Fused sphere surfaces

Fused sphere surfaces, such as fused sphere Van der Waals surfaces (VDWS ) are simple approximations to molecular contour surfaces. By specifying the locations of the centers and the radii of formal atomic spheres in a molecule, the fused sphere surface is fully defined as the envelope surface of the fused spheres and can be easily generated by a computer. Although fused sphere VDW surfaces are not capable of representing the fine details of molecular shape, such surfaces are very useful for an approximate shape representation. 

All of the above tests were for hard chains at surfaces. The only comparison between theory and simulation for various values of fluid-fluid and bulk fluid attractions is that done by Patra and Yethiraj (PY) [137], who presented a simple van der Waals DFT for polymers and compared to simulations of fused-sphere chains. In their theory, PY used the Yethiraj functional [39] for the hard-chain contribution to the free energy and a simple mean-field term for the attractive contribution. Their excess free energy functional is given by... 

An effective approach is to compute V(r) on an appropriately-defined molecular surface, because this is what is seen or felt by the other reactant. Such a surface is of course arbitrary, because there is no rigorous basis for it. A common procedure has been to use a set of fused spheres centered on the individual nuclei, with van der Waals or other suitable radii [34—37]. We prefer, however, to follow the suggestion of Bader et al. [38] and take the molecular surface to correspond to an outer contour of the electronic density. This has the advantage of reflecting features such... 

Figure 4.1 An illustration of a fused sphere Van der Waals surface (VDWS) of a molecule.

In particular, if the atomic radii are taken as some of the recommended values of the atomic Van der Waals radii, then one obtains a fused sphere Van der Waals surface (VDWS) of the molecule. Several different sets of atomic radii have been proposed [85-87,255], and the fused sphere molecular surface obtained depends on this choice. 

The 3D space requirements of most molecules can be represented to a good approximation by such Van der Waals surfaces. Fused sphere VDWS s are used extensively in molecular modeling, especially in the interpretation of biochemical processes and computer aided drug design. These approximate molecular surfaces are conceptually simple, their computation and graphical display on a computer screen take relatively short time, even for large biomolecules. 

In Figure 4.1 an example of a fused spheres Van der Waals surface is shown. This figure illustrates an important difference between a MIDCO and a VDWS at the seam of interpenetration of the spheres the latter surfaces are not differentiable. 

Shape Analysis of Fused Sphere Van der Waals Surfaces and Other Locally Nondifferentiable Molecular Surfaces... 

The sequence of seeing graphs for families of MlDCO s of the ethanol molecule has been used for shape characterization [347], and the method is equally applicable to fused sphere VDW surfaces, and to solvent accessible surfaces. 

The input data for the shape analysis methods are provided by well-established quantum chemical or empirical computational methods for the calculation of electronic charge distributions, electrostatic potentials, fused spheres Van der Waals surfaces, or protein backbones. The subsequent topological shape analysis is equally applicable to any existing molecule or to molecules which have not yet been synthesized. This is precisely where the predictive power of such shape analysis lies based on a detailed shape analysis, a prediction can be made on the expected activity of all molecules in the sequence and these methods can select the most promising candidates from a sequence of thousands of possible molecules. The actual expensive and time-consuming synthetic work and various chemical and biochemical tests of... 

Fortuitously, for most molecules, the MIDCO s G(a) of the chemically most important small density threshold values a are those where the deviations are small from the simple fused sphere model surfaces. The usual Van der Waals surfaces fall within this range. For a molecule containing N nuclei, these VDWS s are obtained as the envelope surfaces of N interpenetrating spheres... 

Figure 7.1 Illustration of the principle of the Fused Spheres Guided Homotopy Method (FSGH), applied for the generation of dot representations of density scalable MIDCO surfaces for the water molecule. Three families of atomic spheres (thin lines) and their envelope surfaces (heavy lines) are shown in the upper part of the figure. In the lower part of the figure, the selected point sets on the innermost family of spheres are connected by interpolating lines to the exposed points (black dots) on the envelope surfaces of two enlarged families of spheres. Linear interpolation along the lines for two selected density values leads to two families of white dots, generating approximations of two MIDCO s (heavy lines in the lower figure).

