the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The solid effect of dynamic nuclear polarization in liquids II: Accounting for gtensor anisotropy at high magnetic fields
Danhua Dai
Thomas F. Prisner
Abstract. In spite of its name, the solid effect of dynamic nuclear polarization (DNP) is operative also in viscous liquids, where the dipolar interaction between the polarized nuclear spins and the polarizing electrons is not completely averaged out by molecular diffusion. Under such slowmotional conditions, it is likely that the tumbling of the polarizing agent is similarly too slow to efficiently average the anisotropies of its magnetic tensors. Here we extend our previous analysis of the solid effect in liquids to account for the effect of gtensor anisotropy at high magnetic fields. Building directly on the mathematical treatment of slowtumbling in electron spin resonance (Freed et al., 1971), we calculate solideffect DNP enhancements in the presence of both translational diffusion of the liquid molecules and rotational diffusion of the polarizing agent. To illustrate the formalism, we analyze highfield (9.4 T) DNP enhancement profiles from nitroxidelabeled lipids in fluid lipid bilayers. By properly accounting for powerbroadening and motionalbroadening, we successfully decompose the measured DNP enhancements into their separate contributions from the solid and Overhauser effects.
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Deniz Sezer et al.
Status: closed

RC1: 'Comment on mr202310', Gunnar Jeschke, 11 Aug 2023
This work extends previous theory on solideffect DNP in viscous liquids by one of the authors to the more complicated situation where g anisotropy is partially averaged by slow tumbling. The derivations provide the basis for a computationally affordable numerical treatment. Field profiles of DNP enhancements (“DNP spectra”) fitted on the basis of this approach are in rather good agreement with experiments. The fitted parameters take physically reasonable values, despite the fact that the model for spin label dynamics is strongly simplified with respect to expected dynamics in the experimentally studied systems. The results constitute a substantial advance in a field of current interest. The manuscript is mostly clear and well written. I recommend publication after minor revision that takes into account the following issues.
 The treatment of the experimental spectra assumes isotropic rotational diffusion (stated, a bit late, in line 602). For 10DoxylPC in DOPC bilayers, this motional model is a strong simplification. Even for 16DoxylPC, which is closer to the chain end and experiences less of an orienting potential, I am not sure how good an isotropicmotion model is. This, and the worsened agreement upon including nitrogen hyperfine coupling/anisotropy for 10DoxylPC (Figure 4A) may indicate error compensation between deficiencies of the motion model and the neglect of ^{14}N hyperfine coupling. In my opinion, this possibility should be mentioned. The difference in parameters between 10DoxylPC and 16DoxylPC (Table 2) might partially be due to the motional model being worse for 10DoxylPC than for 16DoxylPC.
 Motion of spinlabelled lipids in lipid bilayers is a rather well researched topic (see, e.g., Livshits, V.A., Kurad, D., Marsh, D. Multifrequency simulations of the EPR spectra of lipid spin labels in membranes. J. Magn. Reson. 2006, 180, 6371 and references therein). I believe that the motional model of 16DoxylPC and 10DoxylPC in DOPC bilayers could be substantially improved by measuring CW EPR spectra at a few more frequencies and fitting them by the microscopicorder macroscopicdisorder model (MOMD) developed by Freed et al. While this is clearly beyond the scope of the present study, I would like to see some comment whether the approach introduced in this manuscript could be used with a MOMD model or at least with a model of anisotropic rotational diffusion (axial rotational diffusion tensor).
 I wonder how difficult it would really be to extend the approach to including ^{14}N hyperfine anisotropy. I would assume that computation time increases only by a factor of three, because the subspaces with different nuclear magnetic quantum number of ^{14}N could be treated independently. The g and hyperfine tensor being coaxial, each subspace could be computed with an effective tensor. The authors may want to consider whether this would be feasible and comment on it. Again, it may be beyond the scope of this study, but may also be of interest, as neglect of ^{14}N hyperfine coupling does make a difference (Figures A4, A5).
