The dipolar coupling tensor is a uniaxial second rank tensor. Its principal axis frame has the z axis along the internuclear vector. The dipolar coupling strength is proportional to the second order Legendre polynomial 〈 〉, where θ is the angle between the internuclear vector and the magnetic field and inversely proportional to the cube of the internuclear distance 〈 1/r^{3} 〉. In a polycrystalline sample, the solid state NMR spectrum of a pair of dipolar-coupled spin ½ nuclei with negligible CSA values gives rise to a Pake powder pattern, from which the dipolar coupling constant may be extracted. However, if the coupled spin ½ nuclei have CSAs much larger than the dipolar coupling it becomes difficult to extract the dipolar coupling constant from the NMR powder spectrum. The figure shows the powder spectrum of two dipolar coupled ^{13}C spins, each with a CSA on the order of 20 kHz and with a dipolar coupling of 500 Hz corresponding to a ^{13}C pair separated by 2.48 Å. The spectrum is clearly dominated by the CSA and it is very difficult to discern the presence of a dipolar coupling in the spectral data.

In most dipolar recoupling pulse sequences, pulsed irradiations are applied synchronously with the sample rotation. In DRAWS (Dipolar Recoupling with a Windowless Sequence), the eight 360° degree pulses (alternating X and X̄ phases) and two 90° degree pulses (Y phases) that constitute the unit pulse cycle (R) are applied within a single period of the rotor (τ_{rotor}). (a) The unit pulse cycle for DRAWS is supercycled as RR̄R̄R, where R̄ indicates a unit cycle phase-shifted by 180° degrees. (b) DRAWS applied in the context of a CPMAS experiment. (c) Pulse Sequence for Double-Quantum-filtered DRAWS. (d) Pulse sequence for two dimensional double quantum DRAWS (DQDRAWS).

The SEDRA pulse experiment is comprised of a train of 180° pulses applied in synchrony with the rotor cycle, one pulse per rotor cycle. Following cross polarization the SEDRA mixing pulses are applied and the free induction decay of single quantum magnetization is collected. The Fourier transformed signal intensity is recorded as a function of the length of the SEDRA mixing time and analyzed to obtain the dipolar coupling between like spins. The version of SEDRA shown here has a basic mixing period of 8 rotor periods (8τ_{rotor}) and employs an XY-8 phase cycling of the pulses to correct for field inhomogeneity. Proton decoupling is applied through the SEDRA mixing and detection periods.

(a) DQDRAWS buildup curves for hydrated, surface adsorbed pS_{2}pS_{3} statherin (◆), lyophilized, bound pS_{2}pS_{3} Statherin (◇), and simulations for φ torsion angles of −60° (——), −85° () and a combination of 35% α-helix (−57°) and 65% β-sheet (−120°) (– – –). The increase in torsion angle on lyophilization correlates well with the loss of α-helical secondary structure. (b) REDOR dephasing curves for hydrated, surface adsorbed pS_{3}F_{7} Statherin(◆) and simulations for a carbon-nitrogen distance of 4.2 Å (——). The pS_{3}F_{7} distance across the hydrogen bond fits best to a distance of 4.2 Å, indicating that the region is well described by an ideal α-helix.

Modeling statherin structure using the ROSETTA ab initio algorithm. (a) A histogram showing percentage of the 3 basic secondary motifs along the backbone in the predicted 31-model cluster. (b) Contact map for the 31-model cluster. Pink squares, shown along the diagonal, indicate short contacts between adjacent residues characteristic of an α helix structure. Contacts that are shown as colored squares perpendicular to the main diagonal in the lower left triangle indicate long range contacts between residues far apart in the sequence. Common features between the models were used to select sites on the protein for incorporation of NMR labels.

Histograms showing distribution of torsion angles and distances between labels across the 31-model cluster. (a) Φ angle and (b) Ψ angle values for the backbone torsion angles between the two carbonyl labels [1-^{13}C]P33 and [1-^{13}C]Y34. Nitrogen-carbon distances between the (c) [^{15}N]Y38 label and the [1-^{13}C]P33 label and between the (d) [^{15}N]Y38 label and the [1-^{13}C]P34 label. Fluorine-carbon distances between the (e) [4′-^{19}F]P23 label and the [1-^{13}C]P33 label and between the (f) [4′-^{19}F]P23 label and the [1-^{13}C]P34 label.

