#
Stochastic Modelling of ^{13}C NMR Spin Relaxation Experiments in Oligosaccharides

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{13}C nuclear magnetic resonance relaxation parameters, ${T}_{1}$ and ${T}_{2}$, and the heteronuclear NOE of several oligosaccharides, which were previously interpreted on the basis of refined ad hoc modelling. The calculated NMR relaxation parameters were in agreement with the experimental data, showing that this general approach can be applied to diverse classes of molecular systems, with the minimal usage of adjustable parameters.

## 1. Introduction

^{15}N,

^{2}H and

^{13}C nuclei, which are extremely sensitive to molecular motions, leading to the possibility to understand localized dynamics (e.g., studying conformational motions specifically in the active site of a protein) and to build a spatially distributed map of the macromolecule flexibility. However, interpretative tools can be complex due to several factors such as (i) the necessity to take into account diverse kinds of interactions, e.g., dipolar

^{15}N and

^{13}C and quadrupolar

^{2}H interactions, (ii) the coupling between global reorientation and large amplitude motions of entire domains, as well as limited local readjustments and restricted single-residue motions. In general, different spectroscopic techniques probe different physical observables, which, in addition, provide information on motions taking place at different time-scales. It seems therefore particularly important to introduce relevant sets of coordinates that are adapted to the observable involved in a particular experimental approach. This consideration is especially relevant in the case of NMR relaxation [1,10], for which interpretative methods for internal relaxation processes were introduced early on in the form of adaptable simple spectral densities, as in the Lipari–Szabo (LS) approach [11,12], or later, in the form of explicit dynamic models, as for instance in the Slowly Relaxing Local Structure (SRLS) model [13,14].

^{13}C nuclear magnetic resonance relaxation parameters, ${T}_{1}$ and ${T}_{2}$, and the heteronuclear NOE of several oligosaccharides, which were previously interpreted on the basis of ad hoc stochastic modelling. In particular, we discuss the computational strategy and implementation of the method and detailed results, which confirm how the calculated NMR relaxation parameters are in satisfactory agreement with the experimental data, and we suggest that this general approach can be safely applied to diverse classes of molecular systems, with a minimal usage of adjustable parameters.

## 2. Methods

#### 2.1. Observable and Geometric Setup

^{15}N,

^{13}C and

^{2}H nuclei depend on dipolar (

^{15}N and

^{13}C) and quadrupolar (

^{2}H) interactions, on chemical shift anisotropy and cross-correlation effects. In general, we define the following set of reference frames: (i) a Laboratory Frame (LF), i.e., a fixed external frame; (ii) an “Attached” Frame (AF), i.e., a frame attached to the molecule, where the exact way of defining the AF is actually model-dependent and is left temporarily undefined, noting that the choice of the AF while straightforward for a rigid molecule is not trivial for a flexible system; (iii) an interaction frame ($\mu \mathrm{F}$), i.e., a local frame linked to the AF where some specific second-rank tensor spectroscopic property $\mu $ is well represented. Depending on the problem at hand, this could be for instance the frame where the

^{13}C-

^{1}H dipolar or

^{13}C chemical shift (CSA) tensors are diagonal [20,21]. In Figure 1, an example is shown of the frame choice to compute the NMR relaxation data of a

^{13}C-

^{1}H probe. The set of Euler angles ${\Omega}$ or other orientational coordinates transforming from the Laboratory Frame (LF) to the AF is time dependent and linked to the local restricted motions, large amplitude conformational motion and global orientation of the molecule. The dipolar and CSA frames are usually supposed to be rigidly attached to the AF. Here, only the Dipolar Frame (DF) is shown as an example.

