Thesis Title: Subtitle - NMR Spectroscopy Research Group

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34 Chapter 2. Possum: paramagnetically orchestrated spectral solver of unassigned methyls. (2.3) (2.4) where PCS S calc(k,l) is the predicted PCS value for the spin S (S = 13 C or 1 H) of the methyl group in residue k arising from the paramagnetism of the lanthanide l, para δ S exp(j,l) is the chemical shift of the resonance j in the presence of the lanthanide l for the spin S, and dia δ S exp(i) is the diamagnetic resonance i for the spin S. The cost function assigns smaller costs for deviations between calculated and observed PCS when the experimentally observed PCS is large, while the constant e(l) prevents a singularity in the cost function and accounts for the error in measurements when a spin experiences small paramagnetic effects far away from the paramagnetic center. e(l) scales with the magnitude of the tensor of the lanthanide l. Empirically determined values (e(Yb 3+ ) = 1/6 ppm and e(Dy 3+ ) = 1 ppm) were used here 2 . Equations (2.3) and (2.4) ensure that each calculated PCS and each experimental chemical shift are chosen exactly once within the global assignment. Equations (1.1), (2.3) and (2.4) present the formulation of the three-index assignment problem (Schell, 1955) which is the three-dimensional instance of the multidimensional assignment problem (MAP). With D being the number of dimensions of the MAP (D = 3 in the example above) and n being the size of each of the D sets of data, there are (n!) D-1 possible assignments. When D is strictly larger than 2, MAP has been proven to be NP-hard (Karp, 1972) and, as a result, no algorithm can guarantee the best solution to the problem in a polynomial time. An exhaustive 2 The purpose of e(l) is to avoid degenerate cases where the experimental PCS is very close to zero. Its value is not critical for the success of the algorithm. E(l) has however been optimized to yield best possible results.
2.3 Experimental section. 35 search through the (n!) 2 possibilities is impracticable for even the smallest problem sizes. An exact branch and bound algorithm that explores only a part of all possible assignments has been proposed (Balas et al., 1991) and works well for small problem sizes, especially when there is a good agreement between predicted and observed PCS. In the present context, a simulated annealing optimization scheme proved more efficient computationally. The dimensionality D of the assignment problem generated by Possum depends on the residue type, the availability of connectivity information, and the number of different lanthanides used. We have performed calculations with up to 6 dimensions. Examples of 3- and 4-dimensional problems are illustrated in Figure 2.2. Figure 2.2 Formulation of the assignment problem depending on the information available. The columns dia δ S exp and para δ S exp contain the chemical shifts (S = 13 C and S = 1 H as observed for 13 C- HSQC cross-peaks) measured in the presence of a diamagnetic or paramagnetic lanthanide, respectively. The column marked PCS S calc contains the 13 C and 1 H PCS calculated from the tensor and the 3D structure of the protein. (a) Assignment problem for residues with a single methyl group (Ala, Met, Thr). The indices i and j refer to the cross-peak number in the diamagnetic and paramagnetic state, respectively, and the index k is the residue number in the amino-acid sequence, as in equation (1.1). The assignment (i = 1, j = 3, k = 1) is illustrated by connecting lines. The associated cost can be calculated using equation (2.2). The other n-1 assignments necessary to calculate the total cost C(l) according to equation (1.1) are not shown. Overall, this assignment problem is three-dimensional. (b) Simultaneous use of the information from two samples containing the paramagnetic lanthanides l1 or l2 creates a four-dimensional assignment problem. (c) For amino acids with two methyl groups (Ile, Leu, Val), the columns dia δ S exp, para δ S exp, and PCS S calc embed the chemical shifts (and PCS) of two methyl groups (m1 and m2). If the methyl- specificity information is not available in the paramagnetic state (illustrated by m? in the column
Page 1 and 2: Computational study of proteins wit
Page 3 and 4: iii Schmitz C, Stanton-Cook MJ, Su
Page 5 and 6: v Acknowledgements I would like to
Page 7 and 8: vii Keywords paramagnetic nmr, pseu
Page 9 and 10: ix 2.3.3 Manual resonance assignmen
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3.7 Study case. 87 The coordinates
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3.10 References. 95 Galassi M, Davi
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Thesis Title: Subtitle - NMR Spectroscopy Research Group

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