Encoding Atom Identity

Up to this point, molecular graphs have been written down as though each atom is identical. Shape attributes have been interpreted on this basis. For application to chemical and biological problems, account must be made of atom 

Consider the pair of molecules, heptane and dipropyl ether. By any criteria, we would say that the shapes were very similar, though not identical. As derived so far, each k, k, and value would be the same for this pair of molecules, implying shape identity. An additional consideration is necessary if we are to quantitate shape attributes where there are major differences in atoms and/or hybrid states. An approximation of shape equality between cyclohexane and benzene is probably not adequate for many structure-activity analyses. Some account must be taken of the fact that different atoms make different size contributions to a molecule, thereby influencing its overall shape. 

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Encoding atom identity can be accomplished in several ways. We have elected to modify the atom count. A, in Eqs. [50], [52], [54], and [55]. The modification is based on the awareness that a non-C(sp ) atom, counted in arriving at the value of A for the molecular graph, is contributing more or less than a C(sp ) contribution to the shape. Therefore, that particular atom should be counted more or less than 1, the increment or decrement called a, which is based on the size contribution of the atom in question relative to C(sp ). 

Moreover, atom-type -state indices were proposed as molecular descriptors encoding topological and electronic information related to particular atom types in the molecule [Hall and Kier, 1995a Hall et al., 1995b]. They are calculated by summing the -state values of all atoms of the same atom type in the molecule. Each atom type is first defined by atom identity, based on the atomic number Z, and valence state, itself identified by the valence state indicator (VST) defined as ... 

The summation is over the A atoms of the skeleton. The zero order chi index carries a low level of structure information. Little of the connectedness of the skeletal network is encoded only the fact of the presence of the nearest neighbor to each atom is encoded. In the °X index, atom identities are quantitated. 

The first-order chi indexes contain more structure information than do the zero-order indexes. The immediate bonding environment of each atom is encoded by virtue of the edge weight. Further, the number of terms in the sum, P, is dependent on the graph type, especially on the number of cycles or rings. The x index encodes both the atom identities as well as the connectedness in the molecular skeleton. 

The variable A x in QSAR Eq. [33] provides discrimination only among the three molecular classes because the atom contribution to A x is zero for saturated carbon atoms. The A°x provides very little structure information with respect to skeletal variation, but it does encode the atom identities and some of the skeletal environment immediately surrounding the heteroatom. The addition of the A x variable greatly increases the discrimination among the three classes because it encodes skeletal information about the carbon atoms a to the heteroatom. The QSAR is improved considerably by the addition of A x. Finally, the addition of A x further improves the QSAR by adding information about the broader reaches of the skeletal environment of each heteroatom, namely, atoms p to the heteroatoms. It can also be seen that the effects of atoms P to the heteroatom are much less important than the heteroatom itself or the a carbon atoms. By the introduction of the delta-chi indexes, an atom level interpretation is made possible. 

The key to useful topological state values is an appropriate form for the r, values. Hall and Kier have shown that simple forms, such as the graph distance d,j, are not useful because they fail to indicate proper topological equivalence. To ensure representation of topological equivalence, two features of the paths must be encoded (1) atomic identity and (2) the sequence of atoms in each path. It has been shown that both these characteristics can be encoded as follows. Atomic identity can be encoded using the molecular connectivity valence delta value, 8. The discussions concerning chi indexes and related quantities have shown the validity of the valence delta value as a characterization of atoms. 

This pair of delta values is seen as a characterization of the atom in its valence state. The simple delta, 5, describes the role of the atom in the skeleton in terms of its connectedness and count of sigma electrons it could be called the sigma electron descriptor. The valence delta, 8, encodes the electronic identity of the atom in terms of both valence electron count and core electron count. It could be called the valence electron descriptor. The isolated, unbonded atom may be thought of as characterized by its atomic number, Z, and the number of valence electrons, Z. In its valence state, the bonded atom is characterized by 8 and 8. Embedded in the molecular skeleton, the full characterization of the atom in the environment of the whole molecule is given by the topological equivalence value, described in a later section, and the electrotopological state value, presented separately.A representation of the whole molecule is accomplished by the combination of chi, kappa, and topological state indexes. 

A nearly identical polysfyrene sample was prepared, but with imiform labeling of all the polymer backbone carbons. This permitted the use of the pulse sequences shown in Figure 18. The nmning of these sequences follows the convention used by those working in the protein structure area. Listing the atoms in the coherence transfer pathway forms the acronym. Those atoms in the coherence transfer pathway whose chemical shifts are not encoded during... 

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