Distances as Degrees of Freedom

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Abstract
Tertiary contact distance information of varying resolution for large biological molecules abounds in the literature. The results provided herein develop a framework by which information of this type can be used to reduce the allowable configuration space of a macromolecule. The approach combines graph theory and distance geometry. Large molecules are represented as simple, undirected graphs, with atoms, or groups, as vertices, and distances between them as edges. It is shown that determination of the exact structure of a molecule in three dimensions only requires the specification of all the distances in a single tetrahedron, and four distances to every other atom. This is 4N-10 distances which is a subset of the total N(N-l)/2 unique distances in a molecule consisting of N atoms. This requirement for only 4N-10 distances has serious implications for distance geometry implementations in which all N(N-l)/2 distances are specified by bounded random numbers. Such distance matrices represent overspecified systems which when solved lead to non-obvious distribution of any error caused by inherent contradictions in the input data. It is also shown that numerous valid subsets of 4N-10 distances can be constructed. It is thus possible to tailor a subset of distances using all known distances as degrees of freedom, and thereby reduce the configuration space of the molecule. Simple algebraic relationships are derived that relate sets of distances, and complicated rotations are avoided. These relationships are used to construct minimum, complete sets of distances necessary to specify the exact structure of the entire molecule in three dimensions from incomplete distance information, and to identify sets of inconsistent distances. The method is illustrated for the flexible structural types present in large ribosomal RNAs: 1.) A five-membered ring; 2.) a chemically bonded chain with its ends in contact (i.e., a hairpin loop); 3.) the spatial orientation of two separate molecules, and; 4.) an RNA helix that can have variation in individual base pairs, giving rise to global deviation from standardized helical forms.