Joining RDC data from flexible protein domains

We study the inverse problem of determining the conformational freedom of two protein domains from residual dipolar coupling (RDC) measurements. For each paramagnetic ion attached to one of the domains we obtain a magnetic susceptibility tensor ? from the RDC of couples of atoms of that domain, and a mean paramagnetic susceptibility tensor ?¯ from the RDC of couples of atoms of the other domain. The latter is an integral average of rotations of ? which depends on the conformational freedom of the two domains. In this paper we consider the case when we have data from paramagnetic ions attached separately to each of the domains. We prove that in this case not all the elements of ? and ?¯ are independent. We derive the mathematical equations for the compatibility of the measurements and show how these relations can be used in the presence of noisy data to determine a compatible set of ? and ?¯ with an unconstrained minimization. If available, information about the shape of the noise can be included in the target function. We show that in this case the compatible set obtained has a reduced error with respect to the noisy data.