#### 2.2. Dynamic Model

#### 2.3. Parameterization and Implementation

## 3. Results

- $\alpha $-L-Rhap-$\alpha $-(1→2)-$\alpha $-L-Rhap-OMe (two residues,
**R2R**); experimental data: Reference [28] - $\beta $-D-Glcp-(1→6)-$\alpha $-D-[6-${}^{13}$C]-Manp-OMe (two residues,
**BGL**); experimental data: Reference [29] - $\beta $-D-Glcp-(1→3)[$\beta $-D-Glcp-(1→2)]-$\alpha $-D-Manp-OMe (three residues,
**GGM**); experimental data: Reference [30] - $\alpha $-D-Manp-(1→2)-$\alpha $-D-Manp-(1→6)-$\alpha $-D-[6-${}^{13}$C]-Manp-OMe (three residues,
**TRI**); experimental data: Reference [31] - $\alpha $-L-Fucp-(1→2)-$\beta $-D-Galp-(1→3)-$\beta $-D-GlcpNAc-(1→3)-$\beta $-D-Galp-(1→4)-D-Glcp- (five residues,
**LNF**); experimental data: Reference [31]

**BGL**at 253 K (Table 2), which is the case of a lower temperature and a solvent with higher viscosity, we found an agreement with experimental data within 20% of the relative error, which was the worst scenario in all the test calculations presented here. Simulations around 298 K and in solvents with a similar viscosity showed an agreement within 10% with the experiments.

**R2R**Potential Energy Surface (PES) along the $\Psi $ angle on ${T}_{1}$, ${T}_{2}$, and the NOE. From MD simulations, it was found that the PES was bistable. The calculation of the NMR relaxation data with the harmonic approximation around the most important minimum led to a 10% error with respect to the calculation done with the bistable potential. The reader is encouraged to inspect Figure 5 of [31]. It reported the 2D PESs along the four ($\Phi $, $\Psi $) couples of dihedral angles connecting the different sugar units. The PES along the angles connecting rings C and D was bistable, while the other three surfaces could be considered, as first approximation, harmonic. A one-to-one mapping between the probe position and the conformational free energy was not possible. Globally, the approximations done in the SFB model implied that the estimation of a few NMR relaxation data was outside the experimental error (e.g., ${T}_{1}$ for the A and the E rings), and the overall agreement was 5–10% worse than the agreement found with the diffusive chain stochastic model applied in the past [31].

**LNF**, for which a higher value of 3.2 Å was obtained, which could be due to the molecule being particularly hydrophilic. This, in turn, implies that the molecular dimensions should be increased to take into account a layer of water surrounding the penta-saccharide.

## 4. Discussion

**GCY**, as discussed in [32], the rotation of the hydroxymethyl group with respect to the sugar ring described by the torsional angle $\theta $—see Figure 7— featured two energy minima at $\theta \approx -70$ and $\theta \approx +50$, with the former conformation being the predominant one. The Tinker energy minimization led to a structure with six over eight units in the predominant conformation, and the results shown above were those obtained running SALEM on one of these six probes.

#### Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Relaxation Observables: An Example