 Work such as this one is notoriously hard to reproduce without having access to the programs used for the computations. The authors may want to consider putting the program on GitHub.
Details:
 Around line 110: “cwEPR spectrum deviates from a Lorentzian line”. I would have written “cwEPR lineshape is not Lorentzian”, because spectra often consist of several Lorentzian lines, not only one.
 When first discussing the deviation in Fig. 1(d) between experimental data and the decomposition into OE and SE contribution, it would be helpful to tell the reader that the paper addresses this issue.
 Line 201, “When generalizing the Bloch equations to nonisotropic g tensor” I would have written “When generalizing the Bloch equations to an anisotropic g tensor”
 Line 220/221: “For the same analysis to apply to solids, spin diffusion should be much faster”. I do not understand how spin diffusion enters the picture here. In the previous sentence, you mention the correlation time of the (supposedly electronnuclear) dipolar interaction. There is no electronnuclear spin diffusion that could be an analogous phenomenon in solids.
 Line 224/225: “correlation time of the dipolar interaction is infinitely long (but still much shorter than the nuclear T1” Please replace “infinitely long” with “much longer than…”. Probably you compare to relaxation on zeroquantum and doublequantum transitions in the absence of fluctuating electronnuclear dipolar interaction?
 Figure A4: The simulation agrees better with experiment without considering the hyperfine interaction that certainly does exist. Some error compensation must be at play here. This needs to be mentioned.
Citation: https://doi.org/10.5194/mr202310RC1 
RC2: 'Comment on mr202310', Anonymous Referee #2, 16 Aug 2023
The paper is a continuation of the series of theoretical works on the Solid Effect. The paper considers the effect of gfactor anisotropy on the DNP spectra. I believe that this theoretical work deserves to be published in the “Magnetic Resonance”. The work is rather bulk, both in volume and in the cases and approximations considered, and, in my opinion, it would be better to divide it into two publications, for instance, in one to consider the theory and in the second to compare the theory with experiment. But this is at the discretion of the authors. I assume that specialists working in the field of DNP will still be able to understand this paper in detail.
I suggest to accept this paper as it is with the following minor remarks:
Already in the abstract states that "the dipolar interaction between the polarized nuclear spins and the polarizing electrons is not completely averaged out
by molecular diffusion." But in the case of isotropic rotation, which is considered in the paper, the mean value of the dipole interaction is zero, whether the rotation is slow or fast. It is necessary to explain in what sense the dipole interaction is not completely averaged out. Also I did not find (perhaps I did not notice) the criterion of "slowness" of rotation or tumbling. Perhaps dipole interaction is not averaged to zero in nitroxidelabeled lipids in fluid lipid bilayers, where there is preferred direction in the bilayer, but the paper considers, as I understood, isotropic rotation. Further, if the rotation is slow in the sense of adiabatic mode (the criterion of adiabaticity is known), it may be easier to consider SE in adiabatic approximation with solid angle \Omega as an adiabatic parameter.
The second minor remark. I was confused by the statement:
"In (59), the sum over l mixes only expansion coefficients whose values l differ by two from L". Does this mean that l=L+2 or l=L2, rather than the correct condition L2 <=l <=L+2 ?
Citation: https://doi.org/10.5194/mr202310RC2 
RC3: 'Comment on mr202310', Anonymous Referee #3, 26 Aug 2023
The manuscript describes detailed work on simulating DNP features originating from the solid effect, both in solids, and in liquids. The liquids case is the most interesting one, whereby asymmetry in the gtensor leads to sizable solideffect features. The work is truly impressive, providing simplified equations that allow very efficient computation of lineshapes, and thereby also providing an efficient data fitting modality. The work is probably easily accessible to the expert, but the broader readership would appreciate if further helpful comments were included. For example, in each section where a certain calculation is performed, it would be very helpful if at the beginning it were stated what the final goal of this portion is. I also feel that it may be suitable to cite the review article by Atsarkin. Some symbols are introduced in an ad hoc fashion which makes it harder to follow the thought process. For example, the provenance of the i \omega_I terms in Eq. 19 is not explained, and one is left to guess. So some additional guiding statements would greatly improve the article.
