Partitioning space

Finally, 3D pharmacophores can be used to provide a naturally partitioned space. By com bining the pharmacophore keys of a set of molecules one can determine how many of th potential 3- or 4- point pharmacophores are accessible to the set and easily identify thos which are not represented. This use of pharmacophores is the basis of a method namei Pharmacophore-Derived Queries (PDQ) [Pickett et al. 1996]. One feature of this particula method is that most molecules will occupy more than one cell (as nearly all molecules wil contain more than one 3-point pharmacophore due to the functionality present an( conformational flexibility). This contrasts with the usual situation, wherein each molecul occupies just one cell. 

Fig. 2.1. A representation of the Madelung field of rutile (Ti02, 202240) in the (110) plane (x,y,z) x + y — 1. The light lines represent the field lines, the heavy lines show the zero-flux boundary that partitions space into bonds (from Preiser et al. 1999).

An alternative physical observable that has been used to define partial atomic charges is the electron density. In X-ray crystallography, the electron density is direedy measured, and by comparison to, say, spherically symmetric neutral atoms, atomic partial charges may be defined experimentally, following some decisions about what to do with respect to partitioning space between the atoms (Coppens 1992). Bader and co-workers have adopted a particular partitioning scheme for use with electronic structure calculations that defines the atoms-in-molecules (AIM) method (Bader 1990). In particular, an atomic volume is... 

Periodic repclitions of a space lattice cell in three dimensions from the original cell vvill completely partition space without overlapping or omissions. El is possible to develop a limited number of such three-dimensional patterns. Bravais. in 1848. demonsirated geometrically that there were but fourteen types of space lattice cells possible, and that these fourteen types could be subdivided into six groups called systems. Each system may be distinguished hy symmetry features, which can be related lo four symmetry elements ... 

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an atom in molecule as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus ... 

Figure 1.17(c) A single node of the three-periodic D-surface. Four funnels (on one side of the surface) meet at each node, at angles of 109.5. Image courtesy of David Anderson, (d) A model of a portion of the D-surface. The surface partitions space into two interpenetrating open labyrinths, each lying on a diamond lattice. 

Further rotation of the monkey saddle leads to the formation of a continuous hyperbolic surface that partitions space into two interpenetrating networks of tunnels. Each network consists of nodes connecting four tunnels meeting at tetrahedral angles (109.5°) (Fig. 1.17(c)). The nodes are arranged on... 

After developing novel approaches to exhaustive conformational analysis, Crippen [21] at the University of Michigan took a novel approach to pharmacophore discovery, based on Voronoi polyhedra (using hyperplanes to partition space into regions encompassing active molecules). This line of investigation was ultimately abandoned, as Crippen was unable to find a satisfactory resolution to the problem of multiple solutions consistent with the SAR. 

Abstract The paper presents maximum probability domains (MPDs). These are regions of the three dimensional space for which the probability to find a given number of electrons is maximal. In order to clarity issues hidden by numerical uncertainties, some simple models are used. They show that MPDs reproduce features which one would expect using chemical intuition. For a given number of electrons, there can be several solutions, corresponding to different chemical situations (e.g. different bonds). Some of them can be equivalent, by symmetry. Symmetry can produce, however, alternative solutions. The models show that MPDs do not exactly partition space, and they can also be formed by disjoint subdomains. Finally, an example shows that a partition of space, as provided by loge theory, can lead to situations difficult to deal with, not present for MPDs. 

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