Triangle bound smoothing

The most important consequence of bound smoothing is the transfer of infonnation from those atoms for which NMR data are available to those that cannot be observed directly in NMR experiments. Within the original experimental bounds, the minimal distance intervals are identified for which all triangle inequalities can be satisfied. A distance chosen outside these intervals would violate at least one triangle inequality. Eor example, an NOE between protons pi and pj and the covalent bond between pj and carbon Cj imposes upper and lower bounds on the distance between pi and Cy, although this distance is not observable experimentally nor is it part of E hem-... 

Metrization guarantees that all distances satisfy the triangle inequahties by repeating a bound-smoothing step after each distance choice. The order of distance choice becomes important [48,49,51] optimally, the distances are chosen in a completely random sequence... 

Figure 5 (a) Bounds matrix (B) after triangle inequality bounds smoothing for square structure with bond lengths and van der Waals radii of 1.0 and 0.25 A, respectively distance matrix (D) after random seleaion of distances between bounds in... 

Fig. 12.20 4-Acetamido benzoic acid. Triangle smoothing predicts that the lower bound distance between the amide nitrogen and the carbonyl oxygen is equal to the sum of the van der Waals radii. The actual distance is about 6.4A.

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.)...

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.

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