From the description shown in the preceding section, it is clear that RF spatial inhomogeneity affects the dipolar modulation in R-PDLF experiments. Therefore, for accurate fits of experimental data, this effect needs to be taken into account. In this and the following section we demonstrate that accounting for the effect of RF inhomogeneity in time-domain fits of experimental data indeed leads to highly accurate fits of the experimental data and consequently to a considerable improvement of the accuracy of the C–H bond order parameters determined. Instead of in the time domain, the fitting could also be performed in the frequency domain, i.e., by performing the Fourier transform of the experimental data in the indirect dimension and by then fitting these data with the Fourier transform of the numerical simulations. Here, we simply use time-domain fits and avoid performing the additional step of the Fourier transform which is unnecessary and has an extra computational time cost.

Figure  shows the application of this time-domain analysis to a sample of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) multilamellar vesicles (MLVs), illustrating the analysis method on the β carbon of the POPE headgroup. POPE is one of the most abundant lipids in cellular membranes. It has recently been investigated with R-PDLF NMR in the context of the NMRlipids project. This project aims to resolve the atomistic molecular structure of the most abundant lipids in nature. To this end, the project uses a combination of solid-state NMR experiments and all-atom molecular dynamics simulations . Using only the first eight points along the indirect dimension, the time-domain curve fitted to the experimental dipolar modulation measured for the β carbons gives an estimation of SCH=0.0280.0090.034, where the superscript and subscript are upper and lower limits as defined in the Supplement. Such a fit enables us to predict the experimental dipolar spectrum that one would obtain using a higher number of indirect dimension points as shown in Fig. b and c. Figure d shows how the time-domain fit procedure enables the estimation of an accurate order parameter using much shorter experiments in comparison to the use of the typical Fourier transform methodology (gray circles). As it will be shown in a later section, this is particularly useful for complex systems for which only a limited number of indirect dimension points can be measured. Moreover, the fit procedure enables the estimation of the uncertainty of the value measured (the detailed description for defining the errors is given in the Supplement), which is more difficult to define with the Fourier transform “read-off” method alone. Figure d also shows the result of using fits with the ideal model (i.e., that does not include the effect of RF inhomogeneity). One observes that these fits become less accurate with an increase of the number of points used (a comparison with the model including RF inhomogeneity is also given in the Supplement in Fig. S2). This is a consequence of the effect of RF inhomogeneity which becomes more prominent at longer t1 times. However, Fig. d also suggests that fitting with the ideal model may provide good results when analyzing short t1 intervals. Based on previous 2H NMR measurements, the β methylene carbon is expected to have the equivalent order parameters for the two C-H bonds . In the next section we demonstrate the usefulness of R-PDLF time-domain fits to analyze cases of two-component dipolar modulations and estimate the accuracy of the order parameters determined with the proposed method by comparison with 2H NMR quadrupolar splittings.

To further test the accuracy of using a model of R-PDLF experiments that includes RF inhomogeneity, we used a water/DMPC/DMPCd54 liquid crystalline system (Lα phase), such that it was possible to measure the 2H NMR spectrum of the perdeuterated acyl chains of the DMPCd54 molecules and compare it with the 1H-13C dipolar couplings determined from the acyl chains of the molecules with natural abundance of isotopes. Figure  shows the R-PDLF dipolar modulation for a number of resolved carbons from the liquid crystalline system and their corresponding fits with and without accounting for RF inhomogeneity. The improvement of the fits by accounting for RF inhomogeneity is visually evident, especially for the smaller coupling. We stress here that the RF inhomogeneity profile used in the simulations was experimentally measured and modeled by a Gaussian profile. The RF inhomogeneity is therefore an objectively quantified a priori constraint that we have obtained from an independent experiment. Moreover, since each of the three carbon peaks analyzed is a sum of components from the two individual acyl chains, sn-1 and sn-2, we also performed the time-domain fits of the dipolar modulations using two components. In the case of the acyl chain methyl groups ω, equivalent dipolar couplings minimize χ2. On the other hand, for the ω-1 and ω-2 methylenes, two distinct dipolar couplings enable a better fit than a single value which suggests a distinct motional geometry in the sn-1 and sn-2 acyl chains for these methylenes (a detailed analysis is given in the Supplement). The non-equivalence of these order parameters is in agreement with previous investigations of specifically deuterated dipalmitoylphosphatidylcholine (DPPC) molecules .

The 2H NMR spectrum measured from DMPCd54 molecules is shown in Fig. . The five lowest quadrupolar splittings were used to calculate C–D bond order parameters. The results are included in Table  together with the results from the time-domain fits of the dipolar modulations from the molecules with natural abundance of isotopes. The agreement between the values obtained by using the different techniques is striking. The maximum difference between the order parameter magnitudes determined with the methodology proposed and the values determined from the 2H NMR spectrum is below ±0.005, much below the errors reported previously (up to ±0.02) when doing similar comparisons between dipolar recoupling experiments with 2H NMR spectroscopy . Here, we used values taken from previous studies for the rigid 1H-13C dipolar and 2H quadrupolar couplings, namely 22 and 126 kHz, respectively . We may therefore exclude a significant isotope effect on the molecular dynamics of the acyl chains in the water/DMPC/DMPCd54 liquid crystalline system. Moreover, we can unambiguously assign the C–D order parameters measured with 2H NMR based on the two-component fits of the R-PDLF data. The measurement of the 2H NMR spectrum alone (from perdeuterated acyl chains) does not provide information concerning the origin of each of the distinct quadrupolar splittings observed. However, R-PDLF enables chemical shift selectivity in the direct dimension. By fitting the indirect dimension time domain of the R-PDLF experiment for the ω, ω-1, and ω-2 carbons with two components (one per acyl chain), one readily enables the assignment of the five lowest quadrupole couplings in the 2H NMR spectrum.

