CHHC and ¹H-¹H Magnetization Exchange: Analysis by Experimental Solid-State NMR and 11-Spin Density-Matrix Simulations

Abstract

A protocol is presented for correcting the effect of non-specific cross polarization in CHHC solid-state MAS NMR experiments, thus allowing the recovery of the ¹H-¹H magnetization exchange functions from the mixing-time dependent buildup of experimental CHHC peak intensity. The presented protocol also incorporates a scaling procedure to take into account the effect of multiplicity of a CH₂ or CH₃ moiety. Experimental CHHC buildup curves are presented for L-Tyrosine.HCl samples where either all or only one in ten molecules are U-¹³C labeled. Good agreement between experiment and 11-spin SPINEVOLUTION simulation (including only isotropic ¹H chemical shifts) is demonstrated for the initial buildup (t_mix < 100 μs) of CHHC peak intensity corresponding to an intramolecular close (2.5 Å) H-H proximity. Differences in the initial CHHC buildup are observed between the 1 in 10 dilute and 100 % samples for cases where there is a close intermolecular H-H proximity in addition to a close intramolecular H-H proximity. For the dilute sample, CHHC cross peak intensities tended to significantly lower values for long mixing times (500 μs) as compared to the 100 % sample. This difference is explained as being due to the dependence of the limiting total magnetization on the ratio N_obs/N_tot between the number of protons that are directly attached to a ¹³C nucleus and hence contribute significantly to the observed ¹³C CHHC NMR signal, and the total number of ¹H spins into the system. ¹H-¹H magnetization exchange curves extracted from CHHC spectra for the 100 % L-Tyrosine.HCl sample exhibit a clear sensitivity to the root sum squared dipolar coupling, with fast build-up being observed for the shortest intramolecular distances (2.5 Å) and slower, yet observable build-up for the longer intermolecular distances (up to 5 Å).

1. Introduction

Solid-state nuclear magnetic resonance increasingly develops as an attractive tool for investigating complex molecular systems with practical relevance in biology, chemistry and materials science. For instance, structural investigations based on identifying protein intra-residue or neighboring residue ¹³C-¹³C (¹³C-¹⁵N) connectivities, distances, and angles are now almost routinely available from ¹³C(¹⁵N) solid-state NMR experiments optimized to work on multiply labeled samples [1–12], but constraints useful to elucidate the 3D structure of proteins [13], or to characterize supramolecular aggregates and crystal packing, are more difficult to obtain. A promising strategy for this purpose is to use the CHHC experiment [14–17], which makes use of the concept that such constraints are easier to extract using ¹H-¹H magnetization exchange, because protons are closer spaced in regions of interest (folding, or intermolecular contacts), and also more strongly dipolar coupled to each other than low γ nuclei.

¹H-¹H magnetization exchange can be directly probed in a NOESY-type ¹H-¹H spin-diffusion two-dimensional correlation experiment [18]. This has been shown for cases of small and moderately sized organic molecules, where sufficient ¹H resolution is obtained using fast MAS or homonuclear ¹H decoupling [19–24]. In a CHHC experiment (see pulse sequence in Figure 1), ¹H-¹H magnetization exchange is probed indirectly, taking advantage of the better resolution for the X nucleus. NHHC [17], NHHN [25] and PHHP [26] implementations as well as extensions to 3D C(DQ)C(SQ)HHC and NHHCC experiments [10] have also been demonstrated. CHHC, NHHC and NHHN experiments are widely applied to biological systems such as microcrystalline proteins [10, 17, 27–32], fibrils [33–37], aggregates [38, 39], chlorophylls [40–43], ion channels [44–46], RNA [47,48] and an anti-cancer agent [49], where cross peaks observed in spectra recorded with mixing times of typically at least 100 μs are used as distance constraints for structure determination protocols.

Fig. 1.

The CHHC pulse sequence employed in the present work, where, in addition to previous implementations, a 60 kHz continuous-wave decoupling field is applied on the ¹³C channel, in order to reduce the negative influence of the ¹³C-¹H dipolar interaction upon the efficiency of ¹H-¹H magnetization exchange during the mixing time.

The reliability of ¹H-¹H distance constraints determined from observed cross peaks in CHHC-type experiments has been demonstrated for small model compounds, e.g., amino acids [10, 15, 17, 25, 28, 50, 51] as well as recently the microcrystalline CrH protein [30], where ¹H-¹H distances are known from single-crystal diffraction data. Specifically, faster experimental buildup (as a function of the mixing time, t_mix) of CHHC peak intensity is observed for shorter ¹H-¹H distances. Experimental CHHC buildup data has been analysed using a classical spin-diffusion model [15, 16], with the extracted spin diffusion coefficients allowing order-of-magnitude estimates of ¹H-¹H distances. Reif et al. [27] and Lange et al. [50] have shown that a good fit to experimental data is obtained using analytical expressions based on n = 0 rotational resonance [52] and spectral spin diffusion [53, 54], respectively. These expressions have a squared dependence on the dipolar coupling constant (and hence the internuclear distance) as well as a fitted phenomenological zero-quantum dephasing term, with the latter depending on the MAS frequency [50]. Elena et al. have presented a related treatment of direct ¹H-¹H magnetization exchange using a multi-spin kinetic rate matrix approach that considers a sum of all relevant magnetization processes between sites i and j in different molecules in the crystal lattice [22, 23].

Advances in computing hardware and density-matrix simulation methodologies [55, 56] mean that the spin dynamics due to 10+ dipolar-coupled nuclei can be simulated. For example, the dephasing in ¹³C free-induction decays and spin-echo experiments under rotor-synchronised Hahn-echo pulse trains have recently been simulated for 10 coupled spins [57, 58]. In this paper, ¹H-¹H magnetization exchange is simulated for 11-spin systems corresponding to specific proton-proton proximities in L-tyrosine.HCl, for which experimental CHHC buildup data is presented. Specifically, a protocol is introduced to correct for non-specific cross-polarization (CP) and take into account XH_n multiplicity, thus allowing a direct comparison between experimental and simulated data. The effect of inter- and intramolecular ¹H-¹H proximities on ¹H-¹H magnetization exchange is investigated using two L-tyrosine.HCl samples where either all or only one in ten molecules are U-¹³C labeled. For the all U-¹³C sample, differences in the rate of buildup of CHHC peak intensity are explained by considering the root sum squared dipolar couplings.

2. Experimental details

2.1. Sample preparation

U (98 %)-¹³C labeled L-Tyrosine was obtained from Cambridge Isotope Laboratories (Andover, MA, USA). Conversion to the L-Tyrosine·HCl salt was achieved by dissolving L-tyrosine in 1M HCl, followed by freeze-drying using a vacuum-pump. Two samples were used in this study: U-¹³C L-Tyrosine·HCl refers to the sample as prepared above. A second sample was prepared by the recrystallization of the U-¹³C labeled L-Tyrosine with natural abundant L-Tyrosine in 1M HCl, so as to yield a U-¹³C^dil_10% L-Tyrosine·HCl sample, i.e., one in ten L-Tyrosine·HCl molecules are fully ¹³C labeled.

2.2 NMR experiments

Experiments were performed at room temperature on a Bruker AVANCE-400 spectrometer operating at a ¹³C Larmor frequency of 100 MHz, at an MAS frequency of 10.5 kHz using a Bruker 4 mm double-resonance probe.

