Distance bounds matrix

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... 

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. 

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. 

In the last constraint, indicates all atoms within the specified residue. Although this final constraint is not strictly required, it helps define the distance bounds matrix and improves embedding and subsequent refinement. It is usually helpful to provide any and all distance information that is available or can be inferred, even if it seems redundant. Many qualitative distance constraints can be much more powerful than a small number of very precise distances. 

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. 

Intuitively, this can be seen by considering a chain of linked points. After iterating through the upper bounds matrix, there will be an upper bound on every distance corresponding to the number of links between each point. 

A corresponding inverse triangle inequality can be applied to each triplet to raise values in the lower bound matrix L. Now, a distance matrix D, usually referred to as the trial distance matrix, can be constructed by simply choosing elements dt/ randomly between w/ and lif and used to construct a metric matrix G. A matrix so constructed might be some approximation to the distances in the real molecule, but probably not a very good one. Clearly, every time an element d is selected, it puts limits on subsequent selected distances. This problem of correlated distances is discussed further in the section Systematic Errors and Bias. 

Figure 3.34. Distance matrix (a) in which unique interatomic distances for a particular conformation of a molecule are stored. Distance range matrix (b) in which ranges of interatomic distances representing conformational flexibilty of molecule are stored. U = upper bound, L = lower bound.

The basis of the DG method is that each ligand molecule is represented as a collection of points in space, each corresponding to atoms or groups of atoms. For a chosen molecular representation, the conformation of the molecule is described in terms of distance between points, i.e. -> interatomic distances. The matrix containing interatomic distances between all possible pairs of points is the -> geometry matrix of the molecule. To account for molecular flexibility, a matrix of lower bounds on the considered interatomic distances and a matrix of upper bounds are also defined fixed interatomic distances are represented by equal values in these matrices. It should be noted that the calculation of the atomic coordinates is a really difficult problem, and several mathematical techniques have been proposed [Crippen, 1977 Crippen, 1978 Crippen, 1991],... 

To illustrate each of the steps in the distance geometry method, consider its application to a very simple molecule, pentane. Taking the standard carbon-carbon bond length to be 1.54 A and assuming tetrahedral geometry about each such atom, the following initial bounds matrix can be constructed (all values have been rounded to three decimal places) ... 

If we assume that the atoms have van der Waals radii of 0.25 A, then we can set all the other lower bounds to 0.5 (the default lower bound between any two atoms is the sum of their van der Waals radii). However, the remaining upper bounds (between atom pairs 1-3 and 2-4) are not known, so we shall enter a default value of 100. This completes the entries in the bounds matrix, and a distance matrix can be generated from it by choosing distances randomly between the corresponding upper and lower bounds and entering them in the appropriate places in D. 

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... 

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. 

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