However, this sequence of envelope surfaces of gradually enlarged fused spheres will not, in general, approximate the MIDCO s adequately for some practical applications in particular, at the seams of interpenetrating spheres this representation does not follow the corresponding MIDCO G(a) well. 

The FSGH method (Fused Sphere Guided Homotopy method) [43]. This method has been designed for the construction of approximate, density scalable ("inflatable") isodensity contour surfaces and their dot representations (i.e., for continuous transformations between different isodensity surfaces of a given molecule). 

Density Scalable Atomic Sphere (DSAS) surfaces [255]. This technique generates radii for atomic spheres for any desired electron density at the surface. The method is used for inexpensive representations of MIDCO s of large molecules, in combination with the Fused Sphere Guided Homotopy method (FSGH) [43]. 

Du Q, Arteca GA. Derivation of fused-sphere molecular surfaces from properties of the electrostatic potential distribution. J Comput Chem 1996 17 1258-1268. 

We will look in particular at the detailed features of V(r) computed on the surfaces of energetic molecules, which is designated Vs(r). This raises the question of how to define a molecular surface. In the past, this has often been done by means of fused spheres centered at the nuclear positions and having, for example, the corresponding van der Waals atomic radii [112,113]. More recently, however, there has been an increasing tendency to follow Bader et al [WA] and take the surface to be some outer contour of the electronic density. This has the advantage that the surface then reflects the specific features of that molecule, such as lone pairs and strained bonds. We use this approach, with p(r) = 0.001 electrons/bohr other outer contours, e.g. p(r) = 0.002 electrons/bohr, would serve just as well [115]. 

The simplest molecular surface is the van der Waals surface, a fused-sphere envelope resulting from the superposition of atomic spheres with van der Waals radii. This surface models qualitatively the space occupied by a molecule. The interaction with other molecules in the environment can be taken into account by considering the part of the surface accessible to the solvent.The smoothed surface derived from solute—solvent contacts is an improved model.i °... 

More detailed shape descriptions must take into account local surface features. At this point, we can adopt two viewpoints, as we did in the case of molecular chains. On the one hand, we can assume that the surface is everywhere differentiable and proceed to study shape features in terms of local curvatures. This is the strategy chosen for isodensity contours, and their characterization is discussed later. On the other hand, we can view the molecular surface as an object with complexity, where differentiability is not a given. In this case, molecular shape can be studied by using pseudofractal surfaces. This latter approach is most appropriate for fused-sphere models. 

Fujii et al. [13] studied morphological structures of the cross section of various hollow fibers and fiat sheet membranes by high-resolution field emission scanning electron microscopy. Figure 6.8 shows a cross-sectional structure of a flat sheet cellulose acetate RO membrane. The layer near the top surface is composed of a densely packed monolayer of polymeric spheres, which is supported by a layer formed with completely packed spheres. The contours of the spheres in the top layer can be observed. The middle layer is also composed of loosely packed and partly fused spheres, which are larger than the spheres in the surface layer. In the middle layer, there are many microvoids, the sizes of which are the same as the spheres. The layer near the bottom is denser than the middle layer, and the spheres are deformed and fused. Interstitial void spaces between the spheres, which may be called microvoids, are clearly observed. This structure seems common for the flat sheet as well as the hollow fiber membranes. For example. Fig. 6.9 shows a cross section of a hollow fiber made of PMMA B-2 (a copolymer containing methyl methacrylate and a small amount of sulfonate groups). The inside surface layer is composed of the dense structure of compactly packed fine polymeric particles. The particle structure of the middle layer... 

Isolated atoms show spherical symmetry, and it is obvious to consider a molecule as a collection of atomic spheres of some appropriate defined vdW radii. Because the vdW radii of atomic spheres used for the representations of molecular space are usually much too large for modeling molecules by simply placing the atomic hard vdW spheres side by side, commonly one generates various fused sphere models for molecules, that is the atomic vdW spheres are interpenetrated one with another. Positions of these spheres may be described by their Cartesian coordinates according to the 3D stereochemical bond pattern of a particular molecule. The envelope of the outer surface of the vdW atomic fused spheres may be regarded as a formal vdW molecular surface. This envelope embeds a formal vdW molecular volume. 

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