R} t){\text{d}}t} \\ = & 2\int\limits_{{t_{0} }}^{{t_{0} + Tr/2}} {\Uplambda_{20}^{\text{Lab}} (\omega_{R} t){\text{d}}t} = 2\phi (t_{0} ,t_{0} + Tr/2) \\ \end{aligned} $$

(5)

where t ₀ defines the initial rotor phase ω _R t ₀. Therefore, after the N rotor periods during the dephasing or refocusing time, the total phase which the magnetization evolves is

$$ \phi = 2 \times N \times \phi (t_{0} ,t_{0} + Tr/2) = 2N \times {\frac{{ - \mu_{0} \hbar \gamma_{i} \gamma_{j} }}{{4\pi r_{ij}^{3} }}} \times {\frac{2\sqrt 2 }{{\omega_{R} }}}{\text{Sin}}2\beta \times {\text{Sin}}(\omega_{R} t_{0} + \gamma ) $$

(6)

which shows that the phase is scaled by a sine function of the Euler angle β. Thus the two exchange sites, which have different Euler angles {α, β, γ} transforming their dipolar coupling tensor from principal to rotor frame, will develop different phases under the dominant of the dipolar coupling Hamiltonian. In the following, the product operator formalism (Sørensen et al. 1983) and an exchange matrix description are applied to illustrate how this phase difference can detect the dynamics of the two exchange sites.

The dynamical model we used is a two-site exchange model for a single ¹³C–¹⁵N pair (Fig. 5b), and we assume that there is no exchange during the dephasing and refocusing parts (although for numerical simulations the exchange during those periods was included). After cross polarization between proton and carbon, the resulting × magnetizations of the two exchange sites evolve under the action of ¹³C–¹⁵N recoupled heteronuclear dipolar coupling during the dephasing time.

$$ \vec{C}_{x}^{1(2)} \mathop{\rightarrow}\limits^{{\text{dipolar}}\;{\text{coupling}}}\vec{C}_{x}^{1(2)} {\text{Cos}}\phi_{1(2)} + 2\vec{C}_{y}^{1(2)} \vec{N}_{z} {\text{Sin}}\phi_{1(2)} . $$

(7)

Then a 90° pulse (phase cycle is indicated in Table 2) on ¹³C flips the antiphase magnetization to generate longitudinal two spin order. (The remaining in-phase part will disappear during mixing time due to spin-spin relaxation and is therefore omitted.) Then, in the mixing time, the exchange process redistributes the magnetization between two exchange sites. After the exchange process,

$$ \begin{aligned} \left( {\begin{array}{*{20}c} {2\vec{C}_{z}^{1} \vec{N}_{z} (t_{\text{m}} )} \\ {2\vec{C}_{z}^{2} \vec{N}_{z} (t_{\text{m}} )} \\ \end{array} } \right) = & {\frac{1}{2}}\left( {\begin{array}{*{20}c} {(e^{{ - k_{\text{ex}} t_{\text{m}} }} + 1)e^{{ - t_{\text{m}} /T_{1} }} } & {1 - e^{{ - k_{\text{ex}} t_{\text{m}} }} } \\ {1 - e^{{ - k_{\text{ex}} t_{\text{m}} }} } & {(e^{{ - k_{\text{ex}} t_{\text{m}} }} + 1)e^{{ - t_{\text{m}} /T_{1} }} } \\ \end{array} } \right) \\ & \times \left( {\begin{array}{*{20}c} {2\vec{C}_{z}^{1} \vec{N}_{z} {\text{Sin}}\phi_{1} } \\ {2\vec{C}_{z}^{2} \vec{N}_{z} {\text{Sin}}\phi_{2} } \\ \end{array} } \right) \\ \end{aligned} $$

(8)

where t _m is the length of the mixing time, k _ex is the exchange rate which is the summation of the forward and backward rates.

Table 2 Phase cycling for dipolar CODEX

Finally, the 90 pulse on carbon flips the carbon’s part back to the y direction and the refocusing part makes the antiphase part evolve back to x direction,

$$ 2\vec{C}_{y}^{1(2)} \vec{N}_{z} (t_{m} )\mathop{\rightarrow}\limits^{\rm dipolar\;coupling} - \vec{C}_{x}^{1(2)} {\rm Sin}\phi_{1(2)} $$

(9)

Therefore, the final signal should be

$$ S \propto {\frac{1}{2}}e^{{ - t_{\text{m}} /T_{1} }} [({\text{Sin}}\phi_{1} - {\text{Sin}}\phi_{2} )^{2} e^{{ - k_{\text{ex}} t_{\text{m}} }} + ({\text{Sin}}\phi_{1} - {\text{Sin}}\phi_{2} )^{2} ]. $$

(10)

The difference between urea model (Fig. 5a) and the single ¹³C–¹⁵N model (Fig. 5b) is that two nitrogen atoms connected with the center carbon atom simultaneously in urea. Therefore, after dephasing time, the carbon’s magnetization in urea is

$$ \begin{aligned} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{C}_{x} \to & (2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{C}_{y} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{N}_{1z} {\text{Sin}}\varphi_{1} {\text{Cos}}\varphi_{2} + 2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{C}_{y} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{N}_{2z} {\text{Cos}}\varphi_{1} {\text{Sin}}\varphi_{2} ) \\ & + (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{C}_{x} {\text{Cos}}\varphi_{1} {\text{Cos}}\varphi_{2} - 4\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{C}_{x} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{N}_{1z} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{N}_{2z} {\text{Sin}}\varphi_{1} {\text{Sin}}\varphi_{2} ) \\ \end{aligned} $$

(11)

where φ₁ and φ₂ are the phases developed under the two ¹³C–¹⁵N dipolar couplings separately. The terms in the first bracket of Eq. (11) are similar to the “SinSin” portion in original CODEX publication (DeAzevedo et al. 2000, 90 _x or 90_−x pulse can flip them to the z-direction) while those terms in the second bracket are similar to “CosCos” portion (90 _y or 90_−y can flip them to the z-direction). The reorientation process which exchanges N₁ and N₂ only affects the “SinSin” portion. Therefore, only “SinSin” portion was obtained in our experiment. The final signal of “SinSin” portion is

$$ S \propto - {\frac{1}{2}}[{\text{Sin}}^{2} (\varphi_{2} + \varphi_{1} ) + {\text{Sin}}^{2} (\varphi_{2} - \varphi_{1} )e^{{ - k_{\text{ex}} t_{\text{m}} }} ]e^{{ - t_{\text{m}} /T_{1} }} . $$

(12)

Equation (12) is a little different from Eq. (10) but the signal in this equation has the same mixing time dependence with that in Eq. (10).