Distance Geometry

Many physical methods used to examine a biological macromolecule in solution produce information about the distance that separates two parts of the molecule. These methods include fluorescence energy transfer, chemical cross-linking, light scattering, and several kinds of nuclear magnetic resonance (NMR) observations. Any information about distances between different parts of a molecule limits the number of conformations that are possible for that molecule. Distance geometry is a mathematical approach that can be used to help define the conformations of a molecule that are consistent with experimental distance information (1-6).

With X-ray crystallography methods, the tertiary structure of the molecule of interest is defined by analysis of a set of diffraction data. A relation exists between every feature of the diffraction pattern exhibited by the crystal and the positions of the atoms of molecules within the crystal. When this connection is elucidated, the three-dimensional (3D) structure of the molecule is revealed. A different approach is used to find 3D structures using distance information. Consider a polymeric molecule in an extended conformation, represented by the line below. 

If free rotation exists about some or all of the bonds along the molecular backbone, a huge number of conformations are possible for the molecule. If, however, an experimental observation indicates, for example, that the atoms at the position labeled A must be within 0.3 nm of the atoms at the position labeled B, then the number of possible conformations for the molecule is greatly reduced—the only conformations that need further consideration are those that are consistent with the constraint that atoms at A and B must be close to each other. Other experimentally defined distances would further reduce the number of possible 3D structures. If enough such constraints exist, only one or a small family of structures is likely that is consistent with all of them. Generally, the more constraints that can be established by experiment, the better known will be the tertiary structure. NMR methods, particularly those that measure nuclear Overhauser effects (NOEs), can produce a large number of distance constraints on possible conformations of a macromolecule. An advantage in using NMR observations in this way is that the molecule of interest does not have to be in the solid (crystalline( state for successful structure determination.

An N×N matrix of distances can be defined for a molecule made up of N atoms; N (N–1)/2 unique interatomic distances exist in this matrix. If all of the distances are accurately known, mathematical methods are available to define the Cartesian coordinates of the atoms from the known distances (4). Many of the distances in the distance matrix are essentially independent of conformation and are reliably known a priori. Such data would include the distance between two hydrogen atoms attached to an aromatic ring. Other distances can be estimated experimentally. Information about dihedral angles and other bond angles can be used to define other distances. For some distances, however, little or no accurate distance information will be available. Thus, the distance matrix will typically be incomplete, and a unique solution for Cartesian coordinates cannot be obtained. A variety of algorithms have been developed to deal with the missing or incomplete distance information. Regardless of the approach used, distance geometry calculations typically do not produce a single conformation but an ensemble of conformations, all of which are consistent with the available distance constraints. These conformations are used as starting points for additional refinement by conformational energy minimization and simulated annealing procedures.

References

1. L. M. Blumethal (1970) Theory and Applications of Distance Geometry, Chelsea, New York. 

2. G. M. Crippen and T. F. Havel (1978) Acta Crystallogr. 34, 282–284. 

3. G. M. Crippen (1981) Distance Geometry and Conformational Calculations, Wiley, Chichister, England. 

4. W. Braun (1987) Quart. Rev. Biophys. 19, 115–117. 

5. I. D. Kuntz, J. F. Tomason, and C. M. Oshhiro (1989) Methods Enzymol. 177, 159–204. 

6. T. F. Havel (1991) Prog. Biophys. Mol. Biol. 56, 43–78.