^{13}C SEDRA NMR decay curves (obtained at a ^{1}H frequency of 500 MHz) showing the carbon signal intensity as a function of the length of time the pulses are applied. (a)^{13}C signal decay in the REDOR reference experiment showing dephasing behavior due to recoupling of the ^{13}C-^{13}C dipolar interaction. The fit to this decay curve is used to extract the structural parameters. (b) structural parameters that influence the decay curve. Graphic visualization of the relation between the mutual orientation of chemical shift anisotropy tensors (green) and the Φ and Ψ angles in the two carbonyl labels. (c) χ^{2} analysis of data based on simulations utilizing the torsion angle dependence. A contour plot of the χ^{2}(Φ,Ψ) function showing angular values in the SEDRA data are best fit. Confidence level of σ is shown as the lowest contour (white).

^{13}C-^{15}N and ^{13}C-^{19}F REDOR NMR decay curves (obtained at ^{1}H frequencies of 300 MHz and 500 MHz, respectively) showing the carbon signal intensity as a function of the length of time the pulses are applied. (a)^{13}C signal decay from recoupling of the ^{13}C-^{15}N dipolar couplings in a REDOR experiment. (b) Dipolar interaction parameters in the C(P33)-C(Y34)-N(Y38) spin triad. (c) The contour plot of the χ^{2}(r_{C(P33)-N(Y38)},α) function (middle) and (d) the graph of the χ^{2}(r_{C(Y34)-N(Y38)}) function (bottom) showing values for which the CN REDOR data is minimized. (e)^{13}C signal decay from recoupling of the ^{13}C-^{19}F dipolar couplings in a REDOR experiment. The fits to these decay curves are used to extract the structural parameters. Graphic visualization of the structural parameters that influence the decays and χ^{2} analysis of simulation and data based on these parameters. The parameter α represents rotation of the heteronuclear dipolar vectors around the CC vector. (f) Dipolar coupling parameters in the C(P33)-C(Y34)-F(P23) spin triad. Here, we denote the two carbons as C′ and C″ since they are indistinguishable in the ^{13}C spectrum. (g) The contour plot of the χ^{2}(r_{C}′_{-F(P23)},α) function and (h) the graph of the χ^{2}(r_{C}″_{-F(P23)}) function demonstrate values for which the CF REDOR data are fit by simulations. Confidence level of σ is shown as the lowest contour (white).

(a) Statistical χ^{2} analysis of ^{15}N-^{31}P REDOR study of binding of ^{15}N K6 of SN-15 to HAP indicates a well-defined minimum distances of 3.8 A and 4.8 A suggesting a possible hydrogen bond between the K6 side-chain and the HAP atoms. The filled squares are the experimental data while the filled circles denote the simulation that gives rise to the χ^{2} minimum and the pentagons indicate the multiple spin simulation of a ^{15}N K6 approaching the (004) crystal plane of hydroxyapatite (parameters used for the simulation are defined thoroughly in (b)). (b) Model of a ^{15}N K6 sidechain of SN-15 approaching the (004) crystal plane of hydroxyapatite. The figure depicts all the spins that were used the above mentioned simulation. The dark circles denote the spins while the lighter shaded circle depicts the ^{15}N spin on the lysine side-chain. The blocks on the peptide depict the six N-terminal residues of SN-15 that are known to be in an extended conformation.

Three different orientations of the phenyl ring with respect to the HAP surface were considered: (i) the ring plane is parallel to the HAP surface, (ii) a line perpendicular to the C_{2}-C_{3} and C_{5}-C_{6} bonds is also perpendicular to the HAP surface, and (iii) a line from C_{1} to C_{4} is perpendicular to the HAP surface. (b) REDOR analysis of the both samples bound to HAP is shown. The experimental data and best simulated fits are shown for both samples. (c) A χ^{2} map is shown for the F14 sample based on the orientations (i) and (ii) depicted in Fig. 10(a). (d) A modified SN-15 model with peptide bound to HAP is presented, in which the C-terminus loses its α-helicity, and both the K6 and F14 side chains are within close proximity to the HAP surface.

Superposition of the 8 models that agree with the constraints derived from the solid state NMR measurements. These models are derived from the full set of 1,000 models. Alignment of the models in the figure was based on root-mean-square deviation calculation of segments (1–15) and (33–38) between the 8 models.

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