Distance constraint

The situation is more complicated in molecular mechanics optimizations, which use Cartesian coordinates. Internal constraints are now relatively complicated, nonlinear functions of the coordinates, e.g., a distance constraint between atoms andJ in the system is — AjI" + (Vj — + ( , - and this... 

The amount of computation necessary to try many conformers can be greatly reduced if a portion of the structure is known. One way to determine a portion of the structure experimentally is to obtain some of the internuclear distances from two-dimensional NMR experiments, as predicted by the nuclear Over-hauser effect (NOE). Once a set of distances are determined, they can be used as constraints within a conformation search. This has been particularly effective for predicting protein structure since it is very difficult to obtain crystallographic structures of proteins. It is also possible to define distance constraints based on the average bond lengths and angles, if we assume these are fairly rigid while all conformations are accessible. 

In the basic metric matrix implementation of the distance constraint technique [16] one starts by generating a distance bounds matrix. This is an A X y square matrix (N the number of atoms) in which the upper bounds occupy the upper diagonal and the lower bounds are placed in the lower diagonal. The matrix is Ailed by information based on the bond structure, experimental data, or a hypothesis. After smoothing the distance bounds matrix, a new distance matrix is generated by random selection of distances between the bounds. The distance matrix is converted back into a 3D confonnation after the distance matrix has been converted into a metric matrix and diagonalized. A new distance matrix... 

TF Flavel. An evaluation of computational strategies for use m the determination of protein structure from distance constraints obtained by nuclear magnetic resonance. Prog Biophys Mol Biol 56 43, 1991. 

W Braun, N. Go. Calculation of protein conformations by proton—proton distance constraints A new efficient algorithm. J Mol Biol 186 611-626, 1985. 

Distance constraints are used to derive possible structures of a protein molecule... 

Figure 18.20 The two-dimensional NMR spectrum shown in Figure 18.17 was used to derive a number of distance constraints for different hydrogen atoms along the polypeptide chain of the C-terminal domain of a cellulase. The diagram shows 10 superimposed structures that all satisfy the distance constraints equally well. These structures are all quite similar since a large number of constraints were experimentally obtained. (Courtesy of P. Kraulis, Uppsala, from data published in P. Kraulis et ah. Biochemistry 28 7241-7257, 1989, by copyright permission of the American Chemical Society.)...

In NMR the magnetic-spin properties of atomic nuclei within a molecule are used to obtain a list of distance constraints between those atoms in the molecule, from which a three-dimensional structure of the protein molecule can be obtained. The method does not require protein crystals and can be used on protein molecules in concentrated solutions. It is, however, restricted in its use to small protein molecules. 

However, moving the support zone farther from the source of contamination often brings up logistical problems associated with the distance. The logistics should be considered at all times. No matter how open the space is, there are always distance constraints. 

In order to enhance our ligand-based query hypothesis, the structural fragments of the initial query were generalized but linked with the same distance constraints. A search of this final query (see Fig. 4-10) in the same list yielded 690 hits and a statistically significant correlation of the presence of this enantiophore and the enan-tioselectivity of the compounds was found (94 % of those are well resolved on Chi-ralcel OD). Note that out of the 4203 compounds of the Chiralcel OD domain search, a 2D search found 1900 structures that contain the substructural features of the generalized query. 

Plus 52 upper distance limits from the cluster for protons that could not be detected, 126 lower distance limits from the cluster for protons without any appreciable paramagnetic line broadening, and 42 distance constraints derived from hydrogen... 

Plus 241 distance constraints for the unassigned residues close to the iron—sulfur cluster derived from the X-ray structure... 

Plus 38 distance constraints derived from hydrogen bonds 131). 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

---