Multiple discrete sites

To describe such a situation with the minimum number of parameters and to enforce a physically reasonable model, SHELXL allows constraining the occupancies of all atoms in a conformer to be the same while the sum of the occupancies of all conformers works out to 1.0. The non-bonded interactions between conformers (e.g. atoms belonging to the same conformers do not make non-bonded interactions with their equivalents in other conformers) are organized fully automatically by SHELXL once the different sites have been assigned to specific conformers (using the PART instruction as described in Chapter 5). 

Approximately half of the atoms in a protein structure are hydrogen atoms (Andersson and Hovmoller, 1998). However, due to the little electron density associated to them, hydrogen atoms are usually ignored in standard macromolecular refinement. 

Adding hydrogen atoms to a protein model usually reduces / 1 and / lfree by the same amount, typically 0.5-1%. More importantly, the inclusion of hydrogens will improve the geometry of the model as the non-bonded interactions (enforced by BUMP restraints, see Chapter 2 and the SHFLX manual for details) between atoms are described more accurately. However, the effect of the inclusion of hydrogens on the crystallographic phases and electron density maps is usually not very pronounced. 

At atomic resolution, not only fully occupied solvent sites, but also partially occupied solvent sites can be modeled with confidence. To facilitate a stable refinement, the occupancies of partially occupied sites should be coupled to neighbouring protein atoms and/or other partially occupied solvent sites in the neighbourhood using constraints. 

A large observable-to-parameter ratio allows for the estimation of standard uncertainties of the refined parameters by the inversion of the normal matrix of the refinement (Cruickshank, 1970 Press et al, 2002). Due to the rapid progress in computing systems, the huge calculations involved can now be done in hours on modem desktop workstations. 

---

Fraser, N. W., Sehgal, P. B., and Darnell, J. E., 1979, Multiple discrete sites for premature RNA chain termination in adenovirus-2 infection Enhancement by 5,6-dichloro-l-p-ribofuranosylbenzimidazole, Proc. Natl. Acad. Sci. USA 76 2571-2575. 

It is a solvent polarity index defined [Kier and Hall, 1986] as the first-order valence connectivity index divided by the number Nf of discrete, isolated functional groups in order to account for multiple interaction sites as ... 

Apart from the rebuilding of the protein and the solvent model in the same way as it is done for structures at less than atomic resolution, the major part of manual modeling at atomic resolution concerns the introduction of multiple discrete conformations. Accurate modeling of multiple conformations, in particular in active sites, often leads to much cleaner electron density and increased interpretabiUty. 

Batch production is considered by the first seven approaches shown in Table 4.2. Timpe and Kallrath (2000) propose a time-discrete MILP for the IPPSP with multiple production sites/plants, multiple production modes, intermediate storages, sales points, and transports between sites, storages, and sales points. The model uses different time scales for production and sales planning which allows a more precise planning of production. In a production period, a site can produce in different modes whereby changeover times and costs are considered. To store intermediate and raw materials, one-product and multiproduct tanks are available and an assignment of products to tanks is implemented. The... 

The particles of interest to us have both internal and external coordinates. The internal coordinates of the particle provide quantitative characterization of its distinguishing traits other than its location while the external coordinates merely denote the location of the particles in physical space. Thus, a particle is distinguished by its internal and external coordinates. We shall refer to the joint space of internal and external coordinates as the particle state space. One or more of either the internal and/or external coordinates may be discrete while the others may be continuous. Thus, the external coordinates may be discrete if particles can occupy only discrete sites in a lattice. There are several ways in which the internal coordinates may be discrete. A simple example is that of particle size in a population of particles, initially all of uniform size, undergoing pure aggregation, for in this case the particle size can only vary as integral multiples of the initial size. For a more exotic example, let the particle be an emulsion droplet (a liquid) in which a precipitation process is carried out producing a discrete number of precipitate particles. Then the number of precipitate particles may serve to describe the discrete internal coordinate of the droplet, which is the main entity of population balance. 

A metal cluster can be considered as a polynuclear compound which contains at least one metal-metal bond. A better definition of cluster catalysis is a reaction in which at least one site of the cluster molecule is mechanistically necessary. Theoretically, homogeneous clusters should be capable of multiple-site catalysis. Many heterogeneous catalytic reactions require multiple-site catalysis and for these reasons discrete molecular metal clusters are often proposed as models of metal surfaces in the processes of chemisorption and catalysis. The use of carbonyl clusters as catalysts for hydrogenation reactions has been the subject of a number of papers, an important question actually being whether the cluster itself is the species responsible for the hydrogenation. Often the cluster is recovered from the catalytic reaction, or is the only species spectroscopically observed under catalytic conditions. These data have been taken as evidence for cluster catalysis. 

---