Distance matrix

The distance matrix describes the distances between all atoms (or functional groups) of a molecule. It is the three-dimensional counterpart of the molecular topology. A direct comparison of distance matrices is normally not possible because the size and appearance of the matrix depends on the atom numbering in the molecules. 

If the matrix is restricted to a subset of atoms, functional groups or pharmacophore centers shared by all molecules considered the matrices can be compared automatically by computer programs [86] (Fig. 11). However, this implies an atom-by-atom superposition of all atoms (or groups, or pharmacophore centers) that are part of the matrix. 

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We tested our new potential by applying a local optimization procedure to the potential of some proteins, starting with the native structure as given in the Brookhaven Protein Data Bank, and observing how far the coordinates moved through local optimization. For a good potential, one expects the optimizer to be close to the native structure. As in Ulrich et al. [34], we measure the distance between optimizer B and native structure A by the distance matrix error... 

Table 1. Distance matrix errors DME (in A) between optimizers and native structures...

The elements of a distance matrix contain values which specify the shortest distance between the atoms involved. Distances can be expressed either as geometric distances (in A) or as topological distances (in number of bonds) (Figure 2-15a,b). 

The distance matrix D of a graph G with n vertices is a square n x n symmetric matrix as represented by Eq. (13), where is the distance between the vertices Vi and Vj in the graph (i.e., the number of edges on the shortest path). 

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

Because of the symmetry of the distance matrix, the Wiener index can be expressed as Eq. (16). 

With Eq. (16) the Wiener index of compound 2 can be calculated from the distance matrix as shown in Eq, (17)... 

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 ... 

I conformation is illustrated schematically in Figure 9.15. The interatomic distance matrix his conformation is ... 

Distance matrix for eight ribose phosphate fragments. 

Distance matrix used in ensemble distance geometry. There are Ni atoms in the first molecule, N2 in the and so on. 

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 distance matrix A, which holds the relative distances (by whatever similarity measure) between the individual confonnations, is rarely informative by itself. For example, when sampling along a molecular dynamics trajectory, the A matrix can have a block diagonal form, indicating that the trajectory has moved from one conformational basin to another. Nonetheless, even in this case, the matrix in itself does not give reliable information about the size and shape of the respective basins. In general, the distance matrix requires further processing. 

It should be stressed that PCA and PCoorA are dual methods that give the same analytical results. Using one or the other is simply a matter of convenience, whether one prefers to work with the covariance matrix C or with the distance matrix A. 

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-... 

Zhu et al. [15] and Liu and Lawrence [61] formalized this argument with a Bayesian analysis. They are seeking a joint posterior probability for an alignment A, a choice of distance matrix 0, and a vector of gap parameters. A, given the data, i.e., the sequences to be aligned p(A, 0, A / i, R2). The Bayesian likelihood and prior for this posterior distribution is... 

In a simple (nonweighted) connected graph, the graph distance dy between a pair of vertices V and Vj is equal to the length of the shortest path cormecting the two vertices, i.e. the number of edges on the shortest path. The distance between two adjacent vertices is 1. The distance matrix D(G) of a simple graph G with N vertices is the square NxN symmetric matrix in which [D],j=cl,j [9, 10]. 

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. 

Since the starting structure and the initial atom pair was casually selected, distance matrix generation and random metrization should be performed several times in order to get an ensemble of metric matrices. 

The similarities between all pairs of objects are measured using one of the measures described earlier. This yields the similarity matrix or, if the distance is used as measure of (dis)similarity, the distance matrix. It is a symmetrical nx matrix containing the similarities between each pair of objects. Let us suppose, for example, that the meteorites A, B, C, D, and E in Table 30.3 have to be classified and that the distance measure selected is Euclidean distance. Using eq. (30.4), one obtains the similarity matrix in Table 30.4. Because the matrix is symmetrical, only half of this matrix needs to be used. 

Note that dj and represent the row- and global means of the squared distance matrix D, respectively. The latter needs only be computed once and for all. The n distances of the variable point d(/cy) to the n row-points, however, are to be reevaluated for every column-item of X and for every marker which is to appear in the corresponding trajectory in the biplot. Usually, the range of is limited between the minimum and maximum value in the jth column of X or somewhat beyond (say 10% of the range on either side). 

Fig. 2.1. A tree representing the phylogeny of Wolbachia in arthropods (groups A and B) and filarial nematodes (groups C and D). Group designations correspond to those proposed by Werren etal. (1995) and by Bandi etal. (1998). The names at the terminal nodes are those of the host species. The tree is based on the ftsZgene sequence alignment used by Bandi etal. (1998). The tree was obtained using a distance matrix method (Jukes and Cantor correction neighbour-joining method).

Wiener-type index from Z weighted distance matrix (Barysz matrix) Wiener-type index from mass weighted distance matrix Wiener-type index from van der Waals weighted distance matrix Wiener-type index from electronegativity weighted distance matrix Wiener-type index from polarizability weighted distance matrix Balaban J index... 

Balaban-type index from polarizability weighted distance matrix connectivity index chi-0... 

W Wiener index — half-sum of the off-diagonal elements of the distance matrix of a graph... 

IC Information content of the distance matrix partitioned by frequency of occurrences of... 

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