## References

- Cavanagh, J.; Fairbrother, W.J.; Palmer, I.A.G.; Skelton, N.J.; Rance, M. Protein NMR Spectroscopy: Principles and Practice; Elsevier Science: Amsterdam, The Netherlands, 2010. [Google Scholar]
- Loth, K.; Pelupessy, P.; Bodenhausen, G. Chemical shift anisotropy tensors of carbonyl, nitrogen, and amide proton nuclei in proteins through cross-correlated relaxation in NMR spectroscopy. J. Am. Chem. Soc.
**2005**, 127, 6062–6068. [Google Scholar] [CrossRef] - Sheppard, D.; Li, D.W.; Godoy-Ruiz, R.; Brueschweiler, R.; Tugarinov, V. Variation in Quadrupole Couplings of a Deuterons in Ubiquitin Suggests the Presence of C-alpha-H-alpha center dot center dot center dot O=C Hydrogen Bonds. J. Am. Chem. Soc.
**2010**, 132, 7709–7719. [Google Scholar] [CrossRef] [PubMed] - Bucci, E.; Steiner, R.F. Anisotropy Decay of Fluorescence as an Experimental Approach to Protein Dynamics. Biophys. Chem.
**1988**, 30, 199–224. [Google Scholar] [CrossRef] - Vergani, B.; Kintrup, M.; Hillen, W.; Lami, H.; Piemont, E.; Bombarda, E.; Alberti, P.; Doglia, S.M.; Chabbert, M. Backbone dynamics of Tet repressor alpha 8 boolean AND alpha 9 loop. Biochemistry
**2000**, 39, 2759–2768. [Google Scholar] [CrossRef] [PubMed] - Hubbell, W.L.; Mchaourab, H.S.; Altenbach, C.; Lietzow, M.A. Watching proteins move using site-directed spin labeling. Structure
**1996**, 4, 779–783. [Google Scholar] [CrossRef][Green Version] - Zhang, Z.; Fleissner, M.R.; Tipikin, D.S.; Liang, Z.; Moscicki, J.K.; Earle, K.A.; Hubbell, W.L.; Freed, J.H. Multifrequency Electron Spin Resonance Study of the Dynamics of Spin Labeled T4 Lysozyme. J. Phys. Chem. B
**2010**, 114, 5503–5521. [Google Scholar] [CrossRef][Green Version] - Hofmann, H.; Hillger, F.; Pfeil, S.H.; Hoffmann, A.; Streich, D.; Haenni, D.; Nettels, D.; Lipman, E.A.; Schuler, B. Single-molecule spectroscopy of protein folding in a chaperonin cage. Proc. Natl. Acad. Sci. USA
**2010**, 107, 11793–11798. [Google Scholar] [CrossRef][Green Version] - Hamon, L.; Pastre, D.; Dupaigne, P.; Breton, C.L.; Cam, E.L.; Pietrement, O. High-resolution AFM imaging of single-stranded DNA-binding (SSB) protein-DNA complexes. Nucleic Acids Res.
**2007**, 35, e58. [Google Scholar] [CrossRef][Green Version] - Kowalewski, J.; Maler, L. Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications; Taylor & Francis: Abingdon, UK, 2006. [Google Scholar]
- Lipari, G.; Szabo, A. Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity. J. Am. Chem. Soc.
**1982**, 104, 4546–4559. [Google Scholar] [CrossRef] - Lipari, G.; Szabo, A. Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 2. Analysis of experimental results. J. Am. Chem. Soc.
**1982**, 104, 4559–4570. [Google Scholar] [CrossRef] - Shapiro, Y.E.; Kahana, E.; Tugarinov, V.; Liang, Z.C.; Freed, J.H.; Meirovitch, E. Domain flexibility in ligand-free and inhibitor-bound Escherichia coli adenylate kinase based on a mode-coupling analysis of N-15 spin relaxation. Biochemistry
**2002**, 41, 6271–6281. [Google Scholar] [CrossRef][Green Version] - Meirovitch, E.; Shapiro, Y.E.; Polimeno, A.; Freed, J.H. An improved picture of methyl dynamics in proteins from slowly relaxing local structure analysis of H-2 spin relaxation. J. Phys. Chem. B
**2007**, 111, 12865–12875. [Google Scholar] [CrossRef][Green Version] - Abergel, D.; Bodenhausen, G. Predicting internal protein dynamics from structures using coupled networks of hindered rotators. J. Chem. Phys.
**2005**, 123, 204901. [Google Scholar] [CrossRef] - Dhulesia, A.; Bodenhausen, G.; Abergel, D. Predicting conformational entropy of bond vectors in proteins by networks of coupled rotators. J. Chem. Phys.
**2008**, 129, 095107. [Google Scholar] [CrossRef][Green Version] - Calandrini, V.; Abergel, D.; Kneller, G.R. Fractional protein dynamics seen by nuclear magnetic resonance spectroscopy: Relating molecular dynamics simulation and experiment. J. Chem. Phys.
**2010**, 133, 145101. [Google Scholar] [CrossRef] - Polimeno, A.; Zerbetto, M.; Abergel, D. Stochastic modelling of macromolecules in solution. I. Relaxation processes. J. Chem. Phys.
**2019**, 150, 184107. [Google Scholar] [CrossRef] - Polimeno, A.; Zerbetto, M.; Abergel, D. Stochastic modelling of macromolecules in solution. II. Spectral densities. J. Chem. Phys.
**2019**, 150, 184108. [Google Scholar] [CrossRef] - Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, UK, 1961. [Google Scholar]
- Peng, J.W.; Wagner, G. Investigation of Protein Motions via Relaxation Measurements; Methods in Enzymology; Academic Press: Cambridge, MA, USA, 1994; pp. 563–595. [Google Scholar]
