Metric matrix method

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

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]. 

Unfortunately, this method cannot be implemented as described. First, the distance matrix D is neither complete nor accurate. Second, a metric matrix, G, calculated from such a matrix may not have only three positive eigenvalues. These problems were also addressed by Crippen and Havel.21... 

Another algorithm attributed to Crippen, linearized embedding, does actually involve the creation of a trial metric matrix, but is otherwise very different from the standard embed algorithm.29 Its main virtue is the incorporation of covalent restraints, chirality, and ring planarity at a more fundamental level than the original embed algorithm. Unfortunately, there does not yet seem to be much experience with the method. 

Metric Matrix, Linearized Embedding, and Torsional Space Methods... 

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. 

If all N N-1)/2 distances in a figure are given, it is possible to find a set of Cartesian coordinates for the N vertices. The method makes use of the properties of the metric matrix. First, the connectivity matrix C, the symmetric NxN matrix of distances squared is formed. From it we form an (Af-l)x(Af-l) symmetric matrix G by subtracting the first row and column from all other rows and columns, respectively. The diagonal elements are then of the form -2(cfi ) and the off-diagonal ones are which is the same as from the properties of the... 

Most of the methods outlined above are suitable for obtaining information on oscillatory reaction networks. As pointed out in several other chapters in this book, related methods can be used for determination of causal connectivities of species and deduction of mechanims in general nonoscillatory networks. Pulses of species concentration by an arbitrary amount have been proposed (see chapter 5) and experimentally applied to glycolysis (see chapter 6). Random perturbation by a species can be used and the response evaluated by means of correlation functions (see chapter 7) this correlation metric construction method has also been tested (see chapter 8). Another approach to determining reaction mechanisms by finding Jacobian matrix elements is described in Mihaliuk et al. [69]. 

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. 

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