Bounded distance matrix

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

Suppose interatomic distances are now randomly assigned between the lower and uppi bounds to give the following distance 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... 

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]. 

Note that although the bounds on the distances satisfy the triangle inequalities, particular choices of distances between these bounds will in general violate them. Therefore, if all distances are chosen within their bounds independently of each other (the method that is used in most applications of distance geometry for NMR strucmre determination), the final distance matrix will contain many violations of the triangle inequalities. The main consequence is a very limited sampling of the conformational space of the embedded structures for very sparse data sets [48,50,51] despite the intrinsic randomness of the tech-... 

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

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.

Another use of distance matrices has been to look at the changes in protein conformation that accompany binding of a small molecule, for example, a substrate or an inhibitor. If, instead of distances, the differences between distances in the enzyme and enzyme with bound substrate are calculated, the distance matrix will more clearly show movements of portions of the protein upon binding of ligands. On binding substrate, for example, a particular a helix may move relative to another. 

The distance matrix of upper and lower bounds now describes the complete conformation space of the molecule but unfortunately cannot describe its chirality, since the distance matrix is invariant with respect to chirality. Chiral constraints are added to supply this missing information a chiral constraint is specified as the signed volume of the tetrahedron formed by the four substituents attached to a chiral center. The volume is calculated as a vector... 

Distances for each atom pair are randomly chosen between their lower and upper bounds. These distances are then converted into three-dimensional coordinates and refined against a simple error function made up of contributions from upper and lower bound violations and chiral constraint violations to ensure that the structure meets all distance and chiral constraints. The details of converting the distance matrix to three-dimensional coordinates are beyond the scope of this chapter but are provided in Crippen s text (126) and in an upcoming review article (133). 

Distance geometry - is a general method for converting a set of distance ranges or bounds into a set of Cartesian coordinates consistent with these bounds. A molecular structure is described by the set of all pairwise interatomic distances in a distance matrix. Cartesian and internal coordinates have been used historically primarily for mathematical and computational convenience for many modeling applications a distance matrix representation is simpler because chemical struaure information is often described by distances. 

Many other intermolecular and intramolecular contacts are described by distances (hydrogen bond lengths, van der Waals contact, experimentally determined distances from nuclear Overhauser effect (NOE) spectra, fluorescence energy transfer, etc.) so that the distance matrix representation can be used to specify all the known information about a molecular structure. These bounds are entered into a distance geometry program, as are other bounds that specify constraints on modeling problems, such as constraints to superimpose atoms in different molecules. Hypotheses about intra- or intermolecular conformations and interactions are easily specified with distance constraints models can be built quickly to test different hypotheses simply by changing the distance constraints. 

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. 

Generation of a distance matrix by random selection of distances between the bounds. Optionally, smooth the distances (metrization). 

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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