In all experiments, CP transfer was optimized for the first Hartmann-Hahn matching condition (ν_1C = ν_1H − ν_R), using ¹H and ¹³C rf nutation frequencies of 51 and 40 kHz, respectively. For π/2 ¹H rf pulses, a π/2 pulse length of 3.8 μs was used. During ¹³C evolution periods, two-pulse phase-modulated (TPPM) ¹H decoupling [59] was applied at a nutation frequency of 70 kHz (Δϕ = 15° and pulse width of 7.4 μs). A recycle delay of 3 s was used.

The CHHC experiments were performed by using the pulse sequence depicted in Fig. 1, where the unwanted proton polarization left after the first CP contact pulse is removed by phase-cycling the second π/2 pulse on the ¹H channel, as introduced in ref. [40]. A contact pulse of 700 μs was used for the first CP, whereas a much shorter contact pulse- (65 μs) was employed for the next two CP steps in order to favorize polarization transfer between bonded ¹³C-¹H in CH and CH₂ moieties. A 60 kHz continuous wave decoupling field was also applied on the ¹³C channel during the mixing time to reduce the negative effect of the ¹³C-¹H dipolar interaction upon the efficiency of ¹H-¹H polarization transfer. 256 (U-¹³C^dil_10% sample) and 32 (U-¹³C sample) transients were coadded for each of 256 t₁ increments of 30μs, corresponding to a F₁ spectral width of 16.6 kHz which was chosen such as to fit only the six protonated ¹³C resonances of interest. Sign discrimination in t₁ was achieved using the TPPI method. Total acquisition times for each 2D CHHC experiment were 54 hours and 7 hours for the U-¹³C^dil_10% and U-¹³C samples, respectively. The signal to noise ratio in the first row of the CHHC experiment for t_mix = 0 was better than 75:1 (U-¹³C^dil_10%) and 120:1 (U-¹³C samples) for all centreband protonated ¹³C resonances.

2.3 Density matrix simulations

Density-matrix simulations of ¹H-¹H magnetization exchange during the t_mix period of a CHHC experiment were performed using the SPINEVOLUTION program [55] for 10 kHz MAS and a ¹H Larmor frequency of 400 MHz. 11-spin systems based upon the proton coordinates as extracted from the crystal structure of L-Tyrosine·HCl [60] were considered, using experimental ¹H isotropic chemical shift values (see the representative SPINEVOLUTION input files in the Appendix). It was verified that the number of crystallite orientations used was sufficient to ensure convergence. ¹H CSAs were neglected – in separate simulations, it was found that small changes to the magnetization exchange curves only started to occur for CSA anisotropies in excess of 40 ppm (for L alanine, the largest calculated ¹H CSA has an anisotropy of 17 ppm [58]). Each simulation directly provides the magnetization-exchange curves that correspond to the transfer from an initially polarized ¹H site, to the all chemically distinct proton sites in the system, and took approximately 120 hours on an Opteron Linux workstation.

3. Theory

3.1 Magnetization exchange

The transfer of z magnetization between two dipolar coupled ¹H nuclei j and k during a mixing time, t_mix, under the spin-diffusion operator Û is described here by a polarization transfer function, F_jk (t_mix):

$$F_{jk}(t_{\mathit{mix}})=\langle I_z^k\mid \widehat{U}(t_{\mathit{mix}})\mid I_z^j\rangle$$ (1)

In a CHHC experiment, a general ¹³C-¹³C SQ-SQ correlation is established between a ¹³C^LH_p and a ¹³C^MH_q resonance, via ¹H-¹H magnetization exchange between p ¹H nuclei attached to ¹³C^L and q ¹H nuclei attached to ¹³C^M. In such a case, it is convenient to define modified polarization transfer functions for the case of unlike and like spins, i.e., corresponding to off-diagonal and diagonal peaks in a 2D CHHC spectrum:

$$F_{LM}^{\prime }(t_{\mathit{mix}})=\frac{1}{pq}\langle \sum _{k=1}^q{I}_{kz}^M\mid \widehat{U}(t_{\mathit{mix}})\mid \sum _{j=1}^p{I}_{jz}^L\rangle$$ (2)

$$F_{LL}^{\prime }(t_{\mathit{mix}})=\frac{1}{p}\langle \sum _{j=1}^p{I}_{jz}^L\mid \widehat{U}(t_{\mathit{mix}})\mid \sum _{j=1}^p{I}_{jz}^L\rangle$$ (3)

The modified polarization transfer functions correspond to ¹H-¹H magnetization exchange starting from unit polarization on a given ¹³C^LH_p group. For t_mix = 0, $F_{LL}^{\prime }(0)=1$ and $F_{LM}^{\prime }(0)=0$.

In this paper, a normalized polarization transfer function is employed according to the definition:

$$F_{LM}^n(t_{\mathit{mix}})=\frac{F_{LM}^{\prime }(t_{\mathit{mix}})}{F_{LL}^{\prime }(0)}$$ (4)

3.2. Compensating heteronuclear effects in CHHC experiments

The indirect observation of ¹H-¹H magnetization exchange in a 2D CHHC experiment benefits from the considerably better ¹³C as opposed to ¹H resolution. However, the two CP steps flanking the ¹H-¹H magnetization exchange period in the CHHC sequence are usually not fully specific even for the very short CP durations typically used (< 100 μs), i.e., magnetization is not transferred exclusively from a given ¹³C nucleus to only its directly attached ¹H nucleus or nuclei, but rather magnetization “leaks out” onto other ¹H nuclei. As a consequence, the intensity of an experimental CHHC cross peak linking a ¹³C^LH_p and a ¹³C^MH_q resonance, $I_{LM}^{ex}(t_{\mathit{mix}})$, does not correspond precisely to the ¹H-¹H polarization transfer functions described above. This section describes a procedure for recovering the ¹H-¹H polarization transfer functions from the experimental CHHC cross-peak intensities.

The intensity of an experimental CHHC cross peak linking a ¹³C^LH_p and a ¹³C^MH_q resonance, $I_{LM}^{ex}(t_{\mathit{mix}})$, corresponds to:

$$I_{LM}^{ex}(t_{\mathit{mix}})=a_{CP1}^L\langle S_{xy}^M\mid {\widehat{U}}_{IS}({\tau }_{CP})\widehat{U}(t_{\mathit{mix}}){\widehat{U}}_{SI}({\tau }_{CP})\mid S_{xy}^L\rangle$$ (5)

Note that S and I refer to ¹³C and ¹H nuclei, respectively. The $a_{CP1}^L$ coefficient takes into account the variation of initial (i.e., at the start of t₁) ¹³C transverse magnetization for different ¹³C resonances arising from the first CP step. The _xy subscript indicates spin S transverse magnetization present at the end of t₁ or the start of t₂. The CP propagators in Eq. (5) are considered to include the contact pulse applied to both spin species and the ¹H 90° pulse that creates I_z (¹H population state) at the start of t_mix or converts I_z at the end of t_mix into ¹H transverse magnetization before the final CP step, i.e., Û_SI (τ_CP) converts S_xy into I_z, while Û_IS (τ_CP) converts I_z into S_xy. The CP dynamics can be described in terms of the transferred polarization:

$${\widehat{U}}_{SI}({\tau }_{CP})\mid S_{xy}^L\rangle \to \frac{{\eta }_L}{p}\mid \sum _{j=1}^p{I}_{jz}^L\rangle +\frac{{\epsilon }_{LM}}{q}\mid \sum _{k=1}^q{I}_{kz}^M\rangle$$ (6)

where the coefficient η corresponds to the amount of magnetization transferred to proton(s) directly attached to the initial ¹³C resonance, while the coefficient ε corresponds to the amount of magnetization transferred to proton(s) attached to a different ¹³C resonance. Similarly,

$$\langle S_{xy}^M\mid {\widehat{U}}_{IS}({\tau }_{CP})\to \langle \sum _{k=1}^q{I}_{kz}^M\mid \frac{{\eta }_M}{q}+\langle \sum _{j=1}^p{I}_{jz}^L\mid \frac{{\epsilon }_{ML}}{p}$$ (7)

