Rotations in NMR: Part II. Applications of the Euler-Rodrigues parameters

Siminovitch, D.

doi:10.1002/(sici)1099-0534(1997)9:4<211::aid-cmr2>3.0.co;2-1

Cited by 24 publications

(14 citation statements)

References 28 publications

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“…The rotation operator formalism of Sanctuary and coworkers [4] and Siminovitch [5, 6] provides an alternative approach and is the subject of the present work. In this method, the propagator given in Eq.…”

Section: Introductionmentioning

confidence: 99%

Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: Applications to shaped pulses and pulse trains

Li¹,

Rance²,

Palmer³

2014

Journal of Magnetic Resonance

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The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence τ1—Rϕ(β)—τ2, in which Rϕ(β) is a rotation by an angle β around an axis with phase ϕ in the transverse plane and τ1 and τ2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for τ1 and τ2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr-Purcell-Meiboom-Gill sequences identifies an eight-pulse phase-alternating scheme that generates a propagator nearly equal to the identity operator.

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Section: Introductionmentioning

confidence: 99%

Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: Applications to shaped pulses and pulse trains

Li¹,

Rance²,

Palmer³

2014

Journal of Magnetic Resonance

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“…Several excellent discussions of quaternions and their use in molecular modeling and protein studies are available, particularly in relation to rigid body fitting,12–14 structure determination,5, 7, 15, 16 and solid‐state NMR spectroscopy 3, 4. In all such discussions, it is important to define clearly the frame of reference represented by a given quaternion.…”

Section: Methodsmentioning

confidence: 99%

“…Using the method developed by Horn,1 Coutsias et al 2 demonstrated in 2004 that quaternions can be used to find the best‐fit rotation needed to be applied to superimpose one protein structure upon another. Quaternions also have found applications in solid‐state NMR spectroscopy due to their capability of succinctly expressing the orientation of a peptide plane relative to an external magnetic field 3, 4…”

Section: Introductionmentioning

confidence: 99%

Quaternion‐based definition of protein secondary structure straightness and its relationship to Ramachandran angles

2011

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We describe here definitions of "local helical axis" and "straightness" that are developed using a simple quaternion-based analysis of protein structure without resort to least-squares fitting. As part of this analysis, it is shown how quaternion differences can be visualized to depict accurately the local helical axis relating any two adjacent amino acid residues in standard, nonidealized proteins. Three different options for the definition of amino acid residue orientation in terms of quaternion frames are described. Two of these, the "C(α) frame" and the "P frame," are shown to be correlated strongly with a simple approximate measure derived solely from Ramachandran angles. The relationship between quaternion-based straightness and recognized DSSP-derived secondary structure motifs is discussed.

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“…While quaternions have been employed extensively to encode molecular orientations (see [1][2][3][4][5]), and have also been applied to RNA (see [6]), applications of quaternions to protein structures have been limited in scope (see, for example, [7][8][9][10][11][12][13][14]). The most widely used approach to analyzing orientations of amino acid residues is the classic work of Ramachandran [15,16], which encodes only local information about orientation angles, although alternative orientation visualization methods have been proposed, e.g., by Bojovic et al [17].…”

Section: Related Workmentioning

confidence: 99%