Geometry metric

Continuous Solvent—Extrusion Process. A schematic for a typical continuous process, widely used for making solvent propellant for cannons, is shown in Figure 7. This continuous process produces ca 1100 metric tons of single-base propellant per month at the U.S. Army Ammunition Plant (Radford, Virginia). Continuous processes have also been developed for double- and triple-base propellants and for stick as well as granular geometries. A principal aspect of these processes has been the extensive use of single- and double-screw extmders instead of the presses used in the batch process. 

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. 

The solid state structures of these materials suggest that the 2D GS network is more accurately described as an assembly of ID GS ribbons connected to each other via lateral (G)N-H—O(S) H-bonds that serve as flexible hinges. These hinges allow the GS sheet to pucker, like an accordion, without an appreciable change in the near-linear geometries, which are considered to be optimal, of the (G)N-H—O(S) H-bonds. With respect to crystal metrics, the range of repeat distances within the GS... 

M. M. Deza and M. Laurent, Geometry of Cuts and Metrics, Springer-Verlag, Berlin, 1997. 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

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