It thus follows that

$$\begin{array}{l}{I}_{LM}^{ex}(t_{\mathit{mix}})=a_{CP1}^L{\eta }_L{\eta }_M\frac{\langle {\displaystyle \sum _{k=1}^q}{I}_{kz}^M\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\rangle }{pq}+\frac{a_{CP1}^L{\eta }_M{\epsilon }_{LM}}{q}\frac{\langle {\displaystyle \sum _{k=1}^q}{I}_{kz}^M\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{k=1}^q}{I}_{kz}^M\rangle }{q}\\ +\frac{a_{CP1}^L{\eta }_L{\epsilon }_{ML}}{p}\frac{\langle {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\rangle }{p}\end{array}$$ (8)

where the term in ε_LMε_ML has been neglected.

The intensity of an experimental CHHC diagonal peak for a ¹³C^LH_p resonance, $I_{LL}^{ex}(t_{\mathit{mix}})$, corresponds to:

$$I_{LL}^{ex}(t_{\mathit{mix}})=a_{CP1}^L\langle S_{xy}^L\mid {\widehat{U}}_{IS}({\tau }_{CP})\widehat{U}(t_{\mathit{mix}}){\widehat{U}}_{SI}({\tau }_{CP})\mid S_{xy}^L\rangle$$ (9)

where

$$\langle S_{xy}^L\mid {\widehat{U}}_{IS}({\tau }_{CP})\to \langle \sum _{j=1}^p{I}_{jz}^L\mid \frac{{\eta }_L}{p}+\langle \sum _{k=1}^q{I}_{kz}^M\mid \frac{{\epsilon }_{LM}}{q}$$ (10)

i.e.,

$$\begin{array}{l}{I}_{LL}^{ex}(t_{\mathit{mix}})=\frac{a_{CP1}^L{{\eta }_L}^2}{p}\frac{\langle {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\rangle }{p}+a_{CP1}^L{\eta }_L{\epsilon }_{LM}\frac{\langle {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{k=1}^q}{I}_{kz}^M\rangle }{pq}\\ +a_{CP1}^L{\eta }_L{\epsilon }_{LM}\frac{\langle {\displaystyle \sum _{k=1}^q}{I}_{kz}^M\mid \widehat{U}(t_{\mathit{mix}})\mid {\displaystyle \sum _{j=1}^p}{I}_{jz}^L\rangle }{pq}\end{array}$$ (11)

where the term in ε_LM² has been neglected.

Using Eqs. (2) and (3), Eqs. (8) and (11) become:

$$I_{LM}^{ex}(t_{\mathit{mix}})=a_{CP1}^L\left[{\eta }_L{\eta }_M{F}_{LM}^{\prime }(t_{\mathit{mix}})+\frac{{\eta }_M{\epsilon }_{LM}}{q}{F}_{MM}^{\prime }(t_{\mathit{mix}})+\frac{{\eta }_L{\epsilon }_{ML}}{p}{F}_{LL}^{\prime }(t_{\mathit{mix}})\right]$$ (12)

$$I_{LL}^{ex}(t_{\mathit{mix}})=a_{CP1}^L\left[\frac{{{\eta }_L}^2}{p}{F}_{LL}^{\prime }(t_{\mathit{mix}})+{\eta }_L{\epsilon }_{LM}{F}_{ML}^{\prime }(t_{\mathit{mix}})+{\eta }_L{\epsilon }_{LM}{F}_{LM}^{\prime }(t_{\mathit{mix}})\right]$$ (13)

For a short t_mix, $F_{ML}^{\prime }(t_{\mathit{mix}})$ and $F_{LM}^{\prime }(t_{\mathit{mix}})$ are both small, while for a short τ_CP, ε_LM and ε_ML are also small. For such conditions, Eq. (13) simplifies to

$$I_{LL}^{ex}(t_{\mathit{mix}})=\frac{a_{CP1}^L{{\eta }_L}^2}{p}{F}_{LL}^{\prime }(t_{\mathit{mix}})$$ (14)

i.e.,

$$F_{LL}^{\prime }(t_{\mathit{mix}})=\frac{p}{a_{CP1}^L{{\eta }_L}^2}{I}_{LL}^{ex}(t_{\mathit{mix}})$$ (15)

Eq. (12) can, then, be reexpressed as

$$\begin{array}{l}{I}_{LM}^{ex}(t_{\mathit{mix}})=a_{CP1}^L{\eta }_L{\eta }_M{F}_{LM}^{\prime }(t_{\mathit{mix}})+\frac{{\eta }_M{\epsilon }_{LM}}{q}\frac{q}{{{\eta }_M}^2}{I}_{MM}^{ex}(t_{\mathit{mix}})+\frac{{\eta }_L{\epsilon }_{ML}}{p}\frac{p}{{{\eta }_L}^2}{I}_{LL}^{ex}(t_{\mathit{mix}})\\ =a_{CP1}^L{\eta }_L{\eta }_M{F}_{LM}^{\prime }(t_{\mathit{mix}})+\frac{{\epsilon }_{LM}}{{\eta }_M}{I}_{MM}^{ex}(t_{\mathit{mix}})+\frac{{\epsilon }_{ML}}{{\eta }_L}{I}_{LL}^{ex}(t_{\mathit{mix}})\end{array}$$ (16)

By analogy

$$I_{ML}^{ex}(t_{\mathit{mix}})=a_{CP1}^M{\eta }_L{\eta }_M{F}_{ML}^{\prime }(t_{\mathit{mix}})+\frac{{\epsilon }_{ML}}{{\eta }_L}{I}_{LL}^{ex}(t_{\mathit{mix}})+\frac{{\epsilon }_{LM}}{{\eta }_M}{I}_{MM}^{ex}(t_{\mathit{mix}})$$ (17)

The differences between Eqs. (16) and (17) explain why L to M and M to L cross peaks in CHHC experiments can exhibit different intensities for non-zero t_mix.

Rearranging Eq. (16),

$$F_{LM}^{\prime }(t_{\mathit{mix}})=\frac{1}{a_{CP1}^L{\eta }_L{\eta }_M}\left[I_{LM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{LM}}{{\eta }_M}{I}_{MM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{ML}}{{\eta }_L}{I}_{LL}^{ex}(t_{\mathit{mix}})\right]$$ (18)

Using Eqs. (15) and (18), the normalized polarization transfer function defined in Eq. (4) is given as:

$$F_{LM}^n(t_{\mathit{mix}})=\frac{F_{LM}^{\prime }(t_{\mathit{mix}})}{F_{LL}^{\prime }(0)}=f_{LM}\left(\frac{I_{LM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{LM}}{{\eta }_M}{I}_{MM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{ML}}{{\eta }_L}{I}_{LL}^{ex}(t_{\mathit{mix}})}{I_{LL}^{ex}(0)}\right)$$ (19)

where

$$f_{LM}=\frac{{\eta }_L}{p{\eta }_M}$$ (20)

Eq. (19) defines how the normalized polarization transfer functions – corresponding to only ¹H-¹H magnetization exchange, i.e., with no distorting effect from non-specific CP–can be extracted from the experimental CHHC cross peak intensities.

