Superposition Generalized Least Squares

Molecular structures may be described and compared in terms of external or internal coordinates. The question of which is to be preferred depends on the type of problem that is to be solved. For example, one problem that is much easier to solve in a Cartesian system is that of finding the principal inertial axes of a molecule indeed, if only internal coordinates are given then, in general, the first step is to convert them to Cartesian ones and then proceed as described in Section 1.2.4. Similarly, the optimal superposition of two or more similar molecules or molecular fragments, i.e. with the condition of least-squared sums of distances between all pairs of corresponding atoms, is best done in a Cartesian system. On the other hand, systematic trends in a collection of molecular structures and correlations among their structural parameters are more readily detectable in internal coordinates. 

The most straightforward method for comparison of 3-D structures involves rigid-body least-squares superposition of the C positions. We have developed a procedure for alignment of several homologous structures [9, 10] without bias to any one in the set. Divergent proteins usually retain the general arrangement of strands and helices. However, when there are less than 20% sequence identities, the differences in relative orientation and position of the secondary structural elements usually preclude their simultaneous superposition [1,2,11,12). 

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