Random structure methods

Random structure methods have proved useful in solving structures from X-ray powder diffraction patterns. The unit cell can usually be found from these patterns, but the normal single-crystal techniques for solving the structure cannot be used. A variation on this technique, the reverse Monte Carlo method, includes in the cost function the difference between the observed powder diffraction pattern and the powder pattern calculated from the model (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Adams and Swenson 2000b Swenson and Adams 2001). 

NON SELF-CONSISTENT THEORIES A. The Quasi-Random Structure Method... 

The coarse-graining approach is commonly used for thermodynamic properties whereas the systematic or random sampling methods are appropriate for static structural properties such as the radial distribution function. 

Ffom a theoretical point of view, stacking fault energies in metals have been reliably calculated from first-principles with different electronic structure methods [4, 5, 6]. For random alloys, the Layer Korringa Kohn Rostoker method in combination with the coherent potential approximation [7] (LKKR-CPA), was shown to be reliable in the prediction of SFE in fcc-based solid solution [8, 9]. 

B-chains until they are acted on by R-enzyme, when maltose or malto-triose will be produced from the residual A-chain, and linear dextrins from the B-chains. The amount of maltose or maltotriose liberated on treating the /3-limit dextrin with R-enzyme will be a measure of the number of A-chains in the molecule, and from these data, the ratio of A B chains in the molecule can be calculated.220 Peat concluded that multiple branching is an intrinsic part of the amylopectin structure, as the observed yield of these sugars was greater than expected for a singly-branched structure. It should be noted that glycogen has been shown by similar enzymic methods to possess a truly random structure.221... 

Two possible space-based schemes are the random structure approach and the lattice method. In the first, the atoms are placed in random positions to form an initial structure containing no chemical information. The configuration of the atoms is then altered in such a way as to better match a pre-selected set of chemical or physical constraints. In the second approach the cations and anions are separately arranged on lattices that minimize the electrostatic repulsions between the (like) ions. The lattices are then merged by placing the anion lattice in the cavities of the cation lattice and vice versa. 

Computer generation of images from random structures. The stereological measurements described above provide a method for estimating three- dimensional properties from observations on two-dimensional images. Unfortunately, the quality of these estimates is sometimes difficult to assess. In order to verify the correctness of our method and to determine the minimum number of samples which must be observed to obtain satisfactory estimates, computer generated random structures were examined. 

Statistical simulation methods can be basically separated into two approaches. The Monte Carlo (MC) framework [17,18,19] utilises random structural variations of single structural units (atoms, molecules, groups, etc.) followed by an evaluation of energies to decide whether the resulting new arrangement of atoms is accepted or should be discarded. Sampling of molecular dynamics (MD) employs equations... 

Conformational isomers represent minima on an energy surface, and all structures and the corresponding strain energies can be obtained by a careful analysis. This can be performed manually (such as in Sections 17.3 and 17.4) or automatically. An automatic procedure may involve a systematic search (grid search methods), a stochastic search (e.g., torsional Monte Carlo or cartesian stochastic, i.e., the random kick method) or molecular dynamics (see Chapter 5 and Section 16.5). Implemented in MOMEC is a random kick stochastic search module, and this has been shown to lead to excellent results, not only for conformational equilibria, but also for distributions of configurational isomers[37]. 

In contrast to rational design methods, where changes in protein properties may be predicted from sequence changes, random mutation methods have been introduced to empirically improve protein properties in the absence of a priori knowledge of structure-function relationships. Error-prone polymerase chain reaction (PCR), for example, can be used to introduce approximately random mutations in amplified DNA sequences. The protein products of these mutations may then be screened to select successful or improved variants.2... 

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