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 (chi) over bar. The latter is an integral average of rotations of the corresponding magnetic susceptibility tensor chi. The largest fraction of time that the two terminals can stay in a given orientation, still producing the (chi) over bar 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.