- Zare, R. Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics; Wiley: Hoboken, NJ, USA, 1988. [Google Scholar]
- Moro, G.J. A stochastic model for crankshaft transitions. J. Phys. Chem.
**1996**, 100, 16419–16422. [Google Scholar] [CrossRef] - Nigro, B.; Moro, G.J. A stochastic model for crankshaft transitions. II. Analysis of transition dynamics. J. Phys. Chem. B
**2002**, 106, 7365–7375. [Google Scholar] [CrossRef] - Risken, H. The Fokker-Planck Equation: Methods of Solution and Applications; Springer: Berlin/Heidelberg, Germany, 1984. [Google Scholar]
- Barone, V.; Zerbetto, M.; Polimeno, A. Hydrodynamic Modeling of Diffusion Tensor Properties of Flexible Molecules. J. Comput. Chem.
**2009**, 30, 2–13. [Google Scholar] [CrossRef] - Rackers, J.A.; Wang, Z.; Lu, C.; Laury, M.L.; Lagardère, L.; Schnieders, M.J.; Piquemal, J.P.; Ren, P.; Ponder, J.W. Tinker 8: Software Tools for Molecular Design. J. Chem. Theory Comput.
**2018**, 14, 5273–5289. [Google Scholar] [CrossRef] - Pendrill, R.; Engstrom, O.; Volpato, A.; Zerbetto, M.; Polimeno, A.; Widmalm, G. Flexibility at a glycosidic linkage revealed by molecular dynamics, stochastic modelling, and 13C NMR spin relaxation: Conformational preferences of α-l-Rhap-α-(1 → 2)-α-l-Rhap-OMe in water and dimethyl sulfoxide solutions. Phys. Chem. Chem. Phys.
**2016**, 18, 3086–3096. [Google Scholar] [CrossRef][Green Version] - Zerbetto, M.; Polimeno, A.; Kotsyubynskyy, D.; Ghalebani, L.; Kowalewski, J.; Meirovitch, E.; Olsson, U.; Widmalm, G. An integrated approach to NMR spin relaxation in flexible biomolecules: Application to β-D-glucopyranosyl-(1→6)-α-D-mannopyranosyl-OMe. J. Chem. Phys.
**2009**, 131, 234501. [Google Scholar] [CrossRef] - Zerbetto, M.; Angles d’Ortoli, T.; Polimeno, A.; Widmalm, G. Differential Dynamics at Glycosidic Linkages of an Oligosaccharide as Revealed by
^{13}C NMR Spin Relaxation and Stochastic Modeling. J. Phys. Chem. B**2018**, 122, 2287–2294. [Google Scholar] [CrossRef][Green Version] - Kotsyubynskyy, D.; Zerbetto, M.; Soltesova, M.; Engström, O.; Pendrill, R.; Kowalewski, J.; Widmalm, G.; Polimeno, A. Stochastic Modeling of Flexible Biomolecules Applied to NMR Relaxation. 2. Interpretation of Complex Dynamics in Linear Oligosaccharides. J. Phys. Chem. B
**2012**, 116, 14541–14555. [Google Scholar] [CrossRef] - Zerbetto, M.; Kotsyubynskyy, D.; Kowalewski, J.; Widmalm, G.; Polimeno, A. Stochastic Modeling of Flexible Biomolecules Applied to NMR Relaxation. I. Internal Dynamics of Cyclodextrins: γ-Cyclodextrin as a Case Study. J. Phys. Chem. B
**2012**, 116, 13159–13171. [Google Scholar] [CrossRef] [PubMed] - Allinger, N.L.; Yuh, Y.H.; Lii, J.H. Molecular mechanics. The MM3 force field for hydrocarbons. 1. J. Am. Chem. Soc.
**1989**, 111, 8551–8566. [Google Scholar] [CrossRef] - Lii, J.H.; Allinger, N.L. Molecular mechanics. The MM3 force field for hydrocarbons. 2. Vibrational frequencies and thermodynamics. J. Am. Chem. Soc.
**1989**, 111, 8566–8575. [Google Scholar] [CrossRef] - Lii, J.H.; Allinger, N.L. Molecular mechanics. The MM3 force field for hydrocarbons. 3. The van der Waals’ potentials and crystal data for aliphatic and aromatic hydrocarbons. J. Am. Chem. Soc.
**1989**, 111, 8576–8582. [Google Scholar] [CrossRef] - Stroylov, V.; Panova, M.; Toukach, P. Comparison of Methods for Bulk Automated Simulation of Glycosidic Bond Conformations. Int. J. Mol. Sci.
**2020**, 21, 7626. [Google Scholar] [CrossRef] [PubMed] - Foley, B.L.; Tessier, M.B.; Woods, R.J. Carbohydrate force fields. WIREs Comput. Mol. Sci.
**2012**, 2, 652–697. [Google Scholar] [CrossRef] [PubMed] - Zerbetto, M.; Polimeno, A.; Widmalm, G. Glycosidic linkage flexibility: The Ψ torsion angle has a bimodal distribution in α-L-Rhap-(1→2)-α-L-Rhap-OMe as deduced from 13C NMR spin relaxation. J. Chem. Phys.
**2020**, 152, 035103. [Google Scholar] [CrossRef] - Kuprov, I.; Morris, L.C.; Glushka, J.N.; Prestegard, J.H. Using molecular dynamics trajectories to predict nuclear spin relaxation behaviour in large spin systems. J. Magn. Res.
**2021**, 323, 106891. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Choice of the reference atoms in the calculation of NMR relaxation observables. (

