Monte Carlo Searches

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

SKELGEN [78] smaU-fragment based, Monte-Carlo search ... 

FIGURE 21.3 Sampling of confonnation space using a Monte Carlo search (with a small number of iterations). 

Monte Carlo searching becomes more difficult for large molecules. This is because a small change in the middle of the molecule can result in a large displacement of the atoms at the ends of the molecule. One solution to this problem is to hold bond lengths and angles fixed, thus changing conformations only, and to use a small maximum displacement. 

This is a fairly reasonable way to describe man-made amorphous polymers, which had not been given time to anneal. For polymers that form very quickly, a quick Monte Carlo search on addition can insert an amount of nonoptimal randomness, as is expected in the physical system. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

In performing a Monte Carlo sampling procedure we let the dice decide, again and again, how to proceed with the search process. In general, a Monte Carlo search consists of two steps (1) generating a new trial conformation and (2) deciding whether the new confonnation will be accepted or rejected. 

Thermal computation which comprises algorithms inspired by multiparticle systems like BROWNian motion (BROWNian search, Monte Carlo search) on the one hand and simulated annealing which is inspired by Boltzmanns statistics on the other hand. 

We have added a companion option to PBUILD, PRANDOM which eases considerably the problem of finding good conformations of a polymer segment. PRANDOM automatically selects all of the polymer backbone and/or side chain bonds and will randomly select rotations for each bond. In a few minutes, one can not only build a polymer fragment, but also set up a Monte-Carlo search of its conformational space. However, even this cannot solve the problems for large models (pentamer or larger), again due to the number of bonds to be rotated. 

Incorporation of an imine unit into a mPE began with a Monte Carlo search of dodecamer (58e), which indicated that the oligomer adopted a six-turn helical structure [86]. To verify these results solvent denaturation studies were performed on 58e which showed a helix coil transition with a AG(CH3CN) very similar to that of the native oligomers (3.0 0.2 for 58e vs 3.2 0.1 for 15). The small difference in the AG(CH3CN) indicates that the imine bond has a negligible effect on the stability of the folded state of the oligomer. With this... 

Scanning of a potential energy surface (see potential energy surface). The methods currently used include random search (stochastic search, e. g., Monte Carlo methods) and molecular dynamics (see deterministic search, Monte Carlo search, stochastic search, molecular dynamics, scanning an energy surface). 

Random search of an energy surface (see conformational search, deterministic search, Monte Carlo search, molecular dynamics). 

The stereochemistry of nucleophilic additions to cycloheptanone has been studied by quantum mechanical calculations.160 Employing a Monte Carlo search, 1000 conformations were generated and minimized. The five lowest conformers, labelled A, B, C, D and E, are shown below with their relative energies. The global minimum A is a chair with the carbonyl on the two-carbon stern fragment. The twist conformer B is almost as stable. The three remaining conformers can be roughly characterized as chairs with the carbonyl located at the bow. 

For much more flexible molecules than 18, DFT PES scans can be impractical. Currently, for such molecules conformational analysis is most efficiently carried out in two stages first, Monte Carlo searching using a molecular mechanics force field (MMFF) determines the stable conformations predicted by the MMFF second, these conformations are re-optimized using DFT. For example, conformational analysis of the... 

In order to characterize the distributions of selective values in the second and the third model, we explored the value landscape by a Monte Carlo search. We created three random samples of 38,000 different sequences each (one repeat with 76,000 sequences gave essentially the same results) with predetermined ratios of probabilities for (0/1) digits, Pi = 0.2857, p2 = 0.5, and p3 = 0.7143, which led to mutant distributions centered at the 20-, 35-, and 50-error mutants of the all-zero sequence Iq. Three different parts of the value landscapes determined by Eqs. (IV.9)-(IV.ll) were explored in that way. The results are shown in Figure 22. 

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