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]. 

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-... 

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]. 

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... 

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. 

Easthope, P.L. and Havel, T.F. Computational experience with an algorithm for tetrangle inequality bound smoothing. Bull. Math. Biol. 1989, 53, 173-194. 

To start with, we consider steady flows of Maxwell-type fluids in a bounded smooth domain f2 of R, iV = 2,3, and with simple boundary conditions, neunely the system... 

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... 

If/ is a bounded, smooth function on R , we may define a generalized function Tf by reference to its action on a test function e. IT via integration ... 

Let us suppose now that we have such a bounded smooth solution of (4.1) and it is such that the limits... 

Extrapolating a complete set of lower and upper limits on all the distances from the sparse set of lower and upper bounds that are usually available, a process known as bound smoothing. 

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