**b**) Relevant reference frames: Laboratory (LF), Molecular (AF) and Dipolar Frames (DF). The sets of Euler angles to transform among the frames are also shown. The sets ${\Omega}$ and ${{\Omega}}_{\mathrm{D}}$ are time dependent, while ${{\Omega}}_{\mathrm{AD}}$ is time independent.

**Figure 2.**Local-minimum structure of the

**R2R**system. Atoms of the active probe are highlighted as spheres.

**Figure 3.**Local-minimum structure of the

**BGL**system. Atoms of the active probe are highlighted as spheres.

**Figure 4.**Local-minimum structure of the

**GGM**system. Atoms of the active probes are highlighted as spheres.

**Figure 5.**Local-minimum structure of the

**TRI**system. Atoms of the active probe are highlighted as spheres.

**Figure 6.**Local-minimum structure of the

**LNF**system. Atoms of the active probes are highlighted as spheres.

**Figure 7.**Local-minimum structure of the

**GCY**system. Atoms of the active probe are highlighted as spheres. The torsional angle $\theta $ for the rotation of the hydroxymethyl group with respect to the sugar ring is also highlighted.

**Table 1.**Experimental and calculated relaxation parameters for system

**R2R**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$, and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Probe: ${}^{13}$CH, Solvent: DMSO-d${}_{6}$,T/K: 298.2, Visc./(Pa s): 2.19 × 10${}^{-3}$ | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600.1 | exp. | 440.0 | 402.6 | 2.361 | ||||

calc. | 449.9 | 420.5 | 2.308 | 1.6 | 2.3 | 4.4 | 2.3 | |

700.0 | exp. | 475.6 | 432.9 | 2.215 | ||||

calc. | 497.5 | 456.1 | 2.150 | 1.6 | 4.6 | 5.4 | 2.9 |

**Table 2.**Experimental and calculated relaxation parameters for system

**BGL**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$ and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Probe: ${}^{13}$CH${}_{2}$, Solvent: DMSO-d${}_{6}$/D${}_{2}$O 7:3 Molar Ratio,T/K: 253, Visc./(Pa s): 2.82 × 10${}^{-2}$ | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

400 | exp. | 150 | 32.5 | 1.03 | ||||

calc. | 177 | 31.4 | 1.22 | 1.8 | 17.8 | 3.2 | 18.5 | |

600 | exp. | 284 | 30.1 | 1.08 | ||||

calc. | 344 | 33.1 | 1.21 | 1.8 | 21.1 | 10.1 | 12.2 | |

$\mathit{T}$/K: 263, Visc./(Pa s): 1.42 × 10${}^{-\mathbf{2}}$ | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

400 | exp. | 117 | 48.0 | 1.19 | ||||

calc. | 121 | 53.7 | 1.26 | 1.8 | 3.6 | 11.8 | 5.9 | |

600 | exp. | 205 | 55.0 | 1.10 | ||||

calc. | 219 | 61.0 | 1.23 | 1.8 | 6.9 | 10.9 | 12.1 | |

$\mathit{T}$/K: 293, Visc./(Pa s): 4.30 × 10${}^{-\mathbf{3}}$ | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

