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We study the inverse problem of the determination of the most favoured relative orientations of moving protein domains from residual dipolar coupling measurements. We present a numerical procedure based on the simplex method for the efficient determination of the maximum probability of a given relative orientation. We prove the convergence of the algorithm and present the results obtained both on synthetic and on experimental data.

We study the inverse problem of determining the position of the moving C-terminal domain in a metalloprotein from measurements of its mean paramagnetic tensorχ . The latter can be represented as a finite sum involving the corresponding magnetic susceptibility tensor χ and a finite number of rotations. We obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation, and we show that only three rotations are required in the representation ofχ , and that in general two are not enough. We also investigate the situation in which a compatible pair of mean paramagnetic tensors is obtained. Under a mild assumption on the corresponding magnetic susceptibility tensors, justified on physical grounds, we again obtain an optimal estimate for the maximum probability that the C-terminal domain can assume a given orientation. Moreover, we prove that only ten rotations are required in the representation of the compatible pair of mean paramagnetic tensors, and that in general three are not enough. The theoretical investigation is concluded by a study of the coaxial case, when all rotations are assumed to have a common axis. Results are obtained via an interesting connection with another inverse problem, the quadratic complex moment problem. Finally, we describe an application to experimental NMR data.

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