the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Oct 2025
Robust Bilinear Rotations II
Abstract. Bilinear rotations are essential building blocks in modern NMR spectroscopy. They allow the rotation of an isolated spin without couplings, i.e. bilinear intereactions, in one way, while rotating spins with a matched coupling in another way. Different classes of rotations form the different bilinear rotations with acronyms BIRD, TANGO, BANGO, and BIG-BIRD. All original elements have in common that hard pulses limit bandwidths and that defined rotations for coupled spins are only possible for a narrow range of coupling constants. We recently introduced the COB-BIRD with a general optimization procedure to obtain robust bilinear rotations well-compensated with respect to couplings, offsets, and B1-inhomogeneities (Y. T. Woordes et al., Sci. Adv. 11 (2025), eadx7094). Here we show a fundamental principle on how the COB-BIRD can be used to construct all types of bilinear rotations with the same improved robustness covering a coupling range of 120–250 Hz. In addition, a construction principle for universal rotation pulses is adapted to produce bilinear rotations from INEPT-type transfer elements, allowing the construction of bilinear rotations also for higher coupling ranges from e.g. COB3-INEPT with coupling compensation in the range of 120–750 Hz. After introducing the two fundamental design principles, example sequences of the four classes of bilinear rotations and different degrees of robustness are derived and characterized in theory and experiment. In addition, a highly useful HMBC/ASAP-HSQC-IPE-COSY supersequence is introduced with a (COB-)BANGO element for Ernst-angle type excitation. Finally, BIRD-decoupled J-resolved INEPT experiments with extreme compensation for partially aligned samples with with total couplings ranging from 47 Hz up to 434 Hz are demonstrated.
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Status: open (until 03 Nov 2025)
- CC1: 'Comment on mr-2025-13', Tom Barbara, 22 Oct 2025 reply
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RC1: 'Comment on mr-2025-13', Anonymous Referee #1, 24 Oct 2025
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General comments
In NMR spectroscopy, which allows to reveal the atomic resolution information about the chemical structure and molecular dynamics of molecules, the ability to manipulate spin coherences according to the ones desire is of great importance. Pulse sequence elements permitting rotations of chosen spin coherences, while leaving other ones unaffected are therefore highly needed. A class of such pulse sequence elements selectively and concurrently rotating the J-coupled IS spin coherences and non-coupled I spin coherences is usually termed Bilinear Rotation (BR). These elements usually come with inherent limitations in the correct-phase signal-maximized coherence transfer as the actual J-couplings of the system of interest may differ from the one for which the experiment has been designed. Additionally, the effects of hard pulse imperfections and magnetic field inhomogeneity come into play, imposing further bounds on the coherence transfer efficiencies. In this article, these issues have been successfully addressed using the new methods coined “COB”, which stands for Coupling, Offset and B1-inhomogeneity compensated pulse sequence elements. The use of pulse-delay calibrated spin evolution allows compensation with respect to the actual J-coupling modulation, while the application of optimal control derived offset and B1-inhomogeneity compensated uniform rotation or point-to-point pulse shapes, allows to make these pulse sequence elements as robust as physically possible. According to the authors, these elements may tolerate an astounding variation in J-coupling of approximately 100-750 Hz, with chemical shift offset in the range of 10 kHz (37.5 kHz) in 1H (13C) dimension.
Specific comments
In this article, these COB pulse sequence elements have been successfully applied to BR elements such as BIRD, TANGO, BANGO and BIG-BIRD, allowing for a highly robust distinction of J-coupled IS coherences and non-coupled I coherences, permitting on-demand concurrent manipulation of those two types of coherences. These BR elements, in general, come in handy when trying to design no-relaxation delay pulse sequences where evolution of chosen coherences is allowed, whilst suppressed for the other ones, in order to use the latter in the subsequent part of the experiment.
Also, BR elements are of prime importance in the fast and efficient real-time homonuclear decoupling elements like BASEREX. In the latter, the large typically one-bond heteronuclear coupling is used to distinguish J-coupled spin coherences IS from IX, where X is a different spin of the same nucleus as I, distant to heteronuclear spin S (with much smaller J-coupling between the spins S and X than between the spins S and I). In principle, the use of such elements up until now has been limited to low J-coupling range and chemical shift offsets. According to the authors, the use of COB principles allows to efficiently extend those experiments for the cases of high (J-coupling or offset) heterogeneity which may be intrinsic to the sample. Such heterogeneity may be found, for instance, in the 19F-1H coupled systems, which are gradually becoming of high interest in nowadays applications like drug chemistry, or even in biomolecular in vitro and in vivo studies. The NMR methods for these systems are slowly emerging, but until recently, required many experiments to cover the extent of 19F chemical shift range or 19F-1H coupling values. Even though an example of such application would be very welcome, it is hard to show all of the applications of the method in one article and may easily go beyond its scope.
