Triangle Inequality Bounds Smoothing

(b) Eigenvectors (V) and eigenvalues from metric matrix (not shown) calculated from D generate the coordinates in X. 

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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... 

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-... 

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.)...

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