Molecules three-dimensional point grid

To follow to actually carry out a TSA simulation a three-dimensional grid, with grid interval of about 0.2 A (5 -106 equispaced points in (132)) is built and the Helmholtz energies at all grid points are computed. Before this can be done in practice, a value for (A2) must be found. Then, local minima and the crest surfaces must be found, using the procedures given in (130,132,165). To study the dynamics of the penetrant molecules on the network of sites a Monte-Carlo procedure is employed, which is presented is some detail in (97). 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

Autocorrelation coefficients are also calculated on the basis of three-dimensional molecular interaction fields (e.g., MIP, CoMFA field or CoMSIA field). These fields are generated by mapping of atom properties to the spatial neighborhood of the molecule [23]. Distances between grid points located in the space around the molecules are used as input for the autocorrelation algorithm. 

A module of the SYBYL system (see entry under molecular modelling) which characterizes molecules by the calculation of interaction energies of probes at grid points in three-dimensional space. Analysis by PLS and factor analysis. 

The molecular field of a molecule is an ensemble of probe-ligand interaction energies. CoMFA treats each element of the field as an independent descriptor. If one computes steric and electrostatic fields over P grid points, the input table will contain 2 x P columns of explanatory variables. For N compounds, the matrix of molecular descriptors will consist of N x 2 X P cells. The table is three-dimensional because each column points to the coordinates at which its energy was calculated. By traversing the N cells along a single column, one can monitor the extent to which a probe, when placed at the associated point, is attracted or repelled by the various molecules. Each column describes localized differences in molecular fields. 

Ligand structures can be represented by molecular fields (electrostatic or steric), which contain enthalpic contributions to binding when implemented by conventional comparative molecular field analysis (CoMFA) (see Comparative Molecular Field Analysis (CoMFA)). Steric volume incorporated in molecular shape analysis (MSA) (Figure la) is another representation commonly used in SAR studies (see Molecular Surface and Volume) Alternatively, in the hypothetical active site lattice (HASL) approach, molecules are represented by a three-dimensional grid of points (lattice) associated with discrete electronic properties. ... 

In parallel with the two-dimensional case, the value of the dependent variable at the pivot point (/, j, k) is the arithmetic average of the values at the grid points adjacent to the pivot point. The computational molecule for the three-dimensional elliptic equation is shown in Fig. 6.5. 

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