The following describes how the coefficients in Eq. (19) can be experimentally determined. Using the above theoretical model, it can be shown that the ratio β_L of the intensity of a specific ¹³C resonance recorded in a CHHC filtered spectrum (obtained by using the CHHC pulse sequence with t₁ and t_mix set to zero), $a_{\mathit{CHHC}}^L$, and that acquired after the first CP step, $a_{CP1}^L$, can be related to the corresponding CP transfer parameter, η_L, and proton multiplicity, through:

$${\beta }_L=\frac{a_{\mathit{CHHC}}^L}{a_{CP1}^L}=\frac{{\eta }_L^2}{p}$$ (21)

It then follows that

$${\eta }_L=\sqrt{p{\beta }_L}$$ (22)

By inserting this in eq. (20) the following expression is obtained for the f_LM coefficient

$$f_{LM}=\frac{1}{\sqrt{pq}}\sqrt{\frac{{\beta }_L}{{\beta }_M}}$$ (23)

which provides the desired dependence only on the experimentally measured parameters β_L(M).

For t_mix = 0, $F_{LM}^{\prime }(t_{\mathit{mix}})=F_{ML}^{\prime }(t_{\mathit{mix}})=0$, and hence Eq. (16) becomes

$$I_{LM}^{ex}(0)=I_{ML}^{ex}(0)=\frac{{\epsilon }_{ML}}{{\eta }_L}{I}_{LL}^{ex}(0)+\frac{{\epsilon }_{LM}}{{\eta }_M}{I}_{MM}^{ex}(0)$$ (24)

Within the approximation that ε = ε_LM = ε_ML ε can be determined:

$${\epsilon }_{LM}=\frac{I_{LM}^{ex}(0)}{\left[\frac{I_{LL}^{ex}(0)}{{\eta }_L}+\frac{I_{MM}^{ex}(0)}{{\eta }_M}\right]}$$ (25)

where η_L and η_M are determined using Eq. (22).

The above calculation is for the case where all carbon nuclei are ¹³C, i.e., where all molecules are U-¹³C labelled. For a dilute sample, which corresponds to the case where only a proportion ρ of the molecules are U-¹³C labeled, it is necessary to correct for the contribution to the diagonal peak intensity of the natural abundance ¹³C nuclei (denoted here as ξ, with ξ= 0.011 for ¹³C) in the proportion (1 − ρ) of the molecules at natural abundance. This correction is required since only ξ² molecules will have two neighboring ¹³C-labeled atoms, so as to give rise to cross peak intensity $I_{LM}^{ex}(0)$, as compared to the ξ molecules that have a single ¹³C-labeled atom that contributes to the diagonal peak intensity. Considering the diagonal peak intensity, $I_{LL}^{ex}(0)$, there is a contribution ξ (1 − ρ) from molecules at natural abundance in addition to the ρ from the U-¹³C labeled molecules. It is hence necessary to modify Eq. (25)

$${\epsilon }_{LM}=\frac{I_{LM}^{ex}(0)}{\left[\frac{\lambda I_{LL}^{ex}(0)}{{\eta }_L}+\frac{\lambda I_{MM}^{ex}(0)}{{\eta }_M}\right]}$$ (26)

where

$$\lambda =\frac{\rho }{\rho +\xi (1-\rho )}$$ (27)

i.e., for ¹³C, λ = 0.91 when ρ = 0.1. Eq. (26) does not include a correction to the cross peak intensity $I_{LM}^{ex}(0)$ which is given by ρ/[ρ + ξ²(1 − ρ)], since this equals 1.00 to two decimal places for ¹³C when ρ = 0.1.

For a dilute sample, it is also necessary to modify Eq. (19):

$$F_{LM}^n(t_{\mathit{mix}})=\frac{F_{LM}^{\prime }(t_{\mathit{mix}})}{F_{LL}^{\prime }(0)}=f_{LM}\left(\frac{I_{LM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{LM}}{{\eta }_M}\lambda I_{MM}^{ex}(t_{\mathit{mix}})-\frac{{\epsilon }_{ML}}{{\eta }_L}\lambda I_{LL}^{ex}(t_{\mathit{mix}})}{\lambda I_{LL}^{ex}(0)}\right)$$ (28)

4. Results and Discussion

4.1 One-dimensional ¹³C CP MAS and CHHC-filtered spectra

Figure 2 compares ¹³C CP MAS (thin line) and ¹³C CHHC-filtered (t₁ = t_mix = 0, τ_CP = 65 μs for the last two CP steps) spectra (thick line) for (a) U-¹³C and (b) U-¹³C^dil_10% L-Tyrosine·HCl. The η_L CP transfer coefficients as determined from a comparison of the signal intensity in the CP MAS and CHHC-filtered experiments (see Eq. (22)) are listed in Table 1.

Fig. 2.

A comparison of ¹³C CP MAS (thin line) and ¹³C CHHC-filtered (t₁ = t_mix = 0, τ_CP = 65 μs for the last two CP steps) spectra (thick line) for (a) U-¹³C and (b) U-¹³C^dil_10% L-Tyrosine·HCl. The contact pulse for the first CP step in the CHHC experiment (and the only CP step in the CP MAS experiment) was of duration 700 μs. For both experiments, 32(U-¹³C) and 256 (U-¹³C^dil_10%) transients were co-added for a recycle delay of 3 s.

Table 1.

The experimental η_L parameters evaluated using Eq. (22) for the specified ¹³C^L sites in L-Tyrosine·HCl.

	ηL (U-13C)	ηL (U-13C10%_dil)
C7	0.66 ±0.01	0.63 ±0.03
C8	0.52 ±0.01	0.50 ±0.02
C2	0.51 ±0.01	0.51 ±0.03
C3	0.52 ±0.01	0.51 ±0.02
C5	0.55 ±0.02	0.54 ±0.03
C4	0.53 ±0.01	0.52 ±0.04

4.2 Two-dimensional CHHC spectra

Fig. 3 presents 2D CHHC (τ_CP = 65 μs for the last two CP steps) spectra of (a,b) U-¹³C and (c,d) U-¹³C^dil_10% L-Tyrosine·HCl recorded with t_mix equal to (a,c) 0 and (b,d) 100 μs. Fig. 4 presents rows through the F₁ resonances corresponding to the six protonated ¹³C nuclei, as extracted from the CHHC spectra at zero mixing time. It is evident that non specific CP during the last two CP steps of the CHHC experiment gives rise to noticeable CHHC cross peaks between directly bonded ¹³C nuclei (i.e., 2 & 4, 3 & 5, and 7 & 8), even for the case of zero mixing time. This is also evident in Fig. 5(a) and (b) which presents the buildup of C7 (^βCH₂) to C8 (^αCH) (open symbols) and C8 to C7 (filled symbols) cross-peak intensity as a function of t_mix for (a) U-¹³C and (b) U-¹³C^dil_10% L-Tyrosine·HCl. Specifically, the cross-peak intensities are normalized with respect to the intensity of the corresponding diagonal peak for zero mixing time, i.e., $I_{LM}^{ex}(t_{\mathit{mix}})/I_{LL}^{ex}(0)$ such that the C7 to C8 and C8 to C7 cross-peak intensities are divided by the intensity of the C7 and C8 diagonal peak intensity for zero mixing time, respectively. In the case of the U-¹³C^dil_10% sample, a scaling by the factor λ as defined in Eq. (27) of 0.91 was applied to $I_{LL}^{ex}(0)$.