The second example of the use of COB-BR elements may be, as has been shown by the authors in the article, in the studies of conformational dynamics of molecules and macromolecules using Residual Dipolar Couplings. In the latter, the use of an alignment medium to partially reduce the rotational averaging of the dipolar coupling interactions leads to efficient modulation of the so-called total (T) coupling (J-coupling plus Dipolar coupling). As a result, the apparent J-couplings will be largely modified positively or negatively to the extent imposed by the degree of alignment of the sample with typical variation of up to 30-50% of the nominal J-value. This modulation will occur differently for different parts of a molecule, encoding precious information on orientational and conformational averaging of the molecule. Normally, this phenomenon would modify the magnetization transfer efficiencies and if sufficient care is not taken, sometimes even phases of the resultant signals differently for different resonances of the same molecule in experiments based on INEPT or BR elements. This, would in turn require acquisition of many experiments with delays in the INEPT/BR elements designed to maximize the signals of every spin system in the aligned sample. The latter is not only difficult, but also time consuming, as the molecular diversity usually requires a few of such experiments with delays optimized to coupling values in the range of e.g. 90 to 200 Hz (for 13C1H spin system). The latter arises due to the fact that the knowledge of the T-coupling values is targeted and usually only its approximate range may be estimated a priori. The use of COB principles coupled with INEPT and BR elements allows one to efficiently remove the dependence on the coupling and offset, to record all of these responses in one single experiment.
The full understanding of the article requires knowledge of the previous work of the authors to fully appreciate the technical details. It is, however, beyond the scope of the article to fully discuss the functioning of all of the BR elements and COB principles of design, while proper citation is correctly annotated by the references.
Technical comments
In the Figure 1., the discrimination between Uniform Rotation (UR) and Point-to-Point (PP) pulse shapes as implemented in the Figure 1C,E,G is available only at the right-hand side of the figure 1I, when discrimination of those pulse shape characteristics is needed already earlier to fully understand the Figure 1C. I would like to propose to shift the "legend" of UR and PP pulses to the top of the Figure 1, next to Figure 1A, which then may be made consistently 20% smaller to accommodate space for the mentioned change. In the same time, the Figures 1H and 1I could be made slightly bigger, as for the moment it is difficult to read the J-values and offsets in those figures. I am guessing that the “PP” placed not far away from “Phi” in the Figure 1H and 1I was supposed to be a subscript, but at the moment it is a bit too far from it and seems to be without any meaning. Also, it is difficult to distinguish UR 90° pulse shape from the hard 90° pulse shape in Figure 1C,E,G. One could improve this distinction by playing with the shade of grey or other visibly different color of choice (e.g. red) for the UR 90° pulse.
In Figure 2F, there seems to be an excessive amount of tiny black spacers: beneath the second part of the rotation sandwich (x, x, -x, x, y) and beneath the second part of IS rotation sandwich (Delta_4, Delta_3, Delta_2). It also seems that these spacers are lacking in the figures of magnetization profiles as a function of J-value just beneath the Figure 2F. This figure should be revised accordingly. Also, from an aesthetic point of view, it would be useful to decrease the size of the pulse sequence scheme elements in Figure 2C,D,E,F and align their centers with the centers of the corresponding magnetization transfer profiles as a function of J placed just underneath.
In line 128, "In all cases but the BIG-BIRD profiles we overlaid simulated J-profiles as solid red lines over the experimental data for inner (uncoupled) and outer (coupled) multiplet components.", the simulated profiles are referred to red lines, while the green lines are shown in the figure. It would be helpful to make it consistently "red" in the plot and the Figure 3 caption, or change the text in the line 128 to "green".
In lines 31, 39, 156 and 157, a question mark is found instead of a citation. Please, correct it.
In line 163, one may find “would be good to be replaceed” where it should read “would be good to be replaced”, please correct.
In line 190, "lytropic liquid crystal" should read "lyotropic liquid crystal", please correct.
Citation: https://doi.org/10.5194/mr-2025-13-RC1
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The applications for BIRD are extensive and the introduction of Chirp pulses is a significant improvement in the method. Some readers may be interested in another area that was worked on back in the late 80's early 90's and that is the application to deuterium NMR in solids and liquid crystals. One can broaden the bandwidth of double quantum coherence and quadrupolar order excitation. This also came out of the Pines lab with a modest contribution by myself: Barbara, Tycko and and Weitekamp J. Magn. Reson. 62,54 (1985) and other papers by Steve Wimperis.