Refinement of Conformations by Computational Methods

The results of NMR measurements have to be converted into a 3D structure. After establishing the constitution by NMR parameters that are transmitted through bond, i.e. J-coupHng constants, information about the spatial structure is introduced. Here, mainly distances from NOE build-up rates are used to define the configuration and conformation. 

Experimental distances from NOEs/ROEs of small molecules are recommended not to be classified into regions of small, medium, and large as it is often done in the structure determination of large molecules. As opposed to macromolecules, the overall correlation time Tc can be considered constant in small molecules. Thus, it is possible to measure distances in the range between 2 and 5 A with an accuracy of about 10%. Often distances between protons are almost exclusively used for the structure determination. This leads to the fact that molecules with small numbers of hydrogen atoms are more difficult to determine. 

The first step of the structure refinement is the appHcation of distance geometry (DG) calculations which do not use an energy function but only experimentally derived distances and restraints which follow directly from the constitution, the so-caUed holonomic constraints. Those constraints are, for example, distances between geminal protons, which normally are in the range between 1.7 and 1.8 A, or the distance between vicinal protons, which can not exceed 3.1 A when protons are in anti-periplanar orientation. 

The second step is the molecular dynamics (MD) calculation that is based on the solution of the Newtonian equations of motion. An arbitrary starting conformation is chosen and the atoms in the molecule can move under the restriction of a certain force field using the thermal energy, distributed via Boltzmann functions to the atoms in the molecule in a stochastic manner. The aim is to find the conformation with minimal energy when the experimental distances and sometimes simultaneously the bond angles as derived from vicinal or direct coupling constants are used as constraints. 

Restrained MD (rMD) is followed by the use of MD in explicit solvent, i.e. the conformation as determined above is taken into a box containing many solvent molecules around the molecule. Subsequently, simulated annealing (SA) and energy minimizahon steps are performed to draw the molecule into the global energy minimum. An MD run (the so-called trajectory) over at least 150ps to Ins is followed and a mean structure is calculated from such a trajectory. The con-formahon must be stable under this condihon even when the experimental constraints are removed. 

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Refinement of Conformations by Computational Methods 243 9.4.2.6 Simulated Annealing (SA)... 

Results such as these have tended to restrict use of the EH method to qualitative predictions of conformation in molecules too large to be conveniently treated by more accurate methods. However, just as the simple Huckel method underwent various refinements (such as the cjo technique) to patch up certain inadequacies, so has the EH method been rehned. Such refinements have been shown to give marked improvement in numerical predictions of various properties. The EH method has been overtaken in popularity by a host of more sophisticated computational methods. (See Chapter 11.) However, it is still sometimes used as a first step in such methods as a way to produce a starting set of approximate MOs. The EHMO method also continues to be important as the computational equivalent of qualitative MO theory (Chapter 14), which continues to play an important role in theoretical treatments of inorganic and organic chemistries (as, for example, in Walsh s Rules and in Woodward-Hoffmann Rules). 

More than 30 years ago Warshel proposed, on the basis of semiempirical simulations, an isomerization mechanism that could explain how this process can occur in the restricted space of the Rh binding pocket (Warshel 1976). Since two adjacent double bonds were found to isomerize simultaneously the mechanism reveal a so-called bicycle pedal motion. Due to the concerted rotation of two double bonds in opposite directions the overall conformational change is minimized and hence this mechanism was found to be space-saving. The empirical valence bond (EVB) method (Warshel and Levitt 1976) was used to compute the excited state potential energy surface of the chromophore during a trajectory calculation where the steric effects of the protein matrix were modeled by specific restraints on the retinal atoms. Since then, Warshel and his coworkers have improved the model employing better structural data and new computational developments (Warshel and Barboy 1982 Warshel and Chu 2001 Warshel et al. 1991). The main refinement of the bicycle pedal mechanism was that the simultaneous rotation of the adjacent double bonds is aborted at a twist of 40° and leads to the isomerization of only one bond (Warshel and Barboy 1982). 

Much of the machinery for successful refinement of bio-molecular structures from NMR data is now in place. It remains to be seen how much the quality of NMR structures can be further improved by the implementation of relaxation matrix refinement or other procedures discussed above. At the very least, iterative NOE refinement protocols are desirable as aids in eliminating errors or over-interpretations in distance constraints. However, many problems remain to be solved, particularly with regard to conformational heterogeneity and from variable order parameters arising from internal motion. Considerable work will be required before these methods can be applied routinely to NMR structure determination. Further progress in developing methods to asses the accuracy and precision of NMR structures will rest upon careful data collection procedures, theoretical analyses of the connection between structure, dynamics and cross-relaxation rates, and the development of improved computational tools to tie these together. 

The empirical methods of computation of conformational energy are called empirical because they use functions which are not derived from a basic theory and some parameters are evaluated from experimental data obtained from model compounds. The hard sphere approximation was introduced by the Madras group who originated the conformational computation studies on proteins. From this simplest approximation, refinements have been progressively developed. These refinements consist mainly of the different contributions included in the potential energy functions and... 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

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