Fig. 3.

2D CHHC spectra of (a,b) U-¹³C and (c,d) U-¹³C^dil_10% L-Tyrosine·HCl recorded with t_mix equal to (a,c) 0 and (b,d) 100 μs. The base contour level is at 3 % and 5% for the two different mixing times.

Fig. 4.

Rows extracted from the 2D CHHC spectra (see Fig. 3(a) and (c)) of (a) U-¹³C and (b) U-¹³C^dil_10% L-Tyrosine·HCl recorded with t_mix equal to 0 μs.

Fig. 5.

The buildup of C7 (^βCH₂) to C8 (^αCH) (open symbols) and C8 to C7 (filled symbols) cross-peak intensity as a function of t_mix for (a,c) U-¹³C and (b,d) U-¹³C^dil_10% L-Tyrosine·HCl. In (a) and (b), the cross-peak intensities are normalized with respect to the intensity of the corresponding diagonal peak for zero mixing time, i.e., $I_{LM}^{ex}(t_{\mathit{mix}})/I_{LL}^{ex}(0)$. In the case of the U-¹³C^dil_10% sample, a scaling by the factor λ as defined in Eq. (27) of 0.91 was applied to $I_{LL}^{ex}(0)$. (c) and (d): corrected normalized buildup plots given by $F_{LM}^n(t_{\mathit{mix}})$ as defined in Eqs. (19) & (28) of Section 3.2. Tables 1 and 2 list the η_L and ε_LM coefficients.

Section 3.2 (see Eqs. (19) & (28)) describes a procedure for recovering the ¹H-¹H magnetization exchange behavior given by $F_{LM}^n(t_{\mathit{mix}})$ from the experimental CHHC buildup curves. Fig. 5(c) and (d) presents such corrected normalized buildup plots for the C7 to C8 and C8 to C7 cross peaks for (c) U-¹³C and (d) U-¹³C^dil_10% L-Tyrosine·HCl. In the evaluation of the ε_LM coefficients (see Eqs. (25) & (26)), the coefficient was set to zero when $I_{LM}^{ex}(t_{\mathit{mix}})/I_{LL}^{ex}(0)<0.01$, i.e., when a particular cross-peak was less than 1 % of the intensity of the corresponding diagonal peak. The evaluated non-zero ε_LM coefficients for the different CHHC cross peaks are listed in Table 2. Considering Fig. 5(c) and (d), it is evident that all corrected curves now start at zero. As shown below, this allows for a clear comparison with the ¹H-¹H magnetization exchange build-up behavior obtained from numerical density matrix simulations. The corrected experimental C7 to C8 and C8 to C7 transfer functions are observed to be the same within (or close to within) the experimental error bars. In the following, experimental $F_{L,M}^n(t_{\mathit{mix}})$ values for a C^L, C^M pair are shown as an average of $F_{LM}^n(t_{\mathit{mix}})$ and $F_{ML}^n(t_{\mathit{mix}})$.

Table 2.

The ε_LM coefficients as evaluated using Eq. (25) and Eq. (26) for the U-¹³C labeled (top entry), and U-¹³C^10%_dil (bottom entry) L-Tyrosine·HCl samples, respectively.

	C7	C8	C2	C3	C5	C4
C7	-	0.016 ±0.002 0.014 ±0.005	0	0	0	0
C8	0.018 ±0.003 0.015 ±0.005	-	0	0	0	0
C2	0	0	-	0	0	0.012 ±0.002 0.011 ±0.003
C3	0	0	0	-	0.014 ±0.003 0.014 ±0.005	0
C5	0	0	0	0.017 ±0.003 0.013 ±0.004	-	0
C4	0	0	0.016 ±0.002 0.014 ±0.004	0	0	-

4.3 Comparison of experimental and simulated CHHC build-up curves

Corrected normalized buildup plots, i.e., $F_{L,M}^n(t_{\mathit{mix}})$, for U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl are shown in Fig. 6 for the (a) C7-C8 and (b) C2-C4 CHHC cross peaks. The experimental data is compared to a SPINEVOLUTION simulation (solid line) of the $F_{LM}^n(t_{\mathit{mix}})$ ¹H-¹H magnetization function defined in Eq. (4) for 11-spin systems centered around the H7 (two protons) & H8 and H2 & H4 ¹H nuclei, as shown in Fig. 6(c) and (d). Inter-proton distances for the 11-spin systems are given in Tables 3 and 4.

Fig. 6.

Comparison between the corrected normalized experimental CHHC buildup curves, i.e., $F_{L,M}^n(t_{\mathit{mix}})$ as defined in Eqs. (19) & (28), for U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl for (a) C7-C8 and (b) C2-C4. The solid lines in (a) and (b) correspond to SPINEVOLUTION density-matrix simulations of $F_{LM}^n(t_{\mathit{mix}})$ (see Eq. (4)) for 11 spin systems centered around H7, H8 and H2, H4 ¹H nuclei as illustrated in (c) and (d), respectively.

Table 3.

Inter-proton distances for the 11-spin system used in the SPINEVOLUTION simulations of the H7,H8 magnetization exchange

	H2	H4	H5	NHa	NHb	NHc	H7a	H7b	H8
H7a	3.75(C)	2.32(A)	3.78(A) 2.93(B)	3.65(A)	2.63(A)	2.85(A)		1.74(A)	3.06(A) 2.93(B)
H7b	2.31(C)	3.5(A)	2.73(A) 2.8(B)	3.51(A)	3.11(A)	2.32(A)	1.74(A)		2.5(A) 3.4(B)
H8	4.14(C)	4.45(A)	2.55(A) 5.25(B)	3.31(A)	2.96(A)	2.49(A)	3.06(A) 2.93(B)	2.5(A) 3.4(B)	5.1(B)

^a(A), (B) and (C) refer to the labeling of the molecules in Fig. 6(c)

^bIntramolecular distances (within molecule A) are shown in bold

^cThe closest distances to the next nearest eight ¹H nuclei external to the central H7 and H8 nuclei are underlined

^dOnly intramolecular H7-H8 polarization transfer was considered in the simulations, i.e., initial density matrix corresponds to I_z for H7a and H7b, with the read out only on H8 of molecule A. See the representative SPINEVOLUTION input files in the Appendix.

Table 4.