Citation: https://doi.org/10.5194/mr202310RC3 
EC1: 'Comment on mr202310', Geoffrey Bodenhausen, 05 Sep 2023
This paper has been carefully read by no less than three reviewers. The corresponding author can submit his final version provided he takes their comments into account, even if the reviewers have not found enough time to respond.
Citation: https://doi.org/10.5194/mr202310EC1 
AC1: 'Comment on mr202310', Deniz Sezer, 14 Sep 2023
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202310/mr202310AC1supplement.pdf
Status: closed

RC1: 'Comment on mr202310', Gunnar Jeschke, 11 Aug 2023
This work extends previous theory on solideffect DNP in viscous liquids by one of the authors to the more complicated situation where g anisotropy is partially averaged by slow tumbling. The derivations provide the basis for a computationally affordable numerical treatment. Field profiles of DNP enhancements (“DNP spectra”) fitted on the basis of this approach are in rather good agreement with experiments. The fitted parameters take physically reasonable values, despite the fact that the model for spin label dynamics is strongly simplified with respect to expected dynamics in the experimentally studied systems. The results constitute a substantial advance in a field of current interest. The manuscript is mostly clear and well written. I recommend publication after minor revision that takes into account the following issues.
 The treatment of the experimental spectra assumes isotropic rotational diffusion (stated, a bit late, in line 602). For 10DoxylPC in DOPC bilayers, this motional model is a strong simplification. Even for 16DoxylPC, which is closer to the chain end and experiences less of an orienting potential, I am not sure how good an isotropicmotion model is. This, and the worsened agreement upon including nitrogen hyperfine coupling/anisotropy for 10DoxylPC (Figure 4A) may indicate error compensation between deficiencies of the motion model and the neglect of ^{14}N hyperfine coupling. In my opinion, this possibility should be mentioned. The difference in parameters between 10DoxylPC and 16DoxylPC (Table 2) might partially be due to the motional model being worse for 10DoxylPC than for 16DoxylPC.
 Motion of spinlabelled lipids in lipid bilayers is a rather well researched topic (see, e.g., Livshits, V.A., Kurad, D., Marsh, D. Multifrequency simulations of the EPR spectra of lipid spin labels in membranes. J. Magn. Reson. 2006, 180, 6371 and references therein). I believe that the motional model of 16DoxylPC and 10DoxylPC in DOPC bilayers could be substantially improved by measuring CW EPR spectra at a few more frequencies and fitting them by the microscopicorder macroscopicdisorder model (MOMD) developed by Freed et al. While this is clearly beyond the scope of the present study, I would like to see some comment whether the approach introduced in this manuscript could be used with a MOMD model or at least with a model of anisotropic rotational diffusion (axial rotational diffusion tensor).
 I wonder how difficult it would really be to extend the approach to including ^{14}N hyperfine anisotropy. I would assume that computation time increases only by a factor of three, because the subspaces with different nuclear magnetic quantum number of ^{14}N could be treated independently. The g and hyperfine tensor being coaxial, each subspace could be computed with an effective tensor. The authors may want to consider whether this would be feasible and comment on it. Again, it may be beyond the scope of this study, but may also be of interest, as neglect of ^{14}N hyperfine coupling does make a difference (Figures A4, A5).
 Work such as this one is notoriously hard to reproduce without having access to the programs used for the computations. The authors may want to consider putting the program on GitHub.
Details:
 Around line 110: “cwEPR spectrum deviates from a Lorentzian line”. I would have written “cwEPR lineshape is not Lorentzian”, because spectra often consist of several Lorentzian lines, not only one.
 When first discussing the deviation in Fig. 1(d) between experimental data and the decomposition into OE and SE contribution, it would be helpful to tell the reader that the paper addresses this issue.
