Distance bounds

In the basic metric matrix implementation of the distance constraint technique [16] one starts by generating a distance bounds matrix. This is an A X y square matrix (N the number of atoms) in which the upper bounds occupy the upper diagonal and the lower bounds are placed in the lower diagonal. The matrix is Ailed by information based on the bond structure, experimental data, or a hypothesis. After smoothing the distance bounds matrix, a new distance matrix is generated by random selection of distances between the bounds. The distance matrix is converted back into a 3D confonnation after the distance matrix has been converted into a metric matrix and diagonalized. A new distance matrix... 

On the basis of Eq. (1), NOEs are usually treated as upper bounds on interatomic distances rather than as precise distance constraints, because the presence of internal motions and, possibly, chemical exchange may diminish the strength of an NOE [23]. In fact, much of the robustness of the NMR structure determination method is due to the use of upper distance bounds instead of exact distance constraints in conjunction with the observation that internal motions and exchange effects usually reduce rather than increase the NOEs [5]. For the same reason, the absence of an NOE is in general not interpreted as a lower bound on the distance between the two interacting spins. 

Upper and lower bounds, bap, on distances da/3 between two atoms a and b, and constraints on individual torsion angles 0i in the form of allowed intervals [f 111, 13 ] are considered. Iu, h and Iv are the sets of atom pairs (a, /I) with upper, lower or van der Waals distance bounds, respectively, and Ia is the set of restrained torsion angles. wu, Wj, wv and wa are weighting factors for the different types of constraints. 

Throughout the torsion angle dynamics calculation the list of van der Waals lower distance bounds is updated every 50 steps using a cutoff of 4.2 A for the interatomic distance. 

The upper distance bound b for a combined constraint is formed from the two upper distance bounds b1 and b2 of the original constraints either as the r 6 sum, b = (b+ b "<5) 1 6, or as the maximum, b = max (b, b2). The first choice minimizes the loss of information if two already correct constraints are combined, whereas the second choice avoids the introduction of too small an upper bound if a correct and an erroneous constraint are combined. 

To try to get a reasonable starting matrix D, one first builds a matrix L of lower distance bounds and a corresponding matrix U of upper bounds. Both matrices should contain any experimental distances as well as any covalently determined distances. In cases such as bond lengths, elements l,t may nearly equal ubut in the case of undetermined distances between points covalently far from each other, /i may be the sum of the van der Waal radii, whereas u will be some large number. 

If the experimental errors are underestimated, it can lead to tight but inaccurate distance bounds. Conversely, the overestimated errors can lead to unnecessarily imprecise distance bounds. RANDMARDI takes into account two types of experimental errors relative integration errors and absolute errors due to spectral noise. The first kind can be estimated, for example, by comparing intensities of symmetric peaks below and above the diagonal, and the second type can be estimated as 50-200% of the lowest quantifiable peak, depending on the spectrum quality. 

BiCh94 Stefan Brands, David Chaum Distance-Bounding Protocols Eurocrypt 93, LNCS 765, Springer-Verlag, Berlin 1994, 344-359. 

Another useful distance geometry model-building application is the elegant ensemble approach of Sheridan et al. (145), where multiple molecules are entered into a single distance bounds matrix. Intramolecular distance constraints are set as described in Section VII.A and mtermolecular distance constraints are entered to force specific intermolecular interactions to occur, for example, to superimpose a set of molecules using common functional groups. This approach is described in more detail in Section X.B on pharmacophore modeling. 

Distance geometry Distance geometry pioneered by G.M. Crippen is a method for converting a set of distance bounds into a set of coordinates that are consistent with these bounds. In applying distance geometry to conformationally flexible structures the upper and lower bounds to the distance between each pair of points (atoms) are used. This approach is useful for molecular modelbuilding and conformational analysis and has been extended to find a common pharmacophore from a set of biologically active molecules. 

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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