Distance geometry bound smoothing

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

Restraints due to artifacts may, by chance, be completely consistent with the correct structure of the molecule. However, the majority of incorrect restraints will be inconsistent with the correct structural data (i.e., the correct restraints and information from the force field). Inconsistencies in the data produce distortions in the structure and violations in some restraints. Structural consistency is often taken as the final criterion to identify problematic restraints. It is, for example, the central idea in the bound-smoothing part of distance geometry algorithms, and it is intimately related to the way distance data are usually specified The error bounds are set wide enough that all data are geometrically consistent. 

Figure 13 Triangle smoothing of lower and upper distance bounds for use in distance geometry. The top diagram illustrates that the upper bound on the A-C interatomic distance cannot be greater than the sum of the upper bounds on the A-B and B-C distances. The lower diagram shows that the lower bound on the A-C distance cannot be smaller than the difference between the lower bound on the A-B distance and the upper bound on the B-C distance.

---