Cartesian coordinates, chain conformation

The conformational distance does not have to be defined in Cartesian coordinates. Eor comparing polypeptide chains it is likely that similarity in dihedral angle space is more important than similarity in Cartesian space. Two conformations of a linear molecule separated by a single low barrier dihedral torsion in the middle of the molecule would still be considered similar on the basis of dihedral space distance but will probably be considered very different on the basis of their distance in Cartesian space. The RMS distance is dihedral angle space differs from Eq. (12) because it has to take into account the 2n periodicity of the torsion angle. 

To compute the free energy of an oligopeptide, it is first necessary to express the Cartesian coordinates of every atom of the molecule (in any conformation) in the same coordinate system. The coordinates of each amino acid residue and those of the end groups of the chain are first expressed in local coordinate systems. The chain is then built in a prespecified amino acid sequence and conformation by connecting the individual residues and end groups, with proper adjustment of the dihedral angles. 

Equation (6-33) describes the probability density for one end of the chain at the coordinates (x yh z,) in the unstrained state, the other end being at the origin of the Cartesian coordinate system. The chain s end-to-end vector r, has a magnitude of (x,-2 +y2 + z2) A. (Figures 6-14 and 6-15, Appendix 1). Following assumption 3, the total number of conformations available to a network of N such chains is... 

Figure 6-14. Conformation of a polymer chain with one end fixed at the origin of a Cartesian coordinate system.

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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