Torsional angles Subject

The master equation is derived by asymptotic analysis of the multidimensional diffusion equation in the torsional angles subjected to the configurational potential, with the coupling between the variables explicitly taken into account. Molecular definitions of the kinetic rates for the conformational transitions occurring at each segment of the chain are obtained in terms of the dynamic and hydrodynamic properties of the system. The following physical quantities enter in the specification of the problem ... 

Fig. 3.7. A united-atom PE chain constituted of 100 united atoms mapped around a stream line. The bond angles as well as the bond lengths have been kept fixed. The torsion angles have been subject to a minimization considering a three-fold potential accounting for the simplified chemical structure of the chain. An additional proximity function has been used to force the chain to follow the trajectory of the stream line [114]...

Fig. 7.5 Illustration of how dispersion forces affect gauche (G) conformations. Compared to structures with gauche forms devoid of dispersion forces (i.e., HF-optimized), structures with gauche forms subject to dispersion forces (MP2 optimized) contract in such a way that the 1. ..5 nonbonded interactions in an attractive part of the van der Waals potential are shortened. Thus, in GG-pentane (shown above), MP2-optimized torsional angles are contracted by several degrees compared to the HF-optimized geometry, causing a reduction in the 1...5 nonbonded distances by several tenths of an A. For additional details and the numerical values see R. F. Frey, M. Cao, S. Q. Newton, and L. Schafer, J. Mol. Struct. 285 (1993) 99.

Many of the conformational properties of peptide systems, including protein conformation, can be approximated in terms of the local interactions encountered in dipeptides, where the two torsional angles 4> (N-C(a)) and < i (C(a)-C ) are the main conformational variables. N-acetyl N -methyl alanine amide, shown in Fig. 7.11, is a model dipeptide that has been the subject of numerous computational studies. 

Dibenzoylmethane (8b) has been the subject of much interest as regards the possibility that its polymorphism is associated with keto-enol tautomerism. Chemical and spectroscopic studies showed that this is not so (33a). This compound had previously been reported to be trimorphic (33b), but one form appears, in fact, to be a eutectic mixture of the other two. The molecules in these two polymorphs are both in the same state of tautomerism they differ in the torsional angle about the (CH)-(CO) bond and in the type of hydrogen bonding in which they participate. It is noteworthy that solutions prepared from these forms at low temperature have differences in chemical and spectroscopic properties that are maintained for some time. For example, such solutions prepared and held at —35° react at different rates with FeCl3. 

In grid searching (GS), each torsion angle is examined, but the. structure is not subjected to geometry optimization. 

It has been common practice to fit a skeletal model to the electron density map, usually by means of a Richards comparator, a device with a half-silvered mirror that enables an observer to see a reflected image of a wire model projected within the three-dimensional map.14 The model is adjusted by varying the torsional angles and in some cases distorting the interbond angles to allow for departures from ideal values, until the atoms are in positions that appear best to satisfy the electron density. It must be emphasized that the fit is a subjective one and that it is impossible to make an objective assessment. Errors in coordinates from models fit to maps at resolutions of 2-2.5 A are usually estimated to be in the 0.2- to 0.5-A range,s although errors much in excess of this may be made. 

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