Topologically equivalent

A Sail, TL Blundell. Definition of general topological equivalence m protein structures A procedure involving comparison of properties and relationships through simulated annealing and dynamic programming. J Mol Biol 212 403-428, 1990. 

Since all graphs in the same class are in this sense topologically equivalent, a representative member of each class, devoid of explicit vertex (or edge) labels, defines a unique unlabeled graph. Figure 2.3, for example, shows all of the undirected and unlabeled graphs of order 4. 

This may be expressed by saying that all vector norms (in finitedimensional space) are topologically equivalent otherwise, that if the sequence of vectors xi is such that... 

Figure 46-8. Fusion of a vesicle with the plasma membrane preserves the orientation of any integral proteins embedded in the vesicle bilayer. Initially, the amino terminal of the protein faces the lumen, or inner cavity, of such a vesicle. After fusion, the amino terminal is on the exterior surface of the plasma membrane. That the orientation of the protein has not been reversed can be perceived by noting that the other end of the molecule, the carboxyl terminal, is always immersed in the cytoplasm. The lumen of a vesicle and the outside of the cell are topologically equivalent. (Re drawn and modified, with permission, from Lodish HF, Rothman JE The assembly of cell membranes. Sci Am [Jan] 1979 240 43.)...

In one extreme case within the topological framework, the two objects can be brought into a perfect correspondence, demonstrating topological equivalence. In a more precise formulation, two topological spaces (Aj, T,) and (X2, T2) are called topologically equivalent or homeomorphic if there exists a function... 

Compound 6 contains seven iron-based units [ 12], of which the six peripheral ones are chemically and topologically equivalent, whereas that constituting the core (Fe(Cp)(C6Me6)+) has a different chemical nature. Accordingly, two redox processes are observed, i.e., oxidation of the peripheral ferrocene moieties and reduction of the core, whose cyclic voltammetric waves have current intensities in the 6 1 ratio. Clearly, the one-electron process of the core is a convenient internal standard to calibrate the number of electron exchanged in the multi-electron process. In the absence of an internal standard, the number of exchanged electrons has to be obtained by coulometry measurements, or by comparison with the intensity of the wave of an external standard after correction for the different diffusion coefficients [15]. 

Eutactical means topologically equivalent to a close packing of Hg, however, with significantly greater interatomic distances than expected for neutral Hg atoms. 

Zubieta et al. have also demonstrated the formation of a cage topologically equivalent to a truncated tetrahedron, [Moi6(OH)1204o]8. Composed of four Mo(VI) and twelve Mo(V) centers, this host possesses a central [Moi204o]20 core which encapsulates a proton or sodium ion. [40]... 

Buckminsterfullerene, an allotrope of carbon, is topologically equivalent to a truncated icosahedron, an Archimedean solid that possesses 12 pentagons and 20 hexagons (Fig. 9-16). [17] Each carbon atom of this fullerene corresponds to a vertex of the polyhedron. As a result, C6o is held together by 90 covalent bonds, the number of edges of the solid. 

The concept of chemistry space pervades, either explicitly or implicitly, much of the literature in chemoinformatics. As is discussed in Subheading 3., chemistry spaces are induced by various similarity measures. The different similarity measures do not, however, give rise to topologically equivalent chemistry spaces—nearest-neighbor relations are generally not preserved among chemistry spaces induced by different similarity measures. The consequences of this are manifold. An especially egregious consequence is that the results of similarity searches based on different similarity measures can differ substantially. And there is no easy solution to this problem. 

In discussing the topology of such structures, it is particularly appropriate then to focus on the structure of the four-connected nets of metal atoms - a procedure in tune with the philosophy of this paper. It is in fact an approach that has already been actively developed elsewhere (see e.g. Ref. ), so we do not pursue this topic further here, other than to include in Table 3 some examples of alumino-silicates in which the metal atom arrangement is topologically equivalent to corresponding network alloy types. 

The B network derived from the electron density topology links all atoms that share faces. Although the B links are all topologically equivalent, only those... 

A number of theoretical studies have been directly concerned with the mechanism of pseudorotation231,286,298,299. It is found231,299 that the Berry process is more favoured but the turnstile process is not ruled out. Note that these two processes are topologically equivalent but proceed along different pathways on the potential energy surface293. 

From its definition, the P-polynomial appears to depend on the particular projection of the link which we are working with. However, when this polynomial was defined it was proven that given any oriented link, no matter how it is deformed or projected, the link will always have the same P-polynomial [5, 6]. This means that two oriented links which are topologically equivalent have the same P-polynomial. In particular, if an oriented link can be deformed to its minor image then it and its minor image will have the same P-polynomial. 

We illustrate how to compute the P-polynomial by evaluating a couple of examples. First let Lq consist of the oriented unlink of two components which is illustrated in Figure 4. Then L+ and L are as shown in the figure. Now we use the second axiom of the definition of the P-polynomial, together with the observation that both L+ and L are topologically equivalent to the unknot, in order to obtain the equation / + Z1 + mP(Lo) = 0. Hence, we conclude that P(Lo) = -m l + I"1). 

Cyclazinylium salts (233), which following MO considerations should be unstable antiaromatic compounds, have not been described so far. Topologically equivalent -systems containing two additional electrons, however, appear to be closed shell systems and some compounds corroborating this expectation have been prepared. 

Figure 5-19 Topological equivalence of toroidal (solenoidal) and plectonemically interwound forms of a circular DNA. These two forms have a constant value of the linking number Lk (or a), the twist Tw, and writhing number Wr.

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