Inter-proton distances for the 11-spin system used in the SPINEVOLUTION simulations of the H2,H4 magnetization exchange

	H1	H2	H3	H4	H5	H7a	H7b
H2	2.35(A)		4.06(B) 4.30(C) 2.95(D)	2.48(A)	2.76(C) 3.18(D)	4.65(A) 2.31(C)	5.35(A)
H4	4.61(A)	2.48(A)	3.24(B) 2.84(C) 3.19(D)		3.07(C) 4.89(D)	2.32(A) 4.15(C)	3.50(A)

^a(A), (B), (C) and (D) refer to the labeling of the molecules in Fig. 6(d)

^bIntramolecular distances (within molecule A) are shown in bold

^cThe closest distances to the next nearest nine ¹H nuclei external to the central H2 and H4 nuclei are underlined

Comparing the experimental data in Fig. 6(a) and (b) with the simulated 11-spin ¹H-¹H magnetization exchange curves, while there is good agreement for short t_mix (< 100 μs), it is noticeable that the experimental data for the U-¹³C^dil_10% sample, in particular, trends to a markedly lower value than that for the simulation at long mixing times. For the diluted sample, the unlabeled molecules located around a fully ¹³C-labeled molecule can be viewed as a proton bath where a significant amount of the initial polarization is lost, because it cannot be back-transferred to observable ¹³C NMR signal during the last CP block. This is responsible for the much stronger attenuation of the CHHC cross-peak intensities at large mixing times in the 10%-diluted sample compared to the 100% sample. Quantitatively, the efficiency loss by this mechanism is illustrated in Fig. 7. Specifically, Fig. 7 compares for the U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl samples the evolution with the mixing time of the total observable experimental polarization that originates from an initial unit C7 polarization. This is defined as the sum of the normalized I_C7 diagonal peak intensity and the intensities I_C7,Cj within its associated cross-peak patterns (corrected by the procedure described above, i.e., $F_{L,M}^n(t_{\mathit{mix}})$), with j = 2, 3, 4, 5 and 8. Considering that an equal distribution of ¹H polarization is obtained at large mixing times, and neglecting spin-lattice relaxation, an asymptotic evolution would be expected towards a saturation level given by the ratio N_obs/N_tot between the number of protons that are directly attached to a ¹³C nucleus and hence contribute significantly to the observed ¹³C NMR signal, and the total number of ¹H spins into the system. Its value is N_obs/N_tot = 7/12 = 0.58 (NB: there are 7 CH and CH₂ protons and 5 NH₃ and OH protons in the L-Tyrosine·HCl molecule) and 7/120 = 0.06 in the case of the U-¹³C and U-¹³C^dil_10% samples, respectively. As can be seen in Fig. 7, the experimental total polarization trends towards these N_obs/N_tot values.

Fig. 7.

The evolution with the CHHC mixing time of the total C7 polarization in the U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl samples, considering an initial state of unit polarization. The total polarization is expected to trend towards N_obs/N_tot = 7/12 = 0.58, and 7/120 = 0.06 for the U-¹³C (circles) and U-¹³C^dil_10% samples, respectively (see horizontal dashed lines).

Considering the experimental data in Fig. 6(a) & (b), differences are apparent for short mixing times (t_mix < 100 μs) when comparing the buildup for the U-¹³C (circles) and U-¹³C^dil_10% (triangles) samples for (a) the C7 (CH₂) & C8 (^αCH) and (b) the C2 and C4 (directly bonded aromatic carbons) CHHC cross peaks. Specifically, in Fig. 6(a), the observed buildup rate is faster for the U-¹³C sample, while in Fig. 6(b), the buildup is the same within the experimental noise for the U-¹³C and U-¹³C^dil_10% samples. This is a consequence of additional close intermolecular proximities for the H7,H8 case: the intra-and inter-molecular contributions to the total C7,C8 cross-peak buildup curve are of comparable magnitudes, as they correspond to H7-H8 inter-proton average distances of 2.8 and 3.2 Å, respectively (see Table 3). By comparison, for the H2,H4 case, the nearest intermolecular proximity is 4.5 Å as compared to the intramolecular proximity of 2.5 Å (see Table 4). The 11-spin SPINEVOLUTION simulations (solid line in Fig. 6(a) & (b)) only consider intramolecular ¹H-¹H magnetization transfer – for H7,H8, see footnote ^d to Table 3 and the representative SPINEVOLUTION input files in the Appendix. Good agreement between experiment and simulation for short mixing times (< 80 μs) is obtained for the C7,C8 buildup curve for the U-¹³C^dil_10% sample (triangles in Fig. 6(a)) and for the C2,C4 buildup curves in Fig. 6(b) for both samples, i.e., for those cases where intermolecular proximities do not contribute to the experimentally detected ¹H-¹H magnetization exchange. The deviations between experiment and simulation at longer mixing times is a consequence of “loss” of magnetization experimentally to ¹H nuclei that are not bonded to a visible ¹³C nucleus as discussed above (see Fig. 7).

4.4. Analysis of the effect of multiple ¹H-¹H contacts in CHHC experiments

Fig. 6 shows examples of ¹H magnetization exchange observed for two specific cases, namely an isolated single ¹H-¹H contact (H2,H4, Fig. 6(b)) and a relatively tight H7-H8 inter-molecular pair in the close neighborhood of a short intra-molecular H7-H8 pair (see Fig. 6(a) for the U-¹³C sample). This section considers the ¹H magnetization exchange behavior for the different cases of ¹H-¹H contacts found in L-Tyrosine·HCl. Table 5 lists all ¹H-¹H distances under 5 Å. If we define a “close” ¹H-¹H contact as a distance under 3.5 Å, the ¹H-¹H contacts corresponding to the distinct CHHC cross-peaks can be classified either with respect to the number of involved contacts, i.e., single- (H2-H3, H2-H4, H2-H7, H3-H5, H3-H7, H4-H5, H4-H7, H5-H8), double- (H2-H5, H5-H7, H7-H8), and triple contacts (H3-H4), or according to their type, i.e., only intra- (H2-H4, H3-H5, H4-H7, H5-H8), only inter- (H2-H3, H2-H5, H2-H7, H3-H4, H3-H7, H4-H5), and mixed intra- and inter-molecular ¹H-¹H contacts (H5-H7, H7-H8). Note that for the case of H7 that corresponds to the two CH₂ protons, proximities of the same type and from the same other proton to both CH₂^a and CH₂^b are only counted once in the above classification – this is consistent with proton multiplicity being explicitly taken into account in the above analysis (see section 3.1).

Table 5.

Proton-proton contacts shorter than 5 Å extracted from the crystal structure [60] of L-Tyrosine·HCl. Intramolecular contacts are shown in bold. The root-sum-square coupling, d_rss, (in kHz, see Eq. (29)) is given in square brackets (¹H nuclei within 5 Å are considered in the summations).

