Upper distance bounds

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. 

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. 

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.

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... 

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. 

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. 

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. 

The distance bounds matrix concisely describes the complete conformational space of a molecule by entering the maximum possible distance (upper bound) between each atom pair in the upper diagonal and the minimum possible distance (lower bound) in the lower diagonal (Figure 2). 

All possible conformers lie between these upper and lower distance bounds—the task of distance geometry is to convert or embed this usually uncertain distance information into accurate three-dimensional Cartesian coordinates. Crippen and HavePi solved the problem for the case of an exact distance matrix, where all distances are known. Much additional effort has gone into finding efficient and practical methods for solving the general problem of a distance bounds matrix, where only a subset of the distances is known exactly. This remains a very difficult problem for large molecules with more than 1000 atoms. 

The initial Cartesian coordinates generated by embedding reflect the overall shape of the structure, but invariably they are poor quality and do not satisfy the original distance bounds as a result of the compression of the structure during its projeaion from N - 1 dimensions to three dimensions. The coordinates are improved by refinement against an error function that forces all constraints to lie between their lower and upper bounds. The error function F... 

Distance geometry uses a four-stage process to derive a conformation of a molecule [Crippen 1981 Crippen and Havel 1988]. First, a matrix of upper and lower interatomic distance bounds is calculated. This matrix contains the maximum and minimum values periiutted to each interatomic distance in the molecule. Values are then randomly assigned to each interatomic distance between its upper and lower bounds. In the third step, the distance matrix is converted into a trial set of Cartesian coordinates, which in the fourth step are then refined. 

Figure 4 Procedure for upper and lower distance bound calculation in ISIS/3D ...

Typically, a 3D structural database stores (I) the topology of the structures, including atom and bond types, and bond connectivity (see Connection Table Connectivity Table), (2) the topography of the structures, represented by the 3D coordinates of a given conformation, and (3) various types of data relevant to the conformation. This may include whole-molecule as well as per-atom or per-bond information. The database may also store various keys and indices to aid searching of structures and data. This information suffices to perform searches over fixed conformations (static 3D searches). For conformation-ally flexible structure searching (CFS), the database may also include information on upper/lower distance bounds between various types of atoms or it may include information derived from conformational analysis of the stored conformations, enabling reconstruction of various conformers at search time. ... 

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