The agreement between the 2H NMR and R-PDLF values highlights the usefulness of modeling R-PDLF data in contrast to the conventional dipolar splitting read-off from the frequency-domain analysis. Such distinct dipolar couplings cannot be observed in the dipolar spectrum due to the limited number of points measured as described previously. In fact, in this case, from the read-off method one obtains for both the ω-1 and ω-2 carbons the lowest order parameters with great precision. The only fitting parameters used to make all the plots shown in Fig.  were the dipolar couplings (i.e., order parameters), namely just one fitting parameter in case of a one-component fit and two fitting parameters in case of a two-component fit. In the case of the two-component fits we have assumed that the intensity for each component was equal. This is a reasonable assumption in this case, because the expected differences between the order parameters and the dynamics of the two acyl chains are very small. Therefore, the rINEPT intensity of each component should be very similar according to the theoretical model for the dependence of 13C rINEPT intensity on CH order parameter and correlation times described by Topgaard and coworkers . In general though, two-component fits may require fitting also the relative magnitude of each component for obtaining a good fit. Here we opted for a minimal amount of fitting parameters, i.e., the simplest model enabling to match the experimental data within experimental uncertainty.

For determining the numbers shown in Table , we have additionally used a phenomenological single exponential relaxation contribution as described in the Supplement. This did not affect the order parameters of the two-component fits determined for the carbons ω-1 and ω-2. On the other hand, for the ω carbon, without inclusion of the single exponential relaxation, the order parameters calculated were 0.026 and 0.020. The line broadening corresponding to the minimum of χ2 for the single-component fits shown in Table  and in Fig.  using the model that includes RF inhomogeneity was in the range of 4–10 Hz. On the other hand, for the model that does not account for RF inhomogeneity, the resulting line broadening from the minimization of χ2 increases significantly in some cases, especially for low frequency dipolar modulations, to as much as 70 Hz. Such an increased relaxation rate is due to the significant deviation of the ideal model from the experimental data as shown in Fig.  for the ω carbon. For fits with the ideal model that do not lead to such large relaxation rates, although the minimum of χ2 is higher than for the model including RF inhomogeneity (as shown in the Supplement in Figs. S2–S8), the fitting outcome is also good (which is the case for the methylene carbons in Table  and for fits of short t1 intervals as shown in Fig. ).

To showcase the applicability of the presented methodology, we investigated the molecular structure of a complex membrane system composed of a brain lipid extract with several lipid types present. The lipid mixture consists of the chloroform : methanol extract of porcine brain tissue as purchased from Avanti Polar Lipids.

Figure  shows the 13C spectrum of the brain lipid extract at excess hydration together with spectra acquired from single phospholipid model systems. Due to the much higher complexity of the brain lipid extract sample in comparison to model membrane systems (usually composed of one to three distinct components), the number of transients required to achieve a reasonable signal-to-noise ratio is considerably higher. Here, 4096 transients were acquired for each point in the indirect dimension with a total experimental time cost of approximately 48 h (in comparison to the 256 transients used for the simple model systems). The comparison of the 13C spectra measured from the brain extract with the spectra from the simple models enables one to identify a number of peaks from different components known to exist in the extract (according to the chemical composition provided by Avanti Polar Lipids for 41.3 wt % of the lipid extract), namely the α, β and γ segments of the headgroup of phosphatidylcholine (PC), the α and β segments of phosphatidylethanolamine (PE), and the α and β segments of phosphatidylserine (PS). From inspection of the 13C rINEPT spectrum, and knowing the lipid composition of myelin that makes up most of the dry weight of brain tissue , it is clear that most of the remaining unknown 58.7 wt % content of the lipid extract is cholesterol plus a considerable fraction of galactose cerebrosides (GalCer) labeled c and s in Fig. b, respectively.

Figure c shows the indirect time-domain modulation for a number of identified peaks in the spectra measured by R-PDLF spectroscopy. In this case, due to the large number of transients required to achieve a reasonable signal-to-noise ratio (4096 scans), only very short indirect dimension modulations can be measured. Namely, only eight points in the indirect dimension were acquired for the brain lipid extract sample. Such a reduced total indirect dimension time is not sufficient to obtain dipolar splittings after Fourier transformation. However, application of time-domain fitting including RF inhomogeneity enables the determination of the dipolar couplings and, therefore, the estimation of C–H bond order parameters. The estimated order parameters are presented in Table .

The headgroup order parameters determined from the set of different phospholipid types in the brain lipid extract are very much in line with what is observed in corresponding lipid model membranes. The only exception is the PS β order parameter with a rather small order parameter equal to 0.040.0310.052. This suggests that the brain lipid extract may contain calcium ions that are known to significantly affect the PS headgroup orientation. A thorough analysis of the order parameters in terms of the molecular structure in the complex brain lipid extract system will not be given here. Such analysis should be done in combination with experiments and MD simulations on PC/PE/PS/cholesterol/GalCer model membranes which resemble the lipid extract composition, to investigate potential phase separation behavior, effects of distinct ions present in the system and pH, or ultimately the effect of macromolecules such as basic myelin protein, the most abundant protein in myelin. We believe that such a combined approach will lead to a more accurate description of the lipid/protein interactions in myelin, which may be important in the context of a number of diseases related to demyelination. The methodology proposed here will enable one to perform such investigations with higher accuracy using shorter experiments and without the requirement of isotopic labeling. Moreover, we envision applications to many other complex membranes such as bacterial membrane models, eukaryotic membrane models with cell type or organelle-specific compositions, technological systems such as lipid–nanoparticles, and to address the molecular structure/orientation of drugs, peptides, or other molecules with anisotropic motion in lipid membranes.