	H7	H8	H2	H3	H5	H4
H7 (βCH2)	-	(2.50; 3.06) (2.93; 3.40) [10.5]	(4.65; >5) (2.31; 3.75) [10.1]	(4.87; > 5) (3.44; 3.52) (3.95; > 5) (4.68; > 5) [4.6]	(2.72; 3.78) (2.80; 2.93) [9.7]	(3.50; 2.33) (4.15; > 5) [10.1]
H8 (αCH)		-	> 5 4.14 4.96 [1.8]	4.75 4.96 [1.5]	2.56 > 5 [7.2]	4.45 4.82 [1.7]
H2 (CHarom)			-	4.29 2.95 4.06 4.30 [5.5]	4.96 2.76 3.19 [6.9]	2.48 4.49 4.68 4.73 [8.2]
H3(CHarom)				-	2.47 4.49 4.78 4.89 [8.3]	4.93 2.84 3.20 3.24 [7.4]
H5 (CHarom)					-	4.28 3.07 4.10 4.89 [4.9]
H4 (CHarom)						-

^aThe distinct ¹H nuclei are tabulated in the order of the corresponding ¹³C resonances (increasing ppm)

The simplest case corresponds to single ¹H-¹H intra-molecular contacts, i.e., C2-C4, C3-C5, C4-C7 and C5-C8. The example of the C2-C4 CHHC buildup curves (Fig. 6(b) has been discussed above: the observed good correspondence between the two experimental curves for the U-¹³C and U-¹³C^dil_10% samples on the one hand, and between the experiment and simulation, on the other hand, are illustrative for the conditions that must satisfied by a short 2.5 Å contact (i.e., corresponding to directly bonded CH_n moieties) that can be considered as a single contact. It is evident that the added contribution to magnetization exchange dynamics of the closest ¹H-¹H contacts can be safely neglected if they correspond to inter-proton distances larger than about ~4 Å. The identification of such intramolecular single contacts is useful in that they provide (as discussed below) reference curves for the analysis of other buildup curves that depend on contributions from multiple contacts.

In the following, we consider experimental CHHC data for U-¹³C L-Tyrosine·HCl, where close intermolecular ¹H-¹H proximities will affect the observed buildup curves. Specifically, Fig. 8 presents corrected normalized experimental CHHC buildup curves (corresponding to $F_{L,M}^n(t_{\mathit{mix}})$ as defined in Eq. (19)) for magnetization starting on (a) C2, (b) C3, (c) C5 and (d) C8. As noted above, the data associated with single ¹H-¹H intra-molecular contacts, i.e., C2-C4, C3-C5 and C5-C8 constitute single-contact intramolecular reference curves: The C2-C4, C3-C5 and C5-C8 peaks correspond to ¹H-¹H distances of 2.48 (H2-H4) and 2.47 Å (H3-H5) between neighboring aromatic protons and 2.56 Å (H5-H8) between an aromatic proton and the Cα proton. While all curves are converging to a similar plateau intensity at the longest experimental t_mix of 0.5 ms, differences in rate of buildup are clearly evident. Importantly, in agreement with the use of the CHHC experiment to identify structural constraints, it will be shown that the differences in buildup rate are directly related to the root-sum-square coupling, d_rss, [61–63] for the corresponding ¹H-¹H proximities (listed in Table 5):

$$d_{\text{rss}}=\sqrt{\sum d_{jk}^2}$$ (29)

where the dipolar coupling constant, d_jk, is defined:

$$d_{\text{jk}}=\left(-\frac{{\mu }_0}{4\pi }\frac{{\gamma }_H^2\hslash }{r_{jk}^3}\right)/2\pi$$ (30)

Fig. 8.

Corrected normalized experimental CHHC buildup curves corresponding to $F_{L,M}^n(t_{\mathit{mix}})$ as defined in Eq. (19) for U-¹³C L-Tyrosine·HCl for magnetization starting on (a) C2, (b) C3, (c) C5 and (d) C8.

The close resemblance between the C2-Cj and C3-Cj patterns in Fig. 8 reveals the presence of relatively similar proton environments around the H2 and H3 protons. In both cases, the upper limit is provided by the corresponding single-contact intramolecular reference curve (C2-C4 and C3-C5), while the lower limit is established by the contact with the H8 proton. The slower buildup of the C2-C8 and C3-C8 curves is consistent with the closest H-H proximity being over 4 Å (H2-H8: intermolecular 4.14 Å, H3-H8: intramolecular 4.75 Å). The remaining CHHC curves in the C2-Cj and C3-Cj patterns are distributed within the two limiting curves. As a common feature, all of them are encoding significant (in some cases multiple) intermolecular contributions, since their fast buildup is inconsistent with the large intramolecular ¹H-¹H distances (> 4.5 Å). The C2-C3/C3-C2 buildup curve, determined by a single 2.9 Å intermolecular contact, is the only one that can be directly compared with the reference curve in terms of the encoded distances. Nevertheless, the difference between the C2-C7 and C3-C7 buildup curves is consistent with the difference in the corresponding average nearest proton-proton distances of 3.0 and 3.5 Å, respectively and d_rss = 10.1 and 4.6 kHz, respectively.

It is informative to compare the examples of the C2-C5 and C3-C4 buildup curves that correspond to multiple intermolecular proximities under 3.5 Å (H2-H5 2.76 & 3.19 Å, d_rss = 6.9 kHz; H3-H4 2.84, 3.20 & 3.24 Å, d_rss = 7.4 kHz) with the curves corresponding to single H-H proximities, i.e., the two intramolecular C2-C4 and C3-C5 reference curves and the C2-C3/C3-C2 single intermolecular contact, where the closest H-H distances are 2.48 Å (H2-H4), 2.47 Å (H3-H5) and 2.95 Å (H2-H3) and d_rss = 8.2 kHz (H2-H4), d_rss = 8.3 kHz (H3-H5) and d_rss = 5.5 kHz (H2-H3). The closeness of the C3-C4 and C3-C5 buildup curves is consistent with the similar d_rss values. The effect of multiple ¹H-¹H proximities leading to a faster buildup is evident when comparing the C2-C5 and C3-C4 curves on the one hand with the C2-C3 curve on the other hand.

A good correspondence between the experimental CHHC data and the corresponding structural parameters was found also for the C5-Cj and C8-Cj patterns in Fig. 8. Notably, C5-Cj is representative for a ¹H site tightly coupled to its surrounding protons. Together with the intramolecular reference curves (C5-C3, C5-C8), the other three curves in this pattern also encode short inter-molecular contacts, with ¹H-¹H distances between 2.8 and 3.1 Å and similar d_rss values (between 4.9 and 9.7 kHz). At the other extreme, the C8-Cj pattern corresponds to strong couplings of H8 only with intra-molecular protons, whereas all the inter-molecular contacts are larger than 4 Å. This is clearly evidenced by the measured buildup curves.

4.5. Comparison of CHHC build-up data for U-¹³C and U-¹³C^dil_10% samples

Fig. 9 compares corrected normalized buildup plots, i.e., $F_{L,M}^n(t_{\mathit{mix}})$, for U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl for the cases where significant intensity (i.e., above the noise level) is observed for the U-¹³C^dil_10% sample. These cases (C2-C4, C3-C5, C4-C7, C5-C7, C5-C8 and C7-C8) all correspond to closest intramolecular H-H proximities under 3 Å, while for the other cases, the closest intramolecular H-H proximity is over 4 Å. In this context, while Fig. 8 shows CHHC buildup curves for the U-¹³C sample involving C8 that correspond to closest H-H distances of over 4 Å, for the U-¹³C^dil_10% sample, it is to be remembered that much CHHC signal intensity is lost to invisible protons attached to ¹²C nuclei at longer mixing times (see Fig. 7 and section 4.3). Comparing the buildup curves for the U-¹³C and U-¹³C^dil_10% samples in the short t_mix regime (< 80 μs), it is observed that four of them (C2-C4, C3-C5, C4-C7, and C5-C8) are quite similar in shape, while for the C5-C7 and C7-C8 cases, faster buildup is observed for the U-¹³C sample. As was discussed in section 4.3 when comparing the CHHC data for C7-C8 and C2-C4 (see Fig. 6), this difference is a consequence of additional close H5-H7 and H7-H8 intermolecular proximities.

