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Conformational freedom of proteins and the maximal probability of sets of orientations

We study the inverse problem of determining the relative orientations of the moving C- and N-terminal domains in a flexible protein from measurements of its mean magnetic susceptibility tensor . The latter is an integral average of rotations of the corresponding magnetic susceptibility tensor χ. The largest fraction of time that the two terminals can stay in a given orientation, still producing the measurements, is the maximal probability of that orientation. We extend this definition to any measurable subset of the rotation group. This extension permits a quantitative assessment of the results when the generating distribution is either continuous or discrete. We establish some properties of the maximal probability and present some numerical experiments.