Dynamic shape space

Consider two different subsets of the same space D, or subsets of two dynamic shape spaces D and D of two different stoichiometric families of molecules. One may compare those domains of the two subsets that belong to the same shape group H 2. Since within these domains the nuclear configuration is not fully specified, that is, there exists some configurational freedom within these domains, the above approach provides a description of the dynamic similarity of molecular shapes. We shall return to the problems of dynamic shape similarity in Chapter 6. 

Each (a,b)-map can be regarded as a subset of the dynamic shape space D. Such a subset contains all points of D where the internal coordinates corresponding to the nuclear arrangement are fixed. 

If the entire range of curvature parameter b is considered, then a list of the finite number of distinct shape matrices and those curvature values bj where a change of the shape matrix occurs, gives a detailed, numerical shape characterization of the MIDCO surface G(a). In the most general case of variations in the two parameters a and b, as well as in the nuclear configuration K, one can study the dynamic shape space invariance domains, the (a,b)-maps, and various projections of the invariance domains of shape matrices, following the principles [158] applied for the shape group invariance domains of the dynamic shape space D. 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

One may analyse the detailed variations of contributions of various nuclear configurations to each shape type Xj as a function of some continuous parameters, for example, as function of the contour density value a and reference curvature parameter b of isodensity contours G(a). This is equivalent to the analysis of the parameter dependence (for example, (a,b)-dependence) of the Ty subsets within the configuration space M, and in particular, in relaxed cross sections or within each catchment region C(X,i) [44]. These changes can be monitored within the dynamic shape space D, obtained as the product space of the nuclear configuration space M and the space of the actual continuous parameters. This approach has been described in some detail in ref. [44], and various applications can be found in refs. [45-47]. 

We can formulate the above ideas more precisely by considering the dynamic shape properties of molecules within a nuclear configuration space M. 

Mezey, P.G. (1993a). Dynamic Shape Analysis of Molecules in Restricted Domains of a Configuration Space. J.Math.Chem., 13,59-70. 

Abstract As a non-invasive technique, NMR spectroscopy allows the observation of molecular transport in porous media without any disturbance of their intrinsic molecular dynamics. The space scale of the diffusion phenomena accessible by NMR ranges from the elementary steps (as studied, e.g., by line-shape analysis or relaxometry) up to macroscopic dimensions. Being able to follow molecular diffusion paths from ca. 100 nm up to ca. 100 xm, PPG NMR has proven to be a particularly versatile tool for diffusion studies in heterogeneous systems. With respect to zeolites, PFG NMR is able to provide direct information about the rate of molecular migration in the intracrystalline space and through assemblages of zeolite crystallites as well as about possible transport resistances on the outer surface of the crystallites (surface barriers). 

The minimum dynamic work space per person for upper extremity motions has a three dimensional cylindrical shape with a diameter of 69 cm (27"). 

The stored object can be represented either as a wire model or as a half-tone (pseudo-coloured) image. The wire-model permits quantitative measurements whereas the half-tone image uncovers structures. Both representations may be superimposed and, furthermore, combined with functional values to represent a dynamic functional model. The representation as a half-tone image with all its details requires a lot more information than the wire model (Roger Adams, 1976 Wu et al, 1977). Therefore the object is stored in a virtual cube shaped space (Meagher, 1982). This cube can be subdivided down to its atomic elements, the voxels (volume elements as pixels , for picture elements), which finally hold binary object information. 

For a given value of lu, equation (6.9) represents a point in complex space P(lu). When LU is varied from zero to infinity, a locus will be generated in the complex space. This locus, shown in Figure 6.2, is in effect a polar plot, and is sometimes called a harmonic response diagram. An important feature of such a diagram is that its shape is uniquely related to the dynamic characteristics of the system. 

A three-dimensional meshwork of proteinaceous filaments of various sizes fills the space between the organelles of all eukaryotic cell types. This material is known collectively as the cytoskeleton, but despite the static property implied by this name, the cytoskeleton is plastic and dynamic. Not only must the cytoplasm move and modify its shape when a cell changes its position or shape, but the cytoskeleton itself causes these movements. In addition to motility, the cytoskeleton plays a role in metabolism. Several glycolytic enzymes are known to be associated with actin filaments, possibly to concentrate substrate and enzymes locally. Many mRNA species appear to be bound by filaments, especially in egg cells where they may be immobilized in distinct regions thereby becoming concentrated in defined tissues upon subsequent cell divisions. 

Dendrimers with a polyphenyl core around a central biphenyl unit decorated at the rim with peryleneimide chromophores have been investigated both in bulk and at the single-molecule level in order to understand their time and space-resolved behavior [28]. The results obtained have shown that the conformational distribution plays an important role in the dynamics of the photophysical processes. Energy transfer in a series of shape-persistent polyphenylene dendrimers substituted with peryleneimide and terryleneimide chro-mophoric units (4-7) has been investigated in toluene solution [29]. 

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