Fig. 9.

Comparison between the corrected normalized experimental CHHC buildup curves, i.e., $F_{L,M}^n(t_{\mathit{mix}})$ as defined in Eqs. (19) & (28), for U-¹³C (circles) and U-¹³C^dil_10% (triangles) L-Tyrosine·HCl for (a) C2-C4, (b) C3-C5, (c) C4-C7, (d) C5-C8, (e) C5-C7 and (f) C7-C8.

5. Discussion

CHHC-type experiments are being increasingly applied to indirectly probe ¹H-¹H magnetization exchange and hence obtain structural constraints, in particular, for large biomolecules. This paper has presented a protocol for correcting the effect of non-specific cross polarization in CHHC experiments, thus allowing the recovery of the ¹H-¹H magnetization exchange functions from the mixing-time dependent buildup of experimental CHHC peak intensity. The presented protocol also incorporates a scaling procedure to take into account the effect of multiplicity of a CH₂ or CH₃ moiety. In this way, direct comparison can be made between experimentally determined ¹H-¹H magnetization exchange functions and numerical density-matrix simulations, without the requirement for any phenomenological factors. For L-Tyrosine.HCl, good agreement between experiment and 11-spin simulation (including only isotropic ¹H chemical shifts) is demonstrated for the specific case of initial buildup (t_mix < 100 μs) of CHHC peak intensity corresponding to an intramolecular close (2.5 Å) H-H proximity. The derived corrections are not limited to the case of ¹H-¹H magnetization exchange, i.e., to zero-quantum mixing schemes, but they are also valid to CHHC experiments employing homonuclear ¹H-¹H recoupling schemes [17,50].

For small and moderately sized organic molecules such as L-Tyrosine.HCl, the experimentally observed buildup of CHHC peak intensity often corresponds to a ¹H-¹H magnetization exchange behaviour that depends on both intra- and intermolecular proximities. Indeed, it is to be noted that the multi-spin kinetic rate matrix analysis of directly observed ¹H-¹H magnetization exchange by Elena et al. exploits the dependence on intermolecular ¹H-¹H proximities to determine the three-dimensional packing of organic molecules in the crystal lattice [22, 23]. In the CHHC experiment, intermolecular effects can be removed by working with dilute samples where a U-¹³C labeled molecule is recrystallised with an excess of molecules at natural abundance. This approach has been employed in previous studies where distance constraints extracted from CHHC experiments have been used as constraints in the structural determination of the three-dimensional conformation of organic molecules [49–51]. In this paper, experimental CHHC buildup curves were presented for L-Tyrosine.HCl samples where either all or only one in ten molecules are U-¹³C labeled. For the dilute sample, CHHC cross peak intensities tended to significantly lower values for long mixing times (500 μs) than for the 100 % sample. This difference has been explained here as being due to the dependence of the limiting total magnetization on the ratio N_obs/N_tot between the number of protons that are directly attached to a ¹³C nucleus and hence contribute significantly to the observed ¹³C CHHC NMR signal, and the total number of ¹H spins into the system.

It has been shown that insight into ¹H-¹H magnetization exchange under multiple intra- and intermolecular ¹H-¹H dipolar couplings can be obtained by a consideration of the root sum squared dipolar couplings corresponding to specific CHHC cross peaks. (Note that a sum squared dipolar coupling is also inherent to the multi-spin kinetic rate matrix analysis of directly observed ¹H-¹H magnetization exchange by Elena et al. [22, 23]) ¹H-¹H magnetization exchange curves extracted from CHHC spectra for the 100 % L-Tyrosine.HCl sample exhibit a clear sensitivity to the root sum squared dipolar coupling, with fast build-up being observed for the shortest intramolecular distances (2.5 Å) and slower, yet observable build-up for the longer intermolecular distances (up to 5 Å). As is to be expected, differences in the initial CHHC buildup were observed between the 1 in 10 dilute and 100 % samples for cases where there is a close intermolecular H-H proximity in addition to a close intramolecular H-H proximity. The demonstrated usefulness of the CHHC experiment as a valuable and reliable source of quantitative H-H proximity information is consistent with previous studies of other small organic molecules [50, 51] as well the microcrystalline CrH protein [30], for which ¹H-¹H distances are known from single-crystal diffraction data.

Of much current interest is the application of CHHC-type experiments, including the recently developed J-CHHC [64], to large biomolecules, where extracted ¹H-¹H distances are then used as constraints in structural determination protocols [29, 31, 32, 36, 37, 44, 46, 64]. For large biomolecules, the ¹H-¹H magnetization exchange as encoded in CHHC-type peaks is less complex than in the case of small and moderately sized molecules since intermolecular ¹H-¹H proximities do not usually contribute, although NHHC experiments have been used to probe inter monomer contacts for the CrH microcrystalline protein [30]. As noted above, it has been shown here that good agreement between experiment and 11-spin simulation (including only isotropic ¹H chemical shifts) was observed for the initial buildup (t_mix < 100 μs) of CHHC peak intensity corresponding to a single intramolecular H-H proximity. For large biomolecules, most CHHC-type peaks usually correspond to such single intramolecular H-H proximities, thus suggesting that an analysis of CHHC-type buildup curves (using the protocol presented here to correct for non-specific CP and take into account XH_n multiplicity) using multi-spin simulations could be utilised to check and refine H-H distances in as-determined biomolecular structures.

Appendix

Representative SPINEVOLUTION input files used for the numerical simulation of the F_jk (t_mix) polarization transfer functions in Eq. (1).

****** The System ***********************************

spectrometer (MHz) 400

spinning_freq (kHz) 10

channels H1

nuclei H1 H1 H1 H1 H1 H1 H1 H1 H1 H1 H1

atomic_coords cross87_d.cor

cs_isotropic 3 2.8 4.5 8 8 8 7 7 7 7 3 ppm

csa_parameters *

j_coupling *

quadrupole *

dip_switchboard *

csa_switchboard *

exchange_nuclei (4 5 6)

bond_len_nuclei *

bond_ang_nuclei *

tors_ang_nuclei *

groups_nuclei *

******* Pulse Sequence ******************************

CHN 1

timing (usec) (5) 200

power (kHz) 0

phase (deg) 0

freq_offs (kHz) 0

phase_cycling * *(RCV)

***************************VARIABLES**********************

******* Options etc **************************************

rho0 1 1 0 0 0 0 0 0 0 0 0 Iz

observed_spins 1 2 3 4 5 6 7 8 9 10 11 Iz

EulerAngles rep100.dat

n_gamma 10

line_broaden (Hz) *

zerofill *

FFT_dimensions *

cross87_d.cor

8.207 −0.698 −0.686 H7a (CH2 - molecule A)

8.554 0.898 −1.274 H7b (CH2 - A)

7.034 0.260 1.303 H8 (CH - A)

5.976 1.412 −1.213 Ha (NH3 - A)

5.169 0.883 0.100 Hb (NH3 - A)

5.951 −0.187 −0.894 Hc (NH3 - A)

10.630 1.831 −0.798 H4 (CHring - A)

9.091 −1.154 1.852 H5 (CHring - A)

9.394 −2.663 −0.444 H2 (CHring inter - C)

9.253 −1.154 −3.245 H5 (CHring inter - B)

7.196 0.260 −3.793 H8 (CH - B)