Thesis Title: Subtitle - NMR Spectroscopy Research Group

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84 Chapter 3. Numbat: new user-friendly method built for automatic Δχ-tensor determination. Different conventions have been used in the literature to report Δχ-tensor parameters, including different definitions of Euler angles, choice of principal and secondary axis of the Δχ- tensor, and units of Δχ-tensor magnitudes. Numbat can report the Δχ-tensor parameters in many different conventions but uses as a default the following conventions: (i) The axes of the Δχ-tensor frame are labelled such that |Δχzz| ≥ |Δχyy| ≥ |Δχxx| in analogy to alignment tensor conventions (Clore et al., 1998). This ensures that axial and rhombic components are always of the same sign. (ii) The Euler angles α, β and γ are expressed in the ―ZYZ‖ convention, i.e., the first rotation of angle α is around the z axis of the protein frame, the second rotation of angle β is around the new y’ axis and the last rotation of angle γ is around the new z’’ axis (Figure 3.2). While for an asymmetric object the Euler angles are uniquely defined if the angles α, β and γ are taken in the intervals [0, 2π[, [0, π[, [0, 2π[, respectively, ambiguities arise for symmetric objects. Therefore, we chose the interval [0, π[ for all three angles, eliminating the potential ambiguities arising from the four symmetry-related Δχ-tensors that generate the same PCS values. In the case of β = 0, an infinite number of combinations of and would produce the same overall rotation. In this case, we set γ = 0. These two rules ensure that any Δχ-tensor is unambiguously reported as a single set of parameters which is referred to in the GUI as UTR (Unique Δχ-Tensor Representation). Figure 3.2 Euler angle definitions used by Numbat. The relative orientation of the Δχ- tensor frame with respect to the protein frame is defined by Euler rotations of angle α, β and γ in the ZYZ convention. (a) A right-handed rotation of angle α around the z axis is applied to the protein frame xyz to give the frame x’y’z’. (b) A second rotation of angle β around the new axis z’ is applied to the frame x’y’z’ to give x’’y’’z’’. (c) The last rotation of angle γ around the z’’ axis gives the Δχ-tensor frame. 3.6.10 Error analysis The Levenberg-Marquardt algorithm is used to minimize the cost c (equation (3.2)), but the quality of the fit cannot be assessed without further error analysis. Therefore, in addition to the uncertainty values provided by the GSL implementation of the minimiser, Numbat embeds a Monte
3.6 Program Features. 85 Carlo protocol with random Gaussian noise added either to the atomic coordinates of the molecule or to the experimental PCS values. The robustness of the Δχ-tensor fit with respect to the PCS data set can also be tested by random subset selection of the PCS values used. Resulting Δχ-tensor orientations are displayed in a Sanson-Flamsteed projection (Bugayevskiy et al., 1995) using the plotting utility gnuplot. 3.6.11 Visualization Graphical visualization of the Δχ-tensor frame and isosurfaces of PCS values in the structure of the molecule presents a convenient way to assess the similarity of the principal axes of multiple Δχ-tensors and the similarity of their respective isosurfaces. To this end Numbat interfaces with the molecular viewers MOLMOL (Koradi et al., 1996) and PyMOL (DeLano, 2002) by generating suitable macro files and displaying the Δχ-tensor frame and corresponding PCS isosurfaces in superimposition with the protein studied, as illustrated in Figure 3.3. The files of the macros, PCS potential and PDB file containing the coordinates of the protein together with coordinates of the metal ion and Δχ-tensor axes can also be saved for later use. Figure 3.3 Visualisation of the Δχ-tensor in MOLMOL and PyMOL, and display of its orientational uncertainty in a Sanson-Flamsteed projection plot. Numbat can directly call MOLMOL and PyMOL to display the axes of the fitted Δχ-tensor and PCS isosurfaces at user-defined contour levels. The orientational uncertainty of the Δχ-tensor frame can be evaluated by a Monte-Carlo protocol with random additions of noise to the structure coordinates and/or PCS data, with optional random
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Computational study of proteins wit
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iii Schmitz C, Stanton-Cook MJ, Su
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v Acknowledgements I would like to
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vii Keywords paramagnetic nmr, pseu
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ix 2.3.3 Manual resonance assignmen
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xi List of Figures Figure 1.1 NMR e
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xiii
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xv List of Abbreviations α Subunit
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2 Chapter 1. Introduction. The scal
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4 Chapter 1. Introduction. In contr
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6 Chapter 1. Introduction. as the c
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10 Chapter 1. Introduction. Protein
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18 Chapter 1. Introduction. (Bugaye
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22 Chapter 1. Introduction. (ii) It
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26 Chapter 1. Introduction. Su XC,
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28 Chapter 2. Possum: paramagnetica
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5.3 The use of PCS for protein dock
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Thesis Title: Subtitle - NMR Spectroscopy Research Group

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