400 | exp. | 127 | 106 | 1.86 | ||||

calc. | 124 | 107 | 1.78 | 1.8 | 2.0 | 1.0 | 4.5 | |

600 | exp. | 166 | 128 | 1.58 | ||||

calc. | 170 | 133 | 1.51 | 1.8 | 2.6 | 3.8 | 4.5 | |

900 | exp. | 219 | 152 | 1.45 | ||||

calc. | 249 | 153 | 1.33 | 1.8 | 13.6 | 0.6 | 8.3 |

**Table 3.**Experimental and calculated relaxation parameters for system

**GGM**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$ and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Solvent: D${}_{2}$O,T/K: 298.6, Visc./(Pa s): 1.09 × 10${}^{-3}$, Probe:${}^{13}$CH on C-2′ | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600.13 | exp. | 456.2 | 416.6 | 2.398 | ||||

calc. | 453.2 | 425.7 | 2.299 | 1.8 | 0.7 | 2.2 | 4.1 | |

699.87 | exp. | 491.1 | 447.4 | 2.267 | ||||

calc. | 503.8 | 463.6 | 2.148 | 1.8 | 2.6 | 3.6 | 5.3 | |

Probe:${}^{\mathbf{13}}$CH on C-2′ | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600.13 | exp. | 491.6 | 450.2 | 2.466 | ||||

calc. | 470.0 | 444.6 | 2.358 | 1.8 | 4.4 | 1.2 | 4.4 | |

699.87 | exp. | 524.5 | 483.9 | 2.346 | ||||

calc. | 521.5 | 483.6 | 2.203 | 1.8 | 0.6 | 0.1 | 6.1 |

**Table 4.**Experimental and calculated relaxation parameters for system

**TRI**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$ and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Probe: ${}^{13}$CH${}_{2}$, Solvent: DMSO-d${}_{6}$/D${}_{2}$O 7:3 Molar Ratio,T/K: 298, Visc./(Pa s): 3.66 × 10${}^{-3}$ | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

500 | exp. | 144.79 | 111.57 | 1.670 | ||||

calc. | 143.60 | 108.21 | 1.481 | 2.2 | 0.8 | 3.0 | 11.3 | |

600 | exp. | 167.59 | 117.67 | 1.460 | ||||

calc. | 169.40 | 118.20 | 1.422 | 2.2 | 1.1 | 0.4 | 2.6 | |

700 | exp. | 188.36 | 124.55 | 1.320 | ||||

calc. | 197.78 | 127.06 | 1.385 | 2.2 | 5.0 | 2.0 | 4.9 |

**Table 5.**Experimental and calculated relaxation parameters for system

**LNF**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$ and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Solvent: DMSO-d${}_{6}$/D_{2}O 7:3 Molar Ratio,T/K: 303, Visc./(Pa s): 1.40 × 10${}^{-3}$, Probe:^{13}CH on C-1 (A Residue) | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600 | exp. | 305.0 | 222.0 | 1.460 | ||||

calc. | 349.5 | 213.0 | 1.489 | 3.2 | 14.6 | 4.1 | 2.0 | |

700 | exp. | 354.0 | 244.0 | 1.420 | ||||

calc. | 418.6 | 227.9 | 1.451 | 3.2 | 18.2 | 6.6 | 2.2 | |

Probe:${}^{\mathbf{13}}$CH on C-1 (B residue) | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600 | exp. | 319.0 | 241.0 | 1.600 | ||||

calc. | 336.3 | 214.4 | 1.423 | 3.2 | 5.4 | 11.0 | 11.0 | |

700 | exp. | 366.0 | 264.0 | 1.530 | ||||

calc. | 400.3 | 229.9 | 1.386 | 3.2 | 9.4 | 12.9 | 9.4 | |

Probe:${}^{\mathbf{13}}$CH on C-1 (C residue) | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600 | exp. | 318.0 | 225.0 | 1.630 | ||||

