Distance geometry metric 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 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. 

Conformations were generated using metric matrix distance geometry algorithm JG (S. Kearsley, Merck Co., unpublished). The conformations were subjected to energy-minimization within... 

J. Kuszewski, M. Nilges, and A. T. Briinger, ]. Biomol. NMR, in press. Sampling and Efficiency of Metric Matrix Distance Geometry A Novel Partial Metrization Algorithm. 

The key to the distance geometry method is the metric matrix, G. Each element gj of G can be calculated by taking the vector dot product of the coordinates of atoms i and . That is. 

At this point, we have shown only that we can convert three-dimensional coordinates to the metric matrix G and then regenerate them. The key to distance geometry is that G also can be derived directly from the distance matrix, D, in which each component djj is the distance between atoms i and /. D is first converted into D< where each component is the distance between each atom i and the center of mass o. 

Efficiency of Metric Matrix Distance Geometry—A Novel Partial Metrization Algorithm. 

As a structure-generating tool, nab provides three methods for building models. They are rigid-body transformations, metric matrix distance geometry, and molecular mechanics. The first two methods are good initial methods, but almost always create structures with some distortion that must be removed. On the other hand, molecular mechanics is a poor initial method but very good at refinement. Thus the three methods work well together. 

Figure 12 Example of four points in space being determined using the metric matrix method of distance geometry.

When distance geometry is applied to NMR data, the metric matrix equations can be solved repeatedly giving an ensemble of conformations (Figure 13) that satisfies the available constraints as well as possible. All these conformations are equally realistic as far as the experimental NMR data go. The predicted molecular geometries can be subjected to further energy minimization refinement with or without distance constraints. 

The most popular method for carrying out the initial search step is based on a metric matrix or distance geometry approach.If we consider describing a macromolecule in terms of the distances between atoms, it is clear that there are many constraints that these distances must satisfy, since for N atoms there are N N — 1 )/2 distances but only 3N coordinates. General considerations for the conditions required to embed a set of interatomic distances into a realizable three-dimensional object forms the subject of distance geometry. The basic approach starts from the metric matrix that contains the scalar products of the vectors x, that give the positions of the atoms ... 

If the origin ( 0 ) is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e., if it has toee positive and N — 3 zero eigenvalues. This is not a trivial theorem, but it may be made plausible by thinking of the eigenanalysis as a principal component analysis all of the distance properties of the molecule should be describable in terms of three components , which would be the x, y and z coordinates. If we denote the eigenvector matrix as w and the eigenvalues A., the metric matrix can be written in two ways ... 

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