Trial distance matrix

The second step concerns distance selection and metrization. Bound smoothing only reduces the possible intervals for interatomic distances from the original bounds. However, the embedding algorithm demands a specific distance for every atom pair in the molecule. These distances are chosen randomly within the interval, from either a uniform or an estimated distribution [48,49], to generate a trial distance matrix. Unifonn distance distributions seem to provide better sampling for very sparse data sets [48]. 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

We can now proceed to the generation of conformations. First, random values are assigned to all the interatomic distances between the upper and lower bounds to give a trial distance matrix. This distance matrix is now subjected to a process called embedding, in which the distance space representation of the conformation is converted to a set of atomic Cartesian coordinates by performing a series of matrix operations. We calculate the metric matrix, G, each of whose elements (i, j) is equal to the scalar product of the vectors from the origin to atoms i and ... 

Distance geometry uses a four-stage process to derive a conformation of a molecule [Crippen 1981 Crippen and Havel 1988]. First, a matrix of upper and lower interatomic distance bounds is calculated. This matrix contains the maximum and minimum values periiutted to each interatomic distance in the molecule. Values are then randomly assigned to each interatomic distance between its upper and lower bounds. In the third step, the distance matrix is converted into a trial set of Cartesian coordinates, which in the fourth step are then refined. 

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