calc. | 336.6 | 235.9 | 1.438 | 3.2 | 5.9 | 4.8 | 11.8 | |

700 | exp. | 360.0 | 240.0 | 1.560 | ||||

calc. | 395.9 | 254.1 | 1.392 | 3.2 | 10.0 | 6.9 | 10.7 | |

Probe:${}^{\mathbf{13}}$CH on C-1 (D residue) | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600 | exp. | 372.0 | 286.0 | 1.740 | ||||

calc. | 365.8 | 275.2 | 1.525 | 3.2 | 1.7 | 3.8 | 12.3 | |

700 | exp. | 404.0 | 302.0 | 1.690 | ||||

calc. | 421.2 | 296.7 | 1.469 | 3.2 | 4.3 | 1.7 | 13.1 | |

Probe:${}^{\mathbf{13}}$CH on C-1 (E residue) | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

600 | exp. | 325.0 | 259.0 | 1.490 | ||||

calc. | 390.1 | 270.1 | 1.728 | 3.2 | 20.0 | 4.3 | 15.6 | |

700 | exp. | 374.0 | 262.0 | 1.500 | ||||

calc. | 455.5 | 289.9 | 1.677 | 3.2 | 21.8 | 10.6 | 11.8 |

**Table 6.**Experimental and calculated relaxation parameters for system

**GCY**. The optimal ${R}_{\mathrm{eff}}$ and the percentage deviations from the experimental ${T}_{1}$, ${T}_{2}$ and the NOE (e (${T}_{1}$), e (${T}_{2}$), e (NOE), respectively) are also reported.

Probe: ${}^{13}$CH${}_{2}$, Solvent: DMSO-d${}_{6}$/D${}_{2}$O 7:3 Molar Ratio,T/K: 323, Visc./(Pa s): 2.90 × 10${}^{-3}$ | ||||||||
---|---|---|---|---|---|---|---|---|

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

400 | exp. | 123.5 | 75.00 | 1.430 | ||||

calc. | 130.7 | 81.59 | 1.458 | 1.8 | 5.9 | 8.8 | 1.9 | |

600 | exp. | 187.0 | 85.00 | 1.330 | ||||

calc. | 213.3 | 96.58 | 1.400 | 1.8 | 14.1 | 13.6 | 5.3 | |

900 | exp. | 314.5 | 110.9 | 1.250 | ||||

calc. | 370.3 | 109.4 | 1.384 | 1.8 | 17.7 | 1.3 | 10.7 | |

$\mathit{T}$/K: 343, Visc./(Pa s): 2.30 × 10${}^{-\mathbf{3}}$ | ||||||||

Freq./MHz | ${\mathit{T}}_{\mathbf{1}}$/ms | ${\mathit{T}}_{\mathbf{2}}$/ms | NOE | ${\mathit{R}}_{\mathrm{eff}}$ | e (${\mathit{T}}_{\mathbf{1}}$) | e (${\mathit{T}}_{\mathbf{2}}$) | e (NOE) | |

400 | exp. | 134.1 | 107.2 | 1.630 | ||||

calc. | 132.5 | 96.8 | 1.498 | 1.8 | 1.1 | 9.7 | 8.1 | |

600 | exp. | 183.8 | 130.1 | 1.510 | ||||

calc. | 203.6 | 117.3 | 1.405 | 1.8 | 10.8 | 9.9 | 6.9 | |

900 | exp. | 274.0 | 154.8 | 1.330 | ||||

calc. | 335.3 | 137.0 | 1.366 | 1.8 | 22.4 | 11.5 | 2.7 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rampino, S.; Zerbetto, M.; Polimeno, A. Stochastic Modelling of ^{13}C NMR Spin Relaxation Experiments in Oligosaccharides. *Molecules* **2021**, *26*, 2418.
https://doi.org/10.3390/molecules26092418

**AMA Style**

Rampino S, Zerbetto M, Polimeno A. Stochastic Modelling of ^{13}C NMR Spin Relaxation Experiments in Oligosaccharides. *Molecules*. 2021; 26(9):2418.
https://doi.org/10.3390/molecules26092418

**Chicago/Turabian Style**

Rampino, Sergio, Mirco Zerbetto, and Antonino Polimeno. 2021. "Stochastic Modelling of ^{13}C NMR Spin Relaxation Experiments in Oligosaccharides" *Molecules* 26, no. 9: 2418.
https://doi.org/10.3390/molecules26092418