Matrix metric

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 ... 

Ihe metric matrix G is a square symmetric matrix. A general property of such matrices that they can be decomposed as follows ... 

The diagonal elements of are the eigenvalues of G and the columns of V are i eigenvectors. The atomic coordinates can be derived from the metric matrix by rewritir... 

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... 

A molecular dynamics force field is a convenient compilation of these data (see Chapter 2). The data may be used in a much simplified fonn (e.g., in the case of metric matrix distance geometry, all data are converted into lower and upper bounds on interatomic distances, which all have the same weight). Similar to the use of energy parameters in X-ray crystallography, the parameters need not reflect the dynamic behavior of the molecule. The force constants are chosen to avoid distortions of the molecule when experimental restraints are applied. Thus, the force constants on bond angle and planarity are a factor of 10-100 higher than in standard molecular dynamics force fields. Likewise, a detailed description of electrostatic and van der Waals interactions is not necessary and may not even be beneficial in calculating NMR strucmres. 

Finding the minimum of the hybrid energy function is very complex. Similar to the protein folding problem, the number of degrees of freedom is far too large to allow a complete systematic search in all variables. Systematic search methods need to reduce the problem to a few degrees of freedom (see, e.g.. Ref. 30). Conformations of the molecule that satisfy the experimental bounds are therefore usually calculated with metric matrix distance geometry methods followed by optimization or by optimization methods alone. 

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

There are many extensive reviews on metric matrix distance geometry [41-44], some of which provide illustrative examples [45,46]. In total, we can distinguish five steps in a distance geometry calculation ... 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

Note that for general parameterizations this metric matrix is neither skew diagonal nor constant-, see below. The equations of motion expressed in Eq. (2.6) are obtained by using the Principle of Stationary Action, 5A = 0, with Lagrangian... 

The procedure of DG calculations can be subdivided in three separated steps [119-121]. At first, holonomic matrices (see below for explanahon) with pairwise distance upper and lower limits are generated from the topology of the molecule of interest. These limits can be further restrained by NOE-derived distance information which are obtained from NMR experiments. In a second step, random distances within the upper and lower limit are selected and are stored in a metric matrix. This operation is called metrization. Finally, all distances are converted into a complex geometry by mathematical operations. Hereby, the matrix-based distance space is projected into a Gartesian coordinate space (embedding). 

Kuszewski, Nilges, M., Brunger, A. T. Sampling and efficiency of metric matrix distance geometry a novel partial metrization algorithm./. Biomol. NMR 1992, 2, 33 55. 

Fig. 32.3. Effect of a weighted metric on distances, (a) representation of a circle in the space 5 defined by the usual Euclidean metric, (b) representation of the same circle in the space S defined by a weighted Euclidean metric. The metric is defined by the metric matrix W.

Technique 1 Clustering of the Metric Matrix. The first step In evaluating the library was to separate It Into classes by explicitly scanning the printed list of compounds. At this point It became... 

The further evaluation of the library and metric proceeds from the metric matrix. Its elements are the dot products of each spectriun with every other In Its class. 

In contrast to the and metric matrices in 3-positivity, the strength of the T2 matrix as a 2-RDM A -representability condition is not completely invariant upon altering the order of the second-quantized operators in C, j - For example, a slightly different metric matrix T2 can be defined by exchanging the operators a, and a in Eq. (40) to obtain... 

Rearranging the operators C j, to, produces a term with a single annihilation operator. Because this term cannot be expressed in terms of the set of operators i ijki he space spanned by the basis functions in the metric matrix T2 differs slightly from the space spanned by the basis functions in the metric matrix T2. 

A generalized metric matrix f 2, however, can be obtained by supplementing the operators from Eq. (40) with the set of single-particle excitation and deexcitation operators ... 

Because 0 <. S, , < 1, the elements of S are completely analogous to the basis-set overlap integrals familiar in quantum chemistry (50). As the off-diagonal elements of the matrix are non-zero, that is, S , 0 for all i /, the basis is non-orthogonal. In some applications S is equivalent to what is typically called the metric matrix in statistics S is equivalent to the correlation matrix. 

Agreement of (10.7) and (10.22) then requires that the full metric matrix have exactly p independent null eigenvectors satisfying M(c+2)1L = 0 (he., et = 0). We now wish to show that these null eigenvectors can be taken as the vectors of (10.17), namely,... 

The fact that the full metric matrix (10.19) of order c + 2 is singular (with p null eigenvalues) implies that its determinant vanishes,... 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

Generalized scaling behavior has been hypothesized to underlie certain observed critical-state limits (to be described in Chapter 11). We wish to show here how apparent generalized scaling behavior such as (S 10.4-3) may be inferred from the existence of null eigenvectors (r ) of the metric matrix [cf. (10.24)],... 

---