Cartesian coordinates conformational analysis

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

For a quantitative description of molecular geometries (i.e. the fixing of the relative positions of the atomic nuclei) one usually has the choice between two possibilities Cartesian or internal coordinates. Within a force field, the potential energy depends on the internal coordinates in a relatively simple manner, whereas the relationship with the Cartesian nuclear coordinates is more complicated. However, in the calculations described here, Cartesian coordinates are always used, since they offer a number of computational advantages which will be commented on later (Sections 2.3. and 3.). In the following we only wish to say a few words about torsion angles, since it is these parameters that are most important for conformational analysis, a topic often forming the core of force field calculations. 

The best method to estimate the validity of a simulation is the construction of C —C distance map. It allows one to analyze local structures as well as overall conformation. However this method of analysis is rather crude because it gives for each pair of residue one all-or-none answer depending on the allowed C —C distance. Crippen and Kuntz (1977) have introduced a novel representation of the backbone conformation referred to as direction matrices. In such a representation, a matrix is calculated from C atom cartesian X-ray coordinates where the ij elements of the matrix is the cosine of the angle between the direction of the chain at residue i and the residue of the chain at residue j This representation gives distinctive pattern for the most important structural features. 

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