The effect of finite sampling on the determination of orientational properties: A theoretical treatment with application to interatomic vectors in proteins

Second-rank tensor properties such as the overall rotational diffusion tensor and the alignment tensor can be determined by NMR methods measuring orientation of interatomic vectors. Here we examine the effect of incomplete sampling of orientation space by interatomic vectors in a molecule on determination of a second-rank tensor. We have developed a quantitative approach to determine (1) how well orientation space is sampled by a particular protein or substructure, (2) to what extent this particular distribution of bond vectors samples the various components of a second-rank tensor and, (3) the ability of this distribution of bond vectors to completely characterize the tensor. This approach is generally applicable to any second-rank tensor property whose determination relies on the sampling of the angular space by the structure or substructure. The theory permits assessment of the expected degree of accuracy of tensor determination using a selected set of interatomic vectors (e.g., NH or C(α)H(α), etc), for a given molecular structure. The sampling properties of real proteins are analyzed using a database of 1736 structures, representing all experimentally determined protein folds. This theoretical approach is applied to the rotational diffusion and alignment tensors obtained from nuclear magnetic resonance data for several systems, including ubiquitin and βARK PH domain. Finally, the proposed sampling characteristics are related to the accuracy of the determination of the rotational diffusion tensor from spin-relaxation data, as an example of an unknown second-rank tensor. Knowing the accuracy of the tensor quantity derived from experimental data assists in optimizing experimental design.