 Line 201, “When generalizing the Bloch equations to nonisotropic g tensor” I would have written “When generalizing the Bloch equations to an anisotropic g tensor”
 Line 220/221: “For the same analysis to apply to solids, spin diffusion should be much faster”. I do not understand how spin diffusion enters the picture here. In the previous sentence, you mention the correlation time of the (supposedly electronnuclear) dipolar interaction. There is no electronnuclear spin diffusion that could be an analogous phenomenon in solids.
 Line 224/225: “correlation time of the dipolar interaction is infinitely long (but still much shorter than the nuclear T1” Please replace “infinitely long” with “much longer than…”. Probably you compare to relaxation on zeroquantum and doublequantum transitions in the absence of fluctuating electronnuclear dipolar interaction?
 Figure A4: The simulation agrees better with experiment without considering the hyperfine interaction that certainly does exist. Some error compensation must be at play here. This needs to be mentioned.
Citation: https://doi.org/10.5194/mr202310RC1 
RC2: 'Comment on mr202310', Anonymous Referee #2, 16 Aug 2023
The paper is a continuation of the series of theoretical works on the Solid Effect. The paper considers the effect of gfactor anisotropy on the DNP spectra. I believe that this theoretical work deserves to be published in the “Magnetic Resonance”. The work is rather bulk, both in volume and in the cases and approximations considered, and, in my opinion, it would be better to divide it into two publications, for instance, in one to consider the theory and in the second to compare the theory with experiment. But this is at the discretion of the authors. I assume that specialists working in the field of DNP will still be able to understand this paper in detail.
I suggest to accept this paper as it is with the following minor remarks:
Already in the abstract states that "the dipolar interaction between the polarized nuclear spins and the polarizing electrons is not completely averaged out
by molecular diffusion." But in the case of isotropic rotation, which is considered in the paper, the mean value of the dipole interaction is zero, whether the rotation is slow or fast. It is necessary to explain in what sense the dipole interaction is not completely averaged out. Also I did not find (perhaps I did not notice) the criterion of "slowness" of rotation or tumbling. Perhaps dipole interaction is not averaged to zero in nitroxidelabeled lipids in fluid lipid bilayers, where there is preferred direction in the bilayer, but the paper considers, as I understood, isotropic rotation. Further, if the rotation is slow in the sense of adiabatic mode (the criterion of adiabaticity is known), it may be easier to consider SE in adiabatic approximation with solid angle \Omega as an adiabatic parameter.
The second minor remark. I was confused by the statement:
"In (59), the sum over l mixes only expansion coefficients whose values l differ by two from L". Does this mean that l=L+2 or l=L2, rather than the correct condition L2 <=l <=L+2 ?
Citation: https://doi.org/10.5194/mr202310RC2 
RC3: 'Comment on mr202310', Anonymous Referee #3, 26 Aug 2023
The manuscript describes detailed work on simulating DNP features originating from the solid effect, both in solids, and in liquids. The liquids case is the most interesting one, whereby asymmetry in the gtensor leads to sizable solideffect features. The work is truly impressive, providing simplified equations that allow very efficient computation of lineshapes, and thereby also providing an efficient data fitting modality. The work is probably easily accessible to the expert, but the broader readership would appreciate if further helpful comments were included. For example, in each section where a certain calculation is performed, it would be very helpful if at the beginning it were stated what the final goal of this portion is. I also feel that it may be suitable to cite the review article by Atsarkin. Some symbols are introduced in an ad hoc fashion which makes it harder to follow the thought process. For example, the provenance of the i \omega_I terms in Eq. 19 is not explained, and one is left to guess. So some additional guiding statements would greatly improve the article.
Citation: https://doi.org/10.5194/mr202310RC3 
EC1: 'Comment on mr202310', Geoffrey Bodenhausen, 05 Sep 2023
This paper has been carefully read by no less than three reviewers. The corresponding author can submit his final version provided he takes their comments into account, even if the reviewers have not found enough time to respond.
Citation: https://doi.org/10.5194/mr202310EC1 
AC1: 'Comment on mr202310', Deniz Sezer, 14 Sep 2023
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202310/mr202310AC1supplement.pdf
Deniz Sezer et al.
Deniz Sezer et al.
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