Dear reviewer,

I have submitted a revised version of the manuscript which addresses your comments. Please find the the detiled answer in the attached PDF.

kind regards

Philippe

Dear reviewer,

Thanks for the careful reading of the manuscript and your comments, I have addressed the different points in a revised version of the manuscript in the following way:

-I would say, this manuscript does not make it easy for the reader to appreciate what is being found, other than there are clearly some RD effects. It would be helpful if the text could contain an interpretation of what is actually happening other than simply comparing simulation and experiment.

I think that I cannot go much further in the interpretation, other than the fact already stated that TOCSY sequences remove the offset contribution to the effective Hamiltonian, hence the wider reach of the weak RD field. I added a phrase in the conclusions to stress this point. Maybe the additions to the article in answer to your third point (the theoretical investigation of the DIPSI sequence for very high RF amplitudes), does provides a more thorough analysis of the phenomenon.

-The description of the validation / control experiments (modulating the H2O z-magnetization) experiments is also a bit brief, and it would be helpful to have more guidance for the reader.

I have extended the description in the methods and the results sections.

-There are further details, which I feel should be discussed. For example, what is the actual bandwidth of the DIPSI? What would happen if the bandwidth were infinite?

I added simulations with stronger RF amplitudes to the main text and the supporting information and simulations at different offsets of the spin-lock. I rationalized (in an appendix) the observed nutation rates in the simulations with very high RF fields on resonance with the solvent frequency. I prefer not to discuss the bandwidth (which does not have a clearcut definition in this case) and just give the RF amplitude. However, I added simulations in the supporting information with different RF amplitudes.

-Are the RD effects only a consequence of the non-ideality of DIPSI? If that is the case, the effects are perhaps not really so surprising?

Other sequences have been simulated, showing similar effects (added to supporting information). I think that (see also answer to previous comment), rather the opposite is the case: the better (more ideal) the sequence the stronger the effect (see also above and my answer to Tom Barbara in the discussion section, https://doi.org/10.5194/mr-2021-65-AC1). 

-The description of the simulations is also unclear, was it performed for a spin-lock or for the actual multi-pulse sequence?

I think it is now clear that the actual multiple-pulse sequence has been simulated (see also the answer to the first reviewer), the code has also been included into supporting information.

Kind regards,

Philippe

Dear Malcolm Levitt,

Thanks for the careful reading of the manuscript and for pointing out the references. I address your comment in the revised version of my manuscript in the following way.

There is some early work on related effects by the group of Gareth Morris, see https://doi.org/10.1006/jmra.1995.9962 and doi.org/10.1016/j.jmr.2006.08.011. The authors should refer to these papers and other ones by the Morris group and comment on them as appropriate.

Indeed the effects described in this work and the above references (and the one mentioned by the first reviewer) have a common cause (the “non-quadrature” of the RD field). I added the mentioned articles to the references. I have also incorporated a more thorough analysis (including an appendix) for the case of very high spin-lock amplitudes (see also my answer to the other reviewers and to Tom Barbara in the discussion). Reading the articles (and comparing the equations) made me aware that I had forgotten a factor cos(ψ) in the radiation damping constant in equation 5, which has now been corrected.

Kind regards,

Philippe

Dear Tom Barbara (in answer to the two comments),

Thank you for your comments and for pointing out the references (which give analytical solutions to the modified Bloch equations in presence of an RF field for the case that the RD field is exactly perpendicular to the transverse magnetization). I am finishing a revised version which addresses the reviewers’ and your comments in detail. It is an interesting question whether there is a connection between the “imaginary resonance offset” and the effect observed in this article. Another relevant reference in this context may be Warren, Hammes and Bates, J. Chem. Phys. 91, 5895-5904 (1989), which analyzes the behavior of the solvent spins under a continuous wave spin-lock. The net effect of the DIPSI-2 cycle on the passive spins, shown in figures 3-5 of the manuscript, is that the magnetization of these spins precesses away from the much larger spin-lock field (it has the effect of a real offset, an effective Hamiltonian dominated by a term proportional to I_z). In contrast, the magnetization of the solvent (the source of RD) is much better (although not perfectly) controlled by de DIPSI-2 spin lock (as you also stated), i.e. it stays very close to the z-axis after each mixing cycle for the conditions in figure 5 while it stays close to the x-axis for the conditions in figure 6. The analysis can be much simplified when one considers an even stronger field (as reviewer 2 suggests) on-resonance with the solvent. When the initial solvent magnetization is along the x-axis, this magnetization stays continuously almost perfectly locked along this axis and (neglecting relaxation) the RD field is constant, with one component parallel to the magnetization equal to R_Rsin(ψ) and one term perpendicular to it along the y-axis equal to R_Rcos(ψ). The latter component is efficiently (in analogy to the offset term) averaged out from the effective Hamiltonian by the DIPSI-2 sequence, while the first component is unaffected since it commutes with the RF field at all times. Note that ψ must not be zero in order for this effect to be present under these conditions (as reviewer 1 and Malcolm Levitt suggest me to discuss). Hence, the effect of the RD field on the dilute spins (after each completed DIPSI-2 cycle) is expected to be a nutation around the x-axis with a frequency of R_Rsin(ψ)/2π. Simulations (the actual DIPSI-2 sequence has always been considered) with an RF amplitude of 100 kHz confirm this. When the initial solvent magnetization is perpendicular (either along the z- or the y-axis) to the spin-lock field, the situation is different since the RD field becomes time-dependent. The net effect seems to be a nutation around the axis of the initial solvent magnetization with a frequency of about one half compared to the previous one. When the initial magnetization of the solvent is oriented arbitrarily, the situation is much more complex. Contrary to (my) intuition, the evolution of the solvent magnetization can be strongly influenced by the much weaker RD field when the initial magnetization is not close to an orientation parallel to one of the three main axes.

Kind regards,

Philippe

Dear Matt Augustine,

Thank you for your comments and the reference you mentioned, which seems to have anticipated many later works. I prefer not to go into any practical aspects. The effect described in the article strongly depends on the exact instrumentation, the pulse sequence and the mixing time. Moreover, what is a tolerable perturbation depends on the system under study and the information one wants to obtain. TOCSY mixing sequences are mostly used in 2D experiments (briefly discussed in the revised version), which makes the analysis even more complex (and providing recommendations difficult) since one has to take into account the evolution of the solvent during the indirect evolution time. In the revised version, I do give (stimulated by Tom Barbara’s and the reviewers’ comments) a more thorough theoretical analysis of the effect in the limit of very strong RF fields. This could help the reader estimate the maximum possible perturbation for a given sequence and radiation damping parameters. If the TOCSY pulse train keeps control of the solvent, the effect is linear with the concentration. I am considering to add a concise version of this answer to the